CQuaternion.h

Go to the documentation of this file.

/* +---------------------------------------------------------------------------+
   |          The Mobile Robot Programming Toolkit (MRPT) C++ library          |
   |                                                                           |
   |                       http://www.mrpt.org/                                |
   |                                                                           |
   |   Copyright (C) 2005-2011  University of Malaga                           |
   |                                                                           |
   |    This software was written by the Machine Perception and Intelligent    |
   |      Robotics Lab, University of Malaga (Spain).                          |
   |    Contact: Jose-Luis Blanco  <jlblanco@ctima.uma.es>                     |
   |                                                                           |
   |  This file is part of the MRPT project.                                   |
   |                                                                           |
   |     MRPT 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 3 of the License, or     |
   |     (at your option) any later version.                                   |
   |                                                                           |
   |   MRPT 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 MRPT.  If not, see <http://www.gnu.org/licenses/>.         |
   |                                                                           |
   +---------------------------------------------------------------------------+ */

#ifndef CQuaternion_H
#define CQuaternion_H

#include <mrpt/math/CMatrixTemplateNumeric.h>
#include <mrpt/math/CArray.h>

namespace mrpt
{
        namespace math
        {
                // For use with a constructor of CQuaternion
                enum TConstructorFlags_Quaternions
                {
                        UNINITIALIZED_QUATERNION = 0
                };

                /** A quaternion, which can represent a 3D rotation as pair \f$ (r,\mathbf{u}) \f$, with a real part "r" and a 3D vector \f$ \mathbf{u} = (x,y,z) \f$, or alternatively, q = r + ix + jy + kz.
                 *
                 *  The elements of the quaternion can be accessed by either:
                 *              - r(), x(), y(), z(), or
                 *              - the operator [] with indices running from 0 (=r) to 3 (=z).
                 *
                 *  Users will usually employ the typedef "CQuaternionDouble" instead of this template.
                 *
                 * For more information about quaternions, see:
                 *  - http://people.csail.mit.edu/bkph/articles/Quaternions.pdf
                 *  - http://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation
                 *
                 * \ingroup mrpt_base_grp
                 * \sa mrpt::poses::CPose3D
                 */
                template <class T>
                class CQuaternion : public CArrayNumeric<T,4>
                {
                        typedef CArrayNumeric<T,4> Base;
                public:
        /* @{ Constructors
         */

                /**     Can be used with UNINITIALIZED_QUATERNION as argument, does not initialize the 4 elements of the quaternion (use this constructor when speed is critical). */
                inline CQuaternion(TConstructorFlags_Quaternions constructor_dummy_param) { }

                /**     Default constructor: construct a (1, (0,0,0) ) quaternion representing no rotation. */
                inline CQuaternion()
                {
                        (*this)[0] = 1;
                        (*this)[1] = 0;
                        (*this)[2] = 0;
                        (*this)[3] = 0;
                }

                /**     Construct a quaternion from its parameters 'r', 'x', 'y', 'z', with q = r + ix + jy + kz. */
                inline CQuaternion(const T r,const T x,const T y,const T z)
                {
                        (*this)[0] = r;
                        (*this)[1] = x;
                        (*this)[2] = y;
                        (*this)[3] = z;
                        ASSERTDEBMSG_(std::abs(normSqr()-1.0)<1e-5, mrpt::format("Initialization data for quaternion is not normalized: %f %f %f %f -> sqrNorm=%f",r,x,y,z,normSqr()) );
                }

                /* @}
                 */


                inline T  r()const {return (*this)[0];} //!< Return r coordinate of the quaternion
                inline T  x()const {return (*this)[1];} //!< Return x coordinate of the quaternion
                inline T  y()const {return (*this)[2];} //!< Return y coordinate of the quaternion
                inline T  z()const {return (*this)[3];} //!< Return z coordinate of the quaternion
                inline void  r(const T r) {(*this)[0]=r;}       //!< Set r coordinate of the quaternion
                inline void  x(const T x) {(*this)[1]=x;}       //!< Set x coordinate of the quaternion
                inline void  y(const T y) {(*this)[2]=y;}       //!< Set y coordinate of the quaternion
                inline void  z(const T z) {(*this)[3]=z;}       //!< Set z coordinate of the quaternion

