Drake
Drake C++ Documentation Geometry Optimization

## Detailed Description

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

Warning
This feature is considered to be experimental and may change or be removed at any time, without any deprecation notice ahead of time.

### 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.

## Classes

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} We currently require the sets X to be bounded. More...

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

class  GraphOfConvexSets
GraphOfConvexSets 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  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  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...

## ◆ 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.

 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, options.enabled_ibex may have a significant impact on the runtime of the algorithm.
 ConvexSets drake::geometry::optimization::MakeIrisObstacles ( const QueryObject< double > & query_object, std::optional< FrameId > reference_frame = std::nullopt )