Inertia composite-rigid-body

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. 

The fifth section describes efficient computational methods for the transformation of spatial quantities between adjacent cowdinate frames and for the calculation of composite-rigid-body inertias. These techniques are used throughout this book to improve the computational efficiency of the algorithms. Otho-factors which simplify the equations are also discussed. 

Walker and Orin [42] present one of the most familiar and efficient approaches for the computation of the inertia matrix in the so-called Composite-Rigid-Body Method [9]. This algorithm utilizes the concq>t of composite-rigid-body inertias to simplify the calculation of the manipulator inotia matrix. The computational complexity of this approach, 0(N% is significantly reduced compared to those described above, but the restriction to revolute and/w prismatic Joints remains. 

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... 

J.3 Computing Spatial Composite Rigid-Body Inertias... 

The spatial composite-rigid-body inertia of links i through N,K, was defined in Equation 3.36 as ... 

It may be shown that this procedure for ccmiputing the spatial composite-rigid-body inertia is exactly equivalent to the procedure used by Walker and Orin in [42]. That is, if the composite mass, composite center of mass, and composite moment of inertia matrix are computed for links i through iV, they may be combined to obtain the spatial matrix K,-. After studying this equivalent approach, howev, it appears that the congruence transformation method given hoe is more efficioit... 

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. 

It can easily be shown that K,+i has the same mathematical fonn as the spatial inertia of a single rigid body. Thus, the congruence transformation of this matrix also requires (49 multiplications, 49 additions) as described above. The addition of I, requires only an extra 10 additions, if we consido the symmetry and form of Ki and I<. Note that the bottom right submatrix of is simply the diagonal matrix of the composite mass of links t through N. Thus, since this composite... 

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