Statistical analysis of tracking and parametric estimation errors in a 2-link robot based on Lyapunov function

Abstract

A convergence analysis of the angular position and velocity tracking errors and parametric estimation errors for a 2-link robot dynamics in the absence of noise exist in the literature (Slotine and Li in Applied nonlinear control. Prentice- Hall Inc, Englewood Cliffs, NJ, 1991). In this paper, we consider the effect of noise on the robot dynamics as well as on the parameter update equation. Noise in the robot dynamics arises due to errors in reading \(q,\dot{q}\), i.e., the angular positions and velocities of the robot link that are used for calculating the errors e, \(\dot{e}\) used to generate the feedback torque via the PD controller. Noise is chosen as Gaussian because it is produced as the cumulative effect of several small independent random effects like electrons moving in the motor wire and dust friction, and we know from the central limit theorem that the sum of a large number of small independent effects has a Gaussian distribution. Noise in the parametric update equation arises due to finite register effects in the digital computer. Both of these noises are modeled as white Gaussian, and the robot and parametric update dynamics have thus been modeled as nonlinear coupled stochastic differential equations. Using linearization of this coupled SDE system around the non-random components of the trajectory estimation error and parametric estimation errors, we first derive stochastic integral expressions for the random components in these errors. We then evaluate the average rate of change of the Lyapunov energy function using Ito calculus and obtain conditions on the noise amplitude so that the limiting value of this average rate of change is negative, guaranteeing asymptotic average stability of the system and parametric estimation algorithm. One of the novelties of this work is the new algorithm for parameter estimation which enables us to improve on the algorithm convergence rate as compared to previous work (Slotine and Li 1991). Finally, our stochastic algorithm is verified by performing hardware experiments with the actual PHANToM Omni robot.