Estimation of stochastic environment force for master–slave robotic system

The aim of this work is to obtain the maximum likelihood estimate (MLE) of controller-gain parameters K^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{K} $$\end{document} of the slave robot to determine the stochastic environment force. This is accomplished by measuring the joint positions of master and slave for a known master torque using stochastic difference equation. Here, the environmental force is modelled as a zero-mean white Gaussian random process. Therefore, the joint probability distribution function (pdf) of the slave angle over a given time duration can be computed as a function of the parameters ‘K’. This pdf is maximized with respect to ‘K’ to obtain the MLE of controller-gain parameters. Subsequently, convergence analysis of error in the estimates is performed. Also, an expression of the Cramer–Rao lower bound (CRLB) is derived to measure accuracy of the estimation. Comparison of CRLB with variance of MLE supports that our estimates are asymptotically efficient. The estimation performance is validated analytically and through simulations carried out on a two-link master–slave robotic system.