技术领域technical field
本发明涉及一种信息弱化条件下具有连通性保持的拉格朗日系统自适应控制方法，属于智能机器人技术领域。The invention relates to a Lagrangian system adaptive control method with connectivity maintenance under the condition of information weakening, belonging to the technical field of intelligent robots.
背景技术Background technique
近年来，多智能体系统的分布式运动协调控制与应用受到日益广泛的关注，逐渐成为复杂性科学研究的一个焦点问题。其中各智能体仅利用局部信息进行交互,并结合通信等手段发挥分布式资源的优势实现整体规划、解决局部冲突,从而达到整体预期目标。多智能体系统作为一门综合性交叉学科，其在工业和军事等应用领域均具有广泛的应用前景和重要的理论研究价值。同时拉格朗日系统的分布式协同控制研究是多智能体系统协同控制研究领域内的重要组成部分。由于欧拉拉格朗日方程能够用于描述机械臂以及航天器等大量实际机械系统的动力学特性，针对拉格朗日系统的研究具有很强的工程适用性以及广阔的应用前景。In recent years, the distributed motion coordination control and application of multi-agent systems has received more and more attention, and has gradually become a focus of complexity science research. Among them, each agent only uses local information to interact, and combines the advantages of distributed resources with communication and other means to achieve overall planning and solve local conflicts, so as to achieve the overall expected goal. As a comprehensive interdisciplinary subject, multi-agent system has broad application prospects and important theoretical research value in industrial and military application fields. At the same time, the research on distributed cooperative control of Lagrangian systems is an important part of the research field of multi-agent system cooperative control. Since the Euler Lagrangian equation can be used to describe the dynamic characteristics of a large number of practical mechanical systems such as manipulators and spacecraft, the research on Lagrangian systems has strong engineering applicability and broad application prospects.
在多智能体系统中，所有智能体利用自身配置的多种传感器和执行器来感知环境并对环境的变化做出适当的反应，而整个多智能体系统在某种程度上可以视为一个移动传感器网络和执行器网络；同时，整个多智能群体在协同操作过程中通过自身所配备的通信设备进行信息交互和共享，使得整个系统在某种程度上又成为一个通信互联网络。由于拉格朗日系统为实际情况下的真实系统，同时现实情况中往往环境较为复杂，突发情况较多，容易出现诸如传感器故障、少数智能体之间通讯丢失等情况，为了提高多智能体系统对环境的适应能力，在保持整个多智能体系统连通的前提下减少多智能体的连通程度即减少每个智能体所能连通的智能体个数以及减少智能体协同控制所需要的信息量以达到减少智能体所配备的传感器数量，这样可以在保证多智能体网络连通性的前提下，以尽可能低的连通度以及尽可能少的通讯信息使整个多智能体系统实现一致性，以此提高多智能体系统对环境的适应能力。在弱化信息的同时，保持多智能体网络的网络连通性也至关重要。多智能体网络的通信连通性保持作为多智能体网络拓扑控制中最基础和最重要的问题之一，是绝大多数应用于多智能体系统协调控制算法例如一致性控制，群体交汇与聚集，队形保持与变换，协作区域探索与覆盖等实现收敛的必要条件。然而，由于单个智能体受到功能设计和硬件条件等因素的制约，其自身所配置的传感器传播和感应信息能力范围十分有限，使得在多智能体系统的网络拓扑随时间演化的过程受智能体的空间位置的影响非常大，如果不采取适当措施，尤其对于一些突发情况，如多障碍物环境或者通讯收到干扰等，网络连通性可能会被破坏，此时多智能体系统便无法实现一致性以及稳定性。In a multi-agent system, all agents use their own sensors and actuators to perceive the environment and respond appropriately to changes in the environment, and the entire multi-agent system can be regarded as a mobile Sensor network and actuator network; at the same time, the entire multi-intelligence group performs information exchange and sharing through its own communication equipment during the collaborative operation process, making the entire system a communication interconnection network to some extent. Since the Lagrangian system is a real system in actual conditions, and the real environment is often more complex, there are many unexpected situations, such as sensor failure, communication loss between a small number of agents, etc., in order to improve the multi-agent The adaptability of the system to the environment, on the premise of maintaining the connectivity of the entire multi-agent system, reducing the degree of multi-agent connectivity means reducing the number of agents that each agent can connect to and reducing the amount of information required for cooperative control of agents In order to reduce the number of sensors equipped by the agent, it can achieve consistency in the entire multi-agent system with the lowest possible connectivity and as little communication information as possible under the premise of ensuring the connectivity of the multi-agent network. This improves the adaptability of the multi-agent system to the environment. While weakening the information, it is also crucial to maintain the network connectivity of the multi-agent network. As one of the most basic and important issues in multi-agent network topology control, the communication connectivity maintenance of multi-agent networks is used in most coordination control algorithms of multi-agent systems, such as consensus control, group convergence and aggregation, Formation maintenance and transformation, collaborative area exploration and coverage are necessary conditions for convergence. However, because a single agent is constrained by factors such as functional design and hardware conditions, the range of sensor dissemination and sensing information capabilities configured by itself is very limited, so that the network topology evolution process of the multi-agent system over time is affected by the agent. The influence of spatial location is very large. If appropriate measures are not taken, especially for some unexpected situations, such as multi-obstacle environment or communication interference, the network connectivity may be destroyed. At this time, the multi-agent system cannot achieve consensus. sex and stability.
