From sliding–rolling loci to instantaneous kinematics: An adjoint approach

The adjoint approach has proven effective in studying the properties and distribution of coupler curves of crank-rocker linkages and the geometry of a rigid object in spatial motion. This paper extends the adjoint approach to a general surface and investigates kinematics of relative motion of two rigid objects that maintain sliding–rolling contact. We established the adjoint curve to a surface and obtained the fixed-point condition, which yielded the geometric kinematics of an arbitrary point on the moving surface. After time was taken into consideration, the velocity of the arbitrary point was obtained by two different ways. The arbitrariness of the point results in a set of overconstrained equations that give the translational and angular velocities of the moving surface. This novel kinematic formulation is expressed in terms of vectors and the geometry of the contact loci. This classical approach reveals the intrinsic kinematic properties of the moving object. We then revisited the classical example of a unit disc rolling–sliding on a plane. A second example of two general surfaces maintaining rolling–sliding contact was further added to illustrate the proposed approach.