Base class for Lie group types. More...

#include <lie_group_base.hpp>

Inheritance diagram for LieGroupBase< Derived >:

[legend]

Public Types
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.

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

using	PlainObject = CastT< Scalar >
	Plain return type.

Public Member Functions
template<bool = true> requires (is_mutable)
auto &	coeffs () const
	Access underlying storages.

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

template<bool = true> requires (is_mutable)
auto *	data () const
	Access raw pointer.

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

template<typename OtherDerived > requires (is_mutable && std::is_same_v<Impl, typename liebase_info<OtherDerived>::Impl>)
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 \( \mathbb{R}^{\mathtt{dim} \times \mathtt{dim}} \).

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

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

template<typename OtherDerived > requires (std::is_same_v<Impl, typename liebase_info<OtherDerived>::Impl>)
PlainObject	operator* (const LieGroupBase< OtherDerived > &o) const noexcept
	Group binary composition operation.

template<typename OtherDerived > requires (is_mutable && std::is_same_v<Impl, typename liebase_info<OtherDerived>::Impl>)
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.

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

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

template<typename OtherDerived > requires (std::is_same_v<Impl, typename liebase_info<OtherDerived>::Impl>)
Tangent	operator- (const LieGroupBase< OtherDerived > &xo) const noexcept
	Right-minus.

Static Public Member Functions
static PlainObject	Identity () noexcept
	Construct the identity element.

static PlainObject	Random () noexcept
	Construct a random element.

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

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

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

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

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

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

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

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

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

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

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

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

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

Static Public Attributes
static constexpr int	RepSize = Impl::RepSize
	Number of scalars in internal representation.

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

static constexpr int	Dim = Impl::Dim
	Side of Lie group matrix representation.

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

Protected Types
using	traits = liebase_info< Derived >
	CRTP traits.

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

Static Protected Attributes
static constexpr bool	is_mutable = traits::is_mutable
	True if underlying storage supports modification.

Detailed Description

template<typename Derived>
class LieGroupBase< Derived >

Base class for Lie group types.

Definition at line 27 of file lie_group_base.hpp.

Member Typedef Documentation

◆ CastT

template<typename Derived >

template<typename NewScalar >

using LieGroupBase< Derived >::CastT = typename traits::template PlainObject<NewScalar>

Plain return type with different scalar.

Definition at line 67 of file lie_group_base.hpp.

◆ Hessian

template<typename Derived >

using LieGroupBase< Derived >::Hessian = Eigen::Matrix<Scalar, Dof, Dof * Dof>

Plain return type with different scalar.

Definition at line 64 of file lie_group_base.hpp.

◆ Impl

template<typename Derived >

using LieGroupBase< Derived >::Impl = typename traits::Impl

protected

Group-specific Lie group implementation.

Definition at line 40 of file lie_group_base.hpp.

◆ Matrix

template<typename Derived >

using LieGroupBase< Derived >::Matrix = Eigen::Matrix<Scalar, Dim, Dim>

Lie group matrix type.

Definition at line 58 of file lie_group_base.hpp.

◆ PlainObject

template<typename Derived >

using LieGroupBase< Derived >::PlainObject = CastT<Scalar>

Plain return type.

Definition at line 69 of file lie_group_base.hpp.

◆ Scalar

template<typename Derived >

using LieGroupBase< Derived >::Scalar = typename traits::Scalar

Scalar type.

Definition at line 56 of file lie_group_base.hpp.

◆ Tangent

template<typename Derived >

using LieGroupBase< Derived >::Tangent = Eigen::Matrix<Scalar, Dof, 1>

Lie group parameterized tangent type.

Definition at line 60 of file lie_group_base.hpp.

◆ TangentMap

template<typename Derived >

using LieGroupBase< Derived >::TangentMap = Eigen::Matrix<Scalar, Dof, Dof>

Matrix representing map between tangent elements.

Definition at line 62 of file lie_group_base.hpp.

