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In this study, the authors propose an adaptive output-feedback controller for trajectory tracking of electrically driven non-holonomic mobile robots in the presence of parametric uncertainties. A new adaptive observer using the transformation matrices is developed to estimate the unmeasured velocities of the mobile robot. By using the transformation matrices, the designed adaptive observer can deal with quadratic velocity terms caused by the Coriolis matrix in the mobile robot dynamics as well as uncertain parameters in quadratic velocity terms. Based on the designed adaptive observer, a simple tracking controller at the actuator level is induced from the dynamic surface design methodology. Using the Lyapunov stability theory, the authors prove that all errors in a closed-loop system are uniformly ultimately bounded and converge to an adjustable neighbourhood of the origin.