gtsam/gtsam/geometry/Rot3M.cpp

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

 * 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    Rot3M.cpp
 * @brief   Rotation (internal: 3*3 matrix representation*)
 * @author  Alireza Fathi
 * @author  Christian Potthast
 * @author  Frank Dellaert
 * @author  Richard Roberts
 */

#include <gtsam/config.h> // Get GTSAM_USE_QUATERNIONS macro

#ifndef GTSAM_USE_QUATERNIONS

#include <gtsam/geometry/Rot3.h>
#include <boost/math/constants/constants.hpp>
#include <cmath>

using namespace std;

namespace gtsam {

static const Matrix3 I3 = Matrix3::Identity();

/* ************************************************************************* */
Rot3::Rot3() : rot_(Matrix3::Identity()), transpose_(Matrix3::Identity()) {}

/* ************************************************************************* */
Rot3::Rot3(const Point3& col1, const Point3& col2, const Point3& col3) {
  rot_.col(0) = col1.vector();
  rot_.col(1) = col2.vector();
  rot_.col(2) = col3.vector();
  transpose_ = rot_.transpose();
}

/* ************************************************************************* */
Rot3::Rot3(double R11, double R12, double R13,
    double R21, double R22, double R23,
    double R31, double R32, double R33) {
    rot_ << R11, R12, R13,
        R21, R22, R23,
        R31, R32, R33;
    transpose_ = rot_.transpose();
}

/* ************************************************************************* */
Rot3::Rot3(const Matrix3& R) {
  rot_ = R;
  transpose_ = rot_.transpose();
}

/* ************************************************************************* */
Rot3::Rot3(const Matrix& R) {
  if (R.rows()!=3 || R.cols()!=3)
    throw invalid_argument("Rot3 constructor expects 3*3 matrix");
  rot_ = R;
  transpose_ = rot_.transpose();
}

/* ************************************************************************* */
Rot3::Rot3(const Quaternion& q) : rot_(q.toRotationMatrix()) {
  transpose_ = rot_.transpose();
}

/* ************************************************************************* */
Rot3 Rot3::Rx(double t) {
  double st = sin(t), ct = cos(t);
  return Rot3(
      1,  0,  0,
      0, ct,-st,
      0, st, ct);
}

/* ************************************************************************* */
Rot3 Rot3::Ry(double t) {
  double st = sin(t), ct = cos(t);
  return Rot3(
      ct, 0, st,
      0, 1,  0,
      -st, 0, ct);
}

/* ************************************************************************* */
Rot3 Rot3::Rz(double t) {
  double st = sin(t), ct = cos(t);
  return Rot3(
      ct,-st, 0,
      st, ct, 0,
      0,  0, 1);
}

/* ************************************************************************* */
// Considerably faster than composing matrices above !
Rot3 Rot3::RzRyRx(double x, double y, double z) {
  double cx=cos(x),sx=sin(x);
  double cy=cos(y),sy=sin(y);
  double cz=cos(z),sz=sin(z);
  double ss_ = sx * sy;
  double cs_ = cx * sy;
  double sc_ = sx * cy;
  double cc_ = cx * cy;
  double c_s = cx * sz;
  double s_s = sx * sz;
  double _cs = cy * sz;
  double _cc = cy * cz;
  double s_c = sx * cz;
  double c_c = cx * cz;
  double ssc = ss_ * cz, csc = cs_ * cz, sss = ss_ * sz, css = cs_ * sz;
  return Rot3(
      _cc,- c_s + ssc,  s_s + csc,
      _cs,  c_c + sss, -s_c + css,
      -sy,        sc_,        cc_
  );
}

