aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots > Class Template Reference

Utility for fitting splines given constraints on function value, derivative value, and continuity. More...

#include <aikido/common/Spline.hpp>

Public Types
using	Scalar = _Scalar
using	Index = _Index
using	TimeVector = Eigen::Matrix< Scalar, NumKnotsAtCompileTime, 1 >
using	OutputVector = Eigen::Matrix< Scalar, NumOutputsAtCompileTime, 1 >
using	OutputMatrix = Eigen::Matrix< Scalar, NumCoefficientsAtCompileTime, NumOutputsAtCompileTime >
using	CoefficientVector = Eigen::Matrix< Scalar, NumCoefficientsAtCompileTime, 1 >
using	CoefficientMatrix = Eigen::Matrix< Scalar, NumCoefficientsAtCompileTime, NumCoefficientsAtCompileTime >
using	ProblemMatrix = Eigen::SparseMatrix< Scalar, 0, Index >
using	ProblemVector = Eigen::Matrix< Scalar, DimensionAtCompileTime, NumOutputsAtCompileTime >
using	SolutionMatrix = Eigen::Matrix< Scalar, NumOutputsAtCompileTime, NumCoefficientsAtCompileTime >
using	Spline = SplineND< Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots >

Public Member Functions
	SplineProblem (const TimeVector &_times)
	Constructs a spline fitting problem with the knot points at the specified times. More...
	SplineProblem (const TimeVector &_times, Index _numCoefficients, Index _numOutputs)
	Constructs a spline fitting problem with knot points at the specified times and the specified number of coefficients and outputs. More...
	SplineProblem (SplineProblem &&_other)=default
	SplineProblem (const SplineProblem &_other)=default
SplineProblem &	operator= (SplineProblem &&_other)=default
SplineProblem &	operator= (const SplineProblem &_other)=default
void	addConstantConstraint (Index _knot, Index _derivative, const OutputVector &_value)
	Adds a constraint that the _derivative-th order derivative of knot point _knot should equal _value. More...
void	addContinuityConstraint (Index _knot, Index _derivative)
	Adds a continuity constraint on the _derivative-th order derivative at knot point _knot. More...
Spline	fit ()
	Fit a spline given the constraints on added to this object. More...
Index	getNumKnots () const
	Gets the number of knot points. More...
Index	getNumOutputs () const
	Gets the number of outputs points. More...
Scalar	getDuration () const
	Gets the duration of the spline. More...

Static Public Member Functions
static CoefficientVector	createTimeVector (Scalar _t, Index _i, Index _n)
	Creates a vector of the form [ 1, t, t^2, ... t^_n ]. More...
static CoefficientMatrix	createCoefficientMatrix (Index _n)
	Creates the _n by _n matrix of derivative coefficients for a polynomial with _n coefficients. More...

Static Public Attributes
static constexpr Index	NumCoefficientsAtCompileTime = _NumCoefficients
static constexpr Index	NumOutputsAtCompileTime = _NumOutputs
static constexpr Index	NumKnotsAtCompileTime = _NumKnots
static constexpr Index	NumSegmentsAtCompileTime
static constexpr Index	DimensionAtCompileTime

Private Attributes
Index	mNumKnots
Index	mNumSegments
Index	mNumCoefficients
Index	mNumOutputs
Index	mDimension
CoefficientMatrix	mCoefficientMatrix
Index	mRowIndex
TimeVector	mTimes
ProblemMatrix	mA
ProblemVector	mB
std::vector< SolutionMatrix, Eigen::aligned_allocator< SolutionMatrix > >	mSolution

Detailed Description

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>
class aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >

Utility for fitting splines given constraints on function value, derivative value, and continuity.

This class is intended to be used by calling methods to add constraints to the spline, then calling fit() to find spline coefficients that satisfy those constraints.

The number of coefficients, outputs, and knot points may be specified either at compile time (via template parameters) or at runtime (if the template parameters are Eigen::Dynamic). We suggest setting as many of these parameters at compile time as possible for best performance.

