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mixed_integer_rotation_constraint_internal_test.cc
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mixed_integer_rotation_constraint_internal_test.cc
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/* clang-format off to disable clang-format-includes */
#include "drake/solvers/mixed_integer_rotation_constraint_internal.h"
/* clang-format on */
#include <gtest/gtest.h>
#include "drake/common/test_utilities/eigen_matrix_compare.h"
#include "drake/math/random_rotation.h"
#include "drake/solvers/mathematical_program.h"
#include "drake/solvers/solve.h"
using Eigen::Vector3d;
namespace drake {
namespace solvers {
namespace {
void CompareIntersectionResults(const std::vector<Vector3d>& desired,
const std::vector<Vector3d>& actual) {
EXPECT_EQ(desired.size(), actual.size());
Eigen::Matrix<bool, Eigen::Dynamic, 1> used =
Eigen::Matrix<bool, Eigen::Dynamic, 1>::Constant(desired.size(), false);
double tol = 1e-8;
for (int i = 0; i < static_cast<int>(desired.size()); i++) {
// need not be in the same order.
bool found_match = false;
for (int j = 0; j < static_cast<int>(desired.size()); j++) {
if (used(j)) continue;
if ((desired[i] - actual[j]).lpNorm<2>() < tol) {
used(j) = true;
found_match = true;
break;
}
}
EXPECT_TRUE(found_match);
}
}
// For 2 binary variable per half axis, we know it cuts the first orthant into
// 7 regions. 3 of these regions have 4 co-planar vertices; 3 of these regions
// have 4 non-coplanar vertices, and one region has 3 vertices.
GTEST_TEST(RotationTest, TestAreAllVerticesCoPlanar) {
Eigen::Vector3d n;
double d;
// 4 co-planar vertices.
std::array<std::pair<Eigen::Vector3d, Eigen::Vector3d>, 3> bmin_bmax_coplanar{
{{Eigen::Vector3d(0.5, 0.5, 0), Eigen::Vector3d(1, 1, 0.5)},
{Eigen::Vector3d(0.5, 0, 0.5), Eigen::Vector3d(1, 0.5, 1)},
{Eigen::Vector3d(0, 0.5, 0.5), Eigen::Vector3d(0.5, 1, 1)}}};
for (const auto& bmin_bmax : bmin_bmax_coplanar) {
auto pts = internal::ComputeBoxEdgesAndSphereIntersection(bmin_bmax.first,
bmin_bmax.second);
EXPECT_TRUE(internal::AreAllVerticesCoPlanar(pts, &n, &d));
for (int i = 0; i < 4; ++i) {
EXPECT_NEAR(n.norm(), 1, 1E-10);
EXPECT_NEAR(n.dot(pts[i]), d, 1E-10);
EXPECT_TRUE((n.array() > 0).all());
}
}
// 4 non co-planar vertices.
std::array<std::pair<Eigen::Vector3d, Eigen::Vector3d>, 3>
bmin_bmax_non_coplanar{
{{Eigen::Vector3d(0.5, 0, 0), Eigen::Vector3d(1, 0.5, 0.5)},
{Eigen::Vector3d(0, 0.5, 0), Eigen::Vector3d(0.5, 1, 0.5)},
{Eigen::Vector3d(0, 0, 0.5), Eigen::Vector3d(0.5, 0.5, 1)}}};
for (const auto& bmin_bmax : bmin_bmax_non_coplanar) {
auto pts = internal::ComputeBoxEdgesAndSphereIntersection(bmin_bmax.first,
bmin_bmax.second);
EXPECT_FALSE(internal::AreAllVerticesCoPlanar(pts, &n, &d));
EXPECT_TRUE(CompareMatrices(n, Eigen::Vector3d::Zero()));
EXPECT_EQ(d, 0);
}
// 3 vertices
Eigen::Vector3d bmin(0.5, 0.5, 0.5);
Eigen::Vector3d bmax(1, 1, 1);
auto pts = internal::ComputeBoxEdgesAndSphereIntersection(bmin, bmax);
EXPECT_TRUE(internal::AreAllVerticesCoPlanar(pts, &n, &d));
EXPECT_TRUE(CompareMatrices(n, Eigen::Vector3d::Constant(1.0 / std::sqrt(3)),
1E-10, MatrixCompareType::absolute));
EXPECT_NEAR(pts[0].dot(n), d, 1E-10);
}
void CheckInnerFacets(const std::vector<Vector3d>& pts) {
// Compute the inner facets of the convex hull of pts. Make sure for each
// facet, there are three points on the facet, and the facet points
// outward from the origin.
