CN112612279B - Second-order formation control method based on complex Laplace matrix - Google Patents
Second-order formation control method based on complex Laplace matrix Download PDFInfo
- Publication number
- CN112612279B CN112612279B CN202011547885.9A CN202011547885A CN112612279B CN 112612279 B CN112612279 B CN 112612279B CN 202011547885 A CN202011547885 A CN 202011547885A CN 112612279 B CN112612279 B CN 112612279B
- Authority
- CN
- China
- Prior art keywords
- robot
- robots
- matrix
- complex
- formation
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Active
Links
- 230000015572 biosynthetic process Effects 0.000 title claims abstract description 69
- 239000011159 matrix material Substances 0.000 title claims abstract description 51
- 238000000034 method Methods 0.000 title claims abstract description 20
- 230000001133 acceleration Effects 0.000 claims description 8
- 238000010586 diagram Methods 0.000 claims description 6
- 230000004044 response Effects 0.000 claims description 5
- 239000013598 vector Substances 0.000 claims description 5
- 230000003993 interaction Effects 0.000 claims description 4
- 238000005070 sampling Methods 0.000 claims description 3
- 230000009977 dual effect Effects 0.000 claims description 2
- 239000002245 particle Substances 0.000 claims description 2
- 230000017105 transposition Effects 0.000 claims 1
- 238000004891 communication Methods 0.000 abstract description 17
- 238000013461 design Methods 0.000 abstract description 10
- 238000011217 control strategy Methods 0.000 abstract description 6
- 230000000694 effects Effects 0.000 abstract description 3
- 108010023321 Factor VII Proteins 0.000 abstract 1
- 238000004519 manufacturing process Methods 0.000 description 8
- 238000011160 research Methods 0.000 description 6
- 238000005516 engineering process Methods 0.000 description 4
- 230000008569 process Effects 0.000 description 3
- 230000001105 regulatory effect Effects 0.000 description 2
- 241000282461 Canis lupus Species 0.000 description 1
- 206010010144 Completed suicide Diseases 0.000 description 1
- 230000001154 acute effect Effects 0.000 description 1
- 230000003044 adaptive effect Effects 0.000 description 1
- 230000005540 biological transmission Effects 0.000 description 1
- 239000003795 chemical substances by application Substances 0.000 description 1
- 238000004140 cleaning Methods 0.000 description 1
- 238000010276 construction Methods 0.000 description 1
- 230000003247 decreasing effect Effects 0.000 description 1
- 230000007547 defect Effects 0.000 description 1
- 238000001514 detection method Methods 0.000 description 1
- 238000011161 development Methods 0.000 description 1
- 238000005265 energy consumption Methods 0.000 description 1
- 230000007613 environmental effect Effects 0.000 description 1
- 238000003306 harvesting Methods 0.000 description 1
- 230000002452 interceptive effect Effects 0.000 description 1
- 238000012067 mathematical method Methods 0.000 description 1
- 238000012546 transfer Methods 0.000 description 1
Images
Classifications
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
- G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
- G05D1/02—Control of position or course in two dimensions
- G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
- G05D1/0212—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory
- G05D1/0223—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory involving speed control of the vehicle
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
- G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
- G05D1/02—Control of position or course in two dimensions
- G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
- G05D1/0212—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory
- G05D1/0221—Control of position or course in two dimensions specially adapted to land vehicles with means for defining a desired trajectory involving a learning process
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
- G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
- G05D1/02—Control of position or course in two dimensions
- G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
- G05D1/0276—Control of position or course in two dimensions specially adapted to land vehicles using signals provided by a source external to the vehicle
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
- G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
- G05D1/02—Control of position or course in two dimensions
- G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
- G05D1/0287—Control of position or course in two dimensions specially adapted to land vehicles involving a plurality of land vehicles, e.g. fleet or convoy travelling
- G05D1/0289—Control of position or course in two dimensions specially adapted to land vehicles involving a plurality of land vehicles, e.g. fleet or convoy travelling with means for avoiding collisions between vehicles
