CN104865960B - A kind of multiple agent approach to formation control based on plane - Google Patents

A kind of multiple agent approach to formation control based on plane Download PDF

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CN104865960B
CN104865960B CN201510213361.9A CN201510213361A CN104865960B CN 104865960 B CN104865960 B CN 104865960B CN 201510213361 A CN201510213361 A CN 201510213361A CN 104865960 B CN104865960 B CN 104865960B
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intelligent body
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CN104865960A (en
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王强
王玉振
张化祥
谭艳艳
李圣涛
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Shandong Normal University
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Abstract

The invention discloses a kind of multiple agent approach to formation control based on plane, comprise the following steps:Step one:The neighbours for determining each intelligent body according to system topology collect Ni(t) system adjacency matrix A, degree matrix D and Laplacian matrix Ls, and according to topological structure are constructed;Step 2:Multiple agent uniformity control protocol is calculated using the information of neighbor node;Step 3:On the basis of uniformity control protocol, by the stretching of matrix, intelligent body is assigned to different packets, packet synchronization control protocol is obtained;Step 4:By rotation transformation, the intelligent body of different grouping is handled again one by one so that each intelligent body is configured to expected formation position.The present invention can arbitrarily be set as needed to position of the intelligent body in formation, and design procedure is simple, and processing procedure clear layer, geometric meaning clearly, an effective control algolithm is provided for multiple agent formation problem.

Description

A kind of multiple agent approach to formation control based on plane
Technical field
The present invention relates to multiple agent field, the formation formation problem of multiple agent in studying plane, it is proposed that Yi Zhongping The formation control method and order switching method of Arbitrary Formation in face.
Background technology
So-called multi-agent system, i.e., it is multiple have intelligence entities by communicating, cooperating each other, coordinate, dispatch, Control etc. carrys out a class system of the structure, function and behavioral trait of expression system.Because one huge, difficult task usually surpasses Go out the limit of power of single intelligent body, therefore the multiple intelligent bodies of needs are mutually coordinated, cooperate with each other to complete jointly, therefore many intelligence There are wide Research Prospects in energy body field.For multi-agent system, formation problem is a kind of particularly important cooperation Require that multiple intelligent bodies keep specific formation when performing a certain task under mode, many scenes, such as:Dining room multirobot is passed Dish is serviced, and the multiple robots in workshop carry out logistics transport and Soccer robot contest etc..This algorithm is to propose intelligence more than one Formation control method when energy body is planar run.
Formation control is one of consistency problem and expanded with extending, and has inseparable relation with uniformity control. The present invention carries out a series of linear transformations to the quantity of state of intelligent body, proposes many intelligence in a kind of plane from consistency problem Can body formation control protocol algorithm.The algorithm geometric meaning is clear and definite, technology is realized simply, with extremely strong operability.
The content of the invention
To solve the deficiency that prior art is present, the invention discloses a kind of multiple agent formation controlling party based on plane Method, to realize that intelligent body keeps it is expected that formation performs a certain task, the intelligent processing ability of strengthening system.On reaching Effect is stated, each intelligent body needs to keep identical speed when in stable condition, and position spacing need to keep the numerical value specified, The numerical value depends on specific formation.
To achieve the above object, concrete scheme of the invention is as follows:
A kind of multiple agent approach to formation control based on plane, comprises the following steps:
Step one:The initial displacement x (0) and initial velocity v (0) of each intelligent body are obtained, it is true according to system topology Neighbours' collection N of fixed each intelligent bodyi(t) system adjacency matrix A, degree matrix D, and according to topological structure are constructed, and is thus constructed Laplacian matrix Ls;
Step 2:Multiple agent uniformity control protocol u is calculated using the information of neighbor nodei, by control protocol uiBring into Corresponding equation is obtained to single intelligent body motion model;
Step 3:Seek the non-zero characteristics root λ of Laplacian matrix Ls12,...,λr, determine that multiple agent uniformity is controlled Agreement uiIn undetermined parameter k, b value;The number of r representing matrixs L mutually different non-zero characteristics root, undetermined parameter k, B needs to meet stability condition;
Step 4:Structural matrix P meets P-1LP=J, and first diagonal element of matrix J is 0, wherein P-1Representing matrix P Inverse matrix, matrix J representing matrix L Jordans' matrix;
Step 5:In uniformity control protocol uiOn the basis of, by the stretching of matrix, intelligent body position is distributed To different packets, packet synchronization control protocol u is obtainedi';
Step 6:Synchronization Control agreement ui' on the basis of, by the rotation transformation of matrix, one by one by the intelligence of different grouping Can body progress handles and obtains final control protocol u againi" so that each intelligent body is configured to expected formation position.
