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We present the design of a low-cost wheeled mobile robot, and an analytical model for predicting its motion under the influence of motor torques and friction forces. Using our proposed model, we show how to analytically compute the gradient of an appropriate loss function, that measures the deviation between predicted motion trajectories and real-world trajectories, which are estimated using Apriltags and an overhead camera. These analytical gradients allow us to automatically infer the unknown friction coefficients, by minimizing the loss function using gradient descent. Motion trajectories that are predicted by the optimized model are in excellent agreement with their real-world counterparts. Experiments show that our proposed approach is computationally superior to existing black-box system identification methods and other data-driven techniques, and also requires very few real-world samples for accurate trajectory prediction. The proposed approach combines the data efficiency of analytical models based on first principles, with the flexibility of data-driven methods, which makes it appropriate for low-cost robots. Using the learned model and our gradient-based optimization approach, we show how to automatically compute motor control signals for driving the robot along pre-specified curves.