| Controlled robotic manipulator systems, normally operating in an acceptable manner, may become unstable when time delays exist in the feedback loop. Such instability can cause oscillations, and thus, deteriorates performance. The main contribution of this dissertation to existing robotics theory is a new performance analysis of robotic manipulator systems with small time delays in the feedback loop. With this new method, limits of stable system operation, that is, robustness, and expected system error bounds as a function of time delay and system parameters, may be predicted. The analysis technique is based upon nonlinear control theory concepts, such as Lyapunov's direct method, and takes advantage of norms to establish operational bounds. Additionally, it approximates robotic manipulator time-delay functions by a Taylor series expansion.;To arrive at our approach, we review robotic theory and the basics of time-delay systems. We evaluate previous research on time-delay systems analysis in general and robotic controls in particular. We find that most methods are based upon linear systems analysis or assume open-loop stable systems. However, neither premise applies to robotic manipulators with time delays. Consequently, we choose Lyapunov's direct method for stability analysis as it is well-suited to nonlinear systems. Approximation of time-delay functions by Taylor series expansion allows us to group all effects of the time delay system functions in a disturbance term, assuming we observe appropriate bounds. By applying a suitable Lyapunov function to the robotic manipulator system we can determine a region of instability, which is uniformly bounded. Outside this region, the system is asymptotically stable. Graphic display of the analysis results allows us to inspect this region and conservatively estimate error bounds for the manipulator joint variables. Thus, we can insure satisfactory operation of the manipulator system by adjusting controller gains and trajectory commands. Predictably, system response is slowed as time delay increases. We demonstrate the validity of the analysis approach by numerical examples of a two-link manipulator using two types of controllers, a computed-torque controller and a robust computed torque-like controller. Time responses of the examples, run on computer generated models, support the analysis results and the predicted performance. |
