//

// Copyright (c) 2015-2023 CNRS INRIA

// Copyright (c) 2015 Wandercraft, 86 rue de Paris 91400 Orsay, France.

//

#ifndef __pinocchio_spatial_explog_hpp__

#define __pinocchio_spatial_explog_hpp__

#include "pinocchio/fwd.hpp"

#include "pinocchio/utils/static-if.hpp"

#include "pinocchio/math/fwd.hpp"

#include "pinocchio/math/sincos.hpp"

#include "pinocchio/math/taylor-expansion.hpp"

#include "pinocchio/spatial/motion.hpp"

#include "pinocchio/spatial/skew.hpp"

#include "pinocchio/spatial/se3.hpp"

#include <Eigen/Geometry>

#include "pinocchio/spatial/log.hpp"

namespace pinocchio

{

  /// \brief Exp: so3 -> SO3.

  ///

  /// Return the integral of the input angular velocity during time 1.

  ///

  /// \param[in] v The angular velocity vector.

  ///

  /// \return The rotational matrix associated to the integration of the angular velocity during time 1.

  ///

  template<typename Vector3Like>

  typename Eigen::Matrix<typename Vector3Like::Scalar,3,3,PINOCCHIO_EIGEN_PLAIN_TYPE(Vector3Like)::Options>

  {

    typedef typename Vector3Like::Scalar Scalar;

    typedef typename PINOCCHIO_EIGEN_PLAIN_TYPE(Vector3Like) Vector3LikePlain;

    typedef Eigen::Matrix<Scalar,3,3,Vector3LikePlain::Options> Matrix3;

                                ct,

  }

  /// \brief Same as \ref log3

  ///

  /// \param[in] R the rotation matrix.

  /// \param[out] theta the angle value.

  ///

  /// \return The angular velocity vector associated to the rotation matrix.

  ///

  template<typename Matrix3Like>

  Eigen::Matrix<typename Matrix3Like::Scalar,3,1,PINOCCHIO_EIGEN_PLAIN_TYPE(Matrix3Like)::Options>

       typename Matrix3Like::Scalar & theta)

  {

    typedef typename Matrix3Like::Scalar Scalar;

    typedef Eigen::Matrix<Scalar,3,1,

                          PINOCCHIO_EIGEN_PLAIN_TYPE(Matrix3Like)::Options> Vector3;

  }

  /// \brief Log: SO(3)-> so(3).

  ///

  /// Pseudo-inverse of log from \f$ SO3 -> { v \in so3, ||v|| \le pi } \f$.

  ///

  /// \param[in] R The rotation matrix.

  ///

  /// \return The angular velocity vector associated to the rotation matrix.

  ///

  template<typename Matrix3Like>

  Eigen::Matrix<typename Matrix3Like::Scalar,3,1,PINOCCHIO_EIGEN_PLAIN_TYPE(Matrix3Like)::Options>

  {

    typename Matrix3Like::Scalar theta;

  }

  ///

  /// \brief Derivative of \f$ \exp{r} \f$

  /// \f[

  ///     \frac{\sin{||r||}}{||r||}                       I_3

  ///   - \frac{1-\cos{||r||}}{||r||^2}                   \left[ r \right]_x

  ///   + \frac{1}{||n||^2} (1-\frac{\sin{||r||}}{||r||}) r r^T

  /// \f]

  ///

  template<AssignmentOperatorType op, typename Vector3Like, typename Matrix3Like>

             const Eigen::MatrixBase<Matrix3Like> & Jexp)

  {

    typedef typename Matrix3Like::Scalar Scalar;

    switch(op)

      {

      case SETTO:

      case ADDTO:

      case RMTO:

      default:

        assert(false && "Wrong Op requesed value");

        break;

      }

  ///

  /// \brief Derivative of \f$ \exp{r} \f$

  /// \f[

  ///     \frac{\sin{||r||}}{||r||}                       I_3

  ///   - \frac{1-\cos{||r||}}{||r||^2}                   \left[ r \right]_x

  ///   + \frac{1}{||n||^2} (1-\frac{\sin{||r||}}{||r||}) r r^T

  /// \f]

  ///

  template<typename Vector3Like, typename Matrix3Like>

             const Eigen::MatrixBase<Matrix3Like> & Jexp)

  {

  /** \brief Derivative of log3

   *

   * This function is the right derivative of @ref log3, that is, for \f$R \in

   * SO(3)\f$ and \f$\omega t in \mathfrak{so}(3)\f$, it provides the linear

   * approximation:

   *

   * \f[

   * \log_3(R \oplus \omega t) = \log_3(R \exp_3(\omega t)) \approx \log_3(R) + \text{Jlog3}(R) \omega t

   * \f]

   *

   *  \param[in] theta the angle value.

