In this paper, the problem of guidance and motion control of mobile robots is addressed and solved within the novel framework of the mixed finite-time/ \(\mathcal H_\infty \) control theory of nonlinear quadratic systems (NLQSs). Starting from a NLQS describing the dynamics of omnidirectional mobile platforms, the main tasks performed for controlling in closed loop the motion of omnidirectional robots can be conveniently formulated as a mixed finite-time/ \(\mathcal H_\infty \) control problem. A robust motion controller, which can effectively rejects disturbances deviating the robot platform from a planned path, can be designed after choosing a linear state-feedback structure for the controller. The synthesis problem is solved through some sufficient conditions contemplating both norm-bounded disturbances and sets constraining initial and terminal conditions, together with a finite-time bound on the output transient. Therefore, for all the allowable uncertainties, in presence of nonzero initial conditions and exogenous disturbance inputs which are possible within an unstructured environment, the motion control tasks can be accomplished through optimal \(\mathcal H_\infty \) performance by simultaneously guaranteeing that the NLQS, which governs in closed loop the robot platform, is finite-time bounded. Finally, the applicability and control performance of the design approach have been evaluated through numerical simulations.
