Impact dynamics in robotic and mechatronic systems

This paper presents an impact model of robotic manipulators subject to changing topology using the notion of linear projection operators. An oblique projection matrix is introduced in order to directly obtain: i) The acceleration of the constrained robotic system from unconstrained acceleration, ii) post-impact velocity of the robot from pre-impact velocity, and iii) impulse force during impact from pre-impact momentum. All solutions are provided in closed-form with elegant geometrical interpretations, e.g., the energy lost during contact is a quadratic function of pre-impact velocities. Moreover, the formulation seemingly works not only when the unilateral constraints remain active through a finite time interval but when changing topology occurs, i.e., unilateral constraints becomes inactive or vice versa, or even when the overall constraint jacobian becomes singular. The formulation is general enough to be applicable to robotic manipulators with chain, tree and close-loop topologies, or to handle simultaneous multiple contacts with non-identical restitution coefficients. The model is proven to be energetically consistent if a global restitution coefficient bounded between zero and one is assumed.