-
- G—PHYSICS
- G05—CONTROLLING; REGULATING
- G05D—SYSTEMS FOR CONTROLLING OR REGULATING NON-ELECTRIC VARIABLES
- G05D1/00—Control of position, course, altitude or attitude of land, water, air or space vehicles, e.g. using automatic pilots
- G05D1/02—Control of position or course in two dimensions
- G05D1/021—Control of position or course in two dimensions specially adapted to land vehicles
- G05D1/0287—Control of position or course in two dimensions specially adapted to land vehicles involving a plurality of land vehicles, e.g. fleet or convoy travelling
- G05D1/0291—Fleet control
Landscapes
- Engineering & Computer Science (AREA)
- Radar, Positioning & Navigation (AREA)
- Remote Sensing (AREA)
- Aviation & Aerospace Engineering (AREA)
- Physics & Mathematics (AREA)
- General Physics & Mathematics (AREA)
- Automation & Control Theory (AREA)
- Feedback Control In General (AREA)
- Manipulator (AREA)
- Control Of Position, Course, Altitude, Or Attitude Of Moving Bodies (AREA)
Abstract
A second-order 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 topological graph of the multi-robot system. 3) A real laplacian matrix is constructed. 4) The weights of the complex laplacian matrix are configured. 5) Design second order control protocol 6) design velocity attenuation factor 7) design discrete control signal. The second-order formation control strategy provided by the patent accords with the actual kinetic equation of the robot, and can accelerate the formation efficiency of the robot to a certain extent, and the effect is particularly obvious when the whole communication network is huge. The method is distributed, has a simple structure, strong practicability and high efficiency, has certain robustness to communication time delay, and provides a feasible scheme for the formation control of the 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 realizing robot formation by driving robots to achieve a target formation form by using a network communication second-order control protocol based on complex weighting in the process of multiple robots advancing.
Background
With the development of the manufacturing technology, the production cost of the robot is continuously reduced, the performance is gradually enhanced, and the functions are gradually diversified. The cheaper production costs of robots make it more economical to cooperatively control multiple robotic devices to perform certain tasks. The cooperative control of a plurality of movable robot devices has wide application prospect in military and civil use. In production, manufacturing and daily life, the multi-mobile machine crowd with cooperative control has entered some posts and played an important role. In the 'double 11' shopping festival in 2019, the robot company Zhijia (Geek +) deploys large-scale cluster scheduling of more than 4000 robots, and 811 thousands of goods are delivered within 72 hours, which is nearly 1 time higher than the manual efficiency. In indoor places such as libraries, museums, banks and the like, a plurality of mobile robots walk in the indoor places to play roles of a navigator, a self-service teller and the like. In the future, as the population structure is adjusted, it is a foreseeable trend that multiple mobile robots replace part of human work. This trend is particularly acute in labor intensive manufacturing processes such as road cleaning, equipment handling, and agricultural product harvesting. In the military field, the application of the cooperative control multi-mobile robot is wider. The suburb unmanned aerial vehicle cluster developed by the army in 2015 has the advantages of small individual volume, light weight, low manufacturing cost and short production period, can be quickly deployed in battlefield battle environments, realizes scale effects in a short time, and achieves the battle target by suicide attacking and destroying target units. In 2020, the electric department group publishes the air-ground cooperative fixed wing unmanned aerial vehicle combat system, namely the swarm. Compared with a suburb wolf machine group, the unmanned aerial vehicle equipped with the bee colony system is larger in scale and stronger in fighting capacity. The common transporter also has strong attack capability after being equipped with the swarm system. In addition to battlefield operations, the mobile robot cluster can be used for many military research works such as geodetic surveying, meteorological observation, aerial photography, urban environment detection, exploration of earth resources and the like. It can be said that the robot group with the large scale of the structure plays an important role in the information war. In the above applications, the robot groups are generally required to be arranged in a designated formation to cover more areas synchronously, thereby achieving the purpose of improving task efficiency.
