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gtsam/gtsam/geometry/Pose3.cpp

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/* ----------------------------------------------------------------------------
* GTSAM Copyright 2010, Georgia Tech Research Corporation,
* Atlanta, Georgia 30332-0415
* All Rights Reserved
* Authors: Frank Dellaert, et al. (see THANKS for the full author list)
* See LICENSE for the license information
* -------------------------------------------------------------------------- */
/**
* @file Pose3.cpp
* @brief 3D Pose manifold SO(3) x R^3 and group SE(3)
*/
#include <gtsam/base/concepts.h>
#include <gtsam/geometry/Pose2.h>
#include <gtsam/geometry/Pose3.h>
#include <gtsam/geometry/Rot3.h>
#include <gtsam/geometry/concepts.h>
#include <cmath>
#include <iostream>
#include <string>
namespace gtsam {
/** instantiate concept checks */
GTSAM_CONCEPT_POSE_INST(Pose3)
/* ************************************************************************* */
Pose3::Pose3(const Pose2& pose2) :
R_(Rot3::Rodrigues(0, 0, pose2.theta())), t_(
Point3(pose2.x(), pose2.y(), 0)) {
}
/* ************************************************************************* */
Pose3 Pose3::Create(const Rot3& R, const Point3& t, OptionalJacobian<6, 3> HR,
OptionalJacobian<6, 3> Ht) {
if (HR) *HR << I_3x3, Z_3x3;
if (Ht) *Ht << Z_3x3, R.transpose();
return Pose3(R, t);
}
// Pose2 constructor Jacobian is always the same.
static const Matrix63 Hpose2 = (Matrix63() << //
0., 0., 0., //
0., 0., 0.,//
0., 0., 1.,//
1., 0., 0.,//
0., 1., 0.,//
0., 0., 0.).finished();
Pose3 Pose3::FromPose2(const Pose2& p, OptionalJacobian<6, 3> H) {
if (H) *H << Hpose2;
return Pose3(p);
}
/* ************************************************************************* */
Pose3 Pose3::inverse() const {
Rot3 Rt = R_.inverse();
return Pose3(Rt, Rt * (-t_));
}
/* ************************************************************************* */
// Calculate Adjoint map
// Ad_pose is 6*6 matrix that when applied to twist xi, returns Ad_pose(xi)
Matrix6 Pose3::AdjointMap() const {
const Matrix3 R = R_.matrix();
Matrix3 A = skewSymmetric(t_.x(), t_.y(), t_.z()) * R;
Matrix6 adj;
adj << R, Z_3x3, A, R; // Gives [R 0; A R]
return adj;
}
/* ************************************************************************* */
// Calculate AdjointMap applied to xi_b, with Jacobians
Vector6 Pose3::Adjoint(const Vector6& xi_b, OptionalJacobian<6, 6> H_pose,
OptionalJacobian<6, 6> H_xib) const {
const Matrix6 Ad = AdjointMap();
// Jacobians
// D1 Ad_T(xi_b) = D1 Ad_T Ad_I(xi_b) = Ad_T * D1 Ad_I(xi_b) = Ad_T * ad_xi_b
// D2 Ad_T(xi_b) = Ad_T
// See docs/math.pdf for more details.
// In D1 calculation, we could be more efficient by writing it out, but do not
// for readability
if (H_pose) *H_pose = -Ad * adjointMap(xi_b);
if (H_xib) *H_xib = Ad;
return Ad * xi_b;
}
/* ************************************************************************* */
/// The dual version of Adjoint
Vector6 Pose3::AdjointTranspose(const Vector6& x, OptionalJacobian<6, 6> H_pose,
OptionalJacobian<6, 6> H_x) const {
const Matrix6 Ad = AdjointMap();
const Vector6 AdTx = Ad.transpose() * x;
// Jacobians
// See docs/math.pdf for more details.
