CN112558613B - Formation control method based on complex Laplace matrix - Google Patents
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Abstract
A formation control method based on a complex Laplace matrix comprises the following steps: 1) and establishing a motion model of the multi-mobile robot in the complex plane. 2) And establishing a communication network of the multi-robot system and a topological graph corresponding to network communication. 3) And constructing a real Laplace matrix corresponding to the communication network topology. 4) And configuring the weight of the complex Laplace matrix according to the formation requirement and the real Laplace matrix. 5) The distributed discrete control signal is designed and calculated. The method can enable engineers to quickly design the network framework of the robot formation to enable the robots to form the designated formation, is distributed, utilizes only the information of the robots and the information of the adjacent robots which can be measured, is simple and convenient to design and high in practicability, and provides a feasible scheme for efficient formation of multiple mobile robots.
Description
Technical Field
The invention relates to the technical field of multi-mobile robot formation control, in particular to a method for achieving robot formation by using a complex network control communication protocol to assemble robots to achieve a formation shape in the process of robot traveling.
Background
With the continuous improvement of the industrial manufacturing technology in China, the manufacturing cost of various mini sensors such as radars, vision cameras and various ultrasonic sensors is continuously reduced, and robots with moving bases and simple structures and various functions become the mastery force of robots in recent years. Particularly, in China, the cost of human resources is rising continuously, and at the time of industrial upgrading and transformation, a robot needs to replace part of human labor. The current situation is particularly remarkable in the fields of storage, express distribution, logistics transmission, take-out distribution and the like. In recent years, with the development of e-commerce, in the fields of express delivery industry, take-out delivery industry and the like, the cooperative transportation control of multiple mobile robots has gradually replaced manual operation in some fields. However, in these fields, mobile robots can only carry small-mass objects according to their loads, and there is no good method for handling large-volume and heavy-mass objects, because the robots still lack cooperative carrying control technology. The problem of the multi-robot cooperative transportation control mainly comprises robot communication, remote sensing transmission, formation control, map construction, path planning and the like, wherein the formation control is the most basic problem in the field of the multi-mobile robot system cooperative transportation research and is one of the cores for researching the problem.
Formation control, in particular, formation control refers to a team of a plurality of robot devices (e.g., robots, satellites, unmanned aerial vehicles, autonomous underwater vehicles, etc.) with movable bases, which maintain a specific spatial relationship (formation) with each other while meeting the control problem of environmental constraints during moving to a target site or assembling at a specified position. Generally, the formation control solves global tasks by realizing group behaviors of multi-robot systems by means of local interactions between robots. Robot formation control has been studied for more than twenty years. There have been many different control protocols and control methods for two decades. However, if the control protocols are classified only according to the number domain of the controller design, all control protocols can be classified into real number domain and complex number domain control communication protocols. The control protocol of the early years is mostly established in the real domain, and some of the research in recent years is transferred to the complex domain. Compared with a real number field, the controller designed in a complex number field is more beneficial to control of formation rotation and formation scaling in application due to the fact that the complex number self-corresponds to a two-dimensional coordinate system of a complex plane, and the controller is more suitable for describing a kinematic equation in the two-dimensional plane.
In recent years, there have been extensive and intensive studies by many scholars and engineers on formation of multi-robot systems in a plurality of domains. Teaching groups of Zhejiang university Linzhi \36191realizeformation of arbitrary graphs by using Distributed Control through designing an interactive network topology based on a complex Laplace matrix multi-agent (Distributed format Control of multi-agent systems using complex laboratory [ J ]. Automatic Control, IEEE Transactions on,2014,59(7):1765 + 1777.). Researchers of the institute of mathematics and system science of Chinese academy of sciences have already been developed and the like and proposed a formation Control strategy based on the surrounding of convex geometric objects of complex adjoint matrix (Long Y, Hong Y. distributed following design of target area with complex adaptive matrices [ J ]. IEEE Transactions on Automatic Control 2014,60(1): 283-. In terms of patent invention, chinese patent documents CN106647771B and CN105511494A are the closest prior art of the present invention. Patent CN106647771B describes a minimum step formation control technique based on a complex laplacian matrix, and CN105511494A describes a formation technique based on a complex laplacian matrix.
