Detailed Description

Collaboration diagram for Lie Algebra methods for SO(2),SO(3),SE(2),SE(3):

Classes
struct	mrpt::poses::Lie::Euclidean< N >
	Traits for Euclidean R^N space. More...

struct	mrpt::poses::Lie::SE< n >
	Traits for SE(n), rigid-body transformations in R^n space. More...

struct	mrpt::poses::Lie::SE< 3 >
	Traits for SE(3), rigid-body transformations in R^3 space. More...

struct	mrpt::poses::Lie::SE< 2 >
	Traits for SE(2), rigid-body transformations in R^2 space. More...

struct	mrpt::poses::Lie::SO< n >
	Traits for SO(n), rotations in R^n space. More...

struct	mrpt::poses::Lie::SO< 3 >
	Traits for SO(3), rotations in R^3 space. More...

struct	mrpt::poses::Lie::SO< 2 >
	Traits for SO(2), rotations in R^2 space. More...

Typedefs
using	mrpt::poses::Lie::SE< 3 >::tangent_vector = mrpt::math::CVectorFixedDouble< DOFs >

using	mrpt::poses::Lie::SE< 3 >::manifold_vector = mrpt::math::CVectorFixedDouble< MANIFOLD_DIM >

using	mrpt::poses::Lie::SE< 3 >::type = CPose3D

using	mrpt::poses::Lie::SE< 3 >::light_type = mrpt::math::TPose3D

using	mrpt::poses::Lie::SE< 3 >::tang2mat_jacob = mrpt::math::CMatrixDouble12_6
	Type for Jacobian: tangent space -> SO(n) matrix. More...

using	mrpt::poses::Lie::SE< 3 >::mat2tang_jacob = mrpt::math::CMatrixDouble6_12
	Type for Jacobian: SO(n) matrix -> tangent space. More...

using	mrpt::poses::Lie::SE< 3 >::matrix_TxT = mrpt::math::CMatrixDouble66
	Type for Jacobians between so(n) vectors on the tangent space. More...

using	mrpt::poses::Lie::SE< 3 >::matrix_MxM = mrpt::math::CMatrixFixed< double, 12, 12 >
	Type for Jacobians between SO(n) 3x4 (sub)matrices in the manifold. More...

using	mrpt::poses::Lie::SE< 2 >::tangent_vector = mrpt::math::CVectorFixedDouble< DOFs >

using	mrpt::poses::Lie::SE< 2 >::manifold_vector = mrpt::math::CVectorFixedDouble< MANIFOLD_DIM >

using	mrpt::poses::Lie::SE< 2 >::type = CPose2D

using	mrpt::poses::Lie::SE< 2 >::light_type = mrpt::math::TPose2D

using	mrpt::poses::Lie::SE< 2 >::matrix_TxT = mrpt::math::CMatrixDouble33
	Type for Jacobians between SO(n) transformations. More...

using	mrpt::poses::Lie::SE< 2 >::matrix_MxM = mrpt::math::CMatrixDouble33
	In SE(3), this type represents Jacobians between SO(n) (sub)matrices in the manifold, but in SE(2) it simply models Jacobians between (x,y,phi) vectors. More...

using	mrpt::poses::Lie::SE< 2 >::tang2mat_jacob = mrpt::math::CMatrixDouble33
	Type for Jacobian: tangent space -> SO(n) matrix. More...

using	mrpt::poses::Lie::SE< 2 >::mat2tang_jacob = mrpt::math::CMatrixDouble33
	Type for Jacobian: SO(n) matrix -> tangent space. More...

