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global_inverse_kinematics.cc
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global_inverse_kinematics.cc
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#include "drake/multibody/inverse_kinematics/global_inverse_kinematics.h"
#include <array>
#include <limits>
#include <stack>
#include <string>
#include "drake/common/eigen_types.h"
#include "drake/common/scope_exit.h"
#include "drake/math/roll_pitch_yaw.h"
#include "drake/math/rotation_matrix.h"
#include "drake/multibody/tree/revolute_joint.h"
#include "drake/solvers/rotation_constraint.h"
#include "drake/solvers/solve.h"
using Eigen::Matrix3d;
using Eigen::Vector3d;
using std::string;
using drake::math::RigidTransformd;
using drake::solvers::VectorDecisionVariable;
using drake::symbolic::Expression;
namespace drake {
namespace multibody {
namespace {
std::map<BodyIndex, JointIndex> GetBodyToJointMap(
const MultibodyPlant<double>& plant) {
// First loop through each joint, stores the map from the child body to the
// joint.
std::map<BodyIndex, JointIndex> body_to_joint_map;
for (JointIndex joint_index{0}; joint_index < plant.num_joints();
++joint_index) {
body_to_joint_map.emplace(plant.get_joint(joint_index).child_body().index(),
joint_index);
}
return body_to_joint_map;
}
std::unordered_set<BodyIndex> GetWeldToWorldBodyIndexSet(
const MultibodyPlant<double>& plant) {
const std::vector<const Body<double>*> weld_to_world_bodies =
plant.GetBodiesWeldedTo(plant.world_body());
std::unordered_set<BodyIndex> weld_to_world_body_index_set;
for (const auto weld_body : weld_to_world_bodies) {
weld_to_world_body_index_set.insert(weld_body->index());
}
return weld_to_world_body_index_set;
}
bool IsWeld(const Joint<double>& joint) {
const bool is_weld = !joint.can_rotate() && !joint.can_translate() &&
joint.num_positions() == 0 &&
joint.num_velocities() == 0;
if (is_weld) {
DRAKE_THROW_UNLESS(dynamic_cast<const WeldJoint<double>*>(&joint) !=
nullptr);
}
return is_weld;
}
bool IsRevolute(const Joint<double>& joint) {
const bool is_revolute = joint.can_rotate() && !joint.can_translate() &&
joint.num_positions() == 1 &&
joint.num_velocities() == 1;
if (is_revolute) {
DRAKE_THROW_UNLESS(dynamic_cast<const RevoluteJoint<double>*>(&joint) !=
nullptr);
}
return is_revolute;
}
} // namespace
GlobalInverseKinematics::GlobalInverseKinematics(
const MultibodyPlant<double>& plant,
const GlobalInverseKinematics::Options& options)
: prog_{},
plant_(plant),
joint_lower_bounds_{Eigen::VectorXd::Constant(
plant_.num_positions(), -std::numeric_limits<double>::infinity())},
joint_upper_bounds_{Eigen::VectorXd::Constant(
plant_.num_positions(), std::numeric_limits<double>::infinity())} {
const int num_bodies = plant_.num_bodies();
R_WB_.resize(num_bodies);
p_WBo_.resize(num_bodies);
const Eigen::VectorXd q_lower = plant_.GetPositionLowerLimits();
const Eigen::VectorXd q_upper = plant_.GetPositionUpperLimits();
solvers::MixedIntegerRotationConstraintGenerator rotation_generator(
options.approach, options.num_intervals_per_half_axis,
options.interval_binning);
const std::map<BodyIndex, JointIndex> body_to_joint_map =
GetBodyToJointMap(plant_);
// Loop through each body in the robot, to add the constraint that the bodies
// are welded by joints.
const std::unordered_set<BodyIndex> weld_to_world_body_index_set =
GetWeldToWorldBodyIndexSet(plant_);
// This dummy_plant_context is used to compute the pose of a body welded to
// the world.
auto dummy_plant_context = plant_.CreateDefaultContext();
// First go through each body to assign the pose variables.
for (BodyIndex body_idx{1}; body_idx < num_bodies; ++body_idx) {
const Body<double>& body = plant_.get_body(body_idx);
const string body_R_name = body.name() + "_R";
const string body_pos_name = body.name() + "_pos";
p_WBo_[body_idx] = prog_.NewContinuousVariables<3>(body_pos_name);
if (weld_to_world_body_index_set.count(body_idx) > 0) {
// This body is welded to the world.
R_WB_[body_idx] = prog_.NewContinuousVariables<3, 3>(body_R_name);
} else {
// This body is not rigidly fixed to the world.
