Coordinate transformation - A solution algorithm for one class of robots

One of the most important features of an advanced control system for articulated robots is the capability of transforming the work space coordinates, which naturally characterize any robot task, into the corresponding joint coordinates, on which control actions are developed. For each task, the coordinate transformation problem consists in calculating one trajectory in the joint space which corresponds to the end effector trajectory, usually given in the Cartesian space. While simple kinematical structures allow for closed-form solutions, there is a class of robots for which this is not true. Typical articulated robot structures have three revolute joints at the end effector; the geometric parameters of these joints actually determine the spatial configuration of the last axes of motion. The large majority of today's nonredundant structures have three intersecting axes at the end effector, and closed form solutions do exist in this case. If the axes intersect two-by-two, as in some rather common arm design, an exact solution seems not to exist. A quite different solution algorithm is established, as compared to the trigonometric approach widely adopted so far, which yields solutions in case of two-by-two intersecting axes. The convergence of the algorithm along any trajectory is proved. Effectiveness of the proposed technique can be argued by the fact that it only makes use of direct kinematics, thus resulting in a contained computational burden. A robot prototype of the kind described above is taken as a reference in order to discuss digital implementation and develop numerical examples.