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This thesis aims to explain how the concepts of functional modeling are implemented in the development and validation of real-world hybrid dynamic systems. I also discuss how control theory is integrated with the design process in order to understand the significance of periodic orbits on a simple dynamic system. Two hybrid system applications with different levels of complexity will be considered in this thesis – an anthropomorphic Bipedal walking robot and a Double Pendulum with a mechanical stop. The primary objectives of this project are to demonstrate the phenomena of Zeno and zeno periodic orbits in hybrid dynamic systems involving impacts. Initially, I describe the salient features of the product design procedure and then explain the significance of functional modeling as a part of this process. We then discuss hybrid dynamic systems and the occurrence of Zeno behavior in their mathematical form. Also, the necessary conditions for existence of Zeno and zeno equilibrium points are provided. Then the theory of completed Lagrangian hybrid systems is explained in detail. We then examine the two hybrid dynamic systems being considered for this project. Prior research undertaken on bipedal walking is explored to understand their design and achievement of stable walking gaits with appropriate actuation mechanisms. Based on this insight, a suitable design procedure is employed to develop the bipedal robot model. The desired actuation mechanisms for all the configurations considered for this model as well as the challenges faced in employing optimal actuation will be discussed. However, due to the high level of complexity of the bipedal robot model, a simpler hybrid dynamic system is considered to simplify fabrication and control of the model. This is the motivation behind designing and building the Double Pendulum model with a mechanical stop in an attempt to observe zeno behavior in this system. We begin by formally demonstrating that the “constrained” double pendulum model displays Zeno behavior and complete this Zeno hybrid system to allow for solutions to be carried past the Zeno point. The end result is periods of unconstrained and constrained motions in the pendulum, with transitions to the constrained motion occurring at the Zeno point. We then consider the development of a real physical pendulum with a mechanical stop and introduce non-plastic impacts. Later, we verify through experimentation that Zeno behavior provides an accurate description of the behavior of the physical system. This provides evidence to substantiate the claim that Zeno behavior, while it does not technically occur in reality, provides an accurate method for predicting the behavior of systems undergoing impacts and that the theory developed to understand Zeno behavior can be applied to better understand these systems.

Indiana University-Purdue University Indianapolis (IUPUI) / Khakpour, Zahra M.S.M.E., Purdue University, May 2017. Multibody Dynamics Model of A Full Human Body For Simulating Walking, Major Professor: Hazim El-Mounayri. Bipedal robotics is a relatively new research area which is concerned with creating walking robots which have mobility and agility characteristics approaching those of humans. Also, in general, simulation of bipedal walking is important in many other applications such as: design and testing of orthopedic implants; testing human walking rehabilitation strategies and devices; design of equipment and facilities for human/robot use/interaction; design of sports equipment; and improving sports performance & reducing injury. One of the main technical challenges in that bipedal robotics area is developing a walking control strategy which results in a stable and balanced upright walking gait of the robot on level as well as non-level (sloped/rough) terrains. In this thesis the following aspects of the walking control strategy are developed and tested in a high-fidelity multibody dynamics model of a humanoid body model: 1. Kinematic design of a walking gait using cubic Hermite splines to specify the motion of the center of the foot. 2. Inverse kinematics to compute the legs joint angles necessary to generate the walking gait. 3. Inverse dynamics using rotary actuators at the joints with PD (Proportional-Derivative) controllers to control the motion of the leg links. The thee-dimensional multibody dynamics model is built using the DIS (Dynamic Interactions Simulator) code. It consists of 42 rigid bodies representing the legs, hip, spine, ribs, neck, arms, and head. The bodies are connected using 42 revolute joints with a rotational actuator along with a PD controller at each joint. A penalty normal contact force model along with a polygonal contact surface representing the bottom of each foot is used to model contact between the foot and the terrain. Friction is modeled using an asperity-based friction model which approximates Coulomb friction using a variable anchor-point spring in parallel with a velocity dependent friction law. In this thesis, it is assumed in the model that a balance controller already exists to ensure that the walking motion is balanced (i.e. that the robot does not tip over). A multi-body dynamic model of the full human body is developed and the controllers are designed to simulate the walking motion. This includes the design of the geometric model, development of the control system in kinematics approach, and the simulation setup.