注意到在实际环境中突发情况较多，针对信号丢失以及传感器故障等问题，弱化信息可以使多智能体系统在通讯拓扑中有少量网络连接丢失以及智能体的部分信息缺失的情况下依旧保持稳定性以及一致性。同时，由于通信网络拓扑的连通性依赖于智能体的空间位置分布，因此可以考虑通过从虚拟智能体间的相互作用力的角度构造势能函数，使智能体间产生吸引/排斥作用力控制智能体间的间距来控制网络的连通。在保证网络在整个控制过程中始终满足通信连通性这一约束前提下针对弱化信息问题设计自适应分布式运动协调算法实现对拉格朗日系统进行控制，对于大规模实际工程群体系统的应用具有重大的理论意义和实践意义。Note that there are many emergencies in the actual environment. For problems such as signal loss and sensor failure, weakening information can make the multi-agent system maintain stability and consistency. At the same time, since the connectivity of the communication network topology depends on the spatial location distribution of the agents, it can be considered to construct a potential energy function from the perspective of the interaction force between the virtual agents to generate attractive/repulsive forces between the agents to control the agents The distance between them controls the connectivity of the network. Under the premise of ensuring that the network always satisfies the constraint of communication connectivity in the entire control process, an adaptive distributed motion coordination algorithm is designed for the weakened information problem to realize the control of the Lagrangian system, which is of great significance for the application of large-scale practical engineering group systems. Great theoretical and practical significance.
信息弱化下兼具连通性保持针对拉格朗日系统的跟踪控制在国内外均处于探索研究阶段，针对弱化信息下的跟踪控制处理，目前很多成果都是基于集中式或半分布式控制的角度或者通过构造观测器对未知信息进行估计等进行解决，针对的系统也主要为较为理想二阶积分器系统，同时研究中通常人为割裂通信连通性保持和分布式控制之间的联系，大部分研究都是在事先假设连通性保持的前提下进行的，缺乏将连通性作为约束条件与具体的跟踪控制任务的目标相结合的综合控制方案。Maintaining connectivity under information weakening Tracking control for Lagrangian systems is still in the exploratory research stage at home and abroad. For tracking control processing under weakened information, many current achievements are based on the perspective of centralized or semi-distributed control Or solve the problem by constructing an observer to estimate unknown information, etc. The targeted system is mainly a relatively ideal second-order integrator system. At the same time, the research usually artificially separates the connection between communication connectivity maintenance and distributed control. Most of the research They are all carried out on the premise that the connectivity is maintained in advance, and there is no comprehensive control scheme that combines the connectivity as a constraint with the specific goal of the tracking control task.
发明内容Contents of the invention
本发明的目的是为解决在只能获取领航者位置信息的条件时拉格朗日系统在连通性保持的约束条件下的跟踪控制问题，将自适应控制与人工势场法相结合，可以使系统在演化的过程中达到预期的控制目标。The purpose of the present invention is to solve the tracking control problem of the Lagrangian system under the constraint condition of connectivity maintenance when only the position information of the navigator can be obtained, and the adaptive control and the artificial potential field method can be combined to make the system In the process of evolution, the expected control goal is achieved.
