Are rigid robots discrete mechanical systems subject to unilateral constraints?

Robots built from rigid struts and joints that interact with hard terrain are commonly modeled as discrete mechanical systems subject to unilateral constraints. When part of the robot impacts the terrain in such models, the velocity must reset to satisfy the unilateral constraints. Many impact laws have been proposed, but few have been tested empirically. To test a variety of impact laws, we constructed an automated jumping robot testbed with a 1-leg robot constrained to move vertically. The robot’s leg is a pantograph mechanism constructed from rigid struts and rigid pin joints. If the robot were a discrete mechanical system subject to unilateral constraints, limb and body velocities would be discontinuous at touchdown. In contrast, velocities appear continuous when computed using a variety of schemes for signal differentiation from empirical motor angle or motion capture data measured at 1kHz. Nevertheless, by directly measuring touchdown to within 1msec using an electrical switch, we are able to assess the extent to which a variety of impact laws predict how velocities change from moments immediately preceding to moments immediately following touchdown, and compare these results with simulated data from the corresponding models.