Improved screw theory using second order terms

z The local displacement of an object is very useful for deciding grasp stability, generating trajectories, recognizing assembly tasks, and so on. To calculate this displacement, the screw theory is employed. It is equivalent to the first order Taylor expansion of the displacement. The screw theory is very convenient, because the displacement is formulated as simultaneous linear inequalities, and a powerful tool to calculate such inequalities, the theory of the polyhedral convex cones, has already been established. However, truncation errors introduced by first order approximations sometimes cause mistaken results. In this paper, we improve the screw theory by using 2nd order terms and verify the validity of the result. 1 I n t r o d u c t i o n The local displacement of an object is very useful for deciding grasp stability[I, 2], generating trajectories[3, 4], recognizing assembly tasks[5, 6], and so on. The screw theory or tools with equivalent capabilities are employed for calculating the displacement. They are the first order Taylor expansion of the displacement. Therefore, the displacement is being formulated as simultaneous linear inequalities. That is a good characteristic because a powerful tool to calculate such inequalities, the theory of the polyhedral convex cones[7], has already been established. However, truncation errors introduced by first order approximations sometimes cause mistaken results. For example, consider the case shown in Figure 1. We can easily see that the contact relation cannot be maintained if the white object rotates about the yaxis, which is perpendicular to this paper. However, we cannot recognize this with the screw theory. To overcome this problem, this paper considers the use of high order terms, especially second order terms. Rimon and Burdick proposed an index for deciding grasp stability that employs second order terms[I]. However, the index can be employed only for this purpose. Research to employ high order terms does not, to our knowledge, exist. Therefore we propose Iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii Figure 1: Case causing mistaken result