Stabilization of collective motion on a sphere

We provide a Lyapunov-based design of decentralized control laws that stabilize relative equilibria in a model of self-propelled particles that travel on the surface of a sphere. Such control laws have applications in planetary-scale mobile sensing networks in air, sea, and space. Relative equilibria of the closed-loop model include formations in which all of the particles travel around a common circular trajectory. Particle interaction can be time-invariant or time-varying and directed or undirected. The algorithm for time-invariant and undirected particle interaction uses a gradient-like control induced from the associated Laplacian matrix. The algorithm for time-varying and directed interaction replaces average quantities in the control law with dynamic consensus variables. An augmented Laplacian algorithm is also proposed to stabilize symmetric circular formations.     

Decentralized control of vehicle formations

This paper investigates a method for decentralized stabilization of vehicle formations using techniques from algebraic graph theory. The vehicles exchange information according to a pre-specified communication digraph, G. A feedback control is designed using relative information between a vehicle and its in-neighbors in G. We prove that a necessary and sufficient condition for an appropriate decentralized linear stabilizing feedback to exist is that G has a rooted directed spanning tree. We show the direct relationship between the rate of convergence to formation and the eigenvalues of the (directed) Laplacian of G. Various special situations are discussed, including symmetric communication graphs and formations with leaders. Several numerical simulations are used to illustrate the results.     

Formation control of mobile agents based on inter-agent distance dynamics

We propose a novel formation control strategy based on inter-agent distances for single-integrator modeled agents in the plane. Attempting to directly control the inter-agent distances, we derive a control law from the distance dynamics. The proposed control law achieves the local asymptotic stability of infinitesimally rigid formations. Triangular infinitesimally rigid formations are globally asymptotically stable under the proposed control law, with all squared distance errors exponentially and monotonically converging to zero. As an extension of existing results, the stability analysis in this paper reveals that any control laws related with the gradient law by multiplication of a positive matrix ensure the local asymptotic stability of infinitesimally rigid formations.     

Symmetry Invariance of Multiagent Formations in Self-Pursuit

Structure in the interconnection topology among individuals of a multiagent system plays a fundamental role in the system's steady-state and transient behaviors. This paper explores how certain interconnection topologies influence symmetry in a multiagent system's trajectories. It is shown how circulant connectivity preserves rotation, and in particular instances, dihedral group symmetries in a formation of locally interacting planar integrators. Moreover, it is revealed to what extent circulant connectivity is also necessary in order that symmetric formations remain symmetric under the multiagent system's dynamics.     

曲线插值约束的LOOP细分曲面

提出基于Loop细分方法的曲线插值方法,不需要修改细分规则,只需以插值曲线的控制多边形为中心多边形,向其两侧构造对称三角网格带,该对称三角网格带将收敛于插值曲线。因此,包含有该三角网格带的多面体网格的极限曲面将经过插值曲线。若要插值多条相交曲线只需在交点处构造全对称三角网格。运用该方法可在三角网格生成的细分曲面中插值多达六条的相交曲线。     

On the robustness to multiple agent losses in 2D and 3D formations

Multi‐agent formations have been recently the subject of many studies. An important operational challenge, largely unaddressed in the literature, is to ensure that functionality of the formation is retained should agents be lost through misadventure, mission reassignment, and so on. In the context of sensor networks, it is also important to allow for the loss of multiple sensors as the low quality of sensor hardware, common unattended implementations, and so on makes it a common issue. In this paper, we address these issues by proposing information structures that are tolerant to the loss of multiple agents. Using graph theory (and more specifically rigidity theory), we characterize several properties of such formations/networks in a unified framework: We characterize the k‐vertex rigidity property of the underling graph of such formations. This is performed by deriving a set of useful conditions that can form a guideline for designing agent‐loss‐tolerant formations. We elaborate the study by characterizing robust formations with the optimal number of control links. We also propose a set of operations preserving the tolerance to multiple agent losses in such formations. These operations provide flexibility in designing the formations in terms of several designing parameters (e.g., geometry, diameter, and max degree). Especially in the case of formations, the ability to handle controller adjustments in a distributed way is important, and the paper addresses this issue for a number of the robustness problems considered. Copyright © 2014 John Wiley & Sons, Ltd.     