◆ traits

template<typename Derived >

using LieGroupBase< Derived >::traits = liebase_info<Derived>

protected

CRTP traits.

Definition at line 37 of file lie_group_base.hpp.

Member Function Documentation

◆ Ad()

template<typename Derived >

TangentMap LieGroupBase< Derived >::Ad ( ) const

inlinenoexcept

Lie group adjoint.

Returns: Matrix \( \mathbf{Ad}_\mathbf{X} \) s.t. \( \mathbf{Ad_X} \mathbf{a} = ( \mathbf{X} \mathbf{a}^\wedge \mathbf{X}^{-1} )^\vee \)

Definition at line 227 of file lie_group_base.hpp.

◆ ad()

template<typename Derived >

template<typename TangentDerived >

static TangentMap LieGroupBase< Derived >::ad ( const Eigen::MatrixBase< TangentDerived > & a )

inlinestaticnoexcept

Lie algebra adjoint.

Returns: Matrix \( \mathbf{ad}_\mathbf{a} \) s.t. \( \mathbf{ad}_\mathbf{a} \mathbf{b} = [\mathbf{a}, \mathbf{b}] \)

Definition at line 332 of file lie_group_base.hpp.

◆ cast()

template<typename Derived >

template<typename NewScalar >

CastT< NewScalar > LieGroupBase< Derived >::cast ( ) const

inlinenoexcept

Cast to different scalar type.

Definition at line 169 of file lie_group_base.hpp.

◆ coeffs() [1/2]

template<typename Derived >

template<bool = true>
requires (is_mutable)

auto & LieGroupBase< Derived >::coeffs ( ) const

inline

Access underlying storages.

Definition at line 74 of file lie_group_base.hpp.

◆ coeffs() [2/2]

template<typename Derived >

const auto & LieGroupBase< Derived >::coeffs ( ) const

inline

Const access underlying storages.

Definition at line 79 of file lie_group_base.hpp.

◆ d2l_exp()

template<typename Derived >

template<typename TangentDerived >

static Hessian LieGroupBase< Derived >::d2l_exp ( const Eigen::MatrixBase< TangentDerived > & a )

inlinestaticnoexcept

Left Hessian of the exponential map.

Returns: \( \mathrm{d}^{2l} \exp_{aa} \) on horizontally stacked Hessian form.

Definition at line 458 of file lie_group_base.hpp.

◆ d2l_expinv()

template<typename Derived >

template<typename TangentDerived >

static Hessian LieGroupBase< Derived >::d2l_expinv ( const Eigen::MatrixBase< TangentDerived > & a )

inlinestaticnoexcept

Left Hessian of the log map.

Returns: \( \left( \mathrm{d}^l \left( \mathrm{d}^l \exp_a \right)^{-1} \right)_a \) on horizontally stacked Hessian form.

Definition at line 470 of file lie_group_base.hpp.

◆ d2r_exp()

template<typename Derived >

template<typename TangentDerived >

static Hessian LieGroupBase< Derived >::d2r_exp ( const Eigen::MatrixBase< TangentDerived > & a )

inlinestaticnoexcept

Right Hessian of the exponential map.

Returns: \( \mathrm{d}^{2r} \exp_{aa} \) on Horizontally stacked Hessian form.

Definition at line 423 of file lie_group_base.hpp.

◆ d2r_expinv()

template<typename Derived >

template<typename TangentDerived >

static Hessian LieGroupBase< Derived >::d2r_expinv ( const Eigen::MatrixBase< TangentDerived > & a )

inlinestaticnoexcept

Right Hessian of the log map.

Returns: \( \left( \mathrm{d}^r \left( \mathrm{d}^r \exp_a \right)^{-1} \right)_a \) on horizontally stakced Hessian form.

Definition at line 441 of file lie_group_base.hpp.

◆ data() [1/2]

template<typename Derived >

template<bool = true>
requires (is_mutable)

auto * LieGroupBase< Derived >::data ( ) const

inline

Access raw pointer.

