SO3.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/base/Matrix.h>
#include <gtsam/base/Vector.h>
#include <gtsam/base/concepts.h>
#include <gtsam/geometry/Point3.h>
#include <gtsam/geometry/SO3.h>
 
#include <Eigen/SVD>
#include <cmath>
#include <limits>
 
namespace gtsam {
 
//******************************************************************************
namespace so3 {
 
static constexpr double one_6th = 1.0 / 6.0;
static constexpr double one_12th = 1.0 / 12.0;
static constexpr double one_24th = 1.0 / 24.0;
static constexpr double one_60th = 1.0 / 60.0;
static constexpr double one_120th = 1.0 / 120.0;
static constexpr double one_180th = 1.0 / 180.0;
static constexpr double one_720th = 1.0 / 720.0;
static constexpr double one_1260th = 1.0 / 1260.0;
 
GTSAM_EXPORT Matrix99 Dcompose(const SO3& Q) {
  Matrix99 H;
  auto R = Q.matrix();
  H << I_3x3 * R(0, 0), I_3x3 * R(1, 0), I_3x3 * R(2, 0),  //
      I_3x3 * R(0, 1), I_3x3 * R(1, 1), I_3x3 * R(2, 1),   //
      I_3x3 * R(0, 2), I_3x3 * R(1, 2), I_3x3 * R(2, 2);
  return H;
}
 
GTSAM_EXPORT Matrix3 compose(const Matrix3& M, const SO3& R,
                             OptionalJacobian<9, 9> H) {
  Matrix3 MR = M * R.matrix();
  if (H) *H = Dcompose(R);
  return MR;
}
 
void ExpmapFunctor::init(bool nearZeroApprox) {
  nearZero =
      nearZeroApprox || (theta2 <= std::numeric_limits<double>::epsilon());
  if (!nearZero) {
    const double sin_theta = std::sin(theta);
    A = sin_theta / theta;
    const double s2 = std::sin(theta / 2.0);
    const double one_minus_cos =
        2.0 * s2 * s2;  // numerically better than [1 - cos(theta)]
    B = one_minus_cos / theta2;
  } else {
    // Taylor expansion at 0
    A = 1.0 - theta2 * one_6th;
    B = 0.5 - theta2 * one_24th;
  }
}
 
ExpmapFunctor::ExpmapFunctor(const Vector3& omega, bool nearZeroApprox)
    : theta2(omega.dot(omega)),
      theta(std::sqrt(theta2)),
      W(skewSymmetric(omega)),
      WW(W * W) {
  init(nearZeroApprox);
}
 
ExpmapFunctor::ExpmapFunctor(const Vector3& axis, double angle,
                             bool nearZeroApprox)
    : theta2(angle * angle),
      theta(angle),
      W(skewSymmetric(axis * angle)),
      WW(W * W) {
  init(nearZeroApprox);
}
 
SO3 ExpmapFunctor::expmap() const { return SO3(I_3x3 + A * W + B * WW); }
 
DexpFunctor::DexpFunctor(const Vector3& omega, bool nearZeroApprox)
    : ExpmapFunctor(omega, nearZeroApprox), omega(omega) {
  if (!nearZero) {
    C = (1 - A) / theta2;
    D = (1.0 - A / (2.0 * B)) / theta2;
    E = (2.0 * B - A) / theta2;
    F = (3.0 * C - B) / theta2;
  } else {
    // Taylor expansion at 0
    // TODO(Frank): flipping signs here does not trigger any tests: harden!
    C = one_6th - theta2 * one_120th;
    D = one_12th + theta2 * one_720th;
    E = one_12th - theta2 * one_180th;
    F = one_60th - theta2 * one_1260th;
  }
}
 
Vector3 DexpFunctor::crossB(const Vector3& v, OptionalJacobian<3, 3> H) const {
  // Wv = omega x v
  const Vector3 Wv = gtsam::cross(omega, v);
  if (H) {
    // Apply product rule to (B Wv)
    // - E * omega.transpose() is 1x3 Jacobian of B with respect to omega
    // - skewSymmetric(v) is 3x3 Jacobian of Wv = gtsam::cross(omega, v)
    *H = - Wv * E * omega.transpose() - B * skewSymmetric(v);
  }
  return B * Wv;
}
 
