Structural recursion equations

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 ... 

Based on current knowledge of the process and its disturbance characteristics, one may know or choose a reasonable difference equation structure for the controller algorithm. Starting with some assumed initial parameter values in the controller equation, the controller can be implemented on the process as shown. The control algorithm is coupled with an on-line recursive estimation algorithm which receives information on the inputs and outputs at each sampling interval and uses this to recursively estimate the optimal controller parameters on-line and to update the controller accordingly. The idea is to use the data collected from the on-line control manipulations to tune the controller directly. 

The time scales of the structural transitions in (NaCl)35Cl mean that it is impossible to use conventional molecular dynamics to investigate the interfunnel dynamics. Instead we use the master equation method outlined in Section III.D. To reduce the computational expense and numerical difficulties we recursively removed from our sample those minima that are only connected to one other minimum—these dead-end minima do not contribute directly to the probability flow between different regions of the PES. The resulting pruned sample had 1624 minima and 2639 transition states. RRKM theory in the harmonic approximation was used to model the individual rate constants,. ... 

In this equation, M is an A x A nonlinear mass matrix and C is the A-dimensional vector of velocity-dependent (Coriolis and other) forces. 0, 0, and 0 represent the torsional position, velocities and accelerations, respectively. The ability to calculate the accelerations recursively relies on the chainlike structure of the protein, in which each node of the chain represents a rigid body. These rigid bodies consist of one atom or a cluster of atoms whose relative positions are fixed. To simplify the explanation of the algorithm, an unbranched chain will be considered, although the approach can be easily extended to branched systems. For this simple case, the first rigid body, at one end of the chain, defines the base (fe = 0), while the last rigid body, at the other end of the chain, defines the tip k = N). The rotatable torsion angle between bodies k and k — I is defined as 0,. ... 

Proceeding along the structure shown, its recursive nature brings us back to the above equation ... 

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