Targets Assignment for Cooperative Multi-UAVs Based on Chaos Optimization Algorithm

Wei Ou, Fengxing Zou, Xiaohong Xu, Zheng Gao Department of Automatic Control, College of Mechatronic Engineering and Automation,

National University of Defense Technology,410073 Changsha, china ouweiwlmq@163.com

Abstract

Targets assignment is one of crucial problem for multi-UAVs cooperative campaign. By taking furthest campaign benefit and survival probability of UAVs as objective function, the mathematics model of targets assignment is formulated in this paper, and an approach solving targets assignment problem based on chaos optimization is proposed, whereas a new encoding method is introduced, chaos search queues is mapped to solution space of targets assignment problem effectively by defining exchange, shift and insertion operators, and an correcting method is applied to keep the validity of solutions. Finally, the efficiency of the proposed algorithm is demonstrated by computer simulations. Keywords: UAV; targets assignment; chaos optimization algorithm; encoding method 1. Introduction

Unmanned Air Vehicle (UAV) performances multiple military missions, such as reconnaissance, alert, and exactitude attack. Because of the multeity and complexity in campaign task, formation and cooperative campaign for UAVs is needed, how to scheme formation and assign targets becomes the key problem [1][2]. Targets assignment problem (TAP) is a typical NP-hard problem, which has prohibitive computational complexity for classical combinatorial optimization methods [3].

Genetic algorithm (GA) and particle swarm optimization (PSO) has obtained extensive application on UAVs-targets assignment problems [4][5] because of their excellent robustness and flexibility. But some limitations exists currently: (1) the model of TAP is relatively simple, many literature studies the problems assigning one UAV to attack each target, but in fact, to improve the campaign benefits and survival possibility,

we often assign more than one UAVs to attack an important target (with high value or threat); (2) the objective functions are relatively simple, it cannot describe the key objectives of campaign completely [6]; (3) these algorithms may run into local optimums, the optimization of the targets assignment scheme con not be assured.

Chaos is a ubiquitous phenomenon existing in nonlinear system, which has the characteristics of randomicity, ergodicity and regularity. The basic antilogy of chaos optimization algorithm (COA) maps chaos queues from chaos space to solution space, and searches all states in a certain area non-repeatedly according to its owing regularity for the best solution of the problem. COA is commonly sensitive to initial value, easiness of escaping from local optimums, converging to global optimums with high precision [7][8]. This paper copes with the UAVs-targets assignment problem, formulates the mathematical model of TAP, and introduces a new approach to solve TAP based on COA, the efficiency of the proposed model and algorithm is validated by computer simulations.

2. Problem Description

Supposing there is uN UAVs to attack multiple targets in batches, the number of targets in a certain batch is TN , we defines U as the set of UAVs, T as the set of targets. On the assumption that:

(1) Each UAV is allowed to attack multiple targets in different batches, but for targets in a certain batch, one target is allowed.

(2) Assigning multiple UAVs to attack one target is allowed, and at least assigning one UAV to each target is required.

(3) Each UAV is allowed to attack one target at will by a different killing probability, if iU attacks jT , the killing probability is ijP .

The 9th International Conference for Young Computer Scientists

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DOI 10.1109/ICYCS.2008.512

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(4) The value of each target is different, the value of jT is jv .

(5) UAVs is threatened while they attack targets, if iU attacks jT , the survival probability of iU is

ijS , assigning multiple UAVs to attack a target would improve the holistic survival probability of the formation.

Then the UAVs-targets assignment problem can be described as follows [6][9]: to find an optimal scheme to assign UAVs to attack targets, the objective of which is to improve the holistic survival probability and the campaign benefit.

Objective1: 1 1

( (1 (1 )))UT NN

ij ijj i

Maxmize S x= =

− −∑ ∏

(1)

Objective2: 1 1

( (1 (1 )))UT NN

j ij ijj i

Maxmize v P x= =

− −∑ ∏

(2)

Subject to: 1

1n

ijj

x=

=∑

(3)

1

1m

iji

x=

≥∑

(4)

Where 1, 2 , ui N= , 1, 2 , Tj N= . ijx is 0 1− variable, if iU attacks jT , 1ijx = ; else 0ijx = . Equation (1) indicates that one objective of this scheme is to maximize the holistic survival probability; equation (2) indicates that another objective is to maximize the campaign benefit; equation (3) and (4) ensures that, for the targets in a certain batch, each UAV is allowed to attack one target only, and each target should be assigned to not less than one UAVs.

