special-euclidean.hpp

Go to the documentation of this file.

//
// Copyright (c) 2016-2021 CNRS INRIA
//
 
#ifndef __pinocchio_multibody_liegroup_special_euclidean_operation_hpp__
#define __pinocchio_multibody_liegroup_special_euclidean_operation_hpp__
 
#include <limits>
 
#include "pinocchio/macros.hpp"
#include "pinocchio/math/quaternion.hpp"
#include "pinocchio/math/matrix.hpp"
#include "pinocchio/spatial/fwd.hpp"
#include "pinocchio/utils/static-if.hpp"
#include "pinocchio/spatial/se3.hpp"
#include "pinocchio/multibody/liegroup/liegroup-base.hpp"
 
#include "pinocchio/multibody/liegroup/vector-space.hpp"
#include "pinocchio/multibody/liegroup/cartesian-product.hpp"
#include "pinocchio/multibody/liegroup/special-orthogonal.hpp"
 
namespace pinocchio
{
  template<int Dim, typename Scalar, int Options = 0>
  struct SpecialEuclideanOperationTpl
  {
  };
 
  template<int Dim, typename Scalar, int Options>
  struct traits<SpecialEuclideanOperationTpl<Dim, Scalar, Options>>
  {
  };
 
  template<typename _Scalar, int _Options>
  struct traits<SpecialEuclideanOperationTpl<2, _Scalar, _Options>>
  {
    typedef _Scalar Scalar;
    enum
    {
      Options = _Options,
      NQ = 4,
      NV = 3
    };
  };
 
  // SE(2)
  template<typename _Scalar, int _Options>
  struct SpecialEuclideanOperationTpl<2, _Scalar, _Options>
  : public LieGroupBase<SpecialEuclideanOperationTpl<2, _Scalar, _Options>>
  {
    PINOCCHIO_LIE_GROUP_TPL_PUBLIC_INTERFACE(SpecialEuclideanOperationTpl);
 
    typedef VectorSpaceOperationTpl<2, Scalar, Options> R2_t;
    typedef SpecialOrthogonalOperationTpl<2, Scalar, Options> SO2_t;
    typedef CartesianProductOperation<R2_t, SO2_t> R2crossSO2_t;
 
    typedef Eigen::Matrix<Scalar, 2, 2, Options> Matrix2;
    typedef Eigen::Matrix<Scalar, 2, 1, Options> Vector2;
 
    template<typename TangentVector, typename Matrix2Like, typename Vector2Like>
    static void exp(
      const Eigen::MatrixBase<TangentVector> & v,
      const Eigen::MatrixBase<Matrix2Like> & R,
      const Eigen::MatrixBase<Vector2Like> & t)
    {
      EIGEN_STATIC_ASSERT_SAME_VECTOR_SIZE(TangentVector_t, TangentVector);
      EIGEN_STATIC_ASSERT_MATRIX_SPECIFIC_SIZE(Matrix2Like, 2, 2);
      EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(Vector2Like, 2);
 
      typedef typename Matrix2Like::Scalar Scalar;
      const Scalar omega = v(2);
      Scalar cv, sv;
      SINCOS(omega, &sv, &cv);
      PINOCCHIO_EIGEN_CONST_CAST(Matrix2Like, R) << cv, -sv, sv, cv;
      using internal::if_then_else;
 
      {
        typename PINOCCHIO_EIGEN_PLAIN_TYPE(Vector2Like) vcross(-v(1), v(0));
        vcross -= -v(1) * R.col(0) + v(0) * R.col(1);
        vcross /= omega;
        Scalar omega_abs = math::fabs(omega);
        PINOCCHIO_EIGEN_CONST_CAST(Vector2Like, t).coeffRef(0) = if_then_else(
          internal::GT, omega_abs, Scalar(1e-14), // TODO: change hard coded value
          vcross.coeff(0), v.coeff(0));
 
        PINOCCHIO_EIGEN_CONST_CAST(Vector2Like, t).coeffRef(1) = if_then_else(
          internal::GT, omega_abs, Scalar(1e-14), // TODO: change hard coded value
          vcross.coeff(1), v.coeff(1));
      }
    }
 
    template<typename Matrix2Like, typename Vector2Like, typename Matrix3Like>
    static void toInverseActionMatrix(
      const Eigen::MatrixBase<Matrix2Like> & R,
      const Eigen::MatrixBase<Vector2Like> & t,
      const Eigen::MatrixBase<Matrix3Like> & M,
      const AssignmentOperatorType op)
    {
      EIGEN_STATIC_ASSERT_MATRIX_SPECIFIC_SIZE(Matrix2Like, 2, 2);
      EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(Vector2Like, 2);
      PINOCCHIO_ASSERT_MATRIX_SPECIFIC_SIZE(Matrix3Like, M, 3, 3);
      Matrix3Like & Mout = PINOCCHIO_EIGEN_CONST_CAST(Matrix3Like, M);
      typedef typename Matrix3Like::Scalar Scalar;
 
