Drake
PiecewisePolynomial< T > Class Template Referencefinal

## Detailed Description

### template<typename T> class drake::trajectories::PiecewisePolynomial< T >

A scalar multi-variate piecewise polynomial.

PiecewisePolynomial represents a list of contiguous segments in a scalar independent variable (typically corresponding to time) with Polynomials defined at each segment. We call the output from evaluating the PiecewisePolynomial at the scalar independent variable "the output", and that output can be either a Eigen MatrixX<T> (if evaluated using value()) or a scalar (if evaluated using scalar_value()).

An example of a piecewise polynomial is a function of m segments in time, where a different polynomial is defined for each segment. For a specific example, consider the absolute value function over the interval [-1, 1]. We can define a PiecewisePolynomial over this interval using breaks at t = { -1.0, 0.0, 1.0 }, and "samples" of abs(t).

// Construct the PiecewisePolynomial.
const std::vector<double> breaks = { -1.0, 0.0, 1.0 };
std::vector<Eigen::MatrixXd> samples(3);
for (int i = 0; i < static_cast<int>(breaks.size()); ++i) {
samples[i].resize(1, 1);
samples[i](0, 0) = std::abs(breaks[i]);
}
const auto pp =
const int row = 0, col = 0;
// Evaluate the PiecewisePolynomial at some values.
std::cout << pp.value(-.5)(row, col) << std::endl; // Outputs 0.5.
std::cout << pp.value(0.0)(row, col) << std::endl; // Outputs 0.0;
// Show how we can evaluate the first derivative (outputs -1.0).
std::cout << pp.derivative(1).value(-.5)(row, col) << std::endl;

A note on terminology. For piecewise-polynomial interpolation, we use breaks to indicate the scalar (e.g. times) which form the boundary of each segment. We use samples to indicate the function value at the breaks, e.g. p(breaks[i]) = samples[i]. The term knot should be reserved for the "(x,y)" coordinate, here knot[i] = (breaks[i], samples[i]), though it is used inconsistently in the interpolation literature (sometimes for breaks, sometimes for samples), so we try to mostly avoid it here.

PiecewisePolynomial objects can be added, subtracted, and multiplied. They cannot be divided because Polynomials are not closed under division.

Warning
PiecewisePolynomial silently clips input evaluations outside of the range defined by the breaks. So pp.value(-2.0, row, col) in the example above would evaluate to -1.0. See value().

@tparam_default_scalars

#include <drake/common/trajectories/piecewise_polynomial.h>

## Public Types

typedef MatrixX< Polynomial< T > > PolynomialMatrix

## Public Member Functions

PiecewisePolynomial ()=default
Constructs an empty piecewise polynomial. More...

template<typename Derived >
PiecewisePolynomial (const Eigen::MatrixBase< Derived > &constant_value)
Single segment, constant value constructor over the interval [-∞, ∞]. More...

~PiecewisePolynomial () override=default

std::unique_ptr< Trajectory< T > > Clone () const override

PiecewisePolynomial< T > derivative (int derivative_order=1) const
Returns a PiecewisePolynomial where each segment is the specified derivative of the corresponding segment in this. More...

PiecewisePolynomial< T > integral (const T &value_at_start_time=0.0) const
Returns a PiecewisePolynomial that is the indefinite integral of this one. More...

PiecewisePolynomial< T > integral (const Eigen::Ref< MatrixX< T >> &value_at_start_time) const
Returns a PiecewisePolynomial that is the indefinite integral of this one. More...

bool empty () const
Returns true if this trajectory has no breaks/samples/polynomials. More...

scalarValue (const T &t, Eigen::Index row=0, Eigen::Index col=0) const
Evaluates the trajectory at the given time without returning the entire matrix. More...

MatrixX< T > value (const T &t) const override
Evaluates the PiecewisePolynomial at the given time t. More...

const PolynomialMatrixgetPolynomialMatrix (int segment_index) const
Gets the matrix of Polynomials corresponding to the given segment index. More...

const Polynomial< T > & getPolynomial (int segment_index, Eigen::Index row=0, Eigen::Index col=0) const
Gets the Polynomial with the given matrix row and column index that corresponds to the given segment index. More...

int getSegmentPolynomialDegree (int segment_index, Eigen::Index row=0, Eigen::Index col=0) const
Gets the degree of the Polynomial with the given matrix row and column index that corresponds to the given segment index. More...

Eigen::Index rows () const override
Returns the row count of the output matrices. More...

Eigen::Index cols () const override
Returns the column count of the output matrices. More...

void Reshape (int rows, int cols)
Reshapes the dimensions of the Eigen::MatrixX<T> returned by value(), EvalDerivative(), etc. More...

PiecewisePolynomial Block (int start_row, int start_col, int block_rows, int block_cols) const
Extracts a trajectory representing a block of size (block_rows, block_cols) starting at (start_row, start_col) from the PiecewisePolynomial. More...

PiecewisePolynomialoperator+= (const PiecewisePolynomial &other)
Adds each Polynomial in the PolynomialMatrix of other to the corresponding Polynomial in the PolynomialMatrix of this, storing the result in this. More...

PiecewisePolynomialoperator-= (const PiecewisePolynomial &other)
Subtracts each Polynomial in the PolynomialMatrix of other from the corresponding Polynomial in the PolynomialMatrix of this, storing the result in this. More...