In the step one, initial displacement x (0) and initial velocity v (0), wherein, x (0)=[x1(0),x2(0),...xn (0)]T, v (0)=[v1(0),v2(0),...vn(0)]T), n represents the number of intelligent body.
In the step one, adjacency matrix A, degree matrix D and Laplacian matrix Ls relation are:L=D-A.
In the step 2, for intelligent body formation control, single intelligent body motion model is:
Wherein, xi,viThe Position And Velocity of i-th of intelligent body, u are represented respectivelyiRepresent the control to be set of i-th of intelligent body Agreement processed;The first derivative of displacement and speed is represented respectively, and n represents the number of intelligent body, because each intelligent body is equal Planar move, therefore quantity of state xi,viBe two dimension, i.e. xi=[xi1,xi2]T,vi=[vi1,vi2]T
In the step 2, multiple agent uniformity control protocol uiSpecially:
Wherein k, b are undetermined constant, and its value size relies on target formation, xi,viThe position of i-th of intelligent body is represented respectively With speed, xj,vjDisplacement and speed of j-th of intelligent body in moment t are represented respectively.
In the step 2, by control protocol uiIt is brought into single intelligent body motion model and obtains corresponding equation, has Body is:
WhereinRepresent that the first derivative of vector x is led with second order respectively Number, x=[x1,x2,...,xn]T, k, b is undetermined constant, for ensuring the stability of system and adjusting the convergent speed of formation, I2 Represent second order unit matrix, i.e. I2=diag (1,1),Represent kronecker products.
In the step 3, multiple agent uniformity control protocol u is determinediIn undetermined parameter k, b value when,
According to Theory of Stability, work as b, k is metWhen, (2)
System determines k, b value with this up to uniformity;Wherein, λll+iβlFor the non-of Laplacian matrix Ls Zero characteristic root, αlIt is characterized root λlReal part, βlIt is characterized root λlImaginary part, r is mutually different non-zero characteristics root number, and i is full Sufficient i2=-1;
In the step 4, in addition to, work as b, when k meets stability condition, system speed is finally converged onThe first two component, be designated asWherein v (0)=[v1(0),v2(0),...vn(0)]TRepresent all The speed of n intelligent body initial time, it can thus be appreciated that the angle of the intelligent body direction of motion and trunnion axis is
In step 4, matrix P is obtained, for judging the intelligent body direction of motion, direction is vectorBefore The cotangent value of the ratio of two components, direction angle is the reference angle of the rotation transformation of step 6.
In the step 5, packet synchronization control protocolWherein ηjiRepresent The distance between different grouping, k, b is undetermined constant, and its value size relies on target formation, xi,viI-th of intelligent body is represented respectively Position And Velocity, xj,vjDisplacement and speed of j-th of intelligent body in moment t are represented respectively.
In the step 6, final control protocol ui”:
Connect intelligent body xiWith the line of barycenter, its Middle θiFor the line and the angle of the direction of motion, size is dependent on expectation formation, ηjiThe distance between expression different grouping, k, B is undetermined constant, and its value size relies on target formation, xi,viThe Position And Velocity of i-th of intelligent body, x are represented respectivelyj,vjPoint Displacement and speed of j-th of intelligent body in moment t are not represented.
The intelligent body includes robot, intelligent vehicle and other nobody autonomous device.
The Ni(t) neighbours collection of i-th of node in moment t is represented.
Beneficial effects of the present invention:
The present invention gives a kind of control of Arbitrary Formation in plane using the neighbours' speed and position relationship of multiple agent Protocol Design Method, and position of each intelligent body in formation can arbitrarily set, this method design procedure is simple, geometry meaning It is adopted clear and definite, with extremely strong practicality and operability.
