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zmp_planner.cc
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zmp_planner.cc
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#include "drake/systems/controllers/zmp_planner.h"
#include <vector>
#include <unsupported/Eigen/MatrixFunctions>
#include "drake/common/text_logging.h"
#include "drake/systems/controllers/linear_quadratic_regulator.h"
namespace drake {
namespace systems {
namespace controllers {
using trajectories::ExponentialPlusPiecewisePolynomial;
using trajectories::PiecewisePolynomial;
Eigen::Vector2d ZMPPlanner::ComputeOptimalCoMdd(
double time, const Eigen::Vector4d& x) const {
DRAKE_DEMAND(planned_);
// Eq. 20 in [1].
Eigen::Vector2d yf = zmp_d_.value(zmp_d_.end_time());
Eigen::Vector4d x_bar = x;
x_bar.head<2>() -= yf;
return K_ * x_bar + k2_.value(time);
}
bool ZMPPlanner::CheckStationaryEndPoint(
const PiecewisePolynomial<double>& zmp_d) const {
PiecewisePolynomial<double> last_segment =
zmp_d.slice(zmp_d.get_number_of_segments() - 1, 1);
PiecewisePolynomial<double> derivative = last_segment.derivative();
int degree = last_segment.getSegmentPolynomialDegree(0);
for (int d = degree; d >= 0; d--) {
if (derivative.value(derivative.end_time()).norm() >
kStationaryThreshold) {
return false;
}
derivative = derivative.derivative();
}
return true;
}
void ZMPPlanner::Plan(const PiecewisePolynomial<double>& zmp_d,
const Eigen::Vector4d& x0, double height, double gravity,
const Eigen::Matrix2d& Qy, const Eigen::Matrix2d& R) {
// Warn the caller if the last point is not stationary. The math is still
// correct, and this is an allowable (but dangerous) use case.
// If the user use the policy / nominal trajectory past the end point, the
// system diverges exponentially fast.
if (!CheckStationaryEndPoint(zmp_d)) {
drake::log()->warn("ZMPPlanner: The desired zmp trajectory does not end "
"in a stationary condition.");
}
int n_segments = zmp_d.get_number_of_segments();
int zmp_d_degree = zmp_d.getSegmentPolynomialDegree(0);
DRAKE_DEMAND(zmp_d_degree >= 0);
DRAKE_DEMAND(zmp_d.rows() == 2 && zmp_d.cols() == 1);
DRAKE_DEMAND(height > 0);
DRAKE_DEMAND(gravity > 0);
zmp_d_ = zmp_d;
Qy_ = Qy;
R_ = R;
// Eq. 1 and 2 in [1].
A_.setZero();
A_.block<2, 2>(0, 2).setIdentity();
B_.setZero();
B_.block<2, 2>(2, 0).setIdentity();
C_.setZero();
C_.block<2, 2>(0, 0).setIdentity();
D_ = -height / gravity * Eigen::Matrix2d::Identity();
// Eq. 9 - 14 in [1].
Eigen::Matrix<double, 4, 4> Q1 = C_.transpose() * Qy_ * C_;
Eigen::Matrix<double, 2, 2> R1 = R_ + D_.transpose() * Qy_ * D_;
Eigen::Matrix<double, 4, 2> N = C_.transpose() * Qy_ * D_;
Eigen::Matrix<double, 2, 2> R1i = R1.inverse();
R1i_ = R1i;
LinearQuadraticRegulatorResult lqr_result =
LinearQuadraticRegulator(A_, B_, Q1, R1, N);
S1_ = lqr_result.S;
K_ = -lqr_result.K;
// Computes the time varying linear and constant term in the value function
// and linear policy. Also known as the backward pass.
Eigen::Matrix<double, 2, 4> NB = (N.transpose() + B_.transpose() * S1_);
// Eq. 23, 24 in [1].
Eigen::Matrix<double, 4, 4> A2 =
NB.transpose() * R1i * B_.transpose() - A_.transpose();
Eigen::Matrix<double, 4, 2> B2 =
2 * (C_.transpose() - NB.transpose() * R1i * D_) * Qy_;
Eigen::Matrix<double, 4, 4> A2i = A2.inverse();
NB_ = NB;
A2_ = A2;
B2_ = B2;
// Last desired ZMP.
Eigen::Vector2d zmp_tf = zmp_d.value(zmp_d.end_time());
Eigen::Vector4d tmp4;
Eigen::MatrixXd alpha = Eigen::MatrixXd::Zero(4, n_segments);
std::vector<Eigen::MatrixXd> beta(n_segments,
Eigen::MatrixXd::Zero(4, zmp_d_degree + 1));
std::vector<Eigen::MatrixXd> gamma(
n_segments, Eigen::MatrixXd::Zero(2, zmp_d_degree + 1));
std::vector<Eigen::MatrixXd> c(n_segments,
Eigen::MatrixXd::Zero(2, zmp_d_degree + 1));
std::vector<Eigen::Matrix<Polynomial<double>, Eigen::Dynamic, Eigen::Dynamic>>
beta_poly(n_segments);
std::vector<Eigen::Matrix<Polynomial<double>, Eigen::Dynamic, Eigen::Dynamic>>
gamma_poly(n_segments);
Eigen::VectorXd delta_time_vec(zmp_d_degree + 1);
delta_time_vec[0] = 1;
// Algorithm 1 in [1] to solve for parameters of s2 and k2.
