ROBUST POSITIONING OF A MANIPULATOR WITH WEAK INTEGRAL FEEDBACK

The problem of the transfer of the links of a robot manipulator from an arbitrary position to the desired position is considered. This problem can be solved in a number of ways. For example, one can use the proportional differential (PD) controller with a constant additional component compensating for the static error in the desired position due to the gravitational forces [1, 2]. If the gravitational forces acting on the robot are known apart from constant parameters and the model of the manipulator does not take into account the flexibility of its members (rigid model), then the controller can be completed with a component adapting to unknown parameters linearly occurring in the equations of motion [3]. For systems of cyclic operation, learning gravity compensation can be performed [4-6]. However, learning is possible only if the hypothesis of reproducibility [7] is valid. Moreover, the system must possess the property that incorrect compensation of the gravitational forces leads to a deviation of the equilibrium of the system from the desired position but the system remains stable. To avoid this deviation, one usually introduces the integral (1) component into the control law. This component makes the controlled system astatic [8, p. 97]. Positioning systems with the conventional PID controller were studied previously by a number of authors. Nevertheless, the results that have been obtained so far concern the local stability of the rigid model [9-11]. In [12], it is shown that the attraction domain can be extended without limit by changing the feedback gains. In the present paper, we prove the global asymptotic stability of the rigid and elastic models of the robot closed by the PID controller for sufficiently small gains of the integral component. All statements are proved on the basis of the corollary of Hoppensteadt,s theorem [13] concerning stability of singularly perturbed systems of differential equations.