R_WB_[body_idx] = solvers::NewRotationMatrixVars(&prog_, body_R_name);
}
}
// Now go through each body to add the kinematic constraint between each body
// and its parent.
for (BodyIndex body_idx{1}; body_idx < num_bodies; ++body_idx) {
const Body<double>& body = plant_.get_body(body_idx);
// If the body is fixed to the world, then fix the decision variables on
// the body position and orientation.
if (weld_to_world_body_index_set.count(body_idx) > 0) {
// This body is welded to the world.
const math::RigidTransformd X_WB = plant_.CalcRelativeTransform(
*dummy_plant_context, plant_.world_frame(), body.body_frame());
// TODO(hongkai.dai): clean up this for loop using
// elementwise matrix constraint when it is ready.
for (int i = 0; i < 3; ++i) {
prog_.AddBoundingBoxConstraint(X_WB.rotation().matrix().col(i),
X_WB.rotation().matrix().col(i),
R_WB_[body_idx].col(i));
}
prog_.AddBoundingBoxConstraint(X_WB.translation(), X_WB.translation(),
p_WBo_[body_idx]);
} else {
// This body is not rigidly fixed to the world.
if (!options.linear_constraint_only) {
solvers::AddRotationMatrixOrthonormalSocpConstraint(&prog_,
R_WB_[body_idx]);
}
if (body.is_floating()) {
// This is the floating base case, just add the rotation matrix
// constraint.
rotation_generator.AddToProgram(R_WB_[body_idx], &prog_);
// No need to add the kinematic constraint between the parent and the
// child link. Skip the code below.
continue;
}
// If the body has a parent, then add the constraint to connect the
// parent body with this body through a joint.
const auto joint = &(plant_.get_joint(body_to_joint_map.at(body_idx)));
if (joint->parent_body().index().is_valid()) {
// Frame Jp is the inboard frame of the joint, rigidly attached to the
// parent link. Frame Jc is the outboard frame of the joint, rigidly
// attached to the child link.
const int parent_idx = joint->parent_body().index();
const RigidTransformd X_PJp =
joint->frame_on_parent().GetFixedPoseInBodyFrame();
const RigidTransformd X_CJc =
joint->frame_on_child().GetFixedPoseInBodyFrame();
if (IsWeld(*joint)) {
const WeldJoint<double>* weld_joint =
static_cast<const WeldJoint<double>*>(joint);
const RigidTransformd X_JpJc = weld_joint->X_FM();
const RigidTransformd X_PC =
X_PJp * X_JpJc * X_CJc.inverse();
// Fixed to the parent body.
// The position can be computed from the parent body pose.
// p_WC = p_WP + R_WP * p_PC
// where C is the child body frame.
// P is the parent body frame.
// W is the world frame.
prog_.AddLinearEqualityConstraint(
p_WBo_[parent_idx] + R_WB_[parent_idx] * X_PC.translation() -
p_WBo_[body_idx],
Vector3d::Zero());
// The orientation can be computed from the parent body orientation.
// R_WP * R_PC = R_WC
Matrix3<Expression> orient_invariance =
R_WB_[parent_idx] * X_PC.rotation().matrix() - R_WB_[body_idx];
for (int i = 0; i < 3; ++i) {
prog_.AddLinearEqualityConstraint(orient_invariance.col(i),
Vector3d::Zero());
}
} else if (IsRevolute(*joint)) {
const RevoluteJoint<double>* revolute_joint =
static_cast<const RevoluteJoint<double>*>(joint);
// Adding mixed-integer constraint will add binary variables into
// the program.
rotation_generator.AddToProgram(R_WB_[body_idx], &prog_);
// axis is the vector of the rotation axis in the joint
// inboard/outboard frame.
const Vector3d axis = revolute_joint->revolute_axis();
// Add the constraint R_WC * R_CJc * axis_Jc = R_WP * R_PJp * axis_Jp,
// where axis_Jc = axis_Jp since the rotation axis is invaraiant in
// the inboard frame Jp and the outboard frame Jc.
prog_.AddLinearEqualityConstraint(
R_WB_[body_idx] * X_CJc.rotation().matrix() * axis -
R_WB_[parent_idx] * X_PJp.rotation().matrix() * axis,
Vector3d::Zero());
// The position of the rotation axis is the same on both child and
// parent bodies.
prog_.AddLinearEqualityConstraint(
p_WBo_[parent_idx] + R_WB_[parent_idx] * X_PJp.translation() -
p_WBo_[body_idx] - R_WB_[body_idx] * X_CJc.translation(),
Vector3d::Zero());
// Now we process the joint limits constraint.
const double joint_lb = q_lower(revolute_joint->position_start());
const double joint_ub = q_upper(revolute_joint->position_start());
AddJointLimitConstraint(body_idx, joint_lb, joint_ub,
options.linear_constraint_only);
} else {
throw std::runtime_error("Unsupported joint type.");
}
}
}
}
}
const solvers::MatrixDecisionVariable<3, 3>&
GlobalInverseKinematics::body_rotation_matrix(BodyIndex body_index) const {
if (body_index >= plant_.num_bodies() || body_index <= 0) {
throw std::runtime_error("body index out of range.");
}
return R_WB_[body_index];
}
const solvers::VectorDecisionVariable<3>&
GlobalInverseKinematics::body_position(BodyIndex body_index) const {
if (body_index >= plant_.num_bodies() || body_index <= 0) {
throw std::runtime_error("body index out of range.");
}
return p_WBo_[body_index];
}
void GlobalInverseKinematics::ReconstructGeneralizedPositionSolutionForBody(
const solvers::MathematicalProgramResult& result, BodyIndex body_idx,
const std::map<BodyIndex, JointIndex>& body_to_joint_map,
const std::unordered_set<BodyIndex>& weld_to_world_body_index_set,
Eigen::Ref<Eigen::VectorXd> q,
std::vector<Eigen::Matrix3d>* reconstruct_R_WB) const {
const Body<double>& body = plant_.get_body(body_idx);
const Matrix3d R_WC = result.GetSolution(R_WB_[body_idx]);
if (body.is_floating()) {
// p_WBi is the position of the body frame in the world frame.
