Detailed Description

Provides an abstraction for reasoning about geometry in optimization problems, and using optimization problems to solve geometry problems.

Relationship to other components in Drake.

SceneGraph handles many types of geometry (see Shape). It is specialized to 3D and puts a strong emphasis on efficient implementation of a particular subset of geometry queries, like collision detection and signed-distance queries. SceneGraph also provides a lot of valuable tools for content management, including parsing geometries from file (via multibody::Parser) and Role.

MathematicalProgram has many relevant solvers::Cost / solvers::Constraint for reasoning about geometry (e.g., the LorentzCone or even LinearConstraint). The class and methods in this group add a level of modeling power above these individual constraints (there are many different types of constraints one would write given various optimization on these sets).

The geometry::optimization tools support:

The ability to model sets in arbitrary dimensions, not just 3D.
Many advanced queries (intersections, Minkowski sum, set containment, etc).
The ability to use sets in mathematical programs
The ability to use mathematical programs to perform operations on sets.
The ability to include the geometry from a SceneGraph into a MathematicalProgram.
The ability to use SceneGraph queries for some of the geometry operations (signed distance, collision), when the set is in 3D.
The ability to use SceneGraph for rendering the geometries when they are in 3D.

Modules
	Geodesic Convexity
	A robot that has revolute joints without any limits has an inherently non-Euclidean configuration space, but one can still consider "geodesically-convex" sets, akin to convex sets in Euclidean space.

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

class	AffineSubspace
	An affine subspace (also known as a "flat", a "linear variety", or a "linear manifold") is a vector subspace of some Euclidean space, potentially translated so as to not pass through the origin. More...

class	CartesianProduct
	The Cartesian product of convex sets is a convex set: S = X₁ × X₂ × ⋯ × Xₙ = {(x₁, x₂, ..., xₙ) \| x₁ ∈ X₁, x₂ ∈ X₂, ..., xₙ ∈ Xₙ}. More...

class	ConvexHull
	Implements the convex hull of a set of convex sets. More...

class	ConvexSet
	Abstract base class for defining a convex set. More...

class	GraphOfConvexSets
	GraphOfConvexSets (GCS) implements the design pattern and optimization problems first introduced in the paper "Shortest Paths in Graphs of Convex Sets". More...

class	HPolyhedron
	Implements a polyhedral convex set using the half-space representation: `{x\| A x ≤ b}`. More...

class	Hyperellipsoid
	Implements an ellipsoidal convex set represented by the quadratic form `{x \| (x-center)ᵀAᵀA(x-center) ≤ 1}`. More...

class	Hyperrectangle
	Axis-aligned hyperrectangle in Rᵈ defined by its lower bounds and upper bounds as {x\| lb ≤ x ≤ ub}. More...

class	ImplicitGraphOfConvexSets
	A base class to define the interface to an implicit graph of convex sets. More...

class	Intersection
	A convex set that represents the intersection of multiple sets: S = X₁ ∩ X₂ ∩ ... More...

struct	IrisOptions
	Configuration options for the IRIS algorithm. More...

class	MinkowskiSum
	A convex set that represents the Minkowski sum of multiple sets: S = X₁ ⨁ X₂ ⨁ ... More...

class	Point
	A convex set that contains exactly one element. More...

class	Spectrahedron
	Implements a spectrahedron (the feasible set of a semidefinite program). More...

class	VPolytope
	A polytope described using the vertex representation. More...

Functions
HPolyhedron	Iris (const ConvexSets &obstacles, const Eigen::Ref< const Eigen::VectorXd > &sample, const HPolyhedron &domain, const IrisOptions &options=IrisOptions())
	The IRIS (Iterative Region Inflation by Semidefinite programming) algorithm, as described in. More...

ConvexSets	MakeIrisObstacles (const QueryObject< double > &query_object, std::optional< FrameId > reference_frame=std::nullopt)
	Constructs ConvexSet representations of obstacles for IRIS in 3D using the geometry from a SceneGraph QueryObject. More...

HPolyhedron	IrisInConfigurationSpace (const multibody::MultibodyPlant< double > &plant, const systems::Context< double > &context, const IrisOptions &options=IrisOptions())
	A variation of the Iris (Iterative Region Inflation by Semidefinite programming) algorithm which finds collision-free regions in the configuration space of `plant`. More...

void	SetEdgeContainmentTerminationCondition (IrisOptions *iris_options, const Eigen::Ref< const Eigen::VectorXd > &x_1, const Eigen::Ref< const Eigen::VectorXd > &x_2, const double epsilon=1e-3, const double tol=1e-6)
	Modifies the `iris_options` to facilitate finding a region that contains the edge between x_1 and x_2. More...