if (H_pose) {
const auto w_T_hat = skewSymmetric(AdTx.head<3>()),
v_T_hat = skewSymmetric(AdTx.tail<3>());
*H_pose << w_T_hat, v_T_hat, //
/* */ v_T_hat, Z_3x3;
}
if (H_x) {
*H_x = Ad.transpose();
}
return AdTx;
}
/* ************************************************************************* */
Matrix6 Pose3::adjointMap(const Vector6& xi) {
Matrix3 w_hat = skewSymmetric(xi(0), xi(1), xi(2));
Matrix3 v_hat = skewSymmetric(xi(3), xi(4), xi(5));
Matrix6 adj;
adj << w_hat, Z_3x3, v_hat, w_hat;
return adj;
}
/* ************************************************************************* */
Vector6 Pose3::adjoint(const Vector6& xi, const Vector6& y,
OptionalJacobian<6, 6> Hxi, OptionalJacobian<6, 6> H_y) {
if (Hxi) {
Hxi->setZero();
for (int i = 0; i < 6; ++i) {
Vector6 dxi;
dxi.setZero();
dxi(i) = 1.0;
Matrix6 Gi = adjointMap(dxi);
Hxi->col(i) = Gi * y;
}
}
const Matrix6& ad_xi = adjointMap(xi);
if (H_y) *H_y = ad_xi;
return ad_xi * y;
}
/* ************************************************************************* */
Vector6 Pose3::adjointTranspose(const Vector6& xi, const Vector6& y,
OptionalJacobian<6, 6> Hxi, OptionalJacobian<6, 6> H_y) {
if (Hxi) {
Hxi->setZero();
for (int i = 0; i < 6; ++i) {
Vector6 dxi;
dxi.setZero();
dxi(i) = 1.0;
Matrix6 GTi = adjointMap(dxi).transpose();
Hxi->col(i) = GTi * y;
}
}
const Matrix6& adT_xi = adjointMap(xi).transpose();
if (H_y) *H_y = adT_xi;
return adT_xi * y;
}
/* ************************************************************************* */
Matrix4 Pose3::Hat(const Vector6& xi) {
Matrix4 X;
const double wx = xi(0), wy = xi(1), wz = xi(2), vx = xi(3), vy = xi(4), vz = xi(5);
X << 0., -wz, wy, vx, wz, 0., -wx, vy, -wy, wx, 0., vz, 0., 0., 0., 0.;
return X;
}
/* ************************************************************************* */
Vector6 Pose3::Vee(const Matrix4& Xi) {
Vector6 xi;
xi << Xi(2, 1), Xi(0, 2), Xi(1, 0), Xi(0, 3), Xi(1, 3), Xi(2, 3);
return xi;
}
/* ************************************************************************* */
void Pose3::print(const std::string& s) const {
std::cout << (s.empty() ? s : s + " ") << *this << std::endl;
}
/* ************************************************************************* */
bool Pose3::equals(const Pose3& pose, double tol) const {
return R_.equals(pose.R_, tol) && traits<Point3>::Equals(t_, pose.t_, tol);
}
/* ************************************************************************* */
Pose3 Pose3::interpolateRt(const Pose3& T, double t,
OptionalJacobian<6, 6> Hself,
OptionalJacobian<6, 6> Harg,
OptionalJacobian<6, 1> Ht) const {
if(Hself || Harg || Ht){
typename MakeJacobian<Rot3, Rot3>::type HselfRot, HargRot;
typename MakeJacobian<Rot3, double>::type HtRot;
typename MakeJacobian<Point3, Point3>::type HselfPoint, HargPoint;
typename MakeJacobian<Point3, double>::type HtPoint;
Rot3 Rint = interpolate<Rot3>(R_, T.R_, t, HselfRot, HargRot, HtRot);
Point3 Pint = interpolate<Point3>(t_, T.t_, t, HselfPoint, HargPoint, HtPoint);
Pose3 result = Pose3(Rint, Pint);
if(Hself) *Hself << HselfRot, Z_3x3, Z_3x3, Rint.transpose() * R_.matrix() * HselfPoint;
if(Harg) *Harg << HargRot, Z_3x3, Z_3x3, Rint.transpose() * T.R_.matrix() * HargPoint;
if(Ht) *Ht << HtRot, Rint.transpose() * HtPoint;
return result;
}
return Pose3(interpolate<Rot3>(R_, T.R_, t),
interpolate<Point3>(t_, T.t_, t));
}
/* ************************************************************************* */
// Expmap is implemented in so3::ExpmapFunctor::expmap, based on Ethan Eade's
// elegant Lie group document, at https://www.ethaneade.org/lie.pdf.
Pose3 Pose3::Expmap(const Vector6& xi, OptionalJacobian<6, 6> Hxi) {
// Get angular velocity omega and translational velocity v from twist xi
const Vector3 w = xi.head<3>(), v = xi.tail<3>();
// Instantiate functor for Dexp-related operations:
const bool nearZero = (w.dot(w) <= 1e-5);
const so3::DexpFunctor local(w, nearZero);
// Compute rotation using Expmap
#ifdef GTSAM_USE_QUATERNIONS
const Rot3 R = traits<gtsam::Quaternion>::Expmap(w);
#else
const Rot3 R(local.expmap());
#endif
// The translation t = local.leftJacobian() * v.
