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In this paper, the problem of tracking a desired motion trajectory for an underwater vehicle-manipulator system without using direct velocity feedback is addressed. For this purpose, an observer is adopted to provide estimation of the system's velocity needed by a tracking control law. The combined controller-observer scheme is designed so as to achieve exponential convergence to zero of both motion tracking and estimation errors. In order to avoid representation singularities of the orientation, unit quaternions are used to express the vehicle attitude. Implementation issues are also considered and simplified control laws are suggested, aimed at suitably trading off tracking performance against reduced computational load. Simulation case studies are carried out to show the effectiveness of the proposed controller-observer algorithm. The obtained performance is compared to that achieved with a control scheme in which the velocity is reconstructed via numerical differentiation of position measurements. The results confirm that the chattering on the control commands is significantly reduced when the controller-observer strategy is adopted in lieu of raw numerical differentiation; this leads to lower energy consumption at the actuators and increases their lifetime.