Definition at line 84 of file lie_group_base.hpp.

◆ data() [2/2]

template<typename Derived >

const auto * LieGroupBase< Derived >::data ( ) const

inline

Const access raw pointer.

Definition at line 89 of file lie_group_base.hpp.

◆ dl_exp()

template<typename Derived >

template<typename TangentDerived >

static TangentMap LieGroupBase< Derived >::dl_exp ( const Eigen::MatrixBase< TangentDerived > & a )

inlinestaticnoexcept

Left jacobian of the exponential map.

Returns: \( \mathrm{d}^l \exp_a \)

Definition at line 401 of file lie_group_base.hpp.

◆ dl_expinv()

template<typename Derived >

template<typename TangentDerived >

static TangentMap LieGroupBase< Derived >::dl_expinv ( const Eigen::MatrixBase< TangentDerived > & a )

inlinestaticnoexcept

Inverse of left jacobian of the exponential map.

Returns: \( \left( \mathrm{d}^l \exp_a \right)^{-1} \)

Definition at line 412 of file lie_group_base.hpp.

◆ dof()

template<typename Derived >

Eigen::Index LieGroupBase< Derived >::dof ( ) const

inlinenoexcept

Dynamic size (degrees of freedom).

Definition at line 107 of file lie_group_base.hpp.

◆ dr_exp()

template<typename Derived >

template<typename TangentDerived >

static TangentMap LieGroupBase< Derived >::dr_exp ( const Eigen::MatrixBase< TangentDerived > & a )

inlinestaticnoexcept

Right jacobian of the exponential map.

Returns: \( \mathrm{d}^r \exp_a \)

Definition at line 367 of file lie_group_base.hpp.

◆ dr_expinv()

template<typename Derived >

template<typename TangentDerived >

static TangentMap LieGroupBase< Derived >::dr_expinv ( const Eigen::MatrixBase< TangentDerived > & a )

inlinestaticnoexcept

Inverse of right jacobian of the exponential map.

Returns: \( \left( \mathrm{d}^r \exp_a \right)^{-1} \)

Definition at line 384 of file lie_group_base.hpp.

◆ exp()

template<typename Derived >

template<typename TangentDerived >

static PlainObject LieGroupBase< Derived >::exp ( const Eigen::MatrixBase< TangentDerived > & a )

inlinestaticnoexcept

Lie group exponential map.

Definition at line 285 of file lie_group_base.hpp.

◆ hat()

template<typename Derived >

template<typename TangentDerived >

static Matrix LieGroupBase< Derived >::hat ( const Eigen::MatrixBase< TangentDerived > & a )

inlinestaticnoexcept

Lie algebra hat map.

Maps a parameterization \( a \in \mathbb{R}^\mathtt{dof} \) a matrix Lie algebra element to the corresponding matrix Lie algebra element \( A \in \mathbb{R}^{\mathtt{dim} \times \mathtt{dim}} \).

See also: vee for the inverse of hat

Definition at line 302 of file lie_group_base.hpp.

◆ Identity()

template<typename Derived >

static PlainObject LieGroupBase< Derived >::Identity ( )

inlinestaticnoexcept

Construct the identity element.

Definition at line 124 of file lie_group_base.hpp.

◆ inverse()

template<typename Derived >

PlainObject LieGroupBase< Derived >::inverse ( ) const

inlinenoexcept

Group inverse operation.

Definition at line 202 of file lie_group_base.hpp.

◆ isApprox()

template<typename Derived >

template<typename OtherDerived >
requires (std::is_same_v<Impl, typename liebase_info<OtherDerived>::Impl>)

bool LieGroupBase< Derived >::isApprox	(	const LieGroupBase< OtherDerived > &	o,
		const Scalar &	eps = `Eigen::NumTraits<Scalar>::dummy_precision()`
	)		const

inlinenoexcept

Check if (approximately) equal to other element o.

