joint-helical.hpp

//
// Copyright (c) 2022-2023 INRIA
//

#ifndef __pinocchio_multibody_joint_helical_hpp__
#define __pinocchio_multibody_joint_helical_hpp__

#include "pinocchio/math/sincos.hpp"
#include "pinocchio/spatial/inertia.hpp"
#include "pinocchio/multibody/joint-motion-subspace.hpp"
#include "pinocchio/multibody/joint/joint-base.hpp"
#include "pinocchio/spatial/spatial-axis.hpp"
#include "pinocchio/utils/axis-label.hpp"

namespace pinocchio
{

  template<typename Scalar, int Options, int axis>
  struct MotionHelicalTpl;

  template<typename Scalar, int Options, int axis>
  struct SE3GroupAction<MotionHelicalTpl<Scalar, Options, axis>>
  {
    typedef MotionTpl<Scalar, Options> ReturnType;
  };

  template<typename Scalar, int Options, int axis, typename MotionDerived>
  struct MotionAlgebraAction<MotionHelicalTpl<Scalar, Options, axis>, MotionDerived>
  {
    typedef MotionTpl<Scalar, Options> ReturnType;
  };

  template<typename _Scalar, int _Options, int axis>
  struct traits<MotionHelicalTpl<_Scalar, _Options, axis>>
  {
    typedef _Scalar Scalar;
    enum
    {
      Options = _Options
    };
    typedef Eigen::Matrix<Scalar, 3, 1, Options> Vector3;
    typedef Eigen::Matrix<Scalar, 6, 1, Options> Vector6;
    typedef Eigen::Matrix<Scalar, 4, 4, Options> Matrix4;
    typedef Eigen::Matrix<Scalar, 6, 6, Options> Matrix6;
    typedef typename PINOCCHIO_EIGEN_REF_CONST_TYPE(Vector6) ToVectorConstReturnType;
    typedef typename PINOCCHIO_EIGEN_REF_TYPE(Vector6) ToVectorReturnType;
    typedef Vector3 AngularType;
    typedef Vector3 LinearType;
    typedef const Vector3 ConstAngularType;
    typedef const Vector3 ConstLinearType;
    typedef Matrix6 ActionMatrixType;
    typedef MotionTpl<Scalar, Options> MotionPlain;
    typedef MotionPlain PlainReturnType;
    typedef Matrix4 HomogeneousMatrixType;
    enum
    {
      LINEAR = 0,
      ANGULAR = 3
    };
  }; // traits MotionHelicalTpl

  template<typename Scalar, int Options, int axis>
  struct TransformHelicalTpl;

  template<typename _Scalar, int _Options, int _axis>
  struct traits<TransformHelicalTpl<_Scalar, _Options, _axis>>
  {
    enum
    {
      axis = _axis,
      Options = _Options,
      LINEAR = 0,
      ANGULAR = 3
    };
    typedef _Scalar Scalar;
    typedef SE3Tpl<Scalar, Options> PlainType;
    typedef Eigen::Matrix<Scalar, 3, 1, Options> Vector3;
    typedef Eigen::Matrix<Scalar, 3, 3, Options> Matrix3;
    typedef Matrix3 AngularType;
    typedef Matrix3 AngularRef;
    typedef Matrix3 ConstAngularRef;
    typedef Vector3 LinearType;
    typedef typename Vector3::ConstantReturnType LinearRef;
    typedef const typename Vector3::ConstantReturnType ConstLinearRef;
    typedef typename traits<PlainType>::ActionMatrixType ActionMatrixType;
    typedef typename traits<PlainType>::HomogeneousMatrixType HomogeneousMatrixType;
    typedef typename traits<PlainType>::ExprType ExprType;
    typedef typename traits<PlainType>::ConstExprType ConstExprType;
  }; // traits TransformHelicalTpl

  template<typename Scalar, int Options, int axis>
  struct SE3GroupAction<TransformHelicalTpl<Scalar, Options, axis>>
  {
    typedef typename traits<TransformHelicalTpl<Scalar, Options, axis>>::PlainType ReturnType;
  };

  template<typename _Scalar, int _Options, int axis>
  struct TransformHelicalTpl : SE3Base<TransformHelicalTpl<_Scalar, _Options, axis>>
  {
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW
    PINOCCHIO_SE3_TYPEDEF_TPL(TransformHelicalTpl);

    typedef SpatialAxis<axis + LINEAR> AxisLinear;
    typedef typename AxisLinear::CartesianAxis3 CartesianAxis3Linear;

    TransformHelicalTpl()
    {
    }
    TransformHelicalTpl(const Scalar & sin, const Scalar & cos, const Scalar & displacement)
    : m_sin(sin)
    , m_cos(cos)
    , m_displacement(displacement)
    {
    }

