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

 * -------------------------------------------------------------------------- */

#include <gtsam/geometry/Pose3.h>
#include <gtsam/geometry/Pose2.h>
#include <gtsam/geometry/concepts.h>
#include <gtsam/base/concepts.h>

#include <cmath>
#include <iostream>
#include <limits>
#include <string>

namespace gtsam {

using std::vector;

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);
}

/* ************************************************************************* */
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;
}

/* ************************************************************************* */
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;
}

/* ************************************************************************* */
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) const {
  return Pose3(interpolate<Rot3>(R_, T.R_, t),
                interpolate<Point3>(t_, T.t_, t));
}

/* ************************************************************************* */
Pose3 Pose3::Expmap(const Vector6& xi, OptionalJacobian<6, 6> Hxi) {
  if (Hxi) *Hxi = ExpmapDerivative(xi);

  // get angular velocity omega and translational velocity v from twist xi
  Vector3 omega(xi(0), xi(1), xi(2)), v(xi(3), xi(4), xi(5));

  Rot3 R = Rot3::Expmap(omega);
  double theta2 = omega.dot(omega);
  if (theta2 > std::numeric_limits<double>::epsilon()) {
    Vector3 t_parallel = omega * omega.dot(v);  // translation parallel to axis
    Vector3 omega_cross_v = omega.cross(v);     // points towards axis
    Vector3 t = (omega_cross_v - R * omega_cross_v + t_parallel) / theta2;
    return Pose3(R, t);
  } else {
    return Pose3(R, v);
  }
}

/* ************************************************************************* */
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>();
  const Matrix3 V = skewSymmetric(v);
  const Matrix3 W = skewSymmetric(w);

  Matrix3 Q;

#ifdef NUMERICAL_EXPMAP_DERIV
  Matrix3 Qj = Z_3x3;
  double invFac = 1.0;
  Q = Z_3x3;
  Matrix3 Wj = I_3x3;
  for (size_t j=1; j<10; ++j) {
    Qj = Qj*W + Wj*V;
    invFac = -invFac/(j+1);
    Q = Q + invFac*Qj;
    Wj = Wj*W;
  }
#else
  // The closed-form formula in Barfoot14tro eq. (102)
  double phi = w.norm();
  const Matrix3 WVW = W * V * W;
  if (std::abs(phi) > nearZeroThreshold) {
    const double s = sin(phi), c = cos(phi);
    const double phi2 = phi * phi, phi3 = phi2 * phi, phi4 = phi3 * phi,
                 phi5 = phi4 * phi;
    // Invert the sign of odd-order terms to have the right Jacobian
    Q = -0.5 * V + (phi - s) / phi3 * (W * V + V * W - WVW) +
        (1 - phi2 / 2 - c) / phi4 * (W * W * V + V * W * W - 3 * WVW) -
        0.5 * ((1 - phi2 / 2 - c) / phi4 - 3 * (phi - s - phi3 / 6.) / phi5) *
            (WVW * W + W * WVW);
  } else {
    Q = -0.5 * V + 1. / 6. * (W * V + V * W - WVW) -
        1. / 24. * (W * W * V + V * W * W - 3 * WVW) +
        1. / 120. * (WVW * W + W * WVW);
  }
#endif

  return Q;
}

/* ************************************************************************* */
Matrix6 Pose3::ExpmapDerivative(const Vector6& xi) {
  const Vector3 w = xi.head<3>();
  const Matrix3 Jw = Rot3::ExpmapDerivative(w);
  const Matrix3 Q = ComputeQforExpmapDerivative(xi);
  Matrix6 J;
  J << Jw, Z_3x3, Q, Jw;
  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 {
  Matrix13 D_local_point;
  double r = range(pose.translation(), Hself, Hpose ? &D_local_point : 0);
  if (Hpose) *Hpose << Matrix13::Zero(), D_local_point * pose.rotation().matrix();
  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 {
  if (Hpose) {
    Hpose->setZero();
    return bearing(pose.translation(), Hself, Hpose.cols<3>(3));
  }
  return bearing(pose.translation(), Hself, {});
}

/* ************************************************************************* */
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);
}

/* ************************************************************************* */
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