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Combat effectiveness of unmanned aerial vehicle (UAV) formations can be severely affected by the mission execution reliability. During the practical execution phase, there are inevitable risks where UAVs being destroyed or targets failed to be executed. To improve the mission reliability, a resilient mission planning framework integrates task pre- and re-assignment modules is developed in this paper. In the task pre-assignment phase, to guarantee the mission reliability, probability constraints regarding the minimum mission success rate are imposed to establish a multi-objective optimization model. And an improved genetic algorithm with the multi-population mechanism and specifically designed evolutionary operators is used for efficient solution. As in the task-reassignment phase, possible trigger events are first analyzed. A real-time contract net protocol-based algorithm is then proposed to address the corresponding emergency scenario. And the dual objective used in the former phase is adapted into a single objective to keep a consistent combat intention. Three cases of different scales demonstrate that the two modules cooperate well with each other. On the one hand, the pre-assignment module can generate high-reliability mission schedules as an elaborate mathematical model is introduced. On the other hand, the re-assignment module can efficiently respond to various emergencies and adjust the original schedule within a millisecond. The corresponding animation is accessible at bilibili.com/video/ BV12t421w7EE for better illustration.
ARTICLE INFO
Article history:
Received 30 January 2024
Received in revised form 30 June 2024
Accepted 7 August 2024
Available online 13 August 2024
Keywords:
Cooperative mission planning
UAV formation
Mission reliability
Evolutionary algorithm
Contract net protocol
ABSTRACT
Combat effectiveness of unmanned aerial vehicle (UAV) formations can be severely affected by the mission execution reliability. During the practical execution phase, there are inevitable risks where UAVs being destroyed or targets failed to be executed. To improve the mission reliability, a resilient mission planning framework integrates task pre- and re-assignment modules is developed in this paper. In the task pre-assignment phase, to guarantee the mission reliability, probability constraints regarding the minimum mission success rate are imposed to establish a multi-objective optimization model. And an improved genetic algorithm with the multi-population mechanism and specifically designed evolutionary operators is used for efficient solution. As in the task-reassignment phase, possible trigger events are first analyzed. A real-time contract net protocol-based algorithm is then proposed to address the corresponding emergency scenario. And the dual objective used in the former phase is adapted into a single objective to keep a consistent combat intention. Three cases of different scales demonstrate that the two modules cooperate well with each other. On the one hand, the pre-assignment module can generate high-reliability mission schedules as an elaborate mathematical model is introduced. On the other hand, the re-assignment module can efficiently respond to various emergencies and adjust the original schedule within a millisecond. The corresponding animation is accessible at bilibili.com/video/ BV12t421w7EE for better illustration.
© 2024 China Ordnance Society. Publishing services by Elsevier B.V. on behalf of KeAi Communications Co. Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
(ProQuest: ... denotes formulae omitted.)
1. Introduction
Unmanned Aerial Vehicles (UAVs), which have the advantages of low cost and zero casualties, are becoming one of the most important combat forces in the military [1]. Increasingly complex battlefields often contain large-scale and persistent missions that cannot be accomplished by a UAV with limited capabilities and resources, Which has led to research into cooperative UAV formation operations that are more advantageous in terms of operational capabilities and resources [2]. Tasks are not necessarily terminated due to the malfunction of some UAVs in the formation, and a UAV formation has been shown to perform better than a single UAV in terms of time and efficiency [3,4]. The key technology for exploiting the UAV formation advantage is task assignment, which is to provide a mapping of UAVs to targets with priorities [5,6]. Task assignment not only allows for better allocation of the UAV formation's resources, but also improves mission reliability.
The cooperative task assignment problem (CTAP) for UAV formations is an NP-hard combinatorial optimization problem with multiple complex constraints [7]. Based on the phase of performing task assignment, CTAP can be categorized into two scenarios, one is task pre-assignment before combat and the other is task reassignment during combat [8]. Research on CTAP focuses on the model construction and algorithm design. Some classical models can be used for task assignment modeling, such as multiple traveling salesman problem [9], vehicle routing problem [10], mixed integer linear programming problem [11], and network flow optimization [12], etc. The existing solution techniques mainly include deterministic algorithms, reinforcement learning and intelligent algorithms. Deterministic algorithms, such as dynamic programming, branch and bound, etc., are generally used for solving small-scale problems, however, they are computationally inefficient for large-scale problem models that contain many decision variables and constraints. Reinforcement learning has the drawbacks of difficult reward function setting, long training period and poor generalization. In contrast, intelligent algorithms have the advantages of low model dependency and short computation time, such as genetic algorithm (GA), wolf pack algorithm (WPA), and ant colony optimization (ACO), etc, which are widely used for solving large-scale CTAP.
Task pre-assignment is the process of assigning a set of task sequences to each UAV based on known information before combat. For describing and solving the more realistic CTAP, some significant aspects besides limited resources should be considered in the modeling, and more efficient algorithms are also required. Considering the precedence order of tasks execution, Edison et al. [13] studied the CTAP coupled with planning paths, and proposed an GA that could obtain good fast feasible solutions. However, target precedence was ignored. To address this issue, Zhao et al. [14] developed a CTAP model involving the target precedence constraints, and given a modified GA by combining the waitable path coordination algorithm and graph-based method. To improve the allocation efficiency, Wu et al. [15] developed a fusion GA that includes the quadratic selection operation using the modified simulated annealing algorithm, which improves the diversity of populations. For the CTAP in hostile environments, Alhagbani et al. [16] proposed a fish-inspired algorithm based on the fish's adaptive schooling and foraging behaviors. For cross-regional joint operations, Yu et al. [17] developed an improved GA for CTAP in which a new chromosome encoding format and an unlocking mechanism are constructed.