    PlainType plain() const
    {
      PlainType res(PlainType::Identity());
      _setRotation(res.rotation());
      res.translation()[axis] = m_displacement;
      return res;
    }

    operator PlainType() const
    {
      return plain();
    }

    template<typename S2, int O2>
    typename SE3GroupAction<TransformHelicalTpl>::ReturnType
    se3action(const SE3Tpl<S2, O2> & m) const
    {
      typedef typename SE3GroupAction<TransformHelicalTpl>::ReturnType ReturnType;
      ReturnType res;
      switch (axis)
      {
      case 0: {
        res.rotation().col(0) = m.rotation().col(0);
        res.rotation().col(1).noalias() = m_cos * m.rotation().col(1) + m_sin * m.rotation().col(2);
        res.rotation().col(2).noalias() = res.rotation().col(0).cross(res.rotation().col(1));
        break;
      }
      case 1: {
        res.rotation().col(2).noalias() = m_cos * m.rotation().col(2) + m_sin * m.rotation().col(0);
        res.rotation().col(1) = m.rotation().col(1);
        res.rotation().col(0).noalias() = res.rotation().col(1).cross(res.rotation().col(2));
        break;
      }
      case 2: {
        res.rotation().col(0).noalias() = m_cos * m.rotation().col(0) + m_sin * m.rotation().col(1);
        res.rotation().col(1).noalias() = res.rotation().col(2).cross(res.rotation().col(0));
        res.rotation().col(2) = m.rotation().col(2);
        break;
      }
      default: {
        assert(false && "must never happen");
        break;
      }
      }
      res.translation() = m.translation();
      res.translation()[axis] += m_displacement;
      return res;
    }

    const Scalar & sin() const
    {
      return m_sin;
    }
    Scalar & sin()
    {
      return m_sin;
    }

    const Scalar & cos() const
    {
      return m_cos;
    }
    Scalar & cos()
    {
      return m_cos;
    }

    const Scalar & displacement() const
    {
      return m_displacement;
    }
    Scalar & displacement()
    {
      return m_displacement;
    }

    template<typename Scalar1, typename Scalar2, typename Scalar3>
    void setValues(const Scalar1 & sin, const Scalar2 & cos, const Scalar3 & displacement)
    {
      m_sin = sin;
      m_cos = cos;
      m_displacement = displacement;
    }

    LinearType translation() const
    {
      return CartesianAxis3Linear() * displacement();
    }
    AngularType rotation() const
    {
      AngularType m(AngularType::Identity());
      _setRotation(m);
      return m;
    }

    bool isEqual(const TransformHelicalTpl & other) const
    {
      return internal::comparison_eq(m_cos, other.m_cos)
             && internal::comparison_eq(m_sin, other.m_sin)
             && internal::comparison_eq(m_displacement, other.m_displacement);
    }

  protected:
    Scalar m_sin, m_cos, m_displacement;
    inline void _setRotation(typename PlainType::AngularRef & rot) const
    {
      switch (axis)
      {
      case 0: {
        rot.coeffRef(1, 1) = m_cos;
        rot.coeffRef(1, 2) = -m_sin;
        rot.coeffRef(2, 1) = m_sin;
        rot.coeffRef(2, 2) = m_cos;
        break;
      }
      case 1: {
        rot.coeffRef(0, 0) = m_cos;
        rot.coeffRef(0, 2) = m_sin;
        rot.coeffRef(2, 0) = -m_sin;
        rot.coeffRef(2, 2) = m_cos;
        break;
      }
      case 2: {
        rot.coeffRef(0, 0) = m_cos;
        rot.coeffRef(0, 1) = -m_sin;
        rot.coeffRef(1, 0) = m_sin;
        rot.coeffRef(1, 1) = m_cos;
        break;
      }
      default: {
        assert(false && "must never happen");
        break;
      }
      }
    }
  }; // struct TransformHelicalTpl

  template<typename _Scalar, int _Options, int axis>
  struct MotionHelicalTpl : MotionBase<MotionHelicalTpl<_Scalar, _Options, axis>>
  {
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW

    MOTION_TYPEDEF_TPL(MotionHelicalTpl);
    typedef SpatialAxis<axis + ANGULAR> AxisAngular;
    typedef typename AxisAngular::CartesianAxis3 CartesianAxis3Angular;
    typedef SpatialAxis<axis + LINEAR> AxisLinear;
    typedef typename AxisLinear::CartesianAxis3 CartesianAxis3Linear;