Esta tese trata de um robô bípede em caminhar dinâmico. Neste robô, que normalmente é um sistema sub-atuado, fazemos uso de um disco de inércia que funciona num certo sentido como um atuador adicional. Através deste disco, obtém-se mais liberdade para a elaboração de passos repetitivos e um aumento na robustez. Por outro lado, o sistema de controle dos passos deve controlar, além do passo propriamente dito, também a velocidade do disco, de modo que não sejam saturados os atuadores (motores elétricos). Apresentamos então um controlador capaz de realizar estas ações simultaneamente. / This Thesis is about a bipedal robot in a dynamic walking gait. In this robot, which is usually a under-actuated system, a inertial wheel is employed and acts as an additional actuator. By using this wheel, one can design a cyclic walking gait with increased robustness and with more freedom. On the other hand, the control system must take care of the step itself, and also must ensure that the wheel speed does not exceed the actuators (motors) limits. We present a controller able to perform this tasks.

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No description available.

This thesis introduces a new measure of balance for bipedal robotics called the foot placement estimator (FPE). To develop this measure, stability first is defined for a simple biped. A proof of the stability of a simple biped in a controls sense is shown to exist using classical methods for nonlinear systems. With the addition of a contact model, an analytical solution is provided to define the bounds of the region of stability. This provides the basis for the FPE which estimates where the biped must step in order to be stable. By using the FPE in combination with a state machine, complete gait cycles are created without any precalculated trajectories. This includes gait initiation and termination. The bipedal model is then advanced to include more realistic mechanical and environmental models and the FPE approach is verified in a dynamic simulation. From these results, a 5-link, point-foot robot is designed and constructed to provide the final validation that the FPE can be used to provide closed-loop gait control. In addition, this approach is shown to demonstrate significant robustness to external disturbances. Finally, the FPE is shown in experimental results to be an unprecedented estimate of where humans place their feet for walking and jumping, and for stepping in response to an external disturbance.

This thesis introduces a new measure of balance for bipedal robotics called the foot placement estimator (FPE). To develop this measure, stability first is defined for a simple biped. A proof of the stability of a simple biped in a controls sense is shown to exist using classical methods for nonlinear systems. With the addition of a contact model, an analytical solution is provided to define the bounds of the region of stability. This provides the basis for the FPE which estimates where the biped must step in order to be stable. By using the FPE in combination with a state machine, complete gait cycles are created without any precalculated trajectories. This includes gait initiation and termination. The bipedal model is then advanced to include more realistic mechanical and environmental models and the FPE approach is verified in a dynamic simulation. From these results, a 5-link, point-foot robot is designed and constructed to provide the final validation that the FPE can be used to provide closed-loop gait control. In addition, this approach is shown to demonstrate significant robustness to external disturbances. Finally, the FPE is shown in experimental results to be an unprecedented estimate of where humans place their feet for walking and jumping, and for stepping in response to an external disturbance.

D'un point de vue énergétique, les robots marcheurs sont moins performants que les humains. Face à ce défi, cette thèse propose une approche pour contrôler et optimiser les allures de marche des robots bipèdes à la fois en 2D et 3D en considérant les fréquences propres du robot et par ajout de ressorts. L'étude porte essentiellement sur un robot bipède 2D à 5 corps et des pieds ponctuels ainsi qu'un robot bipède 3D à 8 corps avec des pieds sans masse à contact linéique. La commande en boucle fermée considérée est basée sur la méthode des contraintes virtuelles et la linéarisation par retour d'état. Suite à des études précédentes, la stabilité du robot bipède 2D est vérifiée par une section de Poincaré unidimensionnelle et étendue au robot bipède 3D à contact linéique avec le sol. L'optimisation est effectuée en utilisant la programmation quadratique séquentielle. Les paramètres optimisés incluent des coefficients de polynômes de Bézier et des paramètres posturaux. Des contraintes d'optimisation sont imposées pour assurer la validité de l'allure de marche. Pour le robot bipède 2D, deux configurations différentes de ressorts placés aux hanches sont étudiées. Ces deux configurations ont permis de réduire le coût énergétique. Pour le robot bipède 3D, les paramètres d'optimisation sont séparés en deux parties : ceux décrivant le mouvement dans le plan sagittal et ceux du plan frontal. Les résultats de l'optimisation montrent que ces deux types de paramètres doivent être optimisés. Ensuite, des ressorts sont ajoutés respectivement par rapport au plan sagittal, par rapport au plan frontal puis dans les deux plans. Les résultats montrent que l'ajout des ressorts dans le plan sagittal permet de réduire significativement le coût énergétique et que l'association de ressorts dans le plan frontal améliore encore plus la consommation d'énergie. / From an energy standpoint, walking robots are less efficient than humans. In facing this challenge, this study aims to provide an approach for controlling and optimizing the gaits of both 2D and 3D bipedal robots with consideration for exploiting natural dynamics and elastic couplings. A 5-link 2D biped with point feet and an 8-link 3D biped with massless line feet are studied. The control method is based on virtual constraints and feedback linearization. Following previous studies, the stability of the 2D biped is verified by computing scalar Poincaré map in closed form, and now this method also applies to the 3D biped because of its line-foot configuration. The optimization is performed using sequential quadratic programming. The optimization parameters include postural parameters and Bézier coefficients, and the optimization constraints are used to ensure gait validity. For the 2D biped, two different configurations of hip joint springs are investigated and both configurations successfully reduce the energy cost. For the 3D biped, the optimization parameters are further divided into sagittal parameters and coronal parameters, and the optimization results indicate that both these parameters should be optimized. After that, hip joint springs are added respectively to the sagittal plane, the coronal plane and both these planes. The results demonstrate that the elastic couplings in the sagittal plane should be considered first and that the additional couplings in the coronal plane reduce the energy cost even further.

