pydrake.math

Bindings for math, including overloads for scalar types and basic SE(3) representations.

Note that arrays of symbolic scalar types, such as Variable and Expression, are exposed using ndarray[object], and as such logical operations are constrained to return boolean values given NumPy’s implementation; this is not desirable, as one should really get a Formula object. As a workaround, this module provides the following vectorized operators, following suit with the operator builtin module: lt, le, eq, ne, ge, and gt.

As an example:

>>> x = np.array([Variable("x0"), Variable("x1")])
>>> y = np.array([Variable("y0"), Variable("y1")])
>>> x >= y
# This should throw a RuntimeError
>>> ge(x, y)
array([<Formula "(x0 >= y0)">, <Formula "(x1 >= y1)">], dtype=object)
pydrake.math.abs(*args, **kwargs)

Overloaded function.

  1. abs(arg0: float) -> float

  2. abs(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. abs(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.acos(*args, **kwargs)

Overloaded function.

  1. acos(arg0: float) -> float

  2. acos(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. acos(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.arccos(*args, **kwargs)

Overloaded function.

  1. arccos(arg0: float) -> float

  2. arccos(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. arccos(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.arcsin(*args, **kwargs)

Overloaded function.

  1. arcsin(arg0: float) -> float

  2. arcsin(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. arcsin(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.arctan(*args, **kwargs)

Overloaded function.

  1. arctan(arg0: float) -> float

  2. arctan(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. arctan(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.arctan2(*args, **kwargs)

Overloaded function.

  1. arctan2(y: float, x: float) -> float

  2. arctan2(y: pydrake.autodiffutils.AutoDiffXd, x: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. arctan2(y: pydrake.symbolic.Expression, x: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.AreQuaternionsEqualForOrientation(*args, **kwargs)

Overloaded function.

  1. AreQuaternionsEqualForOrientation(quat1: pydrake.common.eigen_geometry.Quaternion, quat2: pydrake.common.eigen_geometry.Quaternion, tolerance: float) -> bool

This function tests whether two quaternions represent the same orientation. This function converts each quaternion to its canonical form and tests whether the absolute value of the difference in corresponding elements of these canonical quaternions is within tolerance.

Parameter quat1:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat is analogous to the rotation matrix R_AB.

Parameter quat2:

Quaternion with a description analogous to quat1.

Parameter tolerance:

Nonnegative real scalar defining the allowable difference in the orientation described by quat1 and quat2.

Returns

True if quat1 and quat2 represent the same orientation (to within tolerance), otherwise False.

  1. AreQuaternionsEqualForOrientation(quat1: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd], quat2: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd], tolerance: pydrake.autodiffutils.AutoDiffXd) -> bool

This function tests whether two quaternions represent the same orientation. This function converts each quaternion to its canonical form and tests whether the absolute value of the difference in corresponding elements of these canonical quaternions is within tolerance.

Parameter quat1:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat is analogous to the rotation matrix R_AB.

Parameter quat2:

Quaternion with a description analogous to quat1.

Parameter tolerance:

Nonnegative real scalar defining the allowable difference in the orientation described by quat1 and quat2.

Returns

True if quat1 and quat2 represent the same orientation (to within tolerance), otherwise False.

  1. AreQuaternionsEqualForOrientation(quat1: pydrake.common.eigen_geometry.Quaternion_[Expression], quat2: pydrake.common.eigen_geometry.Quaternion_[Expression], tolerance: pydrake.symbolic.Expression) -> pydrake.symbolic.Formula

This function tests whether two quaternions represent the same orientation. This function converts each quaternion to its canonical form and tests whether the absolute value of the difference in corresponding elements of these canonical quaternions is within tolerance.

Parameter quat1:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat is analogous to the rotation matrix R_AB.

Parameter quat2:

Quaternion with a description analogous to quat1.

Parameter tolerance:

Nonnegative real scalar defining the allowable difference in the orientation described by quat1 and quat2.

Returns

True if quat1 and quat2 represent the same orientation (to within tolerance), otherwise False.

pydrake.math.asin(*args, **kwargs)

Overloaded function.

  1. asin(arg0: float) -> float

  2. asin(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. asin(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.atan(*args, **kwargs)

Overloaded function.

  1. atan(arg0: float) -> float

  2. atan(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. atan(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.atan2(*args, **kwargs)

Overloaded function.

  1. atan2(y: float, x: float) -> float

  2. atan2(y: pydrake.autodiffutils.AutoDiffXd, x: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. atan2(y: pydrake.symbolic.Expression, x: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.BalanceQuadraticForms(S: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], P: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) → numpy.ndarray[numpy.float64[m, n]]

Given two quadratic forms, x’Sx > 0 and x’Px, (with P symmetric and full rank), finds a change of variables x = Ty, which simultaneously diagonalizes both forms (as inspired by “balanced truncation” in model-order reduction [1]). In this note, we use abs(M) to indicate the elementwise absolute value.

Adapting from [1], we observe that there is a family of coordinate systems that can simultaneously diagonalize T’ST and T’PT. Using D to denote a diagonal matrix, we call the result S-normal if T’ST = I and abs(T’PT) = D⁻², call it P-normal if T’ST = D² and abs(T’PT) = I, and call it “balanced” if T’ST = D and abs(T’PT) = D⁻¹. Note that if P > 0, then T’PT = D⁻¹.

We find x=Ty such that T’ST = D and abs(T’PT) = D⁻¹, where D is diagonal. The recipe is: - Factorize S = LL’, and choose R=L⁻¹. - Take svd(RPR’) = UΣV’, and note that U=V for positive definite matrices, and V is U up to a sign flip of the singular vectors for all symmetric matrices. - Choose T = R’U Σ^{-1/4}, where the matrix exponent can be taken elementwise because Σ is diagonal. This gives T’ST = Σ^{-1/2} (by using U’U=I), and abs(T’PT) = Σ^{1/2}. If P > 0, then T’PT = Σ^{1/2}.

Note that the numerical “balancing” can address the absolute scaling of the quadratic forms, but not the relative scaling. To understand this, consider the scalar case: we have two quadratic functions, sx² and px², with s>0, p>0. We’d like to choose x=Ty so that sT²y² and pT²y² are “balanced” (we’d like them both to be close to y²). We’ll choose T=p^{-1/4}s^{-1/4}, which gives sx² = sqrt(s/p)y², and px² = sqrt(p/s)y². For instance if s=1e8 and p=1e8, then t=1e-4 and st^2 = pt^2 = 1. But if s=10, p=1e7, then t=0.01, and st^2 = 1e-3, pt^2 = 1e3.

In the matrix case, the absolute scaling is important – it ensures that the two quadratic forms have the same matrix condition number and makes them as close as possible to 1. Besides absolute scaling, in the matrix case the balancing transform diagonalizes both quadratic forms.

[1] B. Moore, “Principal component analysis in linear systems: Controllability, observability, and model reduction,” IEEE Trans. Automat. Contr., vol. 26, no. 1, pp. 17–32, Feb. 1981.

class pydrake.math.BarycentricMesh

Represents a multi-linear function (from vector inputs to vector outputs) by interpolating between points on a mesh using (triangular) barycentric interpolation.

For a technical description of barycentric interpolation, see e.g. Remi Munos and Andrew Moore, “Barycentric Interpolators for Continuous Space and Time Reinforcement Learning”, NIPS 1998

__init__(self: pydrake.math.BarycentricMesh, arg0: list[set[float]]) → None

Constructs the mesh.

Eval(self: pydrake.math.BarycentricMesh, arg0: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], arg1: numpy.ndarray[numpy.float64[m, 1]]) → numpy.ndarray[numpy.float64[m, 1]]

Returns the function evaluated at input.

EvalBarycentricWeights(self: pydrake.math.BarycentricMesh, arg0: numpy.ndarray[numpy.float64[m, 1]]) → tuple[numpy.ndarray[numpy.int32[m, 1]], numpy.ndarray[numpy.float64[m, 1]]]

Writes the mesh indices used for interpolation to mesh_indices, and the interpolating coefficients to weights. Inputs that are outside the bounding box of the input_grid are interpolated as though they were projected (elementwise) to the closest face of the defined mesh.

Parameter input:

must be a vector of length get_num_inputs().

Parameter mesh_indices:

is a pointer to a vector of length get_num_interpolants().

Parameter weights:

is a vector of coefficients (which sum to 1) of length get_num_interpolants().

get_all_mesh_points(self: pydrake.math.BarycentricMesh) → numpy.ndarray[numpy.float64[m, n]]

Returns a matrix with all of the mesh points, one per column.

get_input_grid(self: pydrake.math.BarycentricMesh) → list[set[float]]
get_input_size(self: pydrake.math.BarycentricMesh) → int
get_mesh_point(self: pydrake.math.BarycentricMesh, arg0: int) → numpy.ndarray[numpy.float64[m, 1]]

Returns the position of a mesh point in the input space referenced by its scalar index to point.

Parameter index:

must be in [0, get_num_mesh_points).

get_num_interpolants(self: pydrake.math.BarycentricMesh) → int
get_num_mesh_points(self: pydrake.math.BarycentricMesh) → int
MeshValuesFrom(self: pydrake.math.BarycentricMesh, arg0: Callable[[numpy.ndarray[numpy.float64[m, 1]]], numpy.ndarray[numpy.float64[m, 1]]]) → numpy.ndarray[numpy.float64[m, n]]

Evaluates vector_func at all input mesh points and extracts the mesh value matrix that should be used to approximate the function with this barycentric interpolation.

Click to expand C++ code...
MatrixXd mesh_values = bary.MeshValuesFrom(
    [](const auto& x) { return Vector1d(std::sin(x[0])); });
class pydrake.math.BsplineBasis

Given a set of non-descending breakpoints t₀ ≤ t₁ ≤ ⋅⋅⋅ ≤ tₘ, a B-spline basis of order k is a set of n + 1 (where n = m - k) piecewise polynomials of degree k - 1 defined over those breakpoints. The elements of this set are called “B-splines”. The vector (t₀, t₁, …, tₘ)’ is referred to as the “knot vector” of the basis and its elements are referred to as “knots”.

At a breakpoint with multiplicity p (i.e. a breakpoint that appears p times in the knot vector), B-splines are guaranteed to have Cᵏ⁻ᵖ⁻¹ continuity.

A B-spline curve using a B-spline basis B, is a parametric curve mapping parameter values in [tₖ₋₁, tₙ₊₁] to a vector space V. For t ∈ [tₖ₋₁, tₙ₊₁] the value of the curve is given by the linear combination of n + 1 control points, pᵢ ∈ V, with the elements of B evaluated at t.

For more information on B-splines and their uses, see (for example) Patrikalakis et al. [1].

[1] https://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node15.html

Note

This class is templated; see BsplineBasis_ for the list of instantiations.

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.BsplineBasis) -> None

  2. __init__(self: pydrake.math.BsplineBasis, order: int, knots: list[float]) -> None

Constructs a B-spline basis with the specified order and knots.

Precondition:

knots is sorted in non-descending order.

Raises

RuntimeError if knots.size() < 2 * order. –

  1. __init__(self: pydrake.math.BsplineBasis, order: int, num_basis_functions: int, type: pydrake.math.KnotVectorType = <KnotVectorType.kClampedUniform: 1>, initial_parameter_value: float = 0.0, final_parameter_value: float = 1.0) -> None

Constructs a B-spline basis with the specified order, num_basis_functions, initial_parameter_value, final_parameter_value, and an auto-generated knot vector of the specified type.

Raises

RuntimeError if num_basis_functions < order –

Precondition:

initial_parameter_value ≤ final_parameter_value

  1. __init__(self: pydrake.math.BsplineBasis, other: pydrake.math.BsplineBasis) -> None

ComputeActiveBasisFunctionIndices(*args, **kwargs)

Overloaded function.

  1. ComputeActiveBasisFunctionIndices(self: pydrake.math.BsplineBasis, parameter_interval: Annotated[list[float], FixedSize(2)]) -> list[int]

Returns the indices of the basis functions which may evaluate to non-zero values for some parameter value in parameter_interval; all other basis functions are strictly zero over parameter_interval.

Precondition:

parameter_interval[0] ≤ parameter_interval[1]

Precondition:

parameter_interval[0] ≥ initial_parameter_value()

Precondition:

parameter_interval[1] ≤ final_parameter_value()

  1. ComputeActiveBasisFunctionIndices(self: pydrake.math.BsplineBasis, parameter_value: float) -> list[int]

Returns the indices of the basis functions which may evaluate to non-zero values for parameter_value; all other basis functions are strictly zero at this point.

Precondition:

parameter_value ≥ initial_parameter_value()

Precondition:

parameter_value ≤ final_parameter_value()

degree(self: pydrake.math.BsplineBasis) → int

The degree of the piecewise polynomials comprising this B-spline basis (k - 1 in the class description).

EvaluateBasisFunctionI(self: pydrake.math.BsplineBasis, i: int, parameter_value: float) → float

Returns the value of the i-th basis function evaluated at parameter_value.

EvaluateCurve(self: pydrake.math.BsplineBasis, control_points: list[numpy.ndarray[numpy.float64[m, 1]]], parameter_value: float) → numpy.ndarray[numpy.float64[m, 1]]

Evaluates the B-spline curve defined by this and control_points at the given parameter_value.

Parameter control_points:

Control points of the B-spline curve.

Parameter parameter_value:

Parameter value at which to evaluate the B-spline curve defined by this and control_points.

Precondition:

control_points.size() == num_basis_functions()

Precondition:

parameter_value ≥ initial_parameter_value()

Precondition:

parameter_value ≤ final_parameter_value()

final_parameter_value(self: pydrake.math.BsplineBasis) → float

The maximum allowable parameter value for B-spline curves using this basis (tₙ₊₁ in the class description).

FindContainingInterval(self: pydrake.math.BsplineBasis, parameter_value: float) → int

For a parameter_value = t, the interval that contains it is the pair of knot values [tᵢ, tᵢ₊₁] for the greatest i such that tᵢ ≤ t and tᵢ < final_parameter_value(). This function returns that value of i.

Precondition:

parameter_value ≥ initial_parameter_value()

Precondition:

parameter_value ≤ final_parameter_value()

initial_parameter_value(self: pydrake.math.BsplineBasis) → float

The minimum allowable parameter value for B-spline curves using this basis (tₖ₋₁ in the class description).

knots(self: pydrake.math.BsplineBasis) → list[float]

The knot vector of this B-spline basis (the vector (t₀, t₁, …, tₘ)’ in the class description).

num_basis_functions(self: pydrake.math.BsplineBasis) → int

The number of basis functions in this B-spline basis (n + 1 in the class description).

order(self: pydrake.math.BsplineBasis) → int

The order of this B-spline basis (k in the class description).

template pydrake.math.BsplineBasis_

Instantiations: BsplineBasis_[float], BsplineBasis_[AutoDiffXd], BsplineBasis_[Expression]

class pydrake.math.BsplineBasis_[AutoDiffXd]

Given a set of non-descending breakpoints t₀ ≤ t₁ ≤ ⋅⋅⋅ ≤ tₘ, a B-spline basis of order k is a set of n + 1 (where n = m - k) piecewise polynomials of degree k - 1 defined over those breakpoints. The elements of this set are called “B-splines”. The vector (t₀, t₁, …, tₘ)’ is referred to as the “knot vector” of the basis and its elements are referred to as “knots”.

At a breakpoint with multiplicity p (i.e. a breakpoint that appears p times in the knot vector), B-splines are guaranteed to have Cᵏ⁻ᵖ⁻¹ continuity.

A B-spline curve using a B-spline basis B, is a parametric curve mapping parameter values in [tₖ₋₁, tₙ₊₁] to a vector space V. For t ∈ [tₖ₋₁, tₙ₊₁] the value of the curve is given by the linear combination of n + 1 control points, pᵢ ∈ V, with the elements of B evaluated at t.

For more information on B-splines and their uses, see (for example) Patrikalakis et al. [1].

[1] https://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node15.html

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.BsplineBasis_[AutoDiffXd]) -> None

  2. __init__(self: pydrake.math.BsplineBasis_[AutoDiffXd], order: int, knots: list[pydrake.autodiffutils.AutoDiffXd]) -> None

Constructs a B-spline basis with the specified order and knots.

Precondition:

knots is sorted in non-descending order.

Raises

RuntimeError if knots.size() < 2 * order. –

  1. __init__(self: pydrake.math.BsplineBasis_[AutoDiffXd], order: int, num_basis_functions: int, type: pydrake.math.KnotVectorType = <KnotVectorType.kClampedUniform: 1>, initial_parameter_value: pydrake.autodiffutils.AutoDiffXd = 0.0, final_parameter_value: pydrake.autodiffutils.AutoDiffXd = 1.0) -> None

Constructs a B-spline basis with the specified order, num_basis_functions, initial_parameter_value, final_parameter_value, and an auto-generated knot vector of the specified type.

Raises

RuntimeError if num_basis_functions < order –

Precondition:

initial_parameter_value ≤ final_parameter_value

  1. __init__(self: pydrake.math.BsplineBasis_[AutoDiffXd], other: pydrake.math.BsplineBasis_[AutoDiffXd]) -> None

ComputeActiveBasisFunctionIndices(*args, **kwargs)

Overloaded function.

  1. ComputeActiveBasisFunctionIndices(self: pydrake.math.BsplineBasis_[AutoDiffXd], parameter_interval: Annotated[list[pydrake.autodiffutils.AutoDiffXd], FixedSize(2)]) -> list[int]

Returns the indices of the basis functions which may evaluate to non-zero values for some parameter value in parameter_interval; all other basis functions are strictly zero over parameter_interval.

Precondition:

parameter_interval[0] ≤ parameter_interval[1]

Precondition:

parameter_interval[0] ≥ initial_parameter_value()

Precondition:

parameter_interval[1] ≤ final_parameter_value()

  1. ComputeActiveBasisFunctionIndices(self: pydrake.math.BsplineBasis_[AutoDiffXd], parameter_value: pydrake.autodiffutils.AutoDiffXd) -> list[int]

Returns the indices of the basis functions which may evaluate to non-zero values for parameter_value; all other basis functions are strictly zero at this point.

Precondition:

parameter_value ≥ initial_parameter_value()

Precondition:

parameter_value ≤ final_parameter_value()

degree(self: pydrake.math.BsplineBasis_[AutoDiffXd]) → int

The degree of the piecewise polynomials comprising this B-spline basis (k - 1 in the class description).

EvaluateBasisFunctionI(self: pydrake.math.BsplineBasis_[AutoDiffXd], i: int, parameter_value: pydrake.autodiffutils.AutoDiffXd) → pydrake.autodiffutils.AutoDiffXd

Returns the value of the i-th basis function evaluated at parameter_value.

EvaluateCurve(self: pydrake.math.BsplineBasis_[AutoDiffXd], control_points: list[numpy.ndarray[object[m, 1]]], parameter_value: pydrake.autodiffutils.AutoDiffXd) → numpy.ndarray[object[m, 1]]

Evaluates the B-spline curve defined by this and control_points at the given parameter_value.

Parameter control_points:

Control points of the B-spline curve.

Parameter parameter_value:

Parameter value at which to evaluate the B-spline curve defined by this and control_points.

Precondition:

control_points.size() == num_basis_functions()

Precondition:

parameter_value ≥ initial_parameter_value()

Precondition:

parameter_value ≤ final_parameter_value()

final_parameter_value(self: pydrake.math.BsplineBasis_[AutoDiffXd]) → pydrake.autodiffutils.AutoDiffXd

The maximum allowable parameter value for B-spline curves using this basis (tₙ₊₁ in the class description).

FindContainingInterval(self: pydrake.math.BsplineBasis_[AutoDiffXd], parameter_value: pydrake.autodiffutils.AutoDiffXd) → int

For a parameter_value = t, the interval that contains it is the pair of knot values [tᵢ, tᵢ₊₁] for the greatest i such that tᵢ ≤ t and tᵢ < final_parameter_value(). This function returns that value of i.

Precondition:

parameter_value ≥ initial_parameter_value()

Precondition:

parameter_value ≤ final_parameter_value()

initial_parameter_value(self: pydrake.math.BsplineBasis_[AutoDiffXd]) → pydrake.autodiffutils.AutoDiffXd

The minimum allowable parameter value for B-spline curves using this basis (tₖ₋₁ in the class description).

knots(self: pydrake.math.BsplineBasis_[AutoDiffXd]) → list[pydrake.autodiffutils.AutoDiffXd]

The knot vector of this B-spline basis (the vector (t₀, t₁, …, tₘ)’ in the class description).

num_basis_functions(self: pydrake.math.BsplineBasis_[AutoDiffXd]) → int

The number of basis functions in this B-spline basis (n + 1 in the class description).

order(self: pydrake.math.BsplineBasis_[AutoDiffXd]) → int

The order of this B-spline basis (k in the class description).

class pydrake.math.BsplineBasis_[Expression]

Given a set of non-descending breakpoints t₀ ≤ t₁ ≤ ⋅⋅⋅ ≤ tₘ, a B-spline basis of order k is a set of n + 1 (where n = m - k) piecewise polynomials of degree k - 1 defined over those breakpoints. The elements of this set are called “B-splines”. The vector (t₀, t₁, …, tₘ)’ is referred to as the “knot vector” of the basis and its elements are referred to as “knots”.

At a breakpoint with multiplicity p (i.e. a breakpoint that appears p times in the knot vector), B-splines are guaranteed to have Cᵏ⁻ᵖ⁻¹ continuity.

A B-spline curve using a B-spline basis B, is a parametric curve mapping parameter values in [tₖ₋₁, tₙ₊₁] to a vector space V. For t ∈ [tₖ₋₁, tₙ₊₁] the value of the curve is given by the linear combination of n + 1 control points, pᵢ ∈ V, with the elements of B evaluated at t.

For more information on B-splines and their uses, see (for example) Patrikalakis et al. [1].

[1] https://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node15.html

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.BsplineBasis_[Expression]) -> None

  2. __init__(self: pydrake.math.BsplineBasis_[Expression], order: int, knots: list[pydrake.symbolic.Expression]) -> None

Constructs a B-spline basis with the specified order and knots.

Precondition:

knots is sorted in non-descending order.

Raises

RuntimeError if knots.size() < 2 * order. –

  1. __init__(self: pydrake.math.BsplineBasis_[Expression], order: int, num_basis_functions: int, type: pydrake.math.KnotVectorType = <KnotVectorType.kClampedUniform: 1>, initial_parameter_value: pydrake.symbolic.Expression = 0.0, final_parameter_value: pydrake.symbolic.Expression = 1.0) -> None

Constructs a B-spline basis with the specified order, num_basis_functions, initial_parameter_value, final_parameter_value, and an auto-generated knot vector of the specified type.

Raises

RuntimeError if num_basis_functions < order –

Precondition:

initial_parameter_value ≤ final_parameter_value

  1. __init__(self: pydrake.math.BsplineBasis_[Expression], other: pydrake.math.BsplineBasis_[Expression]) -> None

ComputeActiveBasisFunctionIndices(*args, **kwargs)

Overloaded function.

  1. ComputeActiveBasisFunctionIndices(self: pydrake.math.BsplineBasis_[Expression], parameter_interval: Annotated[list[pydrake.symbolic.Expression], FixedSize(2)]) -> list[int]

Returns the indices of the basis functions which may evaluate to non-zero values for some parameter value in parameter_interval; all other basis functions are strictly zero over parameter_interval.

Precondition:

parameter_interval[0] ≤ parameter_interval[1]

Precondition:

parameter_interval[0] ≥ initial_parameter_value()

Precondition:

parameter_interval[1] ≤ final_parameter_value()

  1. ComputeActiveBasisFunctionIndices(self: pydrake.math.BsplineBasis_[Expression], parameter_value: pydrake.symbolic.Expression) -> list[int]

Returns the indices of the basis functions which may evaluate to non-zero values for parameter_value; all other basis functions are strictly zero at this point.

Precondition:

parameter_value ≥ initial_parameter_value()

Precondition:

parameter_value ≤ final_parameter_value()

degree(self: pydrake.math.BsplineBasis_[Expression]) → int

The degree of the piecewise polynomials comprising this B-spline basis (k - 1 in the class description).

EvaluateBasisFunctionI(self: pydrake.math.BsplineBasis_[Expression], i: int, parameter_value: pydrake.symbolic.Expression) → pydrake.symbolic.Expression

Returns the value of the i-th basis function evaluated at parameter_value.

EvaluateCurve(self: pydrake.math.BsplineBasis_[Expression], control_points: list[numpy.ndarray[object[m, 1]]], parameter_value: pydrake.symbolic.Expression) → numpy.ndarray[object[m, 1]]

Evaluates the B-spline curve defined by this and control_points at the given parameter_value.

Parameter control_points:

Control points of the B-spline curve.

Parameter parameter_value:

Parameter value at which to evaluate the B-spline curve defined by this and control_points.

Precondition:

control_points.size() == num_basis_functions()

Precondition:

parameter_value ≥ initial_parameter_value()

Precondition:

parameter_value ≤ final_parameter_value()

final_parameter_value(self: pydrake.math.BsplineBasis_[Expression]) → pydrake.symbolic.Expression

The maximum allowable parameter value for B-spline curves using this basis (tₙ₊₁ in the class description).

FindContainingInterval(self: pydrake.math.BsplineBasis_[Expression], parameter_value: pydrake.symbolic.Expression) → int

For a parameter_value = t, the interval that contains it is the pair of knot values [tᵢ, tᵢ₊₁] for the greatest i such that tᵢ ≤ t and tᵢ < final_parameter_value(). This function returns that value of i.

Precondition:

parameter_value ≥ initial_parameter_value()

Precondition:

parameter_value ≤ final_parameter_value()

initial_parameter_value(self: pydrake.math.BsplineBasis_[Expression]) → pydrake.symbolic.Expression

The minimum allowable parameter value for B-spline curves using this basis (tₖ₋₁ in the class description).

knots(self: pydrake.math.BsplineBasis_[Expression]) → list[pydrake.symbolic.Expression]

The knot vector of this B-spline basis (the vector (t₀, t₁, …, tₘ)’ in the class description).

num_basis_functions(self: pydrake.math.BsplineBasis_[Expression]) → int

The number of basis functions in this B-spline basis (n + 1 in the class description).

order(self: pydrake.math.BsplineBasis_[Expression]) → int

The order of this B-spline basis (k in the class description).

pydrake.math.CalculateAngularVelocityExpressedInBFromQuaternionDt(*args, **kwargs)

Overloaded function.

  1. CalculateAngularVelocityExpressedInBFromQuaternionDt(quat_AB: pydrake.common.eigen_geometry.Quaternion, quatDt: numpy.ndarray[numpy.float64[4, 1]]) -> numpy.ndarray[numpy.float64[3, 1]]

This function calculates angular velocity from a quaternion and its time- derivative. Algorithm from [Kane, 1983] Section 1.13, Pages 58-59.

Parameter quat_AB:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat_AB is analogous to the rotation matrix R_AB.

Parameter quatDt:

Time-derivative of quat_AB, i.e. [ẇ, ẋ, ẏ, ż].

Returns w_AB_B:

B’s angular velocity in A, expressed in B.

  1. CalculateAngularVelocityExpressedInBFromQuaternionDt(quat_AB: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd], quatDt: numpy.ndarray[object[4, 1]]) -> numpy.ndarray[object[3, 1]]

This function calculates angular velocity from a quaternion and its time- derivative. Algorithm from [Kane, 1983] Section 1.13, Pages 58-59.

Parameter quat_AB:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat_AB is analogous to the rotation matrix R_AB.

Parameter quatDt:

Time-derivative of quat_AB, i.e. [ẇ, ẋ, ẏ, ż].

Returns w_AB_B:

B’s angular velocity in A, expressed in B.

  1. CalculateAngularVelocityExpressedInBFromQuaternionDt(quat_AB: pydrake.common.eigen_geometry.Quaternion_[Expression], quatDt: numpy.ndarray[object[4, 1]]) -> numpy.ndarray[object[3, 1]]

This function calculates angular velocity from a quaternion and its time- derivative. Algorithm from [Kane, 1983] Section 1.13, Pages 58-59.

Parameter quat_AB:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat_AB is analogous to the rotation matrix R_AB.

Parameter quatDt:

Time-derivative of quat_AB, i.e. [ẇ, ẋ, ẏ, ż].

Returns w_AB_B:

B’s angular velocity in A, expressed in B.

pydrake.math.CalculateQuaternionDtConstraintViolation(*args, **kwargs)

Overloaded function.

  1. CalculateQuaternionDtConstraintViolation(quat: pydrake.common.eigen_geometry.Quaternion, quatDt: numpy.ndarray[numpy.float64[4, 1]]) -> float

This function calculates how well a quaternion and its time-derivative satisfy the quaternion time-derivative constraint specified in [Kane, 1983] Section 1.13, equations 12-13, page 59. For a quaternion [w, x, y, z], the quaternion must satisfy: w^2 + x^2 + y^2 + z^2 = 1, hence its time-derivative must satisfy: 2*(w*ẇ + x*ẋ + y*ẏ + z*ż) = 0.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: A quaternion like quat_AB is analogous to the rotation matrix R_AB.

Parameter quatDt:

Time-derivative of quat, i.e., [ẇ, ẋ, ẏ, ż].

Returns quaternionDt_constraint_violation:

The amount the time- derivative of the quaternion constraint has been violated, which may be positive or negative (0 means the constraint is perfectly satisfied).

  1. CalculateQuaternionDtConstraintViolation(quat: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd], quatDt: numpy.ndarray[object[4, 1]]) -> pydrake.autodiffutils.AutoDiffXd

This function calculates how well a quaternion and its time-derivative satisfy the quaternion time-derivative constraint specified in [Kane, 1983] Section 1.13, equations 12-13, page 59. For a quaternion [w, x, y, z], the quaternion must satisfy: w^2 + x^2 + y^2 + z^2 = 1, hence its time-derivative must satisfy: 2*(w*ẇ + x*ẋ + y*ẏ + z*ż) = 0.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: A quaternion like quat_AB is analogous to the rotation matrix R_AB.