Vector3 DexpFunctor::doubleCrossC(const Vector3& v,
                                  OptionalJacobian<3, 3> H) const {
  // WWv = omega x (omega x * v)
  Matrix3 doubleCrossJacobian;
  const Vector3 WWv =
      gtsam::doubleCross(omega, v, H ? &doubleCrossJacobian : nullptr);
  if (H) {
    // Apply product rule to (C WWv)
    // - F * omega.transpose() is 1x3 Jacobian of C with respect to omega
    // doubleCrossJacobian is 3x3 Jacobian of WWv = gtsam::doubleCross(omega, v)
    *H = - WWv * F * omega.transpose() + C * doubleCrossJacobian;
  }
  return C * WWv;
}
 
// Multiplies v with left Jacobian through vector operations only.
Vector3 DexpFunctor::applyDexp(const Vector3& v, OptionalJacobian<3, 3> H1,
                               OptionalJacobian<3, 3> H2) const {
  Matrix3 D_BWv_w, D_CWWv_w;
  const Vector3 BWv = crossB(v, H1 ? &D_BWv_w : nullptr);
  const Vector3 CWWv = doubleCrossC(v, H1 ? &D_CWWv_w : nullptr);
  if (H1) *H1 = -D_BWv_w + D_CWWv_w;
  if (H2) *H2 = rightJacobian();
  return v - BWv + CWWv;
}
 
Vector3 DexpFunctor::applyInvDexp(const Vector3& v, OptionalJacobian<3, 3> H1,
                                  OptionalJacobian<3, 3> H2) const {
  const Matrix3 invJr = rightJacobianInverse();
  const Vector3 c = invJr * v;
  if (H1) {
    Matrix3 H;
    applyDexp(c, H);  // get derivative H of forward mapping
    *H1 = -invJr * H;
  }
  if (H2) *H2 = invJr;
  return c;
}
 
Vector3 DexpFunctor::applyLeftJacobian(const Vector3& v,
                                       OptionalJacobian<3, 3> H1,
                                       OptionalJacobian<3, 3> H2) const {
  Matrix3 D_BWv_w, D_CWWv_w;
  const Vector3 BWv = crossB(v, H1 ? &D_BWv_w : nullptr);
  const Vector3 CWWv = doubleCrossC(v, H1 ? &D_CWWv_w : nullptr);
  if (H1) *H1 = D_BWv_w + D_CWWv_w;
  if (H2) *H2 = leftJacobian();
  return v + BWv + CWWv;
}
 
Vector3 DexpFunctor::applyLeftJacobianInverse(const Vector3& v,
                                              OptionalJacobian<3, 3> H1,
                                              OptionalJacobian<3, 3> H2) const {
  const Matrix3 invJl = leftJacobianInverse();
  const Vector3 c = invJl * v;
  if (H1) {
    Matrix3 H;
    applyLeftJacobian(c, H);  // get derivative H of forward mapping
    *H1 = -invJl * H;
  }
  if (H2) *H2 = invJl;
  return c;
}
 
}  // namespace so3
 
//******************************************************************************
template <>
GTSAM_EXPORT
SO3 SO3::AxisAngle(const Vector3& axis, double theta) {
  return so3::ExpmapFunctor(axis, theta).expmap();
}
 
//******************************************************************************
template <>
GTSAM_EXPORT
SO3 SO3::ClosestTo(const Matrix3& M) {
  Eigen::JacobiSVD<Matrix3> svd(M, Eigen::ComputeFullU | Eigen::ComputeFullV);
  const auto& U = svd.matrixU();
  const auto& V = svd.matrixV();
  const double det = (U * V.transpose()).determinant();
  return SO3(U * Vector3(1, 1, det).asDiagonal() * V.transpose());
}
 