    MotionHelicalTpl()
    {
    }

    MotionHelicalTpl(const Scalar & w, const Scalar & v)
    : m_w(w)
    , m_v(v)
    {
    }

    inline PlainReturnType plain() const
    {
      return PlainReturnType(CartesianAxis3Linear() * m_v, CartesianAxis3Angular() * m_w);
    }

    template<typename OtherScalar>
    MotionHelicalTpl __mult__(const OtherScalar & alpha) const
    {
      return MotionHelicalTpl(alpha * m_w, alpha * m_v);
    }

    template<typename MotionDerived>
    void setTo(MotionDense<MotionDerived> & m) const
    {
      for (Eigen::DenseIndex k = 0; k < 3; ++k)
      {
        m.angular()[k] = k == axis ? m_w : (Scalar)0;
        m.linear()[k] = k == axis ? m_v : (Scalar)0;
      }
    }

    template<typename MotionDerived>
    inline void addTo(MotionDense<MotionDerived> & v) const
    {
      typedef typename MotionDense<MotionDerived>::Scalar OtherScalar;
      v.angular()[axis] += (OtherScalar)m_w;
      v.linear()[axis] += (OtherScalar)m_v;
    }

    template<typename S2, int O2, typename D2>
    inline void se3Action_impl(const SE3Tpl<S2, O2> & m, MotionDense<D2> & v) const
    {
      v.angular().noalias() = m.rotation().col(axis) * m_w;
      v.linear().noalias() = m.translation().cross(v.angular()) + m_v * (m.rotation().col(axis));
    }

    template<typename S2, int O2>
    MotionPlain se3Action_impl(const SE3Tpl<S2, O2> & m) const
    {

      MotionPlain res;
      se3Action_impl(m, res);
      return res;
    }

    template<typename S2, int O2, typename D2>
    void se3ActionInverse_impl(const SE3Tpl<S2, O2> & m, MotionDense<D2> & v) const
    {
      // Linear
      CartesianAxis3Linear::alphaCross(m_w, m.translation(), v.angular());
      v.linear().noalias() =
        m.rotation().transpose() * v.angular() + m_v * (m.rotation().transpose().col(axis));

      // Angular
      v.angular().noalias() = m.rotation().transpose().col(axis) * m_w;
    }

    template<typename S2, int O2>
    MotionPlain se3ActionInverse_impl(const SE3Tpl<S2, O2> & m) const
    {
      MotionPlain res;
      se3ActionInverse_impl(m, res);
      return res;
    }

    template<typename M1, typename M2>
    EIGEN_STRONG_INLINE void motionAction(const MotionDense<M1> & v, MotionDense<M2> & mout) const
    {
      // Linear
      CartesianAxis3Linear::alphaCross(-m_w, v.linear(), mout.linear());
      CartesianAxis3Linear::alphaCross(-m_v, v.angular(), mout.angular());
      mout.linear() += mout.angular();
      // Angular
      CartesianAxis3Angular::alphaCross(-m_w, v.angular(), mout.angular());
    }

    template<typename M1>
    MotionPlain motionAction(const MotionDense<M1> & v) const
    {
      MotionPlain res;
      motionAction(v, res);
      return res;
    }

    Scalar & angularRate()
    {
      return m_w;
    }
    const Scalar & angularRate() const
    {
      return m_w;
    }

    Scalar & linearRate()
    {
      return m_v;
    }
    const Scalar & linearRate() const
    {
      return m_v;
    }

    bool isEqual_impl(const MotionHelicalTpl & other) const
    {
      return internal::comparison_eq(m_w, other.m_w) && internal::comparison_eq(m_v, other.m_v);
    }

  protected:
    Scalar m_w, m_v;
  }; // struct MotionHelicalTpl
  template<typename S1, int O1, int axis, typename MotionDerived>
  typename MotionDerived::MotionPlain
  operator+(const MotionHelicalTpl<S1, O1, axis> & m1, const MotionDense<MotionDerived> & m2)
  {
    typename MotionDerived::MotionPlain res(m2);
    res += m1;
    return res;
  }

  template<typename MotionDerived, typename S2, int O2, int axis>
  EIGEN_STRONG_INLINE typename MotionDerived::MotionPlain
  operator^(const MotionDense<MotionDerived> & m1, const MotionHelicalTpl<S2, O2, axis> & m2)
  {
    return m2.motionAction(m1);
  }

  template<typename Scalar, int Options, int axis>
  struct JointMotionSubspaceHelicalTpl;

  template<typename Scalar, int Options, int axis>
  struct SE3GroupAction<JointMotionSubspaceHelicalTpl<Scalar, Options, axis>>
  {
    typedef Eigen::Matrix<Scalar, 6, 1, Options> ReturnType;
  };

  template<typename Scalar, int Options, int axis, typename MotionDerived>
  struct MotionAlgebraAction<JointMotionSubspaceHelicalTpl<Scalar, Options, axis>, MotionDerived>
  {
    typedef Eigen::Matrix<Scalar, 6, 1, Options> ReturnType;
  };

