/
se3.hpp
1082 lines (973 loc) · 34.6 KB
/
se3.hpp
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#ifndef SOPHUS_SE3_HPP
#define SOPHUS_SE3_HPP
#include "so3.hpp"
namespace Sophus {
template <class Scalar_, int Options = 0>
class SE3;
using SE3d = SE3<double>;
using SE3f = SE3<float>;
} // namespace Sophus
namespace Eigen {
namespace internal {
template <class Scalar_, int Options>
struct traits<Sophus::SE3<Scalar_, Options>> {
using Scalar = Scalar_;
using TranslationType = Sophus::Vector3<Scalar, Options>;
using SO3Type = Sophus::SO3<Scalar, Options>;
};
template <class Scalar_, int Options>
struct traits<Map<Sophus::SE3<Scalar_>, Options>>
: traits<Sophus::SE3<Scalar_, Options>> {
using Scalar = Scalar_;
using TranslationType = Map<Sophus::Vector3<Scalar>, Options>;
using SO3Type = Map<Sophus::SO3<Scalar>, Options>;
};
template <class Scalar_, int Options>
struct traits<Map<Sophus::SE3<Scalar_> const, Options>>
: traits<Sophus::SE3<Scalar_, Options> const> {
using Scalar = Scalar_;
using TranslationType = Map<Sophus::Vector3<Scalar> const, Options>;
using SO3Type = Map<Sophus::SO3<Scalar> const, Options>;
};
} // namespace internal
} // namespace Eigen
namespace Sophus {
// SE3 base type - implements SE3 class but is storage agnostic.
//
// SE(3) is the group of rotations and translation in 3d. It is the semi-direct
// product of SO(3) and the 3d Euclidean vector space. The class is represented
// using a composition of SO3 for rotation and a one 3-vector for
// translation.
//
// SE(3) is neither compact, nor a commutative group.
//
// See SO3 for more details of the rotation representation in 3d.
//
template <class Derived>
class SE3Base {
public:
using Scalar = typename Eigen::internal::traits<Derived>::Scalar;
using TranslationType =
typename Eigen::internal::traits<Derived>::TranslationType;
using SO3Type = typename Eigen::internal::traits<Derived>::SO3Type;
using QuaternionType = typename SO3Type::QuaternionType;
// Degrees of freedom of manifold, number of dimensions in tangent space
// (three for translation, three for rotation).
static int constexpr DoF = 6;
// Number of internal parameters used (4-tuple for quaternion, three for
// translation).
static int constexpr num_parameters = 7;
// Group transformations are 4x4 matrices.
static int constexpr N = 4;
using Transformation = Matrix<Scalar, N, N>;
using Point = Vector3<Scalar>;
using HomogeneousPoint = Vector4<Scalar>;
using Line = ParametrizedLine3<Scalar>;
using Tangent = Vector<Scalar, DoF>;
using Adjoint = Matrix<Scalar, DoF, DoF>;
// For binary operations the return type is determined with the
// ScalarBinaryOpTraits feature of Eigen. This allows mixing concrete and Map
// types, as well as other compatible scalar types such as Ceres::Jet and
// double scalars with SE3 operations.
template <typename OtherDerived>
using ReturnScalar = typename Eigen::ScalarBinaryOpTraits<
Scalar, typename OtherDerived::Scalar>::ReturnType;
template <typename OtherDerived>
using SE3Product = SE3<ReturnScalar<OtherDerived>>;
template <typename PointDerived>
using PointProduct = Vector3<ReturnScalar<PointDerived>>;
template <typename HPointDerived>
using HomogeneousPointProduct = Vector4<ReturnScalar<HPointDerived>>;
// Adjoint transformation
//
// This function return the adjoint transformation ``Ad`` of the group
// element ``A`` such that for all ``x`` it holds that
// ``hat(Ad_A * x) = A * hat(x) A^{-1}``. See hat-operator below.
//
SOPHUS_FUNC Adjoint Adj() const {
Sophus::Matrix3<Scalar> const R = so3().matrix();
Adjoint res;
res.block(0, 0, 3, 3) = R;
res.block(3, 3, 3, 3) = R;
res.block(0, 3, 3, 3) = SO3<Scalar>::hat(translation()) * R;
res.block(3, 0, 3, 3) = Matrix3<Scalar>::Zero(3, 3);
return res;
}
// Extract rotation angle about canonical X-axis
//
Scalar angleX() const { return so3().angleX(); }
// Extract rotation angle about canonical Y-axis
//
Scalar angleY() const { return so3().angleY(); }
// Extract rotation angle about canonical Z-axis
//
Scalar angleZ() const { return so3().angleZ(); }
// Returns copy of instance casted to NewScalarType.
//
template <class NewScalarType>
SOPHUS_FUNC SE3<NewScalarType> cast() const {
return SE3<NewScalarType>(so3().template cast<NewScalarType>(),
translation().template cast<NewScalarType>());
}
// Returns derivative of this * exp(x) wrt x at x=0.
