Explore the homological structure of robotic arm motion spaces through this 52-minute mathematical lecture that applies topological methods to robotics. Discover how to compute homology groups for robotic arms—linkages with path-like underlying graphs—when their end effectors are constrained to move along lines or smooth curves. Learn how this topological motion planning problem transforms into a combinatorial one using Morse theory and building upon Walker and Farber's foundational work on polygonal linkages. Gain insights into the intersection of algebraic topology and robotics, understanding how mathematical abstractions provide concrete solutions to motion planning challenges in constrained environments.