      typename PINOCCHIO_EIGEN_PLAIN_TYPE(Vector2Like) tinv((R.transpose() * t).reverse());
      tinv[0] *= Scalar(-1.);
      switch (op)
      {
      case SETTO:
        Mout.template topLeftCorner<2, 2>() = R.transpose();
        Mout.template topRightCorner<2, 1>() = tinv;
        Mout.template bottomLeftCorner<1, 2>().setZero();
        Mout(2, 2) = (Scalar)1;
        break;
      case ADDTO:
        Mout.template topLeftCorner<2, 2>() += R.transpose();
        Mout.template topRightCorner<2, 1>() += tinv;
        Mout(2, 2) += (Scalar)1;
        break;
      case RMTO:
        Mout.template topLeftCorner<2, 2>() -= R.transpose();
        Mout.template topRightCorner<2, 1>() -= tinv;
        Mout(2, 2) -= (Scalar)1;
        break;
      default:
        assert(false && "Wrong Op requesed value");
        break;
      }
    }
 
    template<typename Matrix2Like, typename Vector2Like, typename TangentVector>
    static void log(
      const Eigen::MatrixBase<Matrix2Like> & R,
      const Eigen::MatrixBase<Vector2Like> & p,
      const Eigen::MatrixBase<TangentVector> & v)
    {
      EIGEN_STATIC_ASSERT_MATRIX_SPECIFIC_SIZE(Matrix2Like, 2, 2);
      EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(Vector2Like, 2);
      EIGEN_STATIC_ASSERT_SAME_VECTOR_SIZE(TangentVector_t, TangentVector);
 
      TangentVector & vout = PINOCCHIO_EIGEN_CONST_CAST(TangentVector, v);
 
      typedef typename Matrix2Like::Scalar Scalar1;
 
      Scalar1 t = SO2_t::log(R);
      const Scalar1 tabs = math::fabs(t);
      const Scalar1 t2 = t * t;
      Scalar1 st, ct;
      SINCOS(tabs, &st, &ct);
      Scalar1 alpha;
      alpha = internal::if_then_else(
        internal::LT, tabs, Scalar(1e-4), // TODO: change hard coded value
        static_cast<Scalar>(1 - t2 / 12 - t2 * t2 / 720),
        static_cast<Scalar>(tabs * st / (2 * (1 - ct))));
 
      vout.template head<2>().noalias() = alpha * p;
      vout(0) += t / 2 * p(1);
      vout(1) += -t / 2 * p(0);
      vout(2) = t;
    }
 
    template<typename Matrix2Like, typename Vector2Like, typename JacobianOutLike>
    static void Jlog(
      const Eigen::MatrixBase<Matrix2Like> & R,
      const Eigen::MatrixBase<Vector2Like> & p,
      const Eigen::MatrixBase<JacobianOutLike> & J)
    {
      EIGEN_STATIC_ASSERT_MATRIX_SPECIFIC_SIZE(Matrix2Like, 2, 2);
      EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(Vector2Like, 2);
      EIGEN_STATIC_ASSERT_SAME_MATRIX_SIZE(JacobianOutLike, JacobianMatrix_t);
 
      JacobianOutLike & Jout = PINOCCHIO_EIGEN_CONST_CAST(JacobianOutLike, J);
 
      typedef typename Matrix2Like::Scalar Scalar1;
 
      Scalar1 t = SO2_t::log(R);
      const Scalar1 tabs = math::fabs(t);
      Scalar1 alpha, alpha_dot;
      Scalar1 t2 = t * t;
      Scalar1 st, ct;
      SINCOS(t, &st, &ct);
      Scalar1 inv_2_1_ct = 0.5 / (1 - ct);
 
      alpha = internal::if_then_else(
        internal::LT, tabs, Scalar(1e-4), static_cast<Scalar>(1 - t2 / 12),
        static_cast<Scalar>(t * st * inv_2_1_ct));
      alpha_dot = internal::if_then_else(
        internal::LT, tabs, Scalar(1e-4), static_cast<Scalar>(-t / 6 - t2 * t / 180),
        static_cast<Scalar>((st - t) * inv_2_1_ct));
 
      typename PINOCCHIO_EIGEN_PLAIN_TYPE(Matrix2Like) V;
      V(0, 0) = V(1, 1) = alpha;
      V(1, 0) = -t / 2;
      V(0, 1) = -V(1, 0);
 