Functions
static type	mrpt::poses::Lie::SE< 3 >::exp (const tangent_vector &x)
	Retraction to SE(3), a pseudo-exponential map $x \rightarrow PseudoExp(x^\wedge)$ and its Jacobian. More...

static tangent_vector	mrpt::poses::Lie::SE< 3 >::log (const type &P)
	SE(3) pseudo-logarithm map $\mathbf{R} \rightarrow \log(\mathbf{R}^\vee)$ . More...

static manifold_vector	mrpt::poses::Lie::SE< 3 >::asManifoldVector (const type &pose)
	Returns a vector with all manifold matrix elements in column-major order. More...

static type	mrpt::poses::Lie::SE< 3 >::fromManifoldVector (const manifold_vector &v)
	The inverse operation of asManifoldVector() More...

static tang2mat_jacob	mrpt::poses::Lie::SE< 3 >::jacob_dexpe_de (const tangent_vector &x)
	Jacobian for the pseudo-exponential exp(). More...

static mat2tang_jacob	mrpt::poses::Lie::SE< 3 >::jacob_dlogv_dv (const type &P)
	Jacobian for the pseudo-logarithm log() See 10.3.11 in [1]. More...

static tang2mat_jacob	mrpt::poses::Lie::SE< 3 >::jacob_dexpeD_de (const CPose3D &D)
	Jacobian d (e^eps * D) / d eps , with eps=increment in Lie Algebra. More...

static tang2mat_jacob	mrpt::poses::Lie::SE< 3 >::jacob_dDexpe_de (const CPose3D &D)
	Jacobian d (D * e^eps) / d eps , with eps=increment in Lie Algebra. More...

static tang2mat_jacob	mrpt::poses::Lie::SE< 3 >::jacob_dAexpeD_de (const CPose3D &A, const CPose3D &D)
	Jacobian d (A * e^eps * D) / d eps , with eps=increment in Lie Algebra. More...

static matrix_MxM	mrpt::poses::Lie::SE< 3 >::jacob_dAB_dA (const type &A, const type &B)
	Jacobian of the pose composition A*B for SE(3) 3x4 (sub)matrices, with respect to A. More...

static matrix_MxM	mrpt::poses::Lie::SE< 3 >::jacob_dAB_dB (const type &A, const type &B)
	Jacobian of the pose composition A*B for SE(3) 3x4 (sub)matrices, with respect to B. More...

static void	mrpt::poses::Lie::SE< 3 >::jacob_dDinvP1invP2_de1e2 (const type &Dinv, const type &P1, const type &P2, mrpt::optional_ref< matrix_TxT > df_de1=std::nullopt, mrpt::optional_ref< matrix_TxT > df_de2=std::nullopt)
	Return one or both of the following 6x6 Jacobians, useful in graph-slam problems: $\frac{\partial pseudoLn(D^{-1} P_1^{-1} P_2}{\partial \epsilon_1}$ $\frac{\partial pseudoLn(D^{-1} P_1^{-1} P_2}{\partial \epsilon_1}$ With $\epsilon_1$ and $\epsilon_2$ increments in the linearized manifold for P1 and P2. More...

static type	mrpt::poses::Lie::SE< 2 >::exp (const tangent_vector &x)
	Exponential map in SE(2), takes [x,y,phi] and returns a CPose2D. More...

static tangent_vector	mrpt::poses::Lie::SE< 2 >::log (const type &P)
	Logarithm map in SE(2), takes a CPose2D and returns [X,Y, phi]. More...

static manifold_vector	mrpt::poses::Lie::SE< 2 >::asManifoldVector (const type &pose)
	Returns a vector with all manifold matrix elements in column-major order. More...

static type	mrpt::poses::Lie::SE< 2 >::fromManifoldVector (const manifold_vector &v)
	The inverse operation of asManifoldVector() More...

static matrix_MxM	mrpt::poses::Lie::SE< 2 >::jacob_dAB_dA (const type &A, const type &B)
	Jacobian of the pose composition A*B for SE(2) with respect to A. More...

static matrix_MxM	mrpt::poses::Lie::SE< 2 >::jacob_dAB_dB (const type &A, const type &B)
	Jacobian of the pose composition A*B for SE(2) with respect to B. More...

static tang2mat_jacob	mrpt::poses::Lie::SE< 2 >::jacob_dDexpe_de (const type &D)
	Jacobian d (D * e^eps) / d eps , with eps=increment in Lie Algebra. More...

