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The motion planning problem can be formulated as a Markov decision process (MDP), if the uncertainties in the robot motion and environments can be modeled probabilistically. The complexity of solving these MDPs grow exponentially as the dimension of the problem increases and hence, it is nearly impossible to solve the problem even without constraints. Using hierarchical methods, these MDPs can be transformed into a semi-Markov decision process (SMDP) which only needs to be solved at certain landmark states. In the deterministic robotics motion planning community, sampling based algorithms like probabilistic roadmaps (PRM) and rapidly exploring random trees (RRTs) have been successful in solving very high dimensional deterministic problem. However they are not robust to system with uncertainties in the system dynamics and hence, one of the primary objective of this work is to generalize PRM/RRT to solve motion planning with uncertainty. We first present generalizations of randomized sampling based algorithms PRM and RRT, to incorporate the process uncertainty, and obstacle location uncertainty, termed as "generalized PRM" (GPRM) and "generalized RRT" (GRRT). The controllers used at the lower level of these planners are feedback controllers which ensure convergence of trajectories while mitigating the effects of process uncertainty. The results indicate that the algorithms solve the motion planning problem for a single agent in continuous state/control spaces in the presence of process uncertainty, and constraints such as obstacles and other state/input constraints. Secondly, a novel adaptive sampling technique, termed as "adaptive GPRM" (AGPRM), is proposed for these generalized planners to increase the efficiency and overall success probability of these planners. It was implemented on high-dimensional robot n-link manipulators, with up to 8 links, i.e. in a 16-dimensional state-space. The results demonstrate the ability of the proposed algorithm to handle the motion planning problem for highly non-linear systems in very high-dimensional state space. Finally, a solution methodology, termed the "multi-agent AGPRM" (MAGPRM), is proposed to solve the multi-agent motion planning problem under uncertainty. The technique uses a existing solution technique to the multiple traveling salesman problem (MTSP) in conjunction with GPRM. For real-time implementation, an ?inter-agent collision detection and avoidance? module was designed which ensures that no two agents collide at any time-step. Algorithm was tested on teams of homogeneous and heterogeneous agents in cluttered obstacle space and the algorithm demonstrate the ability to handle such problems in continuous state/control spaces in presence of process uncertainty.

This thesis presents two new routing problems, both with objective functions focused on relative positioning of teams during the routing horizon. The relative positioning results in temporal and spatial dependencies among routes and is quantified with a nonlinear dispersion metric, designed to evaluate the instantaneous distances among teams over a time interval. This metric allows the design of objective functions to approximate teams during routes execution, when minimized, or disperse them, when maximized. Both approximation and dispersion are important routing characteristics in some practical applications, and two new optimization problems are proposed with these opposite objectives. The first one is a variation of the Multiple Traveling Salesman Problem, and its goal is to find a set of tours where the salesmen travel close to each other, minimizing dispersion. A Local Search Genetic Algorithm is proposed to solve the problem. It includes specialized genetic operators and neighborhoods. A new set of benchmark instances is proposed, adapted for the new problem from literature instances. Computational results show that the proposed approach provides solutions with the desired characteristics of minimal dispersion. The second problem is a bi-objective arc routing problem in which routes must be constructed in order to maximize collected profit and dispersion of teams. The maximization of the dispersion metric fosters the scattering of the teams during routing procedure. Usually, profit and dispersion objectives are conflicting, and by using a bi-objective approach the decision maker is able to choose a trade-off between collecting profits and scattering teams. Two solution methods are proposed, a Multi-objective Genetic Algorithm and a Multi-objective Genetic Local Search Algorithm, both specialized in order to exploit the characteristics of the problem. It is demonstrated, by means of computational experiments on a new set of benchmark instances, that the proposed approach provides approximation sets with the desired characteristics. / Esta tese apresenta dois novos problemas de roteamento, ambos com funções objetivo voltadas para o posicionamento relativo das equipes durante o horizonte de roteamento. O posicionamento relativo resulta em uma dependência temporal e espacial entre rotas e é quantificado com uma métrica de dispersão não-linear, projetada para avaliar as distâncias instantâneas entre as equipes ao longo de um intervalo de tempo. Esta métrica permite a concepção de funções objetivo para aproximar as equipes durante a execução das rotas, quando minimizada, ou para dispersá-las, quando maximizada. Tanto a aproximação quanto a dispersão são características importantes de roteamento em algumas aplicações práticas, e dois novos problemas de otimização são propostos com esses objetivos opostos. O primeiro é uma variação do Problema de Múltiplos Caixeiros Viajantes, e seu objetivo é encontrar um conjunto de rotas em que os caixeiros viajam próximos uns dos outros, minimizando a dispersão. Um Algoritmo Genético com Busca Local é proposto para resolver o problema. Ele inclui operadores genéticos e vizinhanças especializados. Um novo conjunto de instâncias é proposto, adaptado para o novo problema de instâncias da literatura. Resultados computacionais mostram que a abordagem proposta proporciona soluções com as características desejadas de dispersão mínima. O segundo problema é um problema de roteamento de arcos biobjetivo em que as rotas devem ser construídas de modo a maximizar o lucro recolhido e o distanciamento entre as equipes. A maximização da métrica promove a dispersão das equipes durante a execução das rotas. Normalmente, os objetivos de lucro e dispersão são conflitantes, e com uma abordagem biobjetivo o tomador de decisão é capaz de avaliar a troca entre a coleta de lucros e a dispersão de equipes. Dois métodos de solução são propostos, um Algoritmo Genético Multiobjetivo e um Algoritmo Genético Multiobjetivo com Busca Local, ambos especializados para explorar as características do problema. É demonstrado, por meio de experimentos computacionais sobre um novo conjunto de instâncias, que a abordagem proposta fornece conjuntos de aproximação com as características desejadas.