Template Parameters

_Scalar	floating type used to represent a scalar value
_Index	integral type used to represent an index
_NumCoefficients	number of polynomial coefficients, or Dynamic
_NumOutputs	number of outputs, or Dynamic
_NumKnots	number of knots, or Dynamic

Member Typedef Documentation

CoefficientMatrix

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

using aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::CoefficientMatrix = Eigen::Matrix< Scalar, NumCoefficientsAtCompileTime, NumCoefficientsAtCompileTime>

CoefficientVector

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

using aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::CoefficientVector = Eigen::Matrix<Scalar, NumCoefficientsAtCompileTime, 1>

Index

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

using aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::Index = _Index

OutputMatrix

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

using aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::OutputMatrix = Eigen:: Matrix<Scalar, NumCoefficientsAtCompileTime, NumOutputsAtCompileTime>

OutputVector

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

using aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::OutputVector = Eigen::Matrix<Scalar, NumOutputsAtCompileTime, 1>

ProblemMatrix

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

using aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::ProblemMatrix = Eigen::SparseMatrix<Scalar, 0, Index>

ProblemVector

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

using aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::ProblemVector = Eigen::Matrix<Scalar, DimensionAtCompileTime, NumOutputsAtCompileTime>

Scalar

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

using aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::Scalar = _Scalar

SolutionMatrix

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

using aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::SolutionMatrix = Eigen:: Matrix<Scalar, NumOutputsAtCompileTime, NumCoefficientsAtCompileTime>

Spline

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

using aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::Spline = SplineND<Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots>

TimeVector

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

using aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::TimeVector = Eigen::Matrix<Scalar, NumKnotsAtCompileTime, 1>

Constructor & Destructor Documentation

SplineProblem() [1/4]

template<class Scalar , class Index , Index _NumCoefficients, Index _NumOutputs, Index _NumKnots>

aikido::common::SplineProblem< Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots >::SplineProblem ( const TimeVector &  _times )

explicit

Constructs a spline fitting problem with the knot points at the specified times.

This overload is only supported if _NumCoefficients and _NumOutputs are specified as template parameters. The length of _times must equal _NumKnots if that template parameter is specified

If _NumCoefficients or _NumOutputs is Eigen::Dynamic, you must use the other constructor overload.

Parameters

_times

list of knot point times, must be monotone increasing

SplineProblem() [2/4]

template<class Scalar , class Index , Index _NumCoefficients, Index _NumOutputs, Index _NumKnots>

aikido::common::SplineProblem< Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots >::SplineProblem	(	const TimeVector &	_times,
		Index	_numCoefficients,
		Index	_numOutputs
	)

Constructs a spline fitting problem with knot points at the specified times and the specified number of coefficients and outputs.

These parameters must match _NumKnots and _NumCoefficients if they are specified at compile time. Additionally, the length of _times must match _NumKnots if that template parameter is specified.

Parameters

_times	list of knot point times, must be monotone increasing
_numCoefficients	number of polynomial coefficients
_numOutputs	number of outputs

SplineProblem() [3/4]

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::SplineProblem ( SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots > &&  _other )

default

SplineProblem() [4/4]

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::SplineProblem ( const SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots > &  _other )

default

Member Function Documentation

addConstantConstraint()

template<class Scalar , class Index , Index _NumCoefficients, Index _NumOutputs, Index _NumKnots>

void aikido::common::SplineProblem< Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots >::addConstantConstraint	(	Index	_knot,
		Index	_derivative,
		const OutputVector &	_value
	)

Adds a constraint that the _derivative-th order derivative of knot point _knot should equal _value.

This adds one constraint if the knot is an end point and two constraints for interior knots. The zero-th derivative is function value.

Parameters

_knot	knot index to constraint
_derivative	order of derivative to constraint
_value	desired value of that knot point's derivative

addContinuityConstraint()

template<class Scalar , class Index , Index _NumCoefficients, Index _NumOutputs, Index _NumKnots>

void aikido::common::SplineProblem< Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots >::addContinuityConstraint	(	Index	_knot,
		Index	_derivative
	)

Adds a continuity constraint on the _derivative-th order derivative at knot point _knot.

This operation is only defined for interior knot points and adds one constraint.

Parameters

_knot	knot index to constraint
_derivative	order of derivative to constraint

createCoefficientMatrix()

template<class Scalar , class Index , Index _NumCoefficients, Index _NumOutputs, Index _NumKnots>

auto aikido::common::SplineProblem< Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots >::createCoefficientMatrix ( Index  _n )

static

Creates the _n by _n matrix of derivative coefficients for a polynomial with _n coefficients.

