Joint space inertia matrix Method

The Structurally Recursive Method is then expanded, and a second, non-recursive algorithm fw the manipulator inertia matrix is derived from it A finite summation, which is a function of individual link inertia matrices and columns of the propriate Jacobian matrices, is defined fw each element of the joint space inertia matrix in the Inertia Projection Method. Further manipulation of this expression and application of the composite-rigid-body inertia concept [42] are used to obtain two additional algwithms, the Modified Composite-Rigid-Body Method and the Spatial Composite-Rigid-Body Method, also in the fourth section. These algorithms do make use of recursive expressions and are more computationally efficient. 

In the sixth section, the computational requirements for the methods presented here are compared with those of existing methods for computing the joint space inertia matrix. Both general and specific cases are considered. It is shown that the Modified Composite-Rigid-Body and Spatial Composite-Rigid-Body Methods are the most computationally efficient of all those compared. 

Parallel computation methods have also been investigated for the Joint space inertia matrix. Amin-Javaheri and Orin [1], as well as Fijany and Bejczy [10], have achieved bett performance by developing parallel and/or pipelined algorithms. In both cases, the parallel forms are based to a great extent on the serial Composite-Rigid-Body Method of Walker and Orin [42], and, of course, the improvement in performance is dependent on an increased number of processes. 

Four algorithms for computing the joint space inertia matrix of a manipulator are presented in this section. We begin with the most physically intuitive algorithm the Structurally Recursive Method. Development of the remaining three methods, namely, the Inertia Projection Method, the Modified Composite-Rigid-Body Method, arid the Spatial Composite-Rigid-Body Method, follows directly from the results of this tot intuitive derivation. 

Methods III and IV have reduced computational complexities of 0(N ). Method III is the most efficient tqrproach for the joint space inertia matrix of all those listed for N < 6, and it computes the manipulator Jacobian matrix as well. Method IV, although the simultaneous Jacobian computation has been eliminated, is the most efficient tqrproach for computing the joint space inertia matrix for N >6. Thus, these two methods together provide the most efficient calculation of H for all N. 

A similar 0(N ) method, presented by Angeles and Ma in [2], uses the concept of an orthogonal complement to construct the joint space inertia matrix. The Cholesky decomposition of this matrix is used in solving the appropriate linear system for the joint accelerations. The computational complexity of this algoithm is slightly better than that in [42], but the algorithm is still not the most efficient It, too, is restricted to configurations of simple revolute and prismatic joints. 

Perhaps the simplest and most obvious way to compute the opmtional space inertia matrix (or its inverse) is by the explicit inversion and multiplication of the Jacobian and joint space inertia matrices as shown in Equations 4.19 and 4.20. We will call this the Explicit Inversion/Multiplication Method. Although we will see that this is not the most efficient approach, it is in standard use today and may serve as a benchmark for new computational approaches. 

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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