Detailed Description

template<typename T>
class drake::math::RotationMatrix< T >

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 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.; When assertions are enabled, several methods in this class perform a validity check and throw std::exception 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.; Drake team (see https://drake.mit.edu/credits).

Template Parameters

T	The scalar type, which must be one of the default scalars.

#include <drake/math/rotation_matrix.h>

Public Member Functions
	RotationMatrix ()
	Constructs a 3x3 identity RotationMatrix – which corresponds to aligning two frames (so that unit vectors Ax = Bx, Ay = By, Az = Bz). More...

	RotationMatrix (const Matrix3< T > &R)
	Constructs a RotationMatrix from a Matrix3. More...

	RotationMatrix (const Eigen::Quaternion< T > &quaternion)
	Constructs a RotationMatrix from an Eigen::Quaternion. More...

	RotationMatrix (const Eigen::AngleAxis< T > &theta_lambda)
	Constructs a RotationMatrix from an Eigen::AngleAxis. More...

	RotationMatrix (const RollPitchYaw< T > &rpy)
	Constructs a RotationMatrix from an RollPitchYaw. More...

template<typename U >
RotationMatrix< U >	cast () const requires is_default_scalar< U >
	Creates a RotationMatrix templatized on a scalar type U from a RotationMatrix templatized on scalar type T. More...

void	set (const Matrix3< T > &R)
	Sets `this` RotationMatrix from a Matrix3. More...

RotationMatrix< T >	inverse () const
	Returns `R_BA = R_AB⁻¹`, the inverse (transpose) of this RotationMatrix. More...

RotationMatrix< T >	transpose () const
	Returns `R_BA = R_AB⁻¹`, the transpose of this RotationMatrix. More...

const Matrix3< T > &	matrix () const
	Returns the Matrix3 underlying a RotationMatrix. More...

const Eigen::Block< const Matrix3< T >, 1, 3, false >	row (int index) const
	Returns `this` rotation matrix's iᵗʰ row (i = 0, 1, 2). More...

const Eigen::Block< const Matrix3< T >, 3, 1, true >	col (int index) const
	Returns `this` rotation matrix's iᵗʰ column (i = 0, 1, 2). More...

RotationMatrix< T > &	operator *= (const RotationMatrix< T > &other)
	In-place multiply of `this` rotation matrix `R_AB` by `other` rotation matrix `R_BC`. More...

RotationMatrix< T >	operator * (const RotationMatrix< T > &other) const
	Calculates `this` rotation matrix `R_AB` multiplied by `other` rotation matrix `R_BC`, returning the composition `R_AB * R_BC`. More...

RotationMatrix< T >	InvertAndCompose (const RotationMatrix< T > &other) const
	Calculates the product of `this` inverted and another RotationMatrix. More...

Vector3< T >	operator * (const Vector3< T > &v_B) const
	Calculates `this` rotation matrix `R_AB` multiplied by an arbitrary Vector3 expressed in the B frame. More...

template<typename Derived >
Eigen::Matrix< typename Derived::Scalar, 3, Derived::ColsAtCompileTime >	operator * (const Eigen::MatrixBase< Derived > &v_B) const
	Multiplies `this` RotationMatrix `R_AB` by the n vectors `v1`, ... More...

boolean< T >	IsValid () const
	Tests if `this` rotation matrix R is a proper orthonormal rotation matrix to within the threshold of get_internal_tolerance_for_orthonormality(). More...

boolean< T >	IsExactlyIdentity () const
	Returns `true` if `this` is exactly equal to the identity matrix. More...

boolean< T >	IsNearlyIdentity (double tolerance=get_internal_tolerance_for_orthonormality()) const
	Returns true if `this` is within tolerance of the identity RigidTransform. More...

boolean< T >	IsNearlyEqualTo (const RotationMatrix< T > &other, double tolerance) const
	Compares each element of `this` to the corresponding element of `other` to check if they are the same to within a specified `tolerance`. More...

boolean< T >	IsExactlyEqualTo (const RotationMatrix< T > &other) const
	Compares each element of `this` to the corresponding element of `other` to check if they are exactly the same. More...