static void	mrpt::poses::Lie::SE< 2 >::jacob_dDinvP1invP2_de1e2 (const type &Dinv, const type &P1, const type &P2, mrpt::optional_ref< matrix_TxT > df_de1=std::nullopt, mrpt::optional_ref< matrix_TxT > df_de2=std::nullopt)
	Return one or both of the following 3x3 Jacobians, useful in graph-slam problems: $\frac{\partial pseudoLn(D^{-1} P_1^{-1} P_2}{\partial \epsilon_1}$ $\frac{\partial pseudoLn(D^{-1} P_1^{-1} P_2}{\partial \epsilon_1}$ With $\epsilon_1$ and $\epsilon_2$ increments in the linearized manifold for P1 and P2. More...

Variables
static constexpr size_t	mrpt::poses::Lie::SE< 3 >::DOFs = 6
	Number of actual degrees of freedom for this transformation. More...

static constexpr size_t	mrpt::poses::Lie::SE< 3 >::MANIFOLD_DIM = 3 * 4
	Dimensionality of the matrix manifold (3x4=12 upper part of the 4x4) More...

static constexpr size_t	mrpt::poses::Lie::SE< 2 >::DOFs = 3
	Number of actual degrees of freedom for this transformation. More...

static constexpr size_t	mrpt::poses::Lie::SE< 2 >::MANIFOLD_DIM = 3
	Dimensionality of the matrix manifold; this should be 3x3=9 for SE(2), but for efficiency, we define it as simply 3x1=3 and use the (x,y,phi) representation as well as the "manifold". More...

Typedef Documentation

◆ light_type [1/2]

using mrpt::poses::Lie::SE< 3 >::light_type = mrpt::math::TPose3D

Definition at line 50 of file SE.h.

◆ light_type [2/2]

using mrpt::poses::Lie::SE< 2 >::light_type = mrpt::math::TPose2D

Definition at line 177 of file SE.h.

◆ manifold_vector [1/2]

using mrpt::poses::Lie::SE< 3 >::manifold_vector = mrpt::math::CVectorFixedDouble<MANIFOLD_DIM>

Definition at line 47 of file SE.h.

◆ manifold_vector [2/2]

using mrpt::poses::Lie::SE< 2 >::manifold_vector = mrpt::math::CVectorFixedDouble<MANIFOLD_DIM>

Definition at line 174 of file SE.h.

◆ mat2tang_jacob [1/2]

using mrpt::poses::Lie::SE< 3 >::mat2tang_jacob = mrpt::math::CMatrixDouble6_12

Type for Jacobian: SO(n) matrix -> tangent space.

Definition at line 56 of file SE.h.

◆ mat2tang_jacob [2/2]

using mrpt::poses::Lie::SE< 2 >::mat2tang_jacob = mrpt::math::CMatrixDouble33

Type for Jacobian: SO(n) matrix -> tangent space.

Definition at line 190 of file SE.h.

◆ matrix_MxM [1/2]

using mrpt::poses::Lie::SE< 3 >::matrix_MxM = mrpt::math::CMatrixFixed<double, 12, 12>

Type for Jacobians between SO(n) 3x4 (sub)matrices in the manifold.

Definition at line 62 of file SE.h.

◆ matrix_MxM [2/2]

using mrpt::poses::Lie::SE< 2 >::matrix_MxM = mrpt::math::CMatrixDouble33

In SE(3), this type represents Jacobians between SO(n) (sub)matrices in the manifold, but in SE(2) it simply models Jacobians between (x,y,phi) vectors.

Definition at line 185 of file SE.h.

◆ matrix_TxT [1/2]

using mrpt::poses::Lie::SE< 3 >::matrix_TxT = mrpt::math::CMatrixDouble66

Type for Jacobians between so(n) vectors on the tangent space.

Definition at line 59 of file SE.h.

◆ matrix_TxT [2/2]

using mrpt::poses::Lie::SE< 2 >::matrix_TxT = mrpt::math::CMatrixDouble33

Type for Jacobians between SO(n) transformations.

