Operational space inertia matrix inverse

Rodriguez, Kreutz, and Jain [37, 38] present a linear recursive algorithm for the operational space inertia matrix, referred to as the operational space mass matrix , as part of an original operate formulation for open- and closed-chain multibody dynamics. In general, this operator apfxoach appears to be quite powerful, especially in matrix factorization and inversion, and with it, the authors... 

Equations 4.S9,4.60,4.67, and 4.70 summarize the fundamental and relevant recursive dynamic equations fw a constrained single chain. These equations wiU now be used to derive the Force Prqtagadon Method for computing the inverse operational space inertia matrix of a single chain. 

Given the recursive dynamic equations for a constrained chain, we will now begin the development of a linear recursive algcxithm fw A the inverse operational space inertia matrix of a single chain. First, we will define a new quantity, (A ) an inertial matrix which relates the spatial acceleration of a link and the propagated spatial contact force exoted at the tip of the same link. We may write a defining equation for this matrix (at link t) as follows ... 

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. 

The two tables differ only in the algorithm used to compute the inverse operational space inertia matrix, A and the coefficient fl. In Chapter 4, the efficient computation of these two quantities was discussed in some detail. It was detomined that the Unit Force Method (Method II) is the most efficient algorithm for these two matrices together for N < 21. The Force Propagation Method (Method ni) is the best solution for and fl for AT > 21. The scalar opmtions required for Method II are used in Table 5.1, while those required for Method III are used in Table 5.2. 

Brandi, Johanni, and Otter [3] computes the articulated-body inertia of each link in the chain, starting at the tip and moving back to the base. This same recursion is the first recursion in the Force Propagation Method for computing A. That is, there is an overlap of computations between the solution for the q)en-chain acceleration terms, tjopen and Xopen. and the calculation of the inverse ( rational space inertia matrix, A for this case. This fact was taken into account when the operations were tabulated. The ( rations listed for SI and A in Table 5.2 include only the second recursion for A and the additional opoations needed for SI. The recursion which computes the articulated-body inertias is included in the computatimis for open and x<,pe . 

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