// Here we call applyLeftJacobian, which is faster if you don't need
// Jacobians, and returns Jacobian of t with respect to w if asked.
// NOTE(Frank): t = applyLeftJacobian(v) does the same as the intuitive formulas
// t_parallel = w * w.dot(v); // translation parallel to axis
// w_cross_v = w.cross(v); // translation orthogonal to axis
// t = (w_cross_v - Rot3::Expmap(w) * w_cross_v + t_parallel) / theta2;
// but functor does not need R, deals automatically with the case where theta2
// is near zero, and also gives us the machinery for the Jacobians.
Matrix3 H;
const Vector3 t = local.applyLeftJacobian(v, Hxi ? &H : nullptr);
if (Hxi) {
// The Jacobian of expmap is given by the right Jacobian of SO(3):
const Matrix3 Jr = local.rightJacobian();
// We are creating a Pose3, so we still need to chain H with R^T, the
// Jacobian of Pose3::Create with respect to t.
const Matrix3 Q = R.matrix().transpose() * H;
*Hxi << Jr, Z_3x3, // Jr here *is* the Jacobian of expmap
Q, Jr; // Here Jr = R^T * J_l, with J_l the Jacobian of t in v.
}
return Pose3(R, t);
}
/* ************************************************************************* */
Vector6 Pose3::Logmap(const Pose3& pose, OptionalJacobian<6, 6> Hpose) {
if (Hpose) *Hpose = LogmapDerivative(pose);
const Vector3 w = Rot3::Logmap(pose.rotation());
const Vector3 T = pose.translation();
const double t = w.norm();
if (t < 1e-10) {
Vector6 log;
log << w, T;
return log;
} else {
const Matrix3 W = skewSymmetric(w / t);
// Formula from Agrawal06iros, equation (14)
// simplified with Mathematica, and multiplying in T to avoid matrix math
const double Tan = tan(0.5 * t);
const Vector3 WT = W * T;
const Vector3 u = T - (0.5 * t) * WT + (1 - t / (2. * Tan)) * (W * WT);
Vector6 log;
log << w, u;
return log;
}
}
/* ************************************************************************* */
Pose3 Pose3::ChartAtOrigin::Retract(const Vector6& xi, ChartJacobian Hxi) {
#ifdef GTSAM_POSE3_EXPMAP
return Expmap(xi, Hxi);
#else
Matrix3 DR;
Rot3 R = Rot3::Retract(xi.head<3>(), Hxi ? &DR : 0);
if (Hxi) {
*Hxi = I_6x6;
Hxi->topLeftCorner<3, 3>() = DR;
}
return Pose3(R, Point3(xi.tail<3>()));
#endif
}
/* ************************************************************************* */
Vector6 Pose3::ChartAtOrigin::Local(const Pose3& pose, ChartJacobian Hpose) {
#ifdef GTSAM_POSE3_EXPMAP
return Logmap(pose, Hpose);
#else
Matrix3 DR;
Vector3 omega = Rot3::LocalCoordinates(pose.rotation(), Hpose ? &DR : 0);
if (Hpose) {
*Hpose = I_6x6;
Hpose->topLeftCorner<3, 3>() = DR;
}
Vector6 xi;
xi << omega, pose.translation();
return xi;
#endif
}
/* ************************************************************************* */
Matrix3 Pose3::ComputeQforExpmapDerivative(const Vector6& xi,
double nearZeroThreshold) {
const auto w = xi.head<3>();
const auto v = xi.tail<3>();
// Instantiate functor for Dexp-related operations:
bool nearZero = (w.dot(w) <= nearZeroThreshold);
so3::DexpFunctor local(w, nearZero);
// Call applyLeftJacobian to get its Jacobian
Matrix3 H;
local.applyLeftJacobian(v, H);
// Multiply with R^T to account for the Pose3::Create Jacobian.