Eigen::Matrix<double, Eigen::Dynamic, 3> A;
Eigen::VectorXd b;
internal::ComputeInnerFacetsForBoxSphereIntersection(pts, &A, &b);
for (int i = 0; i < A.rows(); ++i) {
for (const auto& pt : pts) {
EXPECT_LE((A.row(i) * pt)(0), b(i) + 1E-10);
}
EXPECT_NEAR(A.row(i).norm(), 1, 1E-10);
// A.row(i) is the inverse of the facet normal, that points outward from the
// origin.
EXPECT_TRUE((A.row(i).array() <= 0).all());
int num_pts_on_plane = 0;
for (const auto& pt : pts) {
if (std::abs((A.row(i) * pt)(0) - b(i)) < 1E-10) {
++num_pts_on_plane;
}
}
EXPECT_GE(num_pts_on_plane, 3);
}
}
void CompareHalfspaceRelaxation(const std::vector<Vector3d> &pts) {
// Computes a possibly less tight n and d analytically. For each triangle with
// vertices pts[i], pts[j] and pts[k], determine if the halfspace coinciding
// with the triangle is a cutting plane (namely all vertices in pts are on one
// side of the halfspace). Compute the farthest distance from the cutting
// planes to the origin.
DRAKE_DEMAND(pts.size() >= 3);
double d = -1;
for (int i = 0; i < static_cast<int>(pts.size()); ++i) {
for (int j = i + 1; j < static_cast<int>(pts.size()); ++j) {
for (int k = j + 1; k < static_cast<int>(pts.size()); ++k) {
// Find the normal of the triangle.
Eigen::Vector3d normal_tmp = (pts[k] - pts[i]).cross(pts[j] - pts[i]);
normal_tmp.normalize();
if (normal_tmp(0) < 0) {
normal_tmp = -normal_tmp;
}
double d_tmp = normal_tmp.transpose() * pts[i];
bool is_cutting_plane = true;
for (const auto &pt : pts) {
if (pt.transpose() * normal_tmp < d_tmp - 1E-10) {
is_cutting_plane = false;
break;
}
}
if (is_cutting_plane) {
d = std::max(d, d_tmp);
}
}
}
}
Eigen::Vector3d n_expected;
double d_expected;
internal::ComputeHalfSpaceRelaxationForBoxSphereIntersection(pts, &n_expected,
&d_expected);
if (pts.size() == 3) {
EXPECT_NEAR(d_expected, d, 1E-6);
}
EXPECT_GE(d_expected, d - 1E-8);
for (const auto &pt : pts) {
EXPECT_GE(pt.transpose() * n_expected - d_expected, -1E-6);
}
}
GTEST_TEST(RotationTest, TestHalfSpaceRelaxation) {
// In some cases, the half space relaxation can be computed analytically. We
// compare the analytical result, against
// ComputeHalfSpaceRelaxationForBoxSphereIntersection()
std::vector<Eigen::Vector3d> pts;
Eigen::Vector3d n;
double d;
// For three points case, the half space relaxation is just the plane
// coinciding with the three points.
pts.emplace_back(1.0 / 3.0, 2.0 / 3.0, 2.0 / 3.0);
pts.emplace_back(2.0 / 3.0, 1.0 / 3.0, 2.0 / 3.0);
pts.emplace_back(2.0 / 3.0, 2.0 / 3.0, 1.0 / 3.0);
internal::ComputeHalfSpaceRelaxationForBoxSphereIntersection(pts, &n, &d);
EXPECT_TRUE(CompareMatrices(n, Eigen::Vector3d::Constant(1 / std::sqrt(3)),
10 * std::numeric_limits<double>::epsilon(),
MatrixCompareType::absolute));
EXPECT_NEAR(d, std::sqrt(3) * 5 / 9, 1E-10);
// Four points, symmetric about the plane x = y. The tightest half space
// relaxation is not the plane coinciding with any three of the points, but
// just coinciding with two of the points.
pts.clear();
// The first two points are on the x = y plane.
pts.emplace_back(1.0 / 3.0, 1.0 / 3.0, std::sqrt(7) / 3.0);
pts.emplace_back(2.0 / 3.0, 2.0 / 3.0, 1.0 / 3.0);
// The last two points are symmetric about the x = y plane.
pts.emplace_back(1.0 / 3.0, 2.0 / 3.0, 2.0 / 3.0);
pts.emplace_back(2.0 / 3.0, 1.0 / 3.0, 2.0 / 3.0);
internal::ComputeHalfSpaceRelaxationForBoxSphereIntersection(pts, &n, &d);
// The normal vector should be on the x = y plane.