   *  \param[in] log the output of log3.

   *  \param[out] Jlog the jacobian

   *

   * Equivalently, \f$\text{Jlog3}\f$ is the right Jacobian of \f$\log_3\f$:

   *

   * \f[

   * \text{Jlog3}(R) = \frac{\partial \log_3(R)}{\partial R}

   * \f]

   *

   * Note that this is the right Jacobian: \f$\text{Jlog3}(R) : T_{R} SO(3) \to T_{\log_6(R)} \mathfrak{so}(3)\f$.

   * (By convention, calculations in Pinocchio always perform right differentiation,

   * i.e., Jacobians are in local coordinates (also known as body coordinates), unless otherwise specified.)

   *

   * If we denote by \f$\theta = \log_3(R)\f$ and \f$\log = \log_3(R,

   * \theta)\f$, then \f$\text{Jlog} = \text{Jlog}_3(R)\f$ can be calculated as:

   *

   *  \f[

   *  \begin{align*}

   *  \text{Jlog} & = \frac{\theta \sin(\theta)}{2 (1 - \cos(\theta))} I_3

   *             + \frac{1}{2} \widehat{\log}

   *             + \left(\frac{1}{\theta^2} - \frac{\sin(\theta)}{2\theta(1-\cos(\theta))}\right) \log \log^T \\

   *             & = I_3

   *             + \frac{1}{2} \widehat{\log}

   *             + \left(\frac{1}{\theta^2} - \frac{1 + \cos \theta}{2 \theta \sin \theta}\right) \widehat{\log}^2

   *  \end{align*}

   *  \f]

   *

   *  where \f$\widehat{v}\f$ denotes the skew-symmetric matrix obtained from the 3D vector \f$v\f$.

   *

   *  \note The inputs must be such that \f$ \theta = \Vert \log \Vert \f$.

   */

  template<typename Scalar, typename Vector3Like, typename Matrix3Like>

             const Eigen::MatrixBase<Vector3Like> & log,

             const Eigen::MatrixBase<Matrix3Like> & Jlog)

  {

  /** \brief Derivative of log3

   *

   *  \param[in] R the rotation matrix.

   *  \param[out] Jlog the jacobian

   *

   *  Equivalent to

   *  \code

   *  double theta;

   *  Vector3 log = pinocchio::log3 (R, theta);

   *  pinocchio::Jlog3 (theta, log, Jlog);

   *  \endcode

   */

  template<typename Matrix3Like1, typename Matrix3Like2>

             const Eigen::MatrixBase<Matrix3Like2> & Jlog)

  {

    typedef typename Matrix3Like1::Scalar Scalar;

    typedef Eigen::Matrix<Scalar,3,1,PINOCCHIO_EIGEN_PLAIN_TYPE(Matrix3Like1)::Options> Vector3;

    Scalar t;

  template<typename Scalar, typename Vector3Like1, typename Vector3Like2, typename Matrix3Like>

             const Eigen::MatrixBase<Vector3Like1> & log,

             const Eigen::MatrixBase<Vector3Like2> & v,

             const Eigen::MatrixBase<Matrix3Like> & vt_Hlog)

  {

    typedef Eigen::Matrix<Scalar,3,1,PINOCCHIO_EIGEN_PLAIN_TYPE(Matrix3Like)::Options> Vector3;

    // theta = (log^T * log)^.5

    // dt/dl = .5 * 2 * log^T * (log^T * log)^-.5

    //       = log^T / theta

    // dt_dl = log / theta

           //db_dt = - (2 * (1 - a) / theta + da_dt ) / theta**2; // db / dtheta

    // Compute dl_dv_v = Jlog * v

    // Jlog = a I3 + .5 [ log ] + b * log * log^T

    // if v == log, then Jlog * v == v

    // Derivative of b * log * log^T

    // Derivative of .5 * [ log ]

    // Derivative of a * I3

  /** \brief Second order derivative of log3

   *

   *  This computes \f$ v^T H_{log} \f$.