Parameter quatDt:

Time-derivative of quat, i.e., [ẇ, ẋ, ẏ, ż].

Returns quaternionDt_constraint_violation:

The amount the time- derivative of the quaternion constraint has been violated, which may be positive or negative (0 means the constraint is perfectly satisfied).

  1. CalculateQuaternionDtConstraintViolation(quat: pydrake.common.eigen_geometry.Quaternion_[Expression], quatDt: numpy.ndarray[object[4, 1]]) -> pydrake.symbolic.Expression

This function calculates how well a quaternion and its time-derivative satisfy the quaternion time-derivative constraint specified in [Kane, 1983] Section 1.13, equations 12-13, page 59. For a quaternion [w, x, y, z], the quaternion must satisfy: w^2 + x^2 + y^2 + z^2 = 1, hence its time-derivative must satisfy: 2*(w*ẇ + x*ẋ + y*ẏ + z*ż) = 0.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: A quaternion like quat_AB is analogous to the rotation matrix R_AB.

Parameter quatDt:

Time-derivative of quat, i.e., [ẇ, ẋ, ẏ, ż].

Returns quaternionDt_constraint_violation:

The amount the time- derivative of the quaternion constraint has been violated, which may be positive or negative (0 means the constraint is perfectly satisfied).

pydrake.math.CalculateQuaternionDtFromAngularVelocityExpressedInB(*args, **kwargs)

Overloaded function.

  1. CalculateQuaternionDtFromAngularVelocityExpressedInB(quat_AB: pydrake.common.eigen_geometry.Quaternion, w_AB_B: numpy.ndarray[numpy.float64[3, 1]]) -> numpy.ndarray[numpy.float64[4, 1]]

This function calculates a quaternion’s time-derivative from its quaternion and angular velocity. Algorithm from [Kane, 1983] Section 1.13, Pages 58-59.

Parameter quat_AB:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat_AB is analogous to the rotation matrix R_AB.

Parameter w_AB_B:

B’s angular velocity in A, expressed in B.

Returns quatDt:

Time-derivative of quat_AB, i.e., [ẇ, ẋ, ẏ, ż].

  1. CalculateQuaternionDtFromAngularVelocityExpressedInB(quat_AB: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd], w_AB_B: numpy.ndarray[object[3, 1]]) -> numpy.ndarray[object[4, 1]]

This function calculates a quaternion’s time-derivative from its quaternion and angular velocity. Algorithm from [Kane, 1983] Section 1.13, Pages 58-59.

Parameter quat_AB:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat_AB is analogous to the rotation matrix R_AB.

Parameter w_AB_B:

B’s angular velocity in A, expressed in B.

Returns quatDt:

Time-derivative of quat_AB, i.e., [ẇ, ẋ, ẏ, ż].

  1. CalculateQuaternionDtFromAngularVelocityExpressedInB(quat_AB: pydrake.common.eigen_geometry.Quaternion_[Expression], w_AB_B: numpy.ndarray[object[3, 1]]) -> numpy.ndarray[object[4, 1]]

This function calculates a quaternion’s time-derivative from its quaternion and angular velocity. Algorithm from [Kane, 1983] Section 1.13, Pages 58-59.

Parameter quat_AB:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat_AB is analogous to the rotation matrix R_AB.

Parameter w_AB_B:

B’s angular velocity in A, expressed in B.

Returns quatDt:

Time-derivative of quat_AB, i.e., [ẇ, ẋ, ẏ, ż].

pydrake.math.ceil(*args, **kwargs)

Overloaded function.

  1. ceil(arg0: float) -> float

  2. ceil(arg0: pydrake.autodiffutils.AutoDiffXd) -> float

  3. ceil(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.ClosestQuaternion(*args, **kwargs)

Overloaded function.

  1. ClosestQuaternion(quat1: pydrake.common.eigen_geometry.Quaternion, quat2: pydrake.common.eigen_geometry.Quaternion) -> pydrake.common.eigen_geometry.Quaternion

Returns a unit quaternion that represents the same orientation as quat2, and has the “shortest” geodesic distance on the unit sphere to quat1.

  1. ClosestQuaternion(quat1: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd], quat2: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]) -> pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]

Returns a unit quaternion that represents the same orientation as quat2, and has the “shortest” geodesic distance on the unit sphere to quat1.

  1. ClosestQuaternion(quat1: pydrake.common.eigen_geometry.Quaternion_[Expression], quat2: pydrake.common.eigen_geometry.Quaternion_[Expression]) -> pydrake.common.eigen_geometry.Quaternion_[Expression]

Returns a unit quaternion that represents the same orientation as quat2, and has the “shortest” geodesic distance on the unit sphere to quat1.

pydrake.math.ComputeNumericalGradient(calc_func: Callable[[numpy.ndarray[numpy.float64[m, 1]]], numpy.ndarray[numpy.float64[m, 1]]], x: numpy.ndarray[numpy.float64[m, 1]], option: pydrake.math.NumericalGradientOption = <NumericalGradientOption(<NumericalGradientMethod.kForward: 0>)>) → numpy.ndarray[numpy.float64[m, n]]

Compute the gradient of a function f(x) through numerical difference.

Parameter calc_func:

calc_func(x, &y) computes the value of f(x), and stores the value in y. calc_func is responsible for properly resizing the output y when it consists of an Eigen vector of Eigen::Dynamic size.

Parameter x:

The point at which the numerical gradient is computed.

Parameter option:

The options for computing numerical gradient.

Template parameter DerivedX:

an Eigen column vector.

Template parameter DerivedY:

an Eigen column vector.

Template parameter DerivedCalcX:

The type of x in the calc_func. Must be an Eigen column vector. It is possible to have DerivedCalcX being different from DerivedX, for example, calc_func could be solvers::EvaluatorBase(const Eigen::Ref<const Eigen::VectorXd>&, Eigen::VectorXd*), but x could be of type Eigen::VectorXd. TODO(hongkai.dai): understand why the default template DerivedCalcX = DerivedX doesn’t compile when I instantiate ComputeNumericalGradient<DerivedX, DerivedY>(calc_func, x);

Returns gradient:

a matrix of size x.rows() x y.rows(). gradient(i, j) is ∂f(i) / ∂x(j)

Examples:

Click to expand C++ code...
{cc}
// Create a std::function from a lambda expression.
std::function<void (const Eigen::Vector2d&, Vector3d*)> foo = [](const
Eigen::Vector2d& x, Vector3d*y) { (*y)(0) = x(0); (*y)(1) = x(0) * x(1);
(*y)(2) = x(0) * std::sin(x(1));};
Eigen::Vector3d x_eval(1, 2, 3);
auto J = ComputeNumericalGradient(foo, x_eval);
// Note that if we pass in a lambda to ComputeNumericalGradient, then
// ComputeNumericalGradient has to instantiate the template types explicitly,
// as in this example. The issue of template deduction with std::function is
// explained in
//
https://stackoverflow.com/questions/48529410/template-arguments-deduction-failed-passing-func-pointer-to-stdfunction
auto bar = [](const Eigen::Vector2d& x, Eigen::Vector2d* y) {*y = x; };
auto J2 = ComputeNumericalGradient<Eigen::Vector2d,
Eigen::Vector2d, Eigen::Vector2d>(bar, Eigen::Vector2d(2, 3));
pydrake.math.ContinuousAlgebraicRiccatiEquation(A: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], B: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], Q: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], R: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) → numpy.ndarray[numpy.float64[m, n]]

Computes the unique stabilizing solution S to the continuous-time algebraic Riccati equation:

\[S A + A' S - S B R^{-1} B' S + Q = 0\]
Raises

RuntimeError if the Hamiltanoian matrix –

Click to expand C++ code...
⌈A   BR⁻¹Bᵀ⌉
⌊Q      −Aᵀ⌋

is not invertible.

Raises

RuntimeError if R is not positive definite. –

Note

the pair (A, B) should be stabilizable, and (Q, A) should be detectable. For more information, please refer to page 526-527 of Linear Systems by Thomas Kailath.

Based on the Matrix Sign Function method outlined in this paper: http://www.engr.iupui.edu/~skoskie/ECE684/Riccati_algorithms.pdf

pydrake.math.cos(*args, **kwargs)

Overloaded function.

  1. cos(arg0: float) -> float

  2. cos(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. cos(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.cosh(*args, **kwargs)

Overloaded function.

  1. cosh(arg0: float) -> float

  2. cosh(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. cosh(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.DecomposePositiveQuadraticForm(Q: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], b: numpy.ndarray[numpy.float64[m, 1]], c: float, tol: float = 0) → tuple[numpy.ndarray[numpy.float64[m, n]], numpy.ndarray[numpy.float64[m, n]]]

Rewrite a quadratic form xᵀQx + bᵀx + c to (Rx+d)ᵀ(Rx+d) where

Click to expand C++ code...
RᵀR = Q
Rᵀd = b / 2
dᵀd = c

This decomposition requires the matrix

Click to expand C++ code...
⌈Q     b/2⌉
⌊bᵀ/2    c⌋

to be positive semidefinite.

We return R and d with the minimal number of rows, namely the rows of R and d equal to the rank of the matrix

Click to expand C++ code...
⌈Q     b/2⌉
⌊bᵀ/2    c⌋

Notice that R might have more rows than Q, For example, the quadratic expression x² + 2x + 5 =(x+1)² + 2², it can be decomposed as

Click to expand C++ code...
⎛⌈1⌉ * x + ⌈1⌉⎞ᵀ * ⎛⌈1⌉ * x + ⌈1⌉⎞
⎝⌊0⌋       ⌊2⌋⎠    ⎝⌊0⌋       ⌊2⌋⎠

Here R has 2 rows while Q only has 1 row.

On the other hand the quadratic expression x² + 2x + 1 can be decomposed as (x+1) * (x+1), where R has 1 row, same as Q.

Also notice that this decomposition is not unique. For example, with any permutation matrix P, we can define

Click to expand C++ code...
R₁ = P*R
d₁ = P*d

Then (R₁*x+d₁)ᵀ(R₁*x+d₁) gives the same quadratic form.

Parameter Q:

The square matrix.

Parameter b:

The vector containing the linear coefficients.

Parameter c:

The constant term.

Parameter tol:

We will determine if this quadratic form is always non-negative, by checking the Eigen values of the matrix [Q b/2] [bᵀ/2 c] are all greater than -tol. $*Default:* is 0.

Returns (R:

, d). R and d have the same number of rows. R.cols() == x.rows(). R.rows() equals to the rank of the matrix

Click to expand C++ code...
[Q    b/2]
   [bᵀ/2   c]

Precondition: 1. The quadratic form is always non-negative, namely the matrix

Click to expand C++ code...
[Q    b/2]
        [bᵀ/2   c]

is positive semidefinite. 2. Q and b are of the correct size. 3. tol is non-negative.

Raises

RuntimeError if the precondition is not satisfied. –

pydrake.math.DecomposePSDmatrixIntoXtransposeTimesX(Y: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], zero_tol: float, return_empty_if_not_psd: bool = False) → numpy.ndarray[numpy.float64[m, n]]

For a symmetric positive semidefinite matrix Y, decompose it into XᵀX, where the number of rows in X equals to the rank of Y. Notice that this decomposition is not unique. For any orthonormal matrix U, s.t UᵀU = identity, X_prime = UX also satisfies X_primeᵀX_prime = Y. Here we only return one valid decomposition.

Parameter Y:

A symmetric positive semidefinite matrix.

Parameter zero_tol:

We will need to check if some value (for example, the absolute value of Y’s eigenvalues) is smaller than zero_tol. If it is, then we deem that value as 0.

Parameter return_empty_if_not_psd:

If true, then return an empty matrix of size 0-by-Y.cols() if Y is not PSD (either the decomposition fails or the resulting eigenvalues are less that zero_tol). If false (the default), then throw an exception if Y is not PSD. This option is particularly useful because it is brittle/expensive to test the exact success criteria before calling this function.

Returns X:

. The matrix X satisfies XᵀX = Y and X.rows() = rank(Y).

Precondition: 1. Y is positive semidefinite or return_empty_if_not_psd = true.

  1. zero_tol is non-negative.

$Raises:

RuntimeError when the pre-conditions are not satisfied.

Note

We only use the lower triangular part of Y.

pydrake.math.DiscreteAlgebraicRiccatiEquation(*args, **kwargs)

Overloaded function.

  1. DiscreteAlgebraicRiccatiEquation(A: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], B: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], Q: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], R: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) -> numpy.ndarray[numpy.float64[m, n]]

Computes the unique stabilizing solution X to the discrete-time algebraic Riccati equation:

AᵀXA − X − AᵀXB(BᵀXB + R)⁻¹BᵀXA + Q = 0

Raises

  • RuntimeError if Q is not symmetric positive semidefinite. –

  • RuntimeError if R is not symmetric positive definite. –

  • RuntimeError if (A, B) isn't a stabilizable pair. –

  • RuntimeError if (A, C) isn't a detectable pair where Q = CᵀC. –

  1. DiscreteAlgebraicRiccatiEquation(A: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], B: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], Q: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], R: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], N: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) -> numpy.ndarray[numpy.float64[m, n]]

Computes the unique stabilizing solution X to the discrete-time algebraic Riccati equation:

AᵀXA − X − (AᵀXB + N)(BᵀXB + R)⁻¹(BᵀXA + Nᵀ) + Q = 0

This is equivalent to solving the original DARE:

A₂ᵀXA₂ − X − A₂ᵀXB(BᵀXB + R)⁻¹BᵀXA₂ + Q₂ = 0

where A₂ and Q₂ are a change of variables:

A₂ = A − BR⁻¹Nᵀ and Q₂ = Q − NR⁻¹Nᵀ

This overload of the DARE is useful for finding the control law uₖ that minimizes the following cost function subject to xₖ₊₁ = Axₖ + Buₖ.

Click to expand C++ code...
∞ [xₖ]ᵀ[Q  N][xₖ]
J = Σ [uₖ] [Nᵀ R][uₖ] ΔT
k=0

This is a more general form of the following. The linear-quadratic regulator is the feedback control law uₖ that minimizes the following cost function subject to xₖ₊₁ = Axₖ + Buₖ:

Click to expand C++ code...
∞
J = Σ (xₖᵀQxₖ + uₖᵀRuₖ) ΔT
k=0

This can be refactored as:

Click to expand C++ code...
∞ [xₖ]ᵀ[Q 0][xₖ]
J = Σ [uₖ] [0 R][uₖ] ΔT
k=0
Raises

  • RuntimeError if Q₂ is not symmetric positive semidefinite. –

  • RuntimeError if R is not symmetric positive definite. –

  • RuntimeError if (A₂, B) isn't a stabilizable pair. –

  • RuntimeError if (A₂, C) isn't a detectable pair where Q₂ = CᵀC. –

pydrake.math.exp(*args, **kwargs)

Overloaded function.

  1. exp(arg0: float) -> float

  2. exp(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. exp(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.ExtractPrincipalSubmatrix(matrix: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], indices: set[int]) → numpy.ndarray[numpy.float64[m, n]]

Extracts the principal submatrix from the ordered set of indices. The indices must be in monotonically increasing order and non-empty. This method makes no assumptions about the symmetry of the matrix, nor that the matrix is square. However, all indices must be valid for both rows and columns.

pydrake.math.floor(*args, **kwargs)

Overloaded function.

  1. floor(arg0: float) -> float

  2. floor(arg0: pydrake.autodiffutils.AutoDiffXd) -> float

  3. floor(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.inv(*args, **kwargs)

Overloaded function.

  1. inv(arg0: numpy.ndarray[numpy.float64[m, n]]) -> numpy.ndarray[numpy.float64[m, n]]

  2. inv(arg0: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

  3. inv(arg0: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

pydrake.math.is_quaternion_in_canonical_form(*args, **kwargs)

Overloaded function.

  1. is_quaternion_in_canonical_form(quat: pydrake.common.eigen_geometry.Quaternion) -> bool

This function tests whether a quaternion is in “canonical form” meaning that it tests whether the quaternion [w, x, y, z] has a non-negative w value. Example: [-0.3, +0.4, +0.5, +0.707] is not in canonical form. Example: [+0.3, -0.4, -0.5, -0.707] is in canonical form.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat is analogous to the rotation matrix R_AB.

Returns

True if quat.w() is nonnegative (in canonical form), else False.

  1. is_quaternion_in_canonical_form(quat: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]) -> bool

This function tests whether a quaternion is in “canonical form” meaning that it tests whether the quaternion [w, x, y, z] has a non-negative w value. Example: [-0.3, +0.4, +0.5, +0.707] is not in canonical form. Example: [+0.3, -0.4, -0.5, -0.707] is in canonical form.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat is analogous to the rotation matrix R_AB.

Returns

True if quat.w() is nonnegative (in canonical form), else False.

  1. is_quaternion_in_canonical_form(quat: pydrake.common.eigen_geometry.Quaternion_[Expression]) -> pydrake.symbolic.Formula

This function tests whether a quaternion is in “canonical form” meaning that it tests whether the quaternion [w, x, y, z] has a non-negative w value. Example: [-0.3, +0.4, +0.5, +0.707] is not in canonical form. Example: [+0.3, -0.4, -0.5, -0.707] is in canonical form.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat is analogous to the rotation matrix R_AB.

Returns

True if quat.w() is nonnegative (in canonical form), else False.

pydrake.math.IsBothQuaternionAndQuaternionDtOK(*args, **kwargs)

Overloaded function.

  1. IsBothQuaternionAndQuaternionDtOK(quat: pydrake.common.eigen_geometry.Quaternion, quatDt: numpy.ndarray[numpy.float64[4, 1]], tolerance: float) -> bool

This function tests if a quaternion satisfies the time-derivative constraint specified in [Kane, 1983] Section 1.13, equation 13, page 59. A quaternion [w, x, y, z] must satisfy w^2 + x^2 + y^2 + z^2 = 1, hence its time-derivative must satisfy 2*(w*ẇ + x*ẋ + y*ẏ + z*ż) = 0. Note: To accurately test whether the time-derivative quaternion constraint is satisfied, the quaternion constraint is also tested to be accurate.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: A quaternion like quat_AB is analogous to the rotation matrix R_AB.

Parameter quatDt:

Time-derivative of quat, i.e., [ẇ, ẋ, ẏ, ż].

Parameter tolerance:

Tolerance for quaternion constraints.

Returns

True if both of the two previous constraints are within tolerance.

  1. IsBothQuaternionAndQuaternionDtOK(quat: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd], quatDt: numpy.ndarray[object[4, 1]], tolerance: float) -> bool

This function tests if a quaternion satisfies the time-derivative constraint specified in [Kane, 1983] Section 1.13, equation 13, page 59. A quaternion [w, x, y, z] must satisfy w^2 + x^2 + y^2 + z^2 = 1, hence its time-derivative must satisfy 2*(w*ẇ + x*ẋ + y*ẏ + z*ż) = 0. Note: To accurately test whether the time-derivative quaternion constraint is satisfied, the quaternion constraint is also tested to be accurate.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: A quaternion like quat_AB is analogous to the rotation matrix R_AB.

Parameter quatDt:

Time-derivative of quat, i.e., [ẇ, ẋ, ẏ, ż].

Parameter tolerance:

Tolerance for quaternion constraints.

Returns

True if both of the two previous constraints are within tolerance.

  1. IsBothQuaternionAndQuaternionDtOK(quat: pydrake.common.eigen_geometry.Quaternion_[Expression], quatDt: numpy.ndarray[object[4, 1]], tolerance: float) -> pydrake.symbolic.Formula

This function tests if a quaternion satisfies the time-derivative constraint specified in [Kane, 1983] Section 1.13, equation 13, page 59. A quaternion [w, x, y, z] must satisfy w^2 + x^2 + y^2 + z^2 = 1, hence its time-derivative must satisfy 2*(w*ẇ + x*ẋ + y*ẏ + z*ż) = 0. Note: To accurately test whether the time-derivative quaternion constraint is satisfied, the quaternion constraint is also tested to be accurate.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: A quaternion like quat_AB is analogous to the rotation matrix R_AB.

Parameter quatDt:

Time-derivative of quat, i.e., [ẇ, ẋ, ẏ, ż].

Parameter tolerance:

Tolerance for quaternion constraints.

Returns

True if both of the two previous constraints are within tolerance.

pydrake.math.isnan(*args, **kwargs)

Overloaded function.

  1. isnan(x: float) -> bool

  2. isnan(x: pydrake.autodiffutils.AutoDiffXd) -> bool

  3. isnan(x: pydrake.symbolic.Expression) -> pydrake.symbolic.Formula

pydrake.math.IsPositiveDefinite(matrix: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], tolerance: float = 0.0) → bool

Checks if a matrix is symmetric (with tolerance symmetry_tolerance – see IsSymmetric) and has all eigenvalues greater than eigenvalue_tolerance. eigenvalue_tolerance must be >= 0 – where 0 implies positive semi-definite (but is of course subject to all of the pitfalls of floating point).

To consider the numerical robustness of the eigenvalue estimation, we specifically check that min_eigenvalue >= eigenvalue_tolerance * max(1, max_abs_eigenvalue).

pydrake.math.IsQuaternionValid(*args, **kwargs)

Overloaded function.

  1. IsQuaternionValid(quat: pydrake.common.eigen_geometry.Quaternion, tolerance: float) -> bool

This function tests if a quaternion satisfies the quaternion constraint specified in [Kane, 1983] Section 1.3, equation 4, page 12, i.e., a quaternion [w, x, y, z] must satisfy: w^2 + x^2 + y^2 + z^2 = 1.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: A quaternion like quat_AB is analogous to the rotation matrix R_AB.

Parameter tolerance:

Tolerance for quaternion constraint, i.e., how much is w^2 + x^2 + y^2 + z^2 allowed to differ from 1.

Returns

True if the quaternion constraint is satisfied within tolerance.

  1. IsQuaternionValid(quat: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd], tolerance: float) -> bool

This function tests if a quaternion satisfies the quaternion constraint specified in [Kane, 1983] Section 1.3, equation 4, page 12, i.e., a quaternion [w, x, y, z] must satisfy: w^2 + x^2 + y^2 + z^2 = 1.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: A quaternion like quat_AB is analogous to the rotation matrix R_AB.

Parameter tolerance:

Tolerance for quaternion constraint, i.e., how much is w^2 + x^2 + y^2 + z^2 allowed to differ from 1.

Returns

True if the quaternion constraint is satisfied within tolerance.

  1. IsQuaternionValid(quat: pydrake.common.eigen_geometry.Quaternion_[Expression], tolerance: float) -> pydrake.symbolic.Formula

This function tests if a quaternion satisfies the quaternion constraint specified in [Kane, 1983] Section 1.3, equation 4, page 12, i.e., a quaternion [w, x, y, z] must satisfy: w^2 + x^2 + y^2 + z^2 = 1.

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: A quaternion like quat_AB is analogous to the rotation matrix R_AB.

Parameter tolerance:

Tolerance for quaternion constraint, i.e., how much is w^2 + x^2 + y^2 + z^2 allowed to differ from 1.

Returns

True if the quaternion constraint is satisfied within tolerance.

pydrake.math.IsSymmetric(*args, **kwargs)

Overloaded function.

  1. IsSymmetric(matrix: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) -> bool

Determines if a matrix is symmetric. If std::equal_to<>()(matrix(i, j), matrix(j, i)) is true for all i, j, then the matrix is symmetric.

  1. IsSymmetric(matrix: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], precision: float) -> bool

Determines if a matrix is symmetric based on whether the difference between matrix(i, j) and matrix(j, i) is smaller than precision for all i, j. The precision is absolute. Matrix with nan or inf entries is not allowed.

class pydrake.math.KnotVectorType

Enum representing types of knot vectors. “Uniform” refers to the spacing between the knots. “Clamped” indicates that the first and last knots have multiplicity equal to the order of the spline.

Reference: http://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node17.html

Members:

kUniform :

kClampedUniform :

__init__(self: pydrake.math.KnotVectorType, value: int) → None
kClampedUniform = <KnotVectorType.kClampedUniform: 1>
kUniform = <KnotVectorType.kUniform: 0>
property name
property value
pydrake.math.log(*args, **kwargs)

Overloaded function.

  1. log(arg0: float) -> float

  2. log(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. log(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.matmul(*args, **kwargs)

Overloaded function.

  1. matmul(arg0: numpy.ndarray[numpy.float64[m, n]], arg1: numpy.ndarray[numpy.float64[m, n]]) -> numpy.ndarray[numpy.float64[m, n]]

Matrix product for dtype=float @ dtype=float -> dtype=float. The numpy matmul A @ B is typically slow when multiplying user-defined dtypes such as AutoDiffXd or Expression. Use this function for better performance, e.g., matmul(A, B). For a dtype=float @ dtype=float, this might be a little slower than numpy, but is provided here for convenience so that the user doesn’t need to be overly careful about the dtype of arguments.

  1. matmul(arg0: numpy.ndarray[numpy.float64[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=float @ dtype=AutoDiffXd -> dtype=AutoDiffXd.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[numpy.float64[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=AutoDiffXd @ dtype=float -> dtype=AutoDiffXd.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=AutoDiffXd @ dtype=AutoDiffXd -> dtype=AutoDiffXd.

  1. matmul(arg0: numpy.ndarray[numpy.float64[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=float @ dtype=Variable -> dtype=Expression.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[numpy.float64[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Variable @ dtype=float -> dtype=Expression.

  1. matmul(arg0: numpy.ndarray[numpy.float64[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=float @ dtype=Monomial -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[numpy.float64[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Monomial @ dtype=float -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[numpy.float64[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=float @ dtype=Polynomial -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[numpy.float64[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Polynomial @ dtype=float -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[numpy.float64[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=float @ dtype=Expression -> dtype=Expression.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[numpy.float64[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Expression @ dtype=float -> dtype=Expression.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Variable @ dtype=Variable -> dtype=Expression.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Variable @ dtype=Monomial -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Monomial @ dtype=Variable -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Variable @ dtype=Polynomial -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Polynomial @ dtype=Variable -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Variable @ dtype=Expression -> dtype=Expression.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Expression @ dtype=Variable -> dtype=Expression.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Monomial @ dtype=Monomial -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Monomial @ dtype=Polynomial -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Polynomial @ dtype=Monomial -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Monomial @ dtype=Expression -> dtype=Expression.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Expression @ dtype=Monomial -> dtype=Expression.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Polynomial @ dtype=Polynomial -> dtype=Polynomial.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Polynomial @ dtype=Expression -> dtype=Expression.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Expression @ dtype=Polynomial -> dtype=Expression.

  1. matmul(arg0: numpy.ndarray[object[m, n]], arg1: numpy.ndarray[object[m, n]]) -> numpy.ndarray[object[m, n]]

Matrix product for dtype=Expression @ dtype=Expression -> dtype=Expression.

pydrake.math.max(*args, **kwargs)

Overloaded function.

  1. max(arg0: float, arg1: float) -> float

  2. max(arg0: pydrake.autodiffutils.AutoDiffXd, arg1: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. max(arg0: pydrake.symbolic.Expression, arg1: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.min(*args, **kwargs)

Overloaded function.

  1. min(arg0: float, arg1: float) -> float

  2. min(arg0: pydrake.autodiffutils.AutoDiffXd, arg1: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. min(arg0: pydrake.symbolic.Expression, arg1: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

class pydrake.math.NumericalGradientMethod

Members:

kForward : Compute the gradient as (f(x + Δx) - f(x)) / Δx, with Δx > 0

kBackward : Compute the gradient as (f(x) - f(x - Δx)) / Δx, with Δx > 0

kCentral : Compute the gradient as (f(x + Δx) - f(x - Δx)) / (2Δx), with Δx > 0

__init__(self: pydrake.math.NumericalGradientMethod, value: int) → None
kBackward = <NumericalGradientMethod.kBackward: 1>
kCentral = <NumericalGradientMethod.kCentral: 2>
kForward = <NumericalGradientMethod.kForward: 0>
property name
property value
class pydrake.math.NumericalGradientOption
__init__(self: pydrake.math.NumericalGradientOption, method: pydrake.math.NumericalGradientMethod, function_accuracy: float = 1e-15) → None
Parameter function_accuracy:

The accuracy of evaluating function f(x). For double-valued functions (with magnitude around 1), the accuracy is usually about 1E-15.

NumericalGradientMethod(self: pydrake.math.NumericalGradientOption) → pydrake.math.NumericalGradientMethod
perturbation_size(self: pydrake.math.NumericalGradientOption) → float
pydrake.math.pow(*args, **kwargs)

Overloaded function.

  1. pow(arg0: float, arg1: float) -> float

  2. pow(arg0: pydrake.autodiffutils.AutoDiffXd, arg1: float) -> pydrake.autodiffutils.AutoDiffXd

  3. pow(arg0: pydrake.symbolic.Expression, arg1: float) -> pydrake.symbolic.Expression

  4. pow(arg0: pydrake.symbolic.Expression, arg1: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.QuaternionToCanonicalForm(*args, **kwargs)

Overloaded function.

  1. QuaternionToCanonicalForm(quat: pydrake.common.eigen_geometry.Quaternion) -> pydrake.common.eigen_geometry.Quaternion

This function returns a quaternion in its “canonical form” meaning that it returns a quaternion [w, x, y, z] with a non-negative w. For example, if passed a quaternion [-0.3, +0.4, +0.5, +0.707], the function returns the quaternion’s canonical form [+0.3, -0.4, -0.5, -0.707].

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat is analogous to the rotation matrix R_AB.

Returns

Canonical form of quat, which means that either the original quat is returned or a quaternion representing the same orientation but with negated [w, x, y, z], to ensure a positive w in returned quaternion.

  1. QuaternionToCanonicalForm(quat: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]) -> pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]

This function returns a quaternion in its “canonical form” meaning that it returns a quaternion [w, x, y, z] with a non-negative w. For example, if passed a quaternion [-0.3, +0.4, +0.5, +0.707], the function returns the quaternion’s canonical form [+0.3, -0.4, -0.5, -0.707].

