SpecialEuclideanGroup{T}

The special Euclidean group $\mathrm{SE}(n) = \mathrm{SO}(n) ⋉ \mathcal T(n)$ is the Lie group consisting of the LeftSemidirectProductGroupOperation of the SpecialOrthogonalGroup and the TranslationGroup together with the GroupOperationAction{LeftGroupOperationAction}.

To be precise, the group operation is defined on $\mathrm{SO}(n) ⋉ \mathcal T(n)$ as follows:

\[(r_1, t_1) ⋅ (r_2, t_2) = (r_1∘r_2, t_1 + r_1⋅t_2),\]

where $r_1,r_2 ∈ \mathrm{SO}(n)$ and $t_1,t_2 ∈ \mathcal T(n)$

Analogously you can write this on elements of $\mathrm{SO}(n) ⋊ \mathcal T(n)$ as

\[(t_1, r_1) ⋅ (t_2, r_2) = (t_1 + r_1⋅s_2, r_1∘r_2)\]

Both these cases can be represented in a single matrix in affine form

\[g = \begin{pmatrix} r & t\\ \mathbf{0}_n^{\mathrm{T}} & 1\end{pmatrix}, \qquad r ∈ \mathrm{SO}(n), t ∈ \mathcal T(n),\]

where $\mathbf{0}_n ∈ ℝ^n$ denotes the vector containing zeros.

We refer also in general to elements on $\mathrm{SE}(n)$ as $g$ and their rotation and translation components as $r$ and $t$, respectively.

Note further that in the notation above and in matrix form the default is the ActionActsOnRight action.

Constructor

SpecialEuclideanGroup(n::Int; variant=:left, kwargs...)
SpecialOrthogonalGroup(n; kwargs...) ⋉ TranslationGroup(n; kwargs...)
TranslationGroup(n; kwargs...) ⋊ SpecialOrthogonalGroup(n; kwargs...)

Generate special Euclidean group $\mathrm{SE}(n) = \mathrm{SO}(n) ⋉ \mathcal T(n)$, where the first constructor is equivalent to the second.

All keyword arguments in kwargs... are passed on to Rotations as well.

The default representation for $\mathrm{SE}(n)$ is the affine form. Alternatively you can use the ArrayPartition from RecursiveArrayTools.jl to work on $(r,t)$. or for $\mathcal T(n) ⋊ \mathrm{SO}(n)$ using the ArrayPartitions $(t,r)$; which corresponds to setting variant=:right in the first constructor.

SpecialEuclideanMatrixPoint <: AbstractLieGroupPoint

represent a point on some AbstractLieGroup by an affine matrix.

\[\begin{pmatrix} M & v\\ \mathbf{0}_n^{\mathrm{T}} & 1\end{pmatrix} ∈ ℝ^{(n+1)×(n+1)}, \qquad M ∈ ℝ^{n×n}, v ∈ \mathcal T(n),\]

where $\mathbf{0}_n ∈ ℝ^n$ denotes the vector containing zeros.

SpecialEuclideanMatrixTangentVector <: AbstractLieAlgebraTangentVector

represent a tangent vector on some AbstractLieGroup by a matrix of the form

\[\begin{pmatrix} M & v\\ \mathbf{0}_n^{\mathrm{T}} & 0\end{pmatrix} ∈ ℝ^{(n+1)×(n+1)}, \qquad M ∈ ℝ^{n×n}, v ∈ \mathcal T(n),\]

where $\mathbf{0}_n ∈ ℝ^n$ denotes the vector containing zeros.

While this tangent vector itself is not an affine matrix itself, it can be used for the Lie algebra of the affine group

SpecialEuclideanProductPoint <: AbstractLieGroupPoint

Represent a point on a Lie group (explicitly) as a point that consists of components

SpecialEuclideanProductTangentVector <: AbstractLieAlgebraTangentVector

Represent a point on a Lie algebra (explicitly) as a tangent vector that consists of components.

exp(G::SpecialEuclidean, X)
exp!(G::SpecialEuclidean, g, X)

Compute the Lie group exponential function on the SpecialEuclideanGroup G$=\mathrm{SE}(2)$ using a TypeParameter{Tuple{2}} for dispatch.

The Lie algebra vector $X = (Y, v) ∈ \mathfrak se(2)$ consists of a rotation component $Y ∈ \mathfrak so(2)$ and a translation component $v ∈ \mathfrak t(2)$, so we can use vee on $\mathrm{SO}(2)$ to obtain the angle of rotation $α$ (or alternatively using $\sqrt{2}α = \lVert Y \rVert_{}$)

For $α ≠ 0$ define

\[U_α = \frac{\sinα}{α}I_2 + \frac{1-\cosα}{α^2}Y,\]

and $U_0 = I_2$, where $I_2$ is the identity matrix.