      Jout.template topLeftCorner<2, 2>().noalias() = V * R;
      Jout.template topRightCorner<2, 1>() << alpha_dot * p[0] + p[1] / 2,
        -p(0) / 2 + alpha_dot * p(1);
      Jout.template bottomLeftCorner<1, 2>().setZero();
      Jout(2, 2) = 1;
    }
 
    static Index nq()
    {
      return NQ;
    }
    static Index nv()
    {
      return NV;
    }
 
    static ConfigVector_t neutral()
    {
      ConfigVector_t n;
      n << Scalar(0), Scalar(0), Scalar(1), Scalar(0);
      return n;
    }
 
    static std::string name()
    {
      return std::string("SE(2)");
    }
 
    template<class ConfigL_t, class ConfigR_t, class Tangent_t>
    static void difference_impl(
      const Eigen::MatrixBase<ConfigL_t> & q0,
      const Eigen::MatrixBase<ConfigR_t> & q1,
      const Eigen::MatrixBase<Tangent_t> & d)
    {
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Matrix2 R0, R1;
      Vector2 t0, t1;
      forwardKinematics(R0, t0, q0);
      forwardKinematics(R1, t1, q1);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
      Matrix2 R(R0.transpose() * R1);
      Vector2 t(R0.transpose() * (t1 - t0));
 
      log(R, t, d);
    }
 
    template<ArgumentPosition arg, class ConfigL_t, class ConfigR_t, class JacobianOut_t>
    void dDifference_impl(
      const Eigen::MatrixBase<ConfigL_t> & q0,
      const Eigen::MatrixBase<ConfigR_t> & q1,
      const Eigen::MatrixBase<JacobianOut_t> & J) const
    {
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Matrix2 R0, R1;
      Vector2 t0, t1;
      forwardKinematics(R0, t0, q0);
      forwardKinematics(R1, t1, q1);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
      Matrix2 R(R0.transpose() * R1);
      Vector2 t(R0.transpose() * (t1 - t0));
 
      if (arg == ARG0)
      {
        PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
        PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
        JacobianMatrix_t J1;
        Jlog(R, t, J1);
        PINOCCHIO_COMPILER_DIAGNOSTIC_POP
 
        // pcross = [ y1-y0, - (x1 - x0) ]
        Vector2 pcross(q1(1) - q0(1), q0(0) - q1(0));
 
        JacobianOut_t & J0 = PINOCCHIO_EIGEN_CONST_CAST(JacobianOut_t, J);
        J0.template topLeftCorner<2, 2>().noalias() = -R.transpose();
        J0.template topRightCorner<2, 1>().noalias() = R1.transpose() * pcross;
        J0.template bottomLeftCorner<1, 2>().setZero();
        J0(2, 2) = -1;
        J0.applyOnTheLeft(J1);
      }
      else if (arg == ARG1)
      {
        Jlog(R, t, J);
      }
    }
 
    template<class ConfigIn_t, class Velocity_t, class ConfigOut_t>
    static void integrate_impl(
      const Eigen::MatrixBase<ConfigIn_t> & q,
      const Eigen::MatrixBase<Velocity_t> & v,
      const Eigen::MatrixBase<ConfigOut_t> & qout)
    {
      ConfigOut_t & out = PINOCCHIO_EIGEN_CONST_CAST(ConfigOut_t, qout);
 
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Matrix2 R0, R;
      Vector2 t0, t;
      forwardKinematics(R0, t0, q);
      exp(v, R, t);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
 
      out.template head<2>().noalias() = R0 * t + t0;
      out.template tail<2>().noalias() = R0 * R.col(0);
    }
 
    template<class Config_t, class Jacobian_t>
    static void integrateCoeffWiseJacobian_impl(
      const Eigen::MatrixBase<Config_t> & q, const Eigen::MatrixBase<Jacobian_t> & J)
    {
      assert(J.rows() == nq() && J.cols() == nv() && "J is not of the right dimension");
 
      Jacobian_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(Jacobian_t, J);
      Jout.setZero();
 
      const typename Config_t::Scalar &c_theta = q(2), &s_theta = q(3);
 
      Jout.template topLeftCorner<2, 2>() << c_theta, -s_theta, s_theta, c_theta;
      Jout.template bottomRightCorner<2, 1>() << -s_theta, c_theta;
    }
 
    template<class Config_t, class Tangent_t, class JacobianOut_t>
    static void dIntegrate_dq_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<JacobianOut_t> & J,
      const AssignmentOperatorType op = SETTO)
    {
      JacobianOut_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(JacobianOut_t, J);
 
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Matrix2 R;
      Vector2 t;
      exp(v, R, t);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
 
      toInverseActionMatrix(R, t, Jout, op);
    }
 
    template<class Config_t, class Tangent_t, class JacobianOut_t>
    static void dIntegrate_dv_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<JacobianOut_t> & J,
      const AssignmentOperatorType op = SETTO)
    {
      JacobianOut_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(JacobianOut_t, J);
      // TODO sparse version
      MotionTpl<Scalar, 0> nu;
      nu.toVector() << v.template head<2>(), 0, 0, 0, v[2];
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Eigen::Matrix<Scalar, 6, 6> Jtmp6;
      Jexp6(nu, Jtmp6);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
 