const Matrix3 R = local.expmap();
return R.transpose() * H;
}
/* ************************************************************************* */
Matrix6 Pose3::ExpmapDerivative(const Vector6& xi) {
Matrix6 J;
Expmap(xi, J);
return J;
}
/* ************************************************************************* */
Matrix6 Pose3::LogmapDerivative(const Pose3& pose) {
const Vector6 xi = Logmap(pose);
const Vector3 w = xi.head<3>();
const Matrix3 Jw = Rot3::LogmapDerivative(w);
const Matrix3 Q = ComputeQforExpmapDerivative(xi);
const Matrix3 Q2 = -Jw*Q*Jw;
Matrix6 J;
J << Jw, Z_3x3, Q2, Jw;
return J;
}
/* ************************************************************************* */
const Point3& Pose3::translation(OptionalJacobian<3, 6> Hself) const {
if (Hself) *Hself << Z_3x3, rotation().matrix();
return t_;
}
/* ************************************************************************* */
const Rot3& Pose3::rotation(OptionalJacobian<3, 6> Hself) const {
if (Hself) {
*Hself << I_3x3, Z_3x3;
}
return R_;
}
/* ************************************************************************* */
Matrix4 Pose3::matrix() const {
static const auto A14 = Eigen::RowVector4d(0,0,0,1);
Matrix4 mat;
mat << R_.matrix(), t_, A14;
return mat;
}
/* ************************************************************************* */
Pose3 Pose3::transformPoseFrom(const Pose3& aTb, OptionalJacobian<6, 6> Hself,
OptionalJacobian<6, 6> HaTb) const {
const Pose3& wTa = *this;
return wTa.compose(aTb, Hself, HaTb);
}
/* ************************************************************************* */
Pose3 Pose3::transformPoseTo(const Pose3& wTb, OptionalJacobian<6, 6> Hself,
OptionalJacobian<6, 6> HwTb) const {
if (Hself) *Hself = -wTb.inverse().AdjointMap() * AdjointMap();
if (HwTb) *HwTb = I_6x6;
const Pose3& wTa = *this;
return wTa.inverse() * wTb;
}
/* ************************************************************************* */
Point3 Pose3::transformFrom(const Point3& point, OptionalJacobian<3, 6> Hself,
OptionalJacobian<3, 3> Hpoint) const {
// Only get matrix once, to avoid multiple allocations,
// as well as multiple conversions in the Quaternion case
const Matrix3 R = R_.matrix();
if (Hself) {
Hself->leftCols<3>() = R * skewSymmetric(-point.x(), -point.y(), -point.z());
Hself->rightCols<3>() = R;
}
if (Hpoint) {
*Hpoint = R;
}
return R_ * point + t_;
}
Matrix Pose3::transformFrom(const Matrix& points) const {
if (points.rows() != 3) {
throw std::invalid_argument("Pose3:transformFrom expects 3*N matrix.");
}
const Matrix3 R = R_.matrix();
return (R * points).colwise() + t_; // Eigen broadcasting!
}
/* ************************************************************************* */
Point3 Pose3::transformTo(const Point3& point, OptionalJacobian<3, 6> Hself,
OptionalJacobian<3, 3> Hpoint) const {
// Only get transpose once, to avoid multiple allocations,
// as well as multiple conversions in the Quaternion case
const Matrix3 Rt = R_.transpose();
const Point3 q(Rt*(point - t_));
if (Hself) {
const double wx = q.x(), wy = q.y(), wz = q.z();
(*Hself) <<
0.0, -wz, +wy,-1.0, 0.0, 0.0,
+wz, 0.0, -wx, 0.0,-1.0, 0.0,
-wy, +wx, 0.0, 0.0, 0.0,-1.0;
}
if (Hpoint) {
*Hpoint = Rt;
}
return q;
}
Matrix Pose3::transformTo(const Matrix& points) const {
if (points.rows() != 3) {
throw std::invalid_argument("Pose3:transformTo expects 3*N matrix.");
}
const Matrix3 Rt = R_.transpose();
return Rt * (points.colwise() - t_); // Eigen broadcasting!