PiecewisePolynomialoperator *= (const PiecewisePolynomial &other)
Multiplies each Polynomial in the PolynomialMatrix of other by the corresponding Polynomial in the PolynomialMatrix of this (i.e., a coefficient-wise multiplication), storing the result in this. More...

PiecewisePolynomialoperator+= (const MatrixX< T > &coeff)

PiecewisePolynomialoperator-= (const MatrixX< T > &coeff)

const PiecewisePolynomial operator+ (const PiecewisePolynomial &other) const
Adds each Polynomial in the PolynomialMatrix of other to the corresponding Polynomial in the PolynomialMatrix of this. More...

const PiecewisePolynomial operator- (const PiecewisePolynomial &other) const
Subtracts each Polynomial in the PolynomialMatrix of other from the corresponding Polynomial in the PolynomialMatrix of this. More...

const PiecewisePolynomial operator- () const
Implements unary minus operator. More...

const PiecewisePolynomial operator * (const PiecewisePolynomial &other) const
Multiplies each Polynomial in the PolynomialMatrix of other by the corresponding Polynomial in the PolynomialMatrix of this (i.e., a coefficient-wise multiplication). More...

const PiecewisePolynomial operator+ (const MatrixX< T > &coeff) const

const PiecewisePolynomial operator- (const MatrixX< T > &coeff) const

bool isApprox (const PiecewisePolynomial &other, double tol, const ToleranceType &tol_type=ToleranceType::kRelative) const
Checks whether a PiecewisePolynomial is approximately equal to this one by calling Polynomial<T>::CoefficientsAlmostEqual() on every element of every segment. More...

void ConcatenateInTime (const PiecewisePolynomial &other)
Concatenates other to the end of this. More...

void AppendCubicHermiteSegment (const T &time, const Eigen::Ref< const MatrixX< T >> &sample, const Eigen::Ref< const MatrixX< T >> &sample_dot)
The CubicHermite spline construction has a nice property of being incremental (each segment can be solved independently). More...

void AppendFirstOrderSegment (const T &time, const Eigen::Ref< const MatrixX< T >> &sample)
Given a new sample, this method adds one segment to the end of this using a first-order hold, where the start sample is taken as the value at the final break of this. More...

void RemoveFinalSegment ()
Removes the final segment from the trajectory, reducing the number of segments by 1. More...

void ReverseTime ()
Modifies the trajectory so that pp_after(t) = pp_before(-t). More...

void ScaleTime (const T &scale)
Scales the time of the trajectory by non-negative scale (use ReverseTime() if you want to also negate time). More...

void shiftRight (const T &offset)
Adds offset to all of the breaks. More...

void setPolynomialMatrixBlock (const PolynomialMatrix &replacement, int segment_index, Eigen::Index row_start=0, Eigen::Index col_start=0)
Replaces the specified block of the PolynomialMatrix at the given segment index. More...

PiecewisePolynomial slice (int start_segment_index, int num_segments) const
Returns the PiecewisePolynomial comprising the num_segments segments starting at the specified start_segment_index. More...

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

PiecewisePolynomialoperator= (const PiecewisePolynomial &)=default

PiecewisePolynomial (PiecewisePolynomial &&)=default

PiecewisePolynomialoperator= (PiecewisePolynomial &&)=default

Polynomial-based construction methods.

Various methods for constructing a PiecewisePolynomial using vectors of matrices of polynomials, one for each output dimension. Unlike the coefficient-based methods, the number of polynomials must equal the number of segments, which will be one fewer than the number of breaks.

The following shows how such a PiecewisePolynomial might be constructed and used:

// Construct the PiecewisePolynomial.
const std::vector<double> breaks = { -1.0, 0.0, 1.0 };
Polynomiald t("t");
std::vector<Polynomiald> polynomials = { -(t*t), (t*t) };
const PiecewisePolynomial<double> pp(polynomials, breaks);
// Evaluate the PiecewisePolynomial at some values.
std::cout << pp.scalar_value(-1.0) << std::endl; // Outputs -1.0
std::cout << pp.scalar_value(1.0) << std::endl; // Outputs 1.0

WARNING: For robust floating point arithmetic, the polynomial for a segment will be evaluated (using value()) by first subtracting the break time from the evaluation time. In other words, when t lies in the half-open interval [breaks[i], breaks[i+1]) then:

value(t) == polynomials[i].eval(t - breaks[i])

meaning that constructing the polynomial like:

const std::vector<double> breaks = { 0.0, 1.0, 2.0 };
Polynomiald t("t");
std::vector<Polynomiald> polynomials = { (t*t), (t*t) };
const PiecewisePolynomial<double> pp(polynomials, breaks);

would give the following result:

