| Subgroup | Definition |
|---|---|
| SO(3), group of rotations in 3D (a spherical joint) | Set of all proper orthogonal 3 × 3-rotational matrices |
| SE(2), special Euclidean group in 2D (all planar motions) | Set of all 3 × 3-matrices: 
|
| SO(2), group of rotations in 2D subgroup of SE(2)—group (a revolute joint) | Set of all proper orthogonal 2 × 2-rotational matrices included in SE(2)- group |
 group of translations in 3D (all spatial displacements) |
Euclidean 3D vector space |
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