Kinematics and Workspace of a Rolling Disk between Planar Manipulators

The following properties are well known for systems with rolling constraints: (a) constraints of rolling are expressed in terms of rate variables that describe the system, (b) these constraints are nonintegrable for systems having spatial motion and integrable for systems restricted to a plane. The integrability of the rolling constraint equations for planar systems has the following consequences: (a) the position kinematics equations explicitly contain the initial conditions unlike the position kinematics equations for robots without rolling constraints, (b) the presence of initial conditions in the position kinematics equations makes the joint solutions, workspace, and motion plans dependent on the initial conditions. In other words, the start conditions determine whether the robot will reach a desired goal position. This behavior is quite different from the behavior of industrial robots which have fixed workspace and kinematic solutions independent of start configuration. This paper presents studies on a rolling disk between two planar manipulators. The objectives of this paper are to: (a) present position kinematics, list upper bounds on the number of solutions, and investigate the effects of initial conditions, and (b) compute workspace analytically and study the effects of initial conditions. These studies are worthy of presentation because the work described in this paper on the effects of initial conditions on kinematics and workspace of planar rolling systems is novel. The planar rolling systems also act as an important subset of spatial rolling systems for which a variety of algorithms for motion planning have been proposed.