Base class for Galielei Lie group types. More...

#include <galilei.hpp>

Inheritance diagram for GalileiBase< _Derived >:

Public Member Functions
Map< SO3< Scalar > >	so3 ()
	Access SO(3) part.

Map< const SO3< Scalar > >	so3 () const
	Const access SO(3) part.

Eigen::Map< Eigen::Vector3< Scalar > >	r3_v ()
	Access R3 velocity part.

Eigen::Map< const Eigen::Vector3< Scalar > >	r3_v () const
	Const access R3 velocity part.

Eigen::Map< Eigen::Vector3< Scalar > >	r3_p ()
	Access R3 position part.

Eigen::Map< const Eigen::Vector3< Scalar > >	r3_p () const
	Const access R3 position part.

Eigen::Map< Eigen::Vector< Scalar, 1 > >	r1_t ()
	Access R1 time part.

Eigen::Map< const Eigen::Vector< Scalar, 1 > >	r1_t () const
	Const access R1 time part.

template<typename EigenDerived >
Eigen::Vector4< Scalar >	operator* (const Eigen::MatrixBase< EigenDerived > &v) const
	Tranformation action on (x, y, z, t) space-time vector.

template<typename EigenDerived >
Eigen::Matrix< Scalar, 4, 10 >	dr_action (const Eigen::MatrixBase< EigenDerived > &v) const
	Jacobian of rotation action w.r.t. group.

Public Member Functions inherited from LieGroupBase< _Derived >
auto &	coeffs () const
	Access underlying storages.

const auto &	coeffs () const
	Const access underlying storages.

auto *	data () const
	Access raw pointer.

const auto *	data () const
	Const access raw pointer.

_Derived &	operator= (const LieGroupBase< OtherDerived > &o) noexcept

Eigen::Index	dof () const noexcept
	Dynamic size (degrees of freedom).

void	setIdentity () noexcept
	Set to group identity element.

void	setRandom () noexcept
	Set to a random element.

Matrix	matrix () const noexcept
	Return as matrix Lie group element in \( \mathbb{R}^{\mathtt{dim} \times \mathtt{dim}} \).

bool	isApprox (const LieGroupBase< OtherDerived > &o, const Scalar &eps=Eigen::NumTraits< Scalar >::dummy_precision()) const noexcept
	Check if (approximately) equal to other element `o`.

CastT< NewScalar >	cast () const noexcept
	Cast to different scalar type.

PlainObject	operator* (const LieGroupBase< OtherDerived > &o) const noexcept
	Group binary composition operation.

_Derived &	operator*= (const LieGroupBase< OtherDerived > &o) noexcept
	Inplace group binary composition operation.

PlainObject	inverse () const noexcept
	Group inverse operation.

Tangent	log () const noexcept
	Lie group logarithm.

TangentMap	Ad () const noexcept
	Lie group adjoint.

PlainObject	operator+ (const Eigen::MatrixBase< TangentDerived > &a) const noexcept
	Right-plus.

_Derived &	operator+= (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Inplace right-plus: \( \mathbf{x} \leftarrow \mathbf{x} \circ \exp(\mathbf{a}) \).

Tangent	operator- (const LieGroupBase< OtherDerived > &xo) const noexcept
	Right-minus.

Public Attributes
	SMOOTH_INHERIT_TYPEDEFS

Additional Inherited Members
Public Types inherited from LieGroupBase< _Derived >
using	Scalar = typename traits::Scalar
	Scalar type.

using	Matrix = Eigen::Matrix< Scalar, Dim, Dim >
	Lie group matrix type.

using	Tangent = Eigen::Matrix< Scalar, Dof, 1 >
	Lie group parameterized tangent type.

using	TangentMap = Eigen::Matrix< Scalar, Dof, Dof >
	Matrix representing map between tangent elements.

using	Hessian = Eigen::Matrix< Scalar, Dof, Dof *Dof >
	Plain return type with different scalar.

using	CastT = typename traits::template PlainObject< NewScalar >
	Plain return type with different scalar.

using	PlainObject = CastT< Scalar >
	Plain return type.

Static Public Member Functions inherited from LieGroupBase< _Derived >
static PlainObject	Identity () noexcept
	Construct the identity element.

static PlainObject	Random () noexcept
	Construct a random element.

static PlainObject	exp (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Lie group exponential map.

static Matrix	hat (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Lie algebra hat map.

static Tangent	vee (const Eigen::MatrixBase< MatrixDerived > &A) noexcept
	Lie alebra vee map.

static TangentMap	ad (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Lie algebra adjoint.

static Tangent	lie_bracket (const Eigen::MatrixBase< TangentDerived1 > &a, const Eigen::MatrixBase< TangentDerived2 > &b) noexcept
	Lie algebra bracket.

static TangentMap	dr_exp (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Right jacobian of the exponential map.

static TangentMap	dr_expinv (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Inverse of right jacobian of the exponential map.

static TangentMap	dl_exp (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Left jacobian of the exponential map.

static TangentMap	dl_expinv (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Inverse of left jacobian of the exponential map.

static Hessian	d2r_exp (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Right Hessian of the exponential map.

static Hessian	d2r_expinv (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Right Hessian of the log map.

static Hessian	d2l_exp (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Left Hessian of the exponential map.

static Hessian	d2l_expinv (const Eigen::MatrixBase< TangentDerived > &a) noexcept
	Left Hessian of the log map.

Static Public Attributes inherited from LieGroupBase< _Derived >
static constexpr int	RepSize
	Number of scalars in internal representation.

static constexpr int	Dof
	Degrees of freedom of manifold (equal to tangent space dimension).

static constexpr int	Dim
	Side of Lie group matrix representation.

static constexpr bool	IsCommutative
	Commutativity of group. A commutative group has a zero Lie bracket.

