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The position control problem of wheeled mobile robots (WMR's) has been a heavily researched area due to both the challenging theoretical nature of the problem (i.e., an underactuated nonlinear system under nonholonomic constraints) and its practical importance. Rather than construct a control problem where the mobile robot is required to track an a priori known desired trajectory, this paper formulates a velocity field tracking control problem. Specifically, a velocity field is developed for the constrained WMR trajectory, and a differentiable controller is formulated to prove global asymptotic velocity field tracking. Motivated by the desire to improve the robustness of the system, the developed differentiable kinematic controller is embedded inside of an adaptive controller that fosters global asymptotic tracking despite parametric uncertainty associated with the dynamic model.