The i-th row of this matrix corresponds to coefficients for the i-th derivative. The j-th column of this matrix corresponds to the coefficient on x^j.

Parameters

_n	number of coefficients

Returns

polynomial coefficient matrix

createTimeVector()

template<class Scalar , class Index , Index _NumCoefficients, Index _NumOutputs, Index _NumKnots>

auto aikido::common::SplineProblem< Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots >::createTimeVector ( Scalar  _t, Index  _i, Index  _n  )

static

Creates a vector of the form [ 1, t, t^2, ... t^_n ].

fit()

template<class Scalar , class Index , Index _NumCoefficients, Index _NumOutputs, Index _NumKnots>

auto aikido::common::SplineProblem< Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots >::fit

Fit a spline given the constraints on added to this object.

The behavior of this function is undefined if the problem is over or under-constrained. To avoid this, be sure to only add (num coefficients) * (num knots - 1) constraints to this class.

Returns

spline that satisfies the constraints

getDuration()

template<class Scalar , class Index , Index _NumCoefficients, Index _NumOutputs, Index _NumKnots>

Scalar aikido::common::SplineProblem< Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots >::getDuration

Gets the duration of the spline.

This is the difference between the time of the first and last knot points.

Returns

duration of the spline

getNumKnots()

template<class Scalar , class Index , Index _NumCoefficients, Index _NumOutputs, Index _NumKnots>

Index aikido::common::SplineProblem< Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots >::getNumKnots

Gets the number of knot points.

Returns

number of knot points

getNumOutputs()

template<class Scalar , class Index , Index _NumCoefficients, Index _NumOutputs, Index _NumKnots>

Index aikido::common::SplineProblem< Scalar, Index, _NumCoefficients, _NumOutputs, _NumKnots >::getNumOutputs

Gets the number of outputs points.

Returns

number of outputs points

operator=() [1/2]

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

SplineProblem& aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::operator= ( const SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots > &  _other )

default

operator=() [2/2]

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

SplineProblem& aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::operator= ( SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots > &&  _other )

default

Member Data Documentation

DimensionAtCompileTime

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

constexpr Index aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::DimensionAtCompileTime

staticconstexpr

Initial value:

= (NumSegmentsAtCompileTime != Eigen::Dynamic
         && _NumCoefficients != Eigen::Dynamic)
            ? (NumSegmentsAtCompileTime * NumCoefficientsAtCompileTime)
            : Eigen::Dynamic

mA

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

ProblemMatrix aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::mA

private

mB

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

ProblemVector aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::mB

private

mCoefficientMatrix

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

CoefficientMatrix aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::mCoefficientMatrix

private

mDimension

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

Index aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::mDimension

private

mNumCoefficients

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

Index aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::mNumCoefficients

private

mNumKnots

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

Index aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::mNumKnots

private

mNumOutputs

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

Index aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::mNumOutputs

private

mNumSegments

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

Index aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::mNumSegments

private

mRowIndex

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

Index aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::mRowIndex

private

mSolution

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

std::vector< SolutionMatrix, Eigen::aligned_allocator<SolutionMatrix> > aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::mSolution

private

mTimes

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

TimeVector aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::mTimes

private

NumCoefficientsAtCompileTime

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

constexpr Index aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::NumCoefficientsAtCompileTime = _NumCoefficients

staticconstexpr

NumKnotsAtCompileTime

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

constexpr Index aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::NumKnotsAtCompileTime = _NumKnots

staticconstexpr

NumOutputsAtCompileTime

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

constexpr Index aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::NumOutputsAtCompileTime = _NumOutputs

staticconstexpr

NumSegmentsAtCompileTime

template<class _Scalar = double, class _Index = int, _Index _NumCoefficients = Eigen::Dynamic, _Index _NumOutputs = Eigen::Dynamic, _Index _NumKnots = Eigen::Dynamic>

constexpr Index aikido::common::SplineProblem< _Scalar, _Index, _NumCoefficients, _NumOutputs, _NumKnots >::NumSegmentsAtCompileTime

staticconstexpr

Initial value:

= (_NumKnots != Eigen::Dynamic) ? (NumKnotsAtCompileTime - 1)
                                      : Eigen::Dynamic