   *

   *  \param[in] R the rotation matrix.

   *  \param[in] v the 3D vector.

   *  \param[out] vt_Hlog the product of the Hessian with the input vector

   */

  template<typename Matrix3Like1, typename Vector3Like, typename Matrix3Like2>

             const Eigen::MatrixBase<Vector3Like> & v,

             const Eigen::MatrixBase<Matrix3Like2> & vt_Hlog)

  {

    typedef typename Matrix3Like1::Scalar Scalar;

    typedef Eigen::Matrix<Scalar,3,1,PINOCCHIO_EIGEN_PLAIN_TYPE(Matrix3Like1)::Options> Vector3;

    Scalar t;

  ///

  /// \brief Exp: se3 -> SE3.

  ///

  /// Return the integral of the input twist during time 1.

  ///

  /// \param[in] nu The input twist.

  ///

  /// \return The rigid transformation associated to the integration of the twist during time 1.

  ///

  template<typename MotionDerived>

  SE3Tpl<typename MotionDerived::Scalar,PINOCCHIO_EIGEN_PLAIN_TYPE(typename MotionDerived::Vector3)::Options>

  {

    typedef typename MotionDerived::Scalar Scalar;

    enum { Options = PINOCCHIO_EIGEN_PLAIN_TYPE(typename MotionDerived::Vector3)::Options };

    typedef SE3Tpl<Scalar,Options> SE3;

    Scalar alpha_wxv, alpha_v, alpha_w, diagonal_term;

                                           ct);

    // Linear

    // Rotational

  }

  /// \brief Exp: se3 -> SE3.

  ///

  /// Return the integral of the input spatial velocity during time 1.

  ///

  /// \param[in] v The twist represented by a vector.

  ///

  /// \return The rigid transformation associated to the integration of the twist vector during time 1.

  ///

  template<typename Vector6Like>

  SE3Tpl<typename Vector6Like::Scalar,PINOCCHIO_EIGEN_PLAIN_TYPE(Vector6Like)::Options>

  {

  }

  /// \brief Log: SE3 -> se3.

  ///

  /// Pseudo-inverse of exp from \f$ SE3 \to { v,\omega \in \mathfrak{se}(3), ||\omega|| < 2\pi } \f$.

  ///

  /// \param[in] M The rigid transformation.

  ///

  /// \return The twist associated to the rigid transformation during time 1.

  ///

  template<typename Scalar, int Options>

  MotionTpl<Scalar,Options>

  {

    typedef MotionTpl<Scalar,Options> Motion;

  }

  /// \brief Log: SE3 -> se3.

  ///

  /// Pseudo-inverse of exp from \f$ SE3 \to { v,\omega \in \mathfrak{se}(3), ||\omega|| < 2\pi } \f$.

  ///

  /// \param[in] M The rigid transformation represented as an homogenous matrix.

  ///

  /// \return The twist associated to the rigid transformation during time 1.

  ///

  template<typename Matrix4Like>

  MotionTpl<typename Matrix4Like::Scalar,Eigen::internal::traits<Matrix4Like>::Options>

  {

    typedef typename Matrix4Like::Scalar Scalar;

    enum {Options = Eigen::internal::traits<Matrix4Like>::Options};

    typedef MotionTpl<Scalar,Options> Motion;

    typedef SE3Tpl<Scalar,Options> SE3;

  }

  /// \brief Derivative of exp6

  /// Computed as the inverse of Jlog6

  template<AssignmentOperatorType op, typename MotionDerived, typename Matrix6Like>

             const Eigen::MatrixBase<Matrix6Like> & Jexp)

  {

    typedef typename MotionDerived::Scalar Scalar;

    typedef typename MotionDerived::Vector3 Vector3;

    typedef Eigen::Matrix<Scalar, 3, 3, Vector3::Options> Matrix3;

    switch(op)

      {

      case SETTO:

      {

          - Jout.template topLeftCorner<3,3>() * J;

      }

      case ADDTO:

      {

          - Jtmp3 * J;

      }

      case RMTO:

      {

          - Jtmp3 * J;