//
SOPHUS_FUNC Matrix<Scalar, num_parameters, DoF> Dx_this_mul_exp_x_at_0()
const {
Matrix<Scalar, num_parameters, DoF> J;
Eigen::Quaternion<Scalar> const q = unit_quaternion();
Scalar const c0 = Scalar(0.5) * q.w();
Scalar const c1 = Scalar(0.5) * q.z();
Scalar const c2 = -c1;
Scalar const c3 = Scalar(0.5) * q.y();
Scalar const c4 = Scalar(0.5) * q.x();
Scalar const c5 = -c4;
Scalar const c6 = -c3;
Scalar const c7 = q.w() * q.w();
Scalar const c8 = q.x() * q.x();
Scalar const c9 = q.y() * q.y();
Scalar const c10 = -c9;
Scalar const c11 = q.z() * q.z();
Scalar const c12 = -c11;
Scalar const c13 = Scalar(2) * q.w();
Scalar const c14 = c13 * q.z();
Scalar const c15 = Scalar(2) * q.x();
Scalar const c16 = c15 * q.y();
Scalar const c17 = c13 * q.y();
Scalar const c18 = c15 * q.z();
Scalar const c19 = c7 - c8;
Scalar const c20 = c13 * q.x();
Scalar const c21 = Scalar(2) * q.y() * q.z();
J(0, 0) = 0;
J(0, 1) = 0;
J(0, 2) = 0;
J(0, 3) = c0;
J(0, 4) = c2;
J(0, 5) = c3;
J(1, 0) = 0;
J(1, 1) = 0;
J(1, 2) = 0;
J(1, 3) = c1;
J(1, 4) = c0;
J(1, 5) = c5;
J(2, 0) = 0;
J(2, 1) = 0;
J(2, 2) = 0;
J(2, 3) = c6;
J(2, 4) = c4;
J(2, 5) = c0;
J(3, 0) = 0;
J(3, 1) = 0;
J(3, 2) = 0;
J(3, 3) = c5;
J(3, 4) = c6;
J(3, 5) = c2;
J(4, 0) = c10 + c12 + c7 + c8;
J(4, 1) = -c14 + c16;
J(4, 2) = c17 + c18;
J(4, 3) = 0;
J(4, 4) = 0;
J(4, 5) = 0;
J(5, 0) = c14 + c16;
J(5, 1) = c12 + c19 + c9;
J(5, 2) = -c20 + c21;
J(5, 3) = 0;
J(5, 4) = 0;
J(5, 5) = 0;
J(6, 0) = -c17 + c18;
J(6, 1) = c20 + c21;
J(6, 2) = c10 + c11 + c19;
J(6, 3) = 0;
J(6, 4) = 0;
J(6, 5) = 0;
return J;
}
// Returns group inverse.
//
SOPHUS_FUNC SE3<Scalar> inverse() const {
SO3<Scalar> invR = so3().inverse();
return SE3<Scalar>(invR, invR * (translation() * Scalar(-1)));
}
// Logarithmic map
//
// Computes the logarithm, the inverse of the group exponential which maps
// element of the group (rigid body transformations) to elements of the
// tangent space (twist).
//
// To be specific, this function computes ``vee(logmat(.))`` with
// ``logmat(.)`` being the matrix logarithm and ``vee(.)`` the vee-operator
// of SE(3).
//
SOPHUS_FUNC Tangent log() const {
// For the derivation of the logarithm of SE(3), see
// J. Gallier, D. Xu, "Computing exponentials of skew symmetric matrices and
// logarithms of orthogonal matrices", IJRA 2002.
// https://pdfs.semanticscholar.org/cfe3/e4b39de63c8cabd89bf3feff7f5449fc981d.pdf
// (Sec. 6., pp. 8)
using std::abs;
using std::cos;
using std::sin;
Tangent upsilon_omega;
auto omega_and_theta = so3().logAndTheta();
Scalar theta = omega_and_theta.theta;
upsilon_omega.template tail<3>() = omega_and_theta.tangent;
Matrix3<Scalar> const Omega =
SO3<Scalar>::hat(upsilon_omega.template tail<3>());
if (abs(theta) < Constants<Scalar>::epsilon()) {
Matrix3<Scalar> const V_inv = Matrix3<Scalar>::Identity() -
Scalar(0.5) * Omega +
Scalar(1. / 12.) * (Omega * Omega);
upsilon_omega.template head<3>() = V_inv * translation();
} else {
Scalar const half_theta = Scalar(0.5) * theta;
Matrix3<Scalar> const V_inv =
(Matrix3<Scalar>::Identity() - Scalar(0.5) * Omega +
(Scalar(1) -
theta * cos(half_theta) / (Scalar(2) * sin(half_theta))) /
(theta * theta) * (Omega * Omega));
upsilon_omega.template head<3>() = V_inv * translation();
}
return upsilon_omega;
}
// It re-normalizes the SO3 element.