// Evaluate the PiecewisePolynomial on both sides of a break.
const int row = 0, col = 0;
const double eps = 0.5 * std::numeric_limits<double>::epsilon();
std::cout << pp.value(1.0-eps)(row, col) << std::endl; // Outputs 1.0
std::cout << pp.value(1.0+eps)(row, col) << std::endl; // Outputs 1e-32

because the second polynomial will be evaluated at 1.0+eps minus the break time for that polynomial (1.0), i.e., t=eps. The intended result for the above example can be obtained by shifting the piecewise polynomial like so:

const std::vector<double> breaks = { 0.0, 1.0, 2.0 };
Polynomiald t("t");
std::vector<Polynomiald> polynomials = { (t*t),
((t+breaks[1])*(t+breaks[1])) };
const PiecewisePolynomial<double> pp(polynomials, breaks);
// Evaluate the PiecewisePolynomial on both sides of a break.
const double eps = 0.5 * std::numeric_limits<double>::epsilon();
std::cout << pp.value(1.0-eps)(row, col) << std::endl; // Outputs 1.0
std::cout << pp.value(1.0+eps)(row, col) << std::endl; // Outputs 1.0
PiecewisePolynomial (const std::vector< PolynomialMatrix > &polynomials_matrix, const std::vector< T > &breaks)
Constructs a PiecewisePolynomial using matrix-output Polynomials defined over each segment. More...

PiecewisePolynomial (const std::vector< Polynomial< T >> &polynomials, const std::vector< T > &breaks)
Constructs a PiecewisePolynomial using scalar-output Polynomials defined over each segment. More...

Public Member Functions inherited from PiecewiseTrajectory< T >
~PiecewiseTrajectory () override=default

int get_number_of_segments () const

start_time (int segment_number) const

end_time (int segment_number) const

duration (int segment_number) const

start_time () const override

end_time () const override

boolean< T > is_time_in_range (const T &t) const
Returns true iff t >= getStartTime() && t <= getEndTime(). More...

int get_segment_index (const T &t) const

const std::vector< T > & get_segment_times () const

void segment_number_range_check (int segment_number) const

Public Member Functions inherited from Trajectory< T >
virtual ~Trajectory ()=default

MatrixX< T > vector_values (const std::vector< T > &t) const
If cols()==1, then evaluates the trajectory at each time t, and returns the results as a Matrix with the ith column corresponding to the ith time. More...

bool has_derivative () const
Returns true iff the Trajectory provides and implementation for EvalDerivative() and MakeDerivative(). More...

MatrixX< T > EvalDerivative (const T &t, int derivative_order=1) const
Evaluates the derivative of this at the given time t. More...

std::unique_ptr< Trajectory< T > > MakeDerivative (int derivative_order=1) const
Takes the derivative of this Trajectory. More...

## Static Public Member Functions

Coefficient-based construction methods.

Various methods for constructing a PiecewisePolynomial using samples of coefficient matrices. Under the hood, PiecewisePolynomial constructs interpolating Polynomial objects that pass through the sample points. These methods differ by the continuity constraints that they enforce at break points and whether each sample represents a full matrix (versions taking const std::vector<MatrixX<T>>&) or a column vector (versions taking const Eigen::Ref<const MatrixX<T>>&).

These methods will throw std::exception if:

• the breaks and samples have different length,
• the breaks are not strictly increasing,
• the samples have inconsistent dimensions (i.e., the matrices do not all have identical dimensions),
• the breaks vector has length smaller than 2.
static PiecewisePolynomial< T > ZeroOrderHold (const std::vector< T > &breaks, const std::vector< MatrixX< T >> &samples)
Constructs a piecewise constant PiecewisePolynomial using matrix samples. More...

static PiecewisePolynomial< T > ZeroOrderHold (const Eigen::Ref< const VectorX< T >> &breaks, const Eigen::Ref< const MatrixX< T >> &samples)
Version of ZeroOrderHold(breaks, samples) that uses vector samples and Eigen VectorXd/MatrixX<T> arguments. More...

static PiecewisePolynomial< T > FirstOrderHold (const std::vector< T > &breaks, const std::vector< MatrixX< T >> &samples)
Constructs a piecewise linear PiecewisePolynomial using matrix samples. More...

static PiecewisePolynomial< T > FirstOrderHold (const Eigen::Ref< const VectorX< T >> &breaks, const Eigen::Ref< const MatrixX< T >> &samples)
Version of FirstOrderHold(breaks, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. More...

static PiecewisePolynomial< T > CubicShapePreserving (const std::vector< T > &breaks, const std::vector< MatrixX< T >> &samples, bool zero_end_point_derivatives=false)
Constructs a third order PiecewisePolynomial using vector samples, where each column of samples represents a sample point. More...

static PiecewisePolynomial< T > CubicShapePreserving (const Eigen::Ref< const VectorX< T >> &breaks, const Eigen::Ref< const MatrixX< T >> &samples, bool zero_end_point_derivatives=false)
Version of CubicShapePreserving(breaks, samples, zero_end_point_derivatives) that uses vector samples and Eigen VectorXd and MatrixX<T> arguments. More...

static PiecewisePolynomial< T > CubicWithContinuousSecondDerivatives (const std::vector< T > &breaks, const std::vector< MatrixX< T >> &samples, const MatrixX< T > &sample_dot_at_start, const MatrixX< T > &sample_dot_at_end)
Constructs a third order PiecewisePolynomial using matrix samples. More...

static PiecewisePolynomial< T > CubicWithContinuousSecondDerivatives (const Eigen::Ref< const VectorX< T >> &breaks, const Eigen::Ref< const MatrixX< T >> &samples, const Eigen::Ref< const VectorX< T >> &sample_dot_at_start, const Eigen::Ref< const VectorX< T >> &sample_dot_at_end)
Version of CubicWithContinuousSecondDerivatives() that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. More...

