explog-quaternion.hpp

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//
// Copyright (c) 2018-2020 CNRS INRIA
//

#ifndef __pinocchio_spatial_explog_quaternion_hpp__
#define __pinocchio_spatial_explog_quaternion_hpp__

#include "pinocchio/math/quaternion.hpp"
#include "pinocchio/spatial/explog.hpp"
#include "pinocchio/utils/static-if.hpp"

namespace pinocchio
{
  namespace quaternion
  {
    
    template<typename Vector3Like, typename QuaternionLike>
    void exp3(const Eigen::MatrixBase<Vector3Like> & v,
              Eigen::QuaternionBase<QuaternionLike> & quat_out)
    {
      EIGEN_STATIC_ASSERT_VECTOR_ONLY(Vector3Like);
      assert(v.size() == 3);
      
      typedef typename Vector3Like::Scalar Scalar;
      enum { Options = PINOCCHIO_EIGEN_PLAIN_TYPE(typename QuaternionLike::Coefficients)::Options };
      typedef Eigen::Quaternion<typename QuaternionLike::Scalar,Options> QuaternionPlain;
      
      const Scalar t2 = v.squaredNorm();
      const Scalar t = math::sqrt(t2);
      
      static const Scalar ts_prec = math::sqrt(Eigen::NumTraits<Scalar>::epsilon()); // Precision for the Taylor series expansion.
    
      Eigen::AngleAxis<Scalar> aa(t,v/t);
      QuaternionPlain quat_then(aa);
      
      QuaternionPlain quat_else;
      quat_else.vec() = (Scalar(1)/Scalar(2) - t2/48) * v;
      quat_else.w() = Scalar(1) - t2/8;
      
      using ::pinocchio::internal::if_then_else;
      for(Eigen::DenseIndex k = 0; k < 4; ++k)
      {
        quat_out.coeffs().coeffRef(k) = if_then_else(::pinocchio::internal::GT, t2, ts_prec,
                                                     quat_then.coeffs().coeffRef(k),
                                                     quat_else.coeffs().coeffRef(k));
      }
      
    }
    
    template<typename Vector3Like>
    Eigen::Quaternion<typename Vector3Like::Scalar, PINOCCHIO_EIGEN_PLAIN_TYPE(Vector3Like)::Options>
    exp3(const Eigen::MatrixBase<Vector3Like> & v)
    {
      typedef Eigen::Quaternion<typename Vector3Like::Scalar, PINOCCHIO_EIGEN_PLAIN_TYPE(Vector3Like)::Options> ReturnType;
      ReturnType res; exp3(v,res);
      return res;
    }
    
    template<typename QuaternionLike>
    Eigen::Matrix<typename QuaternionLike::Scalar,3,1,PINOCCHIO_EIGEN_PLAIN_TYPE(typename QuaternionLike::Vector3)::Options>
    log3(const Eigen::QuaternionBase<QuaternionLike> & quat,
         typename QuaternionLike::Scalar & theta)
    {
      typedef typename QuaternionLike::Scalar Scalar;
      enum { Options = PINOCCHIO_EIGEN_PLAIN_TYPE(typename QuaternionLike::Vector3)::Options };
      typedef Eigen::Matrix<Scalar,3,1,Options> Vector3;
      
      Vector3 res;
      const Scalar norm_squared = quat.vec().squaredNorm();

      static const Scalar eps = Eigen::NumTraits<Scalar>::epsilon();
      static const Scalar ts_prec = TaylorSeriesExpansion<Scalar>::template precision<2>();
      const Scalar norm = math::sqrt(norm_squared + eps * eps);

      using ::pinocchio::internal::if_then_else;
      using ::pinocchio::internal::GE;
      using ::pinocchio::internal::LT;

      const Scalar pos_neg = if_then_else(GE, quat.w(), Scalar(0),
                                          Scalar(+1),
                                          Scalar(-1));
        
