IK_Math.h

Go to the documentation of this file.

/*
 * This program is free software; you can redistribute it and/or
 * modify it under the terms of the GNU General Public License
 * as published by the Free Software Foundation; either version 2
 * of the License, or (at your option) any later version.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software Foundation,
 * Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
 *
 * The Original Code is Copyright (C) 2001-2002 by NaN Holding BV.
 * All rights reserved.
 * Original author: Laurence
 */
 
#pragma once
 
#include <Eigen/Core>
#include <Eigen/Geometry>
 
#include <cmath>
 
using Eigen::Affine3d;
using Eigen::Matrix3d;
using Eigen::MatrixXd;
using Eigen::Vector3d;
using Eigen::VectorXd;
 
static const double IK_EPSILON = 1e-20;
 
static inline bool FuzzyZero(double x)
{
  return fabs(x) < IK_EPSILON;
}
 
static inline double Clamp(const double x, const double min, const double max)
{
  return (x < min) ? min : (x > max) ? max : x;
}
 
static inline Eigen::Matrix3d CreateMatrix(double xx,
                                           double xy,
                                           double xz,
                                           double yx,
                                           double yy,
                                           double yz,
                                           double zx,
                                           double zy,
                                           double zz)
{
  Eigen::Matrix3d M;
  M(0, 0) = xx;
  M(0, 1) = xy;
  M(0, 2) = xz;
  M(1, 0) = yx;
  M(1, 1) = yy;
  M(1, 2) = yz;
  M(2, 0) = zx;
  M(2, 1) = zy;
  M(2, 2) = zz;
  return M;
}
 
static inline Eigen::Matrix3d RotationMatrix(double sine, double cosine, int axis)
{
  if (axis == 0)
    return CreateMatrix(1.0, 0.0, 0.0, 0.0, cosine, -sine, 0.0, sine, cosine);
  else if (axis == 1)
    return CreateMatrix(cosine, 0.0, sine, 0.0, 1.0, 0.0, -sine, 0.0, cosine);
  else
    return CreateMatrix(cosine, -sine, 0.0, sine, cosine, 0.0, 0.0, 0.0, 1.0);
}
 
static inline Eigen::Matrix3d RotationMatrix(double angle, int axis)
{
  return RotationMatrix(sin(angle), cos(angle), axis);
}
 
static inline double EulerAngleFromMatrix(const Eigen::Matrix3d &R, int axis)
{
  double t = sqrt(R(0, 0) * R(0, 0) + R(0, 1) * R(0, 1));
 
  if (t > 16.0 * IK_EPSILON) {
    if (axis == 0)
      return -atan2(R(1, 2), R(2, 2));
    else if (axis == 1)
      return atan2(-R(0, 2), t);
    else
      return -atan2(R(0, 1), R(0, 0));
  }
  else {
    if (axis == 0)
      return -atan2(-R(2, 1), R(1, 1));
    else if (axis == 1)
      return atan2(-R(0, 2), t);
    else
      return 0.0f;
  }
}
 
static inline double safe_acos(double f)
{
  // acos that does not return NaN with rounding errors
  if (f <= -1.0)
    return M_PI;
  else if (f >= 1.0)
    return 0.0;
  else
    return acos(f);
}
 
static inline Eigen::Vector3d normalize(const Eigen::Vector3d &v)
{
  // a sane normalize function that doesn't give (1, 0, 0) in case
  // of a zero length vector
  double len = v.norm();
  return FuzzyZero(len) ? Eigen::Vector3d(0, 0, 0) : Eigen::Vector3d(v / len);
}
 
static inline double angle(const Eigen::Vector3d &v1, const Eigen::Vector3d &v2)
{
  return safe_acos(v1.dot(v2));
}
 
static inline double ComputeTwist(const Eigen::Matrix3d &R)
{
  // qy and qw are the y and w components of the quaternion from R
  double qy = R(0, 2) - R(2, 0);
  double qw = R(0, 0) + R(1, 1) + R(2, 2) + 1;
 
  double tau = 2.0 * atan2(qy, qw);
 
  return tau;
}
 
static inline Eigen::Matrix3d ComputeTwistMatrix(double tau)
{
  return RotationMatrix(tau, 1);
}
 
static inline void RemoveTwist(Eigen::Matrix3d &R)
{
  // compute twist parameter
  double tau = ComputeTwist(R);
 
  // compute twist matrix
  Eigen::Matrix3d T = ComputeTwistMatrix(tau);
 
  // remove twist
  R = R * T.transpose();
}
 
static inline Eigen::Vector3d SphericalRangeParameters(const Eigen::Matrix3d &R)
{
  // compute twist parameter
  double tau = ComputeTwist(R);
 
  // compute swing parameters
  double num = 2.0 * (1.0 + R(1, 1));
 
  // singularity at pi
  if (fabs(num) < IK_EPSILON)
    // TODO: this does now rotation of size pi over z axis, but could
    // be any axis, how to deal with this I'm not sure, maybe don't
    // enforce limits at all then.
    return Eigen::Vector3d(0.0, tau, 1.0);
 
  num = 1.0 / sqrt(num);
  double ax = -R(2, 1) * num;
  double az = R(0, 1) * num;
 
  return Eigen::Vector3d(ax, tau, az);
}
 
static inline Eigen::Matrix3d ComputeSwingMatrix(double ax, double az)
{
  // length of (ax, 0, az) = sin(theta/2)
  double sine2 = ax * ax + az * az;
  double cosine2 = sqrt((sine2 >= 1.0) ? 0.0 : 1.0 - sine2);
 
  // compute swing matrix
  Eigen::Matrix3d S(Eigen::Quaterniond(-cosine2, ax, 0.0, az));
 
  return S;
}
 
static inline Eigen::Vector3d MatrixToAxisAngle(const Eigen::Matrix3d &R)
{
  Eigen::Vector3d delta = Eigen::Vector3d(R(2, 1) - R(1, 2), R(0, 2) - R(2, 0), R(1, 0) - R(0, 1));
 
  double c = safe_acos((R(0, 0) + R(1, 1) + R(2, 2) - 1) / 2);
  double l = delta.norm();
 
  if (!FuzzyZero(l))
    delta *= c / l;
 
  return delta;
}
 
static inline bool EllipseClamp(double &ax, double &az, double *amin, double *amax)
{
  double xlim, zlim, x, z;
 
  if (ax < 0.0) {
    x = -ax;
    xlim = -amin[0];
  }
  else {
    x = ax;
    xlim = amax[0];
  }
 
  if (az < 0.0) {
    z = -az;
    zlim = -amin[1];
  }
  else {
    z = az;
    zlim = amax[1];
  }
 
  if (FuzzyZero(xlim) || FuzzyZero(zlim)) {
    if (x <= xlim && z <= zlim)
      return false;
 
    if (x > xlim)
      x = xlim;
    if (z > zlim)
      z = zlim;
  }
  else {
    double invx = 1.0 / (xlim * xlim);
    double invz = 1.0 / (zlim * zlim);
 
    if ((x * x * invx + z * z * invz) <= 1.0)
      return false;
 
    if (FuzzyZero(x)) {
      x = 0.0;
      z = zlim;
    }
    else {
      double rico = z / x;
      double old_x = x;
      x = sqrt(1.0 / (invx + invz * rico * rico));
      if (old_x < 0.0)
        x = -x;
      z = rico * x;
    }
  }
 
  ax = (ax < 0.0) ? -x : x;
  az = (az < 0.0) ? -z : z;
 
  return true;
}