static PiecewisePolynomial< T > CubicHermite (const std::vector< T > &breaks, const std::vector< MatrixX< T >> &samples, const std::vector< MatrixX< T >> &samples_dot)
Constructs a third order PiecewisePolynomial using matrix samples and derivatives of samples (samples_dot); each matrix element of samples_dot represents the derivative with respect to the independent variable (e.g., the time derivative) of the corresponding entry in samples. More...

static PiecewisePolynomial< T > CubicHermite (const Eigen::Ref< const VectorX< T >> &breaks, const Eigen::Ref< const MatrixX< T >> &samples, const Eigen::Ref< const MatrixX< T >> &samples_dot)
Version of CubicHermite(breaks, samples, samples_dot) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. More...

static PiecewisePolynomial< T > CubicWithContinuousSecondDerivatives (const std::vector< T > &breaks, const std::vector< MatrixX< T >> &samples, bool periodic_end_condition=false)
Constructs a third order PiecewisePolynomial using matrix samples. More...

static PiecewisePolynomial< T > CubicWithContinuousSecondDerivatives (const Eigen::Ref< const VectorX< T >> &breaks, const Eigen::Ref< const MatrixX< T >> &samples, bool periodic_end_condition=false)
Version of CubicWithContinuousSecondDerivatives(breaks, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. More...

static PiecewisePolynomial< T > Cubic (const Eigen::Ref< const VectorX< T >> &breaks, const Eigen::Ref< const MatrixX< T >> &samples, bool periodic_end_condition=false)

static PiecewisePolynomial LagrangeInterpolatingPolynomial (const std::vector< T > &times, const std::vector< MatrixX< T >> &samples)
Constructs a polynomial with a single segment of the lowest possible degree that passes through all of the sample points. More...

static PiecewisePolynomial< T > LagrangeInterpolatingPolynomial (const Eigen::Ref< const VectorX< T >> &times, const Eigen::Ref< const MatrixX< T >> &samples)
Version of LagrangeInterpolatingPolynomial(times, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments. More...

Static Public Member Functions inherited from PiecewiseTrajectory< T >
static std::vector< T > RandomSegmentTimes (int num_segments, std::default_random_engine &generator)

Static Public Attributes inherited from PiecewiseTrajectory< T >
static constexpr double kEpsilonTime = std::numeric_limits<double>::epsilon()
Minimum delta quantity used for comparing time. More...

Protected Member Functions inherited from PiecewiseTrajectory< T >
PiecewiseTrajectory ()=default

PiecewiseTrajectory (const std::vector< T > &breaks)
breaks increments must be greater or equal to kEpsilonTime. More...

bool SegmentTimesEqual (const PiecewiseTrajectory &b, double tol=kEpsilonTime) const

const std::vector< T > & breaks () const

std::vector< T > & get_mutable_breaks ()

PiecewiseTrajectory (const PiecewiseTrajectory &)=default

PiecewiseTrajectoryoperator= (const PiecewiseTrajectory &)=default

PiecewiseTrajectory (PiecewiseTrajectory &&)=default

PiecewiseTrajectoryoperator= (PiecewiseTrajectory &&)=default

Protected Member Functions inherited from Trajectory< T >
Trajectory ()=default

Trajectory (const Trajectory &)=default

Trajectoryoperator= (const Trajectory &)=default

Trajectory (Trajectory &&)=default

Trajectoryoperator= (Trajectory &&)=default

## ◆ PolynomialMatrix

 typedef MatrixX > PolynomialMatrix

## ◆ PiecewisePolynomial() [1/6]

 PiecewisePolynomial ( )
default

Constructs an empty piecewise polynomial.

## ◆ PiecewisePolynomial() [2/6]

 PiecewisePolynomial ( const PiecewisePolynomial< T > & )
default

## ◆ PiecewisePolynomial() [3/6]

 PiecewisePolynomial ( PiecewisePolynomial< T > && )
default

## ◆ PiecewisePolynomial() [4/6]

 PiecewisePolynomial ( const Eigen::MatrixBase< Derived > & constant_value )
explicit

Single segment, constant value constructor over the interval [-∞, ∞].

The constructed PiecewisePolynomial will return constant_value at every evaluated point (i.e., value(t) = constant_value ∀t ∈ [-∞, ∞]).

## ◆ PiecewisePolynomial() [5/6]

 PiecewisePolynomial ( const std::vector< PolynomialMatrix > & polynomials_matrix, const std::vector< T > & breaks )

Constructs a PiecewisePolynomial using matrix-output Polynomials defined over each segment.

Precondition
polynomials.size() == breaks.size() - 1

## ◆ PiecewisePolynomial() [6/6]

 PiecewisePolynomial ( const std::vector< Polynomial< T >> & polynomials, const std::vector< T > & breaks )

Constructs a PiecewisePolynomial using scalar-output Polynomials defined over each segment.

Precondition
polynomials.size() == breaks.size() - 1

## ◆ ~PiecewisePolynomial()

 ~PiecewisePolynomial ( )
overridedefault

## ◆ AppendCubicHermiteSegment()

 void AppendCubicHermiteSegment ( const T & time, const Eigen::Ref< const MatrixX< T >> & sample, const Eigen::Ref< const MatrixX< T >> & sample_dot )

The CubicHermite spline construction has a nice property of being incremental (each segment can be solved independently).