Definition at line 180 of file SE.h.

◆ tang2mat_jacob [1/2]

using mrpt::poses::Lie::SE< 3 >::tang2mat_jacob = mrpt::math::CMatrixDouble12_6

Type for Jacobian: tangent space -> SO(n) matrix.

Definition at line 53 of file SE.h.

◆ tang2mat_jacob [2/2]

using mrpt::poses::Lie::SE< 2 >::tang2mat_jacob = mrpt::math::CMatrixDouble33

Type for Jacobian: tangent space -> SO(n) matrix.

Definition at line 188 of file SE.h.

◆ tangent_vector [1/2]

using mrpt::poses::Lie::SE< 3 >::tangent_vector = mrpt::math::CVectorFixedDouble<DOFs>

Definition at line 46 of file SE.h.

◆ tangent_vector [2/2]

using mrpt::poses::Lie::SE< 2 >::tangent_vector = mrpt::math::CVectorFixedDouble<DOFs>

Definition at line 173 of file SE.h.

◆ type [1/2]

using mrpt::poses::Lie::SE< 3 >::type = CPose3D

Definition at line 49 of file SE.h.

◆ type [2/2]

using mrpt::poses::Lie::SE< 2 >::type = CPose2D

Definition at line 176 of file SE.h.

Function Documentation

◆ asManifoldVector() [1/2]

static manifold_vector mrpt::poses::Lie::SE< 3 >::asManifoldVector ( const type & pose )

static

Returns a vector with all manifold matrix elements in column-major order.

For SE(3), it is a 3x4=12 vector.

◆ asManifoldVector() [2/2]

static manifold_vector mrpt::poses::Lie::SE< 2 >::asManifoldVector ( const type & pose )

static

Returns a vector with all manifold matrix elements in column-major order.

For SE(2), though, it directly returns the vector [x,y,phi] for efficiency in comparison to 3x3 homogeneous coordinates.

◆ exp() [1/2]

static type mrpt::poses::Lie::SE< 3 >::exp ( const tangent_vector & x )

static

Retraction to SE(3), a pseudo-exponential map $x \rightarrow PseudoExp(x^\wedge)$ and its Jacobian.

Input: 6-len vector in Lie algebra se(3) [x,y,z, rx,ry,rz]
Output: translation and rotation in SE(3) as CPose3D Note that this method implements retraction via a pseudo-exponential, where only the rotational part undergoes a real matrix exponential, while the translation is left unmodified. This is done for computational efficiency, and does not change the results of optimizations as long as the corresponding local coordinates (pseudo-logarithm) are used as well.

See section 9.4.2 in [1]

◆ exp() [2/2]

static type mrpt::poses::Lie::SE< 2 >::exp ( const tangent_vector & x )

static

Exponential map in SE(2), takes [x,y,phi] and returns a CPose2D.

◆ fromManifoldVector() [1/2]

static type mrpt::poses::Lie::SE< 3 >::fromManifoldVector ( const manifold_vector & v )

static

The inverse operation of asManifoldVector()

◆ fromManifoldVector() [2/2]

static type mrpt::poses::Lie::SE< 2 >::fromManifoldVector ( const manifold_vector & v )

static

The inverse operation of asManifoldVector()

◆ jacob_dAB_dA() [1/2]

static matrix_MxM mrpt::poses::Lie::SE< 3 >::jacob_dAB_dA	(	const type &	A,
		const type &	B
	)

static

Jacobian of the pose composition A*B for SE(3) 3x4 (sub)matrices, with respect to A.

Note: See section 7.3.1 of in [1]

◆ jacob_dAB_dA() [2/2]

static matrix_MxM mrpt::poses::Lie::SE< 2 >::jacob_dAB_dA	(	const type &	A,
		const type &	B
	)

static

Jacobian of the pose composition A*B for SE(2) with respect to A.