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat is analogous to the rotation matrix R_AB.

Returns

Canonical form of quat, which means that either the original quat is returned or a quaternion representing the same orientation but with negated [w, x, y, z], to ensure a positive w in returned quaternion.

  1. QuaternionToCanonicalForm(quat: pydrake.common.eigen_geometry.Quaternion_[Expression]) -> pydrake.common.eigen_geometry.Quaternion_[Expression]

This function returns a quaternion in its “canonical form” meaning that it returns a quaternion [w, x, y, z] with a non-negative w. For example, if passed a quaternion [-0.3, +0.4, +0.5, +0.707], the function returns the quaternion’s canonical form [+0.3, -0.4, -0.5, -0.707].

Parameter quat:

Quaternion [w, x, y, z] that relates two right-handed orthogonal unitary bases e.g., Ax, Ay, Az (A) to Bx, By, Bz (B). Note: quat is analogous to the rotation matrix R_AB.

Returns

Canonical form of quat, which means that either the original quat is returned or a quaternion representing the same orientation but with negated [w, x, y, z], to ensure a positive w in returned quaternion.

pydrake.math.RealContinuousLyapunovEquation(A: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], Q: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) → numpy.ndarray[numpy.float64[m, n]]
Parameter A:

A user defined real square matrix.

Parameter Q:

A user defined real symmetric matrix.

Precondition:

Q is a symmetric matrix.

Computes a unique solution X to the continuous Lyapunov equation: AᵀX + XA + Q = 0, where A is real and square, and Q is real, symmetric and of equal size as A.

Raises

  • RuntimeError if A or Q are not square matrices or do not have the –

  • same size. –

Limitations: Given the Eigenvalues of A as λ₁, …, λₙ, there exists a unique solution if and only if λᵢ + λ̅ⱼ ≠ 0 ∀ i,j, where λ̅ⱼ is the complex conjugate of λⱼ.

Raises

RuntimeError if the solution is not unique. –

There are no further limitations on the eigenvalues of A. Further, if all λᵢ have negative real parts, and if Q is positive semi-definite, then X is also positive semi-definite [1]. Therefore, if one searches for a Lyapunov function V(z) = zᵀXz for the stable linear system ż = Az, then the solution of the Lyapunov Equation AᵀX + XA + Q = 0 only returns a valid Lyapunov function if Q is positive semi-definite.

The implementation is based on SLICOT routine SB03MD [2]. Note the transformation Q = -C. The complexity of this routine is O(n³). If A is larger than 2-by-2, then a Schur factorization is performed.

Raises

RuntimeError if Schur factorization failed. –

A tolerance of ε is used to check if a double variable is equal to zero, where the default value for ε is 1e-10. It has been used to check (1) if λᵢ + λ̅ⱼ = 0, ∀ i,j; (2) if A is a 1-by-1 zero matrix; (3) if A’s trace or determinant is 0 when A is a 2-by-2 matrix.

[1] Bartels, R.H. and G.W. Stewart, “Solution of the Matrix Equation AX + XB = C,” Comm. of the ACM, Vol. 15, No. 9, 1972.

[2] http://slicot.org/objects/software/shared/doc/SB03MD.html

pydrake.math.RealDiscreteLyapunovEquation(A: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous], Q: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) → numpy.ndarray[numpy.float64[m, n]]
Parameter A:

A user defined real square matrix.

Parameter Q:

A user defined real symmetric matrix.

Precondition:

Q is a symmetric matrix.

Computes the unique solution X to the discrete Lyapunov equation: AᵀXA - X + Q = 0, where A is real and square, and Q is real, symmetric and of equal size as A.

Raises

  • RuntimeError if A or Q are not square matrices or do not have the –

  • same size. –

Limitations: Given the Eigenvalues of A as λ₁, …, λₙ, there exists a unique solution if and only if λᵢ * λⱼ ≠ 1 ∀ i,j and λᵢ ≠ ±1, ∀ i [1].

Raises

RuntimeError if the solution is not unique.[3] –

There are no further limitations on the eigenvalues of A. Further, if |λᵢ|<1, ∀ i, and if Q is positive semi-definite, then X is also positive semi-definite [2]. Therefore, if one searches for a Lyapunov function V(z) = zᵀXz for the stable linear system zₙ₊₁ = Azₙ, then the solution of the Lyapunov Equation AᵀXA - X + Q = 0 only returns a valid Lyapunov function if Q is positive semi-definite.

The implementation is based on SLICOT routine SB03MD [2]. Note the transformation Q = -C. The complexity of this routine is O(n³). If A is larger than 2-by-2, then a Schur factorization is performed.

Raises

RuntimeError if Schur factorization fails. –

A tolerance of ε is used to check if a double variable is equal to zero, where the default value for ε is 1e-10. It has been used to check (1) if λᵢ = ±1 ∀ i; (2) if λᵢ * λⱼ = 1, i ≠ j.

[1] Barraud, A.Y., “A numerical algorithm to solve AᵀXA - X = Q,” IEEE® Trans. Auto. Contr., AC-22, pp. 883-885, 1977.

[2] http://slicot.org/objects/software/shared/doc/SB03MD.html

[3] https://www.mathworks.com/help/control/ref/dlyap.html

class pydrake.math.RigidTransform

This class represents a proper rigid transform between two frames which can be regarded in two ways. A rigid transform describes the “pose” between two frames A and B (i.e., the relative orientation and position of A to B). Alternately, it can be regarded as a distance-preserving operator that can rotate and/or translate a rigid body without changing its shape or size (rigid) and without mirroring/reflecting the body (proper), e.g., it can add one position vector to another and express the result in a particular basis as p_AoQ_A = X_AB * p_BoQ_B (Q is any point). In many ways, this rigid transform class is conceptually similar to using a homogeneous matrix as a linear operator. See operator* documentation for an exception.

The class stores a RotationMatrix that relates right-handed orthogonal unit vectors Ax, Ay, Az fixed in frame A to right-handed orthogonal unit vectors Bx, By, Bz fixed in frame B. The class also stores a position vector from Ao (the origin of frame A) to Bo (the origin of frame B). The position vector is expressed in frame A. The monogram notation for the transform relating frame A to B is X_AB. The monogram notation for the rotation matrix relating A to B is R_AB. The monogram notation for the position vector from Ao to Bo is p_AoBo_A. See multibody_quantities for monogram notation for dynamics.

Note

This class does not store the frames associated with the transform and cannot enforce correct usage of this class. For example, it makes sense to multiply RigidTransforms as X_AB * X_BC, but not X_AB * X_CB.

Note

This class is not a 4x4 transformation matrix – even though its operator*() methods act mostly like 4x4 matrix multiplication. Instead, this class contains a 3x3 rotation matrix class and a 3x1 position vector. To convert this to a 3x4 matrix, use GetAsMatrix34(). To convert this to a 4x4 matrix, use GetAsMatrix4(). To convert this to an Eigen::Isometry, use GetAsIsometry().

Note

An isometry is sometimes regarded as synonymous with rigid transform. The RigidTransform class has important advantages over Eigen::Isometry. - RigidTransform is built on an underlying rigorous 3x3 RotationMatrix class that has significant functionality for 3D orientation. - In Debug builds, RigidTransform requires a valid 3x3 rotation matrix and a valid (non-NAN) position vector. Eigen::Isometry does not. - RigidTransform catches bugs that are undetected by Eigen::Isometry. - RigidTransform has additional functionality and ease-of-use, resulting in shorter, easier to write, and easier to read code. - The name Isometry is unfamiliar to many roboticists and dynamicists and for them Isometry.linear() is (for example) a counter-intuitive method name to return a rotation matrix.

Note

One of the constructors in this class provides an implicit conversion from an Eigen Translation to RigidTransform.

Authors:

Paul Mitiguy (2018) Original author.

Authors:

Drake team (see https://drake.mit.edu/credits).

Note

This class is templated; see RigidTransform_ for the list of instantiations.

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.RigidTransform) -> None

Constructs the RigidTransform that corresponds to aligning the two frames so unit vectors Ax = Bx, Ay = By, Az = Bz and point Ao is coincident with Bo. Hence, the constructed RigidTransform contains an identity RotationMatrix and a zero position vector.

  1. __init__(self: pydrake.math.RigidTransform, other: pydrake.math.RigidTransform) -> None

  2. __init__(self: pydrake.math.RigidTransform, R: pydrake.math.RotationMatrix, p: numpy.ndarray[numpy.float64[3, 1]]) -> None

Constructs a RigidTransform from a rotation matrix and a position vector.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

  1. __init__(self: pydrake.math.RigidTransform, rpy: pydrake.math.RollPitchYaw, p: numpy.ndarray[numpy.float64[3, 1]]) -> None

Constructs a RigidTransform from a RollPitchYaw and a position vector.

Parameter rpy:

a RollPitchYaw which is a Space-fixed (extrinsic) X-Y-Z rotation with “roll-pitch-yaw” angles [r, p, y] or equivalently a Body- fixed (intrinsic) Z-Y-X rotation with “yaw-pitch-roll” angles [y, p, r].

See also

RotationMatrix::RotationMatrix(const RollPitchYaw<T>&)

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

  1. __init__(self: pydrake.math.RigidTransform, quaternion: pydrake.common.eigen_geometry.Quaternion, p: numpy.ndarray[numpy.float64[3, 1]]) -> None

Constructs a RigidTransform from a Quaternion and a position vector.

Parameter quaternion:

a non-zero, finite quaternion which may or may not have unit length [i.e., quaternion.norm() does not have to be 1].

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

Raises

  • RuntimeError in debug builds if the rotation matrix that is built –

  • from quaternion is invalid. –

See also

RotationMatrix::RotationMatrix(const Eigen::Quaternion<T>&)

  1. __init__(self: pydrake.math.RigidTransform, theta_lambda: pydrake.common.eigen_geometry.AngleAxis, p: numpy.ndarray[numpy.float64[3, 1]]) -> None

Constructs a RigidTransform from an AngleAxis and a position vector.

Parameter theta_lambda:

an Eigen::AngleAxis whose associated axis (vector direction herein called lambda) is non-zero and finite, but which may or may not have unit length [i.e., lambda.norm() does not have to be 1].

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted ``p_AoBo_A

Raises

  • RuntimeError in debug builds if the rotation matrix that is built –

  • from theta_lambda` is invalid. –

See also

RotationMatrix::RotationMatrix(const Eigen::AngleAxis<T>&)

  1. __init__(self: pydrake.math.RigidTransform, R: pydrake.math.RotationMatrix) -> None

Constructs a RigidTransform with a given RotationMatrix and a zero position vector.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

  1. __init__(self: pydrake.math.RigidTransform, p: numpy.ndarray[numpy.float64[3, 1]]) -> None

Constructs a RigidTransform that contains an identity RotationMatrix and a given position vector p.

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

  1. __init__(self: pydrake.math.RigidTransform, pose: pydrake.common.eigen_geometry.Isometry3) -> None

Constructs a RigidTransform from an Eigen Isometry3.

Parameter pose:

Isometry3 that contains an allegedly valid rotation matrix R_AB and also contains a position vector p_AoBo_A from frame A’s origin to frame B’s origin. p_AoBo_A must be expressed in frame A.

Raises

  • RuntimeError in debug builds if R_AB is not a proper orthonormal –

  • 3x3 rotation matrix. –

Note

No attempt is made to orthogonalize the 3x3 rotation matrix part of pose. As needed, use RotationMatrix::ProjectToRotationMatrix().

  1. __init__(self: pydrake.math.RigidTransform, pose: numpy.ndarray[numpy.float64[m, n]]) -> None

Constructs a RigidTransform from an appropriate Eigen expression.

Parameter pose:

Generic Eigen matrix expression.

Raises

  • RuntimeError if the Eigen expression in pose does not resolve –

  • to a Vector3 or 3x4 matrix or 4x4 matrix or if the rotational part –

  • of pose is not a proper orthonormal 3x3 rotation matrix or if –

  • pose` is a 4x4 matrix whose final row is not [0, 0, 0, 1] –

Note

No attempt is made to orthogonalize the 3x3 rotation matrix part of pose. As needed, use RotationMatrix::ProjectToRotationMatrix().

Note

This constructor prevents ambiguity that would otherwise exist for a RigidTransform constructor whose argument is an Eigen expression.

Click to expand C++ code...
const Vector3<double> position(4, 5, 6);
const RigidTransform<double> X1(3 * position);
----------------------------------------------
const RotationMatrix<double> R(RollPitchYaw<double>(1, 2, 3));
Eigen::Matrix<double, 3, 4> pose34;
pose34 << R.matrix(), position;
const RigidTransform<double> X2(1.0 * pose34);
----------------------------------------------
Eigen::Matrix<double, 4, 4> pose4;
pose4 << R.matrix(), position,
         0, 0, 0, 1;
const RigidTransform<double> X3(pose4 * pose4);
template cast

Instantiations: cast[float], cast[AutoDiffXd], cast[Expression]

cast[AutoDiffXd](self: pydrake.math.RigidTransform) → pydrake.math.RigidTransform_[AutoDiffXd]

Creates a RigidTransform templatized on a scalar type U from a RigidTransform templatized on scalar type T. For example,

Click to expand C++ code...
RigidTransform<double> source = RigidTransform<double>::Identity();
RigidTransform<AutoDiffXd> foo = source.cast<AutoDiffXd>();
Template parameter U:

Scalar type on which the returned RigidTransform is templated.

Note

RigidTransform<From>::cast<To>() creates a new RigidTransform<To> from a RigidTransform<From> but only if type To is constructible from type From. This cast method works in accordance with Eigen’s cast method for Eigen’s objects that underlie this RigidTransform. For example, Eigen currently allows cast from type double to AutoDiffXd, but not vice-versa.

cast[Expression](self: pydrake.math.RigidTransform) → pydrake.math.RigidTransform_[Expression]

Creates a RigidTransform templatized on a scalar type U from a RigidTransform templatized on scalar type T. For example,

Click to expand C++ code...
RigidTransform<double> source = RigidTransform<double>::Identity();
RigidTransform<AutoDiffXd> foo = source.cast<AutoDiffXd>();
Template parameter U:

Scalar type on which the returned RigidTransform is templated.

Note

RigidTransform<From>::cast<To>() creates a new RigidTransform<To> from a RigidTransform<From> but only if type To is constructible from type From. This cast method works in accordance with Eigen’s cast method for Eigen’s objects that underlie this RigidTransform. For example, Eigen currently allows cast from type double to AutoDiffXd, but not vice-versa.

cast[float](self: pydrake.math.RigidTransform) → pydrake.math.RigidTransform

Creates a RigidTransform templatized on a scalar type U from a RigidTransform templatized on scalar type T. For example,

Click to expand C++ code...
RigidTransform<double> source = RigidTransform<double>::Identity();
RigidTransform<AutoDiffXd> foo = source.cast<AutoDiffXd>();
Template parameter U:

Scalar type on which the returned RigidTransform is templated.

Note

RigidTransform<From>::cast<To>() creates a new RigidTransform<To> from a RigidTransform<From> but only if type To is constructible from type From. This cast method works in accordance with Eigen’s cast method for Eigen’s objects that underlie this RigidTransform. For example, Eigen currently allows cast from type double to AutoDiffXd, but not vice-versa.

GetAsIsometry3(self: pydrake.math.RigidTransform) → pydrake.common.eigen_geometry.Isometry3

Returns the isometry in ℜ³ that is equivalent to a RigidTransform.

GetAsMatrix34(self: pydrake.math.RigidTransform) → numpy.ndarray[numpy.float64[3, 4]]

Returns the 3x4 matrix associated with this RigidTransform, i.e., X_AB.

Click to expand C++ code...
┌                ┐
 │ R_AB  p_AoBo_A │
 └                ┘
GetAsMatrix4(self: pydrake.math.RigidTransform) → numpy.ndarray[numpy.float64[4, 4]]

Returns the 4x4 matrix associated with this RigidTransform, i.e., X_AB.

Click to expand C++ code...
┌                ┐
 │ R_AB  p_AoBo_A │
 │                │
 │   0      1     │
 └                ┘
GetMaximumAbsoluteDifference(self: pydrake.math.RigidTransform, other: pydrake.math.RigidTransform) → float

Computes the infinity norm of this - other (i.e., the maximum absolute value of the difference between the elements of this and other).

Parameter other:

RigidTransform to subtract from this.

Returns

‖this - other‖∞

GetMaximumAbsoluteTranslationDifference(self: pydrake.math.RigidTransform, other: pydrake.math.RigidTransform) → float

Returns the maximum absolute value of the difference in the position vectors (translation) in this and other. In other words, returns the infinity norm of the difference in the position vectors.

Parameter other:

RigidTransform whose position vector is subtracted from the position vector in this.

static Identity() → pydrake.math.RigidTransform

Returns the identity RigidTransform (corresponds to coincident frames).

Returns

the RigidTransform that corresponds to aligning the two frames so unit vectors Ax = Bx, Ay = By, Az = Bz and point Ao is coincident with Bo. Hence, the returned RigidTransform contains a 3x3 identity matrix and a zero position vector.

inverse(self: pydrake.math.RigidTransform) → pydrake.math.RigidTransform

Returns X_BA = X_AB⁻¹, the inverse of this RigidTransform.

Note

The inverse of RigidTransform X_AB is X_BA, which contains the rotation matrix R_BA = R_AB⁻¹ = R_ABᵀ and the position vector p_BoAo_B_ (position from B’s origin Bo to A’s origin Ao, expressed in frame B).

Note

: The square-root of a RigidTransform’s condition number is roughly the magnitude of the position vector. The accuracy of the calculation for the inverse of a RigidTransform drops off with the sqrt condition number.

InvertAndCompose(self: pydrake.math.RigidTransform, other: pydrake.math.RigidTransform) → pydrake.math.RigidTransform

Calculates the product of this inverted and another RigidTransform. If you consider this to be the transform X_AB, and other to be X_AC, then this method returns X_BC = X_AB⁻¹ * X_AC. For T==double, this method can be much faster than inverting first and then performing the composition, because it can take advantage of the special structure of a rigid transform to avoid unnecessary memory and floating point operations. On some platforms it can use SIMD instructions for further speedups.

Parameter other:

RigidTransform that post-multiplies this inverted.

Returns X_BC:

where X_BC = this⁻¹ * other.

Note

It is possible (albeit improbable) to create an invalid rigid transform by accumulating round-off error with a large number of multiplies.

IsExactlyEqualTo(self: pydrake.math.RigidTransform, other: pydrake.math.RigidTransform) → bool

Returns true if this is exactly equal to other.

Parameter other:

RigidTransform to compare to this.

Returns

True if each element of this is exactly equal to the corresponding element of other.

IsExactlyIdentity(self: pydrake.math.RigidTransform) → bool

Returns True if this is exactly the identity RigidTransform.

See also

IsNearlyIdentity().

IsNearlyEqualTo(self: pydrake.math.RigidTransform, other: pydrake.math.RigidTransform, tolerance: float) → bool

Compares each element of this to the corresponding element of other to check if they are the same to within a specified tolerance.

Parameter other:

RigidTransform to compare to this.

Parameter tolerance:

maximum allowable absolute difference between the elements in this and other.

Returns

True if ‖this.matrix() - other.matrix()‖∞ <= tolerance.

Note

Consider scaling tolerance with the largest of magA and magB, where magA and magB denoted the magnitudes of this position vector and other position vectors, respectively.

IsNearlyIdentity(self: pydrake.math.RigidTransform, translation_tolerance: float) → bool

Returns true if this is within tolerance of the identity RigidTransform.

Parameter translation_tolerance:

a non-negative number. One way to choose translation_tolerance is to multiply a characteristic length (e.g., the magnitude of a characteristic position vector) by an epsilon (e.g., RotationMatrix::get_internal_tolerance_for_orthonormality()).

Returns

True if the RotationMatrix portion of this satisfies RotationMatrix::IsNearlyIdentity() and if the position vector portion of this is equal to zero vector within translation_tolerance.

See also

IsExactlyIdentity().

static MakeUnchecked(pose: numpy.ndarray[numpy.float64[3, 4]]) → pydrake.math.RigidTransform

(Advanced) Constructs a RigidTransform from a 3x4 matrix, without any validity checks nor orthogonalization.

Parameter pose:

3x4 matrix that contains a 3x3 rotation matrix R_AB and also a 3x1 position vector p_AoBo_A (the position vector from frame A’s origin to frame B’s origin, expressed in frame A).

Click to expand C++ code...
┌                ┐
 │ R_AB  p_AoBo_A │
 └                ┘
multiply(*args, **kwargs)

Overloaded function.

  1. multiply(self: pydrake.math.RigidTransform, other: pydrake.math.RigidTransform) -> pydrake.math.RigidTransform

Multiplies this RigidTransform X_AB by the other RigidTransform X_BC and returns the RigidTransform X_AC = X_AB * X_BC.

  1. multiply(self: pydrake.math.RigidTransform, p_BoQ_B: numpy.ndarray[numpy.float64[3, 1]]) -> numpy.ndarray[numpy.float64[3, 1]]

Multiplies this RigidTransform X_AB by the position vector p_BoQ_B which is from Bo (B’s origin) to an arbitrary point Q.

Parameter p_BoQ_B:

position vector from Bo to Q, expressed in frame B.

Returns p_AoQ_A:

position vector from Ao to Q, expressed in frame A.

  1. multiply(self: pydrake.math.RigidTransform, vec_B: numpy.ndarray[numpy.float64[4, 1]]) -> numpy.ndarray[numpy.float64[4, 1]]

Multiplies this RigidTransform X_AB by the 4-element vector vec_B, equivalent to X_AB.GetAsMatrix4() * vec_B.

Parameter vec_B:

4-element vector whose first 3 elements are the position vector p_BoQ_B from Bo (frame B’s origin) to an arbitrary point Q, expressed in frame B and whose 4ᵗʰ element is 1 𝐨𝐫 whose first 3 elements are a vector (maybe unrelated to Bo or Q) and whose 4ᵗʰ element is 0.

Returns vec_A:

4-element vector whose first 3 elements are the position vector p_AoQ_A from Ao (frame A’s origin) to Q, expressed in frame A and whose 4ᵗʰ element is 1 𝐨𝐫 whose first 3 elements are a vector (maybe unrelated to Bo and Q) and whose 4ᵗʰ element is 0.

Raises

RuntimeError if the 4ᵗʰ element of vec_B is not 0 or 1. –

  1. multiply(self: pydrake.math.RigidTransform, p_BoQ_B: numpy.ndarray[numpy.float64[3, n]]) -> numpy.ndarray[numpy.float64[3, n]]

Multiplies this RigidTransform X_AB by the n position vectors p_BoQ1_B … p_BoQn_B, where p_BoQi_B is the iᵗʰ position vector from Bo (frame B’s origin) to an arbitrary point Qi, expressed in frame B.

Parameter p_BoQ_B:

3 x n matrix with n position vectors p_BoQi_B or an expression that resolves to a 3 x n matrix of position vectors.

Returns p_AoQ_A:

3 x n matrix with n position vectors p_AoQi_A, i.e., n position vectors from Ao (frame A’s origin) to Qi, expressed in frame A. Specifically, this operator* is defined so that X_AB * p_BoQ_B returns p_AoQ_A = p_AoBo_A + R_AB * p_BoQ_B, where p_AoBo_A is the position vector from Ao to Bo expressed in A and R_AB is the rotation matrix relating the orientation of frames A and B.

Note

As needed, use parentheses. This operator* is not associative. To see this, let p = p_AoBo_A, q = p_BoQ_B and note (X_AB * q) * 7 = (p + R_AB * q) * 7 ≠ X_AB * (q * 7) = p + R_AB * (q * 7).

Click to expand C++ code...
const RollPitchYaw<double> rpy(0.1, 0.2, 0.3);
const RigidTransform<double> X_AB(rpy, Vector3d(1, 2, 3));
Eigen::Matrix<double, 3, 2> p_BoQ_B;
p_BoQ_B.col(0) = Vector3d(4, 5, 6);
p_BoQ_B.col(1) = Vector3d(9, 8, 7);
const Eigen::Matrix<double, 3, 2> p_AoQ_A = X_AB * p_BoQ_B;
rotation(self: pydrake.math.RigidTransform) → pydrake.math.RotationMatrix

Returns R_AB, the rotation matrix portion of this RigidTransform.

Returns R_AB:

the rotation matrix portion of this RigidTransform.

set(self: pydrake.math.RigidTransform, R: pydrake.math.RotationMatrix, p: numpy.ndarray[numpy.float64[3, 1]]) → None

Sets this RigidTransform from a RotationMatrix and a position vector.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

set_rotation(*args, **kwargs)

Overloaded function.

  1. set_rotation(self: pydrake.math.RigidTransform, R: pydrake.math.RotationMatrix) -> None

Sets the RotationMatrix portion of this RigidTransform.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

  1. set_rotation(self: pydrake.math.RigidTransform, rpy: pydrake.math.RollPitchYaw) -> None

Sets the rotation part of this RigidTransform from a RollPitchYaw.

Parameter rpy:

“roll-pitch-yaw” angles.

See also

RotationMatrix::RotationMatrix(const RollPitchYaw<T>&) which describes the parameter, preconditions, etc.

  1. set_rotation(self: pydrake.math.RigidTransform, quaternion: pydrake.common.eigen_geometry.Quaternion) -> None

Sets the rotation part of this RigidTransform from a Quaternion.

Parameter quaternion:

a quaternion which may or may not have unit length.

See also

RotationMatrix::RotationMatrix(const Eigen::Quaternion<T>&) which describes the parameter, preconditions, exception conditions, etc.

  1. set_rotation(self: pydrake.math.RigidTransform, theta_lambda: pydrake.common.eigen_geometry.AngleAxis) -> None

Sets the rotation part of this RigidTransform from an AngleAxis.

Parameter theta_lambda:

an angle theta (in radians) and vector lambda.

See also

RotationMatrix::RotationMatrix(const Eigen::AngleAxis<T>&) which describes the parameter, preconditions, exception conditions, etc.

set_translation(self: pydrake.math.RigidTransform, p: numpy.ndarray[numpy.float64[3, 1]]) → None

Sets the position vector portion of this RigidTransform.

Parameter p:

position vector from Ao (frame A’s origin) to Bo (frame B’s origin) expressed in frame A. In monogram notation p is denoted p_AoBo_A.

SetFromIsometry3(self: pydrake.math.RigidTransform, pose: pydrake.common.eigen_geometry.Isometry3) → None

Sets this RigidTransform from an Eigen Isometry3.

Parameter pose:

Isometry3 that contains an allegedly valid rotation matrix R_AB and also contains a position vector p_AoBo_A from frame A’s origin to frame B’s origin. p_AoBo_A must be expressed in frame A.

Raises

  • RuntimeError in debug builds if R_AB is not a proper orthonormal –

  • 3x3 rotation matrix. –

Note

No attempt is made to orthogonalize the 3x3 rotation matrix part of pose. As needed, use RotationMatrix::ProjectToRotationMatrix().

SetIdentity(self: pydrake.math.RigidTransform) → pydrake.math.RigidTransform

Sets this RigidTransform so it corresponds to aligning the two frames so unit vectors Ax = Bx, Ay = By, Az = Bz and point Ao is coincident with Bo. Hence, this RigidTransform contains a 3x3 identity matrix and a zero position vector.

translation(self: pydrake.math.RigidTransform) → numpy.ndarray[numpy.float64[3, 1]]

Returns p_AoBo_A, the position vector portion of this RigidTransform, i.e., position vector from Ao (frame A’s origin) to Bo (frame B’s origin).

template pydrake.math.RigidTransform_

Instantiations: RigidTransform_[float], RigidTransform_[AutoDiffXd], RigidTransform_[Expression]

class pydrake.math.RigidTransform_[AutoDiffXd]

This class represents a proper rigid transform between two frames which can be regarded in two ways. A rigid transform describes the “pose” between two frames A and B (i.e., the relative orientation and position of A to B). Alternately, it can be regarded as a distance-preserving operator that can rotate and/or translate a rigid body without changing its shape or size (rigid) and without mirroring/reflecting the body (proper), e.g., it can add one position vector to another and express the result in a particular basis as p_AoQ_A = X_AB * p_BoQ_B (Q is any point). In many ways, this rigid transform class is conceptually similar to using a homogeneous matrix as a linear operator. See operator* documentation for an exception.

The class stores a RotationMatrix that relates right-handed orthogonal unit vectors Ax, Ay, Az fixed in frame A to right-handed orthogonal unit vectors Bx, By, Bz fixed in frame B. The class also stores a position vector from Ao (the origin of frame A) to Bo (the origin of frame B). The position vector is expressed in frame A. The monogram notation for the transform relating frame A to B is X_AB. The monogram notation for the rotation matrix relating A to B is R_AB. The monogram notation for the position vector from Ao to Bo is p_AoBo_A. See multibody_quantities for monogram notation for dynamics.

Note

This class does not store the frames associated with the transform and cannot enforce correct usage of this class. For example, it makes sense to multiply RigidTransforms as X_AB * X_BC, but not X_AB * X_CB.

Note

This class is not a 4x4 transformation matrix – even though its operator*() methods act mostly like 4x4 matrix multiplication. Instead, this class contains a 3x3 rotation matrix class and a 3x1 position vector. To convert this to a 3x4 matrix, use GetAsMatrix34(). To convert this to a 4x4 matrix, use GetAsMatrix4(). To convert this to an Eigen::Isometry, use GetAsIsometry().

Note

An isometry is sometimes regarded as synonymous with rigid transform. The RigidTransform class has important advantages over Eigen::Isometry. - RigidTransform is built on an underlying rigorous 3x3 RotationMatrix class that has significant functionality for 3D orientation. - In Debug builds, RigidTransform requires a valid 3x3 rotation matrix and a valid (non-NAN) position vector. Eigen::Isometry does not. - RigidTransform catches bugs that are undetected by Eigen::Isometry. - RigidTransform has additional functionality and ease-of-use, resulting in shorter, easier to write, and easier to read code. - The name Isometry is unfamiliar to many roboticists and dynamicists and for them Isometry.linear() is (for example) a counter-intuitive method name to return a rotation matrix.

