  In this paper, we develop the theory of constrained motion spaces of robotic
arms. We compute their homology groups in two cases: when the constraint is a
horizontal line and when it is a smooth curve whose motion space is a smooth
manifold. We show the computation of homology amounts to counting the collinear
configurations, reducing a topological problem to a combinatorial problem. Our
results rely on Morse theory, along with Walker's and Farber's work on
polygonal linkages.