From the perspective of graphical formation control in a two-dimensional plane, most of the existing literature on formation control can be divided into two categories. One is aimed at solving how to form a formation pattern of a specific shape, and the other is aimed at solving how to maintain stability of the formation pattern and meet formation aggregation speed in various interference (such as communication delay, obstacle blocking and the like) environments. How to form a formation figure of a specific shape, also commonly referred to as a clustering problem, is mainly aimed at forming a plurality of robots into a specified shape. In patent CN105511494A, forest shigella 36191et al designs a distributed control technique using complex laplace to solve the problem of formation of two-dimensional planes by combining the concept of double-root graph in graph theory and the mathematical manifold theory. In the technology, a dual-root graph means that at least two nodes exist in a topological structure, and from the two nodes, a path to any other node can be found from the topological graph, and the graph is called as the dual-root graph; the laplacian matrix (laplacian) comes from the concept of graph theory in mathematics, and can be obtained from the corresponding topology of the communication network. In graph theory, the laplacian operator is generally a real number, and zero and one are used to represent the connection and disconnection of two topology nodes, respectively. The complex laplacian is to assign an edge of two connected nodes in the topology by a complex number, so as to generate a corresponding laplacian matrix. Although the invention can ensure that any formation pattern can be formed, the invention has some defects. First, it is difficult to design a specific dual-root graph for a general application scenario because the topology of the dual-root graph is usually complex. Secondly, the patent also relies on configuring a stability control matrix for stabilizing the entire system. When the number of robots is small (10 or less), the stable matrix can be configured with eigenvalues by software such as Matlab. When the number of robots is increased (more than 10), it is difficult to design a stable matrix due to high computational complexity. Furthermore, when there is a delay in the communication in the network, it is difficult to guarantee convergence by this queuing method. In view of the above problems, the present invention proposes a simple and easy-to-implement robot cluster formation control strategy that relies only on the simplest topology. The control strategy provided by the invention only needs a directed spanning tree existing in the communication topology, namely, each robot has at least one individual robot to communicate with the directed spanning tree. When the control strategy technology provided by the invention is used, a complex communication network structure is not required to be designed, a stable control matrix is not required to be configured, the robustness on the network communication delay is very strong, and theoretically, the formation control result cannot be influenced by any long-time communication delay. By the method, a designer can quickly realize the formation control of the robot on the plane.
Disclosure of Invention
Aiming at the problem that a plurality of mobile robots form a specific formation on a two-dimensional plane, the invention aims to overcome the defects of complex topological structure, narrow application area and the like in the prior art, and provides a formation control method of the multiple mobile robots based on a complex Laplace matrix, aiming at helping the robots to form a formation control more simply and conveniently in an actual application scene.
The invention provides a formation control method of multiple mobile robots. First, the motion of the robot in a two-dimensional plane is modeled, and the coordinates of the robot in the two-dimensional plane are represented by complex numbers. The coordinate of the robot in the two-dimensional plane x is expressed by the real part of the complex number, and the coordinate of the robot in the two-dimensional plane y is expressed by the imaginary part of the complex number. Then, the current positions of all the groups of mobile robots are represented by complex vectors. Then, according to the graph theory, an interactive topology of the robot with the directed spanning tree is designed. And finally, representing the target formation of the robot group by using a row of complex vectors, and solving a complex Laplace operator to design a distributed control law. The method comprises the following specific steps:
firstly, a global coordinate system is established for the motion space of the robot. An x-y cartesian coordinate system is established for the robot within its motion space. For each robot, its coordinates (x, y) in this space can be marked, using a complex number (x + yj) for characterizing the robot's position in the plane. j refers to the unit imaginary number in the complex number
Namely, it is
Using symbols
Representing the set of all complex numbers. Without loss of generality, the number of robots participating in formation in a plane is set to be n in total, and the robots are numbered by numbers 1,2 …, n-1 and n respectively. The position of the ith robot in the plane is marked by the symbol xiTo show that all the positions of the robot can be represented by a row of n-dimensional complex vectors
Is represented by the formula, x ═ x1,x2,…,xn)TWherein (·)TIs the transpose of the matrix. In formation control, if collisions are not considered, typically looking at the robots as particles of a collision-free volume, each robot in the system obeys a single integrator motion model:
representing multi-robot systems and their local interactions with each other as a directed topology graph G ═ (V, E), where V ═ V1,v2,…vnDenotes a set of n nodes in the diagram, viThe representation shows that the ith node, i.e. the ith robot,
representing a set of nodes and edges between the nodes, eikE represents that the robot i can measure the relative position of the robot k
Where p represents the distance between two robots,
indicating the angle of robot k relative to robot i. Starting from any robot, a directed spanning tree is established, and other robots are all on nodes of the spanning tree. In short, each robot is capable of measuring at least the relative position of any of the remaining robots.
the generated adjacency matrix W corresponds to the undirected graph G ═ V, E. If the ith robot can measure the relative position of the kth robot, i.e. there is eikE, then w ik1. On the contrary, if the ith robot cannot measure the relative position of the kth robot, i.e. the relative position of the kth robot is determined
Then w ik0. W hereinikRepresenting the ith row and kth column elements of the matrix W.
