Fuzzy Cauchy problem for fuzzy estimation — Application to unicycle-type mobile robots

Abstract

This paper is an attempt to provide a solution to the problem of state estimation of dynamical systems, when no measured output is available, through the example of a unicycle-type mobile robot. The latter is only equipped with encoders on each wheel which provide information only about the system's inputs (see equations (1) and (2)). As is the case in most real situations, in this work, it is also assumed that the initial states are only “approximately” known and we represent them by fuzzy numbers. Fuzzy initial states along with system dynamic equations provide us particular fuzzy differential equations (FDEs), referred as fuzzy Cauchy problem in the literature [7], [5], [4]. The question to which we want to provide an answer is: Knowing the “fuzzy states” of the system at t = 0, are we able to estimate them at any time t >; 0 or, equivalently, starting from the fuzzy initial states, are we able to build the solution of the fuzzy Cauchy problem at any time t >; 0? To solve this problem we consider two approaches: First, solving the robot's crisp (non fuzzy) differential equations with fuzzy initial conditions. In the second, the differential equations themselves are considered as fuzzy and, after discretization, the solution is built step by step using mainly fuzzy arithmetics. Moreover, considering encoders' inherent imprecision, we also assume that the input signals are only approximately known and represent them by fuzzy maps (functions having fuzzy numbers as instantaneous values). We however always make the assumption that the robot wheels do not slip.