                /**     Set this quaternion to the rotation described by a 3D (Rodrigues) rotation vector \f$ \mathbf{v} \f$:
                  *   If \f$ \mathbf{v}=0 \f$, then the quaternion is \f$ \mathbf{q} = [1 ~ 0 ~ 0 ~ 0]^\top \f$, otherwise:
                  *    \f[ \mathbf{q} = \left[ \begin{array}{c}
                  *       \cos(\frac{\theta}{2}) \\
                  *       v_x \frac{\sin(\frac{\theta}{2})}{\theta} \\
                  *       v_y \frac{\sin(\frac{\theta}{2})}{\theta} \\
                  *       v_z \frac{\sin(\frac{\theta}{2})}{\theta}
                  *    \end{array} \right] \f]
                  *    where \f$ \theta = |\mathbf{v}| = \sqrt{v_x^2+v_y^2+v_z^2} \f$.
                  *  \sa "Representing Attitude: Euler Angles, Unit Quaternions, and Rotation Vectors (2006)", James Diebel.
                  */
                template <class ARRAYLIKE3>
                void  fromRodriguesVector(const ARRAYLIKE3 &v)
                {
                        if (v.size()!=3) THROW_EXCEPTION("Vector v must have a length=3");

                        const T x = v[0];
                        const T y = v[1];
                        const T z = v[2];
                        const T angle = std::sqrt(x*x+y*y+z*z);
                        if (angle<1e-7)
                        {
                                (*this)[0] = 1;
                                (*this)[1] = static_cast<T>(0.5)*x;
                                (*this)[2] = static_cast<T>(0.5)*y;
                                (*this)[3] = static_cast<T>(0.5)*z;
                        }
                        else
                        {
                                const T s = (::sin(angle/2))/angle;
                                const T c = ::cos(angle/2);
                                (*this)[0] = c;
                                (*this)[1] = x * s;
                                (*this)[2] = y * s;
                                (*this)[3] = z * s;
                        }
                }


                /** @name Lie Algebra methods
                    @{ */


                /** Logarithm of the 3x3 matrix defined by this pose, generating the corresponding vector in the SO(3) Lie Algebra,
                  *  which coincides with the so-called "rotation vector" (I don't have space here for the proof ;-).
                  *  \param[out] out_ln The target vector, which can be: std::vector<>, or mrpt::vector_double or any row or column Eigen::Matrix<>.
                  *  \sa exp,  mrpt::poses::SE_traits  */
                template <class ARRAYLIKE3>
                inline void ln(ARRAYLIKE3 &out_ln) const
                {
                        if (out_ln.size()!=3) out_ln.resize(3);
                        this->ln_noresize(out_ln);
                }
                /** \overload that returns by value */
                template <class ARRAYLIKE3>
                inline ARRAYLIKE3 ln() const
                {
                        ARRAYLIKE3 out_ln;
                        this->ln(out_ln);
                        return out_ln;
                }
                /** Like ln() but does not try to resize the output vector. */
                template <class ARRAYLIKE3>
                void ln_noresize(ARRAYLIKE3 &out_ln) const
                {
                        using mrpt::utils::square;
                        const T xyz_norm = std::sqrt(square(x())+square(y())+square(z()));
                        const T K = (xyz_norm<1e-7) ?  2 : 2*::acos(r())/xyz_norm;
                        out_ln[0] = K*x();
                        out_ln[1] = K*y();
                        out_ln[2] = K*z();
                }

                /** Exponential map from the SO(3) Lie Algebra to unit quaternions.
                  *  \sa ln,  mrpt::poses::SE_traits  */
                template <class ARRAYLIKE3>
                inline static CQuaternion<T> exp(const ARRAYLIKE3 & v)
                {
                        CQuaternion<T> q(UNINITIALIZED_QUATERNION);
                        q.fromRodriguesVector(v);
                        return q;
                }
                /** \overload */
                template <class ARRAYLIKE3>
                inline static void exp(const ARRAYLIKE3 & v, CQuaternion<T> & out_quat)
                {
                        out_quat.fromRodriguesVector(v);
                }