      switch (op)
      {
      case SETTO:
        Jout << Jtmp6.template topLeftCorner<2, 2>(), Jtmp6.template topRightCorner<2, 1>(),
          Jtmp6.template bottomLeftCorner<1, 2>(), Jtmp6.template bottomRightCorner<1, 1>();
        break;
      case ADDTO:
        Jout.template topLeftCorner<2, 2>() += Jtmp6.template topLeftCorner<2, 2>();
        Jout.template topRightCorner<2, 1>() += Jtmp6.template topRightCorner<2, 1>();
        Jout.template bottomLeftCorner<1, 2>() += Jtmp6.template bottomLeftCorner<1, 2>();
        Jout.template bottomRightCorner<1, 1>() += Jtmp6.template bottomRightCorner<1, 1>();
        break;
      case RMTO:
        Jout.template topLeftCorner<2, 2>() -= Jtmp6.template topLeftCorner<2, 2>();
        Jout.template topRightCorner<2, 1>() -= Jtmp6.template topRightCorner<2, 1>();
        Jout.template bottomLeftCorner<1, 2>() -= Jtmp6.template bottomLeftCorner<1, 2>();
        Jout.template bottomRightCorner<1, 1>() -= Jtmp6.template bottomRightCorner<1, 1>();
        break;
      default:
        assert(false && "Wrong Op requesed value");
        break;
      }
    }
 
    template<class Config_t, class Tangent_t, class JacobianIn_t, class JacobianOut_t>
    static void dIntegrateTransport_dq_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<JacobianIn_t> & Jin,
      const Eigen::MatrixBase<JacobianOut_t> & J_out)
    {
      JacobianOut_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(JacobianOut_t, J_out);
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Matrix2 R;
      Vector2 t;
      exp(v, R, t);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
 
      Vector2 tinv = (R.transpose() * t).reverse();
      tinv[0] *= Scalar(-1.);
 
      Jout.template topRows<2>().noalias() = R.transpose() * Jin.template topRows<2>();
      Jout.template topRows<2>().noalias() += tinv * Jin.template bottomRows<1>();
      Jout.template bottomRows<1>() = Jin.template bottomRows<1>();
    }
 
    template<class Config_t, class Tangent_t, class JacobianIn_t, class JacobianOut_t>
    static void dIntegrateTransport_dv_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<JacobianIn_t> & Jin,
      const Eigen::MatrixBase<JacobianOut_t> & J_out)
    {
      JacobianOut_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(JacobianOut_t, J_out);
      MotionTpl<Scalar, 0> nu;
      nu.toVector() << v.template head<2>(), 0, 0, 0, v[2];
 
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Eigen::Matrix<Scalar, 6, 6> Jtmp6;
      Jexp6(nu, Jtmp6);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
 
      Jout.template topRows<2>().noalias() =
        Jtmp6.template topLeftCorner<2, 2>() * Jin.template topRows<2>();
      Jout.template topRows<2>().noalias() +=
        Jtmp6.template topRightCorner<2, 1>() * Jin.template bottomRows<1>();
      Jout.template bottomRows<1>().noalias() =
        Jtmp6.template bottomLeftCorner<1, 2>() * Jin.template topRows<2>();
      Jout.template bottomRows<1>().noalias() +=
        Jtmp6.template bottomRightCorner<1, 1>() * Jin.template bottomRows<1>();
    }
 
    template<class Config_t, class Tangent_t, class Jacobian_t>
    static void dIntegrateTransport_dq_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<Jacobian_t> & J)
    {
      Jacobian_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(Jacobian_t, J);
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Matrix2 R;
      Vector2 t;
      exp(v, R, t);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
 
      Vector2 tinv = (R.transpose() * t).reverse();
      tinv[0] *= Scalar(-1);
      // TODO: Aliasing
      Jout.template topRows<2>() = R.transpose() * Jout.template topRows<2>();
      // No Aliasing
      Jout.template topRows<2>().noalias() += tinv * Jout.template bottomRows<1>();
    }
 
    template<class Config_t, class Tangent_t, class Jacobian_t>
    static void dIntegrateTransport_dv_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<Jacobian_t> & J)
    {
      Jacobian_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(Jacobian_t, J);
      MotionTpl<Scalar, 0> nu;
      nu.toVector() << v.template head<2>(), 0, 0, 0, v[2];
 
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Eigen::Matrix<Scalar, 6, 6> Jtmp6;
      Jexp6(nu, Jtmp6);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
      // TODO: Remove aliasing
      Jout.template topRows<2>() =
        Jtmp6.template topLeftCorner<2, 2>() * Jout.template topRows<2>();
      Jout.template topRows<2>().noalias() +=
        Jtmp6.template topRightCorner<2, 1>() * Jout.template bottomRows<1>();
      Jout.template bottomRows<1>() =
        Jtmp6.template bottomRightCorner<1, 1>() * Jout.template bottomRows<1>();
      Jout.template bottomRows<1>().noalias() +=
        Jtmp6.template bottomLeftCorner<1, 2>() * Jout.template topRows<2>();
    }
 
    template<class Config_t>
    static void normalize_impl(const Eigen::MatrixBase<Config_t> & qout)
    {
      pinocchio::normalize(qout.const_cast_derived().template tail<2>());
    }
 