3. Solving Approach Based on COA 3.1 Encoding Method

To express the feasible solution, encoding for

TAP is needed. We define the corresponding code for solutions as solution vector. For making solution vectors satisfies to the restrictions, the encoding approach is as following: Given a vector based on the number of targets when encoding, then explain it according to the position of this numbers appearing in the vector when decoding, the numbers in the vector represent targets, positions of the numbers appearing in the vector represent UAVs.

Suppose there are 5 UAVs and 4 targets, the solution vector is (2,3,1,4,2)X = , the assignment scheme is as Table 1.

Table 1. UAVs-targets assignment UAV 1 2 3 4 5

Solution vector 2 3 1 4 2

The vector will be explained as follows: UAV 1 and UAV 5 attack target 2, UAV 2 attacks target 3, UAV 3 attacks target 1, UAV 4 attacks target 4.

3.2 Initialization

The initialization method of the solution vector

X is as following: (1) let all of the positive integral in [1, ]TN appears once from positive 1 to TN of X , array randomly; (2) the elements from positive 1TN + to WN are selected from the integers in [1, ]TN randomly. All of the elements in the vector is belongs to the integers in [1, ]TN , and all of the targets at least appears in the vector once, which ensures the elements in the vector represent the targets validly and the validity of the solution vector.

Chaos phenomenon can experience all the states in a certain area non-repeatedly, we apply the Logistic equation to obtain chaos queues.

1 (1 )n n nζ μζ ζ+ = − (5) Where 0 1nζ≤ ≤ , 1, 2n = , μ is the controls parameter, let 4μ = , then the system will be in completely chaos state. 3.3 Definition of Base Operators

For mapping the chaos queues to feasible solution space, we define operators to actualize exchange, shift and insert elements in the solution vector. The definitions are as following:

Figure 1. The exchange operator

Definition 1 (exchange operator): exchange element i with element j in the solution vector, keep other elements immovable, marked this operation as “ ij exchange”. E.g. (3,1,2, 4,1,2)X = , to actualize “2,4 exchange”, the process is shown in Figure 1.

Figure 2. The shift operator

Definition 2 (shift operator): shift all the elements between element 1i + and j to the position

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onebefore their own position, insert the original element i to position j , marked this operation as “ ij shift”. E.g. (3,1,2, 4,1,2)X = , to actualize “2,4 shift”, the process is shown in Figure 2.

Figure 3. The insertion operator

Definition 3 (insertion operator): shift all the elements between element i and the last element to the position one behind their own (delete the last element), then insert a new element j in position i , marked this operation as “ ij insertion”. E.g.

(3,1,2, 4,1,2)X = , to actualize “2,4 insertion”, the process is shown in Figure 3.

3.4 Validity Correcting

Exchange operator and shift operator change the order of the elements appearing in the vector, but do not influence the times of the elements appearing in the vector, but insertion operator inserts a new element and deletes the last element, the times of the elements appearing in the vector may be influenced, accordingly, the validity of the vector may be violated, so validity correcting is needed.

Figure 4. Validity correcting

The basic steps of correcting method are as follows: （1）find all of the targets never appearing in the vector; (2) find the targets appearing in the vector most frequently, replace the last position of this target appears in the vector by one of the target found in (1); (3) repeat (2), until the vector is valid. E.g. 6uN = , 4TN = , (4,2,1, 4, 4, 2)X = , because target 3 never appears in the vector, the vector is not valid, the correcting method is shown in Figure 4. 3.5 The Fitness Function

To take the two objective of the problem (improve the survival probability and seek for the optimal

campaign benefit) into account, the fitness function is as equation (6) shown.

1 21 11 1(1 (1 )) (1 (1 ))

U UT TN NN N

ij ij j ij ijj ji i

F S x v P xλ λ= == =

= − − + − −∑ ∑∏ ∏

6)

Where 1λ and 2λ are the weight coefficients, which are applied to adjust the aggregation weight between the two objective function.

