Redundant manipulators

Khatib also discusses the operational space formulation for redundant manipulators. In this case, the definition for the operational space inertia matrix... 

Ar(x) - 6x6 operational space inertia matrix for a redundant manipulator, J(q) = 6y.N Jacobian matrix,... 

C. A. Klein and C. H. Huang. Review of Pseudoinvose Control Jin Use with Kinematically Redundant Manipulators. IEEE Transactions on Systems, Man, and Cybernetics, SMC-13(3) 245-250, March/April 1983. 

From the classification it was found that, for this specific problem, there are 10 redundant and 6 nonredundant measured variables, and all the unmeasured process variables are determinable. Symbolic manipulation of the equations allowed us to obtain the three redundant equations used in the reconciliation problem ... 

These redundancies should not be present in SWISS-PROT or TrEMBL thus it was necessary to find methods to manipulate the data from redundant source databases to meet the stringent standards of minimal redundancy. The objective was to recognize and eliminate the redundancy already present in the databases and to prevent further redundancy entering the database. 

In the same context as the heat of formation, the JANAF tables have tabulated most conveniently the equilibrium constants of formation for practically every substance of concern in combustion systems. The equilibrium constant of formation (KPt[) is based on the equilibrium equation of formation of a species from its elements in their normal states. Thus by algebraic manipulation it is possible to determine the equilibrium constant of any reaction. In flame temperature calculations, by dealing only with equilibrium constants of formation, there is no chance of choosing a redundant set of equilibrium reactions. Of course, the equilibrium constant of formation for elements in their normal state is one. 

It should be noted that unnecessary application of the one-branch approach may lead to needless algebraic manipulations. Furthermore, the branching technique often results in redundant terms that must be searched out and deleted. As an example, let us write the determinant for E in Scheme 4 by the one-branch method ... 

Note that within the same brackets, the redundant terms (underlined by solid lines) are readily removed, but the products of consecutive branch terms, 15 21 and 15 31 (underlined by dashed lines) within the second set of brackets cannot be canceled without further manipulation. One can either eliminate them during further expansion of the equation or repeat the consecutive-branch approach until all the consecutive branches end with a branch leading away from a common node. This common-node consecutive branching approach is always valid—even when all the loop-containing terms have been eliminated. Using this approach, we can write... 

Online analysis processing mainly comprises the interactive exploration of multidimensional data sets, or data cubes, which are manipulated by operations from matrix algebra, for example, slice-and-dice, roll-up, and drill-down. Computing performance is related to data warehouse size and also data quality, for example, missing data, unsharpness, and redundancy. The multidimensionality issue is critical for extracting pertinent information and selecting the results to be stored and visualized. 

Now, from our manipulations above we can easily see that F can be constructed from a skeleton matrix obtained using only a P2 list, since the Fock operator is totally symmetric. However, this far from ideal, since we would like to avoid the redundancies that arise unless we use the Pi list of two-electron integrals, or, here, supermatrix elements. We define first a matrix Y(IJKL) with the property that the IJ block is given by... 

Our mini data base contains two factual statements about the same compound aconitine. In a more realistic data base there will be many such cases. Specifying the same compound in several facts is somewhat redundant. It would be more economical to lump together the TLC data for a given compound in just one single statement. We would like to be able to manipulate all Rf data and TLC system codes as one piece of information and still have access to the individual entries. PROLOG supplies such a data structure, the list. A list is an ordered collection of entries, which can be addressed as a whole, and where the first entry (the head of... 

Organism-biased codon usage can be remedied by manipulating the interplay between a codon and its encoded amino acid. It is evident that several codons encode a particular amino acid, but only one amino acid is encoded by a particular codon. This redundancy is, thus, unidirectional in this context. Consequently, one could either increase the availability of the amino acid corresponding to the rare codon or change the transcript sequence to reflect the codon preferences of the host (analogous to market forces, one can either manipulate the supply or the demand sides). 

For the reduction of chemical mechanisms, reaction-rate analysis has probably the largest record of success. A novel way for the inspection of rates is based on the study of algebraic-rate sensitivities and the Jacobian matrix. These methods can be used for the automatic identification of redundant species and reactions, to produce a reduced mechanism consisting of a subset of the original mechanism. The use of algebraic manipulation in techniques such as the QSSA and lumping, make the production of a reduced mechanism essential and make subsequent calculations as simple as possible. 

As discussed in detail earlier, products of normal-ordered operators can be simplified algebraically using Wick s theorem by evaluating pairs of contractions between the component annihilation and creation operators. Many of these contractions produce mathematically redundant terms that can be combined after complicated manipulation to eventually produce a much simpler expression. Diagrams provide a straightforward scheme by which these redundancies may be eliminated. 

Ruess, L., Schmidt, I.K., Michelsen, A. and Jonasson, S. (2001) Manipulations of a microbial based soil food web at two arctic sites - evidence of species redundancy among the nematode fauna Applied Soil Ecology, 17, 19-30. 

The general discussion of redundant variables will be preceded by a detailed discussion of the special case of two-electron wavefunctions. The formal manipulations are easier to understand for this case than in the general N-electron case. Additionally, the exact full-CI wavefunctions may be included in these formal manipulations. Finally, several features of the two-electron case may then be used in understanding the more general cases. For the discussion of this special case, the approach of Mclver is used to define a matrix of determinants of the form... 

We will start this section by introducing the basic issues related to the formulation of the SS-MRCC theory as originally formulated [38,39], and also briefly highlight the manipulations exploiting the redundancy of the cluster amplitudes which bypass the intruders and ensure size-extensivity. 

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