Detailed Description

Optional argument for constructing CspaceFreePolytopeBase.

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

Public Member Functions
	Options ()

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

Public Attributes
bool	with_cross_y {false}
	For non-polytopic collision geometries, we will impose a matrix-sos constraint X(s) being psd, with a slack indeterminates y, such that the polynomial. More...

Constructor & Destructor Documentation

◆ Options()

Options ( )

Member Function Documentation

◆ Serialize()

void Serialize ( Archive * a )

Passes this object to an Archive.

Refer to YAML Serialization for background.

Member Data Documentation

◆ with_cross_y

bool with_cross_y {false}

For non-polytopic collision geometries, we will impose a matrix-sos constraint X(s) being psd, with a slack indeterminates y, such that the polynomial.

p(s, y) = ⌈ 1 ⌉ᵀ * X(s) * ⌈ 1 ⌉
          ⌊ y ⌋           ⌊ y ⌋

is positive. This p(s, y) polynomial doesn't contain the cross term of y (namely it doesn't have y(i)*y(j), i≠j). When we select the monomial basis for this polynomial, we can also exclude the cross term of y in the monomial basis.

To illustrate the idea, let's consider the following toy example: if we want to certify that a(0) + a(1)*y₀ + a(2)*y₁ + a(3)*y₀² + a(4)*y₁² is positive (this polynomial doesn't have the cross term y₀*y₁), we can write it as

⌈ 1⌉ᵀ * A₀ * ⌈ 1⌉ + ⌈ 1⌉ᵀ * A₁ * ⌈ 1⌉
⌊y₀⌋         ⌊y₀⌋   ⌊y₁⌋         ⌊y₁⌋

with two small psd matrices A₀, A₁ Instead of

⌈ 1⌉ᵀ * A * ⌈ 1⌉
|y₀|        |y₀|
⌊y₁⌋        ⌊y₁⌋

with one large psd matrix A. The first parameterization won't have the cross term y₀*y₁ by construction, while the second parameterization requires imposing extra constraints on certain off-diagonal terms in A so that the cross term vanishes.

If we set with_cross_y = false, then we will use the monomial basis that doesn't generate cross terms of y, leading to smaller size sos problems. If we set with_cross_y = true, then we will use the monomial basis that will generate cross terms of y, causing larger size sos problems, but possibly able to certify a larger C-space polytope.

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

drake/geometry/optimization/cspace_free_polytope_base.h