Distance‐based undirected formations of single‐integrator and double‐integrator modeled agents in n‐dimensional space

We study the local asymptotic stability of undirected formations of single‐integrator and double‐integrator modeled agents based on interagent distance control. First, we show that n‐dimensional undirected formations of single‐integrator modeled agents are locally asymptotically stable under a gradient control law. The stability analysis in this paper reveals that the local asymptotic stability does not require the infinitesimal rigidity of the formations. Second, on the basis of the topological equivalence of a dissipative Hamiltonian system and a gradient system, we show that the local asymptotic stability of undirected formations of double‐integrator modeled agents in n‐dimensional space is achieved under a gradient‐like control law. Simulation results support the validity of the stability analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

具有时变时滞耦合的二阶多主体系统的编队控制

多主体系统的编队控制是一类重要的网络协同控制问题.研究了在有向连接拓扑结构下,具有时变时滞耦合的二阶多主体系统的编队控制问题.通过一种多层领导机制的框架建模,得到了时不变编队、时变编队和时变轨迹追踪3种编队问题的充分性条件,并证明了各种预期队列是以指数的收敛速度形成的.数值仿真进一步验证了理论结果的正确性,为该理论在实际中应用起到指导作用.     

Stabilization of sets with application to multi-vehicle coordinated motion

In this paper, we develop stability and control design framework for time-varying and time-invariant sets of nonlinear dynamical systems using vector Lyapunov functions. Several Lyapunov functions arise naturally in multi-agent systems, where each agent can be associated with a generalized energy function which further becomes a component of a vector Lyapunov function. We apply the developed control framework to the problem of multi-vehicle coordinated motion to design distributed controllers for individual vehicles moving in a specified formation. The main idea of our approach is that a moving formation of vehicles can be characterized by a time-varying set in the state space, and hence, the problem of distributed control design for multi-vehicle coordinated motion is equivalent to the design of stabilizing controllers for time-varying sets of nonlinear dynamical systems. The control framework is shown to ensure global exponential stabilization of multi-vehicle formations. Finally, we implement the feedback stabilizing controllers for time-invariant sets to achieve global exponential stabilization of static formations of multiple vehicles.     

基于神经网络的高阶随机非线性系统的状态反馈控制

基于神经网络(NN)研究了一类含有未知非线性项的高阶随机不确定系统的自适应状态反馈控制问题. 通过引入径向基函数神经网络(RBF NN) 逼近方法, 运用 backstepping 技术以及选择合适的 Lyapunov 函数, 我们构造了一个自适应状态反馈控制器使得闭环系统是半全局一致最终有界的. 仿真例子验证了设计方法的有效性.     

Formation shape control based on bearing rigidity

Distance measurements are not the only geometric quantities that can be used for multi-agent formation shape control. Bearing measurements can be used in conjunction with distances. This article employs bearing rigidity for mobile formations, which was developed for robot and sensor network localisation, so that bearings can be used for shape control in mobile formations. The first part of this article examines graph theoretical models for formation network analysis and control law design that are needed to maintain the shape of a formation in two-dimensional space, while the formation moves as a cohesive whole. Bearing-based shape control for a formation of mobile agents involves the design of distributed control laws that ensure the formation moves, so that bearing constraints maintain some desired values. The second part of this article focuses on the design of a distributed control scheme for nonholonomic agents to solve the bearing-based formation shape control problem. In particular, a control law using feedback linearisation is proposed based on shape variables. We simulate the shape control behaviour on differential drive agents for an exemplary bearing rigid formation using the results obtained in the first and second parts of this article.     

