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gtsam/gtsam/geometry/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
* -------------------------------------------------------------------------- */
/**
* @file SO3.cpp
* @brief 3*3 matrix representation of SO(3)
* @author Frank Dellaert
* @author Luca Carlone
* @author Duy Nguyen Ta
* @date December 2014
*/
#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);
}
Matrix3 ExpmapFunctor::expmap() const { return 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(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 local(omega);
*H = local.dexp();
return SO3(local.expmap());
} else {
return SO3(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
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