Brief description of the drawings
Fig. 1 topology diagrams;
Fig. 2 topological diagram adjacency matrix;
Fig. 3 topology scale matrixes;
Fig. 4 topological diagram Laplacian matrixes;
The formation array example of 16 intelligent bodies of Fig. 54 × 4;
Fig. 6 control protocol design flow diagrams
Embodiment:
The present invention is described in detail below in conjunction with the accompanying drawings:
The technical solution of the present invention gives point to propose multiple agent formation control method in a kind of plane The design of relevant parameter in cloth controller.The program, which has, realizes simple, the features such as control performance is excellent.
As shown in figure 1, Fig. 1 give between the topological structure schematic diagram of 6 intelligent bodies, two of which intelligent body if Side is connected, then two intelligent bodies mutually claim neighbours;If the side is unidirectional, then it represents that information transfer is also to be single between intelligent body To now the topology is called digraph, otherwise referred to as non-directed graph.Side between neighbours is if weight, then topology is called weighting Figure, for weighted graph, if weights are not zero between two intelligent bodies, the two intelligent bodies are called neighbours.
Fig. 2 is the adjacency matrix of topological structure, and wherein adjacency matrix A is if weighted graph, then corresponding aijFor specific power Weight values;Fig. 3 is the degree matrix D of topological diagram, and degree matrix is pair of horns matrix, diagonal element diiIt is all equal to the row in adjacency matrix Element sum;Fig. 4 represents the Laplacian matrixes of topological diagram, if topological structure is connected graph, Laplacian matrixes have One zero characteristic roots, and remaining characteristic root has positive real part.
Multiple agent formation control method, step one in a kind of plane:Each intelligent body is determined according to system topology Neighbours collection Ni(t) system adjacency matrix A, degree matrix D and Laplacian matrix Ls, and according to topological structure are constructed;Step 2: Multiple agent uniformity control protocol is calculated using the information of neighbor node;Step 3:On the basis of uniformity control protocol, By the stretching of matrix, intelligent body is assigned to different packets, packet synchronization control protocol is obtained;Step 4:Pass through Rotation transformation, is one by one handled the intelligent body of different grouping again so that each intelligent body is configured to expected formation position Put.
The design cycle of control parameter is as shown in fig. 6, the specific algorithm of control protocol, is specially when realizing:
2-1) obtain the Position And Velocity of each intelligent body of initial time;
2-2) judge whether system topological connects, if can not connect, it is expected that formation can not be realized, if can connect, Then it is transferred to step 2-3)
Control protocol u1 2-3) is set so that system realizes uniformity;
Packet synchronization agreement u2 2-4) is set on the basis of uniformity;
Rotation displacement 2-5) is carried out for the intelligent body in same group, agreement u3 is set;
2-6) Comprehensive Control agreement u1, packet synchronization agreement u2 and agreement u3, obtain final control protocol.
The parameter of final control protocol is when calculating, specially:
3-1) multiple agent uniformity is calculated, and coefficient k (1) in final control protocol is calculated firstji,k(1)ii,b,i,j ∈ 1,2 ..., n so that system realizes that uniformity, i.e. speed are consistent, and position is consistent, wherein k (1)ji,k(1)iiTable Show the k converted for the first time in the final control protocol of formulaji,kiiIt is worth size, k (1)ji,k(1)ii, b size depends on The value of the non-zero characteristics root of Laplacian matrix Ls, is shown in formula (2);
Packet calculating 3-2) is carried out to intelligent body, according to formation needs, intelligent body is grouped, i.e., to different grouping Intelligent body carries out different degrees of stretching k (2)ji,k(2)ii, i, j ∈ 1,2 ..., n, telescopic level depend on specific The distance between formation difference group;
Rotary extension conversion 3-3) is carried out with the intelligent body in group, rotation change is gradually carried out to the intelligent body in same packet Change k (3)ji,k(3)iiSo that the intelligent body in same packet keeps the distance specified;
3-4) make kji=k (3)jik(2)jik(1)ji, kii=k (3)iik(2)iik(1)ii, obtain kii,kji(i, j=1, 2 ..., n), and b, it is brought into the final control protocol of formulaAs Controller parameter to be asked.
The Ni(t) neighbours collection of i-th of node in moment t is represented.
The intelligent body includes robot, intelligent vehicle and other nobody autonomous device.
