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Using results on the topology of moduli space of polygons [Jaggi, 92;Kapovich and Millson, 94], it can be shown that for a planar robot arm with $n$segments there are some values of the base-length, $z$, at which theconfiguration space of the constrained arm (arm with its end effector fixed)has two disconnected components, while at other values the constrainedconfiguration space has one connected component. We first review some of theseknown results.  Then the main design problem addressed in this paper is the construction ofpairs of continuous inverse kinematics for arbitrary robot arms, with theproperty that the two inverse kinematics agree when the constrainedconfiguration space has a single connected component, but they give distinctconfigurations (one in each connected component) when the configuration spaceof the constrained arm has two components. This design is made possible by afundamental theoretical contribution in this paper -- a classification ofconfiguration spaces of robot arms such that the type of path that the system(robot arm) takes through certain critical values of the forward kinematicsfunction is completely determined by the class to which the configuration spaceof the arm belongs. This classification result makes the aforesaid designproblem tractable, making it sufficient to design a pair of inverse kinematicsfor each class of configuration spaces (three of them in total).  We discuss the motivation for this work, which comes from a more extensiveproblem of motion planning for the end effector of a robot arm requiring us tocontinuously sample one configuration from each connected component of theconstrained configuration spaces.  We demonstrate the low complexity of the presented algorithm through aJavascript + HTML5 based implementation available athttp://hans.math.upenn.edu/~subhrabh/nowiki/robot_arm_JS-HTML5/arm.html

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