Force Propagation Method algorithm

The scalar ( rations (multiplications, additions) required to compute A and A using the Force Propagation Method are shown in Table 4.6. These scalar operations are given for ap AT degree-of-freedom manipulator with simple Involute and/or prismatic joints. Note that 1)1, K, and L)), may all be computed off-line, and that the initial condition, (Ag) = 0, allows some simplification in the first iteration of the Forward Recursion. The computational complexity of the complete algorithm is 0(N), an improvement over the previous two algorithms. The efficient coordinate tiansfcMmations described in Chapter 3 are utilized in every case. 

Table 4.S Algorithm for the Force Propagation Method (Method III)...

The two tables differ only in the algorithm used to compute the inverse operational space inertia matrix, A and the coefficient fl. In Chapter 4, the efficient computation of these two quantities was discussed in some detail. It was detomined that the Unit Force Method (Method II) is the most efficient algorithm for these two matrices together for N < 21. The Force Propagation Method (Method ni) is the best solution for and fl for AT > 21. The scalar opmtions required for Method II are used in Table 5.1, while those required for Method III are used in Table 5.2. 

Note that the number of operations listed for fl and A in Table 5.2 is less than the total given for these two quantities in the 0 N) Force Propagation Method in Chapter 4. This reduction was achieved through a little insight First we note that the first recursion in the open-chain Direct Dynamics algorithm of... 

Our AIMD simulations are all-electron and self-consistent at each 0.4 femtoseconds (fs) time step. Variational fitting ensures accurate forces for any finite orbital or fitting basis sets and any finite numerical grid. These forces are used to propagate the nuclear motion according to the velocity Verlet algorithm [22]. The accuracy of these methods is indicated by the fact that during the 500 time-step simulations of methyl iodide dissociation described below, the center of mass moved by less than 10-6 A. 

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