Stepping trajectory-based algorithms

The stepping trajectory-based algorithms, such as simplex [30] and steepest descent [31] methods, are designed as more directed searches than the other types... 

In the first paper, an approximate path planning algorithm first navigates a point robot in polygonal terrains and returns paths with suboptimal lengths. An innovative four-step trajectory planning algorithm is developed in the second paper for a 3-axis articulated robot in which precise joint-level control can be easily achieved, major mechanical vibration can be avoided, and a well tuned speed can be used to maintain the efficiency. Inverse and forward kinematics for the Stewart Platform-based Manipulator are presented in the third paper. 

Combining the steps described above, a simple stochastic algorithm for a trajectory-based propagation of the QCL equation can be constructed as follows ... 

Algorithmically, action-based methods are similar to the NEB method since in both cases a path functional is minimized. They differ, however, in the nature of the particular functional. While in the NEB method a path functional is constructed in an ad hoc way such that the path ttaverses the transition state separating reactants fi om products, the functional minimized in action-based methods corresponds, in principle, to the fully dynamical trajectories of classical mechanics. This property, however, is lost if extremely large time steps are used. In this case, the method yields a possible sequence of events that may be encountered by the system as it evolves fi om its initial to its final state, but a dynamical interpretation of such a sequence of states is not strictly permissible any more. Nevertheless, large time step trajectories that minimize the Gauss (Onsager-Machlup) action can provide possible scenarios for transitions that are computationally untreatable otherwise. 

According to the proposed approach, the integrated process and control design is performed by combining two procedures, one constructive where the robust output-feedback controller is designed via inverse optimality with respect to control robustness, and one based on direct optimization where the batch equipment and motion are obtained. The constructive step yields the controller and the algorithm to draw the closed-loop (i.e. the dynamical inverse or ZD) motion in terms of the tracked output trajectory. In the direct optimization step, the... 

However, dielectric solvation models do suffer from a too simple description of the solute-solvent interactions, and also from a static vision of the solvated system, which is supposed to be at thermodynamic equilibrium. A step forward was done after the publication in 1990 of a paper by Field et al. [19] on the QM/MM model, which was an extension of the seminal work reported by Warshel and Levitt in 1976 [20] (the acronym QM/MM stands for Quanmm Mechanics/Molecular Mechanics). The QM/MM model was initially developed to carry out Molecular Dynamics (MD) simulations of large molecules such as proteins using a semi-empirical description of the active site. But extension to first-principles studies of ions and molecules in solution [21-23] was reported soon after Karplus work and algorithms to simulate chemical reaction trajectories and analyze non-equilibrium effects in solution based on a rare event sampling approach were developed [24, 25]. 

The quantitative analysis of spirals yields, on the one hand, a detailed picture of the concentration distribution inside and outside the spiral core. This information can be used to clarify details in the mechanistic steps of the reaction coupled with diffusion. On the other hand, a large number of topological scenarios in the core region have been explored by now that are based on algorithms for the extraction of the essential dynamic features. These are commonly represented by the trajectory of the spiral tip. Unfortunately, in the experimenal evaluation the precise location of the tip is still a rather ill defined quantity and each experimentalist has his own procedure to trace the tip trajectories. In numerical simulation a clearer definition can be provided in terms of the concentration levels of the two variables. 

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