本发明的技术方案如下：Technical scheme of the present invention is as follows:
一种具有连通性保持的拉格朗日系统自适应控制方法，包括以下步骤：A method for adaptive control of Lagrange systems with connectivity preservation, comprising the following steps:
步骤一、确立智能体的数学模型：考虑具有N个智能体的群组在n维欧式空间中移动，智能体的数学模型为拉格朗日系统模型，即典型的非线性系统模型，拉格朗日系统的数学模型的描述如下：Step 1. Establish the mathematical model of the agent: Consider a group with N agents moving in n-dimensional Euclidean space. The mathematical model of the agent is a Lagrangian system model, that is, a typical nonlinear system model, Lagrange The mathematical model of the daily system is described as follows:
其中是智能体i的位置向量，为qi的一阶导数，也即是智能体i的速度向量，qij＝qi-qj为智能体i与智能体j的相对位置向量，τi∈R2是作用于智能体i的控制输入；分别为惯量矩阵以及科里奥利力的离心矩阵；gi∈R2为智能体i的重力向量；in is the position vector of agent i, is the first-order derivative of qi , that is, the velocity vector of agent i, qij = qi -qj is the relative position vector of agent i and agent j, τi ∈ R2 acts on agent i control input; are the inertia matrix and the centrifugal matrix of Coriolis force respectively; gi ∈ R2 is the gravity vector of agent i;
步骤二、基于步骤一确定的数学模型设计信息弱化下兼具连通性保持的自适应跟踪控制律设计，该控制律包括三个部分：第一部分为连通性保持部分；第二部分为自适应控制律部分；其中自适应控制律部分引入两个自适应变量，其中一个自适应变量为所有伴随者之间的速度一致性增益，也即速度耦合强度，另一个为所有伴随者与领航者之间的位置一致性增益，也即位置导航反馈，其中所述的速度耦合强度的变化速率为使用当前伴随者与其他所有伴随者的速度误差的二次型乘以控制增益，位置导航反馈的变化速率为当前伴随者与领航者之间的位置误差的二次型乘以控制增益；第三部分为控制律余项，用于消除大量与数学模型相关的冗余项；Step 2. Based on the mathematical model determined in step 1, design an adaptive tracking control law with connectivity preservation under weakened information. The control law includes three parts: the first part is the connectivity maintaining part; the second part is the adaptive control The law part; the adaptive control law part introduces two adaptive variables, one of which is the speed consistency gain between all followers, that is, the speed coupling strength, and the other is the speed coupling strength between all followers and the leader. The position consistency gain of , that is, the position navigation feedback, wherein the rate of change of the velocity coupling strength is the quadratic form of the velocity error between the current companion and all other followers multiplied by the control gain, the rate of change of the position navigation feedback The quadratic form of the position error between the current companion and the leader is multiplied by the control gain; the third part is the remainder of the control law, which is used to eliminate a large number of redundant terms related to the mathematical model;
步骤三、自适应跟踪控制律的仿真实验验证：预设实验仿真的固定参数以及选取实验的内容，固定参数分为3个部分：A、步骤一得到的智能体的数学模型参数；B、选取人工势场的参数，包括势能函数的边界值以及极小值的位置；C、系统仿真参数；其次引入步骤二设计的控制律，并调节控制律中的自适应控制变量，在所述实验仿真的固定参数以及实验内容后编写仿真实验的Matlab程序，通过调节所述两个自适应变量值能够加快或者减慢系统收敛的快慢，也即实现伴随者对领航者的跟踪控制的快慢，多次调解该两个变量，得到多个控制律实现系统收敛的时间，将多个变量值以及对应的收敛时间进行记录并比较便可以得到一组相对最优的变量值，即完成所述的自适应控制方法。Step 3. Simulation experiment verification of adaptive tracking control law: preset the fixed parameters of the experimental simulation and select the content of the experiment. The fixed parameters are divided into 3 parts: A. The mathematical model parameters of the agent obtained in step 1; B. Select The parameters of the artificial potential field, including the boundary value of the potential energy function and the position of the minimum value; C, system simulation parameters; secondly, introduce the control law designed in step 2, and adjust the adaptive control variable in the control law, in the experimental simulation Write the Matlab program of the simulation experiment after the fixed parameters and the experimental content. By adjusting the two adaptive variable values, the speed of system convergence can be accelerated or slowed down, that is, the speed of the follower to the leader's tracking control can be realized. By mediating the two variables, the time for multiple control laws to achieve system convergence can be obtained, and a set of relatively optimal variable values can be obtained by recording and comparing the values of multiple variables and the corresponding convergence times, that is, completing the self-adaptive Control Method.
在智能体的运动过程中，上述的两个自适应变量根据所获取的信息不断更新自身的值，进而实现伴随着对领航者的跟踪控制。During the movement of the agent, the above two adaptive variables constantly update their own values according to the acquired information, and then realize the tracking control of the navigator.
采用人工势场函数对所有智能体进行控制约束，当智能体之间的距离达到通讯范围时，人工势场函数会使智能体之间产生一个无穷大的引力，使智能体之间的网络拓扑能够始终保持连通。The artificial potential field function is used to control and constrain all the agents. When the distance between the agents reaches the communication range, the artificial potential field function will generate an infinite gravitational force between the agents, so that the network topology between the agents can be Always stay connected.
