Controlling Swarm Systems and Soft Continuum Robots Based on Partial Differential Equations
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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.
本博士论文致力于解决机器人学中的可扩展性挑战——这一关键难题长期阻碍了先进机器人系统在日常场景中的广泛应用。当系统自由度提升、任务复杂度增加，或受控机器人数量增多时，均可能产生可扩展性问题，此类挑战会对系统的有效性、可靠性及达成运行目标的能力造成负面影响。 本论文重点研究两类场景：一类是子系统数量大幅增长的集群系统（swarm system）场景，另一类是机器人自由度显著提升的连续体机器人（continuum robot）场景。本文对两类集群系统展开研究：第一类为集群机器人系统，旨在探索如何设计大量相对简单的智能体（agent），使其通过自身间及与环境的局部交互涌现出预期的集体行为。随着机器人数量增长，预测与控制该集体行为的难度将不断提升，因此核心挑战在于平衡涌现行为的可保障性与系统的去中心化特性。第二类集群系统场景为：利用多机器人群体与大规模人类人群进行交互，例如开发机器人引导的人群疏散系统。在此场景下，关键挑战在于通过机器人与人类的局部交互，操控一组自主机器人间接调控人群行为。针对由高自由度引发的可扩展性问题，本文聚焦连续体机器人：此类机器人采用柔性材料构建，可沿整体结构连续弯曲，理论上拥有无穷多自由度。以往研究大多依赖将系统简化至低维度的方法，因此核心难点在于平衡模型精度、计算速度与模型复杂度三者间的关系。 为解决可扩展性问题，本论文研究基于连续介质力学的建模与控制方法。连续介质力学是力学的一个分支，聚焦于在空间中连续分布的材料（如固体、液体与气体）的物理特性。该建模策略采用连续介质近似，将系统尺度拓展至无穷大，并通过偏微分方程（Partial Differential Equations，简称PDEs）实现对系统动力学的简洁描述。基于该建模思路，可借鉴连续介质力学中的扩散、能量耗散等概念指导控制器设计。 针对集群机器人的研究，源于对环境管控任务的需求——此类任务要求灵活调控机器人的时空分布。为此，我们采用统计力学中的平均场模型（mean-field model），将集群视为连续介质，通过PDEs描述其密度演化过程。受扩散过程启发，我们设计了反馈速度场与机器人控制器，以调控集群密度的演化；同时提出了基于通信的算法，使单个机器人能够估算当前的集群密度。该方法可突破涌现行为保障与去中心化之间的权衡，为集群控制构建出可验证且具备可扩展性的框架。我们还将类似思路应用于机器人引导的人群疏散任务，通过平均场流体动力学模型对人群密度演化进行建模，并为机器人设计导航控制器，使其在人群中生成局部力场以辅助人群疏散。 针对连续体机器人的研究，源于外科场景下对精准且稳定的感知与控制的需求。为此，我们采用融合固体力学与几何力学的精确几何科瑟拉杆（Cosserat rod）模型，旨在避免任何可能导致显著建模误差的近似或简化操作。在科瑟拉杆框架下，连续体机器人被建模为沿中心轴线堆叠的连续刚性截面序列，其运动学与动力学通过几何李群上的PDEs进行描述。受能量耗散概念的启发，我们开发了仅基于末端测量值即可重构连续体机器人状态的估计算法，以及用于操控机器人构型的控制算法。