Then the result $g=(R,t)$ is given by

\[R = \exp_{\mathrm{SO}(2)}Y ∈ \mathrm{SO}(2), \quad \text{ and } \quad t = U_αv ∈ \mathcal T(2).\]

This result can be computed in-place of g.

exp(G::SpecialEuclidean, X)
exp!(G::SpecialEuclidean, g, X)

Compute the Lie group exponential function on the SpecialEuclideanGroup G$=\mathrm{SE}(3)$ using a TypeParameter{Tuple{3}} for dispatch.

Since $X = (Y, v) ∈ \mathfrak se(3)$ consists of a rotation component $Y ∈ \mathfrak se(3)$ and a translation component $v ∈ \mathfrak t(3)$, we can use vee on $\mathrm{SO}(3)$ computing the coefficients norm to obtain the angle of rotation $α$ (or alternatively using $\sqrt{2}α = \lVert Y \rVert_{}$).

For $α ≠ 0$ define

\[U_α = I_3 + \frac{1-\cosα}{α^2}Y + \frac{α-\sinα}{α^3}Y^2,\]

and $U_0 = I_3$, where $I_2$ is the identity matrix.

Then the result $g=(R,t)$ is given by

\[R = \exp_{\mathrm{SO}(3)}Y ∈ \mathrm{SO}(3), \quad \text{ and } \quad t = U_αv ∈ \mathcal T(3).\]

This result can be computed in-place of g.

g[G::SpecialEuclideanGroup,s]
getindex(g, G::SpecialEuclideanGroup, s)
X[𝔤,s]
getindex(g, 𝔤, s)

Access sub-parts of a SpecialEuclideanGroup G or its Lie algebra 𝔤. where s can be an index :Rotation, :Translation to access the single parts. Use : to access all submanifold components as a unified tuple.

g[G::SpecialEuclideanGroup,s]
getindex(g, G::SpecialEuclideanGroup, s)
X[𝔤,s]
getindex(g, 𝔤, s)

Access sub-parts of a SpecialEuclideanGroup G or its Lie algebra 𝔤. where s can be an index :Rotation, :Translation to access the single parts. Use : to access all submanifold components as a unified tuple.

g[G::SpecialEuclideanGroup,s]
getindex(g, G::SpecialEuclideanGroup, s)
X[𝔤,s]
getindex(g, 𝔤, s)

Access sub-parts of a SpecialEuclideanGroup G or its Lie algebra 𝔤. where s can be an index :Rotation, :Translation to access the single parts. Use : to access all submanifold components as a unified tuple.

inv(G::SpecialEuclideanGroup, g)
inv(G::SpecialEuclideanGroup, h, g)

Compute the inverse element of a $g ∈ \mathrm{SE}(n)$ from the SpecialEuclideanGroup(n).

In affine form, $g = \begin{pmatrix} r & t\\ \mathbf{0}_n^{\mathrm{T}} & 1\end{pmatrix}$, where $\mathbf{0}_n ∈ ℝ^n$ denotes the vector containing zeros.

The inverse reads

\[g^{-1} = \begin{pmatrix} r^{\mathrm{T}} & -r^{\mathrm{T}}t\\ \mathbf{0}_n^{\mathrm{T}} & 1\end{pmatrix}.\]

This computation can be done in-place of h.

log(G::SpecialEuclidean, g)
log!(G::SpecialEuclidean, X, g)

Compute the Lie group logarithm function on the SpecialEuclideanGroup G$=\mathrm{SE}(2)$, and G uses a TypeParameter{Tuple{2}} for dispatch.

Since $g=(R,t) ∈ \mathrm{SE}(2)$ consists of a rotation component $R ∈ \mathrm{SO}(2)$ and a translation component $t ∈ \mathcal T(2)$, we first compute $Y = \log_{\mathrm{SO}(2)}R$. Then we can use vee on $\mathrm{SO}(2)$ to obtain the angle of rotation $α$ (or alternatively using $\sqrt{2}α = \lVert Y \rVert_{}$)

For $α ≠ 0$ define

\[V_α = \frac{α}{2} \begin{pmatrix} \frac{\sinα}{1-\cosα} & 1\\ -1 & \frac{\sinα}{1-\cosα}\end{pmatrix}\]

and $V_0 = I_2$, where $I_2$ is the identity matrix. Note that this is the inverse of $U_α$ as given in the group exponential

