are obtained. In this case the values of S and C are computed and the angle 9 is then given by

 8 = atan2(S, C).



 Exchanginq Base and End-Effector Frames


 The inverse kinematics problem consists of finding joint variables that realize a given relationship between two frames, the base frame F0 and the end-effector frame FnThe roles of these two frames are in fact interchangeable as we illustrate in Figure 4.2. This means that the problem can be viewed as finding the joint variables necessary for the robot to achieve the base frame as viewed from the endeffector frame. This problem reversal requires that the DHparameters be rearranged and intermediate frames be reassigned as illustrated in Figure 4.1 but it can be useful in many ways. For example, several computationally

efficient inverse kinematic techniques have been developed for robots with the last three joint axes intersecting at a common point (Featherstone 1983, Hollerbach and sahar 1983, Paul and Zhang 1986, Low and Dubey 1986). The same techniques can be used for a robot whose first three axes intersect by reversing the roles of end--effector and base frame. In the next chapters, we will use this problem reversal technique to avoid repetitious developments.

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