本发明的有益效果：Beneficial effects of the present invention:
本发明研究了信息弱化条件下具有连通性保持的拉格朗日系统的分布式跟踪控制，针对在多智能体系统分布式跟踪控制中无法获取到领航者速度的问题，提出了将自适应控制律与连通性保持相结合的控制策略，使得在只获取领航者位置信息的情况下，智能体在运动过程中能够始终保持连通性同时能够同时实现对匀速运动的领航者的跟踪控制。本发明中设计的控制方法解决了拉格朗日系统在跟踪控制时无法获取领航者速度信息的问题，为今后多智能体分布式跟踪控制在复杂环境中的应用提供了一条思路，并且通过相应参数的调节，可以使伴随者以较快的收敛速度实现与领航者的速度同步，完成区域探索，编队控制等多种控制目标提供解决办法。The present invention studies the distributed tracking control of the Lagrangian system with connectivity preservation under the condition of information weakening, and aims at the problem that the speed of the navigator cannot be obtained in the distributed tracking control of the multi-agent system, and proposes an adaptive control The control strategy combining law and connectivity maintenance enables the agent to maintain connectivity and simultaneously realize the tracking control of the uniformly moving navigator while only obtaining the position information of the navigator. The control method designed in the present invention solves the problem that the Lagrangian system cannot obtain the speed information of the navigator during tracking control, and provides a way of thinking for the application of multi-agent distributed tracking control in complex environments in the future, and through corresponding The adjustment of the parameters can enable the companion to synchronize with the speed of the leader at a faster convergence speed, and provide solutions for various control objectives such as area exploration and formation control.
附图说明Description of drawings
图1—所有智能体的初始状态(t＝0s)；Figure 1—Initial state of all agents (t=0s);
图2—领航者匀速直线运动时所有智能体在不同时刻的状态；Figure 2—The state of all agents at different moments when the navigator moves in a straight line at a uniform speed;
图3—领航者匀速直线运动时所有智能体的最终状态；Figure 3—The final state of all agents when the navigator moves in a straight line at a uniform speed;
图4—领航者匀速直线运动时伴随者与领航者的速度误差；Figure 4—The speed error between the companion and the leader when the leader moves in a straight line at a uniform speed;
图5—领航者匀速直线运动时的代数连通度；Figure 5—The algebraic connectivity of the navigator when moving in a straight line at a uniform speed;
图6—领航者匀速圆周运动时所有智能体在不同时刻的状态；Figure 6—The state of all agents at different moments when the navigator moves in a uniform circular motion;
图7—领航者匀速圆周运动时所有智能体的最终状态；Figure 7—The final state of all agents when the navigator moves in a uniform circular motion;
图8—领航者匀速圆周运动时伴随者与领航者的速度误差。Figure 8—The speed error between the companion and the leader when the leader moves in a uniform circular motion.
具体实施方式detailed description
下面结合附图和实施例对本发明做进一步说明：Below in conjunction with accompanying drawing and embodiment the present invention will be further described:
考虑具有N个智能体的移动智能体系统，动态特性由式(1)给出。稳定的跟踪控制是指所有伴随者均能够与领航者渐近达到一致的速度，同时控制过程中智能体之间避免碰撞。Considering a mobile agent system with N agents, the dynamic characteristics are given by Equation (1). Stable tracking control means that all followers can asymptotically reach the same speed as the leader, while avoiding collisions between agents during the control process.
为了详细介绍本发明，首先介绍本发明中使用的代数图论方法。In order to introduce the present invention in detail, the algebraic graph theory method used in the present invention is first introduced.
动态通信图G(t)＝{V,E(t)}是一个时变的无向图，由N个顶点的顶点集V＝{n1,n2,...,nN}和边集组成，顶点表示群组中的智能体，边集包含有邻居关系的顶点的无序对。用Ni＝{nj|(ni,nj)∈E(t)}表示智能体i的所有通信邻居。由于不允许自环，所以定义带权重的邻接矩阵为A＝[aij]∈RN×N其中aij∈[0,1]且其大小与通讯距离相关。无向图的度矩阵用D＝diag{di}表示，其中表示该对角阵的每一个元素。以此定义通信图G对应的拉普拉斯矩阵为L＝D-A，该矩阵为半正定矩阵且其特征值满足0＝λ1≤λ2≤...≤λN，拉普拉斯矩阵L还满足L1＝0，其中1元素均为1的列向量。同时当通信图G连通时有N(L)＝span(1)，其中N(□)为核空间。为了描述智能体之间的时变邻居关系，拓扑连通图的边集E(t)必须满足以下条件：The dynamic communication graph G(t)={V,E(t)} is a time-varying undirected graph, which consists of a vertex set of N vertices V={n1 ,n2 ,...,nN } and edges set Composition, the vertices represent the agents in the group, and the edge set contains unordered pairs of vertices with neighbor relations. Let Ni ={nj |(ni ,nj )∈E(t)} represent all communication neighbors of agent i. Since self-loops are not allowed, so Define the weighted adjacency matrix as A=[aij ]∈RN×N where aij ∈[0,1] and its size is related to the communication distance. The degree matrix of an undirected graph is represented by D=diag{di }, where represents each element of the diagonal matrix. In this way, the Laplacian matrix corresponding to the communication graph G is defined as L=DA, which is a positive semi-definite matrix and whose eigenvalues satisfy 0=λ1 ≤λ2 ≤...≤λN , the Laplacian matrix L It also satisfies L1=0, where the 1 elements are all 1 column vectors. At the same time, when the communication graph G is connected, N(L)=span(1), where N(□) is the kernel space. In order to describe the time-varying neighbor relationship between agents, the edge set E(t) of the topologically connected graph must satisfy the following conditions:
1)初始链路E(0)＝{(ni,nj|||xij(0)||＜R,ni,nj∈V}；1) Initial link E(0)={(ni ,nj |||xij (0)||<R,ni ,nj ∈V};
2)如果在时刻t之前智能体i和j不是邻居，并且||xij(t)||＜R-δ(0＜δ＜R)，那么在它们之间加入一条新的链路；2) If agents i and j are not neighbors before time t, and ||xij (t)||<R-δ (0<δ<R), then add a new link between them;
3)如果||xij(t)||＞R，那么从E(t)中删除链路(ni,nj)3) If ||xij (t)||＞R, then delete link (ni ,nj ) from E(t)
定义邻接矩阵A中交互权重aij(t)为：Define the interaction weight aij (t) in the adjacency matrix A as:
其中0＜δ＜R是一个固定的切换阈值。Where 0<δ<R is a fixed switching threshold.
