Dynamic Simulation Of Nonlinear Hybrid Rigid-Object Systems With Intermittent Non-Holonomic Constraints: In Search Of Stable Passive Hopping

Abstract

A methodology is outlined for derivation of equations of motion and development of arbitrarily accurate numerical simulations of nonlinear hybrid rigid-object systems with intermittent non-holonomic constraints. Some recently discovered high-order explicit Runge-Kutta methods are studied; a 10th order method is validated as particularly effective for integrating initial value problems typically encountered in this work. A hybrid simulation code system is developed and used to simulate simple well known mechanical systems, verifying the validity of the methodology and the accuracy of the code. One novel simulation technique demonstrated is a zero-parameter infinite sliding friction model that makes intermittent contact simulation physically self-consistent without introducing arbitrary parameters. Several simple robotics systems are simulated; a simulation of the simplest walker is compared to published results, further increasing confidence in the simulation system code. The simulation system is used to study an unpowered, uncontrolled, passive hopping device that hops down a flat inclined 5 meter ramp but cannot stand up. The hopping device seems to have a stable limit cycle since it can consistently be launched by hand and then hop un-assisted 100 consecutive times down a ramp. However, numerical simulations insist that the passive hopper actually has an unstable limit cycle, predicting exponential growth of the inevitable perturbations from hand launching will cause it to fall down quickly. An optimization-guided search for stable period-1 motions fails when applied to the set of feasible models corresponding to the physical hopping device. The apparent paradox is resolved by simulating a controller that emulates hand launching technique. The simulations convincingly show the launch technique attracts the state to a neighborhood sufficiently close to the unstable limit cycle such that when it is released, it consistently traverses a flat inclined ramp without falling. By-products of the search for stable period-1 motions are models that are substantially different than the hopping device but do have stable period-1 limit cycles; these are the first known stable and physically realizable 2D passive locomotion models that do not have a statically stable configuration.