Given a new sample and it's derivative, this method adds one segment to the end of this where the start sample and derivative are taken as the value and derivative at the final break of this.

Precondition
this is not empty()
time > end_time()
sample and sample_dot must have size rows() x cols().

## ◆ AppendFirstOrderSegment()

 void AppendFirstOrderSegment ( const T & time, const Eigen::Ref< const MatrixX< T >> & sample )

Given a new sample, this method adds one segment to the end of this using a first-order hold, where the start sample is taken as the value at the final break of this.

## ◆ Block()

 PiecewisePolynomial Block ( int start_row, int start_col, int block_rows, int block_cols ) const

Extracts a trajectory representing a block of size (block_rows, block_cols) starting at (start_row, start_col) from the PiecewisePolynomial.

Returns
a PiecewisePolynomial such that ret.value(t) = this.value(t).block(i,j,p,q);

## ◆ Clone()

 std::unique_ptr > Clone ( ) const
overridevirtual
Returns
A deep copy of this Trajectory.

Implements Trajectory< T >.

## ◆ cols()

 Eigen::Index cols ( ) const
overridevirtual

Returns the column count of the output matrices.

Exceptions
 std::exception if empty().

Implements Trajectory< T >.

## ◆ ConcatenateInTime()

 void ConcatenateInTime ( const PiecewisePolynomial< T > & other )

Concatenates other to the end of this.

Warning
The resulting PiecewisePolynomial will only be continuous to the degree that the first Polynomial of other is continuous with the last Polynomial of this. See warning about evaluating discontinuous derivatives at breaks in derivative().
Parameters
 other PiecewisePolynomial instance to concatenate.
Exceptions
 std::exception if trajectories' dimensions do not match each other (either rows() or cols() does not match between this and other). std::exception if this->end_time() and other->start_time() are not within PiecewiseTrajectory::kEpsilonTime from each other.

## ◆ Cubic()

 static PiecewisePolynomial Cubic ( const Eigen::Ref< const VectorX< T >> & breaks, const Eigen::Ref< const MatrixX< T >> & samples, bool periodic_end_condition = false )
static

## ◆ CubicHermite() [1/2]

 static PiecewisePolynomial CubicHermite ( const std::vector< T > & breaks, const std::vector< MatrixX< T >> & samples, const std::vector< MatrixX< T >> & samples_dot )
static

Constructs a third order PiecewisePolynomial using matrix samples and derivatives of samples (samples_dot); each matrix element of samples_dot represents the derivative with respect to the independent variable (e.g., the time derivative) of the corresponding entry in samples.

Each segment is fully specified by samples and sample_dot at both ends. Second derivatives are not continuous.

## ◆ CubicHermite() [2/2]

 static PiecewisePolynomial CubicHermite ( const Eigen::Ref< const VectorX< T >> & breaks, const Eigen::Ref< const MatrixX< T >> & samples, const Eigen::Ref< const MatrixX< T >> & samples_dot )
static

Version of CubicHermite(breaks, samples, samples_dot) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments.

Corresponding columns of samples and samples_dot are used as the sample point and independent variable derivative, respectively.

Precondition
samples.cols() == samples_dot.cols() == breaks.size().

## ◆ CubicShapePreserving() [1/2]

 static PiecewisePolynomial CubicShapePreserving ( const std::vector< T > & breaks, const std::vector< MatrixX< T >> & samples, bool zero_end_point_derivatives = false )
static

Constructs a third order PiecewisePolynomial using vector samples, where each column of samples represents a sample point.

First derivatives are chosen to be "shape preserving", i.e. if samples is monotonic within some interval, the interpolated data will also be monotonic. The second derivative is not guaranteed to be smooth across the entire spline.

MATLAB calls this method "pchip" (short for "Piecewise Cubic Hermite Interpolating Polynomial"), and provides a nice description in their documentation. http://home.uchicago.edu/~sctchoi/courses/cs138/interp.pdf is also a good reference.

If zero_end_point_derivatives is false, the first and last first derivative is chosen using a non-centered, shape-preserving three-point formulae. See equation (2.10) in the following reference for more details. http://www.mi.sanu.ac.rs/~gvm/radovi/mon.pdf If zero_end_point_derivatives is true, they are set to zeros.

If zero_end_point_derivatives is false, breaks and samples must have at least 3 elements for the algorithm to determine the first derivatives.

If zero_end_point_derivatives is true, breaks and samples may have 2 or more elements. For the 2 elements case, the result is equivalent to computing a cubic polynomial whose values are given by samples, and derivatives set to zero.

Exceptions
 std::exception if: breaks has length smaller than 3 and zero_end_point_derivatives is false, breaks has length smaller than 2 and zero_end_point_derivatives is true. std::exception under the conditions specified under coefficient_construction_methods.

## ◆ CubicShapePreserving() [2/2]

 static PiecewisePolynomial CubicShapePreserving ( const Eigen::Ref< const VectorX< T >> & breaks, const Eigen::Ref< const MatrixX< T >> & samples, bool zero_end_point_derivatives = false )
static

Version of CubicShapePreserving(breaks, samples, zero_end_point_derivatives) that uses vector samples and Eigen VectorXd and MatrixX<T> arguments.