/* ************************************************************************* */
Rot3 Rot3::rodriguez(const Vector& w, double theta) {
  // get components of axis \omega
  double wx = w(0), wy=w(1), wz=w(2);
  double wwTxx = wx*wx, wwTyy = wy*wy, wwTzz = wz*wz;
#ifndef NDEBUG
  double l_n = wwTxx + wwTyy + wwTzz;
  if (std::abs(l_n-1.0)>1e-9) throw domain_error("rodriguez: length of n should be 1");
#endif

  double c = cos(theta), s = sin(theta), c_1 = 1 - c;

  double swx = wx * s, swy = wy * s, swz = wz * s;
  double C00 = c_1*wwTxx, C01 = c_1*wx*wy, C02 = c_1*wx*wz;
  double                  C11 = c_1*wwTyy, C12 = c_1*wy*wz;
  double                                   C22 = c_1*wwTzz;

  return Rot3(
        c + C00, -swz + C01,  swy + C02,
      swz + C01,    c + C11, -swx + C12,
     -swy + C02,  swx + C12,    c + C22);
}

/* ************************************************************************* */
Rot3 Rot3::compose (const Rot3& R2) const {
  return *this * R2;
}

/* ************************************************************************* */
Rot3 Rot3::compose (const Rot3& R2,
    boost::optional<Matrix3&> H1, boost::optional<Matrix3&> H2) const {
  if (H1) *H1 = R2.transpose();
  if (H2) *H2 = I3;
  return *this * R2;
}

/* ************************************************************************* */
Rot3 Rot3::compose (const Rot3& R2,
    boost::optional<Matrix&> H1, boost::optional<Matrix&> H2) const {
  if (H1) *H1 = R2.transpose();
  if (H2) *H2 = I3;
  return *this * R2;
}

/* ************************************************************************* */
Rot3 Rot3::operator*(const Rot3& R2) const {
  return Rot3(Matrix3(rot_*R2.rot_));
}

/* ************************************************************************* */
Rot3 Rot3::inverse(boost::optional<Matrix&> H1) const {
  if (H1) *H1 = -rot_;
  return Rot3(transpose());
}

/* ************************************************************************* */
Rot3 Rot3::between (const Rot3& R2,
    boost::optional<Matrix&> H1, boost::optional<Matrix&> H2) const {
  if (H1) *H1 = -(R2.transpose()*rot_);
  if (H2) *H2 = I3;
  Matrix3 R12 = transpose()*R2.rot_;
  return Rot3(R12);
}

/* ************************************************************************* */
Point3 Rot3::rotate(const Point3& p,
    boost::optional<Matrix&> H1,  boost::optional<Matrix&> H2) const {
  if (H1 || H2) {
      if (H1) *H1 = rot_ * skewSymmetric(-p.x(), -p.y(), -p.z());
      if (H2) *H2 = rot_;
    }
  return Point3(rot_ * p.vector());
}

/* ************************************************************************* */
// Log map at identity - return the canonical coordinates of this rotation
Vector3 Rot3::Logmap(const Rot3& R) {

  static const double PI = boost::math::constants::pi<double>();

  const Matrix3& rot = R.rot_;
  // Get trace(R)
  double tr = rot.trace();

  // when trace == -1, i.e., when theta = +-pi, +-3pi, +-5pi, etc.
  // we do something special
  if (std::abs(tr+1.0) < 1e-10) {
    if(std::abs(rot(2,2)+1.0) > 1e-10)
      return (PI / sqrt(2.0+2.0*rot(2,2) )) *
          Vector3(rot(0,2), rot(1,2), 1.0+rot(2,2));
    else if(std::abs(rot(1,1)+1.0) > 1e-10)
      return (PI / sqrt(2.0+2.0*rot(1,1))) *
          Vector3(rot(0,1), 1.0+rot(1,1), rot(2,1));
    else // if(std::abs(R.r1_.x()+1.0) > 1e-10)  This is implicit
      return (PI / sqrt(2.0+2.0*rot(0,0))) *
          Vector3(1.0+rot(0,0), rot(1,0), rot(2,0));
  } else {
    double magnitude;
    double tr_3 = tr-3.0; // always negative
    if (tr_3<-1e-7) {
      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: magnitude \approx 1/2-(t-3)/12 + O((t-3)^2)
      magnitude = 0.5 - tr_3*tr_3/12.0;
    }
    return magnitude*Vector3(
        rot(2,1)-rot(1,2),
        rot(0,2)-rot(2,0),
        rot(1,0)-rot(0,1));
  }
}