Protected Types inherited from LieGroupBase< _Derived >
using	traits = liebase_info< _Derived >
	CRTP traits.

using	Impl = typename traits::Impl
	Group-specific Lie group implementation.

Static Protected Attributes inherited from LieGroupBase< _Derived >
static constexpr bool	is_mutable
	True if underlying storage supports modification.

Detailed Description

template<typename _Derived>
class GalileiBase< _Derived >

Base class for Galielei Lie group types.

Internally represented as \(\mathbb{S}^3 \times \mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R} \).

Memory layout

Group: \( \mathbf{x} = [vx, vy, vz, px, py, pz, t q_x, q_y, q_z, q_w] \)
Tangent: \( \mathbf{a} = [bx, by, bz, tx, ty, tz, s, \omega_x, \omega_y, \omega_z] \)

Constraints

Group: \(q_x^2 + q_y^2 + q_z^2 + q_w^2 = 1 \)
Tangent: \( -\pi < \omega_x, \omega_y, \omega_z \leq \pi \)

Lie group matrix form

\[ \mathbf{X} = \begin{bmatrix} R & v & p \\ 0 & 1 & t \\ 0 & 0 & 1 \end{bmatrix} \in \mathbb{R}^{5 \times 5} \]

where \(R\) is a 3x3 rotation matrix.

Lie algebra matrix form

\[ \mathbf{a}^\wedge = \begin{bmatrix} 0 & -\omega_z & \omega_y & b_x & t_x\\ \omega_z & 0 & -\omega_x & b_y & t_y\\ -\omega_y & \omega_x & 0 & b_y & t_y\\ 0 & 0 & 0 & 0 & s\\ 0 & 0 & 0 & 0 & 1\\ \end{bmatrix} \in \mathbb{R}^{5 \times 5} \]

Definition at line 63 of file galilei.hpp.

Member Function Documentation

◆ dr_action()

template<typename _Derived >

template<typename EigenDerived >

Eigen::Matrix< Scalar, 4, 10 > GalileiBase< _Derived >::dr_action ( const Eigen::MatrixBase< EigenDerived > & v ) const

inline

Jacobian of rotation action w.r.t. group.

\[ \mathrm{d}^r (X v)_X \]

Definition at line 160 of file galilei.hpp.

◆ operator*()

template<typename _Derived >

template<typename EigenDerived >

Eigen::Vector4< Scalar > GalileiBase< _Derived >::operator* ( const Eigen::MatrixBase< EigenDerived > & v ) const

inline

Tranformation action on (x, y, z, t) space-time vector.

(x, t) -> (R x + v t + p, t + tau)

Definition at line 144 of file galilei.hpp.

◆ r1_t() [1/2]

template<typename _Derived >

Eigen::Map< Eigen::Vector< Scalar, 1 > > GalileiBase< _Derived >::r1_t ( )

inline

Access R1 time part.

Definition at line 124 of file galilei.hpp.

◆ r1_t() [2/2]

template<typename _Derived >

Eigen::Map< const Eigen::Vector< Scalar, 1 > > GalileiBase< _Derived >::r1_t ( ) const

inline

Const access R1 time part.

Definition at line 133 of file galilei.hpp.

◆ r3_p() [1/2]

template<typename _Derived >

Eigen::Map< Eigen::Vector3< Scalar > > GalileiBase< _Derived >::r3_p ( )

inline

Access R3 position part.

Definition at line 107 of file galilei.hpp.

◆ r3_p() [2/2]

template<typename _Derived >

Eigen::Map< const Eigen::Vector3< Scalar > > GalileiBase< _Derived >::r3_p ( ) const

inline

Const access R3 position part.

Definition at line 116 of file galilei.hpp.

◆ r3_v() [1/2]

template<typename _Derived >

Eigen::Map< Eigen::Vector3< Scalar > > GalileiBase< _Derived >::r3_v ( )

inline

Access R3 velocity part.

Definition at line 90 of file galilei.hpp.

◆ r3_v() [2/2]

template<typename _Derived >

Eigen::Map< const Eigen::Vector3< Scalar > > GalileiBase< _Derived >::r3_v ( ) const

inline

Const access R3 velocity part.

Definition at line 99 of file galilei.hpp.

◆ so3() [1/2]

template<typename _Derived >

Map< SO3< Scalar > > GalileiBase< _Derived >::so3 ( )

inline

Access SO(3) part.

Definition at line 76 of file galilei.hpp.

◆ so3() [2/2]

template<typename _Derived >

Map< const SO3< Scalar > > GalileiBase< _Derived >::so3 ( ) const

inline

Const access SO(3) part.

Definition at line 85 of file galilei.hpp.

Member Data Documentation

◆ SMOOTH_INHERIT_TYPEDEFS

template<typename _Derived >

GalileiBase< _Derived >::SMOOTH_INHERIT_TYPEDEFS

Definition at line 71 of file galilei.hpp.

The documentation for this class was generated from the following file:

include/smooth/galilei.hpp

Public Member Functions

Public Attributes

Additional Inherited Members

Detailed Description

Memory layout

Constraints

Lie group matrix form

Lie algebra matrix form

Member Function Documentation

◆ dr_action()

◆ operator*()

◆ r1_t() [1/2]

◆ r1_t() [2/2]

◆ r3_p() [1/2]

◆ r3_p() [2/2]

◆ r3_v() [1/2]

◆ r3_v() [2/2]

◆ so3() [1/2]

◆ so3() [2/2]

Member Data Documentation

◆ SMOOTH_INHERIT_TYPEDEFS