Function Documentation

◆ Iris()

HPolyhedron drake::geometry::optimization::Iris	(	const ConvexSets &	obstacles,
		const Eigen::Ref< const Eigen::VectorXd > &	sample,
		const HPolyhedron &	domain,
		const IrisOptions &	options = `IrisOptions()`
	)

The IRIS (Iterative Region Inflation by Semidefinite programming) algorithm, as described in.

R. L. H. Deits and R. Tedrake, “Computing large convex regions of obstacle-free space through semidefinite programming,” Workshop on the Algorithmic Fundamentals of Robotics, Istanbul, Aug. 2014. http://groups.csail.mit.edu/robotics-center/public_papers/Deits14.pdf

This algorithm attempts to locally maximize the volume of a convex polytope representing obstacle-free space given a sample point and list of convex obstacles. Rather than compute the volume of the polytope directly, the algorithm maximizes the volume of an inscribed ellipsoid. It alternates between finding separating hyperplanes between the ellipsoid and the obstacles and then finding a new maximum-volume inscribed ellipsoid.

Parameters

obstacles	is a vector of convex sets representing the occupied space.
sample	provides a point in the space; the algorithm is initialized using a tiny sphere around this point. The algorithm is only guaranteed to succeed if this sample point is collision free (outside of all obstacles), but in practice the algorithm can often escape bad initialization (assuming the require_sample_point_is_contained option is false).
domain	describes the total region of interest; computed IRIS regions will be inside this domain. It must be bounded, and is typically a simple bounding box (e.g. from HPolyhedron::MakeBox).

The obstacles, sample, and the domain must describe elements in the same ambient dimension (but that dimension can be any positive integer).

◆ IrisInConfigurationSpace()

HPolyhedron drake::geometry::optimization::IrisInConfigurationSpace	(	const multibody::MultibodyPlant< double > &	plant,
		const systems::Context< double > &	context,
		const IrisOptions &	options = `IrisOptions()`
	)

A variation of the Iris (Iterative Region Inflation by Semidefinite programming) algorithm which finds collision-free regions in the configuration space of plant.

See also: Iris for details on the original algorithm. This variant uses nonlinear optimization (instead of convex optimization) to find collisions in configuration space; each potential collision is probabilistically "certified" by restarting the nonlinear optimization from random initial seeds inside the candidate IRIS region until it fails to find a collision in options.num_collision_infeasible_samples consecutive attempts.

This method constructs a single Iris region in the configuration space of plant.

See also: planning::IrisInConfigurationSpaceFromCliqueCover for a method to automatically cover the configuration space with multiple Iris regions.

Parameters

plant	describes the kinematics of configuration space. It must be connected to a SceneGraph in a systems::Diagram.
context	is a context of the `plant`. The context must have the positions of the plant set to the initialIRIS seed configuration.
options	provides additional configuration options. In particular, increasing `options.num_collision_infeasible_samples` increases the chances that the IRIS regions are collision free but can also significantly increase the run-time of the algorithm. The same goes for `options.num_additional_constraints_infeasible_samples`.

Exceptions

std::exception	if the sample configuration in `context` is infeasible.
std::exception	if termination_func is invalid on the domain. See IrisOptions.termination_func for more details.

◆ MakeIrisObstacles()

ConvexSets drake::geometry::optimization::MakeIrisObstacles	(	const QueryObject< double > &	query_object,
		std::optional< FrameId >	reference_frame = `std::nullopt`
	)

Constructs ConvexSet representations of obstacles for IRIS in 3D using the geometry from a SceneGraph QueryObject.

All geometry in the scene with a proximity role, both anchored and dynamic, are consider to be fixed obstacles frozen in the poses captured in the context used to create the QueryObject.

When multiple representations are available for a particular geometry (e.g. a Box can be represented as either an HPolyhedron or a VPolytope), then this method will prioritize the representation that we expect is most performant for the current implementation of the IRIS algorithm.

◆ SetEdgeContainmentTerminationCondition()

void drake::geometry::optimization::SetEdgeContainmentTerminationCondition	(	IrisOptions *	iris_options,
		const Eigen::Ref< const Eigen::VectorXd > &	x_1,
		const Eigen::Ref< const Eigen::VectorXd > &	x_2,
		const double	epsilon = `1e-3`,
		const double	tol = `1e-6`
	)

Modifies the iris_options to facilitate finding a region that contains the edge between x_1 and x_2.

It sets iris_options.starting_ellipse to be a hyperellipsoid that contains the edge, is centered at the midpoint of the edge and extends in other directions by epsilon. It also sets iris_options.termination_func such that IRIS iterations terminate when the edge is no longer contained in the IRIS region with tolerance tol.