A Class of Quasi-Natural Potentials and Hyper-Stable PID Servo-Loops for Nonlinear Robotic Systems

Abstract

This paper introduces a quasi-natural potential in robot dynamics, which induces a type of sinusoidal position (joint-angle) feedback with saturation in servo-loops. By means of SP-D (Saturated Proportional and Differential) feedback, not only the robot dynamics incorporated with these servo-loops but also the residual error dynamics between the desired and actual joint trajectories satisfy the passivity with respect to the residual input torque and a linear sum of residual velocity and saturated residual position outputs. It is shown by taking advantage of the hyper-stability theory that in the case of linear sum in addition to the SP-D feedback gives rise to the globally asymptotic stability. It is further shown that in the case of trajectory tracking the passivity of residual robot dynamics with SP-D feedback loops plays a vital role and eventually leads to the adaptability in model-based adaptive control and the learnability in iterative learning.

Learnability and Adaptability from the Viewpoint of Passivity Analysis

“…As is pointed out by the authors 14-17 , such a quadratic potential of difference joint angle vector ∆q is not adequate for establishment of the passivity of a residual error robot dynamics. Instead of it, an alternative potential function S i (θ) called a "quasi-natural potential" is introduced by the authors [14][15][16][17] . This potential S i (θ) has a shape depicted in Figure 4 and in general satisfies the following properties:…”

“…This paper innovates on this passivity argument by renewing basic robot servo-loops on the basis of introduction of a quasi-natural potential [13][14][15][16][17] . This potential induces an SP-D (Saturated Proportional and Differential) feedback loop that enables the robot dynamics to be passive with respect to the residual torque input and the residual output ∆y = ∆ + αs(∆q), where α > 0, s(∆q) = (s q 1 (∆q 1 ), …, s n (∆q n )) T , and s i (∆q i ) is a saturated function of ∆q i .…”

Human-Simulating Intelligent PID Control

“…For example, a saturated-P, and differential feedback plus a PI controller driven by a bounded nonlinear function of position errors [4], a linear PD plus an integral action of a nonlinear function of position errors [5], and a linear PD plus a double integral action driven by the positions error and the filtered position [6] are presented recently. An unconquerable drawback of PID-like controller above is that its tuning procedure is very complex, tedious and difficult to obtain the satisfied transient control performance because these PID-like controllers often produce surge and big overshoot, even may lead to instability.…”

Robustness of PID-controlled manipulators with respect to external disturbances

“…It may be argued that modifications of PID control to achieve global asymptotic stability, such as introducing nonlinear terms (cf. [10]) or making the integrator time varying -cf. [11], may yield better performance; our contribution is not to propose a new robust controller (probably difficult to implement) for robot manipulators but to analyze the robustness of the classical linear time-invariant PID control.…”