Definition at line 159 of file lie_group_base.hpp.

◆ lie_bracket()

template<typename Derived >

template<typename TangentDerived1 , typename TangentDerived2 >

static Tangent LieGroupBase< Derived >::lie_bracket	(	const Eigen::MatrixBase< TangentDerived1 > &	a,
		const Eigen::MatrixBase< TangentDerived2 > &	b
	)

inlinestaticnoexcept

Lie algebra bracket.

\[ [ \mathbf{a}, \mathbf{b}] = \left( \mathbf{a}^\wedge \mathbf{b}^\wedge - \mathbf{b}^\wedge \mathbf{a}^\wedge \right)^\vee. \]

Definition at line 352 of file lie_group_base.hpp.

◆ log()

template<typename Derived >

Tangent LieGroupBase< Derived >::log ( ) const

inlinenoexcept

Lie group logarithm.

Returns: tangent logarithm element.

Definition at line 214 of file lie_group_base.hpp.

◆ matrix()

template<typename Derived >

Matrix LieGroupBase< Derived >::matrix ( ) const

inlinenoexcept

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

Definition at line 147 of file lie_group_base.hpp.

◆ operator*()

template<typename Derived >

template<typename OtherDerived >
requires (std::is_same_v<Impl, typename liebase_info<OtherDerived>::Impl>)

PlainObject LieGroupBase< Derived >::operator* ( const LieGroupBase< OtherDerived > & o ) const

inlinenoexcept

Group binary composition operation.

Definition at line 181 of file lie_group_base.hpp.

◆ operator*=()

template<typename Derived >

template<typename OtherDerived >
requires (is_mutable && std::is_same_v<Impl, typename liebase_info<OtherDerived>::Impl>)

Derived & LieGroupBase< Derived >::operator*= ( const LieGroupBase< OtherDerived > & o )

inlinenoexcept

Inplace group binary composition operation.

Definition at line 193 of file lie_group_base.hpp.

◆ operator+()

template<typename Derived >

template<typename TangentDerived >

PlainObject LieGroupBase< Derived >::operator+ ( const Eigen::MatrixBase< TangentDerived > & a ) const

inlinenoexcept

Right-plus.

Returns: \( \mathbf{x} \oplus \mathbf{a} = \mathbf{x} \circ \exp(\mathbf{a}) \)

Definition at line 244 of file lie_group_base.hpp.

◆ operator+=()

template<typename Derived >

template<typename TangentDerived >
requires (is_mutable)

Derived & LieGroupBase< Derived >::operator+= ( const Eigen::MatrixBase< TangentDerived > & a )

inlinenoexcept

Inplace right-plus: \( \mathbf{x} \leftarrow \mathbf{x} \circ \exp(\mathbf{a}) \).

Returns: Reference to this

Definition at line 256 of file lie_group_base.hpp.

◆ operator-()

template<typename Derived >

template<typename OtherDerived >
requires (std::is_same_v<Impl, typename liebase_info<OtherDerived>::Impl>)

Tangent LieGroupBase< Derived >::operator- ( const LieGroupBase< OtherDerived > & xo ) const

inlinenoexcept

Right-minus.

Returns: Tangent space element \( \mathbf{a} \) such that \( \mathbf{this} = \mathbf{xo} \oplus \mathbf{a} \).

\[ \mathbf{x}_1 \ominus \mathbf{x}_2 := \log(\mathbf{x}_2^{-1} \circ \mathbf{x}_1) \]

Definition at line 274 of file lie_group_base.hpp.

◆ operator=()

template<typename Derived >

template<typename OtherDerived >
requires (is_mutable && std::is_same_v<Impl, typename liebase_info<OtherDerived>::Impl>)

Derived & LieGroupBase< Derived >::operator= ( const LieGroupBase< OtherDerived > & o )

inlinenoexcept

Assignment operation from other storage type.