T	GetMaximumAbsoluteDifference (const RotationMatrix< T > &other) const
	Computes the infinity norm of `this` - `other` (i.e., the maximum absolute value of the difference between the elements of `this` and `other`). More...

RollPitchYaw< T >	ToRollPitchYaw () const
	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`. More...

Eigen::Quaternion< T >	ToQuaternion () const
	Returns a quaternion q that represents `this` RotationMatrix. More...

Vector4< T >	ToQuaternionAsVector4 () const
	Utility method to return the Vector4 associated with ToQuaternion(). More...

Eigen::AngleAxis< T >	ToAngleAxis () const
	Returns an AngleAxis `theta_lambda` containing an angle `theta` and unit vector (axis direction) `lambda` that represents `this` RotationMatrix. More...

	RotationMatrix (internal::DoNotInitializeMemberFields)
	(Internal use only) Constructs a RotationMatrix without initializing the underlying 3x3 matrix. More...

Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
	RotationMatrix (const RotationMatrix &)=default

RotationMatrix &	operator= (const RotationMatrix &)=default

	RotationMatrix (RotationMatrix &&)=default

RotationMatrix &	operator= (RotationMatrix &&)=default

Static Public Member Functions
static RotationMatrix< T >	MakeUnchecked (const Matrix3< T > &R)
	(Advanced) Makes a RotationMatrix from a Matrix3. More...

static RotationMatrix< T >	MakeFromOrthonormalColumns (const Vector3< T > &Bx, const Vector3< T > &By, const Vector3< T > &Bz)
	(Advanced) Makes the RotationMatrix `R_AB` from right-handed orthogonal unit vectors `Bx`, `By`, `Bz` so the columns of `R_AB` are `[Bx, By, Bz]`. More...

static RotationMatrix< T >	MakeFromOrthonormalRows (const Vector3< T > &Ax, const Vector3< T > &Ay, const Vector3< T > &Az)
	(Advanced) Makes the RotationMatrix `R_AB` from right-handed orthogonal unit vectors `Ax`, `Ay`, `Az` so the rows of `R_AB` are `[Ax, Ay, Az]`. More...

static RotationMatrix< T >	MakeXRotation (const T &theta)
	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`. More...

static RotationMatrix< T >	MakeYRotation (const T &theta)
	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`. More...

static RotationMatrix< T >	MakeZRotation (const T &theta)
	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`. More...

static RotationMatrix< T >	MakeFromOneVector (const Vector3< T > &b_A, int axis_index)
	Creates a 3D right-handed orthonormal basis B from a given vector b_A, returned as a rotation matrix R_AB. More...

static RotationMatrix< T >	MakeFromOneUnitVector (const Vector3< T > &u_A, int axis_index)
	(Advanced) Creates a right-handed orthonormal basis B from a given unit vector u_A, returned as a rotation matrix R_AB. More...

static RotationMatrix< T >	MakeClosestRotationToIdentityFromUnitZ (const Vector3< T > &u_A)
	Creates a 3D right-handed orthonormal basis B from a given unit vector u_A, returned as a rotation matrix R_AB. More...

static const RotationMatrix< T > &	Identity ()
	Returns the 3x3 identity RotationMatrix. More...

static T	GetMeasureOfOrthonormality (const Matrix3< T > &R)
	Returns how close the matrix R is to being a 3x3 orthonormal matrix by computing `‖R ⋅ Rᵀ - I‖∞` (i.e., the maximum absolute value of the difference between the elements of R ⋅ Rᵀ and the 3x3 identity matrix). More...

static boolean< T >	IsOrthonormal (const Matrix3< T > &R, double tolerance)
	Tests if a generic Matrix3 has orthonormal vectors to within the threshold specified by `tolerance`. More...

static boolean< T >	IsValid (const Matrix3< T > &R, double tolerance)
	Tests if a generic Matrix3 seems to be a proper orthonormal rotation matrix to within the threshold specified by `tolerance`. More...

static boolean< T >	IsValid (const Matrix3< T > &R)
	Tests if a generic Matrix3 is a proper orthonormal rotation matrix to within the threshold of get_internal_tolerance_for_orthonormality(). More...

static RotationMatrix< T >	ProjectToRotationMatrix (const Matrix3< T > &M, T *quality_factor=nullptr)
	Given an approximate rotation matrix M, finds the RotationMatrix R closest to M. More...

static constexpr double	get_internal_tolerance_for_orthonormality ()
	Returns an internal tolerance that checks rotation matrix orthonormality. More...

static Eigen::Quaternion< T >	ToQuaternion (const Eigen::Ref< const Matrix3< T >> &M)
	Returns a unit quaternion q associated with the 3x3 matrix M. More...

static Vector4< T >	ToQuaternionAsVector4 (const Matrix3< T > &M)
	Utility method to return the Vector4 associated with ToQuaternion(M). More...

