We design an model predictive control (MPC) approach for planning and control of non-holonomic mobile robots. Linearizing the system dynamics around the pre-computed reference trajectory gives a time-varying LQ MPC problem. We analytically show that by specially designing the MPC controller, the time-varying, linearized system can yield asymptotic stability around the origin in the tracking task. We further propose two obstacle avoidance methods. We show that by defining linearized constraint in velocity-space and explicitly coupling the two control inputs based on current state, our second method directly accounts for the non-holonomic property of the system and therefore alleviates infeasibility of the optimization problems. Simulation results suggest that regarding both static and dynamic obstacle avoidance, the planned trajectories by our LQ MPC approach are comparably smooth and effective as solving non-linear programming (NLP) problems, but in a more efficient way.