Lie Group for 3D Rigid Transformation
The modeling and control of mobile robotic systems involve rigid transformations in the 3D geometry. This article addresses two Lie groups for 3D transformations with some basic properties.
The Lie algebra (see the following Figure) is the tangent space associated with a Lie group generated by differentiating the group transformations along with chosen directions at the identity transformation. The Lie algebra \(T_{\varepsilon}\mathcal{M}\) (red plane) is the tangent space to the Lie group’s manifold \(\mathcal{M}\) (here represented as a blue sphere) at the identity \(\varepsilon\). The sphere in \(\mathbb{R}^3\) is in fact not a Lie group but used as an intuitive representation.
Through the exponential map shown in the Figure, each straight path \(\mathbf{v}t\) through the origin on the Lie algebra produces a path \(\exp \left( \mathbf{v}t \right)\) around the manifold which runs along the respective geodesic. Conversely, each element of the group has an equivalent in the Lie algebra. The complicated nonlinear Lie group is nearly completely determined by the associated Lie algebra, which is a linear vector space, and its Lie bracket. This relation is so profound that almost all operations in the group, which is curved and nonlinear, have an exact equivalent in the Lie algebra, which is a linear vector space. For instance, velocities of transformations are well represented in the tangent space. The orientation of a rigid body in 3D space can be described by the special orthogonal group SO(3), and the position and orientation by the special Euclidean group SE(3).
Elements of SO(3) are represented by \(3 \times 3\) rotation matrices. Transformation composition and inversion in the group correspond to matrix multiplication and inversion. Since rotation matrices are orthogonal, transposition is equivalent to inversion. $$ \mathrm{SO}(3) = {\mathbf{R} \in \mathbb{R}^{3\times 3} | \mathbf{R}^\top \mathbf{R} = \mathbf{I}_3, \det(\mathbf{R})=1} $$
Let $ \mathbf{v} \in \mathbb{R}^3 $ denote the position of a point, the rotational transformation between {A} and {B} is expressed as $$ {^{A}}\mathbf{v} = {^{A}{B}}\mathbf{R}\, {^{B}}\mathbf{v} , \quad {^{A}(3) \enspace , $$ where the rotation matrix }}\mathbf{R} \in \mathrm{SO\({^{A}_{B}\negthinspace}\mathbf{R}\) represents the coordinate transformation from the lower index, frame {B}, to the upper index, frame {A}. SO(3) is the three-dimensional special orthogonal group, which consists of all rotations about the origin of \(\mathbb{R}^3\), defined as \(\mathrm{SO}(3) = \{\mathbf{R} \in \mathbb{R}^{3\times 3}\ | \ \mathbf{R}^\top \mathbf{R} = \mathbf{I}_3, \det(\mathbf{R})=1\}\).
The Lie algebra, so(3), is the set of $ 3 \times 3 $ skew-symmetric matrices constructed by a vector \(\mathbf{\omega} \in \mathbb{R}^3\): $$ \left[ \mathbf{\omega} \right]_{\times} = \begin{bmatrix} 0 & -\omega_3 & \omega_2\ \omega_3 & 0 & -\omega_1\ -\omega_2 & \omega_1 & 0 \end{bmatrix} \enspace . $$
SE(3) comprises arbitrary combinations of translations and rotations. The elements of SE(3) are \(4 \times 4\) homogeneous transformation matrices that represent the position and orientation of a body relative to a reference frame. $$ \mathrm{SE}(3) = \begin{bmatrix} \mathrm{SO}(3) & \mathbb{R}^3 \ \mathbf{0}_{1\times3} & 1 \ \end{bmatrix} $$
Same as in SO(3), transformation composition and inversion correspond to matrix multiplication and inversion. $$ \mathbf{T}_1, \mathbf{T}_2 \in \mathrm{SE}(3) $$ $$ \mathbf{T}_1 \cdot \mathbf{T}_2 = \begin{bmatrix} \mathbf{R}_1 & \mathbf{p}_1\ \mathbf{0} & 1\ \end{bmatrix} \cdot \begin{bmatrix} \mathbf{R}_2 & \mathbf{p}_2\ \mathbf{0} & 1\ \end{bmatrix} = \begin{bmatrix} \mathbf{R}_1 \mathbf{R}_2 & \mathbf{R}_1 \mathbf{p}_2 + \mathbf{p}_1\ \mathbf{0} & 1\ \end{bmatrix} $$ $$ \mathbf{T}_1^{-1} = \begin{bmatrix} \mathbf{R}_1^\top & -\mathbf{R}_1^\top \mathbf{p}_1\ \mathbf{0} & 1\ \end{bmatrix} $$
The group action on 3D vector \(\mathbf{v} = \begin{bmatrix} x & y & z & 1 \end{bmatrix}\) is given by $$ {^{A}}\mathbf{v} = {^{A}{B}}\mathbf{T}\, {^{B}}\mathbf{v} , \quad {^{A}(3) \enspace . $$}}\mathbf{T} \in \mathrm{SE
The Lie algebra se(3) is the set of \(4 \times 4\) matrices corresponding to differential translations and rotations: $$ \left[ \begin{array}{c|c} \left[ \mathbf{\omega} \right]{\times} & \mathbf{\upsilon}\ \hline \mathbf{0} & 0 \end{array} \right] \in \mathrm{se(3)} $$ where \(\mathbf{\upsilon}, \ \mathbf{\omega}\) are vectors in \(\mathbb{R}^3\).