假设所有的智能体有相同的感知半径R，通信连接只在距离小于R的智能体间存在，为了描述感知网络的信息流，给出下面的定义：Assuming that all agents have the same perception radius R, communication connections only exist between agents whose distance is less than R, in order to describe the information flow of the perception network, the following definition is given:
图的连通性：若一个动态图G(t)中任意两顶点间存在一条路径，则G(t)在t时刻是连通的。Graph connectivity: If there is a path between any two vertices in a dynamic graph G(t), then G(t) is connected at time t.
接着介绍本发明中使用的人工势能方法。Next, the artificial potential energy method used in the present invention is introduced.
人工势能法在是智能体的运动空间中创建了一个势能。该势能由两部分组成：一个是引力场UA，随着智能体和目标点的距离增加而单调递增，且方向指向目标点；另一个是斥力场UR，在智能体处在障碍物位置时有一极大值，并随着智能体与障碍物距离的增大而单调减小，方向指向远离障碍物方向。整个势能是其引力部分和斥力部分的叠加。The artificial potential energy method creates a potential energy in the agent's motion space. The potential energy consists of two parts: one is the gravitational field UA , which monotonically increases with the increase of the distance between the agent and the target point, and the direction points to the target point; the other is the repulsive field UR, when the agent is in the obstacle position There is a maximum value, and it decreases monotonously with the increase of the distance between the agent and the obstacle, and the direction points away from the obstacle. The whole potential energy is the superposition of its attractive and repulsive parts.
假设智能体在二维空间中运动，则坐标为q的智能体在势能作用下，所受的力F(q)是U(q)的负梯度，如式(3)Assuming that the agent is moving in a two-dimensional space, the force F(q) experienced by the agent with coordinate q under the action of potential energy is the negative gradient of U(q), as shown in formula (3)
F(q)＝FA(q)+FR(q)＝-▽U(q)(3)F(q)＝FA (q)+FR (q)＝-▽U(q)(3)
这个合力决定了智能体的运动。This resultant force determines the agent's motion.
在包含N个智能体的群中，令||xij||(为了简化，只考虑二维的情况)表示智能体i和j的相对距离，智能体i和它的连通邻接集Ni中的智能体j之间的相互作用通过人工势能函数来Vij描述。In a group containing N agents, let ||xij || (for simplicity, only consider the two-dimensional case) denote the relative distance between agent i and j, agent i and its connected adjacency set Ni The interaction between agents j is described by an artificial potential energy function Vij .
为了实现智能体运动时的避碰性指标，要求当机器人i和j的距离很近时，Vij趋于无穷大，表现为很强的排斥力；为了实现连通性保持的目标，要求当智能体i和j距离将要超出感知半径R时，Vij也趋于无穷大，表现为很强的吸引力；而为了实现群集运动队形收敛的指标，要求当距离达到队形的稳定距离d时，Vij存在唯一的极小值。In order to achieve the collision avoidance index when the agent moves, it is required that when the distance between the robot i andj is very close, Vij tends to infinity, showing a strong repulsive force; in order to achieve the goal of maintaining connectivity, it is required that when the robot When the distance between i and j is about to exceed the perception radius R, Vij also tends to infinity, showing a strong attraction; and in order to achieve the index of cluster motion formation convergence, it is required that when the distance reaches the stable distance d of the formation, Vij has a unique minimum value.