Friends
template<typename U >
class	RotationMatrix

template<class HashAlgorithm >
void	hash_append (HashAlgorithm &hasher, const RotationMatrix &R) noexcept
	Implements the hash_append generic hashing concept. More...

Related Functions
(Note that these are not member functions.)
using	RotationMatrixd = RotationMatrix< double >
	Abbreviation (alias/typedef) for a RotationMatrix double scalar type. More...

Constructor & Destructor Documentation

◆ RotationMatrix() [1/8]

RotationMatrix ( const RotationMatrix< T > & )

default

◆ RotationMatrix() [2/8]

RotationMatrix ( RotationMatrix< T > && )

default

◆ RotationMatrix() [3/8]

RotationMatrix ( )

Constructs a 3x3 identity RotationMatrix – which corresponds to aligning two frames (so that unit vectors Ax = Bx, Ay = By, Az = Bz).

◆ RotationMatrix() [4/8]

RotationMatrix ( const Matrix3< T > & R )

explicit

Constructs a RotationMatrix from a Matrix3.

Parameters

[in] R an allegedly valid rotation matrix.

Exceptions

std::exception in debug builds if R fails IsValid(R).

◆ RotationMatrix() [5/8]

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

explicit

Constructs a RotationMatrix from an Eigen::Quaternion.

Parameters

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

Exceptions

std::exception 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.

◆ RotationMatrix() [6/8]

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

explicit

Constructs a RotationMatrix from an Eigen::AngleAxis.

Parameters

[in] 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].

Exceptions

std::exception 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.

◆ RotationMatrix() [7/8]

RotationMatrix ( const RollPitchYaw< T > & rpy )

explicit

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].

Parameters

[in]	rpy	radian measures of three angles [roll, pitch, yaw].
[in]	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]`.

Return values

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.

       ⎡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

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.

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.

◆ RotationMatrix() [8/8]

RotationMatrix ( internal::DoNotInitializeMemberFields )

explicit

(Internal use only) Constructs a RotationMatrix without initializing the underlying 3x3 matrix.

For use by RigidTransform and RotationMatrix only.

Member Function Documentation

◆ cast()

RotationMatrix<U> cast ( ) const

Creates a RotationMatrix templatized on a scalar type U from a RotationMatrix templatized on scalar type T.

For example,

RotationMatrix<double> source = RotationMatrix<double>::Identity();

RotationMatrix<AutoDiffXd> foo = source.cast<AutoDiffXd>();

Template Parameters

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()

const Eigen::Block<const Matrix3<T>, 3, 1, true> col ( int index ) const

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).
Parameters

[in] index requested column index (0 <= index <= 2).

See also
row(), matrix()

Exceptions

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);

◆ get_internal_tolerance_for_orthonormality()

static constexpr double get_internal_tolerance_for_orthonormality ( )

static

Returns an internal tolerance that checks rotation matrix orthonormality.

Returns: internal tolerance (small multiplier of double-precision epsilon) used to check whether or not a rotation matrix is orthonormal.

Note: The tolerance is chosen by developers to ensure a reasonably valid (orthonormal) rotation matrix.; To orthonormalize a 3x3 matrix, use ProjectToRotationMatrix().

◆ GetMaximumAbsoluteDifference()

T GetMaximumAbsoluteDifference ( const RotationMatrix< T > & other ) const

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

Parameters

[in] other RotationMatrix to subtract from this.

