376 lines
12 KiB
C++
376 lines
12 KiB
C++
/* ----------------------------------------------------------------------------
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* GTSAM Copyright 2010, Georgia Tech Research Corporation,
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* Atlanta, Georgia 30332-0415
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* All Rights Reserved
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* Authors: Frank Dellaert, et al. (see THANKS for the full author list)
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* See LICENSE for the license information
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* -------------------------------------------------------------------------- */
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/**
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* @file Pose2.cpp
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* @brief 2D Pose
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*/
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#include <gtsam/geometry/concepts.h>
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#include <gtsam/geometry/Pose2.h>
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#include <gtsam/base/Testable.h>
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#include <gtsam/base/concepts.h>
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#include <cmath>
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#include <iostream>
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#include <iomanip>
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using namespace std;
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namespace gtsam {
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/** instantiate concept checks */
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GTSAM_CONCEPT_POSE_INST(Pose2)
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static const Rot2 R_PI_2(Rot2::fromCosSin(0., 1.));
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/* ************************************************************************* */
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Matrix3 Pose2::matrix() const {
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Matrix2 R = r_.matrix();
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Matrix32 R0;
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R0.block<2,2>(0,0) = R;
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R0.block<1,2>(2,0).setZero();
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Matrix31 T;
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T << t_.x(), t_.y(), 1.0;
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Matrix3 RT_;
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RT_.block<3,2>(0,0) = R0;
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RT_.block<3,1>(0,2) = T;
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return RT_;
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}
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/* ************************************************************************* */
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void Pose2::print(const string& s) const {
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std::cout << (s.empty() ? s : s + " ") << *this << std::endl;
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}
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/* ************************************************************************* */
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std::ostream &operator<<(std::ostream &os, const Pose2& pose) {
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os << "(" << pose.x() << ", " << pose.y() << ", " << pose.theta() << ")";
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return os;
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}
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/* ************************************************************************* */
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bool Pose2::equals(const Pose2& q, double tol) const {
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return equal_with_abs_tol(t_, q.t_, tol) && r_.equals(q.r_, tol);
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}
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/* ************************************************************************* */
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Pose2 Pose2::Expmap(const Vector3& xi, OptionalJacobian<3, 3> H) {
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assert(xi.size() == 3);
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if (H) *H = Pose2::ExpmapDerivative(xi);
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const Point2 v(xi(0),xi(1));
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const double w = xi(2);
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if (std::abs(w) < 1e-10)
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return Pose2(xi[0], xi[1], xi[2]);
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else {
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const Rot2 R(Rot2::fromAngle(w));
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const Point2 v_ortho = R_PI_2 * v; // points towards rot center
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const Point2 t = (v_ortho - R.rotate(v_ortho)) / w;
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return Pose2(R, t);
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}
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}
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/* ************************************************************************* */
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Vector3 Pose2::Logmap(const Pose2& p, OptionalJacobian<3, 3> H) {
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if (H) *H = Pose2::LogmapDerivative(p);
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const Rot2& R = p.r();
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const Point2& t = p.t();
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double w = R.theta();
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if (std::abs(w) < 1e-10)
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return Vector3(t.x(), t.y(), w);
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else {
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double c_1 = R.c()-1.0, s = R.s();
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double det = c_1*c_1 + s*s;
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Point2 p = R_PI_2 * (R.unrotate(t) - t);
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Point2 v = (w/det) * p;
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return Vector3(v.x(), v.y(), w);
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}
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}
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/* ************************************************************************* */
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Pose2 Pose2::ChartAtOrigin::Retract(const Vector3& v, ChartJacobian H) {
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#ifdef GTSAM_SLOW_BUT_CORRECT_EXPMAP
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return Expmap(v, H);
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#else
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if (H) {
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*H = I_3x3;
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H->topLeftCorner<2,2>() = Rot2(-v[2]).matrix();
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}
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return Pose2(v[0], v[1], v[2]);
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#endif
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}
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/* ************************************************************************* */
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Vector3 Pose2::ChartAtOrigin::Local(const Pose2& r, ChartJacobian H) {
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#ifdef GTSAM_SLOW_BUT_CORRECT_EXPMAP
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return Logmap(r, H);
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#else
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if (H) {
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*H = I_3x3;
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H->topLeftCorner<2,2>() = r.rotation().matrix();
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}
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return Vector3(r.x(), r.y(), r.theta());
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#endif
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}
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/* ************************************************************************* */
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// Calculate Adjoint map
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// Ad_pose is 3*3 matrix that when applied to twist xi, returns Ad_pose(xi)
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Matrix3 Pose2::AdjointMap() const {
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double c = r_.c(), s = r_.s(), x = t_.x(), y = t_.y();
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Matrix3 rvalue;
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rvalue <<
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c, -s, y,
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s, c, -x,
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0.0, 0.0, 1.0;
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return rvalue;
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}
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/* ************************************************************************* */
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Matrix3 Pose2::adjointMap(const Vector3& v) {
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// See Chirikjian12book2, vol.2, pg. 36
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Matrix3 ad = Z_3x3;
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ad(0,1) = -v[2];
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ad(1,0) = v[2];
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ad(0,2) = v[1];
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ad(1,2) = -v[0];
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return ad;
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}
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/* ************************************************************************* */
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Matrix3 Pose2::ExpmapDerivative(const Vector3& v) {
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double alpha = v[2];
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Matrix3 J;
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if (std::abs(alpha) > 1e-5) {
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// Chirikjian11book2, pg. 36
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/* !!!Warning!!! Compare Iserles05an, formula 2.42 and Chirikjian11book2 pg.26
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* Iserles' right-trivialization dexpR is actually the left Jacobian J_l in Chirikjian's notation
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* In fact, Iserles 2.42 can be written as:
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* \dot{g} g^{-1} = dexpR_{q}\dot{q}
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* where q = A, and g = exp(A)
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* and the LHS is in the definition of J_l in Chirikjian11book2, pg. 26.