for (int t = n_segments - 1; t >= 0; t--) {
c[t].setZero();
c[t].row(0).head(zmp_d_degree + 1) =
zmp_d.getPolynomial(t, 0, 0).GetCoefficients();
c[t].row(1).head(zmp_d_degree + 1) =
zmp_d.getPolynomial(t, 1, 0).GetCoefficients();
/// switch to zbar coord
c[t].col(0) -= zmp_tf;
// degree 4
beta[t].col(zmp_d_degree) = -A2i * B2 * c[t].col(zmp_d_degree);
gamma[t].col(zmp_d_degree) =
R1i * D_ * Qy_ * c[t].col(zmp_d_degree) -
0.5 * R1i * B_.transpose() * beta[t].col(zmp_d_degree);
for (int d = zmp_d_degree - 1; d >= 0; d--) {
beta[t].col(d) = A2i * ((d + 1) * beta[t].col(d + 1) - B2 * c[t].col(d));
gamma[t].col(d) = R1i * D_ * Qy_ * c[t].col(d) -
0.5 * R1i * B_.transpose() * beta[t].col(d);
}
if (t == n_segments - 1) {
tmp4 = Eigen::Vector4d::Zero();
} else {
tmp4 = alpha.col(t + 1) + beta[t + 1].col(0);
}
double dt = zmp_d.duration(t);
Eigen::Matrix4d A2exp = A2 * dt;
A2exp = A2exp.exp();
for (int i = 0; i < zmp_d_degree + 1; i++)
delta_time_vec[i] = std::pow(dt, i);
tmp4 = tmp4 - beta[t] * delta_time_vec;
alpha.col(t) = A2exp.inverse() * tmp4;
beta_poly[t].resize(4, 1);
for (int n = 0; n < 4; n++) {
beta_poly[t](n, 0) = Polynomial<double>(beta[t].row(n));
}
gamma_poly[t].resize(2, 1);
for (int n = 0; n < 2; n++) {
gamma_poly[t](n, 0) = Polynomial<double>(gamma[t].row(n));
}
}
// Eq. 25 in [1].
PiecewisePolynomial<double> beta_traj(beta_poly, zmp_d.get_segment_times());
s2_ = ExponentialPlusPiecewisePolynomial<double>(Eigen::Matrix4d::Identity(),
A2, alpha, beta_traj);
// Eq. 28 in [1].
PiecewisePolynomial<double> gamma_traj(gamma_poly, zmp_d.get_segment_times());
k2_ = ExponentialPlusPiecewisePolynomial<double>(-0.5 * R1i * B_.transpose(),
A2, alpha, gamma_traj);
// Computes the nominal CoM trajectory. Also known as the forward pass.
// Eq. 35, 36 in [1].
Eigen::Matrix<double, 8, 8> Az, Azi;
Eigen::Matrix<double, 8, 2> Bz;
Az.block<4, 4>(0, 0) = A_ + B_ * K_;
Az.block<4, 4>(0, 4) = -0.5 * B_ * R1i * B_.transpose();
Az.block<4, 4>(4, 0).setZero();
Az.block<4, 4>(4, 4) = A2;
Azi = Az.inverse();
Bz.block<4, 2>(0, 0) = B_ * R1i * D_ * Qy_;
Bz.block<4, 2>(4, 0) = B2;
Eigen::MatrixXd a(8, n_segments);
a.bottomRows<4>() = alpha;
std::vector<Eigen::Matrix<Polynomial<double>, Eigen::Dynamic, Eigen::Dynamic>>
b_poly(n_segments);
std::vector<Eigen::MatrixXd> b(n_segments,
Eigen::MatrixXd(4, zmp_d_degree + 1));
Eigen::Matrix<double, 8, 1> tmp81;
Eigen::Matrix<double, 8, 8> Az_exp;
Eigen::Matrix<double, 4, 8> I48;
I48.block<4, 4>(0, 0).setIdentity();
I48.block<4, 4>(0, 4).setZero();
Eigen::Vector4d x = x0;
x.head<2>() -= zmp_tf;
// Algorithm 2 in [1] to solve for the CoM trajectory.
// Since s2 is already solved above, we only compute the top half for the
// CoM trajectory.
for (int t = 0; t < n_segments; t++) {
double dt = zmp_d.duration(t);
b[t].col(zmp_d_degree) = -Azi.topRows(4) * Bz * c[t].col(zmp_d_degree);
for (int d = zmp_d_degree - 1; d >= 0; d--) {
tmp81.head<4>() = b[t].col(d + 1);
tmp81.tail<4>() = beta[t].col(d + 1);
tmp81 = tmp81 * (d + 1);
b[t].col(d) = Azi.topRows(4) * (tmp81 - Bz * c[t].col(d));
}
a.block<4, 1>(0, t) = x - b[t].col(0);
Az_exp = Az * dt;
Az_exp = Az_exp.exp();
for (int i = 0; i < zmp_d_degree + 1; i++)
delta_time_vec[i] = std::pow(dt, i);
x = I48 * Az_exp * a.col(t) + b[t] * delta_time_vec;
b[t].block<2, 1>(0, 0) += zmp_tf; // Map CoM position back to world frame.
b_poly[t].resize(2, 1);
for (int n = 0; n < 2; n++) {
b_poly[t](n, 0) = Polynomial<double>(b[t].row(n));
}
}
Eigen::Matrix<double, 2, 8> tmp28;
tmp28.block<2, 2>(0, 0).setIdentity();
tmp28.block<2, 6>(0, 2).setZero();
PiecewisePolynomial<double> b_traj(b_poly, zmp_d.get_segment_times());
com_ = ExponentialPlusPiecewisePolynomial<double>(tmp28, Az, a, b_traj);
comd_ = com_.derivative();
// ComputeOptimalCoMdd(t, nominal_x) should equal to comdd_.
comdd_ = comd_.derivative();
planned_ = true;
}
} // namespace controllers
} // namespace systems
} // namespace drake