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,q, i.e., the angular positions and velocities of the robot link that are used for calculating the errors e,ė 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.

Classical robots perturbed by Lévy processes: analysis and Lévy disturbance rejection methods

“…As in the existing literature [1] we define a quadratic Lyapunov function of e(t) and δθ(t) which decreases monotonically to zero as t → ∞ in the noiseless case provided that we assume an appropriate parametric update equation. Like the robot dynamics, we also assume that the parametric update equation also contains a Lévy noise component and then obtain a expression for the average rate of change in the Lyapunov function using Ito's formula for Poisson process.…”

“…X (.) is no longer a Markov process but as in [1], we can transform this SDE in to an infinitedimensional Markovian SDE by introducing a vector…”

“…Apart from motor noise, disturbance can come from hand tremor or jerks if we are dealing with only a master robot or it can come from jerky environment if we are dealing only with slave robots. Modeling the disturbance as white Gaussian noise processes and estimating the parameters of the robot based on such a Gaussian statistical model have been discussed in the existing literature [1,2].…”

“…Some work [1] also exists in proving stability theorems for robots with Gaussian noise using stochastic Lyapunov theory. This analysis is based on calculating the rate of change on the ensemble averaged Lyapunov energy function using Ito's formula for Brownian motion.…”

Estimation of stochastic environment force for master–slave robotic system

“…Therefore, in this work, environmental force has been assumed to follow the structure of Gaussian statistics. The whiteness of the environmental noise is based on the fact that usually the environmental kicks appear randomly and independent of each other, similar to a molecule that kicks a pollen particle in a fluid, leading to the pollen particle executing Brownian motion [19][20][21][22].…”