How to cooperatively drive and control the robot group to complete the designated formation is a very complicated problem, and the content thereof comprises the sub-problems of robot kinematics control, aerodynamics, data transmission interference and counter interference, robot formation control, robot map construction, robot motion track planning and the like. Among these subproblems, how to perform formation control of robots is one of the basic problems of cooperative control of a multi-mobile robot system. The core of the robot formation control problem is a control problem of how to maintain a certain expected geometric relationship (i.e. formation) in space by a team consisting of a plurality of movable robot devices (such as unmanned aerial vehicles, artificial satellites, underwater detectors and the like) and meet environmental constraints at the same time. Generally, formation control refers to a control method for achieving a formation goal of robots through local information interaction between the robots. The formation control problem has been studied for nearly 40 years since the 80's last century since computer engineers simulated the formation flight of birds with locally consistent control protocols. The laplacian matrix-based consistency problem study in 2003 opened up new chapters for multi-robot system control over the next 20 years. In general, these studies derive three types of formation control based on location, relative distance, and distance. Among them, the formation control based on relative distance is also generally regarded as the formation control based on consistency, and is considered as a technology with the most application prospect by many scholars and engineers. The reason is that the control topology is relatively simple, the requirement on the environment is low, and the device can be deployed in common indoor environments such as warehouses, transfer centers and the like, and can also be deployed in some extreme environments such as outer space, deep sea, disaster centers and the like. The technology has good expandability, and the number of the robots can be conveniently increased and decreased without affecting the stability of the whole formation system. The formation control strategies based on consistency are mostly established on a real number domain, and the control strategies are real number polynomials. With the progress of research, the design of controllers in recent years has partially shifted to a controller in a plural domain — the control protocol is a polynomial containing plural numbers. Real numbers are generally considered to correspond to points on a straight line, while complex numbers are considered to correspond to points in a plane. Due to this property of complex numbers, controllers designed from complex numbers are more suitable for describing kinematic equations in two-dimensional planes. The control of formation rotation and formation scaling of the formation is particularly more conveniently and concisely described.
In the multi-robot system formation control of a plurality of fields, many researchers have conducted intensive research in recent years. Forest aspiration 36191. According to the protocol, the mobile robot can realize formation Control in a two-dimensional plane (Distributed format Control of multi-agent systems using complex display [ J ], IEEE Transactions on Automatic Control,2014,59(7): 1765-. Researchers of Chinese academy of sciences have already come up with others and proposed a formation Control strategy based on convex geometric target surrounding of complex adjoint matrix (Long Y, Hong Y. distributed surrounding design of target region with complex adaptive matrix [ 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.
The prior formation control literature mainly researches the formation control technology in two aspects. On one hand, the research of formation control is focused, namely, how to design a simple control protocol to enable the robot to form a target formation in a plane. On the other hand, the research is focused on the performance of robot formation control, namely how the robot formation control resists external interference, including communication delay, operation delay and communication chain breakage, and how the robot ensures the performance of fast convergence speed and robustness. In patent CN105511494A, linkung 36191provides a complex analytic expression of a robot formation control protocol through the theory of manifold in complex domain, and solves the problem of how to design a communication network for robot formation control through the concept of graph theory and the matrix theory. Although the invention constructs a formation frame based on the complex Laplace matrix, the invention has some problems. Firstly, in the system proposed by the invention, the topological graph corresponding to the communication between the robots must be double-rooted. The double roots are the concept of graph theory in mathematics, and mean that two root nodes are necessary in a topological graph, and the nodes can be connected to any node in the graph from the two root nodes. The complexity of the biproot graph makes it difficult for engineers to design the topology. Especially when the whole network is very large (when the number of robots exceeds 100), engineers may need to consider a topological network containing thousands of communication connections at the same time. Secondly, the invention must solve a stability matrix to stabilize the system to reconfigure the system poles. The solution of the stability matrix is not easy, and a numerical solution needs to be solved by a mathematical method such as a Newton homotopy method. In no particular case, the system solution of more than 10 robots is not easy, while the system of more than 100 robots is almost impossible to solve. Aiming at the problems, the invention provides a robot second-order formation control protocol which is simple in structure and convenient to implement, and only a topological graph corresponding to a communication network needs to have a directed spanning tree. Directed spanning tree means that each robot has at least one measurable neighboring robot, which is the simplest of the topological graph in a multi-robot system. In addition, according to the method, a control protocol can be properly regulated and controlled (the regulated control protocol can be faster than a first-order controller) according to performance requirements (convergence speed), and complicated feature value configuration work is not required to be carried out due to the stability of the system.
