An Impact Force Limiting Adaptive Controller for a Robotic System Undergoing a Non-Contact to Contact Transition
The problem of controlling a robot during a non-contact to contact transition has been a historically challenging problem that is practically motivated by applications that require a robotic system to interact with the environment. The control challenge is due, in part, to the impact effects that result in possible high stress, rapid dissipation of energy, and fast acceleration and deceleration. Over the last two decades, results have focused on control designs that can be applied during the non-contact to contact transition. One main theme in these results is the desire to prescribe, reduce, or control the interaction forces during or after the robot impact with the environment because large interaction forces can damage both the robot and/or the environment or lead to degraded performance or instabilities. Two main approaches have been exploited to accommodate for the non-contact to contact transition. The first approach is to exploit kinematic redundancy of the manipulator to reduce the impact force. The second mainstream approach is to exploit a discontinuous controller that switches based on the different phases of the dynamics (i.e., non-contact, robot impact transition, and in-contact coupled manipulator and environment). Typically, these discontinuous controllers consist of a velocity controller in the pre-contact phase that switches to a force controller during the in-contact phase. Motivation exists to explore alternative methods because kinematic redundancy is not always possible, and discontinuous controllers require infinite control frequency (i.e., exhibit chattering) or yield degraded stability results (i.e., uniformly ultimately bounded). In this paper, we consider a two link planar robotic arm that transitions from free motion to contact with an unactuated mass-spring system. The robot/mass-spring system collision is modeled as a differentiable impact. As in our previous efforts, the objective is to control a robot from a non-contact initial condition to a desired (in-contact) position so that the mass-spring system is regulated to a desired compressed state. The focus of our previous work was to develop a continuous exact model knowledge and adaptive controller that could achieve the objective despite the impact collision disturbance. When these results were implemented in the presence of large initial conditions, a violent impact between the robot and the mass-spring system resulted. In fact, the controller was artificially saturated (the saturation effects were not considered in the stability analysis) to reduce the impact forces so that the mass deflection would not destroy the capacitance probe. These results provide the motivation for the current control development. Specifically, the feedback elements for the controller in this paper are contained inside of hyperbolic tangent functions as a means to limit the impact forces resulting from large initial conditions as the robot transitions from non-contact to contact. Although saturating the feedback error is an intuitive solution that has been proposed in previous literature for other types of robotic systems with limited actuation, several new technical challenges arise due to the impact condition. The main challenge is that the use of saturated feedback does not allow some coupling terms to be canceled in the stability analysis, resulting in the need to develop state dependent upper bounds that reduce the stability to a semi-global result (as compared to the global results in our previous papers). The semi-global result is problematic in the current applicative context because certain control terms do not appear in the closed-loop error system during the non-contact condition, resulting in a uniformly ultimately bounded result until the robot makes contact. Hence, the result hinges on new development within the semi-global stability proof for an error system that is only uniformly ultimately bounded during the non-contact phase. This problem is exacerbated by the fact that the Lyapunov function contains radially unbounded hyperbolic functions of some states that only appear inside of saturated hyperbolic terms in the Lyapunov derivative. New control development, closed-loop error systems, and Lyapunov-based stability analysis arguments are used to conclude the result. It is interesting to note that only the position and velocity terms of the spring-mass system and the joint angles and the angular velocities terms of the planar robotic arm are required for the proposed controller (i.e., the controller does not depend on measuring the impact force and the acceleration terms). Experimental results are provided that successfully demonstrate the control objective.
A movie of the experiment is provided below.
Publications
C. Liang, S. Bhasin, K. Dupree, and W. E. Dixon, “An Impact Force Limiting Adaptive Controller for a Robotic System Undergoing a Non-Contact to Contact Transition,” IEEE Transactions on Control Systems Technology, accepted, to appear.
C. Liang, S. Bhasin, K. Dupree and W. E. Dixon, “An Impact Force Limiting Adaptive Controller for a Robotic System Undergoing a Non-Contact to Contact Transition,” Proceedings of the 2007 IEEE Conference on Decision and Controls, New Orleans, Louisiana, 2007, pp. 3555-3560.