const Vector3d p_WBi = result.GetSolution(p_WBo_[body_idx]);
const math::RotationMatrix<double> normalized_rotmat =
math::RotationMatrix<double>::ProjectToRotationMatrix(R_WC);
q.segment<3>(body.floating_positions_start()) = p_WBi;
if (body.has_quaternion_dofs()) {
// The position order is x-y-z-qw-qx-qy-qz, namely translation
// first, and quaternion second.
q.segment<4>(body.floating_positions_start() + 3) =
normalized_rotmat.ToQuaternionAsVector4();
} else {
// The position order is x-y-z-roll-pitch-yaw.
q.segment<3>(body.floating_positions_start() + 3) =
math::RollPitchYaw<double>(normalized_rotmat).vector();
}
reconstruct_R_WB->at(body_idx) = normalized_rotmat.matrix();
return;
}
// This dummy_plant_context is used to compute the pose of a body welded to
// the world.
auto dummy_plant_context = plant_.CreateDefaultContext();
const Joint<double>& joint = plant_.get_joint(body_to_joint_map.at(body_idx));
const Body<double>& parent = joint.parent_body();
if (weld_to_world_body_index_set.count(body_idx) == 0) {
// R_WP is the rotation matrix of parent frame to the world frame.
const Matrix3d& R_WP = reconstruct_R_WB->at(parent.index());
const RigidTransformd X_PJp =
joint.frame_on_parent().GetFixedPoseInBodyFrame();
const RigidTransformd X_CJc =
joint.frame_on_child().GetFixedPoseInBodyFrame();
// For each different type of joints, use a separate branch to compute
// the posture for that joint.
if (joint.num_positions() == 1) {
const int position_idx = joint.position_start();
const double joint_lb = joint_lower_bounds_(position_idx);
const double joint_ub = joint_upper_bounds_(position_idx);
// Should NOT do this evil dynamic cast here, but currently we do
// not have a method to tell if a joint is revolute or not.
if (dynamic_cast<const RevoluteJoint<double>*>(&joint) != nullptr) {
const RevoluteJoint<double>* revolute_joint =
dynamic_cast<const RevoluteJoint<double>*>(&joint);
const Matrix3d R_JpJc = X_PJp.rotation().matrix().transpose() *
R_WP.transpose() * R_WC *
X_CJc.rotation().matrix();
// The matrix R_JpJc is very likely not on SO(3). The reason is
// that we use a relaxation of the rotation matrix, and thus
// R_WC and R_WP might not lie on SO(3) exactly. Here we need to project
// R_JpJc to SO(3), with joint axis as the rotation axis, and
// joint limits as the lower and upper bound on the rotation angle.
const Vector3d axis_F = revolute_joint->revolute_axis();
const double revolute_joint_angle = math::ProjectMatToRotMatWithAxis(
R_JpJc, axis_F, joint_lb, joint_ub);
q(position_idx) = revolute_joint_angle;
reconstruct_R_WB->at(body_idx) =
R_WP * X_PJp.rotation().matrix() *
Eigen::AngleAxisd(revolute_joint_angle, axis_F).toRotationMatrix() *
X_CJc.rotation().matrix().transpose();
} else {
// TODO(hongkai.dai): add prismatic and helical joints.
throw std::runtime_error("Unsupported joint type.");
}
} else if (joint.num_positions() == 0) {
// Deliberately left empty because the joint is removed by welding the
// parent body to the child body.
}
} else {
// The reconstructed body orientation is just the world fixed
// orientation.
const RigidTransformd X_WB = plant_.CalcRelativeTransform(
*dummy_plant_context, plant_.world_frame(), body.body_frame());
reconstruct_R_WB->at(body_idx) = X_WB.rotation().matrix();
}
}
Eigen::VectorXd GlobalInverseKinematics::ReconstructGeneralizedPositionSolution(
const solvers::MathematicalProgramResult& result) const {
Eigen::VectorXd q(plant_.num_positions());
// First loop through each joint, stores the map from the child body to the
// joint.
const std::map<BodyIndex, JointIndex> body_to_joint_map =
GetBodyToJointMap(plant_);
const std::unordered_set<BodyIndex> weld_to_world_body_index_set =
GetWeldToWorldBodyIndexSet(plant_);
// reconstruct_R_WB[i] is the orientation of body i'th body frame expressed in
// the world frame, computed from the reconstructed posture.