Disclosure of Invention
The invention provides a second-order formation method of multiple mobile robots, aiming at overcoming the defects of complex topological structure and difficulty in configuring stable matrix when the multiple mobile robots are controlled in a two-dimensional plane formation manner in the prior art and aiming at enabling the robot formation control to be simpler, more convenient and more efficient.
The invention discloses a second-order formation 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 graph theory, an interactive topology of the robot with the directed spanning tree is designed, and a complex Laplace matrix is formed according to the topology. And a column of complex vectors is used for representing the target formation of the robot group, and a complex Laplacian is solved to design a second-order distributed control law. Finally, a speed attenuation factor is designed, and a discrete time control protocol is designed. 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, the robots are generally considered to be particles of a collision-free volume. Each robot in the system obeys a dual integrator motion model:
Step 2, establishing a topological graph of the multi-robot system;
information interaction among multiple robots is represented as a directed topological graph G ═ (V, E), wherein V ═ V1,v2,…vnDenotes a set of n nodes in the diagram, node v in the diagram GiThe (i) th robot is represented,
representing nodes and nodesSet of edges in between, edge e in graph GikE represents that the robot i can measure the relative position of the robot k
Where p represents the distance between the 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, it is sufficient that each robot can measure at least the relative position of any of the remaining robots.
Step 3, realizing a Laplace matrix according to the topological graph;
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.
Step 4, designing a plurality of Laplace matrixes;
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 theta can be arbitrarily specified, the value of theta can be changed 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 (5)
the second-order control protocol of the robot consists of the position complex weighted sum of the robot and the adjacent robots and the speed of the robot:
wherein u isiRepresenting the acceleration control input of the ith robot,
and
indicating the position of the ith and kth robots, p, respectivelyikThe ith row of the matrix P represents the kth element, viRepresenting the speed of the ith robot, gamma being a positive real number representing a speed decay factor, the formation rate, N, being adjustableiRepresenting the set of the remaining robots i can measure, i.e. Ni＝{vk:eikE.g. E. The global dynamic response of all robots at the input of the control signal (5) can be expressed as:
wherein, On×nRepresenting a matrix of n rows and n columns all zero, In×nA unit array of n rows and n columns is shown.
Step 6, designing a speed attenuation factor gamma;
for a continuous control system, gamma can be any positive integer, and the system is stable. However, in practical applications, the velocity attenuation factor γ needs to be designed specifically for the performance requirements and hardware requirements. The patent provides two ways for the number of the robots in the system:
a) the number of robots in the system is small (less than 10). Calculating the eigenvalues delta of the matrix L1,δ2,…,δnSelecting gamma>1 eigenvalue λ of the configuration matrix BiConfiguring eigenvalues of matrix B
The selection principle of gamma is as small as possible, the value can be gradually increased from 1, the larger the gamma is, the faster the system response speed is (but limited by a communication network), and the larger the gamma value is, the higher the requirement on the acceleration performance of the robot is.
b) When the number of robots in the system is large (10 or more). Suggesting a velocity decay factor
n represents the number of robots in the system.
Step 7, designing a discrete control signal;
since the control signal is usually given as a discrete-time signal in practical applications, the discrete-time control signal corresponding to equation (6) is:
x(k+1)＝(I+εB)x(k)＝Ax(k) (8)
wherein epsilon is sampling time, and the value range is 0.01< epsilon < 0.05.