The step 3-2) in when being grouped to intelligent body, can first horizontal direction be grouped, also can be first in Vertical Square To being grouped.
Step 3-2)-step 3-4) be Protocol Design parser, intelligent body actual motion route will not according to aggregation, point Group, three steps of rotation are run.
In order to realize faster rate of convergence, in k, on the premise of b meets formula (2), it can suitably increase k, b value.
Concrete case:
Next how will be realized by 16 intelligent bodies exemplified by 4 × 4 formation processes (Fig. 5).
1st, uniformity is designed.Make control protocolBy uiBring formula (1) into and whole Li Ke get:
2. it is theoretical according to the stability of a system, work as k, b > 0, andWhen (αllRespectively The characteristic root λ of Laplacian matrix LslReal part and imaginary part, r is the number of non-zero characteristics root that matrix L is not mutually equal, system Speed is with displacement up to uniformity;
3. a pair intelligent body is grouped., will if reaching intended effect after packet according to the correlation theory of linear transformation System equation is expressed as after matrix formWherein matrix The spacing of different grouping passes through η1234Size adjust accordingly (η values are identical in same packet), by the system point Solve that after single subsystem, for system (1), control protocol can be obtainedIf wherein xiWhen belonging to l groups, ηil(l=1,2,3,4), I4For quadravalence unit matrix, i.e. I4=diag [1,1,1,1];
4. carry out rotary extension conversion with the intelligent body in group.The barycenter O under multiagent system stable state is found, and will It sets up rectangular coordinate system, connection intelligent body x as the origin of coordinatesiWith barycenter, the angle of the line and the direction of motion is designated as θi。 Multiple agent equation is expressed as after matrix equation, hadWhereinIt can thus be concluded that control protocol:
Although above-mentioned the embodiment of the present invention is described with reference to accompanying drawing, not to present invention protection model The limitation enclosed, one of ordinary skill in the art should be understood that on the basis of technical scheme those skilled in the art are not Need to pay various modifications or deform still within protection scope of the present invention that creative work can make.

Claims (9)

1. a kind of multiple agent approach to formation control based on plane, it is characterized in that, comprise the following steps:
Step one:The initial displacement x (0) and initial velocity v (0) of each intelligent body are obtained, is determined according to system topology every Neighbours' collection N of individual intelligent bodyi(t) system adjacency matrix A, degree matrix D, and according to topological structure are constructed, and is thus constructed Laplacian matrix Ls;
Step 2:Multiple agent uniformity control protocol u is calculated using the information of neighbor nodei, by control protocol uiIt is brought into list Individual intelligent body motion model obtains corresponding equation;
Step 3:Seek the non-zero characteristics root λ of Laplacian matrix Ls12,...,λr, determine multiple agent uniformity control protocol uiIn undetermined parameter k, b value;The number of r representing matrixs L mutually different non-zero characteristics root, undetermined parameter k, b need to Meet stability of a system condition;
Step 4:Structural matrix P meets P-1LP=J, and first diagonal element of matrix J is 0, wherein P-1Representing matrix P's is inverse Matrix, matrix J representing matrix L Jordans' matrix;
Step 5:In uniformity control protocol uiOn the basis of, by the stretching of matrix, intelligent body position is assigned to not Same packet, obtains packet synchronization control protocol ui';
Step 6:Synchronization Control agreement ui' on the basis of, by the rotation transformation of matrix, one by one by the intelligent body of different grouping again Secondary progress, which is handled, obtains final control protocol ui" so that each intelligent body is configured to expected formation position;
In step 4, in addition to:Work as b, when k meets stability condition, system speed is finally converged onPreceding two Individual component, is designated asWherein v (0)=[v1(0),v2(0),...vn(0)]TRepresent whole n intelligent body initial times Speed, I2Represent second order unit matrix, i.e. I2=diag (1,1), it can thus be appreciated that the angle of the intelligent body direction of motion and trunnion axis is
2. a kind of multiple agent approach to formation control as claimed in claim 1 based on plane, it is characterized in that, the step one In, initial displacement x (0) and initial velocity v (0), wherein, x (0)=[x1(0),x2(0),...xn(0)]T, v (0)=[v1(0), v2(0),...vn(0)]T), n represents the number of intelligent body.
