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The transverse function approach pursues the objective to converge tracking error to a bounded transverse function, not the classical objective of asymptotic stabilization. In this paper a new transverse function is presented and control of a path following wheeled mobile robot using transverse function is studied. The effects of the parameters, including the initial value of the transverse function, the parameters of the transverse function and the feedback control parameters, on the system performances are illustrated and investigated. Finally results of application to time-delay systems are reported.