Note

One of the constructors in this class provides an implicit conversion from an Eigen Translation to RigidTransform.

Authors:

Paul Mitiguy (2018) Original author.

Authors:

Drake team (see https://drake.mit.edu/credits).

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.RigidTransform_[AutoDiffXd]) -> None

Constructs the RigidTransform that corresponds to aligning the two frames so unit vectors Ax = Bx, Ay = By, Az = Bz and point Ao is coincident with Bo. Hence, the constructed RigidTransform contains an identity RotationMatrix and a zero position vector.

  1. __init__(self: pydrake.math.RigidTransform_[AutoDiffXd], other: pydrake.math.RigidTransform_[AutoDiffXd]) -> None

  2. __init__(self: pydrake.math.RigidTransform_[AutoDiffXd], R: pydrake.math.RotationMatrix_[AutoDiffXd], p: numpy.ndarray[object[3, 1]]) -> None

Constructs a RigidTransform from a rotation matrix and a position vector.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

  1. __init__(self: pydrake.math.RigidTransform_[AutoDiffXd], rpy: pydrake.math.RollPitchYaw_[AutoDiffXd], p: numpy.ndarray[object[3, 1]]) -> None

Constructs a RigidTransform from a RollPitchYaw and a position vector.

Parameter rpy:

a RollPitchYaw which is a Space-fixed (extrinsic) X-Y-Z rotation with “roll-pitch-yaw” angles [r, p, y] or equivalently a Body- fixed (intrinsic) Z-Y-X rotation with “yaw-pitch-roll” angles [y, p, r].

See also

RotationMatrix::RotationMatrix(const RollPitchYaw<T>&)

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

  1. __init__(self: pydrake.math.RigidTransform_[AutoDiffXd], quaternion: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd], p: numpy.ndarray[object[3, 1]]) -> None

Constructs a RigidTransform from a Quaternion and a position vector.

Parameter quaternion:

a non-zero, finite quaternion which may or may not have unit length [i.e., quaternion.norm() does not have to be 1].

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

Raises

  • RuntimeError in debug builds if the rotation matrix that is built –

  • from quaternion is invalid. –

See also

RotationMatrix::RotationMatrix(const Eigen::Quaternion<T>&)

  1. __init__(self: pydrake.math.RigidTransform_[AutoDiffXd], theta_lambda: pydrake.common.eigen_geometry.AngleAxis_[AutoDiffXd], p: numpy.ndarray[object[3, 1]]) -> None

Constructs a RigidTransform from an AngleAxis and a position vector.

Parameter theta_lambda:

an Eigen::AngleAxis whose associated axis (vector direction herein called lambda) is non-zero and finite, but which may or may not have unit length [i.e., lambda.norm() does not have to be 1].

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted ``p_AoBo_A

Raises

  • RuntimeError in debug builds if the rotation matrix that is built –

  • from theta_lambda` is invalid. –

See also

RotationMatrix::RotationMatrix(const Eigen::AngleAxis<T>&)

  1. __init__(self: pydrake.math.RigidTransform_[AutoDiffXd], R: pydrake.math.RotationMatrix_[AutoDiffXd]) -> None

Constructs a RigidTransform with a given RotationMatrix and a zero position vector.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

  1. __init__(self: pydrake.math.RigidTransform_[AutoDiffXd], p: numpy.ndarray[object[3, 1]]) -> None

Constructs a RigidTransform that contains an identity RotationMatrix and a given position vector p.

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

  1. __init__(self: pydrake.math.RigidTransform_[AutoDiffXd], pose: pydrake.common.eigen_geometry.Isometry3_[AutoDiffXd]) -> None

Constructs a RigidTransform from an Eigen Isometry3.

Parameter pose:

Isometry3 that contains an allegedly valid rotation matrix R_AB and also contains a position vector p_AoBo_A from frame A’s origin to frame B’s origin. p_AoBo_A must be expressed in frame A.

Raises

  • RuntimeError in debug builds if R_AB is not a proper orthonormal –

  • 3x3 rotation matrix. –

Note

No attempt is made to orthogonalize the 3x3 rotation matrix part of pose. As needed, use RotationMatrix::ProjectToRotationMatrix().

  1. __init__(self: pydrake.math.RigidTransform_[AutoDiffXd], pose: numpy.ndarray[object[m, n]]) -> None

Constructs a RigidTransform from an appropriate Eigen expression.

Parameter pose:

Generic Eigen matrix expression.

Raises

  • RuntimeError if the Eigen expression in pose does not resolve –

  • to a Vector3 or 3x4 matrix or 4x4 matrix or if the rotational part –

  • of pose is not a proper orthonormal 3x3 rotation matrix or if –

  • pose` is a 4x4 matrix whose final row is not [0, 0, 0, 1] –

Note

No attempt is made to orthogonalize the 3x3 rotation matrix part of pose. As needed, use RotationMatrix::ProjectToRotationMatrix().

Note

This constructor prevents ambiguity that would otherwise exist for a RigidTransform constructor whose argument is an Eigen expression.

Click to expand C++ code...
const Vector3<double> position(4, 5, 6);
const RigidTransform<double> X1(3 * position);
----------------------------------------------
const RotationMatrix<double> R(RollPitchYaw<double>(1, 2, 3));
Eigen::Matrix<double, 3, 4> pose34;
pose34 << R.matrix(), position;
const RigidTransform<double> X2(1.0 * pose34);
----------------------------------------------
Eigen::Matrix<double, 4, 4> pose4;
pose4 << R.matrix(), position,
         0, 0, 0, 1;
const RigidTransform<double> X3(pose4 * pose4);
template cast

Instantiations: cast[AutoDiffXd]

cast[AutoDiffXd](self: pydrake.math.RigidTransform_[AutoDiffXd]) → pydrake.math.RigidTransform_[AutoDiffXd]

Creates a RigidTransform templatized on a scalar type U from a RigidTransform templatized on scalar type T. For example,

Click to expand C++ code...
RigidTransform<double> source = RigidTransform<double>::Identity();
RigidTransform<AutoDiffXd> foo = source.cast<AutoDiffXd>();
Template parameter U:

Scalar type on which the returned RigidTransform is templated.

Note

RigidTransform<From>::cast<To>() creates a new RigidTransform<To> from a RigidTransform<From> but only if type To is constructible from type From. This cast method works in accordance with Eigen’s cast method for Eigen’s objects that underlie this RigidTransform. For example, Eigen currently allows cast from type double to AutoDiffXd, but not vice-versa.

GetAsIsometry3(self: pydrake.math.RigidTransform_[AutoDiffXd]) → pydrake.common.eigen_geometry.Isometry3_[AutoDiffXd]

Returns the isometry in ℜ³ that is equivalent to a RigidTransform.

GetAsMatrix34(self: pydrake.math.RigidTransform_[AutoDiffXd]) → numpy.ndarray[object[3, 4]]

Returns the 3x4 matrix associated with this RigidTransform, i.e., X_AB.

Click to expand C++ code...
┌                ┐
 │ R_AB  p_AoBo_A │
 └                ┘
GetAsMatrix4(self: pydrake.math.RigidTransform_[AutoDiffXd]) → numpy.ndarray[object[4, 4]]

Returns the 4x4 matrix associated with this RigidTransform, i.e., X_AB.

Click to expand C++ code...
┌                ┐
 │ R_AB  p_AoBo_A │
 │                │
 │   0      1     │
 └                ┘
GetMaximumAbsoluteDifference(self: pydrake.math.RigidTransform_[AutoDiffXd], other: pydrake.math.RigidTransform_[AutoDiffXd]) → pydrake.autodiffutils.AutoDiffXd

Computes the infinity norm of this - other (i.e., the maximum absolute value of the difference between the elements of this and other).

Parameter other:

RigidTransform to subtract from this.

Returns

‖this - other‖∞

GetMaximumAbsoluteTranslationDifference(self: pydrake.math.RigidTransform_[AutoDiffXd], other: pydrake.math.RigidTransform_[AutoDiffXd]) → pydrake.autodiffutils.AutoDiffXd

Returns the maximum absolute value of the difference in the position vectors (translation) in this and other. In other words, returns the infinity norm of the difference in the position vectors.

Parameter other:

RigidTransform whose position vector is subtracted from the position vector in this.

static Identity() → pydrake.math.RigidTransform_[AutoDiffXd]

Returns the identity RigidTransform (corresponds to coincident frames).

Returns

the RigidTransform that corresponds to aligning the two frames so unit vectors Ax = Bx, Ay = By, Az = Bz and point Ao is coincident with Bo. Hence, the returned RigidTransform contains a 3x3 identity matrix and a zero position vector.

inverse(self: pydrake.math.RigidTransform_[AutoDiffXd]) → pydrake.math.RigidTransform_[AutoDiffXd]

Returns X_BA = X_AB⁻¹, the inverse of this RigidTransform.

Note

The inverse of RigidTransform X_AB is X_BA, which contains the rotation matrix R_BA = R_AB⁻¹ = R_ABᵀ and the position vector p_BoAo_B_ (position from B’s origin Bo to A’s origin Ao, expressed in frame B).

Note

: The square-root of a RigidTransform’s condition number is roughly the magnitude of the position vector. The accuracy of the calculation for the inverse of a RigidTransform drops off with the sqrt condition number.

InvertAndCompose(self: pydrake.math.RigidTransform_[AutoDiffXd], other: pydrake.math.RigidTransform_[AutoDiffXd]) → pydrake.math.RigidTransform_[AutoDiffXd]

Calculates the product of this inverted and another RigidTransform. If you consider this to be the transform X_AB, and other to be X_AC, then this method returns X_BC = X_AB⁻¹ * X_AC. For T==double, this method can be much faster than inverting first and then performing the composition, because it can take advantage of the special structure of a rigid transform to avoid unnecessary memory and floating point operations. On some platforms it can use SIMD instructions for further speedups.

Parameter other:

RigidTransform that post-multiplies this inverted.

Returns X_BC:

where X_BC = this⁻¹ * other.

Note

It is possible (albeit improbable) to create an invalid rigid transform by accumulating round-off error with a large number of multiplies.

IsExactlyEqualTo(self: pydrake.math.RigidTransform_[AutoDiffXd], other: pydrake.math.RigidTransform_[AutoDiffXd]) → bool

Returns true if this is exactly equal to other.

Parameter other:

RigidTransform to compare to this.

Returns

True if each element of this is exactly equal to the corresponding element of other.

IsExactlyIdentity(self: pydrake.math.RigidTransform_[AutoDiffXd]) → bool

Returns True if this is exactly the identity RigidTransform.

See also

IsNearlyIdentity().

IsNearlyEqualTo(self: pydrake.math.RigidTransform_[AutoDiffXd], other: pydrake.math.RigidTransform_[AutoDiffXd], tolerance: float) → bool

Compares each element of this to the corresponding element of other to check if they are the same to within a specified tolerance.

Parameter other:

RigidTransform to compare to this.

Parameter tolerance:

maximum allowable absolute difference between the elements in this and other.

Returns

True if ‖this.matrix() - other.matrix()‖∞ <= tolerance.

Note

Consider scaling tolerance with the largest of magA and magB, where magA and magB denoted the magnitudes of this position vector and other position vectors, respectively.

IsNearlyIdentity(self: pydrake.math.RigidTransform_[AutoDiffXd], translation_tolerance: float) → bool

Returns true if this is within tolerance of the identity RigidTransform.

Parameter translation_tolerance:

a non-negative number. One way to choose translation_tolerance is to multiply a characteristic length (e.g., the magnitude of a characteristic position vector) by an epsilon (e.g., RotationMatrix::get_internal_tolerance_for_orthonormality()).

Returns

True if the RotationMatrix portion of this satisfies RotationMatrix::IsNearlyIdentity() and if the position vector portion of this is equal to zero vector within translation_tolerance.

See also

IsExactlyIdentity().

static MakeUnchecked(pose: numpy.ndarray[object[3, 4]]) → pydrake.math.RigidTransform_[AutoDiffXd]

(Advanced) Constructs a RigidTransform from a 3x4 matrix, without any validity checks nor orthogonalization.

Parameter pose:

3x4 matrix that contains a 3x3 rotation matrix R_AB and also a 3x1 position vector p_AoBo_A (the position vector from frame A’s origin to frame B’s origin, expressed in frame A).

Click to expand C++ code...
┌                ┐
 │ R_AB  p_AoBo_A │
 └                ┘
multiply(*args, **kwargs)

Overloaded function.

  1. multiply(self: pydrake.math.RigidTransform_[AutoDiffXd], other: pydrake.math.RigidTransform_[AutoDiffXd]) -> pydrake.math.RigidTransform_[AutoDiffXd]

Multiplies this RigidTransform X_AB by the other RigidTransform X_BC and returns the RigidTransform X_AC = X_AB * X_BC.

  1. multiply(self: pydrake.math.RigidTransform_[AutoDiffXd], p_BoQ_B: numpy.ndarray[object[3, 1]]) -> numpy.ndarray[object[3, 1]]

Multiplies this RigidTransform X_AB by the position vector p_BoQ_B which is from Bo (B’s origin) to an arbitrary point Q.

Parameter p_BoQ_B:

position vector from Bo to Q, expressed in frame B.

Returns p_AoQ_A:

position vector from Ao to Q, expressed in frame A.

  1. multiply(self: pydrake.math.RigidTransform_[AutoDiffXd], vec_B: numpy.ndarray[object[4, 1]]) -> numpy.ndarray[object[4, 1]]

Multiplies this RigidTransform X_AB by the 4-element vector vec_B, equivalent to X_AB.GetAsMatrix4() * vec_B.

Parameter vec_B:

4-element vector whose first 3 elements are the position vector p_BoQ_B from Bo (frame B’s origin) to an arbitrary point Q, expressed in frame B and whose 4ᵗʰ element is 1 𝐨𝐫 whose first 3 elements are a vector (maybe unrelated to Bo or Q) and whose 4ᵗʰ element is 0.

Returns vec_A:

4-element vector whose first 3 elements are the position vector p_AoQ_A from Ao (frame A’s origin) to Q, expressed in frame A and whose 4ᵗʰ element is 1 𝐨𝐫 whose first 3 elements are a vector (maybe unrelated to Bo and Q) and whose 4ᵗʰ element is 0.

Raises

RuntimeError if the 4ᵗʰ element of vec_B is not 0 or 1. –

  1. multiply(self: pydrake.math.RigidTransform_[AutoDiffXd], p_BoQ_B: numpy.ndarray[object[3, n]]) -> numpy.ndarray[object[3, n]]

Multiplies this RigidTransform X_AB by the n position vectors p_BoQ1_B … p_BoQn_B, where p_BoQi_B is the iᵗʰ position vector from Bo (frame B’s origin) to an arbitrary point Qi, expressed in frame B.

Parameter p_BoQ_B:

3 x n matrix with n position vectors p_BoQi_B or an expression that resolves to a 3 x n matrix of position vectors.

Returns p_AoQ_A:

3 x n matrix with n position vectors p_AoQi_A, i.e., n position vectors from Ao (frame A’s origin) to Qi, expressed in frame A. Specifically, this operator* is defined so that X_AB * p_BoQ_B returns p_AoQ_A = p_AoBo_A + R_AB * p_BoQ_B, where p_AoBo_A is the position vector from Ao to Bo expressed in A and R_AB is the rotation matrix relating the orientation of frames A and B.

Note

As needed, use parentheses. This operator* is not associative. To see this, let p = p_AoBo_A, q = p_BoQ_B and note (X_AB * q) * 7 = (p + R_AB * q) * 7 ≠ X_AB * (q * 7) = p + R_AB * (q * 7).

Click to expand C++ code...
const RollPitchYaw<double> rpy(0.1, 0.2, 0.3);
const RigidTransform<double> X_AB(rpy, Vector3d(1, 2, 3));
Eigen::Matrix<double, 3, 2> p_BoQ_B;
p_BoQ_B.col(0) = Vector3d(4, 5, 6);
p_BoQ_B.col(1) = Vector3d(9, 8, 7);
const Eigen::Matrix<double, 3, 2> p_AoQ_A = X_AB * p_BoQ_B;
rotation(self: pydrake.math.RigidTransform_[AutoDiffXd]) → pydrake.math.RotationMatrix_[AutoDiffXd]

Returns R_AB, the rotation matrix portion of this RigidTransform.

Returns R_AB:

the rotation matrix portion of this RigidTransform.

set(self: pydrake.math.RigidTransform_[AutoDiffXd], R: pydrake.math.RotationMatrix_[AutoDiffXd], p: numpy.ndarray[object[3, 1]]) → None

Sets this RigidTransform from a RotationMatrix and a position vector.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

set_rotation(*args, **kwargs)

Overloaded function.

  1. set_rotation(self: pydrake.math.RigidTransform_[AutoDiffXd], R: pydrake.math.RotationMatrix_[AutoDiffXd]) -> None

Sets the RotationMatrix portion of this RigidTransform.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

  1. set_rotation(self: pydrake.math.RigidTransform_[AutoDiffXd], rpy: pydrake.math.RollPitchYaw_[AutoDiffXd]) -> None

Sets the rotation part of this RigidTransform from a RollPitchYaw.

Parameter rpy:

“roll-pitch-yaw” angles.

See also

RotationMatrix::RotationMatrix(const RollPitchYaw<T>&) which describes the parameter, preconditions, etc.

  1. set_rotation(self: pydrake.math.RigidTransform_[AutoDiffXd], quaternion: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]) -> None

Sets the rotation part of this RigidTransform from a Quaternion.

Parameter quaternion:

a quaternion which may or may not have unit length.

See also

RotationMatrix::RotationMatrix(const Eigen::Quaternion<T>&) which describes the parameter, preconditions, exception conditions, etc.

  1. set_rotation(self: pydrake.math.RigidTransform_[AutoDiffXd], theta_lambda: pydrake.common.eigen_geometry.AngleAxis_[AutoDiffXd]) -> None

Sets the rotation part of this RigidTransform from an AngleAxis.

Parameter theta_lambda:

an angle theta (in radians) and vector lambda.

See also

RotationMatrix::RotationMatrix(const Eigen::AngleAxis<T>&) which describes the parameter, preconditions, exception conditions, etc.

set_translation(self: pydrake.math.RigidTransform_[AutoDiffXd], p: numpy.ndarray[object[3, 1]]) → None

Sets the position vector portion of this RigidTransform.

Parameter p:

position vector from Ao (frame A’s origin) to Bo (frame B’s origin) expressed in frame A. In monogram notation p is denoted p_AoBo_A.

SetFromIsometry3(self: pydrake.math.RigidTransform_[AutoDiffXd], pose: pydrake.common.eigen_geometry.Isometry3_[AutoDiffXd]) → None

Sets this RigidTransform from an Eigen Isometry3.

Parameter pose:

Isometry3 that contains an allegedly valid rotation matrix R_AB and also contains a position vector p_AoBo_A from frame A’s origin to frame B’s origin. p_AoBo_A must be expressed in frame A.

Raises

  • RuntimeError in debug builds if R_AB is not a proper orthonormal –

  • 3x3 rotation matrix. –

Note

No attempt is made to orthogonalize the 3x3 rotation matrix part of pose. As needed, use RotationMatrix::ProjectToRotationMatrix().

SetIdentity(self: pydrake.math.RigidTransform_[AutoDiffXd]) → pydrake.math.RigidTransform_[AutoDiffXd]

Sets this RigidTransform so it corresponds to aligning the two frames so unit vectors Ax = Bx, Ay = By, Az = Bz and point Ao is coincident with Bo. Hence, this RigidTransform contains a 3x3 identity matrix and a zero position vector.

translation(self: pydrake.math.RigidTransform_[AutoDiffXd]) → numpy.ndarray[object[3, 1]]

Returns p_AoBo_A, the position vector portion of this RigidTransform, i.e., position vector from Ao (frame A’s origin) to Bo (frame B’s origin).

class pydrake.math.RigidTransform_[Expression]

This class represents a proper rigid transform between two frames which can be regarded in two ways. A rigid transform describes the “pose” between two frames A and B (i.e., the relative orientation and position of A to B). Alternately, it can be regarded as a distance-preserving operator that can rotate and/or translate a rigid body without changing its shape or size (rigid) and without mirroring/reflecting the body (proper), e.g., it can add one position vector to another and express the result in a particular basis as p_AoQ_A = X_AB * p_BoQ_B (Q is any point). In many ways, this rigid transform class is conceptually similar to using a homogeneous matrix as a linear operator. See operator* documentation for an exception.

The class stores a RotationMatrix that relates right-handed orthogonal unit vectors Ax, Ay, Az fixed in frame A to right-handed orthogonal unit vectors Bx, By, Bz fixed in frame B. The class also stores a position vector from Ao (the origin of frame A) to Bo (the origin of frame B). The position vector is expressed in frame A. The monogram notation for the transform relating frame A to B is X_AB. The monogram notation for the rotation matrix relating A to B is R_AB. The monogram notation for the position vector from Ao to Bo is p_AoBo_A. See multibody_quantities for monogram notation for dynamics.

Note

This class does not store the frames associated with the transform and cannot enforce correct usage of this class. For example, it makes sense to multiply RigidTransforms as X_AB * X_BC, but not X_AB * X_CB.

Note

This class is not a 4x4 transformation matrix – even though its operator*() methods act mostly like 4x4 matrix multiplication. Instead, this class contains a 3x3 rotation matrix class and a 3x1 position vector. To convert this to a 3x4 matrix, use GetAsMatrix34(). To convert this to a 4x4 matrix, use GetAsMatrix4(). To convert this to an Eigen::Isometry, use GetAsIsometry().

Note

An isometry is sometimes regarded as synonymous with rigid transform. The RigidTransform class has important advantages over Eigen::Isometry. - RigidTransform is built on an underlying rigorous 3x3 RotationMatrix class that has significant functionality for 3D orientation. - In Debug builds, RigidTransform requires a valid 3x3 rotation matrix and a valid (non-NAN) position vector. Eigen::Isometry does not. - RigidTransform catches bugs that are undetected by Eigen::Isometry. - RigidTransform has additional functionality and ease-of-use, resulting in shorter, easier to write, and easier to read code. - The name Isometry is unfamiliar to many roboticists and dynamicists and for them Isometry.linear() is (for example) a counter-intuitive method name to return a rotation matrix.

Note

One of the constructors in this class provides an implicit conversion from an Eigen Translation to RigidTransform.

Authors:

Paul Mitiguy (2018) Original author.

Authors:

Drake team (see https://drake.mit.edu/credits).

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.RigidTransform_[Expression]) -> None

Constructs the RigidTransform that corresponds to aligning the two frames so unit vectors Ax = Bx, Ay = By, Az = Bz and point Ao is coincident with Bo. Hence, the constructed RigidTransform contains an identity RotationMatrix and a zero position vector.

  1. __init__(self: pydrake.math.RigidTransform_[Expression], other: pydrake.math.RigidTransform_[Expression]) -> None

  2. __init__(self: pydrake.math.RigidTransform_[Expression], R: pydrake.math.RotationMatrix_[Expression], p: numpy.ndarray[object[3, 1]]) -> None

Constructs a RigidTransform from a rotation matrix and a position vector.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

  1. __init__(self: pydrake.math.RigidTransform_[Expression], rpy: pydrake.math.RollPitchYaw_[Expression], p: numpy.ndarray[object[3, 1]]) -> None

Constructs a RigidTransform from a RollPitchYaw and a position vector.

Parameter rpy:

a RollPitchYaw which is a Space-fixed (extrinsic) X-Y-Z rotation with “roll-pitch-yaw” angles [r, p, y] or equivalently a Body- fixed (intrinsic) Z-Y-X rotation with “yaw-pitch-roll” angles [y, p, r].

See also

RotationMatrix::RotationMatrix(const RollPitchYaw<T>&)

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

  1. __init__(self: pydrake.math.RigidTransform_[Expression], quaternion: pydrake.common.eigen_geometry.Quaternion_[Expression], p: numpy.ndarray[object[3, 1]]) -> None

Constructs a RigidTransform from a Quaternion and a position vector.

Parameter quaternion:

a non-zero, finite quaternion which may or may not have unit length [i.e., quaternion.norm() does not have to be 1].

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

Raises

  • RuntimeError in debug builds if the rotation matrix that is built –

  • from quaternion is invalid. –

See also

RotationMatrix::RotationMatrix(const Eigen::Quaternion<T>&)

  1. __init__(self: pydrake.math.RigidTransform_[Expression], theta_lambda: pydrake.common.eigen_geometry.AngleAxis_[Expression], p: numpy.ndarray[object[3, 1]]) -> None

Constructs a RigidTransform from an AngleAxis and a position vector.

Parameter theta_lambda:

an Eigen::AngleAxis whose associated axis (vector direction herein called lambda) is non-zero and finite, but which may or may not have unit length [i.e., lambda.norm() does not have to be 1].

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted ``p_AoBo_A

Raises

  • RuntimeError in debug builds if the rotation matrix that is built –

  • from theta_lambda` is invalid. –

See also

RotationMatrix::RotationMatrix(const Eigen::AngleAxis<T>&)

  1. __init__(self: pydrake.math.RigidTransform_[Expression], R: pydrake.math.RotationMatrix_[Expression]) -> None

Constructs a RigidTransform with a given RotationMatrix and a zero position vector.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

  1. __init__(self: pydrake.math.RigidTransform_[Expression], p: numpy.ndarray[object[3, 1]]) -> None

Constructs a RigidTransform that contains an identity RotationMatrix and a given position vector p.

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

  1. __init__(self: pydrake.math.RigidTransform_[Expression], pose: pydrake.common.eigen_geometry.Isometry3_[Expression]) -> None

Constructs a RigidTransform from an Eigen Isometry3.

Parameter pose:

Isometry3 that contains an allegedly valid rotation matrix R_AB and also contains a position vector p_AoBo_A from frame A’s origin to frame B’s origin. p_AoBo_A must be expressed in frame A.

Raises

  • RuntimeError in debug builds if R_AB is not a proper orthonormal –

  • 3x3 rotation matrix. –

Note

No attempt is made to orthogonalize the 3x3 rotation matrix part of pose. As needed, use RotationMatrix::ProjectToRotationMatrix().

  1. __init__(self: pydrake.math.RigidTransform_[Expression], pose: numpy.ndarray[object[m, n]]) -> None

Constructs a RigidTransform from an appropriate Eigen expression.

Parameter pose:

Generic Eigen matrix expression.

Raises

  • RuntimeError if the Eigen expression in pose does not resolve –

  • to a Vector3 or 3x4 matrix or 4x4 matrix or if the rotational part –

  • of pose is not a proper orthonormal 3x3 rotation matrix or if –

  • pose` is a 4x4 matrix whose final row is not [0, 0, 0, 1] –

Note

No attempt is made to orthogonalize the 3x3 rotation matrix part of pose. As needed, use RotationMatrix::ProjectToRotationMatrix().

Note

This constructor prevents ambiguity that would otherwise exist for a RigidTransform constructor whose argument is an Eigen expression.

Click to expand C++ code...
const Vector3<double> position(4, 5, 6);
const RigidTransform<double> X1(3 * position);
----------------------------------------------
const RotationMatrix<double> R(RollPitchYaw<double>(1, 2, 3));
Eigen::Matrix<double, 3, 4> pose34;
pose34 << R.matrix(), position;
const RigidTransform<double> X2(1.0 * pose34);
----------------------------------------------
Eigen::Matrix<double, 4, 4> pose4;
pose4 << R.matrix(), position,
         0, 0, 0, 1;
const RigidTransform<double> X3(pose4 * pose4);
template cast

Instantiations: cast[Expression]

cast[Expression](self: pydrake.math.RigidTransform_[Expression]) → pydrake.math.RigidTransform_[Expression]

Creates a RigidTransform templatized on a scalar type U from a RigidTransform templatized on scalar type T. For example,

Click to expand C++ code...
RigidTransform<double> source = RigidTransform<double>::Identity();
RigidTransform<AutoDiffXd> foo = source.cast<AutoDiffXd>();
Template parameter U:

Scalar type on which the returned RigidTransform is templated.

Note

RigidTransform<From>::cast<To>() creates a new RigidTransform<To> from a RigidTransform<From> but only if type To is constructible from type From. This cast method works in accordance with Eigen’s cast method for Eigen’s objects that underlie this RigidTransform. For example, Eigen currently allows cast from type double to AutoDiffXd, but not vice-versa.

GetAsIsometry3(self: pydrake.math.RigidTransform_[Expression]) → pydrake.common.eigen_geometry.Isometry3_[Expression]

Returns the isometry in ℜ³ that is equivalent to a RigidTransform.

GetAsMatrix34(self: pydrake.math.RigidTransform_[Expression]) → numpy.ndarray[object[3, 4]]

Returns the 3x4 matrix associated with this RigidTransform, i.e., X_AB.

Click to expand C++ code...
┌                ┐
 │ R_AB  p_AoBo_A │
 └                ┘
GetAsMatrix4(self: pydrake.math.RigidTransform_[Expression]) → numpy.ndarray[object[4, 4]]

Returns the 4x4 matrix associated with this RigidTransform, i.e., X_AB.

Click to expand C++ code...
┌                ┐
 │ R_AB  p_AoBo_A │
 │                │
 │   0      1     │
 └                ┘
GetMaximumAbsoluteDifference(self: pydrake.math.RigidTransform_[Expression], other: pydrake.math.RigidTransform_[Expression]) → pydrake.symbolic.Expression

Computes the infinity norm of this - other (i.e., the maximum absolute value of the difference between the elements of this and other).

Parameter other:

RigidTransform to subtract from this.