Definition at line 96 of file lie_group_base.hpp.

◆ Random()

template<typename Derived >

static PlainObject LieGroupBase< Derived >::Random ( )

inlinestaticnoexcept

Construct a random element.

Set the seed with std::srand(unsigned).

Definition at line 136 of file lie_group_base.hpp.

◆ setIdentity()

template<typename Derived >

void LieGroupBase< Derived >::setIdentity ( )

inlinenoexcept

Set to group identity element.

Definition at line 112 of file lie_group_base.hpp.

◆ setRandom()

template<typename Derived >

void LieGroupBase< Derived >::setRandom ( )

inlinenoexcept

Set to a random element.

Set the seed with std::srand(unsigned).

Definition at line 119 of file lie_group_base.hpp.

◆ vee()

template<typename Derived >

template<typename MatrixDerived >

static Tangent LieGroupBase< Derived >::vee ( const Eigen::MatrixBase< MatrixDerived > & A )

inlinestaticnoexcept

Lie alebra vee map.

Maps a matrix Lie algebra element \( A \in \mathbb{R}^{\texttt{dim} \times \mathtt{dim}} \) to the corresponding parameterization \( a \in \mathbb{R}^\mathtt{dof} \).

See also: hat for the inverse of vee

Definition at line 318 of file lie_group_base.hpp.

Member Data Documentation

◆ Dim

template<typename Derived >

constexpr int LieGroupBase< Derived >::Dim = Impl::Dim

staticconstexpr

Side of Lie group matrix representation.

Definition at line 51 of file lie_group_base.hpp.

◆ Dof

template<typename Derived >

constexpr int LieGroupBase< Derived >::Dof = Impl::Dof

staticconstexpr

Degrees of freedom of manifold (equal to tangent space dimension).

Definition at line 49 of file lie_group_base.hpp.

◆ is_mutable

template<typename Derived >

constexpr bool LieGroupBase< Derived >::is_mutable = traits::is_mutable

staticconstexprprotected

True if underlying storage supports modification.

Definition at line 43 of file lie_group_base.hpp.

◆ IsCommutative

template<typename Derived >

constexpr bool LieGroupBase< Derived >::IsCommutative = Impl::IsCommutative

staticconstexpr

Commutativity of group. A commutative group has a zero Lie bracket.

Definition at line 53 of file lie_group_base.hpp.

◆ RepSize

template<typename Derived >

constexpr int LieGroupBase< Derived >::RepSize = Impl::RepSize

staticconstexpr

Number of scalars in internal representation.

Definition at line 47 of file lie_group_base.hpp.

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

include/smooth/lie_group_base.hpp

Public Types

Public Member Functions

Static Public Member Functions

Static Public Attributes

Protected Types

Static Protected Attributes

Detailed Description

Member Typedef Documentation

◆ CastT

◆ Hessian

◆ Impl

◆ Matrix

◆ PlainObject

◆ Scalar

◆ Tangent

◆ TangentMap

◆ traits

Member Function Documentation

◆ Ad()

◆ ad()

◆ cast()

◆ coeffs() [1/2]

◆ coeffs() [2/2]

◆ d2l_exp()

◆ d2l_expinv()

◆ d2r_exp()

◆ d2r_expinv()

◆ data() [1/2]

◆ data() [2/2]

◆ dl_exp()

◆ dl_expinv()

◆ dof()

◆ dr_exp()

◆ dr_expinv()

◆ exp()

◆ hat()

◆ Identity()

◆ inverse()

◆ isApprox()

◆ lie_bracket()

◆ log()

◆ matrix()

◆ operator*()

◆ operator*=()

◆ operator+()

◆ operator+=()

◆ operator-()

◆ operator=()

◆ Random()

◆ setIdentity()

◆ setRandom()

◆ vee()

Member Data Documentation

◆ Dim

◆ Dof

◆ is_mutable

◆ IsCommutative

◆ RepSize