BezierCurve< T > Class Template Reference

Detailed Description

template<typename T>
class drake::trajectories::BezierCurve< T >

A Bézier curve is defined by a set of control points p₀ through pₙ, where n is called the order of the curve (n = 1 for linear, 2 for quadratic, 3 for cubic, etc.).

The first and last control points are always the endpoints of the curve; however, the intermediate control points (if any) generally do not lie on the curve, but the curve is guaranteed to stay within the convex hull of the control points.

See also BsplineTrajectory. A B-spline can be thought of as a composition of overlapping Bézier curves (where each evaluation only depends on a local subset of the control points). In contrast, evaluating a Bézier curve will use all of the control points.

Template Parameters

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

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

Public Member Functions
	BezierCurve ()
	Default initializer. More...
	BezierCurve (double start_time, double end_time, const Eigen::Ref< const MatrixX< T >> &control_points)
	Constructs a Bézier curve over the interval t ∈ [start_time, end_time] with control points defined in the columns of control_points. More...
virtual	~BezierCurve ()=default
int	order () const
	Returns the order of the curve (1 for linear, 2 for quadratic, etc.). More...
T	BernsteinBasis (int i, const T &time, std::optional< int > order=std::nullopt) const
	Returns the value of the ith basis function of order (1 for linear, 2 for quadratic, etc) evaluated at time. More...
const MatrixX< T > &	control_points () const
	Returns a const reference to the control points which define the curve. More...
Eigen::SparseMatrix< double >	AsLinearInControlPoints (int derivative_order=1) const
	Supports writing optimizations using the control points as decision variables. More...
std::unique_ptr< trajectories::Trajectory< T > >	Clone () const override
MatrixX< T >	value (const T &time) const override
	Evaluates the curve at the given time. More...
VectorX< symbolic::Expression >	GetExpression (symbolic::Variable time=symbolic::Variable("t")) const
	Extracts the expanded underlying polynomial expression of this curve in terms of variable time. More...
void	ElevateOrder ()
	Increases the order of the curve by 1. More...
Eigen::Index	rows () const override
Eigen::Index	cols () const override
T	start_time () const override
T	end_time () const override
Implements CopyConstructible, CopyAssignable, MoveConstructible, MoveAssignable
	BezierCurve (const BezierCurve &)=default
BezierCurve &	operator= (const BezierCurve &)=default
	BezierCurve (BezierCurve &&)=default
BezierCurve &	operator= (BezierCurve &&)=default
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...
MatrixX< T >	vector_values (const Eigen::Ref< const VectorX< 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...

Additional Inherited Members
Protected Member Functions inherited from Trajectory< T >
	Trajectory ()=default
	Trajectory (const Trajectory &)=default
Trajectory &	operator= (const Trajectory &)=default
	Trajectory (Trajectory &&)=default
Trajectory &	operator= (Trajectory &&)=default

Constructor & Destructor Documentation

BezierCurve() [1/4]

BezierCurve ( const BezierCurve< T > &  )

default

BezierCurve() [2/4]

BezierCurve ( BezierCurve< T > &&  )

default

BezierCurve() [3/4]

BezierCurve

(

)

Default initializer.

Constructs an empty Bézier curve over the interval t ∈ [0, 1].

BezierCurve() [4/4]

BezierCurve	(	double	start_time,
		double	end_time,
		const Eigen::Ref< const MatrixX< T >> &	control_points
	)

Constructs a Bézier curve over the interval t ∈ [start_time, end_time] with control points defined in the columns of control_points.

Precondition

end_time >= start_time.

~BezierCurve()

virtual ~BezierCurve ( )

virtualdefault

Member Function Documentation

AsLinearInControlPoints()

Eigen::SparseMatrix<double> AsLinearInControlPoints

(

int

derivative_order = 1

)

const

Supports writing optimizations using the control points as decision variables.

This method returns the matrix, M, defining the control points of the order derivative in the form:

derivative.control_points() = this.control_points() * M

For instance, since we have

derivative.control_points().col(k) = this.control_points() * M.col(k),

constraining the kth control point of the nth derivative to be in [ub, lb], could be done with:

auto M = curve.AsLinearInControlPoints(n);
for (int i=0; i<curve.rows(); ++i) {
  auto c = std::make_shared<solvers::LinearConstraint>(
    M.col(k).transpose(), Vector1d(lb(i)), Vector1d(ub(i)));
  prog.AddConstraint(c, curve.row(i).transpose());
}

Iterating over the rows of the control points is the natural sparsity pattern here (since M is the same for all rows). For instance, we also have

derivative.control_points().row(k).T = M.T * this.control_points().row(k).T,

or

vec(derivative.control_points().T) = blockMT * vec(this.control_points().T),
blockMT = [ M.T,   0, .... 0 ]
          [   0, M.T, 0, ... ]
          [      ...         ]
          [  ...    , 0, M.T ].

Precondition

derivative_order >= 0.

BernsteinBasis()

T BernsteinBasis	(	int	i,
		const T &	time,
		std::optional< int >	order = std::nullopt
	)		const

Returns the value of the ith basis function of order (1 for linear, 2 for quadratic, etc) evaluated at time.

The default value for the optional argument order is the order() of this.

Clone()

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

overridevirtual

Returns

A deep copy of this Trajectory.

Implements Trajectory< T >.

cols()

Eigen::Index cols ( ) const

overridevirtual

Returns

The number of columns in the matrix returned by value().

Implements Trajectory< T >.

control_points()

const MatrixX<T>& control_points

(

)

const

Returns a const reference to the control points which define the curve.

ElevateOrder()

void ElevateOrder

(

)

Increases the order of the curve by 1.

A Bézier curve of order n can be converted into a Bézier curve of order n + 1 with the same shape. The control points of this are modified to obtain the equivalent curve.

end_time()

T end_time ( ) const

overridevirtual

Implements Trajectory< T >.

GetExpression()

VectorX<symbolic::Expression> GetExpression

(

symbolic::Variable

time = symbolic::Variable("t")

)

const

Extracts the expanded underlying polynomial expression of this curve in terms of variable time.

operator=() [1/2]

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

default

operator=() [2/2]

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

default

order()

int order

(

)

const

Returns the order of the curve (1 for linear, 2 for quadratic, etc.).

rows()

Eigen::Index rows ( ) const

overridevirtual

Returns

The number of rows in the matrix returned by value().

Implements Trajectory< T >.

start_time()

T start_time ( ) const

overridevirtual

Implements Trajectory< T >.

value()

MatrixX<T> value ( const T &  time ) const

overridevirtual

Evaluates the curve at the given time.

Warning

If t does not lie in the range [start_time(), end_time()], the trajectory will silently be evaluated at the closest valid value of time to time. For example, value(-1) will return value(0) for a trajectory defined over [0, 1].

Implements Trajectory< T >.

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

• drake/common/trajectories/bezier_curve.h