Returns: ‖this - other‖∞

◆ GetMeasureOfOrthonormality()

static T GetMeasureOfOrthonormality ( const Matrix3< T > & R )

static

Returns how close the matrix R is to being a 3x3 orthonormal matrix by computing ‖R ⋅ Rᵀ - I‖∞ (i.e., the maximum absolute value of the difference between the elements of R ⋅ Rᵀ and the 3x3 identity matrix).

Parameters

[in] R matrix being checked for orthonormality.

Returns: ‖R ⋅ Rᵀ - I‖∞

◆ Identity()

static const RotationMatrix<T>& Identity ( )

static

Returns the 3x3 identity RotationMatrix.

◆ inverse()

RotationMatrix<T> inverse ( ) const

Returns R_BA = R_AB⁻¹, the inverse (transpose) of this RotationMatrix.

Note: For a valid rotation matrix R_BA = R_AB⁻¹ = R_ABᵀ.

◆ InvertAndCompose()

RotationMatrix<T> InvertAndCompose ( const RotationMatrix< T > & other ) const

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.

Parameters

[in] other RotationMatrix that post-multiplies this inverted.

Return values

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.

◆ IsExactlyEqualTo()

boolean<T> IsExactlyEqualTo ( const RotationMatrix< T > & other ) const

Compares each element of this to the corresponding element of other to check if they are exactly the same.

Parameters

[in] other RotationMatrix to compare to this.

Returns: true if each element of this is exactly equal to the corresponding element in other.

See also: IsNearlyEqualTo().

◆ IsExactlyIdentity()

boolean<T> IsExactlyIdentity ( ) const

Returns true if this is exactly equal to the identity matrix.

See also: IsNearlyIdentity().

◆ IsNearlyEqualTo()

boolean<T> IsNearlyEqualTo	(	const RotationMatrix< T > &	other,
		double	tolerance
	)		const

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

Parameters

[in]	other	RotationMatrix to compare to `this`.
[in]	tolerance	maximum allowable absolute difference between the matrix elements in `this` and `other`.

Returns: true if ‖this - other‖∞ <= tolerance.

See also: IsExactlyEqualTo().

◆ IsNearlyIdentity()

boolean<T> IsNearlyIdentity ( double tolerance = get_internal_tolerance_for_orthonormality() ) const

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

Parameters

[in] tolerance non-negative number that is generally the default value, namely RotationMatrix::get_internal_tolerance_for_orthonormality().

See also: IsExactlyIdentity().

◆ IsOrthonormal()

static boolean<T> IsOrthonormal	(	const Matrix3< T > &	R,
		double	tolerance
	)

static

Tests if a generic Matrix3 has orthonormal vectors to within the threshold specified by tolerance.

Parameters

[in]	R	an allegedly orthonormal rotation matrix.
[in]	tolerance	maximum allowable absolute difference between R * Rᵀ and the identity matrix I, i.e., checks if `‖R ⋅ Rᵀ - I‖∞ <= tolerance`.

Returns: true if R is an orthonormal matrix.

◆ IsValid() [1/3]

static boolean<T> IsValid	(	const Matrix3< T > &	R,
		double	tolerance
	)

static

Tests if a generic Matrix3 seems to be a proper orthonormal rotation matrix to within the threshold specified by tolerance.

Parameters

[in]	R	an allegedly valid rotation matrix.
[in]	tolerance	maximum allowable absolute difference of `R * Rᵀ` and the identity matrix I (i.e., checks if `‖R ⋅ Rᵀ - I‖∞ <= tolerance`).

Returns: true if R is a valid rotation matrix.

◆ IsValid() [2/3]

static boolean<T> IsValid ( const Matrix3< T > & R )

static

Tests if a generic Matrix3 is a proper orthonormal rotation matrix to within the threshold of get_internal_tolerance_for_orthonormality().

Parameters

[in] R an allegedly valid rotation matrix.

Returns: true if R is a valid rotation matrix.

◆ IsValid() [3/3]

boolean<T> IsValid ( ) const

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.

◆ MakeClosestRotationToIdentityFromUnitZ()

static RotationMatrix<T> MakeClosestRotationToIdentityFromUnitZ ( const Vector3< T > & u_A )

static

Creates a 3D right-handed orthonormal basis B from a given unit vector u_A, returned as a rotation matrix R_AB.

