Controlling Swarm Systems and Soft Continuum Robots Based on Partial Differential Equations

This dissertation is dedicated to the scalability challenge in robotics--a crucial issue that has hindered the extensive use of advanced robotic systems in our everyday scenarios. Scalability issues can arise as the system grows in degree of freedom, complexity in tasks, or the number of robots under control. These challenges can influence the system's effectiveness, reliability, and capacity to achieve operational objectives. This dissertation specifically examines two scenarios: one where there is a significant increase in the number of subsystems, as seen in swarm systems, and another where there is a substantial increase in the degrees of freedom of the robot, as observed in continuum robots. We examine two categories of swarm systems. The first one is swarm robotic systems, which investigate how a large number of relatively simple physical agents can be designed to display a desired collective behavior that emerges from local interactions among the agents themselves and with their surroundings. When the number of robots grows, predicting and controlling the collective behavior becomes increasingly challenging. Consequently, the main challenge revolves around balancing the guarantee of emergent behavior and decentralization. The second situation in swarm systems occurs when a group of robots is employed to interact with a large human crowd, for instance, in developing a robot-guided human crowd evacuation system. In this case, the key challenge lies in controlling a set of autonomous robots to indirectly regulate the human swarm through local interactions between humans and robots. When it comes to scalability resulting from a large number of degrees of freedom, we focus on continuum robots. These robots are constructed using flexible materials, can bend continuously throughout their structure, and theoretically have an infinite number of degrees of freedom. Previous studies have mostly depended on simplifying the system into lower dimensions. Therefore, the main difficulty lies in balancing the accuracy of the model and the computational speed in relation to the complexity of the model. In order to tackle the issue of scalability, this dissertation investigates modeling and control approaches rooted in continuum mechanics. Continuum mechanics is a branch of mechanics that focuses on the physical properties of materials that are considered to be continuously distributed throughout their spatial area, such as solids, liquids, and gases. This modeling strategy employs a continuum approximation that expands the system's scale to infinity and achieves a concise description of the system's dynamics using partial differential equations (PDEs). On the basis of this modeling approach, concepts in continuum mechanics, such as diffusion and energy dissipation, can be used to inspire the design of controllers. Our research on swarm robots is driven by the need for environmental control tasks that demand flexible management of robots' spatial and temporal distribution. To address this, we use mean-field models from statistical mechanics, which treat the swarm as a continuum and describe its density evolution through PDEs. Drawing inspiration from diffusion processes, we develop feedback velocity fields and robot controllers to regulate the evolution of the swarm density. Additionally, we introduce communication-based algorithms for individual robots to estimate the current swarm density. This approach allows us to overcome the trade-off between emergent behavior guarantees and decentralization, leading to a verifiable and scalable framework for swarm control. We also apply similar concepts to robot-guided human crowd evacuation, where we model the crowd density evolution using mean-field hydrodynamic models. Then, we design controllers for the robots to navigate within the human crowd, creating local force fields to facilitate crowd evacuation. Our research on continuum robots is driven by the need for precise and stable sensing and control in surgical scenarios. To achieve this, we employ geometrically exact Cosserat rod models from the fusion of solid mechanics and geometric mechanics. By employing these models, we aim to avoid any approximations or simplifications that may lead to significant modeling inaccuracies. In the Cosserat rod framework, a continuum robot is conceptualized as a continuous series of rigid cross-sections stacked along a central axis, with its kinematics and dynamics described through PDEs in geometric Lie groups. Drawing inspiration from the concept of energy dissipation, we develop estimation algorithms to construct the states of the continuum robot based solely on tip measurements, along with control algorithms to manipulate the robot's configuration.