因此人工势能函数Vij是一个关于距离||xij||非负的，分段连续的在(0,R)上可微的径向无界函数，并满足以下条件：Therefore, the artificial potential energy function Vij is a radial unbounded function that is differentiable on (0,R) and is non-negative with respect to the distance ||xij ||, piecewise continuous, and satisfies the following conditions:
1)当||xij||→0或||xij||→R时，V(||xij||)→∞，1) When ||xij ||→0 or ||xij ||→R, V(||xij ||)→∞,
2)当达到预定距离dij时，Vij达到最小。2) When the predetermined distance dij is reached, Vij reaches the minimum.
下面对于本发明中跟踪控制规则的设计进行详细的说明。The design of the tracking control rules in the present invention will be described in detail below.
我们可以将本发明的跟踪控制规则简单表述为自适应控制律、人工势能函数的结合。在自适应控制律主要分为两部分，一部分为针对伴随者与伴随者之间的速度一致性项设计自适应参数，另一部分为针对伴随者与领航者的位置一致性项设计自适应参数。通过误差反馈动态调节两个部分的自适应参数，使系统快速达到一致性。人工势能函数部分则通过设置引力场以及斥力场，使智能体在运行过程中智能体之间维持一定的距离，保持整个多智能体网络的连通性以及实现智能体之间的避碰。We can simply express the tracking control rule of the present invention as a combination of an adaptive control law and an artificial potential energy function. The adaptive control law is mainly divided into two parts, one is to design adaptive parameters for the speed consistency item between the companion and the companion, and the other is to design adaptive parameters for the position consistency item between the companion and the leader. The adaptive parameters of the two parts are dynamically adjusted through error feedback, so that the system can quickly reach consistency. The artificial potential energy function part maintains a certain distance between the agents during the operation process by setting the gravitational field and the repulsive force field, maintains the connectivity of the entire multi-agent network, and realizes collision avoidance between agents.
使用Ψ(□)表示智能体i和智能体j之间的人工势能函数。连通系统内的势能函数Ψ(□)的设计如下：Use Ψ(□) to denote the artificial potential energy function between agent i and agent j. The potential energy function Ψ(□) in the connected system is designed as follows:
其中，||qij||表示智能体i与智能体j之间的距离，R表示智能体的通讯半径。这个势能函数可以保证人工势能函数在(0,R)的范围内连续可微，同时当时智能体i与智能体j之间的距离趋近于0或者趋近于通讯半径R时，势能函数Ψ(□)均趋近于无穷大，确保智能体之间的避碰以及通讯网络的连通性保持。同时(4)式中可以求得当时，Ψ(□)在此处达到最小。因此提出d势能函数可以使多智能体系统在运动过程中始终保持连通以及避碰，同时可以使得整个系统达到稳定的结构。Among them, ||qij || represents the distance between agent i and agent j, and R represents the communication radius of the agent. This potential energy function can ensure that the artificial potential energy function is continuously differentiable in the range of (0, R), and at the same time, when the distance between agent i and agent j approaches 0 or approaches the communication radius R, the potential energy function Ψ (□) are approaching infinity, ensuring the collision avoidance between agents and the connectivity of the communication network. At the same time, in formula (4), it can be obtained that Ψ(□) reaches the minimum here. Therefore, the proposed d potential energy function can keep the multi-agent system always connected and avoid collision during the movement process, and at the same time can make the whole system reach a stable structure.
自适应控制的规则如下：The rules of adaptive control are as follows:
其中，Ni(t)是与时间相关的智能体i的邻居集合。aij为智能体i与智能体j之间权值，已在(2)中定义。hi表示伴随者与领航者的连通情况，hi＝1时表示伴随者与领航者连通，否则hi＝0。pij＝pi-pj表示伴随者i与伴随者j之间的速度误差同时qir分别表示伴随者i与领航者r之间的位置误差。mij以及c均为动态自适应参数，其中mij表示耦合强度c表示位置反馈权重。mij以及c的变化率如下Among them, Ni (t) is the neighbor set of agent i related to time. aij is the weight between agent i and agent j, which has been defined in (2). hi represents the connection between the follower and the leader, when hi =1, it means the connection between the follower and the leader, otherwise hi =0. pij = pi -pj represents the velocity error between companion i and companion j while qir represents the position error between companion i and leader r, respectively. mij and c are dynamic adaptive parameters, where mij represents the coupling strength and c represents the position feedback weight. The rate of change of mij and c is as follows
其中Mij与C均为正实数。Among them, Mij and C are both positive real numbers.
上述控制规则可以描述为：在伴随者对领航者的跟踪过程中，通过(6)中所给出的自适应控制律，不断动态改变伴随者之间的速度耦合强度以及伴随者与领航者之间的位置反馈权重，使得伴随着在对领航者进行跟踪的同时不断伴随者之间也同时进行速度一致性同步，最终实现伴随者与领航者的速度同步。The above control rules can be described as: in the process of the follower tracking the leader, through the adaptive control law given in (6), the velocity coupling strength between the follower and the relationship between the follower and the leader are constantly and dynamically changed. The weight of the position feedback between the companions makes the speed consistency synchronization between the companions while tracking the leader, and finally realizes the speed synchronization between the companion and the leader.