Note: See appendix B of [1]

◆ jacob_dAB_dB() [1/2]

static matrix_MxM mrpt::poses::Lie::SE< 3 >::jacob_dAB_dB	(	const type &	A,
		const type &	B
	)

static

Jacobian of the pose composition A*B for SE(3) 3x4 (sub)matrices, with respect to B.

Note: See section 7.3.1 of in [1]

◆ jacob_dAB_dB() [2/2]

static matrix_MxM mrpt::poses::Lie::SE< 2 >::jacob_dAB_dB	(	const type &	A,
		const type &	B
	)

static

Jacobian of the pose composition A*B for SE(2) with respect to B.

Note: See appendix B of [1]

◆ jacob_dAexpeD_de()

static tang2mat_jacob mrpt::poses::Lie::SE< 3 >::jacob_dAexpeD_de	(	const CPose3D &	A,
		const CPose3D &	D
	)

static

Jacobian d (A * e^eps * D) / d eps , with eps=increment in Lie Algebra.

Note: Eq. 10.3.7 in [1]

◆ jacob_dDexpe_de() [1/2]

static tang2mat_jacob mrpt::poses::Lie::SE< 3 >::jacob_dDexpe_de ( const CPose3D & D )

static

Jacobian d (D * e^eps) / d eps , with eps=increment in Lie Algebra.

Note: Section 10.3.4 in [1]

◆ jacob_dDexpe_de() [2/2]

static tang2mat_jacob mrpt::poses::Lie::SE< 2 >::jacob_dDexpe_de ( const type & D )

static

Jacobian d (D * e^eps) / d eps , with eps=increment in Lie Algebra.

Note: See appendix B in [1]

◆ jacob_dDinvP1invP2_de1e2() [1/2]

static void mrpt::poses::Lie::SE< 3 >::jacob_dDinvP1invP2_de1e2	(	const type &	Dinv,
		const type &	P1,
		const type &	P2,
		mrpt::optional_ref< matrix_TxT >	df_de1 = `std::nullopt`,
		mrpt::optional_ref< matrix_TxT >	df_de2 = `std::nullopt`
	)

static

Return one or both of the following 6x6 Jacobians, useful in graph-slam problems:

$\frac{\partial pseudoLn(D^{-1} P_1^{-1} P_2}{\partial \epsilon_1}$

With $\epsilon_1$ and $\epsilon_2$ increments in the linearized manifold for P1 and P2.

Note: Section 10.3.10 in [1]

◆ jacob_dDinvP1invP2_de1e2() [2/2]

static void mrpt::poses::Lie::SE< 2 >::jacob_dDinvP1invP2_de1e2	(	const type &	Dinv,
		const type &	P1,
		const type &	P2,
		mrpt::optional_ref< matrix_TxT >	df_de1 = `std::nullopt`,
		mrpt::optional_ref< matrix_TxT >	df_de2 = `std::nullopt`
	)

static

Return one or both of the following 3x3 Jacobians, useful in graph-slam problems:

$\frac{\partial pseudoLn(D^{-1} P_1^{-1} P_2}{\partial \epsilon_1}$

With $\epsilon_1$ and $\epsilon_2$ increments in the linearized manifold for P1 and P2.

Note: See appendix B in [1]

◆ jacob_dexpe_de()

static tang2mat_jacob mrpt::poses::Lie::SE< 3 >::jacob_dexpe_de ( const tangent_vector & x )

static

Jacobian for the pseudo-exponential exp().

See 10.3.1 in [1]

Input: 6-len vector in Lie algebra se(3)
Jacobian: Jacobian of the 3x4 matrix (stacked as a column major 12-vector) wrt the vector in the tangent space.

◆ jacob_dexpeD_de()

static tang2mat_jacob mrpt::poses::Lie::SE< 3 >::jacob_dexpeD_de ( const CPose3D & D )

static

Jacobian d (e^eps * D) / d eps , with eps=increment in Lie Algebra.

Note: Section 10.3.3 in [1]

◆ jacob_dlogv_dv()

static mat2tang_jacob mrpt::poses::Lie::SE< 3 >::jacob_dlogv_dv ( const type & P )

static

Jacobian for the pseudo-logarithm log() See 10.3.11 in [1].

