Locomotion via Impact Switching between Decoupling Vector Fields

This paper investigates motion planning for underactuated systems with impacts, as in legged robots. Some such systems admit decoupling vector fields between impacts, and the system can be thought of as a kinematic system as it moves along the integral curves of these decoupling vector fields. This reduces motion planning to choosing a sequence of decoupling vector fields and impact transition times between these vector fields. Each transition must satisfy the condition that the image of the pre-impact state (through the impact mapping) leads to a post-impact state aligned with the new decoupling vector field. We derive this condition, extending previous work where transitions were only allowed at zero velocity. Our approach combines the benefits of computationally efficient kinematic planning with potentially faster execution times due to the fact that the system does not have to be slowed to zero velocity at the switches. Switches are restricted to a lower-dimensional configuration surface, however. We have applied this approach to motion planning for a planar two-link robot which locomotes by using a single revolute actuator at the joint between the links and by alternately clamping one of two possible pivot points to the ground