Returns

‖this - other‖∞

GetMaximumAbsoluteTranslationDifference(self: pydrake.math.RigidTransform_[Expression], other: pydrake.math.RigidTransform_[Expression]) → pydrake.symbolic.Expression

Returns the maximum absolute value of the difference in the position vectors (translation) in this and other. In other words, returns the infinity norm of the difference in the position vectors.

Parameter other:

RigidTransform whose position vector is subtracted from the position vector in this.

static Identity() → pydrake.math.RigidTransform_[Expression]

Returns the identity RigidTransform (corresponds to coincident frames).

Returns

the RigidTransform that corresponds to aligning the two frames so unit vectors Ax = Bx, Ay = By, Az = Bz and point Ao is coincident with Bo. Hence, the returned RigidTransform contains a 3x3 identity matrix and a zero position vector.

inverse(self: pydrake.math.RigidTransform_[Expression]) → pydrake.math.RigidTransform_[Expression]

Returns X_BA = X_AB⁻¹, the inverse of this RigidTransform.

Note

The inverse of RigidTransform X_AB is X_BA, which contains the rotation matrix R_BA = R_AB⁻¹ = R_ABᵀ and the position vector p_BoAo_B_ (position from B’s origin Bo to A’s origin Ao, expressed in frame B).

Note

: The square-root of a RigidTransform’s condition number is roughly the magnitude of the position vector. The accuracy of the calculation for the inverse of a RigidTransform drops off with the sqrt condition number.

InvertAndCompose(self: pydrake.math.RigidTransform_[Expression], other: pydrake.math.RigidTransform_[Expression]) → pydrake.math.RigidTransform_[Expression]

Calculates the product of this inverted and another RigidTransform. If you consider this to be the transform X_AB, and other to be X_AC, then this method returns X_BC = X_AB⁻¹ * X_AC. For T==double, this method can be much faster than inverting first and then performing the composition, because it can take advantage of the special structure of a rigid transform to avoid unnecessary memory and floating point operations. On some platforms it can use SIMD instructions for further speedups.

Parameter other:

RigidTransform that post-multiplies this inverted.

Returns X_BC:

where X_BC = this⁻¹ * other.

Note

It is possible (albeit improbable) to create an invalid rigid transform by accumulating round-off error with a large number of multiplies.

IsExactlyEqualTo(self: pydrake.math.RigidTransform_[Expression], other: pydrake.math.RigidTransform_[Expression]) → pydrake.symbolic.Formula

Returns true if this is exactly equal to other.

Parameter other:

RigidTransform to compare to this.

Returns

True if each element of this is exactly equal to the corresponding element of other.

IsExactlyIdentity(self: pydrake.math.RigidTransform_[Expression]) → pydrake.symbolic.Formula

Returns True if this is exactly the identity RigidTransform.

See also

IsNearlyIdentity().

IsNearlyEqualTo(self: pydrake.math.RigidTransform_[Expression], other: pydrake.math.RigidTransform_[Expression], tolerance: float) → pydrake.symbolic.Formula

Compares each element of this to the corresponding element of other to check if they are the same to within a specified tolerance.

Parameter other:

RigidTransform to compare to this.

Parameter tolerance:

maximum allowable absolute difference between the elements in this and other.

Returns

True if ‖this.matrix() - other.matrix()‖∞ <= tolerance.

Note

Consider scaling tolerance with the largest of magA and magB, where magA and magB denoted the magnitudes of this position vector and other position vectors, respectively.

IsNearlyIdentity(self: pydrake.math.RigidTransform_[Expression], translation_tolerance: float) → pydrake.symbolic.Formula

Returns true if this is within tolerance of the identity RigidTransform.

Parameter translation_tolerance:

a non-negative number. One way to choose translation_tolerance is to multiply a characteristic length (e.g., the magnitude of a characteristic position vector) by an epsilon (e.g., RotationMatrix::get_internal_tolerance_for_orthonormality()).

Returns

True if the RotationMatrix portion of this satisfies RotationMatrix::IsNearlyIdentity() and if the position vector portion of this is equal to zero vector within translation_tolerance.

See also

IsExactlyIdentity().

static MakeUnchecked(pose: numpy.ndarray[object[3, 4]]) → pydrake.math.RigidTransform_[Expression]

(Advanced) Constructs a RigidTransform from a 3x4 matrix, without any validity checks nor orthogonalization.

Parameter pose:

3x4 matrix that contains a 3x3 rotation matrix R_AB and also a 3x1 position vector p_AoBo_A (the position vector from frame A’s origin to frame B’s origin, expressed in frame A).

Click to expand C++ code...
┌                ┐
 │ R_AB  p_AoBo_A │
 └                ┘
multiply(*args, **kwargs)

Overloaded function.

  1. multiply(self: pydrake.math.RigidTransform_[Expression], other: pydrake.math.RigidTransform_[Expression]) -> pydrake.math.RigidTransform_[Expression]

Multiplies this RigidTransform X_AB by the other RigidTransform X_BC and returns the RigidTransform X_AC = X_AB * X_BC.

  1. multiply(self: pydrake.math.RigidTransform_[Expression], p_BoQ_B: numpy.ndarray[object[3, 1]]) -> numpy.ndarray[object[3, 1]]

Multiplies this RigidTransform X_AB by the position vector p_BoQ_B which is from Bo (B’s origin) to an arbitrary point Q.

Parameter p_BoQ_B:

position vector from Bo to Q, expressed in frame B.

Returns p_AoQ_A:

position vector from Ao to Q, expressed in frame A.

  1. multiply(self: pydrake.math.RigidTransform_[Expression], vec_B: numpy.ndarray[object[4, 1]]) -> numpy.ndarray[object[4, 1]]

Multiplies this RigidTransform X_AB by the 4-element vector vec_B, equivalent to X_AB.GetAsMatrix4() * vec_B.

Parameter vec_B:

4-element vector whose first 3 elements are the position vector p_BoQ_B from Bo (frame B’s origin) to an arbitrary point Q, expressed in frame B and whose 4ᵗʰ element is 1 𝐨𝐫 whose first 3 elements are a vector (maybe unrelated to Bo or Q) and whose 4ᵗʰ element is 0.

Returns vec_A:

4-element vector whose first 3 elements are the position vector p_AoQ_A from Ao (frame A’s origin) to Q, expressed in frame A and whose 4ᵗʰ element is 1 𝐨𝐫 whose first 3 elements are a vector (maybe unrelated to Bo and Q) and whose 4ᵗʰ element is 0.

Raises

RuntimeError if the 4ᵗʰ element of vec_B is not 0 or 1. –

  1. multiply(self: pydrake.math.RigidTransform_[Expression], p_BoQ_B: numpy.ndarray[object[3, n]]) -> numpy.ndarray[object[3, n]]

Multiplies this RigidTransform X_AB by the n position vectors p_BoQ1_B … p_BoQn_B, where p_BoQi_B is the iᵗʰ position vector from Bo (frame B’s origin) to an arbitrary point Qi, expressed in frame B.

Parameter p_BoQ_B:

3 x n matrix with n position vectors p_BoQi_B or an expression that resolves to a 3 x n matrix of position vectors.

Returns p_AoQ_A:

3 x n matrix with n position vectors p_AoQi_A, i.e., n position vectors from Ao (frame A’s origin) to Qi, expressed in frame A. Specifically, this operator* is defined so that X_AB * p_BoQ_B returns p_AoQ_A = p_AoBo_A + R_AB * p_BoQ_B, where p_AoBo_A is the position vector from Ao to Bo expressed in A and R_AB is the rotation matrix relating the orientation of frames A and B.

Note

As needed, use parentheses. This operator* is not associative. To see this, let p = p_AoBo_A, q = p_BoQ_B and note (X_AB * q) * 7 = (p + R_AB * q) * 7 ≠ X_AB * (q * 7) = p + R_AB * (q * 7).

Click to expand C++ code...
const RollPitchYaw<double> rpy(0.1, 0.2, 0.3);
const RigidTransform<double> X_AB(rpy, Vector3d(1, 2, 3));
Eigen::Matrix<double, 3, 2> p_BoQ_B;
p_BoQ_B.col(0) = Vector3d(4, 5, 6);
p_BoQ_B.col(1) = Vector3d(9, 8, 7);
const Eigen::Matrix<double, 3, 2> p_AoQ_A = X_AB * p_BoQ_B;
rotation(self: pydrake.math.RigidTransform_[Expression]) → pydrake.math.RotationMatrix_[Expression]

Returns R_AB, the rotation matrix portion of this RigidTransform.

Returns R_AB:

the rotation matrix portion of this RigidTransform.

set(self: pydrake.math.RigidTransform_[Expression], R: pydrake.math.RotationMatrix_[Expression], p: numpy.ndarray[object[3, 1]]) → None

Sets this RigidTransform from a RotationMatrix and a position vector.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

Parameter p:

position vector from frame A’s origin to frame B’s origin, expressed in frame A. In monogram notation p is denoted p_AoBo_A.

set_rotation(*args, **kwargs)

Overloaded function.

  1. set_rotation(self: pydrake.math.RigidTransform_[Expression], R: pydrake.math.RotationMatrix_[Expression]) -> None

Sets the RotationMatrix portion of this RigidTransform.

Parameter R:

rotation matrix relating frames A and B (e.g., R_AB).

  1. set_rotation(self: pydrake.math.RigidTransform_[Expression], rpy: pydrake.math.RollPitchYaw_[Expression]) -> None

Sets the rotation part of this RigidTransform from a RollPitchYaw.

Parameter rpy:

“roll-pitch-yaw” angles.

See also

RotationMatrix::RotationMatrix(const RollPitchYaw<T>&) which describes the parameter, preconditions, etc.

  1. set_rotation(self: pydrake.math.RigidTransform_[Expression], quaternion: pydrake.common.eigen_geometry.Quaternion_[Expression]) -> None

Sets the rotation part of this RigidTransform from a Quaternion.

Parameter quaternion:

a quaternion which may or may not have unit length.

See also

RotationMatrix::RotationMatrix(const Eigen::Quaternion<T>&) which describes the parameter, preconditions, exception conditions, etc.

  1. set_rotation(self: pydrake.math.RigidTransform_[Expression], theta_lambda: pydrake.common.eigen_geometry.AngleAxis_[Expression]) -> None

Sets the rotation part of this RigidTransform from an AngleAxis.

Parameter theta_lambda:

an angle theta (in radians) and vector lambda.

See also

RotationMatrix::RotationMatrix(const Eigen::AngleAxis<T>&) which describes the parameter, preconditions, exception conditions, etc.

set_translation(self: pydrake.math.RigidTransform_[Expression], p: numpy.ndarray[object[3, 1]]) → None

Sets the position vector portion of this RigidTransform.

Parameter p:

position vector from Ao (frame A’s origin) to Bo (frame B’s origin) expressed in frame A. In monogram notation p is denoted p_AoBo_A.

SetFromIsometry3(self: pydrake.math.RigidTransform_[Expression], pose: pydrake.common.eigen_geometry.Isometry3_[Expression]) → None

Sets this RigidTransform from an Eigen Isometry3.

Parameter pose:

Isometry3 that contains an allegedly valid rotation matrix R_AB and also contains a position vector p_AoBo_A from frame A’s origin to frame B’s origin. p_AoBo_A must be expressed in frame A.

Raises

  • RuntimeError in debug builds if R_AB is not a proper orthonormal –

  • 3x3 rotation matrix. –

Note

No attempt is made to orthogonalize the 3x3 rotation matrix part of pose. As needed, use RotationMatrix::ProjectToRotationMatrix().

SetIdentity(self: pydrake.math.RigidTransform_[Expression]) → pydrake.math.RigidTransform_[Expression]

Sets this RigidTransform so it corresponds to aligning the two frames so unit vectors Ax = Bx, Ay = By, Az = Bz and point Ao is coincident with Bo. Hence, this RigidTransform contains a 3x3 identity matrix and a zero position vector.

translation(self: pydrake.math.RigidTransform_[Expression]) → numpy.ndarray[object[3, 1]]

Returns p_AoBo_A, the position vector portion of this RigidTransform, i.e., position vector from Ao (frame A’s origin) to Bo (frame B’s origin).

class pydrake.math.RollPitchYaw

This class represents the orientation between two arbitrary frames A and D associated with a Space-fixed (extrinsic) X-Y-Z rotation by “roll-pitch-yaw” angles [r, p, y], which is equivalent to a Body-fixed (intrinsic) Z-Y-X rotation by “yaw-pitch-roll” angles [y, p, r]. The rotation matrix R_AD associated with this roll-pitch-yaw [r, p, y] rotation sequence is equal to the matrix multiplication shown below.

Click to expand C++ code...
⎡cos(y) -sin(y)  0⎤   ⎡ cos(p)  0  sin(p)⎤   ⎡1      0        0 ⎤
R_AD = ⎢sin(y)  cos(y)  0⎥ * ⎢     0   1      0 ⎥ * ⎢0  cos(r)  -sin(r)⎥
       ⎣    0       0   1⎦   ⎣-sin(p)  0  cos(p)⎦   ⎣0  sin(r)   cos(r)⎦
     =       R_AB          *        R_BC          *        R_CD

Note

In this discussion, A is the Space frame and D is the Body frame. One way to visualize this rotation sequence is by introducing intermediate frames B and C (useful constructs to understand this rotation sequence). Initially, the frames are aligned so Di = Ci = Bi = Ai (i = x, y, z) where Dx, Dy, Dz and Ax, Ay, Az are orthogonal unit vectors fixed in frames D and A respectively. Similarly for Bx, By, Bz and Cx, Cy, Cz in frame B, C. Then D is subjected to successive right-handed rotations relative to A.

  • 1st rotation R_CD: Frame D rotates relative to frames C, B, A by a

roll angle r about Dx = Cx. Note: D and C are no longer aligned.

  • 2nd rotation R_BC: Frames D, C (collectively – as if welded together)

rotate relative to frame B, A by a pitch angle p about Cy = By. Note: C and B are no longer aligned.

  • 3rd rotation R_AB: Frames D, C, B (collectively – as if welded)

rotate relative to frame A by a yaw angle y about Bz = Az. Note: B and A are no longer aligned. The monogram notation for the rotation matrix relating A to D is R_AD.

See also

multibody_quantities for monogram notation for dynamics and orientation_discussion “a discussion on rotation matrices”.

Note

This class does not store the frames associated with this rotation sequence.

Note

This class is templated; see RollPitchYaw_ for the list of instantiations.

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.RollPitchYaw, other: pydrake.math.RollPitchYaw) -> None

  2. __init__(self: pydrake.math.RollPitchYaw, rpy: numpy.ndarray[numpy.float64[3, 1]]) -> None

Constructs a RollPitchYaw from a 3x1 array of angles.

Parameter rpy:

3x1 array with roll, pitch, yaw angles (units of radians).

Raises

RuntimeError in debug builds if !IsValid(rpy) –

  1. __init__(self: pydrake.math.RollPitchYaw, roll: float, pitch: float, yaw: float) -> None

Constructs a RollPitchYaw from roll, pitch, yaw angles (radian units).

Parameter roll:

x-directed angle in SpaceXYZ rotation sequence.

Parameter pitch:

y-directed angle in SpaceXYZ rotation sequence.

Parameter yaw:

z-directed angle in SpaceXYZ rotation sequence.

Raises

  • RuntimeError in debug builds if !IsValid(Vector3<T>(roll, pitch, –

  • yaw)) –

  1. __init__(self: pydrake.math.RollPitchYaw, R: pydrake.math.RotationMatrix) -> None

Uses a RotationMatrix to construct a RollPitchYaw with roll-pitch-yaw angles [r, p, y] in the range -π <= r <= π, -π/2 <= p <= π/2, -π <= y <= π.

Parameter R:

a RotationMatrix.

Note

This new high-accuracy algorithm avoids numerical round-off issues encountered by some algorithms when pitch is within 1E-6 of π/2 or -π/2.

  1. __init__(self: pydrake.math.RollPitchYaw, quaternion: pydrake.common.eigen_geometry.Quaternion) -> None

Uses a Quaternion to construct a RollPitchYaw with roll-pitch-yaw angles [r, p, y] in the range -π <= r <= π, -π/2 <= p <= π/2, -π <= y <= π.

Parameter quaternion:

unit Quaternion.

Note

This new high-accuracy algorithm avoids numerical round-off issues encountered by some algorithms when pitch is within 1E-6 of π/2 or -π/2.

Raises

RuntimeError in debug builds if !IsValid(rpy) –

  1. __init__(self: pydrake.math.RollPitchYaw, matrix: numpy.ndarray[numpy.float64[3, 3]]) -> None

Construct from raw rotation matrix. See RotationMatrix overload for more information.

CalcAngularVelocityInChildFromRpyDt(self: pydrake.math.RollPitchYaw, rpyDt: numpy.ndarray[numpy.float64[3, 1]]) → numpy.ndarray[numpy.float64[3, 1]]

Calculates angular velocity from this RollPitchYaw whose roll-pitch-yaw angles [r; p; y] relate the orientation of two generic frames A and D.

Parameter rpyDt:

Time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Returns

w_AD_D, frame D’s angular velocity in frame A, expressed in “child” frame D. In other words, returns [ω0; ω1; ω2]ᴅ, where w_AD_D = ω0 Dx + ω1 Dy + ω2 Dz, and where [ω0; ω1; ω2]ᴅ is calculated via the 3x3 matrix Nc⁻¹ (the inverse of the matrix Nc documented in CalcMatrixRelatingRpyDtToAngularVelocityInChild()).

Click to expand C++ code...
⌈ ω0 ⌉         ⌈ ṙ ⌉            ⌈ 1      0        -sin(p)    ⌉
| ω1 |  = Nc⁻¹ | ṗ |     Nc⁻¹ = | 0   cos(r)   sin(r)*cos(p) |
⌊ ω2 ⌋ᴅ        ⌊ ẏ ⌋            ⌊ 0  -sin(r)   cos(r)*cos(p) ⌋
CalcAngularVelocityInParentFromRpyDt(self: pydrake.math.RollPitchYaw, rpyDt: numpy.ndarray[numpy.float64[3, 1]]) → numpy.ndarray[numpy.float64[3, 1]]

Calculates angular velocity from this RollPitchYaw whose roll-pitch-yaw angles [r; p; y] relate the orientation of two generic frames A and D.

Parameter rpyDt:

Time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Returns

w_AD_A, frame D’s angular velocity in frame A, expressed in “parent” frame A. In other words, returns [ωx; ωy; ωz]ᴀ, where w_AD_A = ωx Ax + ωy Ay + ωz Az, and where [ωx; ωy; ωz]ᴀ is calculated via the the 3x3 matrix Np⁻¹ (the inverse of the matrix Np documented in CalcMatrixRelatingRpyDtToAngularVelocityInParent()).

Click to expand C++ code...
⌈ ωx ⌉         ⌈ ṙ ⌉            ⌈ cos(y)*cos(p)  -sin(y)  0 ⌉
| ωy |  = Np⁻¹ | ṗ |     Np⁻¹ = | sin(y)*cos(p)   cos(y)  0 |
⌊ ωz ⌋ᴀ        ⌊ ẏ ⌋            ⌊   -sin(p)         0     1 ⌋
CalcRotationMatrixDt(self: pydrake.math.RollPitchYaw, rpyDt: numpy.ndarray[numpy.float64[3, 1]]) → numpy.ndarray[numpy.float64[3, 3]]

Forms Ṙ, the ordinary derivative of the RotationMatrix R with respect to an independent variable t (t usually denotes time) and R is the RotationMatrix formed by this RollPitchYaw. The roll-pitch-yaw angles r, p, y are regarded as functions of t [i.e., r(t), p(t), y(t)].

Parameter rpyDt:

Ordinary derivative of rpy with respect to an independent variable t (t usually denotes time, but not necessarily).

Returns

Ṙ, the ordinary derivative of R with respect to t, calculated as Ṙ = ∂R/∂r * ṙ + ∂R/∂p * ṗ + ∂R/∂y * ẏ. In other words, the returned (i, j) element is ∂Rij/∂r * ṙ + ∂Rij/∂p * ṗ + ∂Rij/∂y * ẏ.

CalcRpyDDtFromAngularAccelInChild(self: pydrake.math.RollPitchYaw, rpyDt: numpy.ndarray[numpy.float64[3, 1]], alpha_AD_D: numpy.ndarray[numpy.float64[3, 1]]) → numpy.ndarray[numpy.float64[3, 1]]

Uses angular acceleration to compute the 2ⁿᵈ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter rpyDt:

time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Parameter alpha_AD_D:

, frame D’s angular acceleration in frame A, expressed in frame D.

Returns

[r̈, p̈, ÿ], the 2ⁿᵈ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

This method has a divide-by-zero error (singularity) when the cosine of the pitch angle p is zero [i.e., cos(p) = 0`]. This problem (called "gimbal lock") occurs when `p = n π + π / 2, where n is any integer. There are associated precision problems (inaccuracies) in the neighborhood of these pitch angles, i.e., when cos(p) ≈ 0.

CalcRpyDDtFromRpyDtAndAngularAccelInParent(self: pydrake.math.RollPitchYaw, rpyDt: numpy.ndarray[numpy.float64[3, 1]], alpha_AD_A: numpy.ndarray[numpy.float64[3, 1]]) → numpy.ndarray[numpy.float64[3, 1]]

Uses angular acceleration to compute the 2ⁿᵈ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter rpyDt:

time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Parameter alpha_AD_A:

, frame D’s angular acceleration in frame A, expressed in frame A.

Returns

[r̈, p̈, ÿ], the 2ⁿᵈ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

This method has a divide-by-zero error (singularity) when the cosine of the pitch angle p is zero [i.e., cos(p) = 0`]. This problem (called "gimbal lock") occurs when `p = n π + π / 2, where n is any integer. There are associated precision problems (inaccuracies) in the neighborhood of these pitch angles, i.e., when cos(p) ≈ 0.

CalcRpyDtFromAngularVelocityInChild(self: pydrake.math.RollPitchYaw, w_AD_D: numpy.ndarray[numpy.float64[3, 1]]) → numpy.ndarray[numpy.float64[3, 1]]

Uses angular velocity to compute the 1ˢᵗ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter w_AD_D:

, frame D’s angular velocity in frame A, expressed in D.

Returns

[ṙ; ṗ; ẏ], the 1ˢᵗ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

Enhanced documentation for this method and its gimbal-lock (divide- by-zero error) is in CalcMatrixRelatingRpyDtToAngularVelocityInChild().

See also

CalcRpyDtFromAngularVelocityInParent()

CalcRpyDtFromAngularVelocityInParent(self: pydrake.math.RollPitchYaw, w_AD_A: numpy.ndarray[numpy.float64[3, 1]]) → numpy.ndarray[numpy.float64[3, 1]]

Uses angular velocity to compute the 1ˢᵗ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter w_AD_A:

, frame D’s angular velocity in frame A, expressed in A.

Returns

[ṙ; ṗ; ẏ], the 1ˢᵗ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

Enhanced documentation for this method and its gimbal-lock (divide- by-zero error) is in CalcMatrixRelatingRpyDtToAngularVelocityInParent().

See also

CalcRpyDtFromAngularVelocityInChild()

pitch_angle(self: pydrake.math.RollPitchYaw) → float

Returns the pitch-angle underlying this RollPitchYaw.

roll_angle(self: pydrake.math.RollPitchYaw) → float

Returns the roll-angle underlying this RollPitchYaw.

ToQuaternion(self: pydrake.math.RollPitchYaw) → pydrake.common.eigen_geometry.Quaternion

Returns a quaternion representation of this RollPitchYaw.

ToRotationMatrix(self: pydrake.math.RollPitchYaw) → pydrake.math.RotationMatrix

Returns the RotationMatrix representation of this RollPitchYaw.

vector(self: pydrake.math.RollPitchYaw) → numpy.ndarray[numpy.float64[3, 1]]

Returns the Vector3 underlying this RollPitchYaw.

yaw_angle(self: pydrake.math.RollPitchYaw) → float

Returns the yaw-angle underlying this RollPitchYaw.

template pydrake.math.RollPitchYaw_

Instantiations: RollPitchYaw_[float], RollPitchYaw_[AutoDiffXd], RollPitchYaw_[Expression]

class pydrake.math.RollPitchYaw_[AutoDiffXd]

This class represents the orientation between two arbitrary frames A and D associated with a Space-fixed (extrinsic) X-Y-Z rotation by “roll-pitch-yaw” angles [r, p, y], which is equivalent to a Body-fixed (intrinsic) Z-Y-X rotation by “yaw-pitch-roll” angles [y, p, r]. The rotation matrix R_AD associated with this roll-pitch-yaw [r, p, y] rotation sequence is equal to the matrix multiplication shown below.

Click to expand C++ code...
⎡cos(y) -sin(y)  0⎤   ⎡ cos(p)  0  sin(p)⎤   ⎡1      0        0 ⎤
R_AD = ⎢sin(y)  cos(y)  0⎥ * ⎢     0   1      0 ⎥ * ⎢0  cos(r)  -sin(r)⎥
       ⎣    0       0   1⎦   ⎣-sin(p)  0  cos(p)⎦   ⎣0  sin(r)   cos(r)⎦
     =       R_AB          *        R_BC          *        R_CD

Note

In this discussion, A is the Space frame and D is the Body frame. One way to visualize this rotation sequence is by introducing intermediate frames B and C (useful constructs to understand this rotation sequence). Initially, the frames are aligned so Di = Ci = Bi = Ai (i = x, y, z) where Dx, Dy, Dz and Ax, Ay, Az are orthogonal unit vectors fixed in frames D and A respectively. Similarly for Bx, By, Bz and Cx, Cy, Cz in frame B, C. Then D is subjected to successive right-handed rotations relative to A.

  • 1st rotation R_CD: Frame D rotates relative to frames C, B, A by a

roll angle r about Dx = Cx. Note: D and C are no longer aligned.

  • 2nd rotation R_BC: Frames D, C (collectively – as if welded together)

rotate relative to frame B, A by a pitch angle p about Cy = By. Note: C and B are no longer aligned.

  • 3rd rotation R_AB: Frames D, C, B (collectively – as if welded)

rotate relative to frame A by a yaw angle y about Bz = Az. Note: B and A are no longer aligned. The monogram notation for the rotation matrix relating A to D is R_AD.

See also

multibody_quantities for monogram notation for dynamics and orientation_discussion “a discussion on rotation matrices”.

Note

This class does not store the frames associated with this rotation sequence.

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.RollPitchYaw_[AutoDiffXd], other: pydrake.math.RollPitchYaw_[AutoDiffXd]) -> None

  2. __init__(self: pydrake.math.RollPitchYaw_[AutoDiffXd], rpy: numpy.ndarray[object[3, 1]]) -> None

Constructs a RollPitchYaw from a 3x1 array of angles.

Parameter rpy:

3x1 array with roll, pitch, yaw angles (units of radians).

Raises

RuntimeError in debug builds if !IsValid(rpy) –

  1. __init__(self: pydrake.math.RollPitchYaw_[AutoDiffXd], roll: pydrake.autodiffutils.AutoDiffXd, pitch: pydrake.autodiffutils.AutoDiffXd, yaw: pydrake.autodiffutils.AutoDiffXd) -> None

Constructs a RollPitchYaw from roll, pitch, yaw angles (radian units).

Parameter roll:

x-directed angle in SpaceXYZ rotation sequence.

Parameter pitch:

y-directed angle in SpaceXYZ rotation sequence.

Parameter yaw:

z-directed angle in SpaceXYZ rotation sequence.

Raises

  • RuntimeError in debug builds if !IsValid(Vector3<T>(roll, pitch, –

  • yaw)) –

  1. __init__(self: pydrake.math.RollPitchYaw_[AutoDiffXd], R: pydrake.math.RotationMatrix_[AutoDiffXd]) -> None

Uses a RotationMatrix to construct a RollPitchYaw with roll-pitch-yaw angles [r, p, y] in the range -π <= r <= π, -π/2 <= p <= π/2, -π <= y <= π.

Parameter R:

a RotationMatrix.

Note

This new high-accuracy algorithm avoids numerical round-off issues encountered by some algorithms when pitch is within 1E-6 of π/2 or -π/2.

  1. __init__(self: pydrake.math.RollPitchYaw_[AutoDiffXd], quaternion: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]) -> None

Uses a Quaternion to construct a RollPitchYaw with roll-pitch-yaw angles [r, p, y] in the range -π <= r <= π, -π/2 <= p <= π/2, -π <= y <= π.

Parameter quaternion:

unit Quaternion.

Note

This new high-accuracy algorithm avoids numerical round-off issues encountered by some algorithms when pitch is within 1E-6 of π/2 or -π/2.

Raises

RuntimeError in debug builds if !IsValid(rpy) –

  1. __init__(self: pydrake.math.RollPitchYaw_[AutoDiffXd], matrix: numpy.ndarray[object[3, 3]]) -> None

Construct from raw rotation matrix. See RotationMatrix overload for more information.

CalcAngularVelocityInChildFromRpyDt(self: pydrake.math.RollPitchYaw_[AutoDiffXd], rpyDt: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Calculates angular velocity from this RollPitchYaw whose roll-pitch-yaw angles [r; p; y] relate the orientation of two generic frames A and D.

Parameter rpyDt:

Time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Returns

w_AD_D, frame D’s angular velocity in frame A, expressed in “child” frame D. In other words, returns [ω0; ω1; ω2]ᴅ, where w_AD_D = ω0 Dx + ω1 Dy + ω2 Dz, and where [ω0; ω1; ω2]ᴅ is calculated via the 3x3 matrix Nc⁻¹ (the inverse of the matrix Nc documented in CalcMatrixRelatingRpyDtToAngularVelocityInChild()).

Click to expand C++ code...
⌈ ω0 ⌉         ⌈ ṙ ⌉            ⌈ 1      0        -sin(p)    ⌉
| ω1 |  = Nc⁻¹ | ṗ |     Nc⁻¹ = | 0   cos(r)   sin(r)*cos(p) |
⌊ ω2 ⌋ᴅ        ⌊ ẏ ⌋            ⌊ 0  -sin(r)   cos(r)*cos(p) ⌋
CalcAngularVelocityInParentFromRpyDt(self: pydrake.math.RollPitchYaw_[AutoDiffXd], rpyDt: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Calculates angular velocity from this RollPitchYaw whose roll-pitch-yaw angles [r; p; y] relate the orientation of two generic frames A and D.