}
/* ************************************************************************* */
double Pose3::range(const Point3& point, OptionalJacobian<1, 6> Hself,
OptionalJacobian<1, 3> Hpoint) const {
Matrix36 D_local_pose;
Matrix3 D_local_point;
Point3 local = transformTo(point, Hself ? &D_local_pose : 0, Hpoint ? &D_local_point : 0);
if (!Hself && !Hpoint) {
return local.norm();
} else {
Matrix13 D_r_local;
const double r = norm3(local, D_r_local);
if (Hself) *Hself = D_r_local * D_local_pose;
if (Hpoint) *Hpoint = D_r_local * D_local_point;
return r;
}
}
/* ************************************************************************* */
double Pose3::range(const Pose3& pose, OptionalJacobian<1, 6> Hself,
OptionalJacobian<1, 6> Hpose) const {
Matrix36 D_point_pose;
Matrix13 D_local_point;
Point3 point = pose.translation(Hpose ? &D_point_pose : 0);
double r = range(point, Hself, Hpose ? &D_local_point : 0);
if (Hpose) *Hpose = D_local_point * D_point_pose;
return r;
}
/* ************************************************************************* */
Unit3 Pose3::bearing(const Point3& point, OptionalJacobian<2, 6> Hself,
OptionalJacobian<2, 3> Hpoint) const {
Matrix36 D_local_pose;
Matrix3 D_local_point;
Point3 local = transformTo(point, Hself ? &D_local_pose : 0, Hpoint ? &D_local_point : 0);
if (!Hself && !Hpoint) {
return Unit3(local);
} else {
Matrix23 D_b_local;
Unit3 b = Unit3::FromPoint3(local, D_b_local);
if (Hself) *Hself = D_b_local * D_local_pose;
if (Hpoint) *Hpoint = D_b_local * D_local_point;
return b;
}
}
/* ************************************************************************* */
Unit3 Pose3::bearing(const Pose3& pose, OptionalJacobian<2, 6> Hself,
OptionalJacobian<2, 6> Hpose) const {
Matrix36 D_point_pose;
Matrix23 D_local_point;
Point3 point = pose.translation(Hpose ? &D_point_pose : 0);
Unit3 b = bearing(point, Hself, Hpose ? &D_local_point : 0);
if (Hpose) *Hpose = D_local_point * D_point_pose;
return b;
}
/* ************************************************************************* */
std::optional<Pose3> Pose3::Align(const Point3Pairs &abPointPairs) {
const size_t n = abPointPairs.size();
if (n < 3) {
return {}; // we need at least three pairs
}
// calculate centroids
const auto centroids = means(abPointPairs);
// Add to form H matrix
Matrix3 H = Z_3x3;
for (const Point3Pair &abPair : abPointPairs) {
const Point3 da = abPair.first - centroids.first;
const Point3 db = abPair.second - centroids.second;
H += da * db.transpose();
}
// ClosestTo finds rotation matrix closest to H in Frobenius sense
const Rot3 aRb = Rot3::ClosestTo(H);
const Point3 aTb = centroids.first - aRb * centroids.second;
return Pose3(aRb, aTb);
}
std::optional<Pose3> Pose3::Align(const Matrix& a, const Matrix& b) {
if (a.rows() != 3 || b.rows() != 3 || a.cols() != b.cols()) {
throw std::invalid_argument(
"Pose3:Align expects 3*N matrices of equal shape.");
}
Point3Pairs abPointPairs;
for (Eigen::Index j = 0; j < a.cols(); j++) {
abPointPairs.emplace_back(a.col(j), b.col(j));
}
return Pose3::Align(abPointPairs);
}
/* ************************************************************************* */
Pose3 Pose3::slerp(double t, const Pose3& other, OptionalJacobian<6, 6> Hx, OptionalJacobian<6, 6> Hy) const {
return interpolate(*this, other, t, Hx, Hy);
}
/* ************************************************************************* */
// Compute vectorized Lie algebra generators for SE(3)
using Matrix16x6 = Eigen::Matrix<double, 16, 6>;
using Vector16 = Eigen::Matrix<double, 16, 1>;
static Matrix16x6 VectorizedGenerators() {
Matrix16x6 G;
for (size_t j = 0; j < 6; j++) {
const Matrix4 X = Pose3::Hat(Vector::Unit(6, j));
G.col(j) = Eigen::Map<const Vector16>(X.data());
}
return G;
}
Vector Pose3::vec(OptionalJacobian<16, 6> H) const {
// Vectorize
const Matrix4 M = matrix();
const Vector X = Eigen::Map<const Vector16>(M.data());
// If requested, calculate H as (I_4 \oplus M) * G.
if (H) {
static const Matrix16x6 G = VectorizedGenerators(); // static to compute only once
for (size_t i = 0; i < 4; i++)
H->block(i * 4, 0, 4, dimension) = M * G.block(i * 4, 0, 4, dimension);
}
return X;
}
/* ************************************************************************* */
std::ostream &operator<<(std::ostream &os, const Pose3& pose) {
// Both Rot3 and Point3 have ostream definitions so we use them.
os << "R: " << pose.rotation() << "\n";
os << "t: " << pose.translation().transpose();
return os;
}
} // namespace gtsam
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