So, I've been thinking about transformation matrices $_{i+1}^iT$ used in robotics engineering, their relationships to the representation of $SE(3) = \mathbb{R}^3 \rtimes SO(3)$ 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 $SE(3)$.
