Learning dynamical systems with bifurcations

Trajectory planning through dynamical systems (DS) provides robust control for robots and has found numerous applications from locomotion to manipulation. However, to date, DS for controlling rhythmic patterns are distinct from DS used to control point to point motion and current approaches switch at run time across these to enable multiple behaviors. This switching can be brittle and subject to instabilities. We present an approach to embed cyclic and point to point dynamics in a single DS. We offer a method to learn the parameters of complete DS through a two-step optimization. By exploiting Hopf bifurcations, we can explicitly and smoothly transit across periodic and non-periodic phases, linear and nonlinear limit cycles, and non-periodic phases, in addition to changing the equilibrium's location and the limit cycle's amplitude. We use diffeomorphism and learn a mapping to modify the learned limit cycle to generate nonlinear limit cycles. The approach is validated with a real 7 DOF KUKA LWR 4+ manipulator to control wiping and with a humanoid robot in simulation.