                /** @} */  // end of Lie algebra


                /**     Calculate the "cross" product (or "composed rotation") of two quaternion: this = q1 x q2
                  *   After the operation, "this" will represent the composed rotations of q1 and q2 (q2 applied "after" q1).
                  */
                inline void  crossProduct(const CQuaternion &q1, const CQuaternion &q2)
                {
                        this->r(  q1.r()*q2.r() - q1.x()*q2.x() - q1.y()*q2.y() - q1.z()*q2.z() );
                        this->x(  q1.r()*q2.x() + q2.r()*q1.x() + q1.y()*q2.z() - q2.y()*q1.z() );
            this->y(  q1.r()*q2.y() + q2.r()*q1.y() + q1.z()*q2.x() - q2.z()*q1.x() );
                        this->z(  q1.r()*q2.z() + q2.r()*q1.z() + q1.x()*q2.y() - q2.x()*q1.y() );
                        this->normalize();
                }

                /** Rotate a 3D point (lx,ly,lz) -> (gx,gy,gz) as described by this quaternion
                  */
                void rotatePoint(const double lx,const double ly,const double lz, double &gx,double &gy,double &gz ) const
                {
                        const double t2 = r()*x(); const double t3 = r()*y(); const double t4 = r()*z(); const double t5 =-x()*x(); const double t6 = x()*y();
                        const double t7 = x()*z(); const double t8 =-y()*y(); const double t9 = y()*z(); const double t10=-z()*z();
                        gx = 2*((t8+ t10)*lx+(t6 - t4)*ly+(t3+t7)*lz)+lx;
                        gy = 2*((t4+  t6)*lx+(t5 +t10)*ly+(t9-t2)*lz)+ly;
                        gz = 2*((t7-  t3)*lx+(t2 + t9)*ly+(t5+t8)*lz)+lz;
                }

                /** Rotate a 3D point (lx,ly,lz) -> (gx,gy,gz) as described by the inverse (conjugate) of this quaternion
                  */
                void inverseRotatePoint(const double lx,const double ly,const double lz, double &gx,double &gy,double &gz ) const
                {
                        const double t2 =-r()*x(); const double t3 =-r()*y(); const double t4 =-r()*z(); const double t5 =-x()*x(); const double t6 = x()*y();
                        const double t7 = x()*z(); const double t8 =-y()*y(); const double t9 = y()*z(); const double t10=-z()*z();
                        gx = 2*((t8+ t10)*lx+(t6 - t4)*ly+(t3+t7)*lz)+lx;
                        gy = 2*((t4+  t6)*lx+(t5 +t10)*ly+(t9-t2)*lz)+ly;
                        gz = 2*((t7-  t3)*lx+(t2 + t9)*ly+(t5+t8)*lz)+lz;
                }

                /** Return the squared norm of the quaternion */
                inline double normSqr() const {
                        using mrpt::utils::square;
                        return square(r()) + square(x()) + square(y()) + square(z());
                }

                /**     Normalize this quaternion, so its norm becomes the unitity.
                  */
                inline void normalize()
                {
                        const T qq = 1.0/std::sqrt( normSqr() );
                        for (unsigned int i=0;i<4;i++)
                                (*this)[i] *= qq;
                }

                /** Calculate the 4x4 Jacobian of the normalization operation of this quaternion.
                  *  The output matrix can be a dynamic or fixed size (4x4) matrix.
                  */
                template <class MATRIXLIKE>
                void  normalizationJacobian(MATRIXLIKE &J) const
                {
                        const T n = 1.0/std::pow(normSqr(),T(1.5));
                        J.setSize(4,4);
                        J.get_unsafe(0,0)=x()*x()+y()*y()+z()*z();
                        J.get_unsafe(0,1)=-r()*x();
                        J.get_unsafe(0,2)=-r()*y();
                        J.get_unsafe(0,3)=-r()*z();

                        J.get_unsafe(1,0)=-x()*r();
                        J.get_unsafe(1,1)=r()*r()+y()*y()+z()*z();
                        J.get_unsafe(1,2)=-x()*y();
                        J.get_unsafe(1,3)=-x()*z();

                        J.get_unsafe(2,0)=-y()*r();
                        J.get_unsafe(2,1)=-y()*x();
                        J.get_unsafe(2,2)=r()*r()+x()*x()+z()*z();
                        J.get_unsafe(2,3)=-y()*z();

                        J.get_unsafe(3,0)=-z()*r();
                        J.get_unsafe(3,1)=-z()*x();
                        J.get_unsafe(3,2)=-z()*y();
                        J.get_unsafe(3,3)=r()*r()+x()*x()+y()*y();
                        J *=n;
                }

