Monte Carlo Localization

In this chapter, we will try to solve this problem of less precise tracking occurring because of the use of a phase loeked loop with the help of Markov Localization and Monte Carlo Localization. 

We will use Monte Carlo Localization to work in multi modal space. Multi modal space is useful for global localization as it uses more than one mode of space also useful for more than multiple instances. Monte Carlo Localization is applied on the received probability density function of each sensor attached on the robot, which then processes all of them and reproduces a final probability density function which is called the final belief of the robot. 

Monte Carlo Localization is used to determine the position of the robot given the map of the environment built up by the Markov Localization. It is a form of particle filter which estimates the sequence of hidden parameters Xj, (where k can be any value), given yj.. 

In Monte Carlo Localization, the space is digitized into some random samples (k). As discussed above, multi modal representation is used because of the global localization. Because of this reason, it requires less memory and is computationally more capable. Grid-based approaches were also used, but more memory was required because of 3-D figures (Oxford, 2012). 

Monte Carlo Localization is trouble-free to put into practice and occupies a smaller amount of memory. A smaller amount of memory is occupied because the belief is updated rather than saved. Only a single last belief is required to calculate the new belief by convolution. Its results are more precise, accurate, and as error free than the feedback/phase locked loop of the Kalman Filter. 

Bodhale, Afzulpurkar, and Thanh (2008) integrate potential fields and use Monte Carlo localization for navigation, obstacle avoidance, and mobile robot localization in a dynamic environment. The path planning algorithm is divided in two submodules, the first includes visibility graph with A search method and the second is the local planning using potential fields. 

The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

A multitude of different variants of this model has been investigated using Monte Carlo simulations (see, for example [M])- The studies aim at correlating the phase behaviour with the molecular architecture and revealing the local structure of the aggregates. This type of model has also proven useful for studying rather complex structures (e.g., vesicles or pores in bilayers). 

Molecular Dynamics and Monte Carlo Simulations. At the heart of the method of molecular dynamics is a simulation model consisting of potential energy functions, or force fields. Molecular dynamics calculations represent a deterministic method, ie, one based on the assumption that atoms move according to laws of Newtonian mechanics. Molecular dynamics simulations can be performed for short time-periods, eg, 50—100 picoseconds, to examine localized very high frequency motions, such as bond length distortions, or, over much longer periods of time, eg, 500—2000 ps, in order to derive equiUbrium properties. It is worthwhile to summarize what properties researchers can expect to evaluate by performing molecular simulations ... 

The hst which follows gives an outline of the properties of a Monte Carlo simulation used in the context of molecular modeling studies for sampling either multiple conformations of smaller, flexible stmctures or multiple local minima of larger macromolecules or polymers ... 

FIG. 1 Total local density p(z) for bulk density p = 0.821 and e /k T = 4.25. The solid line is for PYl theory, the dashed line is for HNCl approximation and the points denote the Monte Carlo simulation results. (Reprinted from S. Sokolowski, D. Henderson, A. Trokhymchuk, O. Pizio. Density profiles of associating fluid near a hard wall PY/EMSA and HNC/EMSA singlet theory, Physica A, 220, 22-32. (1995), with permission from Elsevier Science.)... 

FIG. 6 A comparison of the Monte Carlo (points), HHNCl, HNCl, and PYl density profiles. The results are for bulk density 0.403 (lower group of curves) and 0.741 (upper group of curves). The curves at z — 0, are, from the bottom, from HHNCl and HNCl approximations. (Reprinted from A. Trokhymchuk, D. Henderson, S. Sokolowski. Local density of overlapping spheres near a hard wall A density functional approach. Physics Letters A 209, 317-320. 1995, with permission from Elsevier Science.)... 

FIG. 9 Changes of the monolayer film critical temperature with the concentration of impurities obtained from the Monte Carlo simulations (open circles) and resulting from the mean field theory (solid line). (Reprinted from A. Patrykiejew. Monte Carlo studies of adsorption. II Localized monolayers on randomly heterogeneous surfaces. Thin Solid Films, 205 189-196, with permision from Elsevier Science.)... 

J. J. Luque, F. Jimenez-Morales, M. C. Lemos. Monte Carlo simulation of a surface reaction model with local interaction. J Chem Phys 96 8535-8538, 1992. 

From this short discussion, it is clear that atomistically detailed molecular dynamics or Monte Carlo simulations can provide a wealth of information on systems on a local molecular atomistic level. They can, in particular, address problems where small changes in chemical composition have a drastic effect. Since chemical detail is avoided in mesoscopic models, these can often capture such effects only indirectly. 

M. J. Vlot, S. Claassen, H. E. Huitema, J. P. v. d. Eerden. Monte Carlo simulation of racemic liquid mixtures thermodynamic properties and local structures. Mol Phys 97 19, 1997 M. J. Vlot, J. C. v. Miltenburg, H. A. Oonk, J. P. V. d. Eerden. Phase diagrams of scalemic mixtures. J Chem Phys 707 10102, 1997. 

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