Parameter rpyDt:

Time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Returns

w_AD_A, frame D’s angular velocity in frame A, expressed in “parent” frame A. In other words, returns [ωx; ωy; ωz]ᴀ, where w_AD_A = ωx Ax + ωy Ay + ωz Az, and where [ωx; ωy; ωz]ᴀ is calculated via the the 3x3 matrix Np⁻¹ (the inverse of the matrix Np documented in CalcMatrixRelatingRpyDtToAngularVelocityInParent()).

Click to expand C++ code...
⌈ ωx ⌉         ⌈ ṙ ⌉            ⌈ cos(y)*cos(p)  -sin(y)  0 ⌉
| ωy |  = Np⁻¹ | ṗ |     Np⁻¹ = | sin(y)*cos(p)   cos(y)  0 |
⌊ ωz ⌋ᴀ        ⌊ ẏ ⌋            ⌊   -sin(p)         0     1 ⌋
CalcRotationMatrixDt(self: pydrake.math.RollPitchYaw_[AutoDiffXd], rpyDt: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 3]]

Forms Ṙ, the ordinary derivative of the RotationMatrix R with respect to an independent variable t (t usually denotes time) and R is the RotationMatrix formed by this RollPitchYaw. The roll-pitch-yaw angles r, p, y are regarded as functions of t [i.e., r(t), p(t), y(t)].

Parameter rpyDt:

Ordinary derivative of rpy with respect to an independent variable t (t usually denotes time, but not necessarily).

Returns

Ṙ, the ordinary derivative of R with respect to t, calculated as Ṙ = ∂R/∂r * ṙ + ∂R/∂p * ṗ + ∂R/∂y * ẏ. In other words, the returned (i, j) element is ∂Rij/∂r * ṙ + ∂Rij/∂p * ṗ + ∂Rij/∂y * ẏ.

CalcRpyDDtFromAngularAccelInChild(self: pydrake.math.RollPitchYaw_[AutoDiffXd], rpyDt: numpy.ndarray[object[3, 1]], alpha_AD_D: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Uses angular acceleration to compute the 2ⁿᵈ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter rpyDt:

time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Parameter alpha_AD_D:

, frame D’s angular acceleration in frame A, expressed in frame D.

Returns

[r̈, p̈, ÿ], the 2ⁿᵈ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

This method has a divide-by-zero error (singularity) when the cosine of the pitch angle p is zero [i.e., cos(p) = 0`]. This problem (called "gimbal lock") occurs when `p = n π + π / 2, where n is any integer. There are associated precision problems (inaccuracies) in the neighborhood of these pitch angles, i.e., when cos(p) ≈ 0.

CalcRpyDDtFromRpyDtAndAngularAccelInParent(self: pydrake.math.RollPitchYaw_[AutoDiffXd], rpyDt: numpy.ndarray[object[3, 1]], alpha_AD_A: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Uses angular acceleration to compute the 2ⁿᵈ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter rpyDt:

time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Parameter alpha_AD_A:

, frame D’s angular acceleration in frame A, expressed in frame A.

Returns

[r̈, p̈, ÿ], the 2ⁿᵈ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

This method has a divide-by-zero error (singularity) when the cosine of the pitch angle p is zero [i.e., cos(p) = 0`]. This problem (called "gimbal lock") occurs when `p = n π + π / 2, where n is any integer. There are associated precision problems (inaccuracies) in the neighborhood of these pitch angles, i.e., when cos(p) ≈ 0.

CalcRpyDtFromAngularVelocityInChild(self: pydrake.math.RollPitchYaw_[AutoDiffXd], w_AD_D: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Uses angular velocity to compute the 1ˢᵗ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter w_AD_D:

, frame D’s angular velocity in frame A, expressed in D.

Returns

[ṙ; ṗ; ẏ], the 1ˢᵗ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

Enhanced documentation for this method and its gimbal-lock (divide- by-zero error) is in CalcMatrixRelatingRpyDtToAngularVelocityInChild().

See also

CalcRpyDtFromAngularVelocityInParent()

CalcRpyDtFromAngularVelocityInParent(self: pydrake.math.RollPitchYaw_[AutoDiffXd], w_AD_A: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Uses angular velocity to compute the 1ˢᵗ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter w_AD_A:

, frame D’s angular velocity in frame A, expressed in A.

Returns

[ṙ; ṗ; ẏ], the 1ˢᵗ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

Enhanced documentation for this method and its gimbal-lock (divide- by-zero error) is in CalcMatrixRelatingRpyDtToAngularVelocityInParent().

See also

CalcRpyDtFromAngularVelocityInChild()

pitch_angle(self: pydrake.math.RollPitchYaw_[AutoDiffXd]) → pydrake.autodiffutils.AutoDiffXd

Returns the pitch-angle underlying this RollPitchYaw.

roll_angle(self: pydrake.math.RollPitchYaw_[AutoDiffXd]) → pydrake.autodiffutils.AutoDiffXd

Returns the roll-angle underlying this RollPitchYaw.

ToQuaternion(self: pydrake.math.RollPitchYaw_[AutoDiffXd]) → pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]

Returns a quaternion representation of this RollPitchYaw.

ToRotationMatrix(self: pydrake.math.RollPitchYaw_[AutoDiffXd]) → pydrake.math.RotationMatrix_[AutoDiffXd]

Returns the RotationMatrix representation of this RollPitchYaw.

vector(self: pydrake.math.RollPitchYaw_[AutoDiffXd]) → numpy.ndarray[object[3, 1]]

Returns the Vector3 underlying this RollPitchYaw.

yaw_angle(self: pydrake.math.RollPitchYaw_[AutoDiffXd]) → pydrake.autodiffutils.AutoDiffXd

Returns the yaw-angle underlying this RollPitchYaw.

class pydrake.math.RollPitchYaw_[Expression]

This class represents the orientation between two arbitrary frames A and D associated with a Space-fixed (extrinsic) X-Y-Z rotation by “roll-pitch-yaw” angles [r, p, y], which is equivalent to a Body-fixed (intrinsic) Z-Y-X rotation by “yaw-pitch-roll” angles [y, p, r]. The rotation matrix R_AD associated with this roll-pitch-yaw [r, p, y] rotation sequence is equal to the matrix multiplication shown below.

Click to expand C++ code...
⎡cos(y) -sin(y)  0⎤   ⎡ cos(p)  0  sin(p)⎤   ⎡1      0        0 ⎤
R_AD = ⎢sin(y)  cos(y)  0⎥ * ⎢     0   1      0 ⎥ * ⎢0  cos(r)  -sin(r)⎥
       ⎣    0       0   1⎦   ⎣-sin(p)  0  cos(p)⎦   ⎣0  sin(r)   cos(r)⎦
     =       R_AB          *        R_BC          *        R_CD

Note

In this discussion, A is the Space frame and D is the Body frame. One way to visualize this rotation sequence is by introducing intermediate frames B and C (useful constructs to understand this rotation sequence). Initially, the frames are aligned so Di = Ci = Bi = Ai (i = x, y, z) where Dx, Dy, Dz and Ax, Ay, Az are orthogonal unit vectors fixed in frames D and A respectively. Similarly for Bx, By, Bz and Cx, Cy, Cz in frame B, C. Then D is subjected to successive right-handed rotations relative to A.

  • 1st rotation R_CD: Frame D rotates relative to frames C, B, A by a

roll angle r about Dx = Cx. Note: D and C are no longer aligned.

  • 2nd rotation R_BC: Frames D, C (collectively – as if welded together)

rotate relative to frame B, A by a pitch angle p about Cy = By. Note: C and B are no longer aligned.

  • 3rd rotation R_AB: Frames D, C, B (collectively – as if welded)

rotate relative to frame A by a yaw angle y about Bz = Az. Note: B and A are no longer aligned. The monogram notation for the rotation matrix relating A to D is R_AD.

See also

multibody_quantities for monogram notation for dynamics and orientation_discussion “a discussion on rotation matrices”.

Note

This class does not store the frames associated with this rotation sequence.

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.RollPitchYaw_[Expression], other: pydrake.math.RollPitchYaw_[Expression]) -> None

  2. __init__(self: pydrake.math.RollPitchYaw_[Expression], rpy: numpy.ndarray[object[3, 1]]) -> None

Constructs a RollPitchYaw from a 3x1 array of angles.

Parameter rpy:

3x1 array with roll, pitch, yaw angles (units of radians).

Raises

RuntimeError in debug builds if !IsValid(rpy) –

  1. __init__(self: pydrake.math.RollPitchYaw_[Expression], roll: pydrake.symbolic.Expression, pitch: pydrake.symbolic.Expression, yaw: pydrake.symbolic.Expression) -> None

Constructs a RollPitchYaw from roll, pitch, yaw angles (radian units).

Parameter roll:

x-directed angle in SpaceXYZ rotation sequence.

Parameter pitch:

y-directed angle in SpaceXYZ rotation sequence.

Parameter yaw:

z-directed angle in SpaceXYZ rotation sequence.

Raises

  • RuntimeError in debug builds if !IsValid(Vector3<T>(roll, pitch, –

  • yaw)) –

  1. __init__(self: pydrake.math.RollPitchYaw_[Expression], R: pydrake.math.RotationMatrix_[Expression]) -> None

Uses a RotationMatrix to construct a RollPitchYaw with roll-pitch-yaw angles [r, p, y] in the range -π <= r <= π, -π/2 <= p <= π/2, -π <= y <= π.

Parameter R:

a RotationMatrix.

Note

This new high-accuracy algorithm avoids numerical round-off issues encountered by some algorithms when pitch is within 1E-6 of π/2 or -π/2.

  1. __init__(self: pydrake.math.RollPitchYaw_[Expression], quaternion: pydrake.common.eigen_geometry.Quaternion_[Expression]) -> None

Uses a Quaternion to construct a RollPitchYaw with roll-pitch-yaw angles [r, p, y] in the range -π <= r <= π, -π/2 <= p <= π/2, -π <= y <= π.

Parameter quaternion:

unit Quaternion.

Note

This new high-accuracy algorithm avoids numerical round-off issues encountered by some algorithms when pitch is within 1E-6 of π/2 or -π/2.

Raises

RuntimeError in debug builds if !IsValid(rpy) –

  1. __init__(self: pydrake.math.RollPitchYaw_[Expression], matrix: numpy.ndarray[object[3, 3]]) -> None

Construct from raw rotation matrix. See RotationMatrix overload for more information.

CalcAngularVelocityInChildFromRpyDt(self: pydrake.math.RollPitchYaw_[Expression], rpyDt: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Calculates angular velocity from this RollPitchYaw whose roll-pitch-yaw angles [r; p; y] relate the orientation of two generic frames A and D.

Parameter rpyDt:

Time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Returns

w_AD_D, frame D’s angular velocity in frame A, expressed in “child” frame D. In other words, returns [ω0; ω1; ω2]ᴅ, where w_AD_D = ω0 Dx + ω1 Dy + ω2 Dz, and where [ω0; ω1; ω2]ᴅ is calculated via the 3x3 matrix Nc⁻¹ (the inverse of the matrix Nc documented in CalcMatrixRelatingRpyDtToAngularVelocityInChild()).

Click to expand C++ code...
⌈ ω0 ⌉         ⌈ ṙ ⌉            ⌈ 1      0        -sin(p)    ⌉
| ω1 |  = Nc⁻¹ | ṗ |     Nc⁻¹ = | 0   cos(r)   sin(r)*cos(p) |
⌊ ω2 ⌋ᴅ        ⌊ ẏ ⌋            ⌊ 0  -sin(r)   cos(r)*cos(p) ⌋
CalcAngularVelocityInParentFromRpyDt(self: pydrake.math.RollPitchYaw_[Expression], rpyDt: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Calculates angular velocity from this RollPitchYaw whose roll-pitch-yaw angles [r; p; y] relate the orientation of two generic frames A and D.

Parameter rpyDt:

Time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Returns

w_AD_A, frame D’s angular velocity in frame A, expressed in “parent” frame A. In other words, returns [ωx; ωy; ωz]ᴀ, where w_AD_A = ωx Ax + ωy Ay + ωz Az, and where [ωx; ωy; ωz]ᴀ is calculated via the the 3x3 matrix Np⁻¹ (the inverse of the matrix Np documented in CalcMatrixRelatingRpyDtToAngularVelocityInParent()).

Click to expand C++ code...
⌈ ωx ⌉         ⌈ ṙ ⌉            ⌈ cos(y)*cos(p)  -sin(y)  0 ⌉
| ωy |  = Np⁻¹ | ṗ |     Np⁻¹ = | sin(y)*cos(p)   cos(y)  0 |
⌊ ωz ⌋ᴀ        ⌊ ẏ ⌋            ⌊   -sin(p)         0     1 ⌋
CalcRotationMatrixDt(self: pydrake.math.RollPitchYaw_[Expression], rpyDt: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 3]]

Forms Ṙ, the ordinary derivative of the RotationMatrix R with respect to an independent variable t (t usually denotes time) and R is the RotationMatrix formed by this RollPitchYaw. The roll-pitch-yaw angles r, p, y are regarded as functions of t [i.e., r(t), p(t), y(t)].

Parameter rpyDt:

Ordinary derivative of rpy with respect to an independent variable t (t usually denotes time, but not necessarily).

Returns

Ṙ, the ordinary derivative of R with respect to t, calculated as Ṙ = ∂R/∂r * ṙ + ∂R/∂p * ṗ + ∂R/∂y * ẏ. In other words, the returned (i, j) element is ∂Rij/∂r * ṙ + ∂Rij/∂p * ṗ + ∂Rij/∂y * ẏ.

CalcRpyDDtFromAngularAccelInChild(self: pydrake.math.RollPitchYaw_[Expression], rpyDt: numpy.ndarray[object[3, 1]], alpha_AD_D: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Uses angular acceleration to compute the 2ⁿᵈ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter rpyDt:

time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Parameter alpha_AD_D:

, frame D’s angular acceleration in frame A, expressed in frame D.

Returns

[r̈, p̈, ÿ], the 2ⁿᵈ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

This method has a divide-by-zero error (singularity) when the cosine of the pitch angle p is zero [i.e., cos(p) = 0`]. This problem (called "gimbal lock") occurs when `p = n π + π / 2, where n is any integer. There are associated precision problems (inaccuracies) in the neighborhood of these pitch angles, i.e., when cos(p) ≈ 0.

CalcRpyDDtFromRpyDtAndAngularAccelInParent(self: pydrake.math.RollPitchYaw_[Expression], rpyDt: numpy.ndarray[object[3, 1]], alpha_AD_A: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Uses angular acceleration to compute the 2ⁿᵈ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter rpyDt:

time-derivative of [r; p; y], i.e., [ṙ; ṗ; ẏ].

Parameter alpha_AD_A:

, frame D’s angular acceleration in frame A, expressed in frame A.

Returns

[r̈, p̈, ÿ], the 2ⁿᵈ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

This method has a divide-by-zero error (singularity) when the cosine of the pitch angle p is zero [i.e., cos(p) = 0`]. This problem (called "gimbal lock") occurs when `p = n π + π / 2, where n is any integer. There are associated precision problems (inaccuracies) in the neighborhood of these pitch angles, i.e., when cos(p) ≈ 0.

CalcRpyDtFromAngularVelocityInChild(self: pydrake.math.RollPitchYaw_[Expression], w_AD_D: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Uses angular velocity to compute the 1ˢᵗ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter w_AD_D:

, frame D’s angular velocity in frame A, expressed in D.

Returns

[ṙ; ṗ; ẏ], the 1ˢᵗ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

Enhanced documentation for this method and its gimbal-lock (divide- by-zero error) is in CalcMatrixRelatingRpyDtToAngularVelocityInChild().

See also

CalcRpyDtFromAngularVelocityInParent()

CalcRpyDtFromAngularVelocityInParent(self: pydrake.math.RollPitchYaw_[Expression], w_AD_A: numpy.ndarray[object[3, 1]]) → numpy.ndarray[object[3, 1]]

Uses angular velocity to compute the 1ˢᵗ time-derivative of this RollPitchYaw whose angles [r; p; y] orient two generic frames A and D.

Parameter w_AD_A:

, frame D’s angular velocity in frame A, expressed in A.

Returns

[ṙ; ṗ; ẏ], the 1ˢᵗ time-derivative of this RollPitchYaw.

Raises

RuntimeError if cos(p) ≈ 0 (p is near gimbal-lock) –

Note

Enhanced documentation for this method and its gimbal-lock (divide- by-zero error) is in CalcMatrixRelatingRpyDtToAngularVelocityInParent().

See also

CalcRpyDtFromAngularVelocityInChild()

pitch_angle(self: pydrake.math.RollPitchYaw_[Expression]) → pydrake.symbolic.Expression

Returns the pitch-angle underlying this RollPitchYaw.

roll_angle(self: pydrake.math.RollPitchYaw_[Expression]) → pydrake.symbolic.Expression

Returns the roll-angle underlying this RollPitchYaw.

ToQuaternion(self: pydrake.math.RollPitchYaw_[Expression]) → pydrake.common.eigen_geometry.Quaternion_[Expression]

Returns a quaternion representation of this RollPitchYaw.

ToRotationMatrix(self: pydrake.math.RollPitchYaw_[Expression]) → pydrake.math.RotationMatrix_[Expression]

Returns the RotationMatrix representation of this RollPitchYaw.

vector(self: pydrake.math.RollPitchYaw_[Expression]) → numpy.ndarray[object[3, 1]]

Returns the Vector3 underlying this RollPitchYaw.

yaw_angle(self: pydrake.math.RollPitchYaw_[Expression]) → pydrake.symbolic.Expression

Returns the yaw-angle underlying this RollPitchYaw.

class pydrake.math.RotationMatrix

This class represents a 3x3 rotation matrix between two arbitrary frames A and B and helps ensure users create valid rotation matrices. This class relates right-handed orthogonal unit vectors Ax, Ay, Az fixed in frame A to right-handed orthogonal unit vectors Bx, By, Bz fixed in frame B. The monogram notation for the rotation matrix relating A to B is R_AB. An example that gives context to this rotation matrix is v_A = R_AB * v_B, where v_B denotes an arbitrary vector v expressed in terms of Bx, By, Bz and v_A denotes vector v expressed in terms of Ax, Ay, Az. See multibody_quantities for monogram notation for dynamics. See orientation_discussion “a discussion on rotation matrices”.

Note

This class does not store the frames associated with a rotation matrix nor does it enforce strict proper usage of this class with vectors.

Note

When assertions are enabled, several methods in this class perform a validity check and throw RuntimeError if the rotation matrix is invalid. When assertions are disabled, many of these validity checks are skipped (which helps improve speed). These validity tests are only performed for scalar types for which drake::scalar_predicate<T>::is_bool is True. For instance, validity checks are not performed when T is symbolic::Expression.

Authors:

Paul Mitiguy (2018) Original author.

Authors:

Drake team (see https://drake.mit.edu/credits).

Note

This class is templated; see RotationMatrix_ for the list of instantiations.

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.RotationMatrix) -> None

Constructs a 3x3 identity RotationMatrix – which corresponds to aligning two frames (so that unit vectors Ax = Bx, Ay = By, Az = Bz).

  1. __init__(self: pydrake.math.RotationMatrix, other: pydrake.math.RotationMatrix) -> None

  2. __init__(self: pydrake.math.RotationMatrix, R: numpy.ndarray[numpy.float64[3, 3]]) -> None

Constructs a RotationMatrix from a Matrix3.

Parameter R:

an allegedly valid rotation matrix.

Raises

RuntimeError in debug builds if R fails IsValid(R) –

  1. __init__(self: pydrake.math.RotationMatrix, quaternion: pydrake.common.eigen_geometry.Quaternion) -> None

Constructs a RotationMatrix from an Eigen::Quaternion.

Parameter quaternion:

a non-zero, finite quaternion which may or may not have unit length [i.e., quaternion.norm() does not have to be 1].

Raises

  • RuntimeError in debug builds if the rotation matrix R that is –

  • built from quaternion fails IsValid(R) For example, an –

  • exception is thrown if quaternion is zero or contains a NaN or –

  • infinity. –

Note

This method has the effect of normalizing its quaternion argument, without the inefficiency of the square-root associated with normalization.

  1. __init__(self: pydrake.math.RotationMatrix, theta_lambda: pydrake.common.eigen_geometry.AngleAxis) -> None

Constructs a RotationMatrix from an Eigen::AngleAxis.

Parameter theta_lambda:

an Eigen::AngleAxis whose associated axis (vector direction herein called lambda) is non-zero and finite, but which may or may not have unit length [i.e., lambda.norm() does not have to be 1].

Raises

  • RuntimeError in debug builds if the rotation matrix R that is –

  • built from theta_lambda fails IsValid(R) For example, an –

  • exception is thrown if lambda is zero or contains a NaN or –

  • infinity. –

  1. __init__(self: pydrake.math.RotationMatrix, rpy: pydrake.math.RollPitchYaw) -> None

Constructs a RotationMatrix from an RollPitchYaw. In other words, makes the RotationMatrix for a Space-fixed (extrinsic) X-Y-Z rotation by “roll-pitch-yaw” angles [r, p, y], which is equivalent to a Body-fixed (intrinsic) Z-Y-X rotation by “yaw-pitch-roll” angles [y, p, r].

Parameter rpy:

radian measures of three angles [roll, pitch, yaw].

Parameter rpy:

a RollPitchYaw which is a Space-fixed (extrinsic) X-Y-Z rotation with “roll-pitch-yaw” angles [r, p, y] or equivalently a Body- fixed (intrinsic) Z-Y-X rotation with “yaw-pitch-roll” angles [y, p, r].

Returns R_AD:

, rotation matrix relating frame A to frame D.

Note

Denoting roll r, pitch p, yaw y, this method returns a rotation matrix R_AD equal to the matrix multiplication shown below.

Click to expand C++ code...
⎡cos(y) -sin(y)  0⎤   ⎡ cos(p)  0  sin(p)⎤   ⎡1      0        0 ⎤
R_AD = ⎢sin(y)  cos(y)  0⎥ * ⎢     0   1      0 ⎥ * ⎢0  cos(r)  -sin(r)⎥
       ⎣    0       0   1⎦   ⎣-sin(p)  0  cos(p)⎦   ⎣0  sin(r)   cos(r)⎦
     =       R_AB          *        R_BC          *        R_CD

Note

In this discussion, A is the Space frame and D is the Body frame. One way to visualize this rotation sequence is by introducing intermediate frames B and C (useful constructs to understand this rotation sequence). Initially, the frames are aligned so Di = Ci = Bi = Ai (i = x, y, z). Then D is subjected to successive right-handed rotations relative to A.

  • 1st rotation R_CD: Frame D rotates relative to frames C, B, A by a

roll angle r about Dx = Cx. Note: D and C are no longer aligned.

  • 2nd rotation R_BC: Frames D, C (collectively – as if welded together)

rotate relative to frame B, A by a pitch angle p about Cy = By. Note: C and B are no longer aligned.

  • 3rd rotation R_AB: Frames D, C, B (collectively – as if welded)

rotate relative to frame A by a roll angle y about Bz = Az. Note: B and A are no longer aligned.

Note

This method constructs a RotationMatrix from a RollPitchYaw. Vice-versa, there are high-accuracy RollPitchYaw constructor/methods that form a RollPitchYaw from a rotation matrix.

template cast

Instantiations: cast[float], cast[AutoDiffXd], cast[Expression]

cast[AutoDiffXd](self: pydrake.math.RotationMatrix) → pydrake.math.RotationMatrix_[AutoDiffXd]

Creates a RotationMatrix templatized on a scalar type U from a RotationMatrix templatized on scalar type T. For example,

Click to expand C++ code...
RotationMatrix<double> source = RotationMatrix<double>::Identity();
RotationMatrix<AutoDiffXd> foo = source.cast<AutoDiffXd>();
Template parameter U:

Scalar type on which the returned RotationMatrix is templated.

Note

RotationMatrix<From>::cast<To>() creates a new RotationMatrix<To> from a RotationMatrix<From> but only if type To is constructible from type From. This cast method works in accordance with Eigen’s cast method for Eigen’s Matrix3 that underlies this RotationMatrix. For example, Eigen currently allows cast from type double to AutoDiffXd, but not vice-versa.

cast[Expression](self: pydrake.math.RotationMatrix) → pydrake.math.RotationMatrix_[Expression]

Creates a RotationMatrix templatized on a scalar type U from a RotationMatrix templatized on scalar type T. For example,

Click to expand C++ code...
RotationMatrix<double> source = RotationMatrix<double>::Identity();
RotationMatrix<AutoDiffXd> foo = source.cast<AutoDiffXd>();
Template parameter U:

Scalar type on which the returned RotationMatrix is templated.

Note

RotationMatrix<From>::cast<To>() creates a new RotationMatrix<To> from a RotationMatrix<From> but only if type To is constructible from type From. This cast method works in accordance with Eigen’s cast method for Eigen’s Matrix3 that underlies this RotationMatrix. For example, Eigen currently allows cast from type double to AutoDiffXd, but not vice-versa.

cast[float](self: pydrake.math.RotationMatrix) → pydrake.math.RotationMatrix

Creates a RotationMatrix templatized on a scalar type U from a RotationMatrix templatized on scalar type T. For example,

Click to expand C++ code...
RotationMatrix<double> source = RotationMatrix<double>::Identity();
RotationMatrix<AutoDiffXd> foo = source.cast<AutoDiffXd>();
Template parameter U:

Scalar type on which the returned RotationMatrix is templated.

Note

RotationMatrix<From>::cast<To>() creates a new RotationMatrix<To> from a RotationMatrix<From> but only if type To is constructible from type From. This cast method works in accordance with Eigen’s cast method for Eigen’s Matrix3 that underlies this RotationMatrix. For example, Eigen currently allows cast from type double to AutoDiffXd, but not vice-versa.

col(self: pydrake.math.RotationMatrix, index: int) → numpy.ndarray[numpy.float64[3, 1]]

Returns this rotation matrix’s iᵗʰ column (i = 0, 1, 2). For this rotation matrix R_AB (which relates right-handed sets of orthogonal unit vectors Ax, Ay, Az to Bx, By, Bz), - col(0) returns Bx_A (Bx expressed in terms of Ax, Ay, Az). - col(1) returns By_A (By expressed in terms of Ax, Ay, Az). - col(2) returns Bz_A (Bz expressed in terms of Ax, Ay, Az).

Parameter index:

requested column index (0 <= index <= 2).

See also

row(), matrix()

Raises

In Debug builds, asserts (0 <= index <= 2) –

Note

For efficiency and consistency with Eigen, this method returns the same quantity returned by Eigen’s col() operator. The returned quantity can be assigned in various ways, e.g., as const auto& Bz_A = col(2); or Vector3<T> Bz_A = col(2);

static Identity() → pydrake.math.RotationMatrix
inverse(self: pydrake.math.RotationMatrix) → pydrake.math.RotationMatrix
InvertAndCompose(self: pydrake.math.RotationMatrix, other: pydrake.math.RotationMatrix) → pydrake.math.RotationMatrix

Calculates the product of this inverted and another RotationMatrix. If you consider this to be the rotation matrix R_AB, and other to be R_AC, then this method returns R_BC = R_AB⁻¹ * R_AC. For T==double, this method can be much faster than inverting first and then performing the composition because it can take advantage of the orthogonality of rotation matrices. On some platforms it can use SIMD instructions for further speedups.

Parameter other:

RotationMatrix that post-multiplies this inverted.

Returns R_BC:

where R_BC = this⁻¹ * other.

Note

It is possible (albeit improbable) to create an invalid rotation matrix by accumulating round-off error with a large number of multiplies.

IsExactlyIdentity(self: pydrake.math.RotationMatrix) → bool

Returns True if this is exactly equal to the identity matrix.

See also

IsNearlyIdentity().

IsNearlyIdentity(self: pydrake.math.RotationMatrix, tolerance: float = 2.842170943040401e-14) → bool

Returns true if this is within tolerance of the identity RigidTransform.

Parameter tolerance:

non-negative number that is generally the default value, namely RotationMatrix::get_internal_tolerance_for_orthonormality().

See also

IsExactlyIdentity().

IsValid(self: pydrake.math.RotationMatrix) → bool

Tests if this rotation matrix R is a proper orthonormal rotation matrix to within the threshold of get_internal_tolerance_for_orthonormality().

Returns

True if this is a valid rotation matrix.

static MakeFromOneVector(b_A: numpy.ndarray[numpy.float64[3, 1]], axis_index: int) → pydrake.math.RotationMatrix

Creates a 3D right-handed orthonormal basis B from a given vector b_A, returned as a rotation matrix R_AB. It consists of orthogonal unit vectors u_A, v_A, w_A where u_A is the normalized b_A in the axis_index column of R_AB and v_A has one element which is zero. If an element of b_A is zero, then one element of w_A is 1 and the other two elements are 0.

Parameter b_A:

vector expressed in frame A that when normalized as u_A = b_A.normalized() represents Bx, By, or Bz (depending on axis_index).

Parameter axis_index:

The index ∈ {0, 1, 2} of the unit vector associated with u_A, 0 means u_A is Bx, 1 means u_A is By, and 2 means u_A is Bz.

Precondition:

axis_index is 0 or 1 or 2.

Raises

  • RuntimeError if b_A cannot be made into a unit vector because b_A –

  • contains a NaN or infinity or b_A < 1.0E-10. –

See also

MakeFromOneUnitVector() if b_A is known to already be unit length.