std::vector<Eigen::Matrix3d> reconstruct_R_WB(plant_.num_bodies());
// is_link_visited[i] is set to true, if the angle of the joint on link i has
// been reconstructed.
std::vector<bool> is_link_visited(plant_.num_bodies(), false);
// The first one is the world frame, thus the orientation is identity.
reconstruct_R_WB[0].setIdentity();
is_link_visited[0] = true;
int num_link_visited = 1;
BodyIndex body_idx{1};
while (num_link_visited < plant_.num_bodies()) {
if (!is_link_visited[body_idx]) {
// unvisited_links records all the unvisited links, along the kinematic
// path from the root to the body with index body_idx (including
// body_idx).
std::stack<BodyIndex> unvisited_links;
unvisited_links.push(body_idx);
BodyIndex parent_idx{};
if (plant_.get_body(body_idx).is_floating()) {
parent_idx = plant_.world_body().index();
} else {
parent_idx = plant_.get_joint(body_to_joint_map.at(body_idx))
.parent_body()
.index();
}
while (!is_link_visited[parent_idx]) {
unvisited_links.push(parent_idx);
// Now update parent_idx
if (plant_.get_body(BodyIndex{parent_idx}).is_floating()) {
parent_idx = plant_.world_body().index();
} else {
parent_idx =
plant_.get_joint(body_to_joint_map.at(BodyIndex{parent_idx}))
.parent_body()
.index();
}
}
// Now the link parent_idx has been visited.
while (!unvisited_links.empty()) {
const BodyIndex unvisited_link_idx = unvisited_links.top();
unvisited_links.pop();
ReconstructGeneralizedPositionSolutionForBody(
result, unvisited_link_idx, body_to_joint_map,
weld_to_world_body_index_set, q, &reconstruct_R_WB);
is_link_visited[unvisited_link_idx] = true;
++num_link_visited;
}
}
++body_idx;
}
return q;
}
solvers::Binding<solvers::LinearConstraint>
GlobalInverseKinematics::AddWorldPositionConstraint(
BodyIndex body_idx, const Eigen::Vector3d& p_BQ,
const Eigen::Vector3d& box_lb_F, const Eigen::Vector3d& box_ub_F,
const RigidTransformd& X_WF) {
if (body_idx >= plant_.num_bodies() || body_idx <= 0) {
throw std::runtime_error("body index out of range.");
}
const Vector3<Expression> p_WQ = p_WBo_[body_idx] + R_WB_[body_idx] * p_BQ;
return prog_.AddLinearConstraint(X_WF.inverse().cast<Expression>() * p_WQ,
box_lb_F, box_ub_F);
}
solvers::Binding<solvers::LinearConstraint>
GlobalInverseKinematics::AddWorldRelativePositionConstraint(
BodyIndex body_idx_B, const Eigen::Vector3d& p_BQ,
BodyIndex body_idx_A, const Eigen::Vector3d& p_AP,
const Eigen::Vector3d& box_lb_F, const Eigen::Vector3d& box_ub_F,
const RigidTransformd& X_WF) {
if (body_idx_B >= plant_.num_bodies() || body_idx_B <= 0) {
throw std::runtime_error("body index out of range.");
}
if (body_idx_A >= plant_.num_bodies() || body_idx_A <= 0) {
throw std::runtime_error("body index out of range.");
}
const Vector3<Expression> p_WQ =
p_WBo_[body_idx_B] + R_WB_[body_idx_B] * p_BQ;
const Vector3<Expression> p_WP =
p_WBo_[body_idx_A] + R_WB_[body_idx_A] * p_AP;
return prog_.AddLinearConstraint(
X_WF.rotation().inverse().cast<Expression>() * (p_WQ - p_WP), box_lb_F,
box_ub_F);
}
solvers::Binding<solvers::LinearConstraint>
GlobalInverseKinematics::AddWorldOrientationConstraint(
BodyIndex body_idx, const Eigen::Quaterniond& desired_orientation,
double angle_tol) {
if (body_idx >= plant_.num_bodies() || body_idx <= 0) {
throw std::runtime_error("body index out of range.");
}
// The rotation matrix error R_e satisfies
// trace(R_e) = 2 * cos(θ) + 1, where θ is the rotation angle between
// desired orientation and the current orientation. Thus the constraint is
// 2 * cos(angle_tol) + 1 <= trace(R_e) <= 3
Matrix3<Expression> rotation_matrix_err =
desired_orientation.toRotationMatrix() * R_WB_[body_idx].transpose();
double lb = angle_tol < M_PI ? 2 * cos(angle_tol) + 1 : -1;
return prog_.AddLinearConstraint(rotation_matrix_err.trace(), lb, 3);
}
void GlobalInverseKinematics::AddPostureCost(
const Eigen::Ref<const Eigen::VectorXd>& q_desired,
const Eigen::Ref<const Eigen::VectorXd>& body_position_cost,
const Eigen::Ref<const Eigen::VectorXd>& body_orientation_cost) {
const int num_bodies = plant_.num_bodies();
if (body_position_cost.rows() != num_bodies) {
std::ostringstream oss;
oss << "body_position_cost should have " << num_bodies << " rows.";
throw std::runtime_error(oss.str());
}
if (body_orientation_cost.rows() != num_bodies) {
std::ostringstream oss;
oss << "body_orientation_cost should have " << num_bodies << " rows.";
throw std::runtime_error(oss.str());
}
for (int i = 1; i < num_bodies; ++i) {
if (body_position_cost(i) < 0) {
std::ostringstream oss;
oss << "body_position_cost(" << i << ") is negative.";
throw std::runtime_error(oss.str());
}
if (body_orientation_cost(i) < 0) {
std::ostringstream oss;
oss << "body_orientation_cost(" << i << ") is negative.";
throw std::runtime_error(oss.str());
}
}
auto context = plant_.CreateDefaultContext();
plant_.SetPositions(context.get(), q_desired);
// Sum up the orientation error for each body to orient_err_sum.