  template<typename Scalar, int Options, int axis, typename ForceDerived>
  struct ConstraintForceOp<JointMotionSubspaceHelicalTpl<Scalar, Options, axis>, ForceDerived>
  {
    typedef typename Eigen::Matrix<Scalar, 1, 1> ReturnType;
  };

  template<typename Scalar, int Options, int axis, typename ForceSet>
  struct ConstraintForceSetOp<JointMotionSubspaceHelicalTpl<Scalar, Options, axis>, ForceSet>
  {
    typedef typename Eigen::Matrix<Scalar, Eigen::Dynamic, Eigen::Dynamic> ReturnType;
  };

  template<typename _Scalar, int _Options, int axis>
  struct traits<JointMotionSubspaceHelicalTpl<_Scalar, _Options, axis>>
  {
    typedef _Scalar Scalar;
    enum
    {
      Options = _Options
    };
    enum
    {
      LINEAR = 0,
      ANGULAR = 3
    };

    typedef MotionHelicalTpl<Scalar, Options, axis> JointMotion;
    typedef Eigen::Matrix<Scalar, 1, 1, Options> JointForce;
    typedef Eigen::Matrix<Scalar, 6, 1, Options> DenseBase;
    typedef Eigen::Matrix<Scalar, 1, 1, Options> ReducedSquaredMatrix;

    typedef DenseBase MatrixReturnType;
    typedef const DenseBase ConstMatrixReturnType;

    typedef typename ReducedSquaredMatrix::IdentityReturnType StDiagonalMatrixSOperationReturnType;
  }; // traits JointMotionSubspaceHelicalTpl

  template<class ConstraintDerived>
  struct TransposeConstraintActionConstraint
  {
    typedef
      typename Eigen::Matrix<typename ConstraintDerived::Scalar, 1, 1, ConstraintDerived::Options>
        ReturnType;
  };

  template<typename _Scalar, int _Options, int axis>
  struct JointMotionSubspaceHelicalTpl
  : JointMotionSubspaceBase<JointMotionSubspaceHelicalTpl<_Scalar, _Options, axis>>
  {
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW

    PINOCCHIO_CONSTRAINT_TYPEDEF_TPL(JointMotionSubspaceHelicalTpl)
    enum
    {
      NV = 1
    };

    typedef SpatialAxis<ANGULAR + axis> AxisAngular;
    typedef SpatialAxis<ANGULAR + axis> AxisLinear;

    typedef typename AxisAngular::CartesianAxis3 CartesianAxis3Angular;
    typedef typename AxisLinear::CartesianAxis3 CartesianAxis3Linear;

    JointMotionSubspaceHelicalTpl()
    {
    }

    JointMotionSubspaceHelicalTpl(const Scalar & h)
    : m_pitch(h)
    {
    }

    template<typename Vector1Like>
    JointMotion __mult__(const Eigen::MatrixBase<Vector1Like> & v) const
    {
      EIGEN_STATIC_ASSERT_VECTOR_SPECIFIC_SIZE(Vector1Like, 1);
      assert(v.size() == 1);
      return JointMotion(v[0], v[0] * m_pitch);
    }

    template<typename S1, int O1>
    typename SE3GroupAction<JointMotionSubspaceHelicalTpl>::ReturnType
    se3Action(const SE3Tpl<S1, O1> & m) const
    {
      typedef typename SE3GroupAction<JointMotionSubspaceHelicalTpl>::ReturnType ReturnType;
      ReturnType res;
      res.template segment<3>(LINEAR) =
        m.translation().cross(m.rotation().col(axis)) + m_pitch * (m.rotation().col(axis));
      res.template segment<3>(ANGULAR) = m.rotation().col(axis);
      return res;
    }

    template<typename S1, int O1>
    typename SE3GroupAction<JointMotionSubspaceHelicalTpl>::ReturnType
    se3ActionInverse(const SE3Tpl<S1, O1> & m) const
    {
      typedef typename SE3GroupAction<JointMotionSubspaceHelicalTpl>::ReturnType ReturnType;
      typedef typename AxisAngular::CartesianAxis3 CartesianAxis3;
      ReturnType res;
      res.template segment<3>(LINEAR).noalias() =
        m.rotation().transpose() * CartesianAxis3::cross(m.translation())
        + m.rotation().transpose().col(axis) * m_pitch;
      res.template segment<3>(ANGULAR) = m.rotation().transpose().col(axis);
      return res;
    }

    int nv_impl() const
    {
      return NV;
    }

    // For force T
    struct TransposeConst : JointMotionSubspaceTransposeBase<JointMotionSubspaceHelicalTpl>
    {
      const JointMotionSubspaceHelicalTpl & ref;
      TransposeConst(const JointMotionSubspaceHelicalTpl & ref)
      : ref(ref)
      {
      }

      template<typename ForceDerived>
      typename ConstraintForceOp<JointMotionSubspaceHelicalTpl, ForceDerived>::ReturnType
      operator*(const ForceDense<ForceDerived> & f) const
      {
        return Eigen::Matrix<Scalar, 1, 1>(f.angular()(axis) + f.linear()(axis) * ref.m_pitch);
      }