      }

      default:

        assert(false && "Wrong Op requesed value");

        break;

      }

  /// \brief Derivative of exp6

  /// Computed as the inverse of Jlog6

  template<typename MotionDerived, typename Matrix6Like>

             const Eigen::MatrixBase<Matrix6Like> & Jexp)

  {

  /** \brief Derivative of log6

   *

   * This function is the right derivative of @ref log6, that is, for \f$M \in

   * SE(3)\f$ and \f$\xi in \mathfrak{se}(3)\f$, it provides the linear

   * approximation:

   *

   * \f[

   * \log_6(M \oplus \xi) = \log_6(M \exp_6(\xi)) \approx \log_6(M) + \text{Jlog6}(M) \xi

   * \f]

   *

   * Equivalently, \f$\text{Jlog6}\f$ is the right Jacobian of \f$\log_6\f$:

   *

   * \f[

   * \text{Jlog6}(M) = \frac{\partial \log_6(M)}{\partial M}

   * \f]

   *

   * Note that this is the right Jacobian: \f$\text{Jlog6}(M) : T_{M} SE(3) \to T_{\log_6(M)} \mathfrak{se}(3)\f$.

   * (By convention, calculations in Pinocchio always perform right differentiation,

   * i.e., Jacobians are in local coordinates (also known as body coordinates), unless otherwise specified.)

   *

   * Internally, it is calculated using the following formulas:

   *

   *  \f[

   *  \text{Jlog6}(M) =

   *  \left(\begin{array}{cc}

   *  \text{Jlog3}(R) & J * \text{Jlog3}(R) \\

   *            0     &     \text{Jlog3}(R) \\

   *  \end{array}\right)

   *  \f]

   *  where

   *  \f[ M =

   *  \left(\begin{array}{cc}

   *  \exp(\mathbf{r}) & \mathbf{p} \\

   *             0     & 1          \\

   *  \end{array}\right)

   *  \f]

   *  \f[

   *  \begin{eqnarray}

   *  J &=&

   *  \left.\frac{1}{2}[\mathbf{p}]_{\times} + \beta'(||r||) \frac{\mathbf{r}^T\mathbf{p}}{||r||}\mathbf{r}\mathbf{r}^T

   *  - (||r||\beta'(||r||) + 2 \beta(||r||)) \mathbf{p}\mathbf{r}^T\right.\\

   *  &&\left. + \mathbf{r}^T\mathbf{p}\beta(||r||)I_3 + \beta(||r||)\mathbf{r}\mathbf{p}^T\right.

   *  \end{eqnarray}

   *  \f]

   *  and

   *  \f[ \beta(x)=\left(\frac{1}{x^2} - \frac{\sin x}{2x(1-\cos x)}\right) \f]

   *

   *

   * \cheatsheet For \f$(A,B) \in SE(3)^2\f$, let \f$M_1(A, B) = A B\f$ and

   * \f$m_1 = \log_6(M_1) \f$. Then, we have the following partial (right)

   * Jacobians: \n

   *  - \f$ \frac{\partial m_1}{\partial A} = Jlog_6(M_1) Ad_B^{-1} \f$,

   *  - \f$ \frac{\partial m_1}{\partial B} = Jlog_6(M_1) \f$.

   *

   * \cheatsheet Let \f$A \in SE(3)\f$, \f$M_2(A) = A^{-1}\f$ and \f$m_2 =

   * \log_6(M_2)\f$. Then, we have the following partial (right) Jacobian: \n

   *  - \f$ \frac{\partial m_2}{\partial A} = - Jlog_6(M_2) Ad_A \f$.

   */

  template<typename Scalar, int Options, typename Matrix6Like>

             const Eigen::MatrixBase<Matrix6Like> & Jlog)

  {

  template<typename Scalar, int Options>

  template<typename OtherScalar>

                                                             const SE3Tpl & B,

                                                             const OtherScalar & alpha)

  {

    typedef SE3Tpl<Scalar,Options> ReturnType;

    typedef MotionTpl<Scalar,Options> Motion;

  }

} // namespace pinocchio

#include "pinocchio/spatial/explog-quaternion.hpp"

#include "pinocchio/spatial/log.hxx"

#endif //#ifndef __pinocchio_spatial_explog_hpp__