When a mechanical system is subject to constraints its motion is in some way restricted. In accordance with Newton's second law, motion is a direct result of forces acting on a system; hence, constraint is inextricably linked to force. The presence of a constraint implies the application of particular forces needed to compel motion in accordance with the constraint; absence of a constraint implies the absence of such forces. The objective of this thesis is to formulate a comprehensive, consistent, and concise method for identifying a set of forces needed to constrain the behavior of a mechanical system modeled as a set of particles and rigid bodies. The goal is accomplished in large part by expressing constraint equations in vector form rather than entirely in terms of scalars. The method developed here can be applied whenever constraints can be described at the acceleration level by a set of independent equations that are linear in acceleration. Hence, the range of applicability extends to servo-constraints or program constraints described at the velocity level with relationships that are nonlinear in velocity. All configuration constraints, and an important class of classical motion constraints, can be expressed at the velocity level by using equations that are linear in velocity; therefore, the associated constraint equations are linear in acceleration when written at the acceleration level. Two new approaches are presented for deriving equations governing motion of a system subject to constraints expressed at the velocity level with equations that are nonlinear in velocity. By using partial accelerations instead of the partial velocities normally employed with Kane's method, it is possible to form dynamical equations that either do or do not contain evidence of the constraint forces, depending on the analyst's interests.

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[pt] Robôs móveis autônomos são um grande foco de pesquisa devido à sua aplicabilidade e interdisciplinaridade. Robôs móveis com roda de direção diferencial, além de possuírem alta não-linearidade, detêm uma característica inerente à sua geometria: suas rodas só podem girar em torno de eixos fixos, sem esterçamento. Com isso, o deslizamento longitudinal e lateral é inevitável, principalmente quando o sistema está em movimento sob efeitos dinâmicos significativos. Controle Preditivo baseado em Modelo Não-Linear, Nonlinear Model Predictive Control (NMPC), é amplamente utilizado nesses casos, já que consegue lidar com sistemas com múltiplas restrições. O presente trabalho apresenta modelos matemáticos de um robô móvel com roda do tipo skidsteer, procedente da direção diferencial, incluindo o deslizamento longitudinal, aos quais o NMPC é empregado para seguimento de trajetória, obtendo trajetórias similares à de referência. Verificando que o custo de processamento de tais controladores pode ser muito alto para uso em tempo real, um controle hierárquico é desenvolvido otimizando as forças longitudinais entre as rodas e o solo para encontrar deslizamentos de referência para uma determinada trajetória a ser seguida. Como em um ambiente real nem todos os estados podem ser medidos, o controle necessita também estimar os estados não medidos. A Estimação de Estados por Horizonte Móvel, (Moving Horizon State Estimation (MHSE)), derivada dos fundamentos do NMPC, foi utilizada para realizar a estimativa, já que possui recursos para manter o sistema sob as restrições. Com o MHSE, o deslizamento do sistema pode ser calculado a partir dos estados estimados para as trajetórias obtidas com o Controle Preditivo baseado em Modelo, (Model Predictive Control (MPC)). Por fim, uma rede neural foi treinada com os estados preditos e estimados com o MHSE para que pudesse substituí-lo para que todo o controle fosse utilizado em tempo real. Com isso, o tempo computacional foi reduzido devido a substituição do MHSE. / [en] Autonomous mobile robots are a major focus of research due to their applicability and interdisciplinarity. Depending on the type of locomotion, the system’s controller needs to handle not only trajectory tracking but also the way the system interacts with the ground. Mobile robots with differential drive wheels, in addition to having high nonlinearity, possess an inherent characteristic due to their geometry: their wheels can only rotate around fixed axes, without steering. As a result, longitudinal and lateral slip is inevitable, especially when the system is in motion under significant dynamic effects. Nonlinear Model Predictive Control (NMPC) is widely used in these cases, as it can handle systems with multiple constraints. This work presents mathematical models of a skid-steer mobile robot, derived from differential drive, including longitudinal slip, to which NMPC is applied for trajectory tracking, achieving trajectories similar to the reference. Given that the processing cost of such controllers can be very high for real-time use, a hierarchical control is developed, optimizing the longitudinal forces between the wheels and the ground to find reference slips for a given trajectory to be followed. Since in a real environment not all states can be measured, the control also needs to estimate the unmeasured states. Moving Horizon State Estimation (MHSE), derived from the fundamentals of NMPC, was used to perform the estimation, as it has the resources to keep the system within the constraints. With MHSE, the system’s slip can be calculated from the estimated states for the trajectories obtained with Model Predictive Control (MPC). Finally, a neural network was trained with the predicted and estimated states using MHSE to replace it so that the entire control could be used in real-time. As a result, computational time was reduced due to the replacement of MHSE.