Each column of samples represents a sample point.

Precondition
samples.cols() == breaks.size().
Exceptions
 std::exception under the conditions specified under coefficient_construction_methods.

## ◆ CubicWithContinuousSecondDerivatives() [1/4]

 static PiecewisePolynomial CubicWithContinuousSecondDerivatives ( const std::vector< T > & breaks, const std::vector< MatrixX< T >> & samples, const MatrixX< T > & sample_dot_at_start, const MatrixX< T > & sample_dot_at_end )
static

Constructs a third order PiecewisePolynomial using matrix samples.

The PiecewisePolynomial is constructed such that the interior segments have the same value, first and second derivatives at breaks. sample_dot_at_start and sample_dot_at_end are used for the first and last first derivatives.

Exceptions
 std::exception if sample_dot_at_start or sample_dot_at_end and samples have inconsistent dimensions. std::exception under the conditions specified under coefficient_construction_methods.

## ◆ CubicWithContinuousSecondDerivatives() [2/4]

 static PiecewisePolynomial CubicWithContinuousSecondDerivatives ( const Eigen::Ref< const VectorX< T >> & breaks, const Eigen::Ref< const MatrixX< T >> & samples, const Eigen::Ref< const VectorX< T >> & sample_dot_at_start, const Eigen::Ref< const VectorX< T >> & sample_dot_at_end )
static

Version of CubicWithContinuousSecondDerivatives() that uses vector samples and Eigen VectorXd / MatrixX<T> arguments.

Each column of samples represents a sample point.

Precondition
samples.cols() == breaks.size().
Exceptions
 std::exception under the conditions specified under coefficient_construction_methods.

## ◆ CubicWithContinuousSecondDerivatives() [3/4]

 static PiecewisePolynomial CubicWithContinuousSecondDerivatives ( const std::vector< T > & breaks, const std::vector< MatrixX< T >> & samples, bool periodic_end_condition = false )
static

Constructs a third order PiecewisePolynomial using matrix samples.

The PiecewisePolynomial is constructed such that the interior segments have the same value, first and second derivatives at breaks. If periodic_end_condition is false (default), then the "Not-a-sample" end condition is used here, which means the third derivatives are continuous for the first two and last two segments. If periodic_end_condition is true, then the first and second derivatives between the end of the last segment and the beginning of the first segment will be continuous. Note that the periodic end condition does not require the first and last sample to be collocated, nor does it add an additional sample to connect the first and last segments. Only first and second derivative continuity is enforced. See https://en.wikipedia.org/wiki/Spline_interpolation and https://www.math.uh.edu/~jingqiu/math4364/spline.pdf for more about cubic splines and their end conditions. The MATLAB docs for methods "spline" and "csape" are also good references.

Precondition
breaks and samples must have at least 3 elements. If periodic_end_condition is true, then for two samples, it would produce a straight line (use FirstOrderHold for this instead), and if periodic_end_condition is false the problem is ill-defined.

## ◆ CubicWithContinuousSecondDerivatives() [4/4]

 static PiecewisePolynomial CubicWithContinuousSecondDerivatives ( const Eigen::Ref< const VectorX< T >> & breaks, const Eigen::Ref< const MatrixX< T >> & samples, bool periodic_end_condition = false )
static

Version of CubicWithContinuousSecondDerivatives(breaks, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments.

Each column of samples represents a sample point.

Precondition
samples.cols() == breaks.size().

## ◆ derivative()

 PiecewisePolynomial derivative ( int derivative_order = 1 ) const

Returns a PiecewisePolynomial where each segment is the specified derivative of the corresponding segment in this.

Any rules or limitations of Polynomial::derivative() also apply to this function.

Derivatives evaluated at non-differentiable points return the value at the left hand side of the interval.

Parameters
 derivative_order The order of the derivative, namely, if derivative_order = n, the n'th derivative of the polynomial will be returned.
Warning
In the event of discontinuous derivatives evaluated at breaks, it is not defined which polynomial (i.e., to the left or right of the break) will be the one that is evaluated at the break.

## ◆ empty()

 bool empty ( ) const

Returns true if this trajectory has no breaks/samples/polynomials.

## ◆ FirstOrderHold() [1/2]

 static PiecewisePolynomial FirstOrderHold ( const std::vector< T > & breaks, const std::vector< MatrixX< T >> & samples )
static

Constructs a piecewise linear PiecewisePolynomial using matrix samples.

Exceptions
 std::exception under the conditions specified under coefficient_construction_methods.

## ◆ FirstOrderHold() [2/2]

 static PiecewisePolynomial FirstOrderHold ( const Eigen::Ref< const VectorX< T >> & breaks, const Eigen::Ref< const MatrixX< T >> & samples )
static

Version of FirstOrderHold(breaks, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments.

Each column of samples represents a sample point.

Precondition
samples.cols() == breaks.size()
Exceptions
 std::exception under the conditions specified under coefficient_construction_methods.

## ◆ getPolynomial()

 const Polynomial& getPolynomial ( int segment_index, Eigen::Index row = 0, Eigen::Index col = 0 ) const

Gets the Polynomial with the given matrix row and column index that corresponds to the given segment index.

Equivalent to getPolynomialMatrix(segment_index)(row, col).

Note
Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.