    template<class Config_t>
    static bool isNormalized_impl(const Eigen::MatrixBase<Config_t> & qin, const Scalar & prec)
    {
      const Scalar norm = Scalar(qin.template tail<2>().norm());
      using std::abs;
      return abs(norm - Scalar(1.0)) < prec;
    }
 
    template<class Config_t>
    static void random_impl(const Eigen::MatrixBase<Config_t> & qout)
    {
      R2crossSO2_t().random(qout);
    }
 
    template<class ConfigL_t, class ConfigR_t, class ConfigOut_t>
    static void randomConfiguration_impl(
      const Eigen::MatrixBase<ConfigL_t> & lower,
      const Eigen::MatrixBase<ConfigR_t> & upper,
      const Eigen::MatrixBase<ConfigOut_t> & qout)
    {
      R2crossSO2_t().randomConfiguration(lower, upper, qout);
    }
 
    template<class ConfigL_t, class ConfigR_t>
    static bool isSameConfiguration_impl(
      const Eigen::MatrixBase<ConfigL_t> & q0,
      const Eigen::MatrixBase<ConfigR_t> & q1,
      const Scalar & prec)
    {
      return R2crossSO2_t().isSameConfiguration(q0, q1, prec);
    }
 
  protected:
    template<typename Matrix2Like, typename Vector2Like, typename Vector4Like>
    static void forwardKinematics(
      const Eigen::MatrixBase<Matrix2Like> & R,
      const Eigen::MatrixBase<Vector2Like> & t,
      const Eigen::MatrixBase<Vector4Like> & q)
    {
      EIGEN_STATIC_ASSERT_SAME_MATRIX_SIZE(Matrix2Like, Matrix2);
      EIGEN_STATIC_ASSERT_SAME_VECTOR_SIZE(Vector2Like, Vector2);
      EIGEN_STATIC_ASSERT_SAME_VECTOR_SIZE(ConfigVector_t, Vector4Like);
 
      PINOCCHIO_EIGEN_CONST_CAST(Vector2Like, t) = q.template head<2>();
      const typename Vector4Like::Scalar &c_theta = q(2), &s_theta = q(3);
      PINOCCHIO_EIGEN_CONST_CAST(Matrix2Like, R) << c_theta, -s_theta, s_theta, c_theta;
    }
  }; // struct SpecialEuclideanOperationTpl<2>
 
  template<typename _Scalar, int _Options>
  struct traits<SpecialEuclideanOperationTpl<3, _Scalar, _Options>>
  {
    typedef _Scalar Scalar;
    enum
    {
      Options = _Options,
      NQ = 7,
      NV = 6
    };
  };
 
  template<typename _Scalar, int _Options>
  struct SpecialEuclideanOperationTpl<3, _Scalar, _Options>
  : public LieGroupBase<SpecialEuclideanOperationTpl<3, _Scalar, _Options>>
  {
    PINOCCHIO_LIE_GROUP_TPL_PUBLIC_INTERFACE(SpecialEuclideanOperationTpl);
 
    typedef CartesianProductOperation<
      VectorSpaceOperationTpl<3, Scalar, Options>,
      SpecialOrthogonalOperationTpl<3, Scalar, Options>>
      R3crossSO3_t;
 
    typedef Eigen::Quaternion<Scalar, Options> Quaternion_t;
    typedef Eigen::Map<Quaternion_t> QuaternionMap_t;
    typedef Eigen::Map<const Quaternion_t> ConstQuaternionMap_t;
    typedef SE3Tpl<Scalar, Options> Transformation_t;
    typedef SE3Tpl<Scalar, Options> SE3;
    typedef typename Eigen::NumTraits<Scalar>::Real RealScalar;
 
    static Index nq()
    {
      return NQ;
    }
    static Index nv()
    {
      return NV;
    }
 
    static ConfigVector_t neutral()
    {
      ConfigVector_t n;
      n.template head<6>().setZero();
      n[6] = 1;
      return n;
    }
 
    static std::string name()
    {
      return std::string("SE(3)");
    }
 
    template<class ConfigL_t, class ConfigR_t, class Tangent_t>
    static void difference_impl(
      const Eigen::MatrixBase<ConfigL_t> & q0,
      const Eigen::MatrixBase<ConfigR_t> & q1,
      const Eigen::MatrixBase<Tangent_t> & d)
    {
      typedef typename Tangent_t::Scalar Scalar;
      ConstQuaternionMap_t quat0(q0.derived().template tail<4>().data());
      assert(quaternion::isNormalized(
        quat0, RealScalar(PINOCCHIO_DEFAULT_QUATERNION_NORM_TOLERANCE_VALUE)));
      ConstQuaternionMap_t quat1(q1.derived().template tail<4>().data());
      assert(quaternion::isNormalized(
        quat1, RealScalar(PINOCCHIO_DEFAULT_QUATERNION_NORM_TOLERANCE_VALUE)));
 
