Base class for SO3 Lie group types. More...

#include <so3.hpp>

Inheritance diagram for SO3Base< _Derived >:

Public Member Functions
Eigen::Map< Eigen::Quaternion< Scalar > >	quat ()
	Access quaterion.

Eigen::Map< const Eigen::Quaternion< Scalar > >	quat () const
	Access const quaterion.

Eigen::Vector3< Scalar >	eulerAngles (Eigen::Index i1=2, Eigen::Index i2=1, Eigen::Index i3=0) const
	Return euler angles.

template<typename EigenDerived >
Eigen::Vector3< Scalar >	operator* (const Eigen::MatrixBase< EigenDerived > &v) const
	Rotation action on 3D vector.

template<typename EigenDerived >
Eigen::Matrix3< Scalar >	dr_action (const Eigen::MatrixBase< EigenDerived > &v) const
	Jacobian of rotation action w.r.t. group.

SO2< Scalar >	project_so2 () const
	Project to SO2.

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 SO3Base< _Derived >

Base class for SO3 Lie group types.

Internally represented as a member of \( \mathbb{S}^3 \) (unit quaternions).

Memory layout

Group: \( \mathbf{x} = [q_x, q_y, q_z, q_w] \) (same as Eigen quaternion).
Tangent: \( \mathbf{a} = [\omega_x, \omega_y, \omega_z] \)

Constraints

Group: \( q_x^2 + q_y^2 + q_z^2 + q_w^2 = 1, q_w >= 0 \)
Tangent: \( -\pi < \omega_x, \omega_y, \omega_z \leq \pi \)

Lie group matrix form

3x3 rotation matrix

Lie algebra matrix form

\[ \mathbf{a}^\wedge = \begin{bmatrix} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \\ \end{bmatrix} \in \mathbb{R}^{3 \times 3} \]

Definition at line 54 of file so3.hpp.

Member Function Documentation

◆ dr_action()

template<typename _Derived >

template<typename EigenDerived >

Eigen::Matrix3< Scalar > SO3Base< _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 113 of file so3.hpp.

◆ eulerAngles()

template<typename _Derived >

Eigen::Vector3< Scalar > SO3Base< _Derived >::eulerAngles	(	Eigen::Index	i1 = `2`,
		Eigen::Index	i2 = `1`,
		Eigen::Index	i3 = `0`
	)		const

inline

Return euler angles.

Parameters

i1,i2,i3 euler angle axis convention ( \(0=x, 1=y, 2=z\)). Default values correspond to ZYX rotation.

Returned angles a1, a2, a3 are s.t. rotation is described by \( Rot_{i_1}(a_1) * Rot_{i_2}(a_2) * Rot_{i_3}(a_3). \) where \( Rot_i(a) \) rotates an angle \(a\) around the \(i\):th axis.

Definition at line 91 of file so3.hpp.

◆ operator*()

template<typename _Derived >

template<typename EigenDerived >

Eigen::Vector3< Scalar > SO3Base< _Derived >::operator* ( const Eigen::MatrixBase< EigenDerived > & v ) const

inline

Rotation action on 3D vector.

Definition at line 100 of file so3.hpp.

◆ project_so2()

template<typename _Derived >

SO2< Scalar > SO3Base< _Derived >::project_so2 ( ) const

inline

Project to SO2.

This keeps the yaw/z axis component of the rotation.

Note: SO2 header must be included.

Definition at line 125 of file so3.hpp.

◆ quat() [1/2]

template<typename _Derived >

Eigen::Map< Eigen::Quaternion< Scalar > > SO3Base< _Derived >::quat ( )

inline

Access quaterion.

Definition at line 67 of file so3.hpp.

◆ quat() [2/2]

template<typename _Derived >

Eigen::Map< const Eigen::Quaternion< Scalar > > SO3Base< _Derived >::quat ( ) const

inline

Access const quaterion.

Definition at line 76 of file so3.hpp.

Member Data Documentation

◆ SMOOTH_INHERIT_TYPEDEFS

template<typename _Derived >

SO3Base< _Derived >::SMOOTH_INHERIT_TYPEDEFS

Definition at line 62 of file so3.hpp.

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

include/smooth/so3.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()

◆ eulerAngles()

◆ operator*()

◆ project_so2()

◆ quat() [1/2]

◆ quat() [2/2]

Member Data Documentation

◆ SMOOTH_INHERIT_TYPEDEFS