Structural recursion

In Chapter 3, four efficient serial algorithms for computing the manipulator joint space inotia matrix are developed. The first, the Structurally Recursive... 

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. 

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. 

To begin the development of the Structurally Recursive Method, a one-link manipulator, as shown in Figure 3.1, is examined. A free-body force equation may be written fix the single link as follows ... 

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 ... 

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. 

Rusinko A III, M W Farmen, C G Lambert, P L Brown and S S Young 1999. Analysis of a Larj Structure/Biological Activity Data Set Using Recursive Partitioning. Journal of Chemic Information and Computer Science 39 1017-1026. 

AUGMENTED SPACE RECURSION METHOD FOR THE CALCULATION OF ELECTRONIC STRUCTURE OF RANDOM ALLOYS... 

Consider, for definiteness, a set of otherwise identical lowest-level components of a system, so that the hierarchy is a tree of constant depth. Since we assume that the components are all identical, the only distinction among the various nodes of the hierarchy consists of the structure of the subtrees. Now suppose we have a tree T that consists of /3 subtrees branching out from the root at the top level. We need to determine the number of different interactions that can occur on each level, independent of the structure of each subtree i.e. isomorphic copies of trees do not contribute to our count. We therefore need to find the number of nonisomorphic subtrees. We can do this recursively. 

Software to predict the properties of formulated products is made more powerful by a recursive procedure which can use formulas stored in files as raw materials. Particular care must be taken with program flow control and data structures for the recursion to be effective. This paper illustrates these issues using an example derived from a working formulation system for coatings development. 

Recursive procedures demand special attention to flow control and data structures. For instance, the flow control within the procedure must correctly handle an error, say missing file information or inconsistent data, that is discovered several iterations deep. Should the procedure break and return to the previous level only Should it force return to the level of the original invocation of the procedure Should it allow an interactive user a choice of supplying missing data, and if so, on any level of iteration If files are opened within the procedures, should they be closed when a recursive call is needed, or are new channel numbers to be requested, using up system resources ... 

The table below illustrates these issues by comparing how a recursive subroutine must handle data which is available from a database, such as the cost of a raw material, data that is calculated for the formulated product, such as PBR, and data for intermediate products. (The variable names shown in the table are part of the example procedure given in the appendix.) Compare with the previous table for a non-recursive modelling procedure s data structure. 

The procedure illustrated here, besides containing only trivial technical calculations, lacks important features that are required in production programs. Extensive error checking and recovery must be performed. The procedure must detect the occurrence of a self-referential system of formulas, which would result in attempting endless recursive calls. Access to multiple raw material and formula databases adds power to the program, but must be implemented by complex code to allow flexible control of that access. The structural and input/output statements to support these features may greatly exceed the number of statements that perform modelling calculations. 

Rusinko A 3rd, Farmen MW, Lambert CG, Brown PL, Young SS. Analysis of a large structure/biological activity data set using recursive partitioning. J Chem Inf Comput Sci 1999 39 1017-26. 

Chen X, Rusinko A, Young SS. Recursive partitioning analysis of a large structure-activity data set using three-dimensional descriptors. J Chem Inf Comput Sci 1998 38 1054-62. 

Recursive estimation methods are routinely used in many applications where process measurements become available continuously and we wish to re-estimate or better update on-line the various process or controller parameters as the data become available. Let us consider the linear discrete-time model having the general structure ... 

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