In robot formation control, the second-order formation control strategy provided by the patent accords with the actual kinetic equation of the robot, and can accelerate the robot formation efficiency to a certain extent. The effect is particularly obvious when the robot communication structure is simple and the whole communication network is huge. The design of the speed attenuation factor in the second-order formation control strategy needs to consider more factors, including the number of robots, communication frequency, communication energy consumption, communication topological structure and the like, and needs to be adapted according to local conditions, so that the patent provides two modes for users to select.
The invention has the advantages that: the method is distributed, the information utilized by the robot only needs the information of the robot and the information of a neighbor robot, the method is simple in structure, strong in practicability and high in efficiency, has certain robustness to communication time delay, and provides a feasible scheme for formation control of the multiple mobile robots.
Drawings
FIG. 1 is an exemplary target formation graph in accordance with the present invention
FIG. 2 is a schematic of the 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 eight robots is aimed at. Eight robots are distributed on a two-dimensional plane with coordinates (12.70,2.06), (2.80,9.60), (17.96,15.93), (14.13,19.63), (20.88,12.22), (21.00,24.60), (19.16,21.55), (1.01,29.06) of a formation that needs to constitute a regular octagonal shape as shown in fig. 1, which can be represented by coordinates (0.71 ), (0.00,1.00), (-0.71,0.71), (-1.00,0.00), (-0.71 ), (0.00, -1.00), (0.71), (1.00,0.00) in a two-dimensional plane space, and the algorithmic process is deduced for this 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 8, 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 8-dimensional complex vectors
Is represented by the formula, x ═ x1,x2,…,x8)T. Each robot motion model in the system is:
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,…,v8Denotes a set of eight 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,
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 onlyOne), the two parts are combined.
Step 3, realizing the Laplace matrix according to the topological graph
Generating adjacency matrix W corresponding to (V, E) of undirected graph G
The corresponding real laplacian matrix L,
step 4, designing a plurality of Laplace matrixes
Define the formation as
The diagonal matrix D ═ diag (ξ) ═ diag (0.71+0.71i,0.00+1.00i, -0.71+0.71i, -1.00+0.00i, -0.71-0.71i, -0.00-1.00i,0.71-0.71i,1.00-0.00 i). The complex Laplace matrix is
P＝DLD-1 (5)
The second-order control protocol of the robot consists of the complex weighted sum of the robot and the adjacent robots and the speed of the robot:
wherein u isiRepresenting the acceleration control input of the ith robot,
and
indicating the position of the ith and kth robots, p, respectivelyikThe ith row of the matrix P represents the kth element, viRepresenting the speed of the ith robot, gamma being a positive real number representing a speed decay factor, the formation rate, N, being adjustableiRepresenting the set of the remaining robots i can measure, i.e. Ni＝{vk:eikE.g. E. The global dynamic response of all robots at the input of the control signal (5) can be expressed as:
wherein, On×nRepresenting a matrix of n rows and n columns all zero, In×nA unit array of n rows and n columns is shown.
Step 6, designing a speed attenuation factor gamma
The number of the robots in the system is eight, and the eigenvalue delta of the matrix L is calculated1＝4.45+0.00i，δ2＝3.26+0.00i，δ3＝0.00+0.00i，δ4＝0.66+0.00i，δ5＝1.29+0.00i，δ6＝2.34+0.00i，δ7＝3.62+0.00i，δ81.38+0.00 i. Taking gamma as 2, the eigenvalue lambda of matrix B1,2＝-1±1.86i，λ3,4＝-1±1.50i，λ5,6＝-1.00±1.16i，λ7,8＝-1±0.53i，λ9,10＝-1.00±0.62i，λ11,12＝-1.00±1.62i，λ13＝0.00+0.00i，λ14＝-0.42+0.00i，λ15＝-2.00+0.00i，λ16And the value is-1.58-0.00 i, and the formation control requirement is met.