A complex laplacian matrix L is defined,
in equation (3), Σ (·) is a summer symbol.
defining symbol e as a natural constant and defining the formation as
According to the complex theory, ejθA point of the unit circle on the complex plane is shown. Since the angle θ can be arbitrarily specified, the value of θ can be changed as needed according to the formation. Let D ═ diag (ξ) be a diagonal matrix, the diagonal elements being each element of ξ, i.e.
The complex Laplace matrix can be designed as
P＝DLD-1 (4)
the control signal of the robot is determined by the complex weighted combination of the positions of the robot and the neighboring robots:
wherein u isiRepresenting the speed control input of the ith robot,
and
indicating the position of the ith and kth robots, p, respectivelyikRepresenting the kth element of the ith row of matrix P. N is a radical ofiRepresenting the set of the remaining robots that robot i can measure,namely Ni＝{vk:eikE.g. E. Under the control signal input, the global dynamic response is:
since the control signal is usually given as a discrete-time signal in practical applications, the discrete-time control signal corresponding to the above equation is:
x(k+1)＝(I-εP)x(k)＝Ax(k) (7)
wherein epsilon is sampling time and value range
dmaxIs the in-degree of the network topology, i.e. the maximum number of remaining robots that all robots can measure.
In the actual application process, multiple mobile robots are required to meet various task requirements in various formation shapes, the existing complex robot formation technology is complex on the whole, and the application range of application is influenced. The invention utilizes the characteristic that the complex characteristic value of the complex Rayleigh matrix corresponding to the network topology after the complex weighting is in a complex plane, and the complex characteristic value can just correspond to the position of the robot formation in a two-dimensional space. Therefore, a user can quickly design the network and the communication weight of the robot according to the target through a simple network structure.
The invention has the advantages that: the method has the advantages that engineers can quickly design the network framework of the robot formation, so that the robots form the designated formation, the method is distributed, the utilized information is only the information of the robots and the information of the adjacent robots which can be measured, the design is simple and convenient, the practicability is high, the robustness is high, and a feasible scheme is provided for the efficient formation of the multiple mobile robots.
Drawings
FIG. 1 is an exemplary target formation graph according to the present invention.
Fig. 2 is a schematic topology of the present invention.
Fig. 3 is a robot formation convergence process under the control of the algorithm of the present invention.
Detailed Description
The novel technical solution of the present invention is further described below with reference to the accompanying drawings and the actual formation case.
A multi-mobile-person system consisting of 6 robots is aimed at. 6 robots are distributed on a two-dimensional plane, the coordinates of which are (5.4701,3.6848), (2.9632,6.2562), (7.4469,7.8023), (3.5784, -0.0631), (1.8896,0.8113), (1.8351,7.7571) need to form a regular hexagonal shape formation as shown in fig. 1, the shape can be represented by coordinates of (0.5000,0.8660), (0.5000,0.8660), (1.0000,0.0000), (0.5000, -0.8660), (0.5000, -0.8660), (1.0000,0.0000) in a two-dimensional plane space, and the algorithmic process is deduced for the case:
Firstly, a global coordinate system is established for the motion space of the robot. X-y cartesian coordinates are established for the robot within its motion space. For each robot, its coordinates (x, y) in this space can be marked, using a complex number (x + yj) for characterizing the robot's position in the plane. These robots are numbered with the numbers 1 to 6, respectively. Let the position of the ith robot in the plane be denoted by symbol xi, the positions of all robots can be represented by a row of 6-dimensional complex vectors
Is represented by the formula, x ═ x1,x2,…,x6)T. Each robot motion model in the system is:
Multiple machinesThe human system and its local interactions with each other are represented as a directed topology G ═ (V, E), where V ═ V1,v2,…v6Denotes a set of 6 nodes in the diagram, viThe representation shows that the ith node, i.e. the ith robot,
representing a set of nodes and edges between the nodes, eikE represents that the robot i can measure the relative position of the robot k
Where p represents the distance between two robots,
indicating the angle of robot k relative to robot i. Starting from any robot, a directed spanning tree is established, and other robots are all on nodes of the spanning tree. In short, each robot can measure at least the relative position of a respective robot, resulting in the topological diagram of fig. 2 (not only).