                /** Compute the Jacobian of the rotation composition operation \f$ p = f(\cdot) = q_{this} \times r \f$, that is the 4x4 matrix \f$ \frac{\partial f}{\partial q_{this} }  \f$.
                  *  The output matrix can be a dynamic or fixed size (4x4) matrix.
                  */
                template <class MATRIXLIKE>
                inline void rotationJacobian(MATRIXLIKE &J) const
                {
                        J.setSize(4,4);
                        J.get_unsafe(0,0)=r(); J.get_unsafe(0,1)=-x(); J.get_unsafe(0,2)=-y(); J.get_unsafe(0,3)=-z();
                        J.get_unsafe(1,0)=x(); J.get_unsafe(1,1)= r(); J.get_unsafe(1,2)=-z(); J.get_unsafe(1,3)= y();
                        J.get_unsafe(2,0)=y(); J.get_unsafe(2,1)= z(); J.get_unsafe(2,2)= r(); J.get_unsafe(2,3)=-x();
                        J.get_unsafe(3,0)=z(); J.get_unsafe(3,1)=-y(); J.get_unsafe(3,2)= x(); J.get_unsafe(3,3)= r();
                }

                /** Calculate the 3x3 rotation matrix associated to this quaternion: \f[ \mathbf{R} = \left(
                  *    \begin{array}{ccc}
                  *     q_r^2+q_x^2-q_y^2-q_z^2         &  2(q_x q_y - q_r q_z) &       2(q_z q_x+q_r q_y) \\
                  *     2(q_x q_y+q_r q_z)              & q_r^2-q_x^2+q_y^2-q_z^2       & 2(q_y q_z-q_r q_x) \\
                  *     2(q_z q_x-q_r q_y) & 2(q_y q_z+q_r q_x)  & q_r^2- q_x^2 - q_y^2 + q_z^2
                  *    \end{array}
                  *  \right)\f]
                  *
                  */
                template <class MATRIXLIKE>
        inline void  rotationMatrix(MATRIXLIKE &M) const
                {
                        M.setSize(3,3);
                        rotationMatrixNoResize(M);
                }

                /** Fill out the top-left 3x3 block of the given matrix with the rotation matrix associated to this quaternion (does not resize the matrix, for that, see rotationMatrix). */
                template <class MATRIXLIKE>
        inline void  rotationMatrixNoResize(MATRIXLIKE &M) const
                {
                        M.get_unsafe(0,0)=r()*r()+x()*x()-y()*y()-z()*z();              M.get_unsafe(0,1)=2*(x()*y() -r()*z());                 M.get_unsafe(0,2)=2*(z()*x()+r()*y());
                        M.get_unsafe(1,0)=2*(x()*y()+r()*z());                          M.get_unsafe(1,1)=r()*r()-x()*x()+y()*y()-z()*z();              M.get_unsafe(1,2)=2*(y()*z()-r()*x());
                        M.get_unsafe(2,0)=2*(z()*x()-r()*y());                          M.get_unsafe(2,1)=2*(y()*z()+r()*x());                          M.get_unsafe(2,2)=r()*r()-x()*x()-y()*y()+z()*z();
                }


                /**     Return the conjugate quaternion  */
                inline void conj(CQuaternion &q_out) const
                {
                        q_out.r( r() );
                        q_out.x(-x() );
                        q_out.y(-y() );
                        q_out.z(-z() );
                }

                /**     Return the conjugate quaternion  */
                inline CQuaternion conj() const
                {
                        CQuaternion q_aux;
                        conj(q_aux);
                        return q_aux;
                }

                /**     Return the yaw, pitch & roll angles associated to quaternion
                  *  \sa For the equations, see The MRPT Book, or see http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToEuler/Quaternions.pdf
                  *  \sa rpy_and_jacobian
                */
                inline void rpy(T &roll, T &pitch, T &yaw) const
                {
                        rpy_and_jacobian(roll,pitch,yaw,static_cast<mrpt::math::CMatrixDouble*>(NULL));
                }

                /**     Return the yaw, pitch & roll angles associated to quaternion, and (optionally) the 3x4 Jacobian of the transformation.
                  *  Note that both the angles and the Jacobian have one set of normal equations, plus other special formulas for the degenerated cases of |pitch|=90 degrees.
                  *  \sa For the equations, see The MRPT Book, or http://www.euclideanspace.com/maths/geometry/rotations/conversions/quaternionToEuler/Quaternions.pdf
                  * \sa rpy
                */
                template <class MATRIXLIKE>
                void rpy_and_jacobian(T &roll, T &pitch, T &yaw, MATRIXLIKE *out_dr_dq = NULL, bool resize_out_dr_dq_to3x4 = true ) const
                {
                        using mrpt::utils::square;
                        using std::sqrt;