//
// Note: Because of the class invariant of SO3, there is typically no need to
// call this function directly.
//
SOPHUS_FUNC void normalize() { so3().normalize(); }
// Returns 4x4 matrix representation of the instance.
//
// It has the following form:
//
// | R t |
// | o 1 |
//
// where ``R`` is a 3x3 rotation matrix, ``t`` a translation 3-vector and
// ``o`` a 3-column vector of zeros.
//
SOPHUS_FUNC Transformation matrix() const {
Transformation homogenious_matrix;
homogenious_matrix.template topLeftCorner<3, 4>() = matrix3x4();
homogenious_matrix.row(3) =
Matrix<Scalar, 1, 4>(Scalar(0), Scalar(0), Scalar(0), Scalar(1));
return homogenious_matrix;
}
// Returns the significant first three rows of the matrix above.
//
SOPHUS_FUNC Matrix<Scalar, 3, 4> matrix3x4() const {
Matrix<Scalar, 3, 4> matrix;
matrix.template topLeftCorner<3, 3>() = rotationMatrix();
matrix.col(3) = translation();
return matrix;
}
// Assignment operator.
//
SOPHUS_FUNC SE3Base& operator=(SE3Base const& other) = default;
// Assignment-like operator from OtherDerived.
//
template <class OtherDerived>
SOPHUS_FUNC SE3Base<Derived>& operator=(SE3Base<OtherDerived> const& other) {
so3() = other.so3();
translation() = other.translation();
return *this;
}
// Group multiplication, which is rotation concatenation.
//
template <typename OtherDerived>
SOPHUS_FUNC SE3Product<OtherDerived> operator*(
SE3Base<OtherDerived> const& other) const {
return SE3Product<OtherDerived>(
so3() * other.so3(), translation() + so3() * other.translation());
}
// Group action on 3-points.
//
// This function rotates and translates a three dimensional point ``p`` by the
// SE(3) element ``bar_T_foo = (bar_R_foo, t_bar)`` (= rigid body
// transformation):
//
// ``p_bar = bar_R_foo * p_foo + t_bar``.
//
template <typename PointDerived,
typename = typename std::enable_if<
IsFixedSizeVector<PointDerived, 3>::value>::type>
SOPHUS_FUNC PointProduct<PointDerived> operator*(
Eigen::MatrixBase<PointDerived> const& p) const {
return so3() * p + translation();
}
// Group action on homogeneous 3-points. See above for more details.
//
template <typename HPointDerived,
typename = typename std::enable_if<
IsFixedSizeVector<HPointDerived, 4>::value>::type>
SOPHUS_FUNC HomogeneousPointProduct<HPointDerived> operator*(
Eigen::MatrixBase<HPointDerived> const& p) const {
const PointProduct<HPointDerived> tp =
so3() * p.template head<3>() + p(3) * translation();
return HomogeneousPointProduct<HPointDerived>(tp(0), tp(1), tp(2), p(3));
}
// Group action on lines.
//
// This function rotates and translates a parametrized line
// ``l(t) = o + t * d`` by the SE(3) element:
//
// Origin is transformed using SE(3) action
// Direction is transformed using rotation part
//
SOPHUS_FUNC Line operator*(Line const& l) const {
return Line((*this) * l.origin(), so3() * l.direction());
}
// In-place group multiplication. This method is only valid if the return type
// of the multiplication is compatible with this SE3's Scalar type.
//
template <typename OtherDerived,
typename = typename std::enable_if<
std::is_same<Scalar, ReturnScalar<OtherDerived>>::value>::type>
SOPHUS_FUNC SE3Base<Derived>& operator*=(SE3Base<OtherDerived> const& other) {
*static_cast<Derived*>(this) = *this * other;
return *this;
}
// Returns rotation matrix.
//
SOPHUS_FUNC Matrix3<Scalar> rotationMatrix() const { return so3().matrix(); }
// Mutator of SO3 group.
//
SOPHUS_FUNC SO3Type& so3() { return static_cast<Derived*>(this)->so3(); }
// Accessor of SO3 group.
//
SOPHUS_FUNC SO3Type const& so3() const {
return static_cast<const Derived*>(this)->so3();
}
// Takes in quaternion, and normalizes it.
//
// Precondition: The quaternion must not be close to zero.
//
SOPHUS_FUNC void setQuaternion(Eigen::Quaternion<Scalar> const& quat) {
so3().setQuaternion(quat);
}
// Sets ``so3`` using ``rotation_matrix``.
//
// Precondition: ``R`` must be orthogonal and ``det(R)=1``.