EXPECT_NEAR(n(0), n(1), 1E-8);
EXPECT_NEAR(n.dot(pts[0]), d, 1E-8);
EXPECT_NEAR(n.dot(pts[1]), d, 1E-8);
}
GTEST_TEST(RotationTest, TestInnerFacetsAndHalfSpace) {
// We show that the inner facet is tighter than the half space for some case.
// To this end, we show that for a box [0 0.5 0] <= x <= [0.5 1 0.5], there is
// some point that does not satisfy the inner facets constraint A*x<=b, but
// satisfies the half space constraint nᵀ*x>=d.
const Eigen::Vector3d bmin(0, 0.5, 0);
const Eigen::Vector3d bmax(0.5, 1, 0.5);
const auto intersection_pts =
internal::ComputeBoxEdgesAndSphereIntersection(bmin, bmax);
DRAKE_DEMAND(intersection_pts.size() == 4);
Eigen::Vector3d n;
double d;
internal::ComputeHalfSpaceRelaxationForBoxSphereIntersection(intersection_pts,
&n, &d);
Eigen::Matrix<double, Eigen::Dynamic, 3> A;
Eigen::VectorXd b;
internal::ComputeInnerFacetsForBoxSphereIntersection(intersection_pts, &A,
&b);
// Now form the optimization program
// A.row(i) * x > b(i) + epsilon for at least one i
// nᵀ * x >= d
// bmin <= x <= bmax
// We will show that there is a feasible solution to this program.
MathematicalProgram prog;
auto x = prog.NewContinuousVariables<3>("x");
auto z = prog.NewBinaryVariables(b.rows(), "z");
for (int i = 0; i < b.rows(); ++i) {
prog.AddLinearConstraint((A.row(i) * x)(0) >= b(i) + 1E-5 + (z(i) - 1) * 2);
}
prog.AddLinearConstraint(z.cast<symbolic::Expression>().sum() >= 1);
prog.AddLinearConstraint(n.dot(x) >= d);
prog.AddBoundingBoxConstraint(bmin, bmax, x);
const MathematicalProgramResult result = Solve(prog);
EXPECT_TRUE(result.is_success());
}
// Test a number of closed-form solutions for the intersection of a box in the
// positive orthant with the unit circle.
GTEST_TEST(RotationTest, TestIntersectBoxWithCircle) {
std::vector<Vector3d> desired;
// Entire first octant.
Vector3d box_min(0, 0, 0);
Vector3d box_max(1, 1, 1);
desired.push_back(Vector3d(1, 0, 0));
desired.push_back(Vector3d(0, 1, 0));
desired.push_back(Vector3d(0, 0, 1));
CompareIntersectionResults(
desired,
internal::ComputeBoxEdgesAndSphereIntersection(box_min, box_max));
CompareHalfspaceRelaxation(desired);
CheckInnerFacets(desired);
// Lifts box bottom (in z). Still has 3 solutions.
box_min << 0, 0, 1.0 / 3.0;
desired[0] << std::sqrt(8) / 3.0, 0, 1.0 / 3.0;
desired[1] << 0, std::sqrt(8) / 3.0, 1.0 / 3.0;
CompareIntersectionResults(
desired,
internal::ComputeBoxEdgesAndSphereIntersection(box_min, box_max));
CompareHalfspaceRelaxation(desired);
CheckInnerFacets(desired);
// Lowers box top (in z). Now we have four solutions.
box_max << 1, 1, 2.0 / 3.0;
desired[2] << std::sqrt(5) / 3.0, 0, 2.0 / 3.0;
desired.push_back(Vector3d(0, std::sqrt(5) / 3.0, 2.0 / 3.0));
CompareIntersectionResults(
desired,
internal::ComputeBoxEdgesAndSphereIntersection(box_min, box_max));
CompareHalfspaceRelaxation(desired);
CheckInnerFacets(desired);
// Gets a different four edges by shortening the box (in x).