3.6 Basic Flow of COA

The Basic flow of the approach solving TAP base on COA is presented as follows:

Step 1（Initialization）：Set MG , 1β , 2β , let 0n = , where MG is maximum iteration number, n

is iteration counter; 1β , 2β is used to choose operators;

initialize solution vector X and initial value 1nζ , 2

nζ（ 1 2, (0,1)n nζ ζ ∈ ） randomly; let nAM AN X= = , where nAN denotes current solution and AM denotes the solution best-so-far; calculate nF , let

m nF F= , where nF is the fitness value of nAN , mF is the fitness value of AM .

Step 2（ chaos queues updating） : Calculate 1

1nζ + and 21nζ + by utilizing equation (5); calculate ik

and jk utilizing equation (7) and equation (8), where ( )floor x denotes the integer part of x .

1

1( )i U nk floor N ζ += ∗ （7）

2

1( )j U nk floor N ζ += ∗ （8） Step 3（Current solution updating）: Generate a

number nR in (0,1) randomly, if 1nR β< , choose exchange operator, actualize “ i jk k exchange” in

nAN ; if 1 2nRβ β≤ ≤ , choose shift operator, actualize “ i jk k shift” in nAN ; if 2 nRβ < , choose inserting operator, generate an integer kk randomly in [1, ]TN , actualize “ i kk k insertion” in nAN . Then we can get 1nAN + , examine the validity of 1nAN + , if 1nAN + is not valid, correct the validity by utilizing the correcting method in section 3.4.

Step 4（Best solution updating）: Calculate 1nF + by utilizing equation（6），if 1n mF F+ > , then let 1m nF F += ,

1nAM AN += ；else keep mF and AM invariable. Step 5（Index updating） : 1n n= + ; 1n nF F += ;

1n nAN AN += . Step 6（Stopping criteria）: If n MG≺ , loop to

step2; else stop, take AM as the solution of TAP.

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4．Computer Simulation

On the assumption that there is a formation formed by 10 UAVs, the mission is to strike a certain batch of air targets, the number of targets is 8, the value of each target is shown in Table 2.

Table 2. Value coefficients of targets Target 1 2 3 4 5 6 7 8 Value 0.9 0.4 0.2 0.5 0.3 0.6 0.9 0.7

Supposing the threat of each target to UAVs is evaluated, the survival probability of each UAV (if an UAV is assigned to attack a certain target) is shown in Table 3.

Table 3. Survival probability of UVAs

UAV Target 1 2 3 4 5 6 7 8

1 0.9 0.5 0.6 0.7 0.8 0.5 0.4 0.5 2 0.6 0.4 0.9 0.7 0.6 0.5 0.4 0.3 3 0.4 0.9 0.4 0.7 0.6 0.4 0.5 0.3 4 0.8 0.7 0.6 0.5 0.9 0.4 0.2 0.6 5 0.7 0.2 0.4 0.6 0.9 0.8 0.3 0.4 6 0.3 0.5 0.4 0.2 0.6 0.5 0.7 0.9 7 0.9 0.8 0.3 0.4 0.5 0.2 0.6 0.9 8 0.4 0.3 0.5 0.2 0.6 0.7 0.8 0.9 9 0.4 0.5 0.3 0.4 0.6 0.4 0.9 0.8 10 0.5 0.4 0.3 0.9 0.8 0.9 0.4 0.6 Supposing the attacking feasibility is evaluated,

the killing probabilities are shown in Table 4. Table 4. Killing probability

UAV Target 1 2 3 4 5 6 7 8

1 0.9 0.4 0.9 0.5 0.5 0.7 0.4 0.9 2 0.7 0.8 0.6 0.8 0.5 0.9 0.8 0.4 3 0.9 0.3 0.4 0.5 0.8 0.6 0.9 0.5 4 0.8 0.6 0.4 0.5 0.8 0.9 0.7 0.6 5 0.7 0.3 0.6 0.4 0.9 0.5 0.2 0.8 6 0.9 0.7 0.6 0.5 0.4 0.2 0.8 0.3 7 0.2 0.9 0.6 0.3 0.5 0.4 0.8 0.9 8 0.6 0.5 0.9 0.9 0.6 0.4 0.7 0.6 9 0.7 0.8 0.3 0.9 0.5 0.6 0.4 0.8 Select the parameters as follows: 4000MG = ;

1 0.6β = , 2 0.8β = . Let aggregation coefficients be different values, many groups of experiments are undertaken to get the optimal solutions under different cases, the results are shown in Table 5.