      /// [CRBA]  MatrixBase operator* (Constraint::Transpose S, ForceSet::Block)
      template<typename Derived>
      typename ConstraintForceSetOp<JointMotionSubspaceHelicalTpl, Derived>::ReturnType
      operator*(const Eigen::MatrixBase<Derived> & F) const
      {
        assert(F.rows() == 6);
        return F.row(ANGULAR + axis) + F.row(LINEAR + axis) * ref.m_pitch;
      }
    }; // struct TransposeConst

    TransposeConst transpose() const
    {
      return TransposeConst(*this);
    }

    /* CRBA joint operators
     *   - ForceSet::Block = ForceSet
     *   - ForceSet operator* (Inertia Y,Constraint S)
     *   - MatrixBase operator* (Constraint::Transpose S, ForceSet::Block)
     *   - SE3::act(ForceSet::Block)
     */
    DenseBase matrix_impl() const
    {
      DenseBase S;
      MotionRef<DenseBase> v(S);
      v << AxisAngular();
      S(LINEAR + axis) = m_pitch;
      return S;
    }

    template<typename MotionDerived>
    typename MotionAlgebraAction<JointMotionSubspaceHelicalTpl, MotionDerived>::ReturnType
    motionAction(const MotionDense<MotionDerived> & m) const
    {
      typedef typename MotionAlgebraAction<JointMotionSubspaceHelicalTpl, MotionDerived>::ReturnType
        ReturnType;
      ReturnType res;
      // Linear
      CartesianAxis3Linear::cross(-m.linear(), res.template segment<3>(LINEAR));
      CartesianAxis3Linear::alphaCross(-m_pitch, m.angular(), res.template segment<3>(ANGULAR));
      res.template segment<3>(LINEAR) += res.template segment<3>(ANGULAR);

      // Angular
      CartesianAxis3Angular::cross(-m.angular(), res.template segment<3>(ANGULAR));
      return res;
    }

    bool isEqual(const JointMotionSubspaceHelicalTpl &) const
    {
      return true;
    }

    Scalar & h()
    {
      return m_pitch;
    }
    const Scalar & h() const
    {
      return m_pitch;
    }

  protected:
    Scalar m_pitch;
  }; // struct JointMotionSubspaceHelicalTpl

  template<typename _Scalar, int _Options, int _axis>
  Eigen::Matrix<_Scalar, 1, 1, _Options> operator*(
    const typename JointMotionSubspaceHelicalTpl<_Scalar, _Options, _axis>::TransposeConst &
      S_transpose,
    const JointMotionSubspaceHelicalTpl<_Scalar, _Options, _axis> & S)
  {
    Eigen::Matrix<_Scalar, 1, 1, _Options> res;
    res(0) = 1.0 + S_transpose.ref.h() * S.h();
    return res;
  }

  template<typename _Scalar, int _Options, int _axis>
  struct JointHelicalTpl
  {
    typedef _Scalar Scalar;

    enum
    {
      Options = _Options,
      axis = _axis
    };
  };

  template<typename S1, int O1, typename S2, int O2, int axis>
  struct MultiplicationOp<InertiaTpl<S1, O1>, JointMotionSubspaceHelicalTpl<S2, O2, axis>>
  {
    typedef Eigen::Matrix<S2, 6, 1, O2> ReturnType;
  };

  /* [CRBA] ForceSet operator* (Inertia Y,Constraint S) */
  namespace impl
  {
    template<typename S1, int O1, typename S2, int O2>
    struct LhsMultiplicationOp<InertiaTpl<S1, O1>, JointMotionSubspaceHelicalTpl<S2, O2, 0>>
    {
      typedef InertiaTpl<S1, O1> Inertia;
      typedef JointMotionSubspaceHelicalTpl<S2, O2, 0> Constraint;
      typedef typename MultiplicationOp<Inertia, Constraint>::ReturnType ReturnType;
      static inline ReturnType run(const Inertia & Y, const Constraint & constraint)
      {
        ReturnType res;
        const S2 & m_pitch = constraint.h();