//
SOPHUS_FUNC void setRotationMatrix(Matrix3<Scalar> const& R) {
SOPHUS_ENSURE(isOrthogonal(R), "R is not orthogonal:\n %", R);
SOPHUS_ENSURE(R.determinant() > 0, "det(R) is not positive: %",
R.determinant());
so3().setQuaternion(Eigen::Quaternion<Scalar>(R));
}
// Returns internal parameters of SE(3).
//
// It returns (q.imag[0], q.imag[1], q.imag[2], q.real, t[0], t[1], t[2]),
// with q being the unit quaternion, t the translation 3-vector.
//
SOPHUS_FUNC Sophus::Vector<Scalar, num_parameters> params() const {
Sophus::Vector<Scalar, num_parameters> p;
p << so3().params(), translation();
return p;
}
// Mutator of translation vector.
//
SOPHUS_FUNC TranslationType& translation() {
return static_cast<Derived*>(this)->translation();
}
// Accessor of translation vector
//
SOPHUS_FUNC TranslationType const& translation() const {
return static_cast<Derived const*>(this)->translation();
}
// Accessor of unit quaternion.
//
SOPHUS_FUNC QuaternionType const& unit_quaternion() const {
return this->so3().unit_quaternion();
}
};
// SE3 default type - Constructors and default storage for SE3 Type.
template <class Scalar_, int Options>
class SE3 : public SE3Base<SE3<Scalar_, Options>> {
using Base = SE3Base<SE3<Scalar_, Options>>;
public:
static int constexpr DoF = Base::DoF;
static int constexpr num_parameters = Base::num_parameters;
using Scalar = Scalar_;
using Transformation = typename Base::Transformation;
using Point = typename Base::Point;
using HomogeneousPoint = typename Base::HomogeneousPoint;
using Tangent = typename Base::Tangent;
using Adjoint = typename Base::Adjoint;
using SO3Member = SO3<Scalar, Options>;
using TranslationMember = Vector3<Scalar, Options>;
EIGEN_MAKE_ALIGNED_OPERATOR_NEW
// Default constructor initialize rigid body motion to the identity.
//
SOPHUS_FUNC SE3();
// Copy constructor
//
SOPHUS_FUNC SE3(SE3 const& other) = default;
// Copy-like constructor from OtherDerived.
//
template <class OtherDerived>
SOPHUS_FUNC SE3(SE3Base<OtherDerived> const& other)
: so3_(other.so3()), translation_(other.translation()) {
static_assert(std::is_same<typename OtherDerived::Scalar, Scalar>::value,
"must be same Scalar type");
}
// Constructor from SO3 and translation vector
//
template <class OtherDerived, class D>
SOPHUS_FUNC SE3(SO3Base<OtherDerived> const& so3,
Eigen::MatrixBase<D> const& translation)
: so3_(so3), translation_(translation) {
static_assert(std::is_same<typename OtherDerived::Scalar, Scalar>::value,
"must be same Scalar type");
static_assert(std::is_same<typename D::Scalar, Scalar>::value,
"must be same Scalar type");
}
// Constructor from rotation matrix and translation vector
//
// Precondition: Rotation matrix needs to be orthogonal with determinant of 1.
//
SOPHUS_FUNC
SE3(Matrix3<Scalar> const& rotation_matrix, Point const& translation)
: so3_(rotation_matrix), translation_(translation) {}
// Constructor from quaternion and translation vector.
//
// Precondition: quaternion must not be close to zero.
//
SOPHUS_FUNC SE3(Eigen::Quaternion<Scalar> const& quaternion,
Point const& translation)
: so3_(quaternion), translation_(translation) {}
// Constructor from 4x4 matrix
//
// Precondition: Rotation matrix needs to be orthogonal with determinant of 1.
// The last row must be (0, 0, 0, 1).
//
SOPHUS_FUNC explicit SE3(Matrix4<Scalar> const& T)
: so3_(T.template topLeftCorner<3, 3>()),
translation_(T.template block<3, 1>(0, 3)) {
SOPHUS_ENSURE((T.row(3) - Matrix<Scalar, 1, 4>(Scalar(0), Scalar(0),
Scalar(0), Scalar(1)))
.squaredNorm() < Constants<Scalar>::epsilon(),
"Last row is not (0,0,0,1), but (%).", T.row(3));
}
// This provides unsafe read/write access to internal data. SO(3) is
// represented by an Eigen::Quaternion (four parameters). When using direct
// write access, the user needs to take care of that the quaternion stays
// normalized.
//
SOPHUS_FUNC Scalar* data() {
// so3_ and translation_ are laid out sequentially with no padding
return so3_.data();
}
// Const version of data() above.
//
SOPHUS_FUNC Scalar const* data() const {
// so3_ and translation_ are laid out sequentially with no padding
return so3_.data();
}
// Accessor of SO3
//
SOPHUS_FUNC SO3Member& so3() { return so3_; }
// Mutator of SO3
//
SOPHUS_FUNC SO3Member const& so3() const { return so3_; }
// Mutator of translation vector
//
SOPHUS_FUNC TranslationMember& translation() { return translation_; }
// Accessor of translation vector
//
SOPHUS_FUNC TranslationMember const& translation() const {
return translation_;
}
// Returns derivative of exp(x) wrt. x.