基于上述控制规则，针对拉格朗日系统的控制律为：Based on the above control rules, the control law for the Lagrangian system is:
网络是不连通的并且初始能量有限的系统，则应用控制规则(7)，最终可以使得所有伴随者与领航者能够渐近达到相同的速度，实现速度同步，同时在运动过程智能体间避免碰撞，通讯网络始终保持连通，并且具有良好的稳定性。 If the network is disconnected and the initial energy is limited, the control rule (7) can be applied to eventually make all the companions and the leader asymptotically reach the same speed, realize speed synchronization, and avoid collisions between agents during the movement process , the communication network is always connected and has good stability.
最后对于实验的结果进行进一步的阐述：Finally, the experimental results are further elaborated:
为了清晰地展现实验结果，数值仿真中使用10个伴随者以及一个领航者。系统模型参数如下，假定重力项gi＝0，则有ugi＝0，同时In order to clearly show the experimental results, 10 companions and one leader are used in the numerical simulation. The parameters of the system model are as follows, assuming that the gravity item gi =0, then ugi =0, and at the same time
假定所有智能体均在二维平面上运动，智能体用点表示。初始速度、位置和连接任意给定，但必须满足以下约束：Assume that all agents move on a two-dimensional plane, and the agents are represented by points. The initial velocity, position and connection are given arbitrarily, but the following constraints must be satisfied:
①所有智能体均随机分布于范围在[0,20]m×[0,20]m的正方形内部，邻接图根据智能体之间的距离按照(2)式求得。① All agents are randomly distributed within a square with a range of [0,20]m×[0,20]m, and the adjacency graph is obtained according to formula (2) according to the distance between agents.
②初始速度为随机生成，x与y方向的初始速度均限制在[-2,2]m/s。②The initial speed is randomly generated, and the initial speed in the x and y directions is limited to [-2,2]m/s.
③所有智能体的智能体同样为随机生成，范围为[0.5,2]kg。同时信息更新率γi＝3。③ The agents of all agents are also randomly generated, and the range is [0.5, 2]kg. At the same time, the information update rate γi =3.
此外，智能体的通讯半径均为R＝4m。假设在初始时间为t＝0，实验结果为系统不断迭代离散生成，每次迭代对应的时间为Δt＝0.01s。对于(6)中的自适应系数Mij＝0.7与C＝0.8，同时在初始时速度耦合强度mij(0)＝0以及位置反馈权重c(0)＝0。In addition, the communication radius of the agents is R=4m. Assuming that the initial time is t=0, the experimental result is that the system continuously iterates discrete generation, and the time corresponding to each iteration is Δt=0.01s. For the adaptive coefficient Mij =0.7 and C=0.8 in (6), at the same time, the velocity coupling strength mij (0)=0 and the position feedback weight c(0)=0 at the initial stage.
图1给出了所有智能体的初始位置即(t＝0s)时刻，所有智能体的速度以及位置均为随机选取，其中红点表示伴随者，蓝点表示所有领航者，带箭头的直线表示所有智能体的速度，箭头方向表示智能体的速度方向，箭头的长短表示智能体速度的相对大小，绿色线条则表示智能体的连通性，智能体之间有绿色线条则表示该两个智能体之间连通即可以互相发送数据。本实验中将展示领航者在两种运动轨迹下的跟踪控制效果，两种情况下的初始条件均如图1中所示。第一个运动情况为领航者做匀速直线运动，第二种运动情况为领航者做匀速圆周运动。Figure 1 shows the initial positions of all agents (t=0s). The speed and position of all agents are randomly selected. The red dots represent the companions, the blue dots represent all the navigators, and the straight lines with arrows represent The speed of all agents, the direction of the arrow indicates the direction of the velocity of the agent, the length of the arrow indicates the relative speed of the agent, the green line indicates the connectivity of the agent, and the green line between the agents indicates the two agents Connected to each other can send data to each other. In this experiment, the tracking control effect of the navigator under two kinds of motion trajectories will be demonstrated, and the initial conditions in both cases are shown in Figure 1. The first motion situation is that the navigator is doing uniform linear motion, and the second motion situation is that the navigator is doing uniform circular motion.
首先为领航者做匀速直线运动的情况。Firstly, it is the case that the navigator is moving in a straight line at a uniform velocity.
图2分别给出了所有智能体在趋于速度一致性过程中智能体的状态，图2(a)以及图2(b)分别表示t＝5s以及t＝10s时智能体的状态。如图所示，所有智能体在t＝5s开始已经逐渐实现了速度一致性。Figure 2 shows the state of all agents in the process of speed consistency. Figure 2(a) and Figure 2(b) represent the states of agents at t=5s and t=10s, respectively. As shown in the figure, all agents have gradually achieved speed consistency since t=5s.
