A Topologist's View of Kinematic Maps and Manipulation Complexity

In this paper we combine a survey of the most important topological properties of kinematic maps that appear in robotics, with the exposition of some basic results regarding the topological complexity of a map. In particular, we discuss mechanical devices that consist of rigid parts connected by joints and show how the geometry of the joints determines the forward kinematic map that relates the configuration of joints with the pose of the end-effector of the device. We explain how to compute the dimension of the joint space and describe topological obstructions for a kinematic map to be a fibration or to admit a continuous section. In the second part of the paper we define the complexity of a continuous map and show how the concept can be viewed as a measure of the difficulty to find a robust manipulation plan for a given mechanical device. We also derive some basic estimates for the complexity and relate it to the degree of instability of a manipulation plan.