        /* Y(:,3) = ( 0,-z, y,  I00+yy+zz,  I01-xy   ,  I02-xz   ) */
        /* Y(:,0) = ( 1,0, 0, 0 , z , -y ) */
        const S1 &m = Y.mass(), &x = Y.lever()[0], &y = Y.lever()[1], &z = Y.lever()[2];
        const typename Inertia::Symmetric3 & I = Y.inertia();

        res << m * m_pitch, -m * z, m * y, I(0, 0) + m * (y * y + z * z),
          I(0, 1) - m * x * y + m * z * m_pitch, I(0, 2) - m * x * z - m * y * m_pitch;

        return res;
      }
    };

    template<typename S1, int O1, typename S2, int O2>
    struct LhsMultiplicationOp<InertiaTpl<S1, O1>, JointMotionSubspaceHelicalTpl<S2, O2, 1>>
    {
      typedef InertiaTpl<S1, O1> Inertia;
      typedef JointMotionSubspaceHelicalTpl<S2, O2, 1> Constraint;
      typedef typename MultiplicationOp<Inertia, Constraint>::ReturnType ReturnType;
      static inline ReturnType run(const Inertia & Y, const Constraint & constraint)
      {
        ReturnType res;
        const S2 & m_pitch = constraint.h();

        /* Y(:,4) = ( z, 0,-x,  I10-xy   ,  I11+xx+zz,  I12-yz   ) */
        /* Y(:,1) = ( 0,1, 0, -z , 0 , x) */
        const S1 &m = Y.mass(), &x = Y.lever()[0], &y = Y.lever()[1], &z = Y.lever()[2];
        const typename Inertia::Symmetric3 & I = Y.inertia();

        res << m * z, m * m_pitch, -m * x, I(1, 0) - m * x * y - m * z * m_pitch,
          I(1, 1) + m * (x * x + z * z), I(1, 2) - m * y * z + m * x * m_pitch;

        return res;
      }
    };

    template<typename S1, int O1, typename S2, int O2>
    struct LhsMultiplicationOp<InertiaTpl<S1, O1>, JointMotionSubspaceHelicalTpl<S2, O2, 2>>
    {
      typedef InertiaTpl<S1, O1> Inertia;
      typedef JointMotionSubspaceHelicalTpl<S2, O2, 2> Constraint;
      typedef typename MultiplicationOp<Inertia, Constraint>::ReturnType ReturnType;
      static inline ReturnType run(const Inertia & Y, const Constraint & constraint)
      {
        ReturnType res;
        const S2 & m_pitch = constraint.h();

        /* Y(:,5) = (-y, x, 0,  I20-xz   ,  I21-yz   ,  I22+xx+yy) */
        /* Y(:,2) = ( 0,0, 1, y , -x , 0) */
        const S1 &m = Y.mass(), &x = Y.lever()[0], &y = Y.lever()[1], &z = Y.lever()[2];
        const typename Inertia::Symmetric3 & I = Y.inertia();

        res << -m * y, m * x, m * m_pitch, I(2, 0) - m * x * z + m * y * m_pitch,
          I(2, 1) - m * y * z - m * x * m_pitch, I(2, 2) + m * (x * x + y * y);

        return res;
      }
    };
  } // namespace impl

  template<typename M6Like, typename S2, int O2, int axis>
  struct MultiplicationOp<Eigen::MatrixBase<M6Like>, JointMotionSubspaceHelicalTpl<S2, O2, axis>>
  {
    typedef Eigen::Matrix<S2, 6, 1> ReturnType;
  };

  /* [ABA] operator* (Inertia Y,Constraint S) */
  namespace impl
  {
    template<typename M6Like, typename Scalar, int Options, int axis>
    struct LhsMultiplicationOp<
      Eigen::MatrixBase<M6Like>,
      JointMotionSubspaceHelicalTpl<Scalar, Options, axis>>
    {
      typedef JointMotionSubspaceHelicalTpl<Scalar, Options, axis> Constraint;
      typedef Eigen::Matrix<Scalar, 6, 1> ReturnType;
      static inline ReturnType
      run(const Eigen::MatrixBase<M6Like> & Y, const Constraint & constraint)
      {
        EIGEN_STATIC_ASSERT_MATRIX_SPECIFIC_SIZE(M6Like, 6, 6);
        return (Y.col(Inertia::ANGULAR + axis) + Y.col(Inertia::LINEAR + axis) * constraint.h());
      }
    };
  } // namespace impl

  template<typename _Scalar, int _Options, int axis>
  struct traits<JointHelicalTpl<_Scalar, _Options, axis>>
  {
    enum
    {
      NQ = 1,
      NV = 1
    };
    typedef _Scalar Scalar;
    enum
    {
      Options = _Options
    };
    typedef JointDataHelicalTpl<Scalar, Options, axis> JointDataDerived;
    typedef JointModelHelicalTpl<Scalar, Options, axis> JointModelDerived;
    typedef JointMotionSubspaceHelicalTpl<Scalar, Options, axis> Constraint_t;
    typedef TransformHelicalTpl<Scalar, Options, axis> Transformation_t;
    typedef MotionHelicalTpl<Scalar, Options, axis> Motion_t;
    typedef MotionZeroTpl<Scalar, Options> Bias_t;

