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Abstract-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.

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.

To my wife Yen-Chen and son Hsu-Chen who constantly filled me with love and joy. ACKNOWLEDGMENTSI would like to express my gratitude to my advisor, mentor, and friend, Dr.Warren E. Dixon for introducing me with the interesting field of Lyapunov-based control. As an advisor, he provided the necessary guidance and allowed me to try some stupid ideas during my research. As a mentor, he helped me understand the high pressure of working in a professional environment and was willing to give me time to learn and adjust. I feel fortunate in getting the opportunity to work with him.

A sufficient condition to solve an optimal control problem is to solve the Hamilton-Jacobi-Bellman (HJB) equation. However, finding a value function that satisfies the HJB equation for a nonlinear systems is challenging. Previous efforts have utilized feedback linearization methods which assume exact model knowledge, or have developed neural network (NN) approximations of the HJB value function. The current effort builds on our previous efforts to illustrate how a NN can be combined with a recent robust feedback method to asymptotically minimize a given quadratic performance index as the generalized coordinates of a nonlinear Euler-Lagrange system asymptotically track a desired time-varying trajectory despite general uncertainty in the dynamics. A Lyapunov analysis is provided to examine the stability of the developed optimal controller.