Operational space inertia matrix Method

The second quantity of interest, the operational space inertia matrix (O.S.I.M.) of a manipulator, is a newer subject of investigation. It was introduced by Khatib [19] as part of the operational space dynamic formulation, in which manipula-Ux control is carried out in end effector variables. The operational space inertia matrix defines the relationship between the gen lized forces and accelerations of the end effectw, effectively reflecting the dynamics of an actuated chain to its tip. This book will demonstrate its value as a tool in the development of Direct Dynamics algorithms for closed-chain configurations. In addition, a number of efficient algorithms, including two linear recursive methods, are derived for its computation. 

Because the operationa] space fonnulation for robot dynamics is fairly new, few efficient methods exist fw computing its components. The conceptual framework for the operational space fomulation was presented by Khatib in [17,18,19], where he established a basic definition for the operational space inertia matrix. In [19], Khatib shows that the operational space inertia matrix of a 6 degree-of-freedom manipulator may be computed as follows ... 

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. 

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. 

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 . 

---