So, I've been thinking about transformation matrices ii+1T used in robotics engineering, their relationships to the representation of SE(3)=R3⋊SO(3) as ii+1T=[U→v→01], how one would multiply 03T=01T12T23T, 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 Q1=S1⊆SE(3), for the "second" manifold Q2=[0,l]⊆SE(3), and for the "third" manifold Q3=S1⊆SE(3). One would then have for the "composition" manifolds Q11, Q212=Q1×Q2, and then for the overall configuration manifold Q3=Q3123=Q1×Q2×Q3. Then one would have for any coordinate patches ϕ1:I1→Q1, ϕ2:I2→Q2, and ϕ3:I3→Q3, an "overall" coordinate patch ϕ:I1×I2×I3→Q1↪Q1×Q2↪Q1×Q2×Q3, and the transformation matrices "come" from the initial coordinate patch of I1 about Q1, then the composition of the inclusion maps ι1:Q1↪Q1×Q2 and ι2:Q1×Q2↪Q1×Q2×Q3 from each partial product into the larger partial product, together with the representation of the embeddings of the Qi's in SE(3).
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