Three and higher dimensional autonomous formations: Rigidity, persistence and structural persistence

In this paper, we generalize the notion of persistence, which has been originally introduced for two-dimensional formations, to R^d for d?3, seeking to provide a theoretical framework for real world applications, which often are in three-dimensional space as opposed to the plane. Persistence captures the desirable property that a formation moves as a cohesive whole when certain agents maintain their distances from certain other agents. We verify that many of the properties of rigid and/or persistent formations established in R² are also valid for higher dimensions. Analysing the closed subgraphs and directed paths in persistent graphs, we derive some further properties of persistent formations. We also provide an easily checkable necessary condition for persistence. We then turn our attention to consider some practical issues raised in multi-agent formation control in three-dimensional space. We display a new phenomenon, not present in R², whereby subsets of agents can behave in a problematic way. When this behaviour is precluded, we say that the graph depicting the multi-agent formation has structural persistence. In real deployment of controlled multi-agent systems, formations with underlying structurally persistent graphs are of interest. We analyse the characteristics of structurally persistent graphs and provide a streamlined test for structural persistence. We study the connections between the allocation of degrees of freedom (DOFs) across agents and the characteristics of persistence and/or structural persistence of a directed graph. We also show how to transfer DOFs among agents, when the formation changes with new agent(s) added, to preserve persistence and/or structural persistence.     

Controllability and motion algorithms for underactuated Lagrangiansystems on Lie groups

We provide controllability tests and motion control algorithms for underactuated mechanical control systems on Lie groups with Lagrangian equal to kinetic energy. Examples include satellite and underwater vehicle control systems with the number of control inputs less than the dimension of the configuration space. Local controllability properties of these systems are characterized, and two algebraic tests are derived in terms of the symmetric product and the Lie bracket of the input vector fields. Perturbation theory is applied to compute approximate solutions for the system under small-amplitude forcing; in-phase signals play a crucial role in achieving motion along symmetric product directions. Motion control algorithms are then designed to solve problems of point-to-point reconfiguration, static interpolation and exponential stabilization. We illustrate the theoretical results and the algorithms with applications to models of planar rigid bodies, satellites and underwater vehicles     

Using Sensor Morphology for Multirobot Formations

In formation-maintenance (formation control) tasks, robots maintain their relative position with respect to their peers, according to a desired geometric shape. Previous work has examined formation-maintenance algorithms, based on formation control graphs, that ensure the theoretical stability of the formation. However, an exponential number of stable controllers exists. Thus a key question is how to select (construct) a formation controller that optimizes desired properties, such as sensor usage. We present a novel representation of the sensing capabilities of robots in formations, using a monitoring multigraph. We first show that graph-theoretic techniques can then be used to efficiently compute optimal sensing policies that maintain a given formation, while minimizing sensing costs. In particular, separation-bearing (distance-angle) control targets are automatically constructed for each individual robot in the formation, taking into account its specific sensor morphology. Then, we present a protocol allowing control graphs to be switched on line, to allow robots to adjust to sensory failures. We report on results from comprehensive experiments with physical and simulated robots. The results show that the use of the dynamic protocol allows formations of real robots to move significantly faster and with greater precision, while reducing the number of formation failures, due to sensor limitations. We also evaluate the sensitivity of our approach to communication reliability, and discuss opportunities and challenges raised by our approach.     

A Graph Theoretic Approach for Modeling Mobile Robot Team Formations

This paper addresses a new approach for modeling and control of multiple teams of mobile robots navigating in a terrain with obstacles, while maintaining a desired formation and changing formations when required. We model each team as a triple, (g,r, ?? ), consisting of a group element, g∈SE(2), that describes the gross position of the lead robot, a set of shape variables, r, that describe the relative positions of robots, and a control graph, ??, that describes the behaviors of the robots in the formation. We assume that all the robots are equipped with the appropriate sensors to detect and avoid other robots and obstacles in the environment. Our framework enables the representation and enumeration of possible control graphs, and the coordination of transitions between any two control graphs. Further, we describe an algorithm that allows each team of robots to move between any two formations, while avoiding obstacles. As the number of robots increases, the number of possible control graphs increases. However, because the control computations are decentralized, the algorithms scale with the number of robots. We present examples to illustrate the control graphs and the algorithm for transitioning between them in the presence and absence of sensor noise. © 2002 Wiley Periodicals, Inc.     

