Steinkamp's Toy Can Hop 100 Times But Can't Stand Up

Abstract

We have experimented with and simulated Steinkamp's passive-dynamic hopper. This hopper cannot stand up (it is statically unstable), yet it can hop the length of a 5 m 0.079 rad sloped ramp, with n≈100 hops. Because, for an unstable periodic motion, a perturbation Δx0 grows exponentially with the number of steps (Δxn≈Δx0×λn), where λ is the system eigenvalue with largest magnitude, one expects that if λ>1 that the amplification after 100 steps, λ100, would be large enough to cause robot failure. So, the experiments seem to indicate that the largest eigenvalue magnitude of the linearized return map is less than one, and the hopper is dynamically stable. However, two independent simulations show more subtlety. Both simulations correctly predict the period of the basic motion, the kinematic details, and the existence of the experimentally observed period ∼11 solutions. However, both simulations also predict that the hopper is slightly unstable (|λ|max>1). This theoretically predicted instability superficially contradicts the experimental observation of 100 hops. Nor do the simulations suggest a stable attractor near the periodic motion. Instead, the conflict between the linearized stability analysis and the experiments seems to be resolved by the details of the launch: a simulation of the hand-holding during launch suggests that experienced launchers use the stability of the loosely held hopper to find a motion that is almost on the barely unstable limit cycle of the free device.