Returns R_AB:

the rotation matrix with properties as described above.

static MakeUnchecked(R: numpy.ndarray[numpy.float64[3, 3]]) → pydrake.math.RotationMatrix

(Advanced) Makes a RotationMatrix from a Matrix3. No check is performed to test whether or not the parameter R is a valid rotation matrix.

static MakeXRotation(theta: float) → pydrake.math.RotationMatrix

Makes the RotationMatrix R_AB associated with rotating a frame B relative to a frame A by an angle theta about unit vector Ax = Bx.

Parameter theta:

radian measure of rotation angle about Ax.

Note

Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitX().

Note

R_AB relates two frames A and B having unit vectors Ax, Ay, Az and Bx, By, Bz. Initially, Bx = Ax, By = Ay, Bz = Az, then B undergoes a right-handed rotation relative to A by an angle theta about Ax = Bx.

Click to expand C++ code...
⎡ 1       0                 0  ⎤
R_AB = ⎢ 0   cos(theta)   -sin(theta) ⎥
       ⎣ 0   sin(theta)    cos(theta) ⎦
static MakeYRotation(theta: float) → pydrake.math.RotationMatrix

Makes the RotationMatrix R_AB associated with rotating a frame B relative to a frame A by an angle theta about unit vector Ay = By.

Parameter theta:

radian measure of rotation angle about Ay.

Note

Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitY().

Note

R_AB relates two frames A and B having unit vectors Ax, Ay, Az and Bx, By, Bz. Initially, Bx = Ax, By = Ay, Bz = Az, then B undergoes a right-handed rotation relative to A by an angle theta about Ay = By.

Click to expand C++ code...
⎡  cos(theta)   0   sin(theta) ⎤
R_AB = ⎢          0    1           0  ⎥
       ⎣ -sin(theta)   0   cos(theta) ⎦
static MakeZRotation(theta: float) → pydrake.math.RotationMatrix

Makes the RotationMatrix R_AB associated with rotating a frame B relative to a frame A by an angle theta about unit vector Az = Bz.

Parameter theta:

radian measure of rotation angle about Az.

Note

Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitZ().

Note

R_AB relates two frames A and B having unit vectors Ax, Ay, Az and Bx, By, Bz. Initially, Bx = Ax, By = Ay, Bz = Az, then B undergoes a right-handed rotation relative to A by an angle theta about Az = Bz.

Click to expand C++ code...
⎡ cos(theta)  -sin(theta)   0 ⎤
R_AB = ⎢ sin(theta)   cos(theta)   0 ⎥
       ⎣         0            0    1 ⎦
matrix(self: pydrake.math.RotationMatrix) → numpy.ndarray[numpy.float64[3, 3]]

Returns the Matrix3 underlying a RotationMatrix.

See also

col(), row()

multiply(*args, **kwargs)

Overloaded function.

  1. multiply(self: pydrake.math.RotationMatrix, other: pydrake.math.RotationMatrix) -> pydrake.math.RotationMatrix

Calculates this rotation matrix R_AB multiplied by other rotation matrix R_BC, returning the composition R_AB * R_BC.

Parameter other:

RotationMatrix that post-multiplies this.

Returns

rotation matrix that results from this multiplied by other.

Note

It is possible (albeit improbable) to create an invalid rotation matrix by accumulating round-off error with a large number of multiplies.

  1. multiply(self: pydrake.math.RotationMatrix, v_B: numpy.ndarray[numpy.float64[3, 1]]) -> numpy.ndarray[numpy.float64[3, 1]]

Calculates this rotation matrix R_AB multiplied by an arbitrary Vector3 expressed in the B frame.

Parameter v_B:

3x1 vector that post-multiplies this.

Returns

3x1 vector v_A = R_AB * v_B.

  1. multiply(self: pydrake.math.RotationMatrix, v_B: numpy.ndarray[numpy.float64[3, n]]) -> numpy.ndarray[numpy.float64[3, n]]

Multiplies this RotationMatrix R_AB by the n vectors v1, … vn, where each vector has 3 elements and is expressed in frame B.

Parameter v_B:

3 x n matrix whose n columns are regarded as arbitrary vectors v1, … vn expressed in frame B.

Returns v_A:

3 x n matrix whose n columns are vectors v1, … vn expressed in frame A.

Click to expand C++ code...
const RollPitchYaw<double> rpy(0.1, 0.2, 0.3);
const RotationMatrix<double> R_AB(rpy);
Eigen::Matrix<double, 3, 2> v_B;
v_B.col(0) = Vector3d(4, 5, 6);
v_B.col(1) = Vector3d(9, 8, 7);
const Eigen::Matrix<double, 3, 2> v_A = R_AB * v_B;
static ProjectToRotationMatrix(M: numpy.ndarray[numpy.float64[3, 3]]) → pydrake.math.RotationMatrix

Given an approximate rotation matrix M, finds the RotationMatrix R closest to M. Closeness is measured with a matrix-2 norm (or equivalently with a Frobenius norm). Hence, this method creates a RotationMatrix R from a 3x3 matrix M by minimizing ‖R - M‖₂ (the matrix-2 norm of (R-M)) subject to R * Rᵀ = I, where I is the 3x3 identity matrix. For this problem, closeness can also be measured by forming the orthonormal matrix R whose elements minimize the double-summation ∑ᵢ ∑ⱼ (R(i,j) - M(i,j))² where i = 1:3, j = 1:3, subject to R * Rᵀ = I. The square-root of this double-summation is called the Frobenius norm.

Parameter M:

a 3x3 matrix.

Parameter quality_factor:

. The quality of M as a rotation matrix. quality_factor = 1 is perfect (M = R). quality_factor = 1.25 means that when M multiplies a unit vector (magnitude 1), a vector of magnitude as large as 1.25 may result. quality_factor = 0.8 means that when M multiplies a unit vector, a vector of magnitude as small as 0.8 may result. quality_factor = 0 means M is singular, so at least one of the bases related by matrix M does not span 3D space (when M multiples a unit vector, a vector of magnitude as small as 0 may result).

Returns

proper orthonormal matrix R that is closest to M.

Raises

RuntimeError if R fails IsValid(R) –

Note

William Kahan (UC Berkeley) and Hongkai Dai (Toyota Research Institute) proved that for this problem, the same R that minimizes the Frobenius norm also minimizes the matrix-2 norm (a.k.a an induced-2 norm), which is defined [Dahleh, Section 4.2] as the column matrix u which maximizes ‖(R - M) u‖ / ‖u‖, where u ≠ 0. Since the matrix-2 norm of any matrix A is equal to the maximum singular value of A, minimizing the matrix-2 norm of (R - M) is equivalent to minimizing the maximum singular value of (R - M).

  • [Dahleh] “Lectures on Dynamic Systems and Controls: Electrical

Engineering and Computer Science, Massachusetts Institute of Technology” https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-241j-dynamic-systems-and-control-spring-2011/readings/MIT6_241JS11_chap04.pdf

row(self: pydrake.math.RotationMatrix, index: int) → numpy.ndarray[numpy.float64[1, 3]]

Returns this rotation matrix’s iᵗʰ row (i = 0, 1, 2). For this rotation matrix R_AB (which relates right-handed sets of orthogonal unit vectors Ax, Ay, Az to Bx, By, Bz), - row(0) returns Ax_B (Ax expressed in terms of Bx, By, Bz). - row(1) returns Ay_B (Ay expressed in terms of Bx, By, Bz). - row(2) returns Az_B (Az expressed in terms of Bx, By, Bz).

Parameter index:

requested row index (0 <= index <= 2).

See also

col(), matrix()

Raises

In Debug builds, asserts (0 <= index <= 2) –

Note

For efficiency and consistency with Eigen, this method returns the same quantity returned by Eigen’s row() operator. The returned quantity can be assigned in various ways, e.g., as const auto& Az_B = row(2); or RowVector3<T> Az_B = row(2);

set(self: pydrake.math.RotationMatrix, R: numpy.ndarray[numpy.float64[3, 3]]) → None

Sets this RotationMatrix from a Matrix3.

Parameter R:

an allegedly valid rotation matrix.

Raises

RuntimeError in debug builds if R fails IsValid(R) –

ToAngleAxis(self: pydrake.math.RotationMatrix) → pydrake.common.eigen_geometry.AngleAxis

Returns an AngleAxis theta_lambda containing an angle theta and unit vector (axis direction) lambda that represents this RotationMatrix.

Note

The orientation and RotationMatrix associated with theta * lambda is identical to that of (-theta) * (-lambda). The AngleAxis returned by this method chooses to have 0 <= theta <= pi.

Returns

an AngleAxis with 0 <= theta <= pi and a unit vector lambda.

ToQuaternion(self: pydrake.math.RotationMatrix) → pydrake.common.eigen_geometry.Quaternion

Returns a quaternion q that represents this RotationMatrix. Since the quaternion q and -q represent the same RotationMatrix, this method chooses to return a canonical quaternion, i.e., with q(0) >= 0.

ToRollPitchYaw(self: pydrake.math.RotationMatrix) → pydrake.math.RollPitchYaw

Returns a RollPitchYaw that represents this RotationMatrix, with roll-pitch-yaw angles [r, p, y] in the range -π <= r <= π, -π/2 <= p <= π/2, -π <= y <= π.

Note

This new high-accuracy algorithm avoids numerical round-off issues encountered by some algorithms when pitch is within 1E-6 of π/2 or -π/2.

transpose(self: pydrake.math.RotationMatrix) → pydrake.math.RotationMatrix
template pydrake.math.RotationMatrix_

Instantiations: RotationMatrix_[float], RotationMatrix_[AutoDiffXd], RotationMatrix_[Expression]

class pydrake.math.RotationMatrix_[AutoDiffXd]

This class represents a 3x3 rotation matrix between two arbitrary frames A and B and helps ensure users create valid rotation matrices. This class relates right-handed orthogonal unit vectors Ax, Ay, Az fixed in frame A to right-handed orthogonal unit vectors Bx, By, Bz fixed in frame B. The monogram notation for the rotation matrix relating A to B is R_AB. An example that gives context to this rotation matrix is v_A = R_AB * v_B, where v_B denotes an arbitrary vector v expressed in terms of Bx, By, Bz and v_A denotes vector v expressed in terms of Ax, Ay, Az. See multibody_quantities for monogram notation for dynamics. See orientation_discussion “a discussion on rotation matrices”.

Note

This class does not store the frames associated with a rotation matrix nor does it enforce strict proper usage of this class with vectors.

Note

When assertions are enabled, several methods in this class perform a validity check and throw RuntimeError if the rotation matrix is invalid. When assertions are disabled, many of these validity checks are skipped (which helps improve speed). These validity tests are only performed for scalar types for which drake::scalar_predicate<T>::is_bool is True. For instance, validity checks are not performed when T is symbolic::Expression.

Authors:

Paul Mitiguy (2018) Original author.

Authors:

Drake team (see https://drake.mit.edu/credits).

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.RotationMatrix_[AutoDiffXd]) -> None

Constructs a 3x3 identity RotationMatrix – which corresponds to aligning two frames (so that unit vectors Ax = Bx, Ay = By, Az = Bz).

  1. __init__(self: pydrake.math.RotationMatrix_[AutoDiffXd], other: pydrake.math.RotationMatrix_[AutoDiffXd]) -> None

  2. __init__(self: pydrake.math.RotationMatrix_[AutoDiffXd], R: numpy.ndarray[object[3, 3]]) -> None

Constructs a RotationMatrix from a Matrix3.

Parameter R:

an allegedly valid rotation matrix.

Raises

RuntimeError in debug builds if R fails IsValid(R) –

  1. __init__(self: pydrake.math.RotationMatrix_[AutoDiffXd], quaternion: pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]) -> None

Constructs a RotationMatrix from an Eigen::Quaternion.

Parameter quaternion:

a non-zero, finite quaternion which may or may not have unit length [i.e., quaternion.norm() does not have to be 1].

Raises

  • RuntimeError in debug builds if the rotation matrix R that is –

  • built from quaternion fails IsValid(R) For example, an –

  • exception is thrown if quaternion is zero or contains a NaN or –

  • infinity. –

Note

This method has the effect of normalizing its quaternion argument, without the inefficiency of the square-root associated with normalization.

  1. __init__(self: pydrake.math.RotationMatrix_[AutoDiffXd], theta_lambda: pydrake.common.eigen_geometry.AngleAxis_[AutoDiffXd]) -> None

Constructs a RotationMatrix from an Eigen::AngleAxis.

Parameter theta_lambda:

an Eigen::AngleAxis whose associated axis (vector direction herein called lambda) is non-zero and finite, but which may or may not have unit length [i.e., lambda.norm() does not have to be 1].

Raises

  • RuntimeError in debug builds if the rotation matrix R that is –

  • built from theta_lambda fails IsValid(R) For example, an –

  • exception is thrown if lambda is zero or contains a NaN or –

  • infinity. –

  1. __init__(self: pydrake.math.RotationMatrix_[AutoDiffXd], rpy: pydrake.math.RollPitchYaw_[AutoDiffXd]) -> None

Constructs a RotationMatrix from an RollPitchYaw. In other words, makes the RotationMatrix for a Space-fixed (extrinsic) X-Y-Z rotation by “roll-pitch-yaw” angles [r, p, y], which is equivalent to a Body-fixed (intrinsic) Z-Y-X rotation by “yaw-pitch-roll” angles [y, p, r].

Parameter rpy:

radian measures of three angles [roll, pitch, yaw].

Parameter rpy:

a RollPitchYaw which is a Space-fixed (extrinsic) X-Y-Z rotation with “roll-pitch-yaw” angles [r, p, y] or equivalently a Body- fixed (intrinsic) Z-Y-X rotation with “yaw-pitch-roll” angles [y, p, r].

Returns R_AD:

, rotation matrix relating frame A to frame D.

Note

Denoting roll r, pitch p, yaw y, this method returns a rotation matrix R_AD equal to the matrix multiplication shown below.

Click to expand C++ code...
⎡cos(y) -sin(y)  0⎤   ⎡ cos(p)  0  sin(p)⎤   ⎡1      0        0 ⎤
R_AD = ⎢sin(y)  cos(y)  0⎥ * ⎢     0   1      0 ⎥ * ⎢0  cos(r)  -sin(r)⎥
       ⎣    0       0   1⎦   ⎣-sin(p)  0  cos(p)⎦   ⎣0  sin(r)   cos(r)⎦
     =       R_AB          *        R_BC          *        R_CD

Note

In this discussion, A is the Space frame and D is the Body frame. One way to visualize this rotation sequence is by introducing intermediate frames B and C (useful constructs to understand this rotation sequence). Initially, the frames are aligned so Di = Ci = Bi = Ai (i = x, y, z). Then D is subjected to successive right-handed rotations relative to A.

  • 1st rotation R_CD: Frame D rotates relative to frames C, B, A by a

roll angle r about Dx = Cx. Note: D and C are no longer aligned.

  • 2nd rotation R_BC: Frames D, C (collectively – as if welded together)

rotate relative to frame B, A by a pitch angle p about Cy = By. Note: C and B are no longer aligned.

  • 3rd rotation R_AB: Frames D, C, B (collectively – as if welded)

rotate relative to frame A by a roll angle y about Bz = Az. Note: B and A are no longer aligned.

Note

This method constructs a RotationMatrix from a RollPitchYaw. Vice-versa, there are high-accuracy RollPitchYaw constructor/methods that form a RollPitchYaw from a rotation matrix.

template cast

Instantiations: cast[AutoDiffXd]

cast[AutoDiffXd](self: pydrake.math.RotationMatrix_[AutoDiffXd]) → pydrake.math.RotationMatrix_[AutoDiffXd]

Creates a RotationMatrix templatized on a scalar type U from a RotationMatrix templatized on scalar type T. For example,

Click to expand C++ code...
RotationMatrix<double> source = RotationMatrix<double>::Identity();
RotationMatrix<AutoDiffXd> foo = source.cast<AutoDiffXd>();
Template parameter U:

Scalar type on which the returned RotationMatrix is templated.

Note

RotationMatrix<From>::cast<To>() creates a new RotationMatrix<To> from a RotationMatrix<From> but only if type To is constructible from type From. This cast method works in accordance with Eigen’s cast method for Eigen’s Matrix3 that underlies this RotationMatrix. For example, Eigen currently allows cast from type double to AutoDiffXd, but not vice-versa.

col(self: pydrake.math.RotationMatrix_[AutoDiffXd], index: int) → numpy.ndarray[object[3, 1]]

Returns this rotation matrix’s iᵗʰ column (i = 0, 1, 2). For this rotation matrix R_AB (which relates right-handed sets of orthogonal unit vectors Ax, Ay, Az to Bx, By, Bz), - col(0) returns Bx_A (Bx expressed in terms of Ax, Ay, Az). - col(1) returns By_A (By expressed in terms of Ax, Ay, Az). - col(2) returns Bz_A (Bz expressed in terms of Ax, Ay, Az).

Parameter index:

requested column index (0 <= index <= 2).

See also

row(), matrix()

Raises

In Debug builds, asserts (0 <= index <= 2) –

Note

For efficiency and consistency with Eigen, this method returns the same quantity returned by Eigen’s col() operator. The returned quantity can be assigned in various ways, e.g., as const auto& Bz_A = col(2); or Vector3<T> Bz_A = col(2);

static Identity() → pydrake.math.RotationMatrix_[AutoDiffXd]
inverse(self: pydrake.math.RotationMatrix_[AutoDiffXd]) → pydrake.math.RotationMatrix_[AutoDiffXd]
InvertAndCompose(self: pydrake.math.RotationMatrix_[AutoDiffXd], other: pydrake.math.RotationMatrix_[AutoDiffXd]) → pydrake.math.RotationMatrix_[AutoDiffXd]

Calculates the product of this inverted and another RotationMatrix. If you consider this to be the rotation matrix R_AB, and other to be R_AC, then this method returns R_BC = R_AB⁻¹ * R_AC. For T==double, this method can be much faster than inverting first and then performing the composition because it can take advantage of the orthogonality of rotation matrices. On some platforms it can use SIMD instructions for further speedups.

Parameter other:

RotationMatrix that post-multiplies this inverted.

Returns R_BC:

where R_BC = this⁻¹ * other.

Note

It is possible (albeit improbable) to create an invalid rotation matrix by accumulating round-off error with a large number of multiplies.

IsExactlyIdentity(self: pydrake.math.RotationMatrix_[AutoDiffXd]) → bool

Returns True if this is exactly equal to the identity matrix.

See also

IsNearlyIdentity().

IsNearlyIdentity(self: pydrake.math.RotationMatrix_[AutoDiffXd], tolerance: float = 2.842170943040401e-14) → bool

Returns true if this is within tolerance of the identity RigidTransform.

Parameter tolerance:

non-negative number that is generally the default value, namely RotationMatrix::get_internal_tolerance_for_orthonormality().

See also

IsExactlyIdentity().

IsValid(self: pydrake.math.RotationMatrix_[AutoDiffXd]) → bool

Tests if this rotation matrix R is a proper orthonormal rotation matrix to within the threshold of get_internal_tolerance_for_orthonormality().

Returns

True if this is a valid rotation matrix.

static MakeFromOneVector(b_A: numpy.ndarray[object[3, 1]], axis_index: int) → pydrake.math.RotationMatrix_[AutoDiffXd]

Creates a 3D right-handed orthonormal basis B from a given vector b_A, returned as a rotation matrix R_AB. It consists of orthogonal unit vectors u_A, v_A, w_A where u_A is the normalized b_A in the axis_index column of R_AB and v_A has one element which is zero. If an element of b_A is zero, then one element of w_A is 1 and the other two elements are 0.

Parameter b_A:

vector expressed in frame A that when normalized as u_A = b_A.normalized() represents Bx, By, or Bz (depending on axis_index).

Parameter axis_index:

The index ∈ {0, 1, 2} of the unit vector associated with u_A, 0 means u_A is Bx, 1 means u_A is By, and 2 means u_A is Bz.

Precondition:

axis_index is 0 or 1 or 2.

Raises

  • RuntimeError if b_A cannot be made into a unit vector because b_A –

  • contains a NaN or infinity or b_A < 1.0E-10. –

See also

MakeFromOneUnitVector() if b_A is known to already be unit length.

Returns R_AB:

the rotation matrix with properties as described above.

static MakeUnchecked(R: numpy.ndarray[object[3, 3]]) → pydrake.math.RotationMatrix_[AutoDiffXd]

(Advanced) Makes a RotationMatrix from a Matrix3. No check is performed to test whether or not the parameter R is a valid rotation matrix.

static MakeXRotation(theta: pydrake.autodiffutils.AutoDiffXd) → pydrake.math.RotationMatrix_[AutoDiffXd]

Makes the RotationMatrix R_AB associated with rotating a frame B relative to a frame A by an angle theta about unit vector Ax = Bx.

Parameter theta:

radian measure of rotation angle about Ax.

Note

Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitX().

Note

R_AB relates two frames A and B having unit vectors Ax, Ay, Az and Bx, By, Bz. Initially, Bx = Ax, By = Ay, Bz = Az, then B undergoes a right-handed rotation relative to A by an angle theta about Ax = Bx.

Click to expand C++ code...
⎡ 1       0                 0  ⎤
R_AB = ⎢ 0   cos(theta)   -sin(theta) ⎥
       ⎣ 0   sin(theta)    cos(theta) ⎦
static MakeYRotation(theta: pydrake.autodiffutils.AutoDiffXd) → pydrake.math.RotationMatrix_[AutoDiffXd]

Makes the RotationMatrix R_AB associated with rotating a frame B relative to a frame A by an angle theta about unit vector Ay = By.

Parameter theta:

radian measure of rotation angle about Ay.

Note

Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitY().

Note

R_AB relates two frames A and B having unit vectors Ax, Ay, Az and Bx, By, Bz. Initially, Bx = Ax, By = Ay, Bz = Az, then B undergoes a right-handed rotation relative to A by an angle theta about Ay = By.

Click to expand C++ code...
⎡  cos(theta)   0   sin(theta) ⎤
R_AB = ⎢          0    1           0  ⎥
       ⎣ -sin(theta)   0   cos(theta) ⎦
static MakeZRotation(theta: pydrake.autodiffutils.AutoDiffXd) → pydrake.math.RotationMatrix_[AutoDiffXd]

Makes the RotationMatrix R_AB associated with rotating a frame B relative to a frame A by an angle theta about unit vector Az = Bz.

Parameter theta:

radian measure of rotation angle about Az.

Note

Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitZ().

Note

R_AB relates two frames A and B having unit vectors Ax, Ay, Az and Bx, By, Bz. Initially, Bx = Ax, By = Ay, Bz = Az, then B undergoes a right-handed rotation relative to A by an angle theta about Az = Bz.

Click to expand C++ code...
⎡ cos(theta)  -sin(theta)   0 ⎤
R_AB = ⎢ sin(theta)   cos(theta)   0 ⎥
       ⎣         0            0    1 ⎦
matrix(self: pydrake.math.RotationMatrix_[AutoDiffXd]) → numpy.ndarray[object[3, 3]]

Returns the Matrix3 underlying a RotationMatrix.

See also

col(), row()

multiply(*args, **kwargs)

Overloaded function.

  1. multiply(self: pydrake.math.RotationMatrix_[AutoDiffXd], other: pydrake.math.RotationMatrix_[AutoDiffXd]) -> pydrake.math.RotationMatrix_[AutoDiffXd]

Calculates this rotation matrix R_AB multiplied by other rotation matrix R_BC, returning the composition R_AB * R_BC.

Parameter other:

RotationMatrix that post-multiplies this.

Returns

rotation matrix that results from this multiplied by other.

Note

It is possible (albeit improbable) to create an invalid rotation matrix by accumulating round-off error with a large number of multiplies.

  1. multiply(self: pydrake.math.RotationMatrix_[AutoDiffXd], v_B: numpy.ndarray[object[3, 1]]) -> numpy.ndarray[object[3, 1]]

Calculates this rotation matrix R_AB multiplied by an arbitrary Vector3 expressed in the B frame.

Parameter v_B:

3x1 vector that post-multiplies this.

Returns

3x1 vector v_A = R_AB * v_B.

  1. multiply(self: pydrake.math.RotationMatrix_[AutoDiffXd], v_B: numpy.ndarray[object[3, n]]) -> numpy.ndarray[object[3, n]]

Multiplies this RotationMatrix R_AB by the n vectors v1, … vn, where each vector has 3 elements and is expressed in frame B.

Parameter v_B:

3 x n matrix whose n columns are regarded as arbitrary vectors v1, … vn expressed in frame B.

Returns v_A:

3 x n matrix whose n columns are vectors v1, … vn expressed in frame A.

Click to expand C++ code...
const RollPitchYaw<double> rpy(0.1, 0.2, 0.3);
const RotationMatrix<double> R_AB(rpy);
Eigen::Matrix<double, 3, 2> v_B;
v_B.col(0) = Vector3d(4, 5, 6);
v_B.col(1) = Vector3d(9, 8, 7);
const Eigen::Matrix<double, 3, 2> v_A = R_AB * v_B;
static ProjectToRotationMatrix(M: numpy.ndarray[object[3, 3]]) → pydrake.math.RotationMatrix_[AutoDiffXd]

Given an approximate rotation matrix M, finds the RotationMatrix R closest to M. Closeness is measured with a matrix-2 norm (or equivalently with a Frobenius norm). Hence, this method creates a RotationMatrix R from a 3x3 matrix M by minimizing ‖R - M‖₂ (the matrix-2 norm of (R-M)) subject to R * Rᵀ = I, where I is the 3x3 identity matrix. For this problem, closeness can also be measured by forming the orthonormal matrix R whose elements minimize the double-summation ∑ᵢ ∑ⱼ (R(i,j) - M(i,j))² where i = 1:3, j = 1:3, subject to R * Rᵀ = I. The square-root of this double-summation is called the Frobenius norm.

Parameter M:

a 3x3 matrix.

Parameter quality_factor:

. The quality of M as a rotation matrix. quality_factor = 1 is perfect (M = R). quality_factor = 1.25 means that when M multiplies a unit vector (magnitude 1), a vector of magnitude as large as 1.25 may result. quality_factor = 0.8 means that when M multiplies a unit vector, a vector of magnitude as small as 0.8 may result. quality_factor = 0 means M is singular, so at least one of the bases related by matrix M does not span 3D space (when M multiples a unit vector, a vector of magnitude as small as 0 may result).

Returns

proper orthonormal matrix R that is closest to M.

Raises

RuntimeError if R fails IsValid(R) –

Note

William Kahan (UC Berkeley) and Hongkai Dai (Toyota Research Institute) proved that for this problem, the same R that minimizes the Frobenius norm also minimizes the matrix-2 norm (a.k.a an induced-2 norm), which is defined [Dahleh, Section 4.2] as the column matrix u which maximizes ‖(R - M) u‖ / ‖u‖, where u ≠ 0. Since the matrix-2 norm of any matrix A is equal to the maximum singular value of A, minimizing the matrix-2 norm of (R - M) is equivalent to minimizing the maximum singular value of (R - M).

  • [Dahleh] “Lectures on Dynamic Systems and Controls: Electrical

Engineering and Computer Science, Massachusetts Institute of Technology” https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-241j-dynamic-systems-and-control-spring-2011/readings/MIT6_241JS11_chap04.pdf

row(self: pydrake.math.RotationMatrix_[AutoDiffXd], index: int) → numpy.ndarray[object[1, 3]]

Returns this rotation matrix’s iᵗʰ row (i = 0, 1, 2). For this rotation matrix R_AB (which relates right-handed sets of orthogonal unit vectors Ax, Ay, Az to Bx, By, Bz), - row(0) returns Ax_B (Ax expressed in terms of Bx, By, Bz). - row(1) returns Ay_B (Ay expressed in terms of Bx, By, Bz). - row(2) returns Az_B (Az expressed in terms of Bx, By, Bz).

Parameter index:

requested row index (0 <= index <= 2).

See also

col(), matrix()

Raises

In Debug builds, asserts (0 <= index <= 2) –

Note

For efficiency and consistency with Eigen, this method returns the same quantity returned by Eigen’s row() operator. The returned quantity can be assigned in various ways, e.g., as const auto& Az_B = row(2); or RowVector3<T> Az_B = row(2);

set(self: pydrake.math.RotationMatrix_[AutoDiffXd], R: numpy.ndarray[object[3, 3]]) → None

Sets this RotationMatrix from a Matrix3.

Parameter R:

an allegedly valid rotation matrix.

Raises

RuntimeError in debug builds if R fails IsValid(R) –

ToAngleAxis(self: pydrake.math.RotationMatrix_[AutoDiffXd]) → pydrake.common.eigen_geometry.AngleAxis_[AutoDiffXd]

Returns an AngleAxis theta_lambda containing an angle theta and unit vector (axis direction) lambda that represents this RotationMatrix.

Note

The orientation and RotationMatrix associated with theta * lambda is identical to that of (-theta) * (-lambda). The AngleAxis returned by this method chooses to have 0 <= theta <= pi.

Returns

an AngleAxis with 0 <= theta <= pi and a unit vector lambda.

ToQuaternion(self: pydrake.math.RotationMatrix_[AutoDiffXd]) → pydrake.common.eigen_geometry.Quaternion_[AutoDiffXd]

Returns a quaternion q that represents this RotationMatrix. Since the quaternion q and -q represent the same RotationMatrix, this method chooses to return a canonical quaternion, i.e., with q(0) >= 0.

ToRollPitchYaw(self: pydrake.math.RotationMatrix_[AutoDiffXd]) → pydrake.math.RollPitchYaw_[AutoDiffXd]

Returns a RollPitchYaw that represents this RotationMatrix, with roll-pitch-yaw angles [r, p, y] in the range -π <= r <= π, -π/2 <= p <= π/2, -π <= y <= π.

