CQuaternion.h

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/* +------------------------------------------------------------------------+
   |                     Mobile Robot Programming Toolkit (MRPT)            |
   |                          https://www.mrpt.org/                         |
   |                                                                        |
   | Copyright (c) 2005-2019, Individual contributors, see AUTHORS file     |
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   +------------------------------------------------------------------------+ */

#pragma once

#include <mrpt/core/bits_math.h>  // square()
#include <mrpt/core/exceptions.h>
#include <mrpt/math/CMatrixDynamic.h>
#include <mrpt/math/CVectorFixed.h>

namespace mrpt::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 type `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_math_grp
 * \sa mrpt::poses::CPose3D
 */
template <class T>
class CQuaternion : public CVectorFixed<T, 4>
{
    using Base = CVectorFixed<T, 4>;

   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) {}
    /** 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;
        ASSERTMSG_(
            std::abs(normSqr() - 1.0) < 1e-3,
            mrpt::format(
                "Initialization data for quaternion is not normalized: %f %f "
                "%f %f -> sqrNorm=%f",
                r, x, y, z, normSqr()));
    }

    /* @}
     */

    /** Return r coordinate of the quaternion */
    inline T r() const { return (*this)[0]; }
    /** Return x coordinate of the quaternion */
    inline T x() const { return (*this)[1]; }
    /** Return y coordinate of the quaternion */
    inline T y() const { return (*this)[2]; }
    /** Return z coordinate of the quaternion */
    inline T z() const { return (*this)[3]; }
    /** Set r coordinate of the quaternion */
    inline void r(const T r) { (*this)[0] = r; }
    /** Set x coordinate of the quaternion */
    inline void x(const T x) { (*this)[1] = x; }
    /** Set y coordinate of the quaternion */
    inline void y(const T y) { (*this)[2] = y; }
    /** Set z coordinate of the quaternion */
    inline void z(const T z) { (*this)[3] = z; }
    /** 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)
    {
        MRPT_START
        ASSERT_(v.size() == 3);
        const T x = v[0], y = v[1], z = v[2];
        const T theta_sq = x * x + y * y + z * z, theta = std::sqrt(theta_sq);
        T r, i;
        if (theta < 1e-6)
        {
            // Taylor series approximation:
            const T theta_po4 = theta_sq * theta_sq;
            i = T(0.5) - T(1.0 / 48.0) * theta_sq + T(1.0 / 3840.0) * theta_po4;
            r = T(1.0) - T(0.5) * theta_sq + T(1.0 / 384.0) * theta_po4;
        }
        else
        {
            i = (std::sin(theta / 2)) / theta;
            r = std::cos(theta / 2);
        }
        (*this)[0] = r;
        (*this)[1] = x * i;
        (*this)[2] = y * i;
        (*this)[3] = z * i;
        ASSERTMSG_(
            normSqr() - 1.0 < 1e-6,
            mrpt::format(
                "fromRodriguesVector() failed, tangent_vector=[%g %g %g]", v[0],
                v[1], v[2]));
        MRPT_END
    }

    /** @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::math::CVectorDouble 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::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)
    {
        // First: compute result, then save in this object. In this way we avoid
        // problems when q1 or q2 == *this !!
        const T new_r = q1.r() * q2.r() - q1.x() * q2.x() - q1.y() * q2.y() -
                        q1.z() * q2.z();
        const T new_x = q1.r() * q2.x() + q2.r() * q1.x() + q1.y() * q2.z() -
                        q2.y() * q1.z();
        const T new_y = q1.r() * q2.y() + q2.r() * q1.y() + q1.z() * q2.x() -
                        q2.z() * q1.x();
        const T new_z = q1.r() * q2.z() + q2.r() * q1.z() + q1.x() * q2.y() -
                        q2.x() * q1.y();
        this->r(new_r);
        this->x(new_x);
        this->y(new_y);
        this->z(new_z);
        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::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(0, 0) = x() * x() + y() * y() + z() * z();
        J(0, 1) = -r() * x();
        J(0, 2) = -r() * y();
        J(0, 3) = -r() * z();

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

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

        J(3, 0) = -z() * r();
        J(3, 1) = -z() * x();
        J(3, 2) = -z() * y();
        J(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(0, 0) = r();
        J(0, 1) = -x();
        J(0, 2) = -y();
        J(0, 3) = -z();
        J(1, 0) = x();
        J(1, 1) = r();
        J(1, 2) = -z();
        J(1, 3) = y();
        J(2, 0) = y();
        J(2, 1) = z();
        J(2, 2) = r();
        J(2, 3) = -x();
        J(3, 0) = z();
        J(3, 1) = -y();
        J(3, 2) = x();
        J(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(0, 0) = r() * r() + x() * x() - y() * y() - z() * z();
        M(0, 1) = 2 * (x() * y() - r() * z());
        M(0, 2) = 2 * (z() * x() + r() * y());
        M(1, 0) = 2 * (x() * y() + r() * z());
        M(1, 1) = r() * r() - x() * x() + y() * y() - z() * z();
        M(1, 2) = 2 * (y() * z() - r() * x());
        M(2, 0) = 2 * (z() * x() - r() * y());
        M(2, 1) = 2 * (y() * z() + r() * x());
        M(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*>(nullptr));
    }

    /** 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 = nullptr,
        bool resize_out_dr_dq_to3x4 = true) const
    {
        using mrpt::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->setZero();
                (*out_dr_dq)(0, 0) = +2 / x();
                (*out_dr_dq)(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->setZero();
                (*out_dr_dq)(0, 0) = -2 / x();
                (*out_dr_dq)(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)(0, 0) = -2 * z() / val4;
                (*out_dr_dq)(0, 1) = -2 * y() / val4;
                (*out_dr_dq)(0, 2) = -(2 * x() / val3 - y() * val5) * val6;
                (*out_dr_dq)(0, 3) = -(2 * r() / val3 - z() * val5) * val6;
                // row 2:
                (*out_dr_dq)(1, 0) = y() * val7;
                (*out_dr_dq)(1, 1) = -z() * val7;
                (*out_dr_dq)(1, 2) = r() * val7;
                (*out_dr_dq)(1, 3) = -x() * val7;
                // row 3:
                (*out_dr_dq)(2, 0) = val9 * x() / val1;
                (*out_dr_dq)(2, 1) =
                    val9 * (r() / val1 - (2 * x() * val2) / val12);
                (*out_dr_dq)(2, 2) =
                    val9 * (z() / val1 - (2 * y() * val2) / val12);
                (*out_dr_dq)(2, 3) = val9 * y() / val1;
            }
        }
    }

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

};  // end class

/** A quaternion of data type "double" */
using CQuaternionDouble = CQuaternion<double>;
/** A quaternion of data type "float" */
using CQuaternionFloat = CQuaternion<float>;

}  // namespace mrpt::math