      Eigen::Quaternion<Scalar, Options> quat_pos;
      quat_pos.w() = pos_neg * quat.w();
      quat_pos.vec() = pos_neg * quat.vec();

      const Scalar theta_2 = math::atan2(norm,quat_pos.w());  // in [0,pi]
      const Scalar y_x = norm / quat_pos.w();  // nonnegative
      const Scalar y_x_sq = norm_squared / (quat_pos.w() * quat_pos.w());

      theta = if_then_else(LT, norm_squared, ts_prec,
                           Scalar(2.)*(Scalar(1) - y_x_sq / Scalar(3)) * y_x,
                           Scalar(2.)*theta_2);

      const Scalar th2_2 = theta * theta / Scalar(4);
      const Scalar inv_sinc = if_then_else(LT, norm_squared, ts_prec,
                                           Scalar(2) * (Scalar(1) + th2_2 / Scalar(6) + Scalar(7)/Scalar(360) * th2_2*th2_2),
                                           theta / math::sin(theta_2));

      for(Eigen::DenseIndex k = 0; k < 3; ++k)
      {
        // res[k] = if_then_else(LT, norm_squared, ts_prec,
        //                       Scalar(2) * (Scalar(1) + y_x_sq / Scalar(6) - y_x_sq*y_x_sq / Scalar(9)) * pos_neg * quat.vec()[k],
        //                       inv_sinc * pos_neg * quat.vec()[k]);
        res[k] = inv_sinc * quat_pos.vec()[k];
      }
      return res;
    }
    
    template<typename QuaternionLike>
    Eigen::Matrix<typename QuaternionLike::Scalar,3,1,PINOCCHIO_EIGEN_PLAIN_TYPE(typename QuaternionLike::Vector3)::Options>
    log3(const Eigen::QuaternionBase<QuaternionLike> & quat)
    {
      typename QuaternionLike::Scalar theta;
      return log3(quat.derived(),theta);
    }
    
    template<typename Vector3Like, typename Matrix43Like>
    void Jexp3CoeffWise(const Eigen::MatrixBase<Vector3Like> & v,
                        const Eigen::MatrixBase<Matrix43Like> & Jexp)
    {
//      EIGEN_STATIC_ASSERT_MATRIX_SPECIFIC_SIZE(Matrix43Like,4,3);
      assert(Jexp.rows() == 4 && Jexp.cols() == 3 && "Jexp does have the right size.");
      Matrix43Like & Jout = PINOCCHIO_EIGEN_CONST_CAST(Matrix43Like,Jexp);
      
      typedef typename Vector3Like::Scalar Scalar;
      
      const Scalar n2 = v.squaredNorm();
      const Scalar n = math::sqrt(n2);
      const Scalar theta = Scalar(0.5) * n;
      const Scalar theta2 = Scalar(0.25) * n2;
      
      if(n2 > math::sqrt(Eigen::NumTraits<Scalar>::epsilon()))
      {
        Scalar c, s;
        SINCOS(theta,&s,&c);
        Jout.template topRows<3>().noalias() = ((Scalar(0.5)/n2) * (c - 2*s/n)) * v * v.transpose();
        Jout.template topRows<3>().diagonal().array() += s/n;
        Jout.template bottomRows<1>().noalias() = -s/(2*n) * v.transpose();
      }
      else
      {
        Jout.template topRows<3>().noalias() =  (-Scalar(1)/Scalar(12) + n2/Scalar(480)) * v * v.transpose();
        Jout.template topRows<3>().diagonal().array() += Scalar(0.5) * (1 - theta2/6);
        Jout.template bottomRows<1>().noalias() = (Scalar(-0.25) * (Scalar(1) - theta2/6)) * v.transpose();
        
      }
    }
    
    template<typename QuaternionLike, typename Matrix3Like>
    void Jlog3(const Eigen::QuaternionBase<QuaternionLike> & quat,
               const Eigen::MatrixBase<Matrix3Like> & Jlog)
    {
      typedef typename QuaternionLike::Scalar Scalar;
      typedef Eigen::Matrix<Scalar,3,1,PINOCCHIO_EIGEN_PLAIN_TYPE(typename QuaternionLike::Coefficients)::Options> Vector3;
      
      Scalar t;
      Vector3 w(log3(quat,t));
      pinocchio::Jlog3(t,w,PINOCCHIO_EIGEN_CONST_CAST(Matrix3Like,Jlog));
    }
  } // namespace quaternion
}

#endif // ifndef __pinocchio_spatial_explog_quaternion_hpp__