一类分布式队形控制及其稳定性分析

针对一类具有二阶动态行为的多机器人系统的队形控制问题, 提出了一种分布式离散协同控制算法, 并应用代数图论和矩阵论的方法对该系统的渐近稳定性和算法的一致收敛性进行分析. 应用上述方法, 证明了确保多智能体系统渐近收敛的充要条件,得到了反馈控制参数的取值范围. 同时证明了在该充要条件下多机器人将逐步收敛到期望队形和同一运动速度. 仿真部分通过一个六机器人系统的队形控制验证了本文研究结果的正确性.     

Nonlinear control strategy development for asymmetric actuators

We develop a nonlinear control method for asymmetric actuators. Asymmetric actuators do not generate symmetric power during an application. Thrusters and shape memory alloy wires are examples of this class of actuators that produce only unidirectional forces. Hydraulic and pneumatic cylinders are generally asymmetric because, under constant pressure, the generated force in the forward direction is different than that in the reverse direction. Existing control methods assume a symmetric actuation, and therefore, application of asymmetric actuators calls for new control schemes. Current control techniques implement a bias in utilizing asymmetric actuators. However, the amount of the bias depends on the control effort and is not constant. Also, the use of a bias changes the system equilibrium point and introduces a steady-state error. We propose a control scheme capable of producing any biased input. The controller is a second-order system coupled to the system through quadratic terms. The application of quadratic terms for the control input enables us to generate any biased control input, which can be utilized by asymmetric actuators     

Spectral-Based Group Formation Control

Given a pair of keyframe formations for a group consisting of multiple individuals, we present a spectral-based approach to smoothly transforming a source group formation into a target formation while respecting the clusters of the involved individuals. The proposed method provides an effective means for controlling the macroscopic spatiotemporal arrangement of individuals for applications such as expressive formations in mass performances and tactical formations in team sports. Our main idea is to formulate this problem as rotation interpolation of the eigenbases for the Laplacian matrices, each of which represents how the individuals are clustered in a given keyframe formation. A stream of time-varying formations is controlled by editing the underlying adjacency relationships among individuals as well as their spatial positions at each keyframe, and interpolating the keyframe formations while producing plausible collective behaviors over a period of time. An interactive system of editing existing group behaviors in a hierarchical fashion has been implemented to provide flexible formation control of large crowds.     

Stabilizability of the angular velocity of a rigid body revisited

We prove that the jurdjevic-Quinn techniques for smooth stabilization can be applied to the equation of the angular velocity of a rigid body. An explicit polynomial control is given for the symmetric case.     

Stability and robustness of large platoons of vehicles with double‐integrator models and nearest neighbor interaction

We study the stability and robustness of a large platoon of vehicles, where each vehicle is modeled as a double integrator, for two decentralized control architectures: predecessor following and symmetric bidirectional. In the predecessor‐following architecture, the control action on each agent only depends on the information from its immediate front neighbor, whereas in the symmetric bidirectional architecture, it depends equally on the information from both its immediate front neighbor and back neighbor. We prove asymptotic stability of the formation for a class of nonlinear controllers with sector nonlinearity, with the linear controller as a special case. We show that the convergence rate of the predecessor‐following architecture is much faster than that of the symmetric bidirectional architecture. However, the predecessor‐following architecture suffers high algebraic growth of initial errors. We also establish scaling laws (with N) of certain H _∞ norms of the formation that measure its robustness to external disturbances for the linear case. It is shown that the robustness performance grows geometrically in N for predecessor‐following architecture but only polynomially in N for symmetric‐bidirectional architecture. Extensive numerical simulations are conducted to verify the predictions for the linear case and empirically estimate the corresponding performance metrics for a saturation‐type nonlinear controller. On the basis of the analytical and numerical results, it is seen that the symmetric bidirectional architecture outperforms the predecessor‐following architecture in all measures of performance. Within the predecessor‐following architecture, the nonlinear controller is seen to perform better in general than the linear one. A number of design guidelines are provided on the basis of these conclusions. Copyright © 2012 John Wiley & Sons, Ltd.