Step 7, designing discrete control signals
Since the control signal is usually given as a discrete-time signal in practical applications, the discrete-time control signal corresponding to equation (6) is:
x(k+1)＝(I+εB)x(k)＝Ax(k) (8)
the sampling time is equal to 0.05.
The method is distributed, has a simple structure, strong practicability and high efficiency, has certain robustness to communication time delay, and provides a feasible scheme for the formation control of the multiple mobile robots.
Claims (1)
1. The second-order 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; establishing an x-y Cartesian coordinate system in a motion space of the robot; 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
Represents the set of all complex numbers; the number of robots participating in formation in a plane is set to be n, and the robots are respectively numbered by using numbers 1,2, n-1 and n; 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 transposition of the matrix; in the formation control, if collision is not considered, the robot is considered as a particle with a collision-free volume; each robot in the system obeys a dual integrator motion model:
step 2, establishing a topological graph of the multi-robot system;
information interaction among multiple robots is represented as a directed topological graph G ═ (V, E), wherein V ═ V1，v2，...vnDenotes a set of n nodes in the diagram, node v in the diagram GiThe (i) th robot is represented,
representing a set of nodes and edges between them, edge e in graph GikE 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 other robots to be 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;
step 3, realizing a Laplace matrix according to the topological graph;
generating an adjacency matrix W corresponding to the directed topology 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 kth robotRelative position, i.e.
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 (3), Σ (·) is a summer symbol;
step 4, designing a plurality of Laplace matrixes;
defining symbol e as a natural constant and defining the formation as
According to the complex theory, ejθRepresenting a point of the unit circle on the complex plane; because the angle theta can be randomly specified, the value of the theta can be changed 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 (5)
Step 5, designing a second-order control protocol;
the second-order control protocol of the robot consists of the position complex weighted sum of the robot and the adjacent robots and the speed of the robot:
wherein u isiRepresenting the acceleration control input of the ith robot,
and
indicating the position of the ith and kth robots, p, respectivelyikThe ith row of the matrix P represents the kth element, viRepresenting the speed of the ith robot, gamma being a positive real number representing a speed decay factor, the formation rate, N, being adjustableiRepresenting the set of the remaining robots i can measure, i.e. Ni＝{vk：eikE is formed; the global dynamic response of all robots at the input of the control signal (5) can be expressed as:
wherein, On×nRepresenting a matrix of n rows and n columns all zero, In×nA unit array representing n rows and n columns;
step 6, designing a speed attenuation factor gamma;
in a continuous control system, gamma can be any positive integer, and the system is stable; in practical application, however, the speed attenuation factor gamma needs to be specially designed due to performance requirements and hardware equipment requirements; two ways are provided for how many the number of individual robots in the system are:
a) the number of robots in the system is less than 10; calculating the eigenvalues delta of the matrix L1，δ2，...，δnSelecting the characteristic value lambda of the configuration matrix B with gamma > 1iConfiguring eigenvalues of matrix B
The selection principle of gamma is as small as possible, the value is gradually increased from 1, the system response speed is higher when the gamma is larger, and the acceleration performance requirement of the robot is higher when the gamma value is larger;
b) when the number of robots in the system is 10 or more; suggesting a velocity decay factor
n represents the number of robots in the system;
step 7, designing a discrete control signal;
since the control signal is usually given as a discrete-time signal in practical applications, the discrete-time control signal corresponding to equation (6) is:
x(k+1)＝(I+εＢ)x(k)＝Ax(k) (8)