      typedef Eigen::Matrix<Scalar, 3, 1, PINOCCHIO_EIGEN_PLAIN_TYPE(Tangent_t)::Options> Vector3;
      const Vector3 dv_pre = q1.derived().template head<3>() - q0.derived().template head<3>();
      const Vector3 dv = quat0.conjugate() * dv_pre;
      PINOCCHIO_EIGEN_CONST_CAST(Tangent_t, d).noalias() =
        log6(quat0.conjugate() * quat1, dv).toVector();
    }
 
    template<ArgumentPosition arg, class ConfigL_t, class ConfigR_t, class JacobianOut_t>
    static void dDifference_impl(
      const Eigen::MatrixBase<ConfigL_t> & q0,
      const Eigen::MatrixBase<ConfigR_t> & q1,
      const Eigen::MatrixBase<JacobianOut_t> & J)
    {
      typedef typename SE3::Vector3 Vector3;
 
      ConstQuaternionMap_t quat0(q0.derived().template tail<4>().data());
      assert(quaternion::isNormalized(
        quat0, RealScalar(PINOCCHIO_DEFAULT_QUATERNION_NORM_TOLERANCE_VALUE)));
      ConstQuaternionMap_t quat1(q1.derived().template tail<4>().data());
      assert(quaternion::isNormalized(
        quat1, RealScalar(PINOCCHIO_DEFAULT_QUATERNION_NORM_TOLERANCE_VALUE)));
 
      const Vector3 dv_pre = q1.derived().template head<3>() - q0.derived().template head<3>();
      const Vector3 trans = quat0.conjugate() * dv_pre;
 
      const Quaternion_t quat_diff = quat0.conjugate() * quat1;
 
      const SE3 M(quat_diff, trans);
 
      if (arg == ARG0)
      {
        PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
        PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
        JacobianMatrix_t J1;
        Jlog6(M, J1);
        PINOCCHIO_COMPILER_DIAGNOSTIC_POP
 
        const Vector3 p1_p0 = quat1.conjugate() * dv_pre;
 
        JacobianOut_t & J0 = PINOCCHIO_EIGEN_CONST_CAST(JacobianOut_t, J);
        J0.template bottomRightCorner<3, 3>() = J0.template topLeftCorner<3, 3>() =
          -M.rotation().transpose();
        J0.template topRightCorner<3, 3>().noalias() =
          skew(p1_p0) * M.rotation().transpose(); // = R1.T * skew(q1_t - q0_t) * R0;
        J0.template bottomLeftCorner<3, 3>().setZero();
        J0.applyOnTheLeft(J1);
      }
      else if (arg == ARG1)
      {
        Jlog6(M, J);
      }
    }
 
    template<class ConfigIn_t, class Velocity_t, class ConfigOut_t>
    static void integrate_impl(
      const Eigen::MatrixBase<ConfigIn_t> & q,
      const Eigen::MatrixBase<Velocity_t> & v,
      const Eigen::MatrixBase<ConfigOut_t> & qout)
    {
      ConfigOut_t & out = PINOCCHIO_EIGEN_CONST_CAST(ConfigOut_t, qout);
      Quaternion_t const quat(q.derived().template tail<4>());
      assert(quaternion::isNormalized(
        quat, RealScalar(PINOCCHIO_DEFAULT_QUATERNION_NORM_TOLERANCE_VALUE)));
      QuaternionMap_t res_quat(out.template tail<4>().data());
 
      using internal::if_then_else;
 
      typedef typename ConfigOut_t::Scalar Scalar;
      enum
      {
        Options = PINOCCHIO_EIGEN_PLAIN_TYPE(ConfigOut_t)::Options
      };
 
      Eigen::Matrix<Scalar, 7, 1, Options> expv;
      quaternion::exp6(v, expv);
 
      out.template head<3>() = (quat * expv.template head<3>()) + q.derived().template head<3>();
 
      ConstQuaternionMap_t quat1(expv.template tail<4>().data());
      res_quat = quat * quat1;
 
      const Scalar dot_product = res_quat.dot(quat);
      for (Eigen::DenseIndex k = 0; k < 4; ++k)
      {
        res_quat.coeffs().coeffRef(k) = if_then_else(
          internal::LT, dot_product, Scalar(0), static_cast<Scalar>(-res_quat.coeffs().coeff(k)),
          res_quat.coeffs().coeff(k));
      }
 
      // Norm of qs might be epsilon-different to 1, so M1.rotation might be epsilon-different to a
      // rotation matrix. It is then safer to re-normalized after converting M1.rotation to
      // quaternion.
      quaternion::firstOrderNormalize(res_quat);
      assert(quaternion::isNormalized(
        res_quat, RealScalar(PINOCCHIO_DEFAULT_QUATERNION_NORM_TOLERANCE_VALUE)));
    }
 
    template<class Config_t, class Jacobian_t>
    static void integrateCoeffWiseJacobian_impl(
      const Eigen::MatrixBase<Config_t> & q, const Eigen::MatrixBase<Jacobian_t> & J)
    {
      assert(J.rows() == nq() && J.cols() == nv() && "J is not of the right dimension");
 
      typedef typename PINOCCHIO_EIGEN_PLAIN_TYPE(Config_t) ConfigPlainType;
      typedef typename PINOCCHIO_EIGEN_PLAIN_TYPE(Jacobian_t) JacobianPlainType;
      typedef typename ConfigPlainType::Scalar Scalar;
 
