By D.Sc., Ph. D. Miomir Vukobratović, Ph. D. Nenad Kirćanski (auth.)

This is the fourth booklet from the sequence "Scientific basics of Ro botics". the 1st volumes have demonstrated abackqround for learning the dynamics and keep an eye on of robots. whereas the 1st publication used to be particular ly dedicated to the dynamics of lively spatial mechanisms, the second one handled the issues of the dynamic regulate of manipulation robots. not like the 1st books, the place recursive computer-aided me thods for environment robotic dynamic equations the place defined, this mono graph offers a brand new method of the formation of robotic dynamics. The objective is to accomplish the real-time version computation utilizing updated mi crocomputers. The offered notion will be known as a numeric-symbolic, or analytic, method of robotic modelling. it is going to be proven that the iteration of analytical robotic version can give new first-class possibili ties relating real-time functions. it really is of crucial value in synthesizing the algorithms for nonadaptive and adaptive keep an eye on of manipulation robots. If may be mentioned that the excessive computational potency has been completed by way of off-line computer-aided coaching of robotic equations. The parameters of a distinct robotic has to be given upfront. This, af ter every one major version in robotic constitution (geometrical and dy namical parameters) ,we needs to repeat the off-line degree. therefore is why the numerical techniques will consistently have their position in learning the dy namic houses of robot platforms. This monograph is geared up in five chapters.

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Cl2A~-1 ClA~-l ClA~-l .. j-l ·2 + _J_ J + 2W. 1 - 1 - qj + W. 1 W. lA. W. 1 qj 2 J- J JJJClqj Clqj qj Clqj W. J These recursive relations reduce the number of multiplications/additions required to compute the driving forces to the dependence on n 2 (for example, the number of additions equals' 82n 2 + 5Un - 384) • The reduced number of operations as compared with Uicker-Kahn's method evidently results from the fact that the inertial system matrix is not explicitly calculated, so it is not required to compute the partial derivatives of ClWj/ClqkClq2 type.

5) i...... +0 Lb .. 0 ing recursive relations. Let us note that coefficients b .. and b. are ~J ~ obtained from Euler's dynamic equations and that they emloy only the principal moments of inertia and no inertia tensor. ] (i ,k=1 , ... 6) a vector with n elements: h. ' ~L. r .. ~ ~ j=1 F where -+ -+ -+ ( b·k+r .. 7) is the gravity vector of the j-th link. The preceding rela- tions are written for revolute joints, but can easily be modified for sliding ones as well [22, 411. 29 The number of numerical operations needed for computing the matrices H(q) and h(q, 23 q) is n 3 + 28n2 + --2525 n i3 n 3 + 20n 2 + --3530 n where n M denotes the number of multiplications, and n A the number of additions.

O "T Accelerations Wi are calculated by forward recursion (from i as in the recur sion Waters' =1 to n) , algorithm. Di and c i are calculated by backward (from i = n to i = 1). Thus, one obtains that the number of 23 multiplications is n M = 830n - 592, and the number of additions n A = 675n- 464. Hollerbach has also recognized that it is possible to reduce the number of operations further, by using 3x3 - instead of 4x4 - matrices. As we have seen, 4x4 - matrices simultaneously describe rotation and translation thus introducing considerable computational redundancy, since rotation may be described by a 3x3 - matrix and translation by a position vector.