So, I've been thinking about transformation matrices ii+1T used in robotics engineering, their relationships to the representation of SE(3)=R3⋊ as _{i+1}^iT = \begin{bmatrix} \begin{array}{c|c} U & \mathbf{\vec{v}} \\ \hline \mathbf{\vec{0}} & 1 \end{array}\end{bmatrix}, how one would multiply _{3}^0T = _{1}^0T _{2}^1T _{3}^2T, for instance, and how this relates to the underlying manifold topology.
So, if the first joint attached to the is a revolute joint (rotational along one axis only), the second one is a prismatic joint (translational along one axis only) and the third one is a again a revolute joint, one would have for the "first" manifold Q_1 = S^1 \subseteq SE(3), for the "second" manifold Q_2 = [0,l] \subseteq SE(3), and for the "third" manifold Q_3 = S^1 \subseteq SE(3). One would then have for the "composition" manifolds Q_1^1, Q_{12}^2 = Q_1 \times Q_2, and then for the overall configuration manifold Q^3 = Q_{123}^3 = Q_1 \times Q_2 \times Q_3. Then one would have for any coordinate patches \phi_1: I_1 \to Q_1, \phi_2: I_2 \to Q_2, and \phi_3: I_3 \to Q_3, an "overall" coordinate patch \phi: I_1 \times I_2 \times I_3 \to Q_1 \hookrightarrow Q_1 \times Q_2 \hookrightarrow Q_1 \times Q_2 \times Q_3, and the transformation matrices "come" from the initial coordinate patch of I_1 about Q_1, then the composition of the inclusion maps \iota_1: Q_1 \hookrightarrow Q_1 \times Q_2 and \iota_2: Q_1 \times Q_2 \hookrightarrow Q_1 \times Q_2 \times Q_3 from each partial product into the larger partial product, together with the representation of the embeddings of the Q_i's in SE(3).