However, the single-objective optimization in the above study cannot simultaneously optimize conflicting indicators of complex combat requirements, such as minimizing the total range, maximizing the number of tasks completed, minimizing the value of destroyed UAVs, etc. Therefore, multi-objective СТАР (MOCTAP) is gaining increasing attention [18]. For the mission planning with ground control stations (GCSs), a novel multi-objective GA that takes constraints as penalty function was presented by Atencia et al. [19]. But the increase in the number of GCSs leads to a large computation burden. For compensation, a weighted random individual generator that guides the algorithm to valid solutions was proposed in Ref. [20]. Focusing on load balancing in CTAP of multirobot systems, Wang et al. [21] proposed a multi-objective ACO where the individual encoding method and the pheromone update rule are designed. For the UAV formation emergency logistic task assignment problem, Tang et al. [22] proposed a multi-objective teaching-learning-based optimization algorithm, in which a variable neighborhood descent search is introduced to improve the local search capability. It should be noted that no assessment of UAV destruction and mission completion is done in the above studies, which leads to mission reliability being neglected. To evaluate the mission reliability of the UAV swarm, Wang et al. [23] established a multi-layer network model and a mission network model, and proposed corresponding evaluation methods for vulnerability and connectivity. In Ref. [24], Kong et al. further extended the work to the scenario with resource supplementation.
In actual combat, unexpected events, such as task execution failure or even UAV damage, may occur due to the highly changing combat situation. Emergencies may make the given combat plan less optimal or even unfeasible. Under this circumstance, people generally tend to use task re-assignment for help. One task reassignment idea is to successively conduct the task assignment in a rolling optimization manner. For the mission planning in dynamic and adversarial environments, Liu et al. [25] designed an intelligent multi-objective receding horizon framework with the ability to online receding and adaptively adjust objectives based on current situations. Huang et al. [26] proposed a heuristic framework for stochastic event scheduling, which can periodically collect new tasks and old uncompleted tasks, and schedule the collected tasks using the proposed dynamic task scheduling. For the dynamic target tracking problem, Hao et al. [27] given a framework that integrates a distributed selection algorithm and a centralized task assignment algorithm. In most cases, this idea is to keep doing optimization at fixed intervals. For situations where the battlefield situation does not change, repetitive calculations can waste lots of communication and computational resources. To avoid this issue, another re-assignment idea, ie, event-triggered task reassignment, is considered. In large-scale scenarios, adjusting the current combat plan based on trigger events is more efficient than reallocating all tasks [28]. In large-scale scenarios, adjusting the current combat plan is more efficient than reassigning all tasks, although some of the global optimization capability is lost [28,29]. To assign emergency tasks, Fei et al. [30] designed a task grouping strategy, and specially constructed a mechanism to ensure that UAVs prioritize tasks with shorter deadlines. It is worth mentioning that for large-scale scenarios containing numerous decision variables, reallocating tasks in real-time is difficult with the centralized framework. For this reason, a series of distributed algorithms such as the methods under the market mechanism are designed [31]. To solve the drawbacks of traditional CNP such as large communication volume and poor timeliness, Zhang et al. [32] introduced credit mechanism and selection mechanism in CNP. In Ref. [33], Wang et al. proposed a partial task reallocation algorithm based on CNP by designing task reallocation strategy and bidding strategies. For simultaneously achieving dynamic task assignment and path planning for multiple UAVs, Duan et al. [34] developed a new hybrid two-stage auction algorithm. However, most CNP-based methods focus mainly on the allocation of new tasks, but how new tasks participate in the original task sequence is not well studied.
In this paper, we focus on resilient UAV formation mission planning. A mission planning framework involving task preassignment and re-assignment modules is developed. As for the term resilience, it reflects two aspects: one is reflecting mission reliability by introducing probability constraints in the task preassignment, and the other is introducing the distributed algorithm to handle task re-assignment triggered by emergencies during combat to ensure better execution of tasks. In detail, the task pre-assignment module and the re-assignment module are described below
(1) For the task pre-assignment module, in order to simultaneously optimize conflict indicators, a multi-objective model that simultaneously optimizes the value of successfully attacked targets and the combat cost (i.e., total range and the value of destroyed UAVs) is established. Besides resource and timing constraints, probabilistic constraints are introduced to ensure mission reliability by setting a threshold for each target. To solve the model, a multi-population optimization algorithm based on evolutionary ideas is proposed. Genetic operators are constructed by changing the UAV, target, or task order in chromosomes. And a migration operator is designed by replacing large rank chromosomes with minimal rank ones in neighboring populations. Furthermore, a strategy is given for selecting a solution from the Paretooptimal set.
(II) For the task re-assignment module, three conditions that trigger the re-assignment process are considered, i.e., the target is not completed, the UAV is destroyed, and the target is completed but other tasks exist for the target. To improve the mission reliability under the trigger conditions, the multi-objective optimization is converted into a singleobjective optimization problem by the weighted sum method, and a CNP-based algorithm is constructed. Different contracts are constructed in the algorithm based on different trigger conditions, which enables the algorithm to real-time adjust the task scheme according to the current battle situation.
The main contributions of this study are as follows
1. A unified framework