/* ************************************************************************* */
Rot3 Rot3::retractCayley(const Vector& omega) const {
  const double x = omega(0), y = omega(1), z = omega(2);
  const double x2 = x * x, y2 = y * y, z2 = z * z;
  const double xy = x * y, xz = x * z, yz = y * z;
  const double f = 1.0 / (4.0 + x2 + y2 + z2), _2f = 2.0 * f;
  return (*this)
      * Rot3((4 + x2 - y2 - z2) * f, (xy - 2 * z) * _2f, (xz + 2 * y) * _2f,
          (xy + 2 * z) * _2f, (4 - x2 + y2 - z2) * f, (yz - 2 * x) * _2f,
          (xz - 2 * y) * _2f, (yz + 2 * x) * _2f, (4 - x2 - y2 + z2) * f);
}

/* ************************************************************************* */
Rot3 Rot3::retract(const Vector& omega, Rot3::CoordinatesMode mode) const {
  if(mode == Rot3::EXPMAP) {
    return (*this)*Expmap(omega);
  } else if(mode == Rot3::CAYLEY) {
    return retractCayley(omega);
  } else if(mode == Rot3::SLOW_CAYLEY) {
    Matrix Omega = skewSymmetric(omega);
    return (*this)*CayleyFixed<3>(-Omega/2);
  } else {
    assert(false);
    exit(1);
  }
}

/* ************************************************************************* */
Vector3 Rot3::localCoordinates(const Rot3& T, Rot3::CoordinatesMode mode) const {
  if(mode == Rot3::EXPMAP) {
    return Logmap(between(T));
  } else if(mode == Rot3::CAYLEY) {
    // Create a fixed-size matrix
    Eigen::Matrix3d A(between(T).matrix());
    // Mathematica closed form optimization (procrastination?) gone wild:
    const double a=A(0,0),b=A(0,1),c=A(0,2);
    const double d=A(1,0),e=A(1,1),f=A(1,2);
    const double g=A(2,0),h=A(2,1),i=A(2,2);
    const double di = d*i, ce = c*e, cd = c*d, fg=f*g;
    const double M = 1 + e - f*h + i + e*i;
    const double K = 2.0 / (cd*h + M + a*M -g*(c + ce) - b*(d + di - fg));
    const double x = (a * f - cd + f) * K;
    const double y = (b * f - ce - c) * K;
    const double z = (fg - di - d) * K;
    return -2 * Vector3(x, y, z);
  } else if(mode == Rot3::SLOW_CAYLEY) {
    // Create a fixed-size matrix
    Eigen::Matrix3d A(between(T).matrix());
    // using templated version of Cayley
    Eigen::Matrix3d Omega = CayleyFixed<3>(A);
    return -2*Vector3(Omega(2,1),Omega(0,2),Omega(1,0));
  } else {
    assert(false);
    exit(1);
  }
}

/* ************************************************************************* */
Matrix3 Rot3::matrix() const {
  return rot_;
}

/* ************************************************************************* */
Point3 Rot3::r1() const { return Point3(rot_.col(0)); }

/* ************************************************************************* */
Point3 Rot3::r2() const { return Point3(rot_.col(1)); }

/* ************************************************************************* */
Point3 Rot3::r3() const { return Point3(rot_.col(2)); }

/* ************************************************************************* */
Quaternion Rot3::toQuaternion() const {
  return Quaternion(rot_);
}

/* ************************************************************************* */
Point3 Rot3::unrotate(const Point3& p) const {
  // Eigen expression
  return Point3(transpose()*p.vector()); // q = Rt*p
}

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

} // namespace gtsam

#endif