math/quaternion.hpp

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//
// Copyright (c) 2016-2020 CNRS INRIA
//
 
#ifndef __pinocchio_math_quaternion_hpp__
#define __pinocchio_math_quaternion_hpp__
 
#ifndef PINOCCHIO_DEFAULT_QUATERNION_NORM_TOLERANCE_VALUE
  #define PINOCCHIO_DEFAULT_QUATERNION_NORM_TOLERANCE_VALUE 1e-8
#endif
 
#include "pinocchio/math/fwd.hpp"
#include "pinocchio/math/comparison-operators.hpp"
#include "pinocchio/math/matrix.hpp"
#include "pinocchio/math/sincos.hpp"
#include "pinocchio/utils/static-if.hpp"
 
#include <boost/type_traits.hpp>
#include <Eigen/Geometry>
 
namespace pinocchio
{
  namespace quaternion
  {
    template<typename D1, typename D2>
    typename D1::Scalar angleBetweenQuaternions(
      const Eigen::QuaternionBase<D1> & q1, const Eigen::QuaternionBase<D2> & q2)
    {
      typedef typename D1::Scalar Scalar;
      const Scalar innerprod = q1.dot(q2);
      Scalar theta = math::acos(innerprod);
      static const Scalar PI_value = PI<Scalar>();
 
      theta = internal::if_then_else(
        internal::LT, innerprod, Scalar(0), static_cast<Scalar>(PI_value - theta), theta);
      return theta;
    }
 
    template<typename D1, typename D2>
    bool defineSameRotation(
      const Eigen::QuaternionBase<D1> & q1,
      const Eigen::QuaternionBase<D2> & q2,
      const typename D1::RealScalar & prec =
        Eigen::NumTraits<typename D1::Scalar>::dummy_precision())
    {
      return (q1.coeffs().isApprox(q2.coeffs(), prec) || q1.coeffs().isApprox(-q2.coeffs(), prec));
    }
 
    template<typename D>
    void firstOrderNormalize(const Eigen::QuaternionBase<D> & q)
    {
      typedef typename D::Scalar Scalar;
      const Scalar N2 = q.squaredNorm();
#ifndef NDEBUG
      const Scalar epsilon = sqrt(sqrt(Eigen::NumTraits<Scalar>::epsilon()));
      typedef apply_op_if<less_than_or_equal_to_op, is_floating_point<Scalar>::value, true>
        static_leq;
      assert(static_leq::op(math::fabs(static_cast<Scalar>(N2 - Scalar(1))), epsilon));
#endif
      const Scalar alpha = ((Scalar)3 - N2) / Scalar(2);
      PINOCCHIO_EIGEN_CONST_CAST(D, q).coeffs() *= alpha;
#ifndef NDEBUG
      const Scalar M =
        Scalar(3) * math::pow(Scalar(1) - epsilon, ((Scalar)-Scalar(5)) / Scalar(2)) / Scalar(4);
      assert(static_leq::op(
        math::fabs(static_cast<Scalar>(q.norm() - Scalar(1))),
        math::max(
          M * sqrt(N2) * (N2 - Scalar(1)) * (N2 - Scalar(1)) / Scalar(2),
          Eigen::NumTraits<Scalar>::dummy_precision())));
#endif
    }
 
    template<typename Derived>
    void uniformRandom(Eigen::QuaternionBase<Derived> & q)
    {
      typedef typename Derived::Scalar Scalar;
 
      // Rotational part
      const Scalar u1 = (Scalar)rand() / RAND_MAX;
      const Scalar u2 = (Scalar)rand() / RAND_MAX;
      const Scalar u3 = (Scalar)rand() / RAND_MAX;
 
      const Scalar mult1 = sqrt(Scalar(1) - u1);
      const Scalar mult2 = sqrt(u1);
 
      static const Scalar PI_value = PI<Scalar>();
      Scalar s2, c2;
      SINCOS(Scalar(2) * PI_value * u2, &s2, &c2);
      Scalar s3, c3;
      SINCOS(Scalar(2) * PI_value * u3, &s3, &c3);
 
      q.w() = mult1 * s2;
      q.x() = mult1 * c2;
      q.y() = mult2 * s3;
      q.z() = mult2 * c3;
    }
 
    namespace internal
    {
 
      template<typename Scalar, bool value = is_floating_point<Scalar>::value>
      struct quaternionbase_assign_impl;
 
      template<Eigen::DenseIndex i>
      struct quaternionbase_assign_impl_if_t_negative
      {
        template<typename Scalar, typename Matrix3, typename QuaternionDerived>
        static inline void
        run(Scalar t, Eigen::QuaternionBase<QuaternionDerived> & q, const Matrix3 & mat)
        {
          using pinocchio::math::sqrt;
 