Then the result $X = (Y, v) ∈ \mathfrak se(2)$ is given by $Y ∈ \mathfrak so(2)$ as computed above and

\[v = V_αr ∈ \mathcal T(2),\]

where $v$ is computed in-place without setting up $V_α$

This result can be computed in-place of g.

log(G::SpecialEuclidean, g)
log!(G::SpecialEuclidean, X, g)

Compute the Lie group logarithm function on the SpecialEuclideanGroup G$=\mathrm{SE}(3)$, where e is the Identity on $\mathrm{SE}(3)$ G uses a TypeParameter{Tuple{3}} for dispatch.

Since $g=(R,t) ∈ \mathrm{SE}(3)$ consists of a rotation component $R ∈ \mathrm{SO}(3)$ and a translation component $t ∈ \mathcal T(2)$, we first compute $Y = \log_{\mathrm{SO}(3)}R$. Then we can use vee on $\mathrm{SO}(3)$ to obtain the angle of rotation $α$ (or alternatively using $\sqrt{2}α = \lVert Y \rVert_{}$)

For $α ≠ 0$ define

\[V_α = I_3 - \frac{1}{2}Y + β Y^2, \quad\text{ where } β = \frac{1}{α^2} - \frac{1 + \cos(α)}{2α\sin(α)}\]

and $V_0 = I_3$, where $I_3$ is the identity matrix. Note that this is the inverse of $U_α$ as given in the group exponential

Then the result $X = (Y, v) ∈ \mathfrak se(3)$ is given by $Y ∈ \mathfrak so(3)$ as computed above and $v = V_α t ∈ \mathfrak t(3)$.

This result can be computed in-place of X.

inv(G::SpecialEuclideanGroup, g)
inv(G::SpecialEuclideanGroup, h, g)

Compute the inverse element of a $g ∈ \mathrm{SE}(n)$ from the SpecialEuclideanGroup(n).

In affine form, $g = \begin{pmatrix} r & t\\ \mathbf{0}_n^{\mathrm{T}} & 1\end{pmatrix}$, where $\mathbf{0}_n ∈ ℝ^n$ denotes the vector containing zeros.

The inverse reads

\[g^{-1} = \begin{pmatrix} r^{\mathrm{T}} & -r^{\mathrm{T}}t\\ \mathbf{0}_n^{\mathrm{T}} & 1\end{pmatrix}.\]

This computation can be done in-place of h.

apply(::GroupAction{LeftMultiplicationGroupAction, <:SpecialEuclideanGroup, <:Euclidean}, g, p)
apply!(::GroupAction{LeftMultiplicationGroupAction, <:SpecialEuclideanGroup, <:Euclidean}, q, g, p)

Given the Lie group SpecialEuclideanGroup and the Euclidean manifold $ℝ^n$, this action performs both the rotation and translation on a vector $p ∈ ℝ^n$, that is, for $g = (R, t) ∈ \mathrm{SE}(n)$, the action is given by

\[q = σ_g(p) = g ⋅ p = Rp + t,\]

where the name of the action, LeftMultiplicationGroupAction, indicates that the group element g acts on the left of the vector p, and directly yields a multiplication if interpreted in homogeneous coordinates.

apply(::GroupAction{LeftMultiplicationGroupAction, <:SpecialEuclideanGroup, <:Euclidean}, g, p)
apply!(::GroupAction{LeftMultiplicationGroupAction, <:SpecialEuclideanGroup, <:Euclidean}, q, g, p)

Given the Lie group SpecialEuclideanGroup and the Euclidean manifold $ℝ^n$, this action performs both the rotation and translation on a vector $p ∈ ℝ^n$, that is, for $g = (R, t) ∈ \mathrm{SE}(n)$, the action is given by

\[q = σ_g(p) = g ⋅ p = Rp + t,\]

where the name of the action, LeftMultiplicationGroupAction, indicates that the group element g acts on the left of the vector p, and directly yields a multiplication if interpreted in homogeneous coordinates.

default_left_action(::SpecialOrthogonalGroup, ::TranslationGroup)

Return the default AbstractGroupActionType for the special Euclidean group $\mathrm{SO}(n) ⋉ \mathcal T(n)$, which is the LeftMultiplicationGroupAction

default_right_action(::TranslationGroup, ::SpecialOrthogonalGroup)