Input: a SE(3) pose as a CPose3D object
Jacobian: Jacobian of the tangent space vector wrt the 3x4 matrix elements (stacked as a column major 12-vector).

◆ log() [1/2]

static tangent_vector mrpt::poses::Lie::SE< 3 >::log ( const type & P )

static

SE(3) pseudo-logarithm map $\mathbf{R} \rightarrow \log(\mathbf{R}^\vee)$ .

Input: translation and rotation in SE(3) as CPose3D
Output: 6-len vector in Lie algebra se(3) [x,y,z, rx,ry,rz]

See exp() for the explanation about the "pseudo" name. For the formulas, see section 9.4.2 in [1]

◆ log() [2/2]

static tangent_vector mrpt::poses::Lie::SE< 2 >::log ( const type & P )

static

Logarithm map in SE(2), takes a CPose2D and returns [X,Y, phi].

Variable Documentation

◆ DOFs [1/2]

constexpr size_t mrpt::poses::Lie::SE< 3 >::DOFs = 6

static

Number of actual degrees of freedom for this transformation.

Definition at line 41 of file SE.h.

◆ DOFs [2/2]

constexpr size_t mrpt::poses::Lie::SE< 2 >::DOFs = 3

static

Number of actual degrees of freedom for this transformation.

Definition at line 165 of file SE.h.

◆ MANIFOLD_DIM [1/2]

constexpr size_t mrpt::poses::Lie::SE< 3 >::MANIFOLD_DIM = 3 * 4

static

Dimensionality of the matrix manifold (3x4=12 upper part of the 4x4)

Definition at line 44 of file SE.h.

◆ MANIFOLD_DIM [2/2]

constexpr size_t mrpt::poses::Lie::SE< 2 >::MANIFOLD_DIM = 3

static

Dimensionality of the matrix manifold; this should be 3x3=9 for SE(2), but for efficiency, we define it as simply 3x1=3 and use the (x,y,phi) representation as well as the "manifold".

This is done for API consistency with SE(3), where the actual matrix is used instead.

Definition at line 171 of file SE.h.

Detailed Description

Classes

Typedefs

Functions

Variables

Typedef Documentation

◆ light_type [1/2]

◆ light_type [2/2]

◆ manifold_vector [1/2]

◆ manifold_vector [2/2]

◆ mat2tang_jacob [1/2]

◆ mat2tang_jacob [2/2]

◆ matrix_MxM [1/2]

◆ matrix_MxM [2/2]

◆ matrix_TxT [1/2]

◆ matrix_TxT [2/2]

◆ tang2mat_jacob [1/2]

◆ tang2mat_jacob [2/2]

◆ tangent_vector [1/2]

◆ tangent_vector [2/2]

◆ type [1/2]

◆ type [2/2]

Function Documentation

◆ asManifoldVector() [1/2]

◆ asManifoldVector() [2/2]

◆ exp() [1/2]

◆ exp() [2/2]

◆ fromManifoldVector() [1/2]

◆ fromManifoldVector() [2/2]

◆ jacob_dAB_dA() [1/2]

◆ jacob_dAB_dA() [2/2]

◆ jacob_dAB_dB() [1/2]

◆ jacob_dAB_dB() [2/2]

◆ jacob_dAexpeD_de()

◆ jacob_dDexpe_de() [1/2]

◆ jacob_dDexpe_de() [2/2]

◆ jacob_dDinvP1invP2_de1e2() [1/2]

◆ jacob_dDinvP1invP2_de1e2() [2/2]

◆ jacob_dexpe_de()

◆ jacob_dexpeD_de()

◆ jacob_dlogv_dv()

◆ log() [1/2]

◆ log() [2/2]

Variable Documentation

◆ DOFs [1/2]

◆ DOFs [2/2]

◆ MANIFOLD_DIM [1/2]

◆ MANIFOLD_DIM [2/2]