      Jacobian_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(Jacobian_t, J);
      Jout.setZero();
 
      ConstQuaternionMap_t quat_map(q.derived().template tail<4>().data());
      assert(quaternion::isNormalized(
        quat_map, RealScalar(PINOCCHIO_DEFAULT_QUATERNION_NORM_TOLERANCE_VALUE)));
      Jout.template topLeftCorner<3, 3>() = quat_map.toRotationMatrix();
      //      Jexp3(quat,Jout.template bottomRightCorner<4,3>());
 
      typedef Eigen::Matrix<Scalar, 4, 3, JacobianPlainType::Options | Eigen::RowMajor> Jacobian43;
      typedef SE3Tpl<Scalar, ConfigPlainType::Options> SE3;
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Jacobian43 Jexp3QuatCoeffWise;
 
      Scalar theta;
      typename SE3::Vector3 v = quaternion::log3(quat_map, theta);
      quaternion::Jexp3CoeffWise(v, Jexp3QuatCoeffWise);
      typename SE3::Matrix3 Jlog;
      Jlog3(theta, v, Jlog);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
 
      //      std::cout << "Jexp3QuatCoeffWise\n" << Jexp3QuatCoeffWise << std::endl;
      //      std::cout << "Jlog\n" << Jlog << std::endl;
 
      //      if(quat_map.w() >= 0.) // comes from the log3 for quaternions which may change the
      //      sign.
      if (quat_map.coeffs()[3] >= Scalar(0)) // comes from the log3 for quaternions which may change
                                             // the sign.
        PINOCCHIO_EIGEN_CONST_CAST(Jacobian_t, J).template bottomRightCorner<4, 3>().noalias() =
          Jexp3QuatCoeffWise * Jlog;
      else
        PINOCCHIO_EIGEN_CONST_CAST(Jacobian_t, J).template bottomRightCorner<4, 3>().noalias() =
          -Jexp3QuatCoeffWise * Jlog;
    }
 
    template<class Config_t, class Tangent_t, class JacobianOut_t>
    static void dIntegrate_dq_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<JacobianOut_t> & J,
      const AssignmentOperatorType op = SETTO)
    {
      JacobianOut_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(JacobianOut_t, J);
 
      switch (op)
      {
      case SETTO:
        Jout = exp6(MotionRef<const Tangent_t>(v.derived())).toDualActionMatrix().transpose();
        break;
      case ADDTO:
        Jout += exp6(MotionRef<const Tangent_t>(v.derived())).toDualActionMatrix().transpose();
        break;
      case RMTO:
        Jout -= exp6(MotionRef<const Tangent_t>(v.derived())).toDualActionMatrix().transpose();
        break;
      default:
        assert(false && "Wrong Op requesed value");
        break;
      }
    }
 
    template<class Config_t, class Tangent_t, class JacobianOut_t>
    static void dIntegrate_dv_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<JacobianOut_t> & J,
      const AssignmentOperatorType op = SETTO)
    {
      switch (op)
      {
      case SETTO:
        Jexp6<SETTO>(MotionRef<const Tangent_t>(v.derived()), J.derived());
        break;
      case ADDTO:
        Jexp6<ADDTO>(MotionRef<const Tangent_t>(v.derived()), J.derived());
        break;
      case RMTO:
        Jexp6<RMTO>(MotionRef<const Tangent_t>(v.derived()), J.derived());
        break;
      default:
        assert(false && "Wrong Op requesed value");
        break;
      }
    }
 
    template<class Config_t, class Tangent_t, class JacobianIn_t, class JacobianOut_t>
    static void dIntegrateTransport_dq_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<JacobianIn_t> & Jin,
      const Eigen::MatrixBase<JacobianOut_t> & J_out)
    {
      JacobianOut_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(JacobianOut_t, J_out);
      Eigen::Matrix<Scalar, 6, 6> Jtmp6;
      Jtmp6 = exp6(MotionRef<const Tangent_t>(v.derived())).toDualActionMatrix().transpose();
 
      Jout.template topRows<3>().noalias() =
        Jtmp6.template topLeftCorner<3, 3>() * Jin.template topRows<3>();
      Jout.template topRows<3>().noalias() +=
        Jtmp6.template topRightCorner<3, 3>() * Jin.template bottomRows<3>();
      Jout.template bottomRows<3>().noalias() =
        Jtmp6.template bottomRightCorner<3, 3>() * Jin.template bottomRows<3>();
    }
 
    template<class Config_t, class Tangent_t, class JacobianIn_t, class JacobianOut_t>
    static void dIntegrateTransport_dv_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<JacobianIn_t> & Jin,
      const Eigen::MatrixBase<JacobianOut_t> & J_out)
    {
      JacobianOut_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(JacobianOut_t, J_out);
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Eigen::Matrix<Scalar, 6, 6> Jtmp6;
      Jexp6<SETTO>(MotionRef<const Tangent_t>(v.derived()), Jtmp6);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
 
