The Inertia Propagation Method

By definition, the operational space inertia matrix of an t-link manipulatcM , A, is the matrix which relates the spatial acceleration of link i and the spatial force vector exoled at the tip of link i. Thus, we may write  

Ihble 4.8 Additional Computations for O, Force Prc agation Method 

Equation 4.103 is the initial condition for this recursive algorithm. Having defined an initial condition to A at the base member, we may proceed to link 1. Once again, we will assume that the joint velocities, actuator torques and/or forces, and gravity forces ate all zero, and all vectors and matrices are initially defined in absolute coordinates. As before, we may write the ftee-body dynamic equation to link 1 as follows  

Combining this result with Equation 4.106, we may express fi as follows  

In this simple recursion, the operational space inertia matrix of the base member, Ao, is propagated across joint 1 by La a new spatial articulated transformation which is very similar in form to the acceloation propagator of the previous section. The propagated matrix is combined with Ii, the spatial inertia of link 1 to form Ai, the operational space inertia matrix of the two-link partial chain comprised of links 0 and 1. Note the similarity between this recursive procedure and the structural recursion used to derive the Structurally Recursive Method (Method I) in Ch t 3. 

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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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