## ◆ getPolynomialMatrix()

 const PolynomialMatrix& getPolynomialMatrix ( int segment_index ) const

Gets the matrix of Polynomials corresponding to the given segment index.

Warning
segment_index is not checked for validity.

## ◆ getSegmentPolynomialDegree()

 int getSegmentPolynomialDegree ( int segment_index, Eigen::Index row = 0, Eigen::Index col = 0 ) const

Gets the degree of the Polynomial with the given matrix row and column index that corresponds to the given segment index.

Equivalent to getPolynomial(segment_index, row, col).GetDegree().

## ◆ integral() [1/2]

 PiecewisePolynomial integral ( const T & value_at_start_time = 0.0 ) const

Returns a PiecewisePolynomial that is the indefinite integral of this one.

Any rules or limitations of Polynomial::integral() also apply to this function.

If value_at_start_time is given, it does the following only for the first segment: adds that constant as the constant term (zeroth-order coefficient) of the resulting Polynomial.

## ◆ integral() [2/2]

 PiecewisePolynomial integral ( const Eigen::Ref< MatrixX< T >> & value_at_start_time ) const

Returns a PiecewisePolynomial that is the indefinite integral of this one.

Any rules or limitations of Polynomial::integral() also apply to this function.

If value_at_start_time is given, it does the following only for the first segment: adds value_at_start_time(row,col) as the constant term (zeroth-order coefficient) of the resulting Polynomial.

## ◆ isApprox()

 bool isApprox ( const PiecewisePolynomial< T > & other, double tol, const ToleranceType & tol_type = ToleranceType::kRelative ) const

Checks whether a PiecewisePolynomial is approximately equal to this one by calling Polynomial<T>::CoefficientsAlmostEqual() on every element of every segment.

Polynomial<T>::CoefficientsAlmostEqual().

## ◆ LagrangeInterpolatingPolynomial() [1/2]

 static PiecewisePolynomial LagrangeInterpolatingPolynomial ( const std::vector< T > & times, const std::vector< MatrixX< T >> & samples )
static

Constructs a polynomial with a single segment of the lowest possible degree that passes through all of the sample points.

Precondition
times must be monotonically increasing.
samples.size() == times.size().

## ◆ LagrangeInterpolatingPolynomial() [2/2]

 static PiecewisePolynomial LagrangeInterpolatingPolynomial ( const Eigen::Ref< const VectorX< T >> & times, const Eigen::Ref< const MatrixX< T >> & samples )
static

Version of LagrangeInterpolatingPolynomial(times, samples) that uses vector samples and Eigen VectorXd / MatrixX<T> arguments.

Each column of samples represents a sample point.

Precondition
samples.cols() == times.size().

## ◆ operator *()

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

Multiplies each Polynomial in the PolynomialMatrix of other by the corresponding Polynomial in the PolynomialMatrix of this (i.e., a coefficient-wise multiplication).

If this corresponds to t² and other corresponds to t³, this *= other will correspond to t⁵.

Exceptions
 std::exception if every element of other.get_segment_times() is not within PiecewiseTrajectory::kEpsilonTime from this->get_segment_times()1.

## ◆ operator *=()

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

Multiplies each Polynomial in the PolynomialMatrix of other by the corresponding Polynomial in the PolynomialMatrix of this (i.e., a coefficient-wise multiplication), storing the result in this.

If this corresponds to t² and other corresponds to t³, this *= other will correspond to t⁵.

Exceptions
 std::exception if every element of other.get_segment_times() is not within PiecewiseTrajectory::kEpsilonTime from this->get_segment_times().

## ◆ operator+() [1/2]

 const PiecewisePolynomial operator+ ( const PiecewisePolynomial< T > & other ) const

Adds each Polynomial in the PolynomialMatrix of other to the corresponding Polynomial in the PolynomialMatrix of this.

If this corresponds to t² and other corresponds to t³, this + other will correspond to t³ + t².

Exceptions
 std::exception if every element of other.get_segment_times() is not within PiecewiseTrajectory::kEpsilonTime from this->get_segment_times().

## ◆ operator+() [2/2]

 const PiecewisePolynomial operator+ ( const MatrixX< T > & coeff ) const

## ◆ operator+=() [1/2]

 PiecewisePolynomial& operator+= ( const PiecewisePolynomial< T > & other )

Adds each Polynomial in the PolynomialMatrix of other to the corresponding Polynomial in the PolynomialMatrix of this, storing the result in this.

If this corresponds to t² and other corresponds to t³, this += other will correspond to t³ + t².

Exceptions
 std::exception if every element of other.get_segment_times() is not within PiecewiseTrajectory::kEpsilonTime from this->get_segment_times().

## ◆ operator+=() [2/2]

 PiecewisePolynomial& operator+= ( const MatrixX< T > & coeff )

## ◆ operator-() [1/3]

 const PiecewisePolynomial operator- ( const PiecewisePolynomial< T > & other ) const

Subtracts each Polynomial in the PolynomialMatrix of other from the corresponding Polynomial in the PolynomialMatrix of this.

If this corresponds to t² and other corresponds to t³, this - other will correspond to t² - t³.

Exceptions
 std::exception if every element of other.get_segment_times() is not within PiecewiseTrajectory::kEpsilonTime from this->get_segment_times().

## ◆ operator-() [2/3]

 const PiecewisePolynomial operator- ( ) const

Implements unary minus operator.