Expression orient_err_sum(0);
// p_WBo_err(i) is the slack variable, representing the position error for
// the (i+1)'th body, which is the Euclidean distance from the body origin
// position, to the desired position.
solvers::VectorXDecisionVariable p_WBo_err =
prog_.NewContinuousVariables(num_bodies - 1, "p_WBo_error");
for (int i = 1; i < num_bodies; ++i) {
// body 0 is the world. There is no position or orientation error on the
// world, so we just skip i = 0 and start from i = 1.
const auto& X_WB_desired = plant_.CalcRelativeTransform(
*context, plant_.world_frame(),
plant_.get_body(BodyIndex{i}).body_frame());
// Add the constraint p_WBo_err(i-1) >= body_position_cost(i) *
// |p_WBo(i) - p_WBo_desired(i) |
Vector4<symbolic::Expression> pos_error_expr;
pos_error_expr << p_WBo_err(i - 1),
body_position_cost(i) * (p_WBo_[i] - X_WB_desired.translation());
prog_.AddLorentzConeConstraint(pos_error_expr);
// The orientation error is on the angle θ between the body orientation and
// the desired orientation, namely 1 - cos(θ).
// cos(θ) can be computed as (trace( R_WB_desired * R_WB_[i]ᵀ) - 1) / 2.
// To see how the angle is computed from a rotation matrix, please refer to
// http://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToAngle/
orient_err_sum +=
body_orientation_cost(i) *
(1 -
((X_WB_desired.rotation().matrix() * R_WB_[i].transpose()).trace() -
1) /
2);
}
// The total cost is the summation of the position error and the orientation
// error.
prog_.AddCost(p_WBo_err.cast<Expression>().sum() + orient_err_sum);
}
solvers::VectorXDecisionVariable
GlobalInverseKinematics::BodyPointInOneOfRegions(
BodyIndex body_index, const Eigen::Ref<const Eigen::Vector3d>& p_BQ,
const std::vector<Eigen::Matrix3Xd>& region_vertices) {
const auto& R_WB = body_rotation_matrix(body_index);
const auto& p_WBo = body_position(body_index);
const int num_regions = region_vertices.size();
const string& body_name = plant_.get_body(body_index).name();
solvers::VectorXDecisionVariable z =
prog_.NewBinaryVariables(num_regions, "z_" + body_name);
std::vector<solvers::VectorXDecisionVariable> w(num_regions);
// We will write p_WQ in two ways, we first write p_WQ as
// sum_i (w_i1 * v_i1 + w_i2 * v_i2 + ... + w_in * v_in). As the convex
// combination of vertices in one of the regions.
Vector3<symbolic::Expression> p_WQ;
p_WQ << 0, 0, 0;
for (int i = 0; i < num_regions; ++i) {
const int num_vertices_i = region_vertices[i].cols();
if (num_vertices_i < 3) {
throw std::runtime_error("Each region should have at least 3 vertices.");
}
w[i] = prog_.NewContinuousVariables(
num_vertices_i, "w_" + body_name + "_region_" + std::to_string(i));
prog_.AddLinearConstraint(w[i].cast<symbolic::Expression>().sum() - z(i) ==
0);
prog_.AddBoundingBoxConstraint(Eigen::VectorXd::Zero(num_vertices_i),
Eigen::VectorXd::Ones(num_vertices_i), w[i]);
p_WQ += region_vertices[i] * w[i];
}
prog_.AddLinearConstraint(z.cast<symbolic::Expression>().sum() == 1);
// p_WQ must match the body pose, as p_WQ = p_WBo + R_WB * p_BQ
prog_.AddLinearEqualityConstraint(p_WBo + R_WB * p_BQ - p_WQ,
Eigen::Vector3d::Zero());
return z;
}
solvers::VectorXDecisionVariable
GlobalInverseKinematics::BodySphereInOneOfPolytopes(
BodyIndex body_index, const Eigen::Ref<const Eigen::Vector3d>& p_BQ,
double radius,
const std::vector<GlobalInverseKinematics::Polytope3D>& polytopes) {
if (body_index >= plant_.num_bodies() || body_index <= 0) {
throw std::runtime_error("body index out of range.");
}
DRAKE_DEMAND(radius >= 0);
const int num_polytopes = static_cast<int>(polytopes.size());
const auto z = prog_.NewBinaryVariables(num_polytopes, "z");
// z1 + ... + zn = 1
prog_.AddLinearEqualityConstraint(Eigen::RowVectorXd::Ones(num_polytopes), 1,
z);
const auto y =
prog_.NewContinuousVariables<3, Eigen::Dynamic>(3, num_polytopes, "y");
const Vector3<symbolic::Expression> p_WQ =
p_WBo_[body_index] + R_WB_[body_index] * p_BQ;
// p_WQ = y.col(0) + ... + y.col(n)
prog_.AddLinearEqualityConstraint(
p_WQ - y.cast<symbolic::Expression>().rowwise().sum(),
Eigen::Vector3d::Zero());
for (int i = 0; i < num_polytopes; ++i) {
DRAKE_DEMAND(polytopes[i].A.rows() == polytopes[i].b.rows());
prog_.AddLinearConstraint(
polytopes[i].A * y.col(i) <=
(polytopes[i].b - polytopes[i].A.rowwise().norm() * radius) * z(i));
}
return z;
}
// Approximate a quadratic constraint (which could be formulated as a Lorentz
// cone constraint) xᵀx ≤ c² by
// -c ≤ xᵢ ≤ c
// ± xᵢ ± xⱼ ≤ √2 * c
// ± x₀ ± x₁ ± x₂ ≤ √3 * c
// These linear approximation are obtained as the tangential planes at some
// points on the surface of the sphere xᵀx ≤ c².
void ApproximateBoundedNormByLinearConstraints(
const Eigen::Ref<const Vector3<symbolic::Expression>>& x, double c,
solvers::MathematicalProgram* prog) {
DRAKE_DEMAND(c >= 0);
// -c ≤ xᵢ ≤ c
prog->AddLinearConstraint(x, Eigen::Vector3d::Constant(-c),
Eigen::Vector3d::Constant(c));
const double sqrt2_c = std::sqrt(2) * c;
const double sqrt3_c = std::sqrt(3) * c;
// ± xᵢ ± xⱼ ≤ √2 * c
for (int i = 0; i < 3; ++i) {
for (int j = i + 1; j < 3; ++j) {
prog->AddLinearConstraint(x(i) + x(j), -sqrt2_c, sqrt2_c);
prog->AddLinearConstraint(x(i) - x(j), -sqrt2_c, sqrt2_c);
}
}
// ± x₀ ± x₁ ± x₂ ≤ √3 * c
prog->AddLinearConstraint(x(0) + x(1) + x(2), -sqrt3_c, sqrt3_c);
prog->AddLinearConstraint(x(0) + x(1) - x(2), -sqrt3_c, sqrt3_c);
prog->AddLinearConstraint(x(0) - x(1) + x(2), -sqrt3_c, sqrt3_c);
prog->AddLinearConstraint(x(0) - x(1) - x(2), -sqrt3_c, sqrt3_c);
}
void GlobalInverseKinematics::AddJointLimitConstraint(
BodyIndex body_index, double joint_lower_bound, double joint_upper_bound,
bool linear_constraint_approximation) {
if (joint_lower_bound > joint_upper_bound) {
throw std::runtime_error(
"The joint lower bound should be no larger than the upper bound.");
}
if (plant_.get_body(body_index).is_floating()) {
throw std::runtime_error(
"The body is floating, do not use AddJointLimitConstraint(), impose "
"the bounds on R_WB and p_WB directly.");
}
const Joint<double>* joint{nullptr};
for (JointIndex joint_index{0}; joint_index < plant_.num_joints();
++joint_index) {
if (plant_.get_joint(joint_index).child_body().index() == body_index) {
joint = &(plant_.get_joint(joint_index));
break;
}
}
if (joint == nullptr) {
throw std::runtime_error(
fmt::format("The body {} is not the child of any joint in the plant.",
plant_.get_body(body_index).name()));
}
const Body<double>& parent = joint->parent_body();
const int parent_idx = parent.index();
const RigidTransformd X_PJp =
joint->frame_on_parent().GetFixedPoseInBodyFrame();
const RigidTransformd X_CJc =
joint->frame_on_child().GetFixedPoseInBodyFrame();
switch (joint->num_velocities()) {
case 0: {
// Fixed to the parent body.
throw std::runtime_error("Cannot impose joint limits for a fixed joint.");
}
case 1: {
// If the new bound [joint_lower_bound joint_upper_bound] is not tighter
// than the existing bound, then we ignore it, without adding new
// constraints.
bool is_limits_tightened = false;
const int position_idx = joint->position_start();
if (joint_lower_bound > joint_lower_bounds_(position_idx)) {
joint_lower_bounds_(position_idx) = joint_lower_bound;
is_limits_tightened = true;
}
if (joint_upper_bound < joint_upper_bounds_(position_idx)) {
joint_upper_bounds_(position_idx) = joint_upper_bound;
is_limits_tightened = true;
}
if (is_limits_tightened) {
// Should NOT do this evil dynamic cast here, but currently we do
// not have a method to tell if a joint is revolute or not.
if (dynamic_cast<const RevoluteJoint<double>*>(joint) != nullptr) {
const auto* revolute_joint =
dynamic_cast<const RevoluteJoint<double>*>(joint);
// axis_F is the vector of the rotation axis in the joint
// inboard/outboard frame.
const Vector3d axis_F = revolute_joint->revolute_axis();
// Now we process the joint limits constraint.
const double joint_bound = (joint_upper_bounds_[position_idx] -
joint_lower_bounds_[position_idx]) /
2;
if (joint_bound < M_PI) {
// We use the fact that if the angle between two unit length
// vectors u and v is smaller than α, it is equivalent to
// |u - v| <= 2*sin(α/2)
// which is a second order cone constraint.
// If the rotation angle θ satisfies
// a <= θ <= b
// This is equivalent to
// -α <= θ - (a+b)/2 <= α
// where α = (b-a) / 2, (a+b) / 2 is the joint offset, such that
// the bounds on β = θ - (a+b)/2 are symmetric.
// We use the following notation:
// R_WP The rotation matrix of parent frame `P` to world
// frame `W`.
// R_WC The rotation matrix of child frame `C` to world
// frame `W`.
// R_PJp The rotation matrix of joint frame `Jp` to parent
// frame `P`.
// R(k, β) The rotation matrix along joint axis k by angle β.
// R_JcC The rotation matrix of child frame `C` to the
// outboard frame `Jc`.
// The kinematics constraint is
// R_WP * R_PJp * R(k, θ) * R_JcC = R_WC.
// This is equivalent to
// R_WP * R_PJp * R(k, (a+b)/2) * R(k, β)) = R_WC * R_CJc.
// So to constrain that -α <= β <= α,
// we can constrain the angle between the two vectors
// R_WC * R_CJc * v and R_WP * R_PJp * R(k,(a+b)/2) * v is no larger
// than α, where v is a unit length vector perpendicular to the
// rotation axis k, in the joint frame. Thus we can constrain that
// |R_WC*R_CJc*v - R_WP * R_PJp * R(k,(a+b)/2)*v | <= 2*sin (α / 2)
// as we explained above.
// First generate a vector v_C that is perpendicular to rotation
// axis, in child frame.
Vector3d v_C = axis_F.cross(Vector3d(1, 0, 0));
double v_C_norm = v_C.norm();
if (v_C_norm < sqrt(2) / 2) {
// axis_F is almost parallel to [1; 0; 0]. Try another axis
// [0, 1, 0]
v_C = axis_F.cross(Vector3d(0, 1, 0));
v_C_norm = v_C.norm();
}
// Normalizes the revolute vector.
v_C /= v_C_norm;
// The constraint would be tighter, if we choose many unit
// length vector `v`, perpendicular to the joint axis, in the
// joint frame. Here to balance between the size of the
// optimization problem, and the tightness of the convex
// relaxation, we just use four vectors in `v`. Notice that
// v_basis contains the orthonormal basis of the null space
// null(axis_F).
std::array<Eigen::Vector3d, 2> v_basis = {{v_C, axis_F.cross(v_C)}};
v_basis[1] /= v_basis[1].norm();
std::array<Eigen::Vector3d, 4> v_samples;
v_samples[0] = v_basis[0];
v_samples[1] = v_basis[1];
v_samples[2] = v_basis[0] + v_basis[1];
v_samples[2] /= v_samples[2].norm();
v_samples[3] = v_basis[0] - v_basis[1];
v_samples[3] /= v_samples[3].norm();
// rotmat_joint_offset is R(k, (a+b)/2) explained above.
const Matrix3d rotmat_joint_offset =
Eigen::AngleAxisd((joint_lower_bounds_[position_idx] +
joint_upper_bounds_[position_idx]) /
2,
axis_F)
.toRotationMatrix();
// joint_limit_expr is going to be within the Lorentz cone.
Eigen::Matrix<Expression, 4, 1> joint_limit_expr;
const double joint_limit_lorentz_rhs = 2 * sin(joint_bound / 2);
joint_limit_expr(0) = joint_limit_lorentz_rhs;
for (const auto& v : v_samples) {
// joint_limit_expr.tail<3> is
// R_WC * R_CJc * v - R_WP * R_PJp * R(k,(a+b)/2) * v mentioned
// above.
joint_limit_expr.tail<3>() =
R_WB_[body_index] * X_CJc.rotation().matrix() * v -
R_WB_[parent_idx] * X_PJp.rotation().matrix() *
rotmat_joint_offset * v;
if (linear_constraint_approximation) {
ApproximateBoundedNormByLinearConstraints(
joint_limit_expr.tail<3>(), joint_limit_lorentz_rhs,
&prog_);
} else {
prog_.AddLorentzConeConstraint(joint_limit_expr);
}
}
const std::unordered_set<BodyIndex> weld_to_world_body_index_set =
GetWeldToWorldBodyIndexSet(plant_);
// This dummy_plant_context is used to compute the pose of a body
// welded to the world.