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* Hence, to compute ExpmapDerivative, we have to use the formula of J_r Chirikjian11book2, pg.36
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*/
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double sZalpha = sin(alpha)/alpha, c_1Zalpha = (cos(alpha)-1)/alpha;
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double v1Zalpha = v[0]/alpha, v2Zalpha = v[1]/alpha;
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J << sZalpha, -c_1Zalpha, v1Zalpha + v2Zalpha*c_1Zalpha - v1Zalpha*sZalpha,
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c_1Zalpha, sZalpha, -v1Zalpha*c_1Zalpha + v2Zalpha - v2Zalpha*sZalpha,
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0, 0, 1;
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}
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else {
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// Thanks to Krunal: Apply L'Hospital rule to several times to
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// compute the limits when alpha -> 0
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J << 1,0,-0.5*v[1],
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0,1, 0.5*v[0],
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0,0, 1;
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}
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return J;
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}
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/* ************************************************************************* */
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Matrix3 Pose2::LogmapDerivative(const Pose2& p) {
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Vector3 v = Logmap(p);
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double alpha = v[2];
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Matrix3 J;
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if (std::abs(alpha) > 1e-5) {
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double alphaInv = 1/alpha;
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double halfCotHalfAlpha = 0.5*sin(alpha)/(1-cos(alpha));
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double v1 = v[0], v2 = v[1];
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J << alpha*halfCotHalfAlpha, -0.5*alpha, v1*alphaInv - v1*halfCotHalfAlpha + 0.5*v2,
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0.5*alpha, alpha*halfCotHalfAlpha, v2*alphaInv - 0.5*v1 - v2*halfCotHalfAlpha,
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0, 0, 1;
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}
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else {
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J << 1,0, 0.5*v[1],
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0,1, -0.5*v[0],
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0,0, 1;
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}
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return J;
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}
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/* ************************************************************************* */
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Pose2 Pose2::inverse() const {
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return Pose2(r_.inverse(), r_.unrotate(Point2(-t_.x(), -t_.y())));
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}
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/* ************************************************************************* */
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// see doc/math.lyx, SE(2) section
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Point2 Pose2::transformTo(const Point2& point,
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OptionalJacobian<2, 3> Hpose, OptionalJacobian<2, 2> Hpoint) const {
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OptionalJacobian<2, 2> Htranslation = Hpose.cols<2>(0);
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OptionalJacobian<2, 1> Hrotation = Hpose.cols<1>(2);
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const Point2 q = r_.unrotate(point - t_, Hrotation, Hpoint);
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if (Htranslation) *Htranslation << -1.0, 0.0, 0.0, -1.0;
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return q;
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}
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Matrix Pose2::transformTo(const Matrix& points) const {
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if (points.rows() != 2) {
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throw std::invalid_argument("Pose2:transformTo expects 2*N matrix.");
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}
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const Matrix2 Rt = rotation().transpose();
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return Rt * (points.colwise() - t_); // Eigen broadcasting!
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}
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/* ************************************************************************* */
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// see doc/math.lyx, SE(2) section
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Point2 Pose2::transformFrom(const Point2& point,
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OptionalJacobian<2, 3> Hpose, OptionalJacobian<2, 2> Hpoint) const {
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OptionalJacobian<2, 2> Htranslation = Hpose.cols<2>(0);
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OptionalJacobian<2, 1> Hrotation = Hpose.cols<1>(2);
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const Point2 q = r_.rotate(point, Hrotation, Hpoint);
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if (Htranslation) *Htranslation = (Hpoint ? *Hpoint : r_.matrix());
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return q + t_;
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}
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Matrix Pose2::transformFrom(const Matrix& points) const {
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if (points.rows() != 2) {
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throw std::invalid_argument("Pose2:transformFrom expects 2*N matrix.");
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}
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const Matrix2 R = rotation().matrix();
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return (R * points).colwise() + t_; // Eigen broadcasting!