    // [ABA]
    typedef Eigen::Matrix<Scalar, 6, NV, Options> U_t;
    typedef Eigen::Matrix<Scalar, NV, NV, Options> D_t;
    typedef Eigen::Matrix<Scalar, 6, NV, Options> UD_t;

    typedef Eigen::Matrix<Scalar, NQ, 1, Options> ConfigVector_t;
    typedef Eigen::Matrix<Scalar, NV, 1, Options> TangentVector_t;

    PINOCCHIO_JOINT_DATA_BASE_ACCESSOR_DEFAULT_RETURN_TYPE
  };

  template<typename _Scalar, int _Options, int axis>
  struct traits<JointDataHelicalTpl<_Scalar, _Options, axis>>
  {
    typedef JointHelicalTpl<_Scalar, _Options, axis> JointDerived;
    typedef _Scalar Scalar;
  };

  template<typename _Scalar, int _Options, int axis>
  struct traits<JointModelHelicalTpl<_Scalar, _Options, axis>>
  {
    typedef JointHelicalTpl<_Scalar, _Options, axis> JointDerived;
    typedef _Scalar Scalar;
  };

  template<typename _Scalar, int _Options, int axis>
  struct JointDataHelicalTpl : public JointDataBase<JointDataHelicalTpl<_Scalar, _Options, axis>>
  {
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW
    typedef JointHelicalTpl<_Scalar, _Options, axis> JointDerived;
    PINOCCHIO_JOINT_DATA_TYPEDEF_TEMPLATE(JointDerived);
    PINOCCHIO_JOINT_DATA_BASE_DEFAULT_ACCESSOR

    ConfigVector_t joint_q;
    TangentVector_t joint_v;

    Constraint_t S;
    Transformation_t M;
    Motion_t v;
    Bias_t c;

    // [ABA] specific data
    U_t U;
    D_t Dinv;
    UD_t UDinv;
    D_t StU;

    JointDataHelicalTpl()
    : joint_q(ConfigVector_t::Zero())
    , joint_v(TangentVector_t::Zero())
    , S((Scalar)0)
    , M((Scalar)0, (Scalar)1, (Scalar)0)
    , v((Scalar)0, (Scalar)0)
    , U(U_t::Zero())
    , Dinv(D_t::Zero())
    , UDinv(UD_t::Zero())
    , StU(D_t::Zero())
    {
    }

    static std::string classname()
    {
      return std::string("JointDataH") + axisLabel<axis>();
    }
    std::string shortname() const
    {
      return classname();
    }

  }; // struct JointDataHelicalTpl

  template<typename NewScalar, typename Scalar, int Options, int axis>
  struct CastType<NewScalar, JointModelHelicalTpl<Scalar, Options, axis>>
  {
    typedef JointModelHelicalTpl<NewScalar, Options, axis> type;
  };

  template<typename _Scalar, int _Options, int axis>
  struct JointModelHelicalTpl : public JointModelBase<JointModelHelicalTpl<_Scalar, _Options, axis>>
  {
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW
    typedef JointHelicalTpl<_Scalar, _Options, axis> JointDerived;
    PINOCCHIO_JOINT_TYPEDEF_TEMPLATE(JointDerived);

    typedef JointModelBase<JointModelHelicalTpl> Base;
    using Base::id;
    using Base::idx_q;
    using Base::idx_v;
    using Base::setIndexes;

    typedef Eigen::Matrix<Scalar, 3, 1, _Options> Vector3;

    JointDataDerived createData() const
    {
      return JointDataDerived();
    }

    JointModelHelicalTpl()
    {
    }

    explicit JointModelHelicalTpl(const Scalar & h)
    : m_pitch(h)
    {
    }

    const std::vector<bool> hasConfigurationLimit() const
    {
      return {true, true};
    }

    const std::vector<bool> hasConfigurationLimitInTangent() const
    {
      return {true, true};
    }

    template<typename ConfigVector>
    EIGEN_DONT_INLINE void
    calc(JointDataDerived & data, const typename Eigen::MatrixBase<ConfigVector> & qs) const
    {
      data.joint_q[0] = qs[idx_q()];
      Scalar ca, sa;
      SINCOS(data.joint_q[0], &sa, &ca);
      data.M.setValues(sa, ca, data.joint_q[0] * m_pitch);
      data.S.h() = m_pitch;
    }