wherein epsilon is sampling time, the numeric area is more than 0.01 and less than 0.05, and I is an identity matrix with 2n rows and 2n columns.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202011547885.9A CN112612279B (en) | 2020-12-24 | 2020-12-24 | Second-order formation control method based on complex Laplace matrix |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN202011547885.9A CN112612279B (en) | 2020-12-24 | 2020-12-24 | Second-order formation control method based on complex Laplace matrix |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN112612279A CN112612279A (en) | 2021-04-06 |
| CN112612279B true CN112612279B (en) | 2022-03-18 |
Family
ID=75244636
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN202011547885.9A Active CN112612279B (en) | 2020-12-24 | 2020-12-24 | Second-order formation control method based on complex Laplace matrix |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN112612279B (en) |
Families Citing this family (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN112558609A (en) * | 2020-12-14 | 2021-03-26 | 北京理工大学 | Queue control method of multi-robot system based on ring topology |
Family Cites Families (5)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN106054875B (en) * | 2016-05-25 | 2019-01-04 | 北京航空航天大学 | A kind of distributed robots dynamic network connectivity control method |
| CN106647771B (en) * | 2017-01-24 | 2019-09-03 | 浙江工业大学 | The most small step formation method of multiple mobile robot |
| CN109032138B (en) * | 2018-07-25 | 2021-10-12 | 南京理工大学 | Consistency algorithm-based multi-robot formation control system and method |
| US11513533B2 (en) * | 2019-02-24 | 2022-11-29 | Technion Research & Development Foundation Limited | Topological belief space planning |
| CN110673649B (en) * | 2019-11-12 | 2022-09-09 | 哈尔滨工业大学（深圳） | Unmanned aerial vehicle formation consistency control method, system and device under time-varying channel based on topology optimization and storage medium |
-
2020
- 2020-12-24 CN CN202011547885.9A patent/CN112612279B/en active Active
Also Published As
| Publication number | Publication date |
|---|---|
| CN112612279A (en) | 2021-04-06 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Wu et al. | Distributed trajectory optimization for multiple solar-powered UAVs target tracking in urban environment by Adaptive Grasshopper Optimization Algorithm | |
| Yaoming et al. | A newly bio-inspired path planning algorithm for autonomous obstacle avoidance of UAV | |
| Wang et al. | Car-like mobile robot path planning in rough terrain using multi-objective particle swarm optimization algorithm | |
| Wang et al. | A hybrid metaheuristic DE/CS algorithm for UCAV three-dimension path planning | |
| Cao et al. | Multi-base multi-UAV cooperative reconnaissance path planning with genetic algorithm | |
| CN101188025B (en) | A high-efficiency real time group animation system | |
| CN106647771B (en) | The most small step formation method of multiple mobile robot | |
| Corner et al. | Parallel simulation of UAV swarm scenarios | |
| CN105511494A (en) | Method for multi unmanned aerial vehicle distributed formation control | |
| CN107065859A (en) | The trajectory predictions method of multiple mobile robot | |
| CN106705975A (en) | Method for planning route of unmanned surface vehicle based on universal gravitation search | |
| CN110181508A (en) | Underwater robot three-dimensional Route planner and system | |
| CN112612279B (en) | Second-order formation control method based on complex Laplace matrix | |
| Lobo et al. | Evolutionary development of tensegrity structures | |
| Gong et al. | Optimized layout methods based on optimization algorithms for DPOS | |
| CN113805609A (en) | Unmanned aerial vehicle group target searching method based on chaos lost pigeon group optimization mechanism | |
| Mengying et al. | Online path planning algorithms for unmanned air vehicle | |
| CN116642491A (en) | Ground unmanned platform field maneuvering path planning method | |
| Huang et al. | An artificial swan formation using the finsler measure in the dynamic window control | |
| Mou et al. | Three-Dimensional Area Coverage with UAV Swarm based on Deep Reinforcement Learning | |
| Patley et al. | Modified particle swarm optimization based path planning for multi-UAV formation | |
| Liu et al. | The continuous-discrete PSO algorithm for shape formation problem of multiple agents in two and three dimensional space | |
| Nishikawa et al. | Coordination control design of heterogeneous swarm robots by means of task-oriented optimization | |
| Zhou et al. | Study on composing dense formations in a dynamic environment of multirotor UAVs by distributed control | |
| Li et al. | Leader-Follower UAV FormationModel Based on R5DOS-Intersection Model. |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant |