Detailed Description

Implements an ellipsoidal convex set represented as an affine scaling of the unit ball {Bu + center | |u|₂ ≤ 1}.

B must be a square matrix.

Compare this with an alternative parametrization of the ellipsoid: {x | (x-center)ᵀAᵀA(x-center) ≤ 1}, which utilizes a quadratic form. The two representations are related by B = A⁻¹ if A and B are invertible.

The quadratic form parametrization is implemented in Hyperellipsoid. It can represent unbounded sets, but not sets along a lower-dimensional affine subspace. The AffineBall parametrization can represent sets along a lower-dimensional affine subspace, but not unbounded sets.

An AffineBall can never be empty – it always contains its center. This includes the zero-dimensional case.

#include <drake/geometry/optimization/affine_ball.h>

Public Member Functions
	AffineBall ()
	Constructs a default (zero-dimensional, nonempty) set. More...

	AffineBall (const Eigen::Ref< const Eigen::MatrixXd > &B, const Eigen::Ref< const Eigen::VectorXd > &center)
	Constructs the ellipsoid from a transformation matrix B and translation center. More...

	AffineBall (const Hyperellipsoid &ellipsoid)
	Constructs an AffineBall from a Hyperellipsoid. More...

	~AffineBall () final

const Eigen::MatrixXd &	B () const
	Returns the affine transformation matrix B. More...

const Eigen::VectorXd &	center () const
	Returns the center of the ellipsoid. More...

template<typename Archive >
void	Serialize (Archive *a)
	Passes this object to an Archive. More...

bool	IsBounded (Parallelism parallelism=Parallelism::None()) const
	Every AffineBall is bounded by construction. More...

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

AffineBall &	operator= (const AffineBall &)=default

	AffineBall (AffineBall &&)=default

AffineBall &	operator= (AffineBall &&)=default

Public Member Functions inherited from ConvexSet
virtual	~ConvexSet ()

std::unique_ptr< ConvexSet >	Clone () const
	Creates a unique deep copy of this set. More...

int	ambient_dimension () const
	Returns the dimension of the vector space in which the elements of this set are evaluated. More...

bool	IntersectsWith (const ConvexSet &other) const
	Returns true iff the intersection between `this` and `other` is non-empty. More...

bool	IsBounded (Parallelism parallelism=Parallelism::None()) const
	Returns true iff the set is bounded, e.g., there exists an element-wise finite lower and upper bound for the set. More...

bool	IsEmpty () const
	Returns true iff the set is empty. More...

std::optional< Eigen::VectorXd >	MaybeGetPoint () const
	If this set trivially contains exactly one point, returns the value of that point. More...

std::optional< Eigen::VectorXd >	MaybeGetFeasiblePoint () const
	Returns a feasible point within this convex set if it is nonempty, and nullopt otherwise. More...

bool	PointInSet (const Eigen::Ref< const Eigen::VectorXd > &x, double tol=0) const
	Returns true iff the point x is contained in the set. More...

std::pair< VectorX< symbolic::Variable >, std::vector< solvers::Binding< solvers::Constraint > > >	AddPointInSetConstraints (solvers::MathematicalProgram *prog, const Eigen::Ref< const solvers::VectorXDecisionVariable > &vars) const
	Adds a constraint to an existing MathematicalProgram enforcing that the point defined by vars is inside the set. More...

std::vector< solvers::Binding< solvers::Constraint > >	AddPointInNonnegativeScalingConstraints (solvers::MathematicalProgram *prog, const Eigen::Ref< const solvers::VectorXDecisionVariable > &x, const symbolic::Variable &t) const
	Let S be this convex set. More...

std::vector< solvers::Binding< solvers::Constraint > >	AddPointInNonnegativeScalingConstraints (solvers::MathematicalProgram *prog, const Eigen::Ref< const Eigen::MatrixXd > &A, const Eigen::Ref< const Eigen::VectorXd > &b, const Eigen::Ref< const Eigen::VectorXd > &c, double d, const Eigen::Ref< const solvers::VectorXDecisionVariable > &x, const Eigen::Ref< const solvers::VectorXDecisionVariable > &t) const
	Let S be this convex set. More...

std::pair< std::unique_ptr< Shape >, math::RigidTransformd >	ToShapeWithPose () const
	Constructs a Shape and a pose of the set in the world frame for use in the SceneGraph geometry ecosystem. More...

double	CalcVolume () const
	Computes the exact volume for the convex set. More...

SampledVolume	CalcVolumeViaSampling (RandomGenerator *generator, const double desired_rel_accuracy=1e-2, const int max_num_samples=1e4) const
	Calculates an estimate of the volume of the convex set using sampling and performing Monte Carlo integration. More...

std::optional< std::pair< std::vector< double >, Eigen::MatrixXd > >	Projection (const Eigen::Ref< const Eigen::MatrixXd > &points) const
	Computes in the L₂ norm the distance and the nearest point in this convex set to every column of `points`. More...

bool	has_exact_volume () const
	Returns true if the exact volume can be computed for this convex set instance. More...

Static Public Member Functions
static AffineBall	MakeAxisAligned (const Eigen::Ref< const Eigen::VectorXd > &radius, const Eigen::Ref< const Eigen::VectorXd > &center)
	Constructs an axis-aligned AffineBall with the implicit form (x₀-c₀)²/r₀² + (x₁-c₁)²/r₁² + ... More...

static AffineBall	MakeHypersphere (double radius, const Eigen::Ref< const Eigen::VectorXd > &center)
	Constructs a hypersphere with `radius` and `center`. More...

static AffineBall	MakeUnitBall (int dim)
	Constructs the L₂-norm unit ball in `dim` dimensions, {x \| \|x\|₂ <= 1 }. More...

static AffineBall	MinimumVolumeCircumscribedEllipsoid (const Eigen::Ref< const Eigen::MatrixXd > &points, double rank_tol=1e-6)
	Constructs the minimum-volume ellipsoid which contains all of the given points. More...

static AffineBall	MakeAffineBallFromLineSegment (const Eigen::Ref< const Eigen::VectorXd > &x_1, const Eigen::Ref< const Eigen::VectorXd > &x_2, const double epsilon=1e-3)
	Constructs an affine ball such that its main diameter is the line segment from `x_1` to `x_2`, and the length of all other diameters are 2 * `epsilon`. More...

Additional Inherited Members
Protected Member Functions inherited from ConvexSet
	ConvexSet (int ambient_dimension, bool has_exact_volume)
	For use by derived classes to construct a ConvexSet. More...

template<typename Archive >
void	Serialize (Archive *a)
	Implements non-virtual base class serialization. More...

virtual std::optional< bool >	DoIsBoundedShortcutParallel (Parallelism) const
	Non-virtual interface implementation for DoIsBoundedShortcutParallel(). More...

virtual std::vector< std::optional< double > >	DoProjectionShortcut (const Eigen::Ref< const Eigen::MatrixXd > &points, EigenPtr< Eigen::MatrixXd > projected_points) const
	Non-virtual interface implementation for DoProjectionShortcut(). More...

virtual bool	DoPointInSet (const Eigen::Ref< const Eigen::VectorXd > &x, double tol) const
	Non-virtual interface implementation for PointInSet(). More...

std::optional< symbolic::Variable >	HandleZeroAmbientDimensionConstraints (solvers::MathematicalProgram prog, const ConvexSet &set, std::vector< solvers::Binding< solvers::Constraint >> constraints) const
	Instances of subclasses such as CartesianProduct and MinkowskiSum can have constituent sets with zero ambient dimension, which much be handled in a special manner when calling methods such as DoAddPointInSetConstraints. More...

	ConvexSet (const ConvexSet &)=default

ConvexSet &	operator= (const ConvexSet &)=default

	ConvexSet (ConvexSet &&)=default

ConvexSet &	operator= (ConvexSet &&)=default

Static Protected Member Functions inherited from ConvexSet
static std::unique_ptr< ConvexSet >	AffineHullShortcut (const ConvexSet &self, std::optional< double > tol)
	When there is a more efficient strategy to compute the affine hull of this set, returns affine hull as an AffineSubspace. More...

Constructor & Destructor Documentation

◆ AffineBall() [1/5]

AffineBall ( const AffineBall & )

default

◆ AffineBall() [2/5]

AffineBall ( AffineBall && )

default

◆ AffineBall() [3/5]

AffineBall ( )

Constructs a default (zero-dimensional, nonempty) set.

◆ AffineBall() [4/5]

AffineBall	(	const Eigen::Ref< const Eigen::MatrixXd > &	B,
		const Eigen::Ref< const Eigen::VectorXd > &	center
	)

Constructs the ellipsoid from a transformation matrix B and translation center.

B describes the linear transformation that is applied to the unit ball in order to produce the ellipsoid, and center describes the translation of the center of the ellipsoid from the origin.

Precondition: B.rows() == B.cols().; B.cols() == center.size().

◆ AffineBall() [5/5]

AffineBall ( const Hyperellipsoid & ellipsoid )

explicit

Constructs an AffineBall from a Hyperellipsoid.

Precondition: ellipsoid.IsBounded().

◆ ~AffineBall()

~AffineBall ( )

final

Member Function Documentation

◆ B()

const Eigen::MatrixXd& B ( ) const

Returns the affine transformation matrix B.

◆ center()

const Eigen::VectorXd& center ( ) const

Returns the center of the ellipsoid.

◆ IsBounded()

bool IsBounded

Every AffineBall is bounded by construction.

Parameters

parallelism Ignored – no parallelization is used.

Note: See parent class's documentation for more details.

◆ MakeAffineBallFromLineSegment()

static AffineBall MakeAffineBallFromLineSegment	(	const Eigen::Ref< const Eigen::VectorXd > &	x_1,
		const Eigen::Ref< const Eigen::VectorXd > &	x_2,
		const double	epsilon = `1e-3`
	)

static

Constructs an affine ball such that its main diameter is the line segment from x_1 to x_2, and the length of all other diameters are 2 * epsilon.

Precondition: x_1.size() == x_2.size().; epsilon >= 0.

Exceptions

std::runtime_error if ‖x_1 - x_2‖₂ is less than 1e-9.

◆ MakeAxisAligned()

static AffineBall MakeAxisAligned	(	const Eigen::Ref< const Eigen::VectorXd > &	radius,
		const Eigen::Ref< const Eigen::VectorXd > &	center
	)

static

Constructs an axis-aligned AffineBall with the implicit form (x₀-c₀)²/r₀² + (x₁-c₁)²/r₁² + ...

(x_N - c_N)²/r_N² ≤ 1, where c is shorthand for center and r is shorthand for radius.
Precondition
radius.size() == center.size().

radius[i] >= 0, for all i.

◆ MakeHypersphere()

static AffineBall MakeHypersphere	(	double	radius,
		const Eigen::Ref< const Eigen::VectorXd > &	center
	)

static

Constructs a hypersphere with radius and center.

Precondition: radius >= 0.

◆ MakeUnitBall()

static AffineBall MakeUnitBall ( int dim )

static

Constructs the L₂-norm unit ball in dim dimensions, {x | |x|₂ <= 1 }.

Precondition: dim >= 0.

◆ MinimumVolumeCircumscribedEllipsoid()

static AffineBall MinimumVolumeCircumscribedEllipsoid	(	const Eigen::Ref< const Eigen::MatrixXd > &	points,
		double	rank_tol = `1e-6`
	)

static

Constructs the minimum-volume ellipsoid which contains all of the given points.

This is commonly referred to as the outer Löwner-John ellipsoid.

If all of the points lie along a proper affine subspace, this method instead computes the minimum-k-volume ellipsoid, where k is the affine dimension of the convex hull of points.

Parameters

points	is a d-by-n matrix, where d is the ambient dimension and each column represents one point.
rank_tol	the tolerance used to detect if the data lies in an affine subspace. The affine ball is computed in the subspace spanned by the left singular vectors of the data matrix whose associated singular values are larger than `rank_tol` * `max_singular_value`. The default is 1e-6 to be compatible with common solver tolerances.

Exceptions

std::exception	if the MathematicalProgram fails to solve. This can happen due to numerical issues caused by `rank_tol` being set too low.
std::exception	if points includes NaNs or infinite values.

Precondition: points.rows() >= 1.; points.cols() >= 1.

◆ operator=() [1/2]

AffineBall& operator= ( AffineBall && )

default

◆ operator=() [2/2]

AffineBall& operator= ( const AffineBall & )

default

◆ Serialize()

void Serialize ( Archive * a )

Passes this object to an Archive.

Refer to YAML Serialization for background.

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

drake/geometry/optimization/affine_ball.h

Detailed Description

Public Member Functions

Static Public Member Functions

Additional Inherited Members

Constructor & Destructor Documentation

◆ AffineBall() [1/5]

◆ AffineBall() [2/5]

◆ AffineBall() [3/5]

◆ AffineBall() [4/5]

◆ AffineBall() [5/5]

◆ ~AffineBall()

Member Function Documentation

◆ B()

◆ center()

◆ IsBounded()

◆ MakeAffineBallFromLineSegment()

◆ MakeAxisAligned()

◆ MakeHypersphere()

◆ MakeUnitBall()

◆ MinimumVolumeCircumscribedEllipsoid()

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ Serialize()