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;
 }; // 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);
 }

 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,
 NVExtended = 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;

 typedef boost::mpl::true_ is_mimicable_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::idx_vExtended;
 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>
 PINOCCHIO_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>
 PINOCCHIO_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>
 PINOCCHIO_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;
 }
 }

 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(), idx_vExtended());
 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;

 template<typename Scalar, int Options, int axis>
 struct ConfigVectorAffineTransform<JointHelicalTpl<Scalar, Options, axis>>
 {
 typedef LinearAffineTransform Type;
 };

} // 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__