Return the default AbstractGroupActionType for the special Euclidean group $\mathcal T(n) ⋊ \mathrm{SO}(n)$, which is the LeftMultiplicationGroupAction

diff_apply(::GroupAction{LeftMultiplicationGroupAction,SpecialEuclideanGroup,Euclidean}, g, p, X)
diff_apply!(::GroupAction{LeftMultiplicationGroupAction,SpecialEuclideanGroup,Euclidean}, Y, g, p, X)

Given the Lie group SpecialEuclideanGroup and the Euclidean manifold $ℝ^n$, the differential of the group action this action performs both the rotation and translation on a vector $p ∈ ℝ^n$, that is, for $g = (R, t) ∈ \mathrm{SE}(n)$, the differential is given by

\[Y = Dσ_g(p)[X] = RX,\]

where the name of the action, LeftMultiplicationGroupAction, indicates that the group element g acts on the left of the vector p, and directly yields a multiplication if interpreted in homogeneous coordinates.

diff_apply(::GroupAction{LeftMultiplicationGroupAction,SpecialEuclideanGroup,Euclidean}, g, p, X)
diff_apply!(::GroupAction{LeftMultiplicationGroupAction,SpecialEuclideanGroup,Euclidean}, Y, g, p, X)

Given the Lie group SpecialEuclideanGroup and the Euclidean manifold $ℝ^n$, the differential of the group action this action performs both the rotation and translation on a vector $p ∈ ℝ^n$, that is, for $g = (R, t) ∈ \mathrm{SE}(n)$, the differential is given by

\[Y = Dσ_g(p)[X] = RX,\]

where the name of the action, LeftMultiplicationGroupAction, indicates that the group element g acts on the left of the vector p, and directly yields a multiplication if interpreted in homogeneous coordinates.

init_constants!(G::SpecialEuclidean, g)
init_Constants!(𝔰𝔢::LieAlgebra{ℝ, SpecialEuclideanGroupOperation, SpecialEuclidean}, X)

Initialize the constant elements of g or X.

The matrix representation of $g∈\mathrm{SE}(n)$ has a last row, that contains zeros, besides the diagonal element, which is $g_{n+1,n+1} = 1$.

The matrix representation of $X∈\mathfrak se(n)$ has a last row that contains zeros.

this function sets these entries accordingly.

Per default for other representations, this function does not change entries for them.

init_constants!(G::SpecialEuclidean, g)
init_Constants!(𝔰𝔢::LieAlgebra{ℝ, SpecialEuclideanGroupOperation, SpecialEuclidean}, X)

Initialize the constant elements of g or X.

The matrix representation of $g∈\mathrm{SE}(n)$ has a last row, that contains zeros, besides the diagonal element, which is $g_{n+1,n+1} = 1$.

The matrix representation of $X∈\mathfrak se(n)$ has a last row that contains zeros.

this function sets these entries accordingly.

Per default for other representations, this function does not change entries for them.

lie_bracket(𝔰𝔢::LieAlgebra{ℝ, SpecialEuclideanGroupOperation, SpecialEuclideanGroup}, X, Y)
lie_bracket!(𝔰𝔢::LieAlgebra{ℝ, SpecialEuclideanGroupOperation, SpecialEuclideanGroup}, Z, X, Y)

Calculate the Lie bracket between elements X and Y of the Lie algebra of the SpecialEuclideanGroup. For the matrix representation, cf. SpecialEuclideanMatrixTangentVector or a <:AbstractMatrix, the formula reads

\[[X, Y] = XY-YX\]

lie_bracket(𝔰𝔢::LieAlgebra{ℝ, SpecialEuclideanGroupOperation, SpecialEuclideanGroup}, X, Y)
lie_bracket!(𝔰𝔢::LieAlgebra{ℝ, SpecialEuclideanGroupOperation, SpecialEuclideanGroup}, Z, X, Y)

Calculate the Lie bracket between elements X and Y of the Lie algebra of the SpecialEuclideanGroup. For the matrix representation, cf. SpecialEuclideanMatrixTangentVector or a <:AbstractMatrix, the formula reads

\[[X, Y] = XY-YX\]

exp(G::SpecialEuclidean, X)
exp!(G::SpecialEuclidean, g, X)

Compute the Lie group exponential function on the SpecialEuclideanGroup G$=\mathrm{SE}(3)$ using a TypeParameter{Tuple{3}} for dispatch.

Since $X = (Y, v) ∈ \mathfrak se(3)$ consists of a rotation component $Y ∈ \mathfrak se(3)$ and a translation component $v ∈ \mathfrak t(3)$, we can use vee on $\mathrm{SO}(3)$ computing the coefficients norm to obtain the angle of rotation $α$ (or alternatively using $\sqrt{2}α = \lVert Y \rVert_{}$).

