Inertia recursive equation

This is the basic recursive matrix equation for the manipulator joint space inertia matrix. 

In the next analysis, we will examine the components of successive inotia matrices as defined by the algorithm given in Table 3.1. First, the expansion of the equations for the Structurally Recursive Method leads to an exjnession for H,j, the Tii X itj submatrix of H, in the form of a summation. Its terms involve projections of individual link inertias onto the preceding joint axis vectors, which... 

Extrapolating from these expanded versions of the equations for the Structurally Recursive Method, a general expression fw the (i,j) submatrix of the 7 T-link manipulator inertia matrix, Hat (or simply H), may be written as follows ... 

Kj is the spatial composile-rigid-body inertia for bodies i through N, ex-pressed in the ith coordinate system. It may be computed recursively using the equation ... 

If Equation 3.32 is examined once again, it is possible to formulate yet another method for computing H. In Method III, after K,- is found, the transformed joint axes are used to obtain the elements of H. In a fourth approach, the composite-rigid-body inertia, K<, is again calculated recursively for all N. However, in contrast to Method III, the joint axes are not transformed to any other coordinate system. Instead, the portion of Ki associated with joint i is transfomed back... 

Note the symmetry of both K and I. The quantity K, is, in essence, an inverse inertia , while the form of L makes it dimensionless. The recursive acceleration equation. Equation 4.37, may also be generalized 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. 

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 shown in Equation 4.75, the final iteration of this recursion at link N gives the inverse opoational space inertia matrix, A for the chain. That is. 

Note the position of the coordinate transformations in this equation compared to the link coordinate version of the articulated-body inertia equation given in Thble 4.S. This difference in position is due to a difference in the order of tq)plication of the transformations across joints and links in the two algorithms. As a chain is traversed in the recursion for A, the ( rational space inertia matrix of a link is first transformed across the present link (via X) and then transfomed across the next joint (via L). In contrast, as a chain is traversed in the recursion for I. the articulated-body in a is first transfomed across a joint and then transfnmed back across the preceding link. 

Note that when the recursion of Equation 4.129 reaches link N, the operational space inertia matrix of the entire chain. A, is known. That is. 

The coefficient matrix of ao in Equation 6.51 rq>resoits the effective simple closed-chain mechanism as seen by the reference member at the origin of its own coordinate system. The operational space inertia of the reference member is just its spatial inertia matrix, lo. Note that the operational space inertias of the augmented chains (acting in parallel on the reference member) add in a simple sum. This is a general rule for inertia matrices. For actuated chains connected in series, the combination rule is not as simple. In this case, extended versions of the recursive algorithms of Chapter 4 may be applied. 

Starting from the terminal bodies and working inward, the composite inertia values as well as the active force and inertia remainder term portions of the inertia forces are determined for the triangularized equations by recursively using the relationships... 

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