    template<typename TangentVector>
    EIGEN_DONT_INLINE void
    calc(JointDataDerived & data, const Blank, const typename Eigen::MatrixBase<TangentVector> & vs)
      const
    {
      data.joint_v[0] = vs[idx_v()];
      data.v.angularRate() = data.joint_v[0];
      data.v.linearRate() = data.joint_v[0] * m_pitch;
    }

    template<typename ConfigVector, typename TangentVector>
    EIGEN_DONT_INLINE void calc(
      JointDataDerived & data,
      const typename Eigen::MatrixBase<ConfigVector> & qs,
      const typename Eigen::MatrixBase<TangentVector> & vs) const
    {
      calc(data, qs.derived());

      data.joint_v[0] = vs[idx_v()];
      data.v.angularRate() = data.joint_v[0];
      data.v.linearRate() = data.joint_v[0] * m_pitch;
    }

    template<typename VectorLike, typename Matrix6Like>
    void calc_aba(
      JointDataDerived & data,
      const Eigen::MatrixBase<VectorLike> & armature,
      const Eigen::MatrixBase<Matrix6Like> & I,
      const bool update_I) const
    {
      data.U = I.col(Inertia::ANGULAR + axis) + m_pitch * I.col(Inertia::LINEAR + axis);
      data.StU[0] =
        data.U(Inertia::ANGULAR + axis) + m_pitch * data.U(Inertia::LINEAR + axis) + armature[0];
      data.Dinv[0] = Scalar(1) / data.StU[0];
      data.UDinv.noalias() = data.U * data.Dinv;

      if (update_I)
        PINOCCHIO_EIGEN_CONST_CAST(Matrix6Like, I).noalias() -= data.UDinv * data.U.transpose();
    }

    static std::string classname()
    {
      return std::string("JointModelH") + axisLabel<axis>();
    }
    std::string shortname() const
    {
      return classname();
    }

    Vector3 getMotionAxis() const
    {
      switch (axis)
      {
      case 0:
        return Vector3::UnitX();
      case 1:
        return Vector3::UnitY();
      case 2:
        return Vector3::UnitZ();
      default:
        assert(false && "must never happen");
        break;
      }
    }

    /// \returns An expression of *this with the Scalar type casted to NewScalar.
    template<typename NewScalar>
    JointModelHelicalTpl<NewScalar, Options, axis> cast() const
    {
      typedef JointModelHelicalTpl<NewScalar, Options, axis> ReturnType;
      ReturnType res(ScalarCast<NewScalar, Scalar>::cast(m_pitch));
      res.setIndexes(id(), idx_q(), idx_v());
      return res;
    }

    Scalar m_pitch;

  }; // struct JointModelHelicalTpl

  typedef JointHelicalTpl<context::Scalar, context::Options, 0> JointHX;
  typedef JointDataHelicalTpl<context::Scalar, context::Options, 0> JointDataHX;
  typedef JointModelHelicalTpl<context::Scalar, context::Options, 0> JointModelHX;

  typedef JointHelicalTpl<context::Scalar, context::Options, 1> JointHY;
  typedef JointDataHelicalTpl<context::Scalar, context::Options, 1> JointDataHY;
  typedef JointModelHelicalTpl<context::Scalar, context::Options, 1> JointModelHY;

  typedef JointHelicalTpl<context::Scalar, context::Options, 2> JointHZ;
  typedef JointDataHelicalTpl<context::Scalar, context::Options, 2> JointDataHZ;
  typedef JointModelHelicalTpl<context::Scalar, context::Options, 2> JointModelHZ;

} // namespace pinocchio

#include <boost/type_traits.hpp>

namespace boost
{
  template<typename Scalar, int Options, int axis>
  struct has_nothrow_constructor<::pinocchio::JointModelHelicalTpl<Scalar, Options, axis>>
  : public integral_constant<bool, true>
  {
  };

  template<typename Scalar, int Options, int axis>
  struct has_nothrow_copy<::pinocchio::JointModelHelicalTpl<Scalar, Options, axis>>
  : public integral_constant<bool, true>
  {
  };

  template<typename Scalar, int Options, int axis>
  struct has_nothrow_constructor<::pinocchio::JointDataHelicalTpl<Scalar, Options, axis>>
  : public integral_constant<bool, true>
  {
  };

  template<typename Scalar, int Options, int axis>
  struct has_nothrow_copy<::pinocchio::JointDataHelicalTpl<Scalar, Options, axis>>
  : public integral_constant<bool, true>
  {
  };
} // namespace boost

#endif // ifndef __pinocchio_multibody_joint_helical_hpp__