Algebraic Topology and Differential Geometry

This lecture explores the applications of algebraic topology and differential geometry in understanding the intrinsic dynamics and global behavior of robots. Topics include cyclic elastic movements in biology, co-design of robot body and control, biologically inspired robot design, classical derivation of robot dynamics, geodesics on manifolds, and the differential formulation of geodesics equations. The lecture delves into the principles of least action, manifolds as geometrical tools for describing robot motion, and the concept of geodesics. It also covers the motion of a multi-body system on a manifold, periodic trajectories, and the design of locomotion based on brake orbits.