Some topological problems in robotics

This paper is intended to be an exposition for mathematicians on some problems in engineering, and, before introducing these problems, it seems worthwhile to first arm the reader with at least one concrete example of the objects under discussion. Figure l(a) shows a picture of a type of robot commonly used in many robotics applications, for example, in the automobile industry. The robot consists of a three-link arm (with joints 01, 02, and 03), and a wrist attached to the end of the arm. The wrist also has three joints, 04 , 05 , and 06 , which can be used to orient a tool mounted on the end of the arm. Because the space of orientations, SO(3), is three-dimensional, a minimum of three joints is needed to orient the tool in an arbitrary fashion. Such robots are typically used for jobs such as parts assembly, welding, or spray painting (see Figure l(b)). The robot can be programmed to do such work by specifying a time-dependent trajectory it should follow in the so-called operational space, which, in the case of parts assembly, might be the space R 3 x SO(3). The R 3 component of the t ra jec tory speci f ies the pos i t ion of the tool mounted on the end of the robot, and the SO(3) component specifies the orientation of the tool. Although the task of a robot is specified by giving a trajectory in the operational space, this trajectory must be followed by giving commands to the joint motors to execute an associated trajectory in the joint space. For the robot pictured in Figure l(a), all joints are rotational, so that this joint space can be viewed as a sixdimensional toms, T 6. There is clearly a function As is the case for many conventional robots, this one is equipped with the minimum number of joints (six) needed to be able to move around freely in its six-dimensional operational space. This economy in engineering design seems natural at first, but it also has disadvantages. If the robot works in a crowded environment, it might be useful to have extra, redundant joints which enable the arm to move around to avoid obstacles while holding the endpoint of the arm fixed. These redundant joints also offer another potential advantage, which is not so obvious at first glance. They can be used to avoid singular joint configurations, i.e., joint positions at which the Jacobian df of the forward kinematics map does not have maximal rank. Singular configurations cause the mecha-