It consists of orthogonal unit vectors [Bx, By, Bz] where Bz is u_A. The angle-axis representation of the resulting rotation is the one with the minimum rotation angle that rotates A to B. When u_A is not parallel or antiparallel to [0, 0, 1], such rotation is unique.

Parameters

[in] u_A unit vector expressed in frame A that represents Bz.

Exceptions

std::exception if u_A is not a unit vector.

Return values

R_AB	the rotation matrix with properties as described above.

◆ MakeFromOneUnitVector()

static RotationMatrix<T> MakeFromOneUnitVector	(	const Vector3< T > &	u_A,
		int	axis_index
	)

static

(Advanced) Creates a right-handed orthonormal basis B from a given unit vector u_A, returned as a rotation matrix R_AB.

Parameters

[in]	u_A	unit vector which is expressed in frame A and is either Bx or By or Bz (depending on the value of axis_index).
[in]	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.

Exceptions

std::exception if u_A is not a unit vector, i.e., |u_A| is not within a tolerance of 4ε ≈ 8.88E-16 to 1.0.

Note: This method is designed for speed and does not normalize u_A to ensure it is a unit vector. Alternatively, consider MakeFromOneVector().

Return values

R_AB	the rotation matrix whose properties are described in MakeFromOneVector().

◆ MakeFromOneVector()

static RotationMatrix<T> MakeFromOneVector	(	const Vector3< T > &	b_A,
		int	axis_index
	)

static

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.

Parameters

[in]	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).
[in]	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.

Exceptions

std::exception 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.

Return values

R_AB	the rotation matrix with properties as described above.

◆ MakeFromOrthonormalColumns()

static RotationMatrix<T> MakeFromOrthonormalColumns	(	const Vector3< T > &	Bx,
		const Vector3< T > &	By,
		const Vector3< T > &	Bz
	)

static

(Advanced) Makes the RotationMatrix R_AB from right-handed orthogonal unit vectors Bx, By, Bz so the columns of R_AB are [Bx, By, Bz].

Parameters

[in]	Bx	first unit vector in right-handed orthogonal set.
[in]	By	second unit vector in right-handed orthogonal set.
[in]	Bz	third unit vector in right-handed orthogonal set.

Exceptions

std::exception in debug builds if R_AB fails IsValid(R_AB).

Note: In release builds, the caller can subsequently test if R_AB is, in fact, a valid RotationMatrix by calling R_AB.IsValid().; The rotation matrix R_AB relates two sets of right-handed orthogonal unit vectors, namely Ax, Ay, Az and Bx, By, Bz. The rows of R_AB are Ax, Ay, Az expressed in frame B (i.e.,Ax_B, Ay_B, Az_B). The columns of R_AB are Bx, By, Bz expressed in frame A (i.e., Bx_A, By_A, Bz_A).

◆ MakeFromOrthonormalRows()

static RotationMatrix<T> MakeFromOrthonormalRows	(	const Vector3< T > &	Ax,
		const Vector3< T > &	Ay,
		const Vector3< T > &	Az
	)

static

(Advanced) Makes the RotationMatrix R_AB from right-handed orthogonal unit vectors Ax, Ay, Az so the rows of R_AB are [Ax, Ay, Az].

Parameters

[in]	Ax	first unit vector in right-handed orthogonal set.
[in]	Ay	second unit vector in right-handed orthogonal set.
[in]	Az	third unit vector in right-handed orthogonal set.

Exceptions

std::exception in debug builds if R_AB fails IsValid(R_AB).

Note: In release builds, the caller can subsequently test if R_AB is, in fact, a valid RotationMatrix by calling R_AB.IsValid().; The rotation matrix R_AB relates two sets of right-handed orthogonal unit vectors, namely Ax, Ay, Az and Bx, By, Bz. The rows of R_AB are Ax, Ay, Az expressed in frame B (i.e.,Ax_B, Ay_B, Az_B). The columns of R_AB are Bx, By, Bz expressed in frame A (i.e., Bx_A, By_A, Bz_A).

◆ MakeUnchecked()

static RotationMatrix<T> MakeUnchecked ( const Matrix3< T > & R )

static

(Advanced) Makes a RotationMatrix from a Matrix3.