//
SOPHUS_FUNC static Sophus::Matrix<Scalar, num_parameters, DoF> Dx_exp_x(
Tangent const& upsilon_omega) {
using std::cos;
using std::pow;
using std::sin;
using std::sqrt;
Sophus::Matrix<Scalar, num_parameters, DoF> J;
Sophus::Vector<Scalar, 3> upsilon = upsilon_omega.template head<3>();
Sophus::Vector<Scalar, 3> omega = upsilon_omega.template tail<3>();
Scalar const c0 = omega[0] * omega[0];
Scalar const c1 = omega[1] * omega[1];
Scalar const c2 = omega[2] * omega[2];
Scalar const c3 = c0 + c1 + c2;
Scalar const o(0);
Scalar const h(0.5);
Scalar const i(1);
if (c3 < Constants<Scalar>::epsilon()) {
Scalar const ux = Scalar(0.5) * upsilon[0];
Scalar const uy = Scalar(0.5) * upsilon[1];
Scalar const uz = Scalar(0.5) * upsilon[2];
// clang-format off
J << o, o, o, h, o, o,
o, o, o, o, h, o,
o, o, o, o, o, h,
o, o, o, o, o, o,
i, o, o, o, uz, -uy,
o, i, o, -uz, o, ux,
o, o, i, uy, -ux, o;
// clang-format on
return J;
}
Scalar const c4 = sqrt(c3);
Scalar const c5 = Scalar(1.0) / c4;
Scalar const c6 = Scalar(0.5) * c4;
Scalar const c7 = sin(c6);
Scalar const c8 = c5 * c7;
Scalar const c9 = pow(c3, -3.0L / 2.0L);
Scalar const c10 = c7 * c9;
Scalar const c11 = Scalar(1.0) / c3;
Scalar const c12 = cos(c6);
Scalar const c13 = Scalar(0.5) * c11 * c12;
Scalar const c14 = c7 * c9 * omega[0];
Scalar const c15 = Scalar(0.5) * c11 * c12 * omega[0];
Scalar const c16 = -c14 * omega[1] + c15 * omega[1];
Scalar const c17 = -c14 * omega[2] + c15 * omega[2];
Scalar const c18 = omega[1] * omega[2];
Scalar const c19 = -c10 * c18 + c13 * c18;
Scalar const c20 = c5 * omega[0];
Scalar const c21 = Scalar(0.5) * c7;
Scalar const c22 = c5 * omega[1];
Scalar const c23 = c5 * omega[2];
Scalar const c24 = -c1;
Scalar const c25 = -c2;
Scalar const c26 = c24 + c25;
Scalar const c27 = sin(c4);
Scalar const c28 = -c27 + c4;
Scalar const c29 = c28 * c9;
Scalar const c30 = cos(c4);
Scalar const c31 = -c30 + Scalar(1);
Scalar const c32 = c11 * c31;
Scalar const c33 = c32 * omega[2];
Scalar const c34 = c29 * omega[0];
Scalar const c35 = c34 * omega[1];
Scalar const c36 = c32 * omega[1];
Scalar const c37 = c34 * omega[2];
Scalar const c38 = pow(c3, -5.0L / 2.0L);
Scalar const c39 = Scalar(3) * c28 * c38 * omega[0];
Scalar const c40 = c26 * c9;
Scalar const c41 = -c20 * c30 + c20;
Scalar const c42 = c27 * c9 * omega[0];
Scalar const c43 = c42 * omega[1];
Scalar const c44 = pow(c3, -2);
Scalar const c45 = Scalar(2) * c31 * c44 * omega[0];
Scalar const c46 = c45 * omega[1];
Scalar const c47 = c29 * omega[2];
Scalar const c48 = c43 - c46 + c47;
Scalar const c49 = Scalar(3) * c0 * c28 * c38;
Scalar const c50 = c9 * omega[0] * omega[2];
Scalar const c51 = c41 * c50 - c49 * omega[2];
Scalar const c52 = c9 * omega[0] * omega[1];
Scalar const c53 = c41 * c52 - c49 * omega[1];
Scalar const c54 = c42 * omega[2];
Scalar const c55 = c45 * omega[2];
Scalar const c56 = c29 * omega[1];
Scalar const c57 = -c54 + c55 + c56;
Scalar const c58 = Scalar(-2) * c56;
Scalar const c59 = Scalar(3) * c28 * c38 * omega[1];