      Jout.template topRows<3>().noalias() =
        Jtmp6.template topLeftCorner<3, 3>() * Jin.template topRows<3>();
      Jout.template topRows<3>().noalias() +=
        Jtmp6.template topRightCorner<3, 3>() * Jin.template bottomRows<3>();
      Jout.template bottomRows<3>().noalias() =
        Jtmp6.template bottomRightCorner<3, 3>() * Jin.template bottomRows<3>();
    }
 
    template<class Config_t, class Tangent_t, class Jacobian_t>
    static void dIntegrateTransport_dq_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<Jacobian_t> & J_out)
    {
      Jacobian_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(Jacobian_t, J_out);
      Eigen::Matrix<Scalar, 6, 6> Jtmp6;
      Jtmp6 = exp6(MotionRef<const Tangent_t>(v.derived())).toDualActionMatrix().transpose();
 
      // Aliasing
      Jout.template topRows<3>() =
        Jtmp6.template topLeftCorner<3, 3>() * Jout.template topRows<3>();
      Jout.template topRows<3>().noalias() +=
        Jtmp6.template topRightCorner<3, 3>() * Jout.template bottomRows<3>();
      Jout.template bottomRows<3>() =
        Jtmp6.template bottomRightCorner<3, 3>() * Jout.template bottomRows<3>();
    }
 
    template<class Config_t, class Tangent_t, class Jacobian_t>
    static void dIntegrateTransport_dv_impl(
      const Eigen::MatrixBase<Config_t> & /*q*/,
      const Eigen::MatrixBase<Tangent_t> & v,
      const Eigen::MatrixBase<Jacobian_t> & J_out)
    {
      Jacobian_t & Jout = PINOCCHIO_EIGEN_CONST_CAST(Jacobian_t, J_out);
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      Eigen::Matrix<Scalar, 6, 6> Jtmp6;
      Jexp6<SETTO>(MotionRef<const Tangent_t>(v.derived()), Jtmp6);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
 
      Jout.template topRows<3>() =
        Jtmp6.template topLeftCorner<3, 3>() * Jout.template topRows<3>();
      Jout.template topRows<3>().noalias() +=
        Jtmp6.template topRightCorner<3, 3>() * Jout.template bottomRows<3>();
      Jout.template bottomRows<3>() =
        Jtmp6.template bottomRightCorner<3, 3>() * Jout.template bottomRows<3>();
    }
 
    template<class ConfigL_t, class ConfigR_t>
    static Scalar squaredDistance_impl(
      const Eigen::MatrixBase<ConfigL_t> & q0, const Eigen::MatrixBase<ConfigR_t> & q1)
    {
      PINOCCHIO_COMPILER_DIAGNOSTIC_PUSH
      PINOCCHIO_COMPILER_DIAGNOSTIC_IGNORED_MAYBE_UNINITIALIZED
      TangentVector_t t;
      difference_impl(q0, q1, t);
      PINOCCHIO_COMPILER_DIAGNOSTIC_POP
      return t.squaredNorm();
    }
 
    template<class Config_t>
    static void normalize_impl(const Eigen::MatrixBase<Config_t> & qout)
    {
      pinocchio::normalize(qout.const_cast_derived().template tail<4>());
    }
 
    template<class Config_t>
    static bool isNormalized_impl(const Eigen::MatrixBase<Config_t> & qin, const Scalar & prec)
    {
      Scalar norm = Scalar(qin.template tail<4>().norm());
      using std::abs;
      return abs(norm - Scalar(1.0)) < prec;
    }
 
    template<class Config_t>
    static void random_impl(const Eigen::MatrixBase<Config_t> & qout)
    {
      R3crossSO3_t().random(qout);
    }
 
    template<class ConfigL_t, class ConfigR_t, class ConfigOut_t>
    static void randomConfiguration_impl(
      const Eigen::MatrixBase<ConfigL_t> & lower,
      const Eigen::MatrixBase<ConfigR_t> & upper,
      const Eigen::MatrixBase<ConfigOut_t> & qout)
    {
      R3crossSO3_t().randomConfiguration(lower, upper, qout);
    }
 
    template<class ConfigL_t, class ConfigR_t>
    static bool isSameConfiguration_impl(
      const Eigen::MatrixBase<ConfigL_t> & q0,
      const Eigen::MatrixBase<ConfigR_t> & q1,
      const Scalar & prec)
    {
      return R3crossSO3_t().isSameConfiguration(q0, q1, prec);
    }
  }; // struct SpecialEuclideanOperationTpl<3>
 
} // namespace pinocchio
 
#endif // ifndef __pinocchio_multibody_liegroup_special_euclidean_operation_hpp__