                        if (out_dr_dq && resize_out_dr_dq_to3x4)
                                out_dr_dq->setSize(3,4);
                        const T discr = r()*y()-x()*z();
                        if (fabs(discr)>0.49999)
                        {       // pitch = 90 deg
                                pitch =  0.5*M_PI;
                                yaw   =-2*atan2(x(),r());
                                roll  = 0;
                                if (out_dr_dq) {
                                        out_dr_dq->zeros();
                                        out_dr_dq->get_unsafe(0,0) = +2/x();
                                        out_dr_dq->get_unsafe(0,2) = -2*r()/(x()*x());
                                }
                        }
                        else if (discr<-0.49999)
                        {       // pitch =-90 deg
                                pitch = -0.5*M_PI;
                                yaw   =+2*atan2(x(),r());
                                roll  = 0;
                                if (out_dr_dq) {
                                        out_dr_dq->zeros();
                                        out_dr_dq->get_unsafe(0,0) = -2/x();
                                        out_dr_dq->get_unsafe(0,2) = +2*r()/(x()*x());
                                }
                        }
                        else
                        {       // Non-degenerate case:
                                yaw   = ::atan2( 2*(r()*z()+x()*y()), 1-2*(y()*y()+z()*z()) );
                                pitch = ::asin ( 2*discr );
                                roll  = ::atan2( 2*(r()*x()+y()*z()), 1-2*(x()*x()+y()*y()) );
                                if (out_dr_dq) {
                                        // Auxiliary terms:
                                        const double val1=(2*x()*x() + 2*y()*y() - 1);
                                        const double val12=square(val1);
                                        const double val2=(2*r()*x() + 2*y()*z());
                                        const double val22=square(val2);
                                        const double xy2 = 2*x()*y();
                                        const double rz2 = 2*r()*z();
                                        const double ry2 = 2*r()*y();
                                        const double val3 = (2*y()*y() + 2*z()*z() - 1);
                                        const double val4 = ((square(rz2 + xy2)/square(val3) + 1)*val3);
                                        const double val5 = (4*(rz2 + xy2))/square(val3);
                                        const double val6 = 1.0/(square(rz2 + xy2)/square(val3) + 1);
                                        const double val7 = 2.0/ sqrt(1 - square(ry2 - 2*x()*z()));
                                        const double val8 = (val22/val12 + 1);
                                        const double val9 = -2.0/val8;
                                        // row 1:
                                        out_dr_dq->get_unsafe(0,0) = -2*z()/val4;
                                        out_dr_dq->get_unsafe(0,1) = -2*y()/val4;
                                        out_dr_dq->get_unsafe(0,2) = -(2*x()/val3 - y()*val5)*val6 ;
                                        out_dr_dq->get_unsafe(0,3) = -(2*r()/val3 - z()*val5)*val6;
                                        // row 2:
                                        out_dr_dq->get_unsafe(1,0) = y()*val7  ;
                                        out_dr_dq->get_unsafe(1,1) = -z()*val7 ;
                                        out_dr_dq->get_unsafe(1,2) = r()*val7 ;
                                        out_dr_dq->get_unsafe(1,3) = -x()*val7 ;
                                        // row 3:
                                        out_dr_dq->get_unsafe(2,0) = val9*x()/val1 ;
                                        out_dr_dq->get_unsafe(2,1) = val9*(r()/val1 - (2*x()*val2)/val12) ;
                                        out_dr_dq->get_unsafe(2,2) = val9*(z()/val1 - (2*y()*val2)/val12) ;
                                        out_dr_dq->get_unsafe(2,3) = val9*y()/val1 ;
                                }
                        }
                }

                inline CQuaternion  operator * (const T &factor)
                {
                        CQuaternion q = *this;
                        q*=factor;
                        return q;
                }

                };      // end class

                typedef CQuaternion<double> CQuaternionDouble;  //!< A quaternion of data type "double"
                typedef CQuaternion<float>  CQuaternionFloat;   //!< A quaternion of data type "float"

        }       // end namespace

} // end namespace mrpt

#endif