//******************************************************************************
template <>
GTSAM_EXPORT
SO3 SO3::ChordalMean(const std::vector<SO3>& rotations) {
  // See Hartley13ijcv:
  // Cost function C(R) = \sum sqr(|R-R_i|_F)
  // Closed form solution = ClosestTo(C_e), where C_e = \sum R_i !!!!
  Matrix3 C_e{Z_3x3};
  for (const auto& R_i : rotations) {
    C_e += R_i.matrix();
  }
  return ClosestTo(C_e);
}
 
//******************************************************************************
template <>
GTSAM_EXPORT
Matrix3 SO3::Hat(const Vector3& xi) {
  return skewSymmetric(xi);
}
 
//******************************************************************************
template <>
GTSAM_EXPORT
Vector3 SO3::Vee(const Matrix3& X) {
  Vector3 xi;
  xi(0) = -X(1, 2);
  xi(1) = +X(0, 2);
  xi(2) = -X(0, 1);
  return xi;
}
 
//******************************************************************************
template <>
GTSAM_EXPORT
Matrix3 SO3::AdjointMap() const {
  return matrix_;
}
 
//******************************************************************************
template <>
GTSAM_EXPORT
SO3 SO3::Expmap(const Vector3& omega, ChartJacobian H) {
  if (H) {
    so3::DexpFunctor impl(omega);
    *H = impl.dexp();
    return impl.expmap();
  } else {
    return so3::ExpmapFunctor(omega).expmap();
  }
}
 
template <>
GTSAM_EXPORT
Matrix3 SO3::ExpmapDerivative(const Vector3& omega) {
  return so3::DexpFunctor(omega).dexp();
}
 
//******************************************************************************
/* Right Jacobian for Log map in SO(3) - equation (10.86) and following
 equations in G.S. Chirikjian, "Stochastic Models, Information Theory, and Lie
 Groups", Volume 2, 2008.
 
   logmap( Rhat * expmap(omega) ) \approx logmap(Rhat) + Jrinv * omega
 
 where Jrinv = LogmapDerivative(omega). This maps a perturbation on the
 manifold (expmap(omega)) to a perturbation in the tangent space (Jrinv *
 omega)
 */
template <>
GTSAM_EXPORT
Matrix3 SO3::LogmapDerivative(const Vector3& omega) {
  using std::cos;
  using std::sin;
 
  double theta2 = omega.dot(omega);
  if (theta2 <= std::numeric_limits<double>::epsilon()) return I_3x3;
  double theta = std::sqrt(theta2);  // rotation angle
 
  // element of Lie algebra so(3): W = omega^
  const Matrix3 W = Hat(omega);
  return I_3x3 + 0.5 * W +
         (1 / (theta * theta) - (1 + cos(theta)) / (2 * theta * sin(theta))) *
             W * W;
}
 
//******************************************************************************
template <>
GTSAM_EXPORT
Vector3 SO3::Logmap(const SO3& Q, ChartJacobian H) {
  using std::sin;
  using std::sqrt;
 
  // note switch to base 1
  const Matrix3& R = Q.matrix();
  const double &R11 = R(0, 0), R12 = R(0, 1), R13 = R(0, 2);
  const double &R21 = R(1, 0), R22 = R(1, 1), R23 = R(1, 2);
  const double &R31 = R(2, 0), R32 = R(2, 1), R33 = R(2, 2);
 
  // Get trace(R)
  const double tr = R.trace();
 
  Vector3 omega;
 