No check is performed to test whether or not the parameter R is a valid rotation matrix.

◆ MakeXRotation()

static RotationMatrix<T> MakeXRotation ( const T & theta )

static

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.

Parameters

[in] theta radian measure of rotation angle about Ax.

Note: Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitX().; 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.
⎡ 1 0 0 ⎤
R_AB = ⎢ 0 cos(theta) -sin(theta) ⎥
⎣ 0 sin(theta) cos(theta) ⎦

◆ MakeYRotation()

static RotationMatrix<T> MakeYRotation ( const T & theta )

static

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.

Parameters

[in] theta radian measure of rotation angle about Ay.

Note: Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitY().; 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.
⎡ cos(theta) 0 sin(theta) ⎤
R_AB = ⎢ 0 1 0 ⎥
⎣ -sin(theta) 0 cos(theta) ⎦

◆ MakeZRotation()

static RotationMatrix<T> MakeZRotation ( const T & theta )

static

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.

Parameters

[in] theta radian measure of rotation angle about Az.

Note: Orientation is same as Eigen::AngleAxis<T>(theta, Vector3d::UnitZ().; 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.
⎡ cos(theta) -sin(theta) 0 ⎤
R_AB = ⎢ sin(theta) cos(theta) 0 ⎥
⎣ 0 0 1 ⎦

◆ matrix()

const Matrix3<T>& matrix ( ) const

Returns the Matrix3 underlying a RotationMatrix.

See also: col(), row()

◆ operator *() [1/3]

RotationMatrix<T> operator * ( const RotationMatrix< T > & other ) const

Calculates this rotation matrix R_AB multiplied by other rotation matrix R_BC, returning the composition R_AB * R_BC.

Parameters

[in] 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.

◆ operator *() [2/3]

Vector3<T> operator * ( const Vector3< T > & v_B ) const

Calculates this rotation matrix R_AB multiplied by an arbitrary Vector3 expressed in the B frame.

Parameters

[in] v_B 3x1 vector that post-multiplies this.

Returns: 3x1 vector v_A = R_AB * v_B.

◆ operator *() [3/3]

Eigen::Matrix<typename Derived::Scalar, 3, Derived::ColsAtCompileTime> operator * ( const Eigen::MatrixBase< Derived > & v_B ) const

Multiplies this RotationMatrix R_AB by the n vectors v1, ...

vn, where each vector has 3 elements and is expressed in frame B.

Parameters

[in] v_B 3 x n matrix whose n columns are regarded as arbitrary vectors v1, ... vn expressed in frame B.

Return values

v_A

3 x n matrix whose n columns are vectors v1, ... vn expressed in frame A.

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;

◆ operator *=()

RotationMatrix<T>& operator *= ( const RotationMatrix< T > & other )

In-place multiply of this rotation matrix R_AB by other rotation matrix R_BC.

On return, this is set to equal R_AB * R_BC.

Parameters

[in] other RotationMatrix that post-multiplies this.

Returns: this rotation matrix which has been 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.

◆ operator=() [1/2]

RotationMatrix& operator= ( RotationMatrix< T > && )

default

◆ operator=() [2/2]

RotationMatrix& operator= ( const RotationMatrix< T > & )

default

◆ ProjectToRotationMatrix()

static RotationMatrix<T> ProjectToRotationMatrix	(	const Matrix3< T > &	M,
		T *	quality_factor = `nullptr`
	)

static

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.

Parameters

[in]	M	a 3x3 matrix.
[out]	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.

Exceptions

std::exception 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()

const Eigen::Block<const Matrix3<T>, 1, 3, false> row ( int index ) const

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).
Parameters

[in] index requested row index (0 <= index <= 2).

See also
col(), matrix()

Exceptions

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()

void set ( const Matrix3< T > & R )

Sets this RotationMatrix from a Matrix3.

Parameters

[in] R an allegedly valid rotation matrix.

Exceptions

std::exception in debug builds if R fails IsValid(R).

◆ ToAngleAxis()

Eigen::AngleAxis<T> ToAngleAxis ( ) const

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() [1/2]

Eigen::Quaternion<T> ToQuaternion ( ) const

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.