Scalar const c60 = -c22 * c30 + c22;
Scalar const c61 = -c18 * c39;
Scalar const c62 = c32 + c61;
Scalar const c63 = c27 * c9;
Scalar const c64 = c1 * c63;
Scalar const c65 = Scalar(2) * c31 * c44;
Scalar const c66 = c1 * c65;
Scalar const c67 = c50 * c60;
Scalar const c68 = -c1 * c39 + c52 * c60;
Scalar const c69 = c18 * c63;
Scalar const c70 = c18 * c65;
Scalar const c71 = c34 - c69 + c70;
Scalar const c72 = Scalar(-2) * c47;
Scalar const c73 = Scalar(3) * c28 * c38 * omega[2];
Scalar const c74 = -c23 * c30 + c23;
Scalar const c75 = -c32 + c61;
Scalar const c76 = c2 * c63;
Scalar const c77 = c2 * c65;
Scalar const c78 = c52 * c74;
Scalar const c79 = c34 + c69 - c70;
Scalar const c80 = -c2 * c39 + c50 * c74;
Scalar const c81 = -c0;
Scalar const c82 = c25 + c81;
Scalar const c83 = c32 * omega[0];
Scalar const c84 = c18 * c29;
Scalar const c85 = Scalar(-2) * c34;
Scalar const c86 = c82 * c9;
Scalar const c87 = c0 * c63;
Scalar const c88 = c0 * c65;
Scalar const c89 = c9 * omega[1] * omega[2];
Scalar const c90 = c41 * c89;
Scalar const c91 = c54 - c55 + c56;
Scalar const c92 = -c1 * c73 + c60 * c89;
Scalar const c93 = -c43 + c46 + c47;
Scalar const c94 = -c2 * c59 + c74 * c89;
Scalar const c95 = c24 + c81;
Scalar const c96 = c9 * c95;
J(0, 0) = o;
J(0, 1) = o;
J(0, 2) = o;
J(0, 3) = -c0 * c10 + c0 * c13 + c8;
J(0, 4) = c16;
J(0, 5) = c17;
J(1, 0) = o;
J(1, 1) = o;
J(1, 2) = o;
J(1, 3) = c16;
J(1, 4) = -c1 * c10 + c1 * c13 + c8;
J(1, 5) = c19;
J(2, 0) = o;
J(2, 1) = o;
J(2, 2) = o;
J(2, 3) = c17;
J(2, 4) = c19;
J(2, 5) = -c10 * c2 + c13 * c2 + c8;
J(3, 0) = o;
J(3, 1) = o;
J(3, 2) = o;
J(3, 3) = -c20 * c21;
J(3, 4) = -c21 * c22;
J(3, 5) = -c21 * c23;
J(4, 0) = c26 * c29 + Scalar(1);
J(4, 1) = -c33 + c35;
J(4, 2) = c36 + c37;
J(4, 3) = upsilon[0] * (-c26 * c39 + c40 * c41) + upsilon[1] * (c53 + c57) +
upsilon[2] * (c48 + c51);
J(4, 4) = upsilon[0] * (-c26 * c59 + c40 * c60 + c58) +
upsilon[1] * (c68 + c71) + upsilon[2] * (c62 + c64 - c66 + c67);
J(4, 5) = upsilon[0] * (-c26 * c73 + c40 * c74 + c72) +
upsilon[1] * (c75 - c76 + c77 + c78) + upsilon[2] * (c79 + c80);
J(5, 0) = c33 + c35;
J(5, 1) = c29 * c82 + Scalar(1);
J(5, 2) = -c83 + c84;
J(5, 3) = upsilon[0] * (c53 + c91) +
upsilon[1] * (-c39 * c82 + c41 * c86 + c85) +
upsilon[2] * (c75 - c87 + c88 + c90);
J(5, 4) = upsilon[0] * (c68 + c79) + upsilon[1] * (-c59 * c82 + c60 * c86) +
upsilon[2] * (c92 + c93);
J(5, 5) = upsilon[0] * (c62 + c76 - c77 + c78) +
upsilon[1] * (c72 - c73 * c82 + c74 * c86) +
upsilon[2] * (c57 + c94);
J(6, 0) = -c36 + c37;
J(6, 1) = c83 + c84;
J(6, 2) = c29 * c95 + Scalar(1);
J(6, 3) = upsilon[0] * (c51 + c93) + upsilon[1] * (c62 + c87 - c88 + c90) +
upsilon[2] * (-c39 * c95 + c41 * c96 + c85);
J(6, 4) = upsilon[0] * (-c64 + c66 + c67 + c75) + upsilon[1] * (c48 + c92) +
upsilon[2] * (c58 - c59 * c95 + c60 * c96);
J(6, 5) = upsilon[0] * (c71 + c80) + upsilon[1] * (c91 + c94) +
upsilon[2] * (-c73 * c95 + c74 * c96);
return J;
}
// Returns derivative of exp(x) wrt. x_i at x=0.
//
SOPHUS_FUNC static Sophus::Matrix<Scalar, num_parameters, DoF>
Dx_exp_x_at_0() {
Sophus::Matrix<Scalar, num_parameters, DoF> J;
Scalar const o(0);
Scalar const h(0.5);
Scalar const i(1);
// clang-format off
J << o, o, o, h, o, o,
o, o, o, o, h, o,
o, o, o, o, o, h,
o, o, o, o, o, o,
i, o, o, o, o, o,
o, i, o, o, o, o,
o, o, i, o, o, o;
// clang-format on
return J;
}
// Returns derivative of exp(x).matrix() wrt. x_i at x=0.
//
SOPHUS_FUNC static Transformation Dxi_exp_x_matrix_at_0(int i) {
return generator(i);
}
// Group exponential
//
// This functions takes in an element of tangent space (= twist ``a``) and
// returns the corresponding element of the group SE(3).
//
// The first three components of ``a`` represent the translational part
// ``upsilon`` in the tangent space of SE(3), while the last three components
// of ``a`` represents the rotation vector ``omega``.
// To be more specific, this function computes ``expmat(hat(a))`` with
// ``expmat(.)`` being the matrix exponential and ``hat(.)`` the hat-operator
// of SE(3), see below.
//
SOPHUS_FUNC static SE3<Scalar> exp(Tangent const& a) {
using std::cos;
using std::sin;
Vector3<Scalar> const omega = a.template tail<3>();
Scalar theta;
SO3<Scalar> const so3 = SO3<Scalar>::expAndTheta(omega, &theta);
Matrix3<Scalar> const Omega = SO3<Scalar>::hat(omega);
Matrix3<Scalar> const Omega_sq = Omega * Omega;
Matrix3<Scalar> V;
if (theta < Constants<Scalar>::epsilon()) {
V = so3.matrix();
// Note: That is an accurate expansion!
} else {
Scalar theta_sq = theta * theta;
V = (Matrix3<Scalar>::Identity() +
(Scalar(1) - cos(theta)) / (theta_sq)*Omega +
(theta - sin(theta)) / (theta_sq * theta) * Omega_sq);
}
return SE3<Scalar>(so3, V * a.template head<3>());
}
// Returns closest SE3 given arbirary 4x4 matrix.
//
template <class S = Scalar>
SOPHUS_FUNC static enable_if_t<std::is_floating_point<S>::value, SE3>
fitToSE3(Matrix4<Scalar> const& T) {
return SE3(SO3<Scalar>::fitToSO3(T.template block<3, 3>(0, 0)),
T.template block<3, 1>(0, 3));
}
// Returns the ith infinitesimal generators of SE(3).
//
// The infinitesimal generators of SE(3) are:
//
// | 0 0 0 1 |
// G_0 = | 0 0 0 0 |
// | 0 0 0 0 |
// | 0 0 0 0 |
//
// | 0 0 0 0 |
// G_1 = | 0 0 0 1 |
// | 0 0 0 0 |
// | 0 0 0 0 |
//
// | 0 0 0 0 |
// G_2 = | 0 0 0 0 |
// | 0 0 0 1 |
// | 0 0 0 0 |
//
// | 0 0 0 0 |
// G_3 = | 0 0 -1 0 |
// | 0 1 0 0 |
// | 0 0 0 0 |
//
// | 0 0 1 0 |
// G_4 = | 0 0 0 0 |
// | -1 0 0 0 |
// | 0 0 0 0 |
//
// | 0 -1 0 0 |
// G_5 = | 1 0 0 0 |
// | 0 0 0 0 |
// | 0 0 0 0 |
//
// Precondition: ``i`` must be in [0, 5].
//
SOPHUS_FUNC static Transformation generator(int i) {
SOPHUS_ENSURE(i >= 0 && i <= 5, "i should be in range [0,5].");
Tangent e;
e.setZero();
e[i] = Scalar(1);
return hat(e);
}
// hat-operator
//
// It takes in the 6-vector representation (= twist) and returns the
// corresponding matrix representation of Lie algebra element.
//
// Formally, the ``hat()`` operator of SE(3) is defined as
//
// ``hat(.): R^6 -> R^{4x4}, hat(a) = sum_i a_i * G_i`` (for i=0,...,5)
//
// with ``G_i`` being the ith infinitesimal generator of SE(3).
//
SOPHUS_FUNC static Transformation hat(Tangent const& a) {
Transformation Omega;
Omega.setZero();
Omega.template topLeftCorner<3, 3>() =
SO3<Scalar>::hat(a.template tail<3>());
Omega.col(3).template head<3>() = a.template head<3>();
return Omega;
}
// Lie bracket
//
// It computes the Lie bracket of SE(3). To be more specific, it computes
//
// ``[omega_1, omega_2]_se3 := vee([hat(omega_1), hat(omega_2)])``
//
// with ``[A,B] := AB-BA`` being the matrix commutator, ``hat(.) the
// hat-operator and ``vee(.)`` the vee-operator of SE(3).
//
SOPHUS_FUNC static Tangent lieBracket(Tangent const& a, Tangent const& b) {
Vector3<Scalar> const upsilon1 = a.template head<3>();
Vector3<Scalar> const upsilon2 = b.template head<3>();
Vector3<Scalar> const omega1 = a.template tail<3>();
Vector3<Scalar> const omega2 = b.template tail<3>();
Tangent res;
res.template head<3>() = omega1.cross(upsilon2) + upsilon1.cross(omega2);
res.template tail<3>() = omega1.cross(omega2);
return res;
}
// Construct x-axis rotation.
//
static SOPHUS_FUNC SE3 rotX(Scalar const& x) {
return SE3(SO3<Scalar>::rotX(x), Sophus::Vector3<Scalar>::Zero());
}
// Construct y-axis rotation.
//
static SOPHUS_FUNC SE3 rotY(Scalar const& y) {
return SE3(SO3<Scalar>::rotY(y), Sophus::Vector3<Scalar>::Zero());
}
// Construct z-axis rotation.
//
static SOPHUS_FUNC SE3 rotZ(Scalar const& z) {
return SE3(SO3<Scalar>::rotZ(z), Sophus::Vector3<Scalar>::Zero());
}
// Draw uniform sample from SE(3) manifold.
//
// Translations are drawn component-wise from the range [-1, 1].
//
template <class UniformRandomBitGenerator>
static SE3 sampleUniform(UniformRandomBitGenerator& generator) {
std::uniform_real_distribution<Scalar> uniform(Scalar(-1), Scalar(1));
return SE3(SO3<Scalar>::sampleUniform(generator),
Vector3<Scalar>(uniform(generator), uniform(generator),
uniform(generator)));
}
// Construct a translation only SE3 instance.
//
template <class T0, class T1, class T2>
static SOPHUS_FUNC SE3 trans(T0 const& x, T1 const& y, T2 const& z) {
return SE3(SO3<Scalar>(), Vector3<Scalar>(x, y, z));
}
static SOPHUS_FUNC SE3 trans(Vector3<Scalar> const& xyz) {
return SE3(SO3<Scalar>(), xyz);
}
// Construct x-axis translation.
//
static SOPHUS_FUNC SE3 transX(Scalar const& x) {
return SE3::trans(x, Scalar(0), Scalar(0));
}
// Construct y-axis translation.
//
static SOPHUS_FUNC SE3 transY(Scalar const& y) {
return SE3::trans(Scalar(0), y, Scalar(0));
}
// Construct z-axis translation.
//
static SOPHUS_FUNC SE3 transZ(Scalar const& z) {
return SE3::trans(Scalar(0), Scalar(0), z);
}
// vee-operator
//
// It takes 4x4-matrix representation ``Omega`` and maps it to the
// corresponding 6-vector representation of Lie algebra.
//
// This is the inverse of the hat-operator, see above.
//
// Precondition: ``Omega`` must have the following structure:
//
// | 0 -f e a |
// | f 0 -d b |
// | -e d 0 c |
// | 0 0 0 0 | .
//
SOPHUS_FUNC static Tangent vee(Transformation const& Omega) {
Tangent upsilon_omega;
upsilon_omega.template head<3>() = Omega.col(3).template head<3>();
upsilon_omega.template tail<3>() =
SO3<Scalar>::vee(Omega.template topLeftCorner<3, 3>());
return upsilon_omega;
}
protected:
SO3Member so3_;
TranslationMember translation_;
};
template <class Scalar, int Options>
SE3<Scalar, Options>::SE3() : translation_(TranslationMember::Zero()) {
static_assert(std::is_standard_layout<SE3>::value,
"Assume standard layout for the use of offsetof check below.");
static_assert(
offsetof(SE3, so3_) + sizeof(Scalar) * SO3<Scalar>::num_parameters ==
offsetof(SE3, translation_),
"This class assumes packed storage and hence will only work "
"correctly depending on the compiler (options) - in "
"particular when using [this->data(), this-data() + "
"num_parameters] to access the raw data in a contiguous fashion.");
}
} // namespace Sophus
namespace Eigen {
// Specialization of Eigen::Map for ``SE3``.
//
// Allows us to wrap SE3 objects around POD array.
template <class Scalar_, int Options>
class Map<Sophus::SE3<Scalar_>, Options>
: public Sophus::SE3Base<Map<Sophus::SE3<Scalar_>, Options>> {
using Base = Sophus::SE3Base<Map<Sophus::SE3<Scalar_>, Options>>;
public:
using Scalar = Scalar_;
using Transformation = typename Base::Transformation;
using Point = typename Base::Point;
using HomogeneousPoint = typename Base::HomogeneousPoint;
using Tangent = typename Base::Tangent;
using Adjoint = typename Base::Adjoint;