The algebraic synthesis of a spatial TS chain for a prescribed acceleration task

In this paper, we formulate and solve the algebraic synthesis equations for a TS serial chain, which is a 5R chain formed so that the first two joint axes intersect at right angles, the T joint, and the last three joints have axes that intersect in a point, forming the S joint. The task for this chain is to have a prescribed acceleration for the end-effector movement. In this case, we show that there are two basic cases for the synthesis problem: (i) tasks in which one acceleration is specified; and (ii) tasks that have two specified accelerations. The polynomial systems for these two cases are shown to have 30 and 38 roots, and we present a polynomial elimination procedure that yields in 30 and 44 roots, respectively. An example is presented that positions a robot arm, in which the elbow joint has failed, in order to accomplish a given task.