Generating adjacency matrix W corresponding to (V, E) of undirected graph G
The corresponding real laplacian matrix L,
Define the formation as
D ═ diag (ξ) ═ diag (0.5000+0.8660i, -0.5000+0.8660i, -1.0000+0.0000i, -0.5000-0.8660i,0.5000-0.8660i,1.0000-0.0000i) can be obtained. The complex Laplace matrix is
P＝DLD-1 (4)
The control signal of the robot is determined by the complex weighted combination of the positions of the robot and the neighboring robots:
wherein u isiRepresenting the speed control input of the ith robot,
and
indicating the position of the ith and kth robots, p, respectivelyikRepresenting the kth element of the ith row of matrix P. N is a radical ofiRepresenting the set of the remaining robots i can measure, i.e. Ni＝{vk:eikE.g. E. Under the control signal input, the global dynamic response is:
calculating its corresponding discrete-time control signal:
x(k+1)＝(I-εP)x(k)＝Ax(k) (7)
where ε is taken to be 0.05 and the actual queuing effect is shown in FIG. 3.
Claims (1)
1. The formation control method based on the complex Laplace matrix comprises the following specific steps:
step 1, establishing a motion model;
firstly, establishing a global coordinate system for the moving space of the robot; for moving space of robotIn the process, an x-y Cartesian coordinate system is established; for each robot, its position in this space is represented in coordinates (x, y), and a complex number (x + yj) is used to characterize the robot's position in the plane, j referring to the unit imaginary number in the complex number
Namely, it is
Using symbols
Represents the set of all complex numbers; without loss of generality, the number of robots participating in formation in a plane can be set to be n, and the robots are respectively numbered by numbers 1,2 …, n-1 and n; the position of the ith robot in the plane is marked by the symbol xiThe positions of all robots can be expressed by a row of n-dimensional complex vectors
Is represented by the formula, x ═ x1,x2,…,xn)TWherein (·)TIs the transposition of the matrix; in formation control, if collisions are not considered, typically looking at the robots as particles of a collision-free volume, each robot in the system obeys a single integrator motion model:
step 2, establishing a topological graph of the multi-robot system;
representing multi-robot systems and their local interactions with each other as a directed topology graph G ═ (V, E), where V ═ V1,v2,…vnDenotes a set of n nodes in the diagram, viThe representation shows that the ith node, i.e. the ith robot,
representing a set of nodes and edges between the nodes, eikE represents that the robot i can measure the relative position of the robot k
Where p represents the distance between the two robots,
represents the angle of robot k relative to robot i; starting from any robot, establishing a directed spanning tree, and enabling the other robots to be on nodes of the spanning tree; in short, each robot is able to measure the relative position of at least one other robot;
step 3, realizing a Laplace matrix according to the topological graph;
generating an adjacency matrix W corresponding to the undirected graph G ═ V, E; if the ith robot can measure the relative position of the kth robot, i.e. there is eikE, then wik1 is ═ 1; on the contrary, if the ith robot cannot measure the relative position of the kth robot, i.e. the relative position of the kth robot is determined
Then wik0; w hereinikRepresents the ith row and the kth column of the matrix W;
a complex laplacian matrix L is defined,
in the formula (2), Σ (·) is a summer symbol;
step 4, designing a plurality of Laplace matrixes;
defining the symbol e asConstant, define the formation as
According to the complex theory, ejθRepresenting a point of the unit circle on the complex plane; the angle theta can be randomly specified, so the formation can change the value of theta according to the use requirement; let D ═ diag (ξ) be a diagonal matrix, the diagonal elements being each element of ξ, i.e.
The complex Laplace matrix can be designed as
P＝DLD-1 (4)
Step 5, converting the continuous system into a discrete system;
the control signal of the robot is determined by the complex weighted combination of the positions of the robot and the neighboring robots:
wherein u isiRepresenting the speed control input of the ith robot,
and
indicating the position of the ith and kth robots, p, respectivelyikRepresents the kth element of the ith row of matrix P; n is a radical ofiRepresenting the set of the remaining robots i can measure, i.e. Ni＝{vk:eikE is formed; under the control signal input, the global dynamic response is:
since the control signal is given as a discrete-time signal in practical application, the corresponding discrete-time dynamic response is as follows:
x(k+1)＝(I-εP)x(k)＝Ax(k) (7)
wherein epsilon is sampling time and value range
dmaxIs the in-degree of the network topology, i.e. the maximum number of robots that can measure the rest of the robots.
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