A Classification of Configuration Spaces of Planar Robot Arms with Application to a Continuous Inverse Kinematics Problem

Using results on the topology of moduli space of polygons [Jag92, KM94], it can be shown that for a planar robot arm with n segments there are some values of the base-length, z (i.e., length of line joining the base of the arm with its end-effector), at which the configuration space of the constrained arm (arm with its end effector fixed) has two disconnected components, while there are other values at which the constrained configuration space has one connected component. We first review some of these known results relating the value of z with the connectivity of the constrained configuration space with end-effector fixed. Then the main design problem addressed in this paper is the construction of pairs of continuous inverse kinematics for arbitrary robot arms, with the property that the two inverse kinematics agree (i.e. return the same configuration) when the constrained configuration space has a single connected component, but they give distinct configurations (one in each connected component) when the configuration space of the constrained arm has two components. This design is made possible by a fundamental theoretical contribution in this paper – a classification of configuration spaces of robot arms such that the type of path that the system (robot arm) takes through certain critical values of the forward kinematics function is completely determined by the class to which the configuration space of the arm belongs. This classification result makes the aforesaid design proble