auto dummy_plant_context = plant_.CreateDefaultContext();
if (weld_to_world_body_index_set.count(BodyIndex{parent_idx}) > 0) {
const RigidTransformd X_WP = plant_.CalcRelativeTransform(
*dummy_plant_context, plant_.world_frame(),
plant_.get_body(BodyIndex{parent_idx}).body_frame());
// If the parent body is rigidly fixed to the world. Then we
// can impose a tighter constraint. Based on the derivation
// above, we have
// R(k, β) = [R_WP * R_PJp * R(k, (a+b)/2)]ᵀ * R_WC * R_CJc
// as a linear expression of the decision variable R_WC
// (notice that R_WP is constant, since the parent body is
// rigidly fixed to the world.
// Any unit length vector `v` that is perpendicular to
// joint axis `axis_F` in the joint Frame, can be written as
// v = V * u, uᵀ * u = 1
// where V = [v_basis[0] v_basis[1]] containing the basis
// vectors for the linear space Null(axis_F).
// On the other hand, we know
// vᵀ * R(k, β) * v = cos(β) >= cos(α)
// due to the joint limits constraint
// -α <= β <= α.
// So we have the condition that
// uᵀ * u = 1
// => uᵀ * Vᵀ * R(k, β) * V * u >= cos(α)
// Using S-lemma, we know this implication is equivalent to
// Vᵀ * [R(k, β) + R(k, β)ᵀ]/2 * V - cos(α) * I is p.s.d
// We let a 2 x 2 matrix
// M = Vᵀ * [R(k, β) + R(k, β)ᵀ]/2 * V - cos(α) * I
// A 2 x 2 matrix M being positive semidefinite (p.s.d) is
// equivalent to the condition that
// [M(0, 0), M(1, 1), M(1, 0)] is in the rotated Lorentz cone.
// R_joint_beta is R(k, β) in the documentation.
Eigen::Matrix<symbolic::Expression, 3, 3> R_joint_beta =
(X_WP.rotation().matrix() * X_PJp.rotation().matrix() *
rotmat_joint_offset)
.transpose() *
R_WB_[body_index] * X_CJc.rotation().matrix();
const double joint_bound_cos{std::cos(joint_bound)};
if (!linear_constraint_approximation) {
Eigen::Matrix<double, 3, 2> V;
V << v_basis[0], v_basis[1];
const Eigen::Matrix<symbolic::Expression, 2, 2> M =
V.transpose() * (R_joint_beta + R_joint_beta.transpose()) /
2 * V -
joint_bound_cos * Eigen::Matrix2d::Identity();
prog_.AddRotatedLorentzConeConstraint(
Vector3<symbolic::Expression>(M(0, 0), M(1, 1), M(1, 0)));
}
// From Rodriguez formula, we know that -α <= β <= α implies
// trace(R(k, β)) = 1 + 2 * cos(β) >= 1 + 2*cos(α)
// So we can impose the constraint
// 1+2*cos(α) ≤ trace(R(k, β))
const symbolic::Expression R_joint_beta_trace{
R_joint_beta.trace()};
prog_.AddLinearConstraint(R_joint_beta_trace >=
1 + 2 * joint_bound_cos);
}
}
} else {
// TODO(hongkai.dai): add prismatic and helical joint.
throw std::runtime_error("Unsupported joint type.");
}
}
break;
}
case 6: {
break;
}
default:
throw std::runtime_error("Unsupported joint type.");
}
}
// TODO(russt): This method is currently calling Solve in order to compute the
// integer variables (given the poses). This is an easy, but relatively
// expensive way; we could instead compute them in closed form based on the
// implementation of the McCormick envelope constraints.
void GlobalInverseKinematics::SetInitialGuess(
const Eigen::Ref<const Eigen::VectorXd>& q) {
auto context = plant_.CreateDefaultContext();
plant_.SetPositions(context.get(), q);
std::vector<solvers::Binding<solvers::BoundingBoxConstraint>> bindings;
// body 0 is the world. There is no position or orientation error on the
// world, so we just skip i = 0 and start from i = 1.
for (int i = 1; i < plant_.num_bodies(); ++i) {
const auto& X_WB = plant_.CalcRelativeTransform(
*context, plant_.world_frame(),
plant_.get_body(BodyIndex{i}).body_frame());
bindings.push_back(prog_.AddBoundingBoxConstraint(
X_WB.translation(), X_WB.translation(), p_WBo_[i]));
bindings.push_back(prog_.AddBoundingBoxConstraint(
X_WB.rotation().matrix(), X_WB.rotation().matrix(), R_WB_[i]));
}
ScopeExit guard([&]() {
for (const auto& b : bindings) {
prog_.RemoveConstraint(b);
}
});
const auto result = solvers::Solve(prog_);
if (!result.is_success()) {
throw std::runtime_error(
"SetInitialGuess tried to solve a variant of your IK problem, but "
"failed.");
}
prog_.SetInitialGuessForAllVariables(result.GetSolution());
}
} // namespace multibody
} // namespace drake