For $α ≠ 0$ define

\[U_α = I_3 + \frac{1-\cosα}{α^2}Y + \frac{α-\sinα}{α^3}Y^2,\]

and $U_0 = I_3$, where $I_2$ is the identity matrix.

Then the result $g=(R,t)$ is given by

\[R = \exp_{\mathrm{SO}(3)}Y ∈ \mathrm{SO}(3), \quad \text{ and } \quad t = U_αv ∈ \mathcal T(3).\]

This result can be computed in-place of g.

exp(G::SpecialEuclidean, X)
exp!(G::SpecialEuclidean, g, X)

Compute the Lie group exponential function on the SpecialEuclideanGroup G$=\mathrm{SE}(2)$ using a TypeParameter{Tuple{2}} for dispatch.

The Lie algebra vector $X = (Y, v) ∈ \mathfrak se(2)$ consists of a rotation component $Y ∈ \mathfrak so(2)$ and a translation component $v ∈ \mathfrak t(2)$, so we can use vee on $\mathrm{SO}(2)$ to obtain the angle of rotation $α$ (or alternatively using $\sqrt{2}α = \lVert Y \rVert_{}$)

For $α ≠ 0$ define

\[U_α = \frac{\sinα}{α}I_2 + \frac{1-\cosα}{α^2}Y,\]

and $U_0 = I_2$, where $I_2$ is the identity matrix.

Then the result $g=(R,t)$ is given by

\[R = \exp_{\mathrm{SO}(2)}Y ∈ \mathrm{SO}(2), \quad \text{ and } \quad t = U_αv ∈ \mathcal T(2).\]

This result can be computed in-place of g.

log(G::SpecialEuclidean, g)
log!(G::SpecialEuclidean, X, g)

Compute the Lie group logarithm function on the SpecialEuclideanGroup G$=\mathrm{SE}(2)$, and G uses a TypeParameter{Tuple{2}} for dispatch.

Since $g=(R,t) ∈ \mathrm{SE}(2)$ consists of a rotation component $R ∈ \mathrm{SO}(2)$ and a translation component $t ∈ \mathcal T(2)$, we first compute $Y = \log_{\mathrm{SO}(2)}R$. Then we can use vee on $\mathrm{SO}(2)$ to obtain the angle of rotation $α$ (or alternatively using $\sqrt{2}α = \lVert Y \rVert_{}$)

For $α ≠ 0$ define

\[V_α = \frac{α}{2} \begin{pmatrix} \frac{\sinα}{1-\cosα} & 1\\ -1 & \frac{\sinα}{1-\cosα}\end{pmatrix}\]

and $V_0 = I_2$, where $I_2$ is the identity matrix. Note that this is the inverse of $U_α$ as given in the group exponential

Then the result $X = (Y, v) ∈ \mathfrak se(2)$ is given by $Y ∈ \mathfrak so(2)$ as computed above and

\[v = V_αr ∈ \mathcal T(2),\]

where $v$ is computed in-place without setting up $V_α$

This result can be computed in-place of g.

log(G::SpecialEuclidean, g)
log!(G::SpecialEuclidean, X, g)

Compute the Lie group logarithm function on the SpecialEuclideanGroup G$=\mathrm{SE}(3)$, where e is the Identity on $\mathrm{SE}(3)$ G uses a TypeParameter{Tuple{3}} for dispatch.

Since $g=(R,t) ∈ \mathrm{SE}(3)$ consists of a rotation component $R ∈ \mathrm{SO}(3)$ and a translation component $t ∈ \mathcal T(2)$, we first compute $Y = \log_{\mathrm{SO}(3)}R$. Then we can use vee on $\mathrm{SO}(3)$ to obtain the angle of rotation $α$ (or alternatively using $\sqrt{2}α = \lVert Y \rVert_{}$)

For $α ≠ 0$ define

\[V_α = I_3 - \frac{1}{2}Y + β Y^2, \quad\text{ where } β = \frac{1}{α^2} - \frac{1 + \cos(α)}{2α\sin(α)}\]

and $V_0 = I_3$, where $I_3$ is the identity matrix. Note that this is the inverse of $U_α$ as given in the group exponential

Then the result $X = (Y, v) ∈ \mathfrak se(3)$ is given by $Y ∈ \mathfrak so(3)$ as computed above and $v = V_α t ∈ \mathfrak t(3)$.

This result can be computed in-place of X.