Note

This new high-accuracy algorithm avoids numerical round-off issues encountered by some algorithms when pitch is within 1E-6 of π/2 or -π/2.

transpose(self: pydrake.math.RotationMatrix_[AutoDiffXd]) → pydrake.math.RotationMatrix_[AutoDiffXd]
class pydrake.math.RotationMatrix_[Expression]

This class represents a 3x3 rotation matrix between two arbitrary frames A and B and helps ensure users create valid rotation matrices. This class relates right-handed orthogonal unit vectors Ax, Ay, Az fixed in frame A to right-handed orthogonal unit vectors Bx, By, Bz fixed in frame B. The monogram notation for the rotation matrix relating A to B is R_AB. An example that gives context to this rotation matrix is v_A = R_AB * v_B, where v_B denotes an arbitrary vector v expressed in terms of Bx, By, Bz and v_A denotes vector v expressed in terms of Ax, Ay, Az. See multibody_quantities for monogram notation for dynamics. See orientation_discussion “a discussion on rotation matrices”.

Note

This class does not store the frames associated with a rotation matrix nor does it enforce strict proper usage of this class with vectors.

Note

When assertions are enabled, several methods in this class perform a validity check and throw RuntimeError if the rotation matrix is invalid. When assertions are disabled, many of these validity checks are skipped (which helps improve speed). These validity tests are only performed for scalar types for which drake::scalar_predicate<T>::is_bool is True. For instance, validity checks are not performed when T is symbolic::Expression.

Authors:

Paul Mitiguy (2018) Original author.

Authors:

Drake team (see https://drake.mit.edu/credits).

__init__(*args, **kwargs)

Overloaded function.

  1. __init__(self: pydrake.math.RotationMatrix_[Expression]) -> None

Constructs a 3x3 identity RotationMatrix – which corresponds to aligning two frames (so that unit vectors Ax = Bx, Ay = By, Az = Bz).

  1. __init__(self: pydrake.math.RotationMatrix_[Expression], other: pydrake.math.RotationMatrix_[Expression]) -> None

  2. __init__(self: pydrake.math.RotationMatrix_[Expression], R: numpy.ndarray[object[3, 3]]) -> None

Constructs a RotationMatrix from a Matrix3.

Parameter R:

an allegedly valid rotation matrix.

Raises

RuntimeError in debug builds if R fails IsValid(R) –

  1. __init__(self: pydrake.math.RotationMatrix_[Expression], quaternion: pydrake.common.eigen_geometry.Quaternion_[Expression]) -> None

Constructs a RotationMatrix from an Eigen::Quaternion.

Parameter quaternion:

a non-zero, finite quaternion which may or may not have unit length [i.e., quaternion.norm() does not have to be 1].

Raises

  • RuntimeError in debug builds if the rotation matrix R that is –

  • built from quaternion fails IsValid(R) For example, an –

  • exception is thrown if quaternion is zero or contains a NaN or –

  • infinity. –

Note

This method has the effect of normalizing its quaternion argument, without the inefficiency of the square-root associated with normalization.

  1. __init__(self: pydrake.math.RotationMatrix_[Expression], theta_lambda: pydrake.common.eigen_geometry.AngleAxis_[Expression]) -> None

Constructs a RotationMatrix from an Eigen::AngleAxis.

Parameter theta_lambda:

an Eigen::AngleAxis whose associated axis (vector direction herein called lambda) is non-zero and finite, but which may or may not have unit length [i.e., lambda.norm() does not have to be 1].

Raises

  • RuntimeError in debug builds if the rotation matrix R that is –

  • built from theta_lambda fails IsValid(R) For example, an –

  • exception is thrown if lambda is zero or contains a NaN or –

  • infinity. –

  1. __init__(self: pydrake.math.RotationMatrix_[Expression], rpy: pydrake.math.RollPitchYaw_[Expression]) -> None

Constructs a RotationMatrix from an RollPitchYaw. In other words, makes the RotationMatrix for a Space-fixed (extrinsic) X-Y-Z rotation by “roll-pitch-yaw” angles [r, p, y], which is equivalent to a Body-fixed (intrinsic) Z-Y-X rotation by “yaw-pitch-roll” angles [y, p, r].

Parameter rpy:

radian measures of three angles [roll, pitch, yaw].

Parameter rpy:

a RollPitchYaw which is a Space-fixed (extrinsic) X-Y-Z rotation with “roll-pitch-yaw” angles [r, p, y] or equivalently a Body- fixed (intrinsic) Z-Y-X rotation with “yaw-pitch-roll” angles [y, p, r].

Returns R_AD:

, rotation matrix relating frame A to frame D.

Note

Denoting roll r, pitch p, yaw y, this method returns a rotation matrix R_AD equal to the matrix multiplication shown below.

Click to expand C++ code...
⎡cos(y) -sin(y)  0⎤   ⎡ cos(p)  0  sin(p)⎤   ⎡1      0        0 ⎤
R_AD = ⎢sin(y)  cos(y)  0⎥ * ⎢     0   1      0 ⎥ * ⎢0  cos(r)  -sin(r)⎥
       ⎣    0       0   1⎦   ⎣-sin(p)  0  cos(p)⎦   ⎣0  sin(r)   cos(r)⎦
     =       R_AB          *        R_BC          *        R_CD

Note

In this discussion, A is the Space frame and D is the Body frame. One way to visualize this rotation sequence is by introducing intermediate frames B and C (useful constructs to understand this rotation sequence). Initially, the frames are aligned so Di = Ci = Bi = Ai (i = x, y, z). Then D is subjected to successive right-handed rotations relative to A.

  • 1st rotation R_CD: Frame D rotates relative to frames C, B, A by a

roll angle r about Dx = Cx. Note: D and C are no longer aligned.

  • 2nd rotation R_BC: Frames D, C (collectively – as if welded together)

rotate relative to frame B, A by a pitch angle p about Cy = By. Note: C and B are no longer aligned.

  • 3rd rotation R_AB: Frames D, C, B (collectively – as if welded)

rotate relative to frame A by a roll angle y about Bz = Az. Note: B and A are no longer aligned.

Note

This method constructs a RotationMatrix from a RollPitchYaw. Vice-versa, there are high-accuracy RollPitchYaw constructor/methods that form a RollPitchYaw from a rotation matrix.

template cast

Instantiations: cast[Expression]

cast[Expression](self: pydrake.math.RotationMatrix_[Expression]) → pydrake.math.RotationMatrix_[Expression]

Creates a RotationMatrix templatized on a scalar type U from a RotationMatrix templatized on scalar type T. For example,

Click to expand C++ code...
RotationMatrix<double> source = RotationMatrix<double>::Identity();
RotationMatrix<AutoDiffXd> foo = source.cast<AutoDiffXd>();
Template parameter U:

Scalar type on which the returned RotationMatrix is templated.

Note

RotationMatrix<From>::cast<To>() creates a new RotationMatrix<To> from a RotationMatrix<From> but only if type To is constructible from type From. This cast method works in accordance with Eigen’s cast method for Eigen’s Matrix3 that underlies this RotationMatrix. For example, Eigen currently allows cast from type double to AutoDiffXd, but not vice-versa.

col(self: pydrake.math.RotationMatrix_[Expression], index: int) → numpy.ndarray[object[3, 1]]

Returns this rotation matrix’s iᵗʰ column (i = 0, 1, 2). For this rotation matrix R_AB (which relates right-handed sets of orthogonal unit vectors Ax, Ay, Az to Bx, By, Bz), - col(0) returns Bx_A (Bx expressed in terms of Ax, Ay, Az). - col(1) returns By_A (By expressed in terms of Ax, Ay, Az). - col(2) returns Bz_A (Bz expressed in terms of Ax, Ay, Az).

Parameter index:

requested column index (0 <= index <= 2).

See also

row(), matrix()

Raises

In Debug builds, asserts (0 <= index <= 2) –

Note

For efficiency and consistency with Eigen, this method returns the same quantity returned by Eigen’s col() operator. The returned quantity can be assigned in various ways, e.g., as const auto& Bz_A = col(2); or Vector3<T> Bz_A = col(2);

static Identity() → pydrake.math.RotationMatrix_[Expression]
inverse(self: pydrake.math.RotationMatrix_[Expression]) → pydrake.math.RotationMatrix_[Expression]
InvertAndCompose(self: pydrake.math.RotationMatrix_[Expression], other: pydrake.math.RotationMatrix_[Expression]) → pydrake.math.RotationMatrix_[Expression]

Calculates the product of this inverted and another RotationMatrix. If you consider this to be the rotation matrix R_AB, and other to be R_AC, then this method returns R_BC = R_AB⁻¹ * R_AC. For T==double, this method can be much faster than inverting first and then performing the composition because it can take advantage of the orthogonality of rotation matrices. On some platforms it can use SIMD instructions for further speedups.

Parameter other:

RotationMatrix that post-multiplies this inverted.

Returns R_BC:

where R_BC = this⁻¹ * other.

Note

It is possible (albeit improbable) to create an invalid rotation matrix by accumulating round-off error with a large number of multiplies.

IsExactlyIdentity(self: pydrake.math.RotationMatrix_[Expression]) → pydrake.symbolic.Formula

Returns True if this is exactly equal to the identity matrix.

See also

IsNearlyIdentity().

IsNearlyIdentity(self: pydrake.math.RotationMatrix_[Expression], tolerance: float = 2.842170943040401e-14) → pydrake.symbolic.Formula

Returns true if this is within tolerance of the identity RigidTransform.

Parameter tolerance:

non-negative number that is generally the default value, namely RotationMatrix::get_internal_tolerance_for_orthonormality().

See also

IsExactlyIdentity().

IsValid(self: pydrake.math.RotationMatrix_[Expression]) → pydrake.symbolic.Formula

Tests if this rotation matrix R is a proper orthonormal rotation matrix to within the threshold of get_internal_tolerance_for_orthonormality().

Returns

True if this is a valid rotation matrix.

static MakeFromOneVector(b_A: numpy.ndarray[object[3, 1]], axis_index: int) → pydrake.math.RotationMatrix_[Expression]

Creates a 3D right-handed orthonormal basis B from a given vector b_A, returned as a rotation matrix R_AB. It consists of orthogonal unit vectors u_A, v_A, w_A where u_A is the normalized b_A in the axis_index column of R_AB and v_A has one element which is zero. If an element of b_A is zero, then one element of w_A is 1 and the other two elements are 0.

Parameter b_A:

vector expressed in frame A that when normalized as u_A = b_A.normalized() represents Bx, By, or Bz (depending on axis_index).

Parameter axis_index:

The index ∈ {0, 1, 2} of the unit vector associated with u_A, 0 means u_A is Bx, 1 means u_A is By, and 2 means u_A is Bz.

Precondition:

axis_index is 0 or 1 or 2.

Raises

  • RuntimeError if b_A cannot be made into a unit vector because b_A –

  • contains a NaN or infinity or b_A < 1.0E-10. –

See also

MakeFromOneUnitVector() if b_A is known to already be unit length.

Returns R_AB:

the rotation matrix with properties as described above.

static MakeUnchecked(R: numpy.ndarray[object[3, 3]]) → pydrake.math.RotationMatrix_[Expression]

(Advanced) Makes a RotationMatrix from a Matrix3. No check is performed to test whether or not the parameter R is a valid rotation matrix.

static MakeXRotation(theta: pydrake.symbolic.Expression) → pydrake.math.RotationMatrix_[Expression]

Makes the RotationMatrix R_AB associated with rotating a frame B relative to a frame A by an angle theta about unit vector Ax = Bx.

Parameter theta:

radian measure of rotation angle about Ax.

Note

Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitX().

Note

R_AB relates two frames A and B having unit vectors Ax, Ay, Az and Bx, By, Bz. Initially, Bx = Ax, By = Ay, Bz = Az, then B undergoes a right-handed rotation relative to A by an angle theta about Ax = Bx.

Click to expand C++ code...
⎡ 1       0                 0  ⎤
R_AB = ⎢ 0   cos(theta)   -sin(theta) ⎥
       ⎣ 0   sin(theta)    cos(theta) ⎦
static MakeYRotation(theta: pydrake.symbolic.Expression) → pydrake.math.RotationMatrix_[Expression]

Makes the RotationMatrix R_AB associated with rotating a frame B relative to a frame A by an angle theta about unit vector Ay = By.

Parameter theta:

radian measure of rotation angle about Ay.

Note

Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitY().

Note

R_AB relates two frames A and B having unit vectors Ax, Ay, Az and Bx, By, Bz. Initially, Bx = Ax, By = Ay, Bz = Az, then B undergoes a right-handed rotation relative to A by an angle theta about Ay = By.

Click to expand C++ code...
⎡  cos(theta)   0   sin(theta) ⎤
R_AB = ⎢          0    1           0  ⎥
       ⎣ -sin(theta)   0   cos(theta) ⎦
static MakeZRotation(theta: pydrake.symbolic.Expression) → pydrake.math.RotationMatrix_[Expression]

Makes the RotationMatrix R_AB associated with rotating a frame B relative to a frame A by an angle theta about unit vector Az = Bz.

Parameter theta:

radian measure of rotation angle about Az.

Note

Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitZ().

Note

R_AB relates two frames A and B having unit vectors Ax, Ay, Az and Bx, By, Bz. Initially, Bx = Ax, By = Ay, Bz = Az, then B undergoes a right-handed rotation relative to A by an angle theta about Az = Bz.

Click to expand C++ code...
⎡ cos(theta)  -sin(theta)   0 ⎤
R_AB = ⎢ sin(theta)   cos(theta)   0 ⎥
       ⎣         0            0    1 ⎦
matrix(self: pydrake.math.RotationMatrix_[Expression]) → numpy.ndarray[object[3, 3]]

Returns the Matrix3 underlying a RotationMatrix.

See also

col(), row()

multiply(*args, **kwargs)

Overloaded function.

  1. multiply(self: pydrake.math.RotationMatrix_[Expression], other: pydrake.math.RotationMatrix_[Expression]) -> pydrake.math.RotationMatrix_[Expression]

Calculates this rotation matrix R_AB multiplied by other rotation matrix R_BC, returning the composition R_AB * R_BC.

Parameter other:

RotationMatrix that post-multiplies this.

Returns

rotation matrix that results from this multiplied by other.

Note

It is possible (albeit improbable) to create an invalid rotation matrix by accumulating round-off error with a large number of multiplies.

  1. multiply(self: pydrake.math.RotationMatrix_[Expression], v_B: numpy.ndarray[object[3, 1]]) -> numpy.ndarray[object[3, 1]]

Calculates this rotation matrix R_AB multiplied by an arbitrary Vector3 expressed in the B frame.

Parameter v_B:

3x1 vector that post-multiplies this.

Returns

3x1 vector v_A = R_AB * v_B.

  1. multiply(self: pydrake.math.RotationMatrix_[Expression], v_B: numpy.ndarray[object[3, n]]) -> numpy.ndarray[object[3, n]]

Multiplies this RotationMatrix R_AB by the n vectors v1, … vn, where each vector has 3 elements and is expressed in frame B.

Parameter v_B:

3 x n matrix whose n columns are regarded as arbitrary vectors v1, … vn expressed in frame B.

Returns v_A:

3 x n matrix whose n columns are vectors v1, … vn expressed in frame A.

Click to expand C++ code...
const RollPitchYaw<double> rpy(0.1, 0.2, 0.3);
const RotationMatrix<double> R_AB(rpy);
Eigen::Matrix<double, 3, 2> v_B;
v_B.col(0) = Vector3d(4, 5, 6);
v_B.col(1) = Vector3d(9, 8, 7);
const Eigen::Matrix<double, 3, 2> v_A = R_AB * v_B;
static ProjectToRotationMatrix(M: numpy.ndarray[object[3, 3]]) → pydrake.math.RotationMatrix_[Expression]

Given an approximate rotation matrix M, finds the RotationMatrix R closest to M. Closeness is measured with a matrix-2 norm (or equivalently with a Frobenius norm). Hence, this method creates a RotationMatrix R from a 3x3 matrix M by minimizing ‖R - M‖₂ (the matrix-2 norm of (R-M)) subject to R * Rᵀ = I, where I is the 3x3 identity matrix. For this problem, closeness can also be measured by forming the orthonormal matrix R whose elements minimize the double-summation ∑ᵢ ∑ⱼ (R(i,j) - M(i,j))² where i = 1:3, j = 1:3, subject to R * Rᵀ = I. The square-root of this double-summation is called the Frobenius norm.

Parameter M:

a 3x3 matrix.

Parameter quality_factor:

. The quality of M as a rotation matrix. quality_factor = 1 is perfect (M = R). quality_factor = 1.25 means that when M multiplies a unit vector (magnitude 1), a vector of magnitude as large as 1.25 may result. quality_factor = 0.8 means that when M multiplies a unit vector, a vector of magnitude as small as 0.8 may result. quality_factor = 0 means M is singular, so at least one of the bases related by matrix M does not span 3D space (when M multiples a unit vector, a vector of magnitude as small as 0 may result).

Returns

proper orthonormal matrix R that is closest to M.

Raises

RuntimeError if R fails IsValid(R) –

Note

William Kahan (UC Berkeley) and Hongkai Dai (Toyota Research Institute) proved that for this problem, the same R that minimizes the Frobenius norm also minimizes the matrix-2 norm (a.k.a an induced-2 norm), which is defined [Dahleh, Section 4.2] as the column matrix u which maximizes ‖(R - M) u‖ / ‖u‖, where u ≠ 0. Since the matrix-2 norm of any matrix A is equal to the maximum singular value of A, minimizing the matrix-2 norm of (R - M) is equivalent to minimizing the maximum singular value of (R - M).

  • [Dahleh] “Lectures on Dynamic Systems and Controls: Electrical

Engineering and Computer Science, Massachusetts Institute of Technology” https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-241j-dynamic-systems-and-control-spring-2011/readings/MIT6_241JS11_chap04.pdf

row(self: pydrake.math.RotationMatrix_[Expression], index: int) → numpy.ndarray[object[1, 3]]

Returns this rotation matrix’s iᵗʰ row (i = 0, 1, 2). For this rotation matrix R_AB (which relates right-handed sets of orthogonal unit vectors Ax, Ay, Az to Bx, By, Bz), - row(0) returns Ax_B (Ax expressed in terms of Bx, By, Bz). - row(1) returns Ay_B (Ay expressed in terms of Bx, By, Bz). - row(2) returns Az_B (Az expressed in terms of Bx, By, Bz).

Parameter index:

requested row index (0 <= index <= 2).

See also

col(), matrix()

Raises

In Debug builds, asserts (0 <= index <= 2) –

Note

For efficiency and consistency with Eigen, this method returns the same quantity returned by Eigen’s row() operator. The returned quantity can be assigned in various ways, e.g., as const auto& Az_B = row(2); or RowVector3<T> Az_B = row(2);

set(self: pydrake.math.RotationMatrix_[Expression], R: numpy.ndarray[object[3, 3]]) → None

Sets this RotationMatrix from a Matrix3.

Parameter R:

an allegedly valid rotation matrix.

Raises

RuntimeError in debug builds if R fails IsValid(R) –

ToAngleAxis(self: pydrake.math.RotationMatrix_[Expression]) → pydrake.common.eigen_geometry.AngleAxis_[Expression]

Returns an AngleAxis theta_lambda containing an angle theta and unit vector (axis direction) lambda that represents this RotationMatrix.

Note

The orientation and RotationMatrix associated with theta * lambda is identical to that of (-theta) * (-lambda). The AngleAxis returned by this method chooses to have 0 <= theta <= pi.

Returns

an AngleAxis with 0 <= theta <= pi and a unit vector lambda.

ToQuaternion(self: pydrake.math.RotationMatrix_[Expression]) → pydrake.common.eigen_geometry.Quaternion_[Expression]

Returns a quaternion q that represents this RotationMatrix. Since the quaternion q and -q represent the same RotationMatrix, this method chooses to return a canonical quaternion, i.e., with q(0) >= 0.

ToRollPitchYaw(self: pydrake.math.RotationMatrix_[Expression]) → pydrake.math.RollPitchYaw_[Expression]

Returns a RollPitchYaw that represents this RotationMatrix, with roll-pitch-yaw angles [r, p, y] in the range -π <= r <= π, -π/2 <= p <= π/2, -π <= y <= π.

Note

This new high-accuracy algorithm avoids numerical round-off issues encountered by some algorithms when pitch is within 1E-6 of π/2 or -π/2.

transpose(self: pydrake.math.RotationMatrix_[Expression]) → pydrake.math.RotationMatrix_[Expression]
pydrake.math.sin(*args, **kwargs)

Overloaded function.

  1. sin(arg0: float) -> float

  2. sin(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. sin(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.sinh(*args, **kwargs)

Overloaded function.

  1. sinh(arg0: float) -> float

  2. sinh(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. sinh(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.SoftOverMax(*args, **kwargs)

Overloaded function.

  1. SoftOverMax(x: list[float], alpha: float = 1.0) -> float

Computes a smooth over approximation of max function, namely SoftOverMax(x) >= max(x). Mathematically we compute this as (log (∑ᵢ exp(αxᵢ))) / α.

Parameter x:

The vector for which we want to compute its soft max.

Parameter alpha:

α in the documentation above. Larger α makes the soft max more similar to max, with a sharper corner. Must be strictly positive and finite.

Default: is 1.

$Raises:

RuntimeError if α <= 0.

Raises

RuntimeError if α is non-finite. –

  1. SoftOverMax(x: list[pydrake.autodiffutils.AutoDiffXd], alpha: float = 1.0) -> pydrake.autodiffutils.AutoDiffXd

Computes a smooth over approximation of max function, namely SoftOverMax(x) >= max(x). Mathematically we compute this as (log (∑ᵢ exp(αxᵢ))) / α.

Parameter x:

The vector for which we want to compute its soft max.

Parameter alpha:

α in the documentation above. Larger α makes the soft max more similar to max, with a sharper corner. Must be strictly positive and finite.

Default: is 1.

$Raises:

RuntimeError if α <= 0.

Raises

RuntimeError if α is non-finite. –

pydrake.math.SoftOverMin(*args, **kwargs)

Overloaded function.

  1. SoftOverMin(x: list[float], alpha: float = 1.0) -> float

Computes a smooth over approximation of min function, namely SoftOverMin(x) >= min(x). Mathematically we compute this as ∑ᵢ exp(-αxᵢ)*xᵢ / d, where d = ∑ⱼ exp(-αxⱼ)

Parameter x:

The vector for which we want to compute its soft min.

Parameter alpha:

α in the documentation above. Larger α makes the soft min more similar to min, with a sharper corner. Must be strictly positive and finite.

Default: is 1.

$Raises:

RuntimeError if α <= 0.

Raises

RuntimeError if α is non-finite. –

  1. SoftOverMin(x: list[pydrake.autodiffutils.AutoDiffXd], alpha: float = 1.0) -> pydrake.autodiffutils.AutoDiffXd

Computes a smooth over approximation of min function, namely SoftOverMin(x) >= min(x). Mathematically we compute this as ∑ᵢ exp(-αxᵢ)*xᵢ / d, where d = ∑ⱼ exp(-αxⱼ)

Parameter x:

The vector for which we want to compute its soft min.

Parameter alpha:

α in the documentation above. Larger α makes the soft min more similar to min, with a sharper corner. Must be strictly positive and finite.

Default: is 1.

$Raises:

RuntimeError if α <= 0.

Raises

RuntimeError if α is non-finite. –

pydrake.math.SoftUnderMax(*args, **kwargs)

Overloaded function.

  1. SoftUnderMax(x: list[float], alpha: float = 1.0) -> float

Computes a smooth under approximation of max function, namely SoftUnderMax(x) <= max(x). Mathematically we compute this as ∑ᵢ exp(αxᵢ)*xᵢ / d, where d = ∑ⱼ exp(αxⱼ)

Parameter x:

The vector for which we want to compute its soft max.

Parameter alpha:

α in the documentation above. Larger α makes the soft max more similar to max, with a sharper corner. Must be strictly positive and finite.

Default: is 1.

$Raises:

RuntimeError if α <= 0.

Raises

RuntimeError if α is non-finite. –

  1. SoftUnderMax(x: list[pydrake.autodiffutils.AutoDiffXd], alpha: float = 1.0) -> pydrake.autodiffutils.AutoDiffXd

Computes a smooth under approximation of max function, namely SoftUnderMax(x) <= max(x). Mathematically we compute this as ∑ᵢ exp(αxᵢ)*xᵢ / d, where d = ∑ⱼ exp(αxⱼ)

Parameter x:

The vector for which we want to compute its soft max.

Parameter alpha:

α in the documentation above. Larger α makes the soft max more similar to max, with a sharper corner. Must be strictly positive and finite.

Default: is 1.

$Raises:

RuntimeError if α <= 0.

Raises

RuntimeError if α is non-finite. –

pydrake.math.SoftUnderMin(*args, **kwargs)

Overloaded function.

  1. SoftUnderMin(x: list[float], alpha: float = 1.0) -> float

Computes a smooth under approximation of min function, namely SoftUnderMin(x) <= min(x). Mathematically we compute this as -(log (∑ᵢ exp(-αxᵢ))) / α

Parameter x:

The vector for which we want to compute its soft min.

Parameter alpha:

α in the documentation above. Larger α makes the soft min more similar to min, with a sharper corner. Must be strictly positive and finite.

Default: is 1.

$Raises:

RuntimeError if α <= 0.

Raises

RuntimeError if α is non-finite. –

  1. SoftUnderMin(x: list[pydrake.autodiffutils.AutoDiffXd], alpha: float = 1.0) -> pydrake.autodiffutils.AutoDiffXd

Computes a smooth under approximation of min function, namely SoftUnderMin(x) <= min(x). Mathematically we compute this as -(log (∑ᵢ exp(-αxᵢ))) / α

Parameter x:

The vector for which we want to compute its soft min.

Parameter alpha:

α in the documentation above. Larger α makes the soft min more similar to min, with a sharper corner. Must be strictly positive and finite.

Default: is 1.

$Raises:

RuntimeError if α <= 0.

Raises

RuntimeError if α is non-finite. –

pydrake.math.sqrt(*args, **kwargs)

Overloaded function.

  1. sqrt(arg0: float) -> float

  2. sqrt(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. sqrt(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.tan(*args, **kwargs)

Overloaded function.

  1. tan(arg0: float) -> float

  2. tan(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. tan(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.tanh(*args, **kwargs)

Overloaded function.

  1. tanh(arg0: float) -> float

  2. tanh(arg0: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

  3. tanh(arg0: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

pydrake.math.ToLowerTriangularColumnsFromMatrix(matrix: numpy.ndarray[numpy.float64[m, n], flags.f_contiguous]) → numpy.ndarray[numpy.float64[m, 1]]

Given a square matrix, extract the lower triangular part as a stacked column vector. This is a particularly useful operation when vectorizing symmetric matrices.

pydrake.math.ToSymmetricMatrixFromLowerTriangularColumns(lower_triangular_columns: numpy.ndarray[numpy.float64[m, 1]]) → numpy.ndarray[numpy.float64[m, n]]

Given a column vector containing the stacked columns of the lower triangular part of a square matrix, returning a symmetric matrix whose lower triangular part is the same as the original matrix.

pydrake.math.UniformlyRandomAngleAxis(generator: pydrake.common.RandomGenerator) → pydrake.common.eigen_geometry.AngleAxis

Generates a rotation (in the axis-angle representation) that rotates a point on the unit sphere to another point on the unit sphere with a uniform distribution over the sphere.

pydrake.math.UniformlyRandomQuaternion(generator: pydrake.common.RandomGenerator) → pydrake.common.eigen_geometry.Quaternion

Generates a rotation (in the quaternion representation) that rotates a point on the unit sphere to another point on the unit sphere with a uniform distribution over the sphere. This method is briefly explained in http://planning.cs.uiuc.edu/node198.html, a full explanation can be found in K. Shoemake. Uniform Random Rotations in D. Kirk, editor, Graphics Gems III, pages 124-132. Academic, New York, 1992.

pydrake.math.UniformlyRandomRotationMatrix(generator: pydrake.common.RandomGenerator) → pydrake.math.RotationMatrix

Generates a rotation (in the rotation matrix representation) that rotates a point on the unit sphere to another point on the unit sphere with a uniform distribution over the sphere.

pydrake.math.UniformlyRandomRPY(generator: pydrake.common.RandomGenerator) → numpy.ndarray[numpy.float64[3, 1]]

Generates a rotation (in the roll-pitch-yaw representation) that rotates a point on the unit sphere to another point on the unit sphere with a uniform distribution over the sphere.

pydrake.math.VectorToSkewSymmetric(*args, **kwargs)

Overloaded function.

  1. VectorToSkewSymmetric(p: numpy.ndarray[numpy.float64[3, 1]]) -> numpy.ndarray[numpy.float64[3, 3]]

  2. VectorToSkewSymmetric(p: numpy.ndarray[object[3, 1]]) -> numpy.ndarray[object[3, 3]]

  3. VectorToSkewSymmetric(p: numpy.ndarray[object[3, 1]]) -> numpy.ndarray[object[3, 3]]

pydrake.math.wrap_to(*args, **kwargs)

Overloaded function.

  1. wrap_to(value: float, low: float, high: float) -> float

For variables that are meant to be periodic, (e.g. over a 2π interval), wraps value into the interval [low, high). Precisely, wrap_to returns: value + k*(high-low) for the unique integer value k that lands the output in the desired interval. low and high must be finite, and low < high.

  1. wrap_to(value: pydrake.autodiffutils.AutoDiffXd, low: pydrake.autodiffutils.AutoDiffXd, high: pydrake.autodiffutils.AutoDiffXd) -> pydrake.autodiffutils.AutoDiffXd

For variables that are meant to be periodic, (e.g. over a 2π interval), wraps value into the interval [low, high). Precisely, wrap_to returns: value + k*(high-low) for the unique integer value k that lands the output in the desired interval. low and high must be finite, and low < high.

  1. wrap_to(value: pydrake.symbolic.Expression, low: pydrake.symbolic.Expression, high: pydrake.symbolic.Expression) -> pydrake.symbolic.Expression

For variables that are meant to be periodic, (e.g. over a 2π interval), wraps value into the interval [low, high). Precisely, wrap_to returns: value + k*(high-low) for the unique integer value k that lands the output in the desired interval. low and high must be finite, and low < high.