          Eigen::DenseIndex j = (i + 1) % 3;
          Eigen::DenseIndex k = (j + 1) % 3;
 
          t = sqrt(mat.coeff(i, i) - mat.coeff(j, j) - mat.coeff(k, k) + Scalar(1.0));
          q.coeffs().coeffRef(i) = Scalar(0.5) * t;
          t = Scalar(0.5) / t;
          q.w() = (mat.coeff(k, j) - mat.coeff(j, k)) * t;
          q.coeffs().coeffRef(j) = (mat.coeff(j, i) + mat.coeff(i, j)) * t;
          q.coeffs().coeffRef(k) = (mat.coeff(k, i) + mat.coeff(i, k)) * t;
        }
      };
 
      struct quaternionbase_assign_impl_if_t_positive
      {
        template<typename Scalar, typename Matrix3, typename QuaternionDerived>
        static inline void
        run(Scalar t, Eigen::QuaternionBase<QuaternionDerived> & q, const Matrix3 & mat)
        {
          using pinocchio::math::sqrt;
 
          t = sqrt(t + Scalar(1.0));
          q.w() = Scalar(0.5) * t;
          t = Scalar(0.5) / t;
          q.x() = (mat.coeff(2, 1) - mat.coeff(1, 2)) * t;
          q.y() = (mat.coeff(0, 2) - mat.coeff(2, 0)) * t;
          q.z() = (mat.coeff(1, 0) - mat.coeff(0, 1)) * t;
        }
      };
 
      template<typename Scalar>
      struct quaternionbase_assign_impl<Scalar, true>
      {
        template<typename Matrix3, typename QuaternionDerived>
        static inline void run(Eigen::QuaternionBase<QuaternionDerived> & q, const Matrix3 & mat)
        {
          using pinocchio::math::sqrt;
 
          Scalar t = mat.trace();
          if (t > Scalar(0.))
            quaternionbase_assign_impl_if_t_positive::run(t, q, mat);
          else
          {
            Eigen::DenseIndex i = 0;
            if (mat.coeff(1, 1) > mat.coeff(0, 0))
              i = 1;
            if (mat.coeff(2, 2) > mat.coeff(i, i))
              i = 2;
 
            if (i == 0)
              quaternionbase_assign_impl_if_t_negative<0>::run(t, q, mat);
            else if (i == 1)
              quaternionbase_assign_impl_if_t_negative<1>::run(t, q, mat);
            else
              quaternionbase_assign_impl_if_t_negative<2>::run(t, q, mat);
          }
        }
      };
 
    } // namespace internal
 
    template<typename D, typename Matrix3>
    void assignQuaternion(Eigen::QuaternionBase<D> & quat, const Eigen::MatrixBase<Matrix3> & R)
    {
      internal::quaternionbase_assign_impl<typename Matrix3::Scalar>::run(
        quat.derived(), R.derived());
    }
 
    template<typename Quaternion>
    inline bool isNormalized(
      const Eigen::QuaternionBase<Quaternion> & quat,
      const typename Quaternion::Coefficients::RealScalar & prec =
        Eigen::NumTraits<typename Quaternion::Coefficients::RealScalar>::dummy_precision())
    {
      return pinocchio::isNormalized(quat.coeffs(), prec);
    }
 
    template<typename Quaternion>
    inline void normalize(const Eigen::QuaternionBase<Quaternion> & quat)
    {
      return pinocchio::normalize(quat.const_cast_derived().coeffs());
    }
 
    template<
      typename Scalar,
      typename QuaternionIn1,
      typename QuaternionIn2,
      typename QuaternionOut>
    inline void slerp(
      const Scalar & u,
      const Eigen::QuaternionBase<QuaternionIn1> & quat0,
      const Eigen::QuaternionBase<QuaternionIn2> & quat1,
      const Eigen::QuaternionBase<QuaternionOut> & res)
    {
      const Scalar one = Scalar(1) - Eigen::NumTraits<Scalar>::epsilon();
      const Scalar d = quat0.dot(quat1);
      const Scalar absD = fabs(d);
 
      const Scalar theta = acos(absD);
      const Scalar sinTheta = sin(theta);
 
      using namespace pinocchio::internal;
 
      const Scalar scale0 = if_then_else(
        pinocchio::internal::GE, absD, one,
        static_cast<Scalar>(Scalar(1) - u),                          // then
        static_cast<Scalar>(sin((Scalar(1) - u) * theta) / sinTheta) // else
      );
 
      const Scalar scale1_factor =
        if_then_else(pinocchio::internal::LT, d, Scalar(0), Scalar(-1), Scalar(1));
      const Scalar scale1 = if_then_else(
                              pinocchio::internal::GE, absD, one,
                              u,                                               // then
                              static_cast<Scalar>(sin((u * theta)) / sinTheta) // else
                              )
                            * scale1_factor;
 
      PINOCCHIO_EIGEN_CONST_CAST(QuaternionOut, res.derived()).coeffs() =
        scale0 * quat0.coeffs() + scale1 * quat1.coeffs();
    }
 
  } // namespace quaternion
 
} // namespace pinocchio
#endif // #ifndef __pinocchio_math_quaternion_hpp__