Direct Dynamics open-chain

Although the details may be quite different, every research effort in the area of dynamic simulation faces a common task — the efficient and accurate solution of the Direct Dynamics problem. In the development of algorithms for Direct Dynamics, two basic approaches have emerged for both open- and closed-chain systems. The first utilizes the inversion of the x manipulator joint space inertia matrix to solve for the joint accelerations. More accurately, the accelerations are found via the solution of a system of linear algebraic equations, but the... 

In an attempt to circumvent the computational complexity of matrix inversion, some researchers are pursuing solutions for the joint accelerations which have a linear recursive form. The inversion of the in tia matrix is explicitly avoided. The resulting linear recursive algorithms have a reduced computational complexity which is 0(N). This is the second basic solution approach to the Direct Dynamics problem, and it has been rqjplied to serial open chains [3,7], single closed chains [22], and some more genoal multibody systems [4, 37]. It is believed that the structure of linear recursive algorithms may also facilitate their implementation on parallel computer systems. 

Given the p sent state of the manipulator, q and q, the input joint actuator traques and forces, r, and the motion of the base, the open-chain term, ijopen. is completely defined. An appropiate Direct Dynamics algorithm for < )en-chain manipulator may be used to detomine its value. With the same givoi information, the coefficient of f in the constrained term, G, is also known. The efficient computation of G was discussed in Chapter 4. 

As was true for the joint accelerations, if the present state, driving actuator tffl ques and/or forces, and motion of the base are known, the open-chain term, Xopenf is known. Its value may also be determined using an tq>pr(piate Direct Dynamics algorithm for ( n-chain manipulators. Because the joint positions are assumed known, the inverse operational space inertia matrix, A is defined. The efficient computation of A and its inverse was the primary topic of Ch ter 4. 

Note that the number of operations listed for fl and A in Table 5.2 is less than the total given for these two quantities in the 0 N) Force Propagation Method in Chapter 4. This reduction was achieved through a little insight First we note that the first recursion in the open-chain Direct Dynamics algorithm of... 

As in the single closed chain case, the open-chain terms, (qt)open and (Xik)open, are completely defined for each chain given the present state genial joint positions and rates, qt and qt, the applied graeral joint torques/forces in the free directions, n, and the motion of the supprat surface. Any appropriate open-chain Direct Dynamics algorithm may be used to calculate these terms. Because the general joint positions are known, fit and Aj are also defined. The efficient computation of fit and for a single serial-link chain was discussed in detail in Chapter 4. 

In Stq> 1, the Direct Dynamics problem is solved for each chain of the mechanism assuming that the reference memba has beoi removed and each chain is in an open, unconstrained state. The general open-chain acceleration veclws, (4t)open and (itk)open, are computed fw each chain, along with the position-dependent matrices, ilk and Aj . 

In this paper, an algorithm for dynamic simulation based on the concept of velocity transformations is presented. This algorithm may be applied to the analysis of open and closed-chain systems. The equations of motion for open chain systems are derived using a direct velocity transformation, called open chain velocity trarvrformation. Closed chain systems are analyzed in two steps. First, they are converted into open chain systems by removing some joints and the open chain velocity transformation is applied then, the closed loop conditions are imposed through a second velocity transformation. The implementation of the proposed algorithm was carried out on a SGI 4D/240 workstation and the results obtained for a series of illustrative examples are presented. 

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