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This paper develops an adaptive dynamic motion controller for position control and trajectory tracking of the omnidirectional mobile robot equipped with four independent omnidirectional wheels equally spaced at 90 degrees from one to another. Such a controller is synthesized by backstepping and will be is proven globally asymptotically stable via the Lyapunov stability theory. The controller is particularly adequate for the robot while navigating over its working environment at any speeds. The effectiveness and merit of the proposed control method is exemplified by conducting several simulations.