Table 5. Best solutions under different cases Case 1λ 2λ Optimal Solution

1 1 0 (1,3,2,5,6,8,2,6,7, 4) 2 0 1 (8,7,7,6,1,1, 2,3, 4,5) 3 0.4 0.6 (1,3,1,6,5,7, 2,7,8, 4) Where case 1 takes survival probability into

account only, the corresponding assignment scheme of the optimal solution is as following: UAV 1 attacks

target 1, UAV 2 attacks target 3, UAV 3 and UAV 7 attack target 2, etc.. Comparing with Table 3, we can make out that this scheme either assigns UAVs to attack the targets that UAVs can gain the maximum survival probability, or assigns multiple UAVs to attack one target to improve survival probability.

Case 2 takes campaign benefit into account only, the corresponding assignment scheme of the optimal solution is as following: UAV 1 attacks target 8, UAV 2 and UAV 3 attack target 7, UAV 4 attacks target 6, etc.. Comparing with Table 2 and Table 4, we can make out that this scheme assigns two UAVs to attack the most valuable targets 1 and 7 respectively, while assigns the other UAVs with largest killing probability to attacks other targets.

Case 3 simultaneously takes survival probability and campaign benefit into account, the corresponding assignment scheme of the optimal solution is shown in Figure 5. Comparing with Tables 2 to 4, we can make out that this scheme assigns two UAVs to attack the most valuable targets 1 and 7 respectively, and the killing probability to targets and survival probability of UAVs are optimized simultaneously, so this solution is a reasonable solution of TAP.

Figure 5. UAV-target assignment ( 1 20.4, 0.6λ λ= = )

Convergence is one of most important evaluation criterion for optimization algorithms, for the proposed three cases, let evaluation function ( ) 1 ( )eval n F n= , where ( )F n denotes the fitness value of the solution best-so-far when the iteration number equals n . Figure 6 is the convergence curve of the evaluation value gained by a group of random experiment.

Figure 6 indicates that the evaluation functions can converge to a steady value for the proposed three cases. Though the iteration number is a little large (for all cases, the algorithm converges to optimal solutions within 3000 times iterations), comparing with evolutionary algorithms (e.g. GA or PSO), if the population size is 40, evolution generation is 200, then more than 8000 times iterations are needed. It indicates

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that the proposed algorithm has high convergence efficiency.

Figure6. Convergence curve of the three cases For the proposed three cases, 50 times of

experiments are undertaken respectively and randomly for the proposed COA and PSO, the population size of PSO is 40, the evolution generation is 200, and utilizes the same encoding and Validity Correcting method as the COA, the statistical result of the final solutions are shown in Table 5.

Table 5. Statistical result of two algorithms

case Algorithm Number of solutions Optimal Suboptimal other

1 COA 9 38 3 PSO 3 27 20

2 COA 11 39 0 PSO 2 31 17

3 COA 9 40 1 PSO 4 28 18

The optimal and suboptimal solutions in Table 5 show the characteristic of the solutions found by the proposed COA and PSO, Which is an evidence for the high rate of convergence and excellent searching efficiency of COA, and indicates that the proposed COA has higher rate of convergence and searching efficiency than PSO.

5. Conclusion

In this paper, a new chaos optimization algorithm (CAO) is proposed to solve UAVs-targets assignment problem. To map the chaos queues to the solution space of TAP, exchange, shift and inserting operators are introduced simultaneously, and a correcting method is presented to ensure the validity of solutions. The proposed algorithm can find the reasonable and optimal solutions of targets assignment problems with high rate of convergence and excellent searching performance, which was illustrated by the computer simulations.

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