box_max(0) = .5;
desired[0] << .5, std::sqrt(23.0) / 6.0, 1.0 / 3.0;
desired[2] << .5, std::sqrt(11.0) / 6.0, 2.0 / 3.0;
CompareIntersectionResults(
desired,
internal::ComputeBoxEdgesAndSphereIntersection(box_min, box_max));
CompareHalfspaceRelaxation(desired);
CheckInnerFacets(desired);
// Now three edges again as we shorten the box (in y).
box_max(1) = .6;
desired.pop_back();
desired[0] << .5, std::sqrt(11.0) / 6.0, 2.0 / 3.0;
desired[1] << 2 * std::sqrt(11.0) / 15.0, .6, 2.0 / 3.0;
desired[2] << .5, .6, std::sqrt(39.0) / 10.0;
CompareIntersectionResults(
desired,
internal::ComputeBoxEdgesAndSphereIntersection(box_min, box_max));
CompareHalfspaceRelaxation(desired);
CheckInnerFacets(desired);
// All four intersections are on the vertical edges.
box_min << 1.0 / 3.0, 1.0 / 3.0, 0;
box_max << 2.0 / 3.0, 2.0 / 3.0, 1;
desired[0] << 1.0 / 3.0, 1.0 / 3.0, std::sqrt(7.0) / 3.0;
desired[1] << 2.0 / 3.0, 1.0 / 3.0, 2.0 / 3.0;
desired[2] << 1.0 / 3.0, 2.0 / 3.0, 2.0 / 3.0;
desired.push_back(Vector3d(2.0 / 3.0, 2.0 / 3.0, 1.0 / 3.0));
CompareIntersectionResults(
desired,
internal::ComputeBoxEdgesAndSphereIntersection(box_min, box_max));
CompareHalfspaceRelaxation(desired);
CheckInnerFacets(desired);
// box_max right on the unit sphere.
box_max << 1.0 / 3.0, 2.0 / 3.0, 2.0 / 3.0;
box_min << 0, 1.0 / 3.0, 0;
// Should return just the single point.
desired.erase(desired.begin() + 1, desired.end());
desired[0] = box_max;
CompareIntersectionResults(
desired,
internal::ComputeBoxEdgesAndSphereIntersection(box_min, box_max));
CheckInnerFacets(desired);
// Multiple vertices are on the sphere.
box_min << 1.0 / 3.0, 1.0 / 3.0, 1.0 / 3.0;
box_max << 2.0 / 3.0, 2.0 / 3.0, 2.0 / 3.0;
desired.clear();
desired.push_back(Eigen::Vector3d(1.0 / 3, 2.0 / 3, 2.0 / 3));
desired.push_back(Eigen::Vector3d(2.0 / 3, 1.0 / 3, 2.0 / 3));
desired.push_back(Eigen::Vector3d(2.0 / 3, 2.0 / 3, 1.0 / 3));
CompareIntersectionResults(
desired,
internal::ComputeBoxEdgesAndSphereIntersection(box_min, box_max));
CheckInnerFacets(desired);
// Six intersections.
box_min = Eigen::Vector3d::Constant(1.0 / 3.0);
box_max = Eigen::Vector3d::Constant(sqrt(6) / 3.0);
desired.clear();
// The intersecting points are the 6 permutations of
// (1.0 / 3.0, sqrt(2) / 3.0, sqrt(6) / 3.0)
desired.resize(6);
desired[0] << 1.0 / 3.0, sqrt(2) / 3.0, sqrt(6) / 3.0;
desired[1] << 1.0 / 3.0, sqrt(6) / 3.0, sqrt(2) / 3.0;
desired[2] << sqrt(2) / 3.0, 1.0 / 3.0, sqrt(6) / 3.0;
desired[3] << sqrt(2) / 3.0, sqrt(6) / 3.0, 1.0 / 3.0;
desired[4] << sqrt(6) / 3.0, 1.0 / 3.0, sqrt(2) / 3.0;
desired[5] << sqrt(6) / 3.0, sqrt(2) / 3.0, 1.0 / 3.0;
CompareIntersectionResults(
desired,
internal::ComputeBoxEdgesAndSphereIntersection(box_min, box_max));
CompareHalfspaceRelaxation(desired);
CheckInnerFacets(desired);
}
} // namespace
} // namespace solvers
} // namespace drake