explog-quaternion.hpp

//
// Copyright (c) 2018-2021 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 eps = Eigen::NumTraits<Scalar>::epsilon();

 const Scalar t2 = v.squaredNorm();
 const Scalar t = math::sqrt(t2 + eps * eps);

 static const Scalar ts_prec =
 TaylorSeriesExpansion<Scalar>::template precision<3>(); // Precision for the Taylor series
 // expansion.

 Eigen::AngleAxis<Scalar> aa(t, v / t);
 QuaternionPlain quat_then(aa);

 // order 4 Taylor expansion in theta / (order 2 in t2)
 QuaternionPlain quat_else;
 const Scalar t2_2 = t2 / 4; // theta/2 squared
 quat_else.vec() =
 Scalar(0.5) * (Scalar(1) - t2_2 / Scalar(6) + t2_2 * t2_2 / Scalar(120)) * v;
 quat_else.w() = Scalar(1) - t2_2 / 2 + t2_2 * t2_2 / 24;

 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 MotionDerived, typename Config_t>
 void exp6(const MotionDense<MotionDerived> & motion, Eigen::MatrixBase<Config_t> & qout)
 {
 enum
 {
 Options = PINOCCHIO_EIGEN_PLAIN_TYPE(Config_t)::Options
 };
 typedef typename Config_t::Scalar Scalar;
 typedef typename MotionDerived::Vector3 Vector3;
 typedef Eigen::Quaternion<Scalar, Options> Quaternion_t;
 const Scalar eps = Eigen::NumTraits<Scalar>::epsilon();

 const typename MotionDerived::ConstAngularType & w = motion.angular();
 const typename MotionDerived::ConstLinearType & v = motion.linear();

 const Scalar t2 = w.squaredNorm() + eps * eps;
 const Scalar t = math::sqrt(t2);

 Scalar ct, st;
 SINCOS(t, &st, &ct);

 const Scalar inv_t2 = Scalar(1) / t2;
 const Scalar ts_prec =
 TaylorSeriesExpansion<Scalar>::template precision<3>(); // Taylor expansion precision

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

 const Scalar alpha_wxv = if_then_else(
 LT, t, ts_prec,
 Scalar(0.5) - t2 / Scalar(24), // then: use Taylor expansion
 (Scalar(1) - ct) * inv_t2 // else
 );

 const Scalar alpha_w2 = if_then_else(
 LT, t, ts_prec, Scalar(1) / Scalar(6) - t2 / Scalar(120), (t - st) * inv_t2 / t);

 // linear part
 Eigen::Map<Vector3> trans_(qout.derived().template head<3>().data());
 trans_.noalias() = v + alpha_wxv * w.cross(v) + alpha_w2 * w.cross(w.cross(v));

 // quaternion part
 typedef Eigen::Map<Quaternion_t> QuaternionMap_t;
 QuaternionMap_t quat_(qout.derived().template tail<4>().data());
 exp3(w, quat_);
 }

 template<typename MotionDerived>
 Eigen::Matrix<
 typename MotionDerived::Scalar,
 7,
 1,
 PINOCCHIO_EIGEN_PLAIN_TYPE(typename MotionDerived::Vector3)::Options>
 exp6(const MotionDense<MotionDerived> & motion)
 {
 typedef typename MotionDerived::Scalar Scalar;
 enum
 {
 Options = PINOCCHIO_EIGEN_PLAIN_TYPE(typename MotionDerived::Vector3)::Options
 };
 typedef Eigen::Matrix<Scalar, 7, 1, Options> ReturnType;

 ReturnType qout;
 exp6(motion, qout);
 return qout;
 }

 template<typename Vector6Like, typename Config_t>
 void exp6(const Eigen::MatrixBase<Vector6Like> & vec6, Eigen::MatrixBase<Config_t> & qout)
 {
 MotionRef<const Vector6Like> nu(vec6.derived());
 ::pinocchio::quaternion::exp6(nu, qout);
 }

 template<typename Vector6Like>
 Eigen::
 Matrix<typename Vector6Like::Scalar, 7, 1, PINOCCHIO_EIGEN_PLAIN_TYPE(Vector6Like)::Options>
 exp6(const Eigen::MatrixBase<Vector6Like> & vec6)
 {
 typedef typename Vector6Like::Scalar Scalar;
 enum
 {
 Options = PINOCCHIO_EIGEN_PLAIN_TYPE(Vector6Like)::Options
 };
 typedef Eigen::Matrix<Scalar, 7, 1, Options> ReturnType;

 ReturnType qout;
 ::pinocchio::quaternion::exp6(vec6, qout);
 return qout;
 }

 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::GE;
 using ::pinocchio::internal::if_then_else;
 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
} // namespace pinocchio

#endif // ifndef __pinocchio_spatial_explog_quaternion_hpp__