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}
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/* ************************************************************************* */
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Rot2 Pose2::bearing(const Point2& point,
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OptionalJacobian<1, 3> Hpose, OptionalJacobian<1, 2> Hpoint) const {
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// make temporary matrices
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Matrix23 D_d_pose; Matrix2 D_d_point;
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Point2 d = transformTo(point, Hpose ? &D_d_pose : 0, Hpoint ? &D_d_point : 0);
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if (!Hpose && !Hpoint) return Rot2::relativeBearing(d);
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Matrix12 D_result_d;
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Rot2 result = Rot2::relativeBearing(d, Hpose || Hpoint ? &D_result_d : 0);
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if (Hpose) *Hpose = D_result_d * D_d_pose;
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if (Hpoint) *Hpoint = D_result_d * D_d_point;
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return result;
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}
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/* ************************************************************************* */
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Rot2 Pose2::bearing(const Pose2& pose,
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OptionalJacobian<1, 3> Hpose, OptionalJacobian<1, 3> Hother) const {
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Matrix12 D2;
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Rot2 result = bearing(pose.t(), Hpose, Hother ? &D2 : 0);
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if (Hother) {
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Matrix12 H2_ = D2 * pose.r().matrix();
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*Hother << H2_, Z_1x1;
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}
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return result;
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}
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/* ************************************************************************* */
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double Pose2::range(const Point2& point,
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OptionalJacobian<1,3> Hpose, OptionalJacobian<1,2> Hpoint) const {
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Point2 d = point - t_;
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if (!Hpose && !Hpoint) return d.norm();
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Matrix12 D_r_d;
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double r = norm2(d, D_r_d);
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if (Hpose) {
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Matrix23 D_d_pose;
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D_d_pose << -r_.c(), r_.s(), 0.0,
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-r_.s(), -r_.c(), 0.0;
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*Hpose = D_r_d * D_d_pose;
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}
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if (Hpoint) *Hpoint = D_r_d;
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return r;
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}
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/* ************************************************************************* */
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double Pose2::range(const Pose2& pose,
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OptionalJacobian<1,3> Hpose,
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OptionalJacobian<1,3> Hother) const {
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Point2 d = pose.t() - t_;
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if (!Hpose && !Hother) return d.norm();
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Matrix12 D_r_d;
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double r = norm2(d, D_r_d);
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if (Hpose) {
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Matrix23 D_d_pose;
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D_d_pose <<
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-r_.c(), r_.s(), 0.0,
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-r_.s(), -r_.c(), 0.0;
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*Hpose = D_r_d * D_d_pose;
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}
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if (Hother) {
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Matrix23 D_d_other;
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D_d_other <<
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pose.r_.c(), -pose.r_.s(), 0.0,
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pose.r_.s(), pose.r_.c(), 0.0;
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*Hother = D_r_d * D_d_other;
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}
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return r;
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}
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/* *************************************************************************
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* Align finds the angle using a linear method:
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* a = Pose2::transformFrom(b) = t + R*b
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* We need to remove the centroids from the data to find the rotation
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* using db=[dbx;dby] and a=[dax;day] we have
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* |dax| |c -s| |dbx| |dbx -dby| |c|
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* | | = | | * | | = | | * | | = H_i*cs
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* |day| |s c| |dby| |dby dbx| |s|
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* where the Hi are the 2*2 matrices. Then we will minimize the criterion
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* J = \sum_i norm(a_i - H_i * cs)
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* Taking the derivative with respect to cs and setting to zero we have
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* cs = (\sum_i H_i' * a_i)/(\sum H_i'*H_i)
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* The hessian is diagonal and just divides by a constant, but this
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* normalization constant is irrelevant, since we take atan2.
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* i.e., cos ~ sum(dbx*dax + dby*day) and sin ~ sum(-dby*dax + dbx*day)
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* The translation is then found from the centroids
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* as they also satisfy ca = t + R*cb, hence t = ca - R*cb
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*/
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std::optional<Pose2> Pose2::Align(const Point2Pairs &ab_pairs) {
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const size_t n = ab_pairs.size();
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if (n < 2) {
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return {}; // we need at least 2 pairs
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}
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// calculate centroids
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Point2 ca(0, 0), cb(0, 0);
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for (const Point2Pair& pair : ab_pairs) {
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ca += pair.first;
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cb += pair.second;
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}
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const double f = 1.0/n;
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ca *= f;
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cb *= f;
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// calculate cos and sin
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double c = 0, s = 0;
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for (const Point2Pair& pair : ab_pairs) {
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Point2 da = pair.first - ca;
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Point2 db = pair.second - cb;
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c += db.x() * da.x() + db.y() * da.y();
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s += -db.y() * da.x() + db.x() * da.y();
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}
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// calculate angle and translation
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const double theta = atan2(s, c);
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const Rot2 R = Rot2::fromAngle(theta);
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const Point2 t = ca - R*cb;
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return Pose2(R, t);
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}
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std::optional<Pose2> Pose2::Align(const Matrix& a, const Matrix& b) {
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if (a.rows() != 2 || b.rows() != 2 || a.cols() != b.cols()) {
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throw std::invalid_argument(
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"Pose2:Align expects 2*N matrices of equal shape.");
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}
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Point2Pairs ab_pairs;
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for (Eigen::Index j = 0; j < a.cols(); j++) {
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ab_pairs.emplace_back(a.col(j), b.col(j));
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}
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return Pose2::Align(ab_pairs);
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}
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/* ************************************************************************* */
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} // namespace gtsam
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