  // when trace == -1, i.e., when theta = +-pi, +-3pi, +-5pi, etc.
  // we do something special
  if (tr + 1.0 < 1e-3) {
    if (R33 > R22 && R33 > R11) {
      // R33 is the largest diagonal, a=3, b=1, c=2
      const double W = R21 - R12;
      const double Q1 = 2.0 + 2.0 * R33;
      const double Q2 = R31 + R13;
      const double Q3 = R23 + R32;
      const double r = sqrt(Q1);
      const double one_over_r = 1 / r;
      const double norm = sqrt(Q1*Q1 + Q2*Q2 + Q3*Q3 + W*W);
      const double sgn_w = W < 0 ? -1.0 : 1.0;
      const double mag = M_PI - (2 * sgn_w * W) / norm;
      const double scale = 0.5 * one_over_r * mag;
      omega = sgn_w * scale * Vector3(Q2, Q3, Q1);
    } else if (R22 > R11) {
      // R22 is the largest diagonal, a=2, b=3, c=1
      const double W = R13 - R31;
      const double Q1 = 2.0 + 2.0 * R22;
      const double Q2 = R23 + R32;
      const double Q3 = R12 + R21;
      const double r = sqrt(Q1);
      const double one_over_r = 1 / r;
      const double norm = sqrt(Q1*Q1 + Q2*Q2 + Q3*Q3 + W*W);
      const double sgn_w = W < 0 ? -1.0 : 1.0;
      const double mag = M_PI - (2 * sgn_w * W) / norm;
      const double scale = 0.5 * one_over_r * mag;
      omega = sgn_w * scale * Vector3(Q3, Q1, Q2);
    } else {
      // R11 is the largest diagonal, a=1, b=2, c=3
      const double W = R32 - R23;
      const double Q1 = 2.0 + 2.0 * R11;
      const double Q2 = R12 + R21;
      const double Q3 = R31 + R13;
      const double r = sqrt(Q1);
      const double one_over_r = 1 / r;
      const double norm = sqrt(Q1*Q1 + Q2*Q2 + Q3*Q3 + W*W);
      const double sgn_w = W < 0 ? -1.0 : 1.0;
      const double mag = M_PI - (2 * sgn_w * W) / norm;
      const double scale = 0.5 * one_over_r * mag;
      omega = sgn_w * scale * Vector3(Q1, Q2, Q3);
    }
  } else {
    double magnitude;
    const double tr_3 = tr - 3.0; // could be non-negative if the matrix is off orthogonal
    if (tr_3 < -1e-6) {
      // this is the normal case -1 < trace < 3
      double theta = acos((tr - 1.0) / 2.0);
      magnitude = theta / (2.0 * sin(theta));
    } else {
      // when theta near 0, +-2pi, +-4pi, etc. (trace near 3.0)
      // use Taylor expansion: theta \approx 1/2-(t-3)/12 + O((t-3)^2)
      // see https://github.com/borglab/gtsam/issues/746 for details
      magnitude = 0.5 - tr_3 / 12.0 + tr_3*tr_3/60.0;
    }
    omega = magnitude * Vector3(R32 - R23, R13 - R31, R21 - R12);
  }
 
  if (H) *H = LogmapDerivative(omega);
  return omega;
}
 
//******************************************************************************
// Chart at origin for SO3 is *not* Cayley but actual Expmap/Logmap
 
template <>
GTSAM_EXPORT
SO3 SO3::ChartAtOrigin::Retract(const Vector3& omega, ChartJacobian H) {
  return Expmap(omega, H);
}
 
template <>
GTSAM_EXPORT
Vector3 SO3::ChartAtOrigin::Local(const SO3& R, ChartJacobian H) {
  return Logmap(R, H);
}
 
//******************************************************************************
// local vectorize
static Vector9 vec3(const Matrix3& R) {
  return Eigen::Map<const Vector9>(R.data());
}
 
// so<3> generators
static std::vector<Matrix3> G3({SO3::Hat(Vector3::Unit(0)),
                                SO3::Hat(Vector3::Unit(1)),
                                SO3::Hat(Vector3::Unit(2))});
 
// vectorized generators
static const Matrix93 P3 =
    (Matrix93() << vec3(G3[0]), vec3(G3[1]), vec3(G3[2])).finished();
 
//******************************************************************************
template <>
GTSAM_EXPORT
Vector9 SO3::vec(OptionalJacobian<9, 3> H) const {
  const Matrix3& R = matrix_;
  if (H) {
    // As Luca calculated (for SO4), this is (I3 \oplus R) * P3
    *H << R * P3.block<3, 3>(0, 0), R * P3.block<3, 3>(3, 0),
        R * P3.block<3, 3>(6, 0);
  }
  return gtsam::vec3(R);
}
//******************************************************************************
 
}  // end namespace gtsam