Quasi-static models of robotic motion with frictional contact provide a\ncomputationally efficient framework for analysis and have been widely used for\nplanning and control of non-prehensile manipulation. In this work, we present a\nnovel quasi-static model of planar manipulation that directly maps commanded\nmanipulator velocities to object motion. While quasi-static models have\ntraditionally been unable to capture grasping and jamming behaviors, our\napproach solves this issue by explicitly modeling the limiting behavior of a\nvelocity-controlled manipulator. We retain the precise modeling of surface\ncontact pressure distributions and efficient computation of contact-rich\nbehaviors of previous methods and additionally prove existence of solutions for\nany desired manipulator motion. We derive continuous and time-stepping\nformulations, both posed as tractable Linear Complementarity Problems (LCPs).\n