Multiplies each Polynomial in this by -1.

## ◆ operator-() [3/3]

 const PiecewisePolynomial operator- ( const MatrixX< T > & coeff ) const

## ◆ operator-=() [1/2]

 PiecewisePolynomial& operator-= ( const PiecewisePolynomial< T > & other )

Subtracts each Polynomial in the PolynomialMatrix of other from the corresponding Polynomial in the PolynomialMatrix of this, storing the result in this.

If this corresponds to t² and other corresponds to t³, this -= other will correspond to t² - t³.

Exceptions
 std::exception if every element of other.get_segment_times() is not within PiecewiseTrajectory::kEpsilonTime from this->get_segment_times().

## ◆ operator-=() [2/2]

 PiecewisePolynomial& operator-= ( const MatrixX< T > & coeff )

## ◆ operator=() [1/2]

 PiecewisePolynomial& operator= ( const PiecewisePolynomial< T > & )
default

## ◆ operator=() [2/2]

 PiecewisePolynomial& operator= ( PiecewisePolynomial< T > && )
default

## ◆ RemoveFinalSegment()

 void RemoveFinalSegment ( )

Removes the final segment from the trajectory, reducing the number of segments by 1.

Precondition
this is not empty()

## ◆ Reshape()

 void Reshape ( int rows, int cols )

Reshapes the dimensions of the Eigen::MatrixX<T> returned by value(), EvalDerivative(), etc.

Precondition
rows x cols must equal this.rows() * this.cols().
Eigen::PlainObjectBase::resize().

## ◆ ReverseTime()

 void ReverseTime ( )

Modifies the trajectory so that pp_after(t) = pp_before(-t).

Note
The new trajectory will evaluate differently at precisely the break points if the original trajectory was discontinuous at the break points. This is because the segments are defined on the half-open intervals [breaks(i), breaks(i+1)), and the order of the breaks have been reversed.

## ◆ rows()

 Eigen::Index rows ( ) const
overridevirtual

Returns the row count of the output matrices.

Exceptions
 std::exception if empty().

Implements Trajectory< T >.

## ◆ scalarValue()

 T scalarValue ( const T & t, Eigen::Index row = 0, Eigen::Index col = 0 ) const

Evaluates the trajectory at the given time without returning the entire matrix.

Equivalent to value(t)(row, col).

Warning
See warnings in value().

## ◆ ScaleTime()

 void ScaleTime ( const T & scale )

Scales the time of the trajectory by non-negative scale (use ReverseTime() if you want to also negate time).

The resulting polynomial evaluates to pp_after(t) = pp_before(t/scale).

As an example, scale=2 will result in a trajectory that is twice as long (start_time() and end_time() have both doubled).

## ◆ setPolynomialMatrixBlock()

 void setPolynomialMatrixBlock ( const PolynomialMatrix & replacement, int segment_index, Eigen::Index row_start = 0, Eigen::Index col_start = 0 )

Replaces the specified block of the PolynomialMatrix at the given segment index.

Note
Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.
Warning
This code relies upon Eigen to verify that the replacement block is not too large.

## ◆ shiftRight()

 void shiftRight ( const T & offset )

Adds offset to all of the breaks.

offset need not be a non-negative number. The resulting polynomial will evaluate to pp_after(t) = pp_before(t-offset).

As an example, offset=2 will result in the start_time() and end_time() being 2 seconds later.

## ◆ slice()

 PiecewisePolynomial slice ( int start_segment_index, int num_segments ) const

Returns the PiecewisePolynomial comprising the num_segments segments starting at the specified start_segment_index.

Note
Calls PiecewiseTrajectory<T>::segment_number_range_check() to validate segment_index.

## ◆ value()

 MatrixX value ( const T & t ) const
overridevirtual

Evaluates the PiecewisePolynomial at the given time t.

Parameters
 t The time at which to evaluate the PiecewisePolynomial.
Returns
The matrix of evaluated values.
Precondition
If T == symbolic::Expression, t.is_constant() must be true.
Warning
If t does not lie in the range that the polynomial is defined over, the polynomial will silently be evaluated at the closest point to t. For example, value(-1) will return value(0) for a polynomial defined over [0, 1].
See warning in the constructor overview above.

Implements Trajectory< T >.

## ◆ ZeroOrderHold() [1/2]

 static PiecewisePolynomial ZeroOrderHold ( const std::vector< T > & breaks, const std::vector< MatrixX< T >> & samples )
static

Constructs a piecewise constant PiecewisePolynomial using matrix samples.

Note that constructing a PiecewisePolynomial requires at least two sample points, although in this case, the second sample point's value is ignored, and only its break time is used.

Exceptions
 std::exception under the conditions specified under coefficient_construction_methods.

## ◆ ZeroOrderHold() [2/2]

 static PiecewisePolynomial ZeroOrderHold ( const Eigen::Ref< const VectorX< T >> & breaks, const Eigen::Ref< const MatrixX< T >> & samples )
static

Version of ZeroOrderHold(breaks, samples) that uses vector samples and Eigen VectorXd/MatrixX<T> arguments.

Each column of samples represents a sample point.

Precondition
samples.cols() == breaks.size()
Exceptions
 std::exception under the conditions specified under coefficient_construction_methods.

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