[pt] A aprendizagem por reforço, uma técnica influente para treinar sistemas inteligentes, ganhou destaque na academia e na indústria devido à sua capacidade de resolver problemas complexos sem modelos pré-existentes. No entanto, sua aplicação a sistemas do mundo real é desafiadora devido à sua complexidade causada por altas não linearidades, amostras limitadas e restrições. Consequentemente, a pesquisa nessa área tem se concentrado principalmente em simulação, onde os modelos podem ser facilmente testados e refinados. Neste trabalho, foi proposta uma estratégia de aprendizagem por reforço para um robô bípede do mundo real aprender o comportamento de caminhada do zero. Também é apresentado um desenho de sistema focado na redução de estresse e simplicidade, garantindo um desempenho robusto, incluindo uma placa de circuito impresso personalizada para o manuseio eficiente dos componentes elétricos. O software do sistema é dividido entre a placa do sistema mestre e o sistema baseado em ROS, permitindo a comunicação entre os componentes e resolvendo o problema de perda de dados e atraso na comunicação. A simulação do modelo do robô é desenvolvida na plataforma Mujoco, incorporando propriedades físicas e parâmetros ambientais. Os algoritmos DDPG, TD3 e SAC foram utilizados para aprendizado e avaliação da técnica de destilação de política para transferência de conhecimento para uma rede mais eficiente. Finalmente, foi avaliada a transferência do aprendizado para o mundo real apresentando um experimento preliminar de aprendizado do zero no mundo real. Os resultados demonstram a eficácia do projeto do sistema robótico e dos algoritmos de aprendizado, alcançando uma caminhada estável na simulação e um máximo de catorze passos na vida real com a destilação de política do algoritmo SAC. / [en] Reinforcement learning, an influential technique for training intelligent systems, has gained prominence in academia and industry due to its ability to solve complex problems without pre-existing models. However, its application to real-world systems is challenging due to its complexity caused by high non-linearities, limited samples, and constraints. Consequently, research in this area has mainly focused on simulation, where models can be easily tested and refined. In this work, a reinforcement learning strategy towards a real-world bipedal robot to learn walking behavior from scratch was proposed. We present a robot system design focused on stress reduction and simplicity, ensuring robust performance, including a custom printed circuit board (PCB) for efficient handling of electrical components. The system s software is divided into the master system board and the ROS system, allowing communication between components and addressing data loss and communication delay issues. The robot model simulation is developed on the Mujoco platform, incorporating physical properties and environmental parameters. We utilize Deep Deterministic Policy Gradient (DDPG), Twin-Delayed Deep Deterministic Policy Gradient (TD3), and Soft Actor-Critic (SAC) algorithms for learning and evaluating the policy distillation technique for transferring knowledge to a more efficient network. Finally, we evaluate the transfer of learning to the real world and present a preliminary experiment of learning from scratch in the real world. Our results demonstrate the effectiveness of the robotic system design and the learning algorithms, achieving stable walking in simulation and a maximum of fourteen steps in real life with the policy distillation of the SAC algorithm.