图3给出了所有智能体在最终状态即t＝13s时的状态，此时所有的伴随者已经与领航者实现了速度一致性，图中所示红点，蓝点，箭头以及绿色线条均与之前一直，青色线条为所有伴随者的行进轨迹，蓝色线条为领航者的行进路线。从轨迹可以很清晰看出所有智能体在运动过程中均未发生碰撞，并且最终所有智能体均保持一定的距离以相同的朝向前进。Figure 3 shows the state of all agents in the final state at t=13s. At this time, all the companions have achieved speed consistency with the leader. The red dots, blue dots, arrows and green lines shown in the figure are all As before, the cyan line is the trajectory of all the followers, and the blue line is the trajectory of the leader. From the trajectories, it can be clearly seen that all agents did not collide during the movement, and eventually all agents kept a certain distance and moved forward in the same direction.
图4给出所有智能体运动过程中伴随者与领航者之间在x方向以及y方向的速度误差，由于初始速度均为随机给定，在开始的误差较大，图中可以看出，智能体的速度误差很快就收敛到了一个极小的值，并且在迭代次数到达600次左右即t＝6s左右就完成了速度一致性。Figure 4 shows the speed errors between the companion and the leader in the x direction and y direction during the movement of all agents. Since the initial speed is randomly given, the initial error is relatively large. It can be seen from the figure that the intelligence The velocity error of the body quickly converges to a very small value, and the velocity consistency is completed when the number of iterations reaches about 600 times, that is, t=6s.
图5给出了智能体运动过程中的代数连通度的变化图，如图所示，代数连通度随着智能体的运动逐渐增加并最终稳定在一个恒值，表明智能体在势能函数的作用下，在收敛过程中智能体之间会逐渐向通讯范围的其他智能体靠近，并最后与其维持一个恒定的距离，之后所有智能体在该种拓扑条件下以相同的速度大小和方向运动Figure 5 shows the change diagram of the algebraic connectivity during the movement of the agent. As shown in the figure, the algebraic connectivity gradually increases with the movement of the agent and finally stabilizes at a constant value, indicating that the role of the agent in the potential energy function Under this condition, during the convergence process, the agents will gradually approach other agents in the communication range, and finally maintain a constant distance with them, and then all agents will move at the same speed and direction under this topology condition
其次为第二种情况即领航者做匀速圆周运动的情形。Next is the situation that the second kind of situation is the navigator doing uniform circular motion.
图6分别给出了所有智能体在趋于速度一致性过程中智能体的状态，图6(a)-(d)分别表示t＝0.5，t＝5s，t＝10s，t＝30s时智能体的状态。如图所示，所有智能体在t＝5s开始已经逐渐实现了速度一致性。Figure 6 shows the states of all agents in the process of speed consistency. Figure 6(a)-(d) respectively represent the intelligence at t=0.5, t=5s, t=10s, and t=30s state of the body. As shown in the figure, all agents have gradually achieved speed consistency since t=5s.
图7给出了所有智能体在最终状态即t＝115s时的状态，此时所有的伴随者已经与领航者实现了速度一致性，并且均运行了3/4个圆周，由于相对整个圆轨迹而言，智能体部分显得过小，将智能体所在位置放大之后放置于轨迹中央，从中可以看出，所有智能体均已相同的速度大小以及相同的朝向运动。Figure 7 shows the state of all agents in the final state at t=115s. At this time, all the companions have achieved speed consistency with the leader, and they all run 3/4 of the circle. Due to the relative trajectory of the entire circle From the point of view, the part of the agent is too small, and the position of the agent is enlarged and placed in the center of the trajectory. It can be seen from it that all the agents have moved at the same speed and direction.
图8给出所有智能体运动过程中伴随者与领航者之间的x方向以及y方向速度误差，同样由于初始速度均为随机给定，在开始的误差会非常大，从图中可以看出，智能体的速度误差很快就收敛到了一个极小的值。并实现速度一致性。Figure 8 shows the x-direction and y-direction velocity errors between the companion and the leader during the movement of all agents. Also, since the initial velocity is randomly given, the initial error will be very large, as can be seen from the figure , the speed error of the agent quickly converges to a very small value. And achieve speed consistency.
以上所述的仅为本发明的较佳实施例而已，本发明不仅仅局限于上述实施例，凡在本发明的精神和原则之内所做的局部改动、等同替换、改进等均应包含在本发明的保护范围之内。What has been described above is only a preferred embodiment of the present invention, and the present invention is not limited to the above-mentioned embodiment, and all local changes, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention should be included in within the protection scope of the present invention.
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