3. a kind of multiple agent approach to formation control as claimed in claim 1 or 2 based on plane, it is characterized in that, the step In rapid one, adjacency matrix A, degree matrix D and Laplacian matrix Ls relation are:L=D-A.
4. a kind of multiple agent approach to formation control as claimed in claim 1 based on plane, it is characterized in that, the step 2 In, for intelligent body formation control, single intelligent body motion model is:
<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>x</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>v</mi> <mi>i</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>v</mi> <mo>&amp;CenterDot;</mo> </mover> <mi>i</mi> </msub> <mo>=</mo> <msub> <mi>u</mi> <mi>i</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>...</mo> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>
Wherein, xi,viThe Position And Velocity of i-th of intelligent body, u are represented respectivelyiRepresent i-th of intelligent body control association to be set View;The first derivative of displacement and speed is represented respectively, n represents the number of intelligent body, because each intelligent body is flat In-plane moving, therefore quantity of state xi,viBe two dimension, i.e. xi=[xi1,xi2]T,vi=[vi1,vi2]T
5. a kind of multiple agent approach to formation control as claimed in claim 1 based on plane, it is characterized in that, the step 2 In, multiple agent uniformity control protocol uiSpecially:
<mrow> <msub> <mi>u</mi> <mi>i</mi> </msub> <mo>=</mo> <munder> <mi>&amp;Sigma;</mi> <mrow> <mi>j</mi> <mo>&amp;Element;</mo> <msub> <mi>N</mi> <mi>i</mi> </msub> </mrow> </munder> <mo>&amp;lsqb;</mo> <mi>b</mi> <mrow> <mo>(</mo> <msub> <mi>v</mi> <mi>j</mi> </msub> <mo>-</mo> <msub> <mi>v</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>+</mo> <mi>k</mi> <mrow> <mo>(</mo> <msub> <mi>x</mi> <mi>j</mi> </msub> <mo>-</mo> <msub> <mi>x</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> <mo>&amp;rsqb;</mo> </mrow>
Wherein k, b are undetermined constant, and its value size relies on target formation, xi,viPosition and the speed of i-th of intelligent body are represented respectively Degree, xj,vjDisplacement and speed of j-th of intelligent body in moment t are represented respectively.
6. a kind of multiple agent approach to formation control as claimed in claim 1 based on plane, it is characterized in that, the step 2 In, by control protocol uiIt is brought into single intelligent body motion model and obtains corresponding equation, is specially:
WhereinThe first derivative and second dervative of vector x, x=are represented respectively [x1,x2,...,xn]T, k, b is undetermined constant, for ensuring the stability of system and adjusting the convergent speed of formation, I2Represent two Rank unit matrix, i.e. I2=diag (1,1),Represent kronecker products.
7. a kind of multiple agent approach to formation control as claimed in claim 1 based on plane, it is characterized in that, the step 3 In, determine multiple agent uniformity control protocol uiIn undetermined parameter k, b value when,
According to Theory of Stability, work as b, k is metWhen, (2) system is determined up to uniformity with this K, b value;Wherein, λll+iβlFor the non-zero characteristics root of Laplacian matrix Ls, αlIt is characterized the real part of root, βlIt is characterized The imaginary part of root, r is mutually different non-zero characteristics root number, and i meets i2=-1.
8. a kind of multiple agent approach to formation control as claimed in claim 7 based on plane, it is characterized in that, the step 5 In, packet synchronization control protocolWherein ηjiThe distance between different grouping is represented, K, b are undetermined constant, and its value size relies on target formation, xi,viThe Position And Velocity of i-th of intelligent body, x are represented respectivelyj,vj Displacement and speed of j-th of intelligent body in moment t are represented respectively.
9. a kind of multiple agent approach to formation control as claimed in claim 8 based on plane, it is characterized in that, the step 6 In, final control protocol ui”:
Connect intelligent body xiWith the line of barycenter, wherein θiFor The angle of the line and the direction of motion, size is dependent on expectation formation, ηjiThe distance between different grouping, k are represented, b is to treat Permanent number, its value size relies on target formation, xi,viThe Position And Velocity of i-th of intelligent body, x are represented respectivelyj,vjRepresent respectively Displacement and speed of j-th of intelligent body in moment t.
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