Base class for SE_k(3) Lie group types. More...

#include <se_k_3.hpp>

Inheritance diagram for SE_K_3Base< _Derived >:

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

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

template<int Ksel> requires (is_mutable && Ksel < K)
Eigen::Map< Eigen::Vector3< Scalar > >	r3 ()
	Access R3 parts.

Eigen::Map< Eigen::Vector3< Scalar > >	r3 (int k)
	Access R3 parts.

template<int Ksel> requires (Ksel < K)
Eigen::Map< const Eigen::Vector3< Scalar > >	r3 () const
	Const access R3 parts.

Eigen::Map< const Eigen::Vector3< Scalar > >	r3 (int k) const
	Const access R3 parts.

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

Static Public Attributes
static constexpr auto	K = Impl::K
	Number of R3 variables.

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.

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.

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 SE_K_3Base< _Derived >

Base class for SE_k(3) Lie group types.

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

Memory layout

Group: \( \mathbf{x} = [p1, ..., pk, q_x, q_y, q_z, q_w] \)
Tangent: \( \mathbf{a} = [v1, ..., vk,, \omega_x, \omega_y, \omega_z] \)

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

\[ \mathbf{X} = \begin{bmatrix} R & P_1 & ... & P_k \\ 0 & 1 & ... & 0 \\ \vdots 0 & 0 & ... & 1 \\ \end{bmatrix} \in \mathbb{R}^{3+k \times 3+k} \]

where \(R\) is a 3x3 rotation matrix and \( P_k = [x_k, y_k, z_k]^T \).

Lie algebra matrix form

\[ \mathbf{a}^\wedge = \begin{bmatrix} 0 & -\omega_z & \omega_y & u_1 & ... & u_k \\ \omega_z & 0 & -\omega_x & v_1 & ... & v_k \\ -\omega_y & \omega_x & 0 & w_1 & ... & w_k \\ 0 & 0 & 0 & 0 & ... & 0 \\ \vdots 0 & 0 & 0 & 0 & ... & 0 \\ \end{bmatrix} \in \mathbb{R}^{3+k \times 3+k} \]

Definition at line 65 of file se_k_3.hpp.

Member Function Documentation

◆ r3() [1/4]

template<typename _Derived >

template<int Ksel>
requires (is_mutable && Ksel < K)

Eigen::Map< Eigen::Vector3< Scalar > > SE_K_3Base< _Derived >::r3 ( )

inline

Access R3 parts.

Template Parameters

Ksel	select part

Definition at line 104 of file se_k_3.hpp.

◆ r3() [2/4]

template<typename _Derived >

template<int Ksel>
requires (Ksel < K)

Eigen::Map< const Eigen::Vector3< Scalar > > SE_K_3Base< _Derived >::r3 ( ) const

inline

Const access R3 parts.

Template Parameters

Ksel	select part

Definition at line 128 of file se_k_3.hpp.

◆ r3() [3/4]

template<typename _Derived >

Eigen::Map< Eigen::Vector3< Scalar > > SE_K_3Base< _Derived >::r3 ( int k )

inline

Access R3 parts.

Parameters

k	select part

Definition at line 115 of file se_k_3.hpp.

◆ r3() [4/4]

template<typename _Derived >

Eigen::Map< const Eigen::Vector3< Scalar > > SE_K_3Base< _Derived >::r3 ( int k ) const

inline

Const access R3 parts.

Parameters

k	select part

Definition at line 139 of file se_k_3.hpp.

◆ so3() [1/2]

template<typename _Derived >

Map< SO3< Scalar > > SE_K_3Base< _Derived >::so3 ( )

inline

Access SO(3) part.

Definition at line 84 of file se_k_3.hpp.

◆ so3() [2/2]

template<typename _Derived >

Map< const SO3< Scalar > > SE_K_3Base< _Derived >::so3 ( ) const

inline

Const access SO(3) part.

Definition at line 93 of file se_k_3.hpp.

Member Data Documentation

◆ K

template<typename _Derived >

constexpr auto SE_K_3Base< _Derived >::K = Impl::K

staticconstexpr

Number of R3 variables.

Definition at line 79 of file se_k_3.hpp.

◆ SMOOTH_INHERIT_TYPEDEFS

template<typename _Derived >

SE_K_3Base< _Derived >::SMOOTH_INHERIT_TYPEDEFS

Definition at line 74 of file se_k_3.hpp.

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

include/smooth/se_k_3.hpp

Public Member Functions

Public Attributes

Static Public Attributes

Additional Inherited Members

Detailed Description

Memory layout

Constraints

Lie group matrix form

Lie algebra matrix form

Member Function Documentation

◆ r3() [1/4]

◆ r3() [2/4]

◆ r3() [3/4]

◆ r3() [4/4]

◆ so3() [1/2]

◆ so3() [2/2]

Member Data Documentation

◆ K

◆ SMOOTH_INHERIT_TYPEDEFS