◆ ToQuaternion() [2/2]

static Eigen::Quaternion<T> ToQuaternion ( const Eigen::Ref< const Matrix3< T >> & M )

static

Returns a unit quaternion q associated with the 3x3 matrix M.

Since the quaternion q and -q represent the same RotationMatrix, this method chooses to return a canonical quaternion, i.e., with q(0) >= 0.

Parameters

[in] M 3x3 matrix to be made into a quaternion.

Returns: a unit quaternion q in canonical form, i.e., with q(0) >= 0.

Exceptions

std::exception in debug builds if the quaternion q returned by this method cannot construct a valid RotationMatrix. For example, if M contains NaNs, q will not be a valid quaternion.

◆ ToQuaternionAsVector4() [1/2]

Vector4<T> ToQuaternionAsVector4 ( ) const

Utility method to return the Vector4 associated with ToQuaternion().

See also: ToQuaternion().

◆ ToQuaternionAsVector4() [2/2]

static Vector4<T> ToQuaternionAsVector4 ( const Matrix3< T > & M )

static

Utility method to return the Vector4 associated with ToQuaternion(M).

Parameters

[in] M 3x3 matrix to be made into a quaternion.

See also: ToQuaternion().

◆ ToRollPitchYaw()

RollPitchYaw<T> ToRollPitchYaw ( ) const

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()

RotationMatrix<T> transpose ( ) const

Returns R_BA = R_AB⁻¹, the transpose of this RotationMatrix.

Note: For a valid rotation matrix R_BA = R_AB⁻¹ = R_ABᵀ.

Friends And Related Function Documentation

◆ hash_append

void hash_append	(	HashAlgorithm &	hasher,
		const RotationMatrix< T > &	R
	)

friend

Implements the hash_append generic hashing concept.

Precondition: T implements the hash_append concept.

◆ RotationMatrix

friend class RotationMatrix

friend

◆ RotationMatrixd

using RotationMatrixd = RotationMatrix<double>

The documentation for this class was generated from the following files:

drake/math/roll_pitch_yaw.h
drake/math/rotation_matrix.h

Detailed Description

template<typename T> class drake::math::RotationMatrix< T >

Public Member Functions

Static Public Member Functions

Friends

Related Functions

Constructor & Destructor Documentation

◆ RotationMatrix() [1/8]

◆ RotationMatrix() [2/8]

◆ RotationMatrix() [3/8]

◆ RotationMatrix() [4/8]

◆ RotationMatrix() [5/8]

◆ RotationMatrix() [6/8]

◆ RotationMatrix() [7/8]

◆ RotationMatrix() [8/8]

Member Function Documentation

◆ cast()

◆ col()

◆ get_internal_tolerance_for_orthonormality()

◆ GetMaximumAbsoluteDifference()

◆ GetMeasureOfOrthonormality()

◆ Identity()

◆ inverse()

◆ InvertAndCompose()

◆ IsExactlyEqualTo()

◆ IsExactlyIdentity()

◆ IsNearlyEqualTo()

◆ IsNearlyIdentity()

◆ IsOrthonormal()

◆ IsValid() [1/3]

◆ IsValid() [2/3]

◆ IsValid() [3/3]

◆ MakeClosestRotationToIdentityFromUnitZ()

◆ MakeFromOneUnitVector()

◆ MakeFromOneVector()

◆ MakeFromOrthonormalColumns()

◆ MakeFromOrthonormalRows()

◆ MakeUnchecked()

◆ MakeXRotation()

◆ MakeYRotation()

◆ MakeZRotation()

◆ matrix()

◆ operator *() [1/3]

◆ operator *() [2/3]

◆ operator *() [3/3]

◆ operator *=()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ ProjectToRotationMatrix()

◆ row()

◆ set()

◆ ToAngleAxis()

◆ ToQuaternion() [1/2]

◆ ToQuaternion() [2/2]

◆ ToQuaternionAsVector4() [1/2]

◆ ToQuaternionAsVector4() [2/2]

◆ ToRollPitchYaw()

◆ transpose()

Friends And Related Function Documentation

◆ hash_append

◆ RotationMatrix

◆ RotationMatrixd

template<typename T>
class drake::math::RotationMatrix< T >