Adaptive Lyapunov-based control of a robot and mass-spring system undergoing an impact collision

The control of dynamic systems that undergo an impact collision is both theoretically challenging and of practical importance. An appeal of studying systems that undergo an impact is that short-duration effects such as high stresses, rapid dissipation of energy, and fast acceleration and deceleration may be achieved from low-energy sources. However, colliding systems present a difficult control challenge because the equations of motion are different when the system suddenly transitions from a noncontact state to a contact state. In this paper, an adaptive nonlinear controller is designed to regulate the states of two dynamic systems that collide. The academic example of a planar robot colliding with an unactuated mass-spring system is used to represent a broader class of such systems. The control objective is defined as the desire to command a robot to collide with an unactuated system and regulate the mass to a desired compressed state while compensating for the unknown constant system parameters. Lyapunov-based methods are used to develop a continuous adaptive controller that yields asymptotic regulation of the mass and robot links. It is interesting to note that one controller is responsible for achieving the control objective when the robot is in free motion (i.e., decoupled from the mass-spring system), when the systems collide, and when the system dynamics are coupled.