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We prove that quasi-static pushing with a sticking contact and ellipsoid approximation of the limit surface is differential flat. Both graphical and algebraic derivations are given. A major conclusion is that the pusher–slider system is reducible to the Dubins car problem where the sticking contact constraints translate to bounded curvature. Planning is as easy as computing a Dubins curve with the additional benefit of time-optimality. For trajectory stabilization, we design closed-loop control using dynamic feedback linearization or open-loop control using two contact points as a form of mechanical feedback. We conduct robotic experiments using objects with different pressure distributions, shape, and contact materials placed at different initial poses that require difficult switching action maneuvers to the goal pose. The average error is within 1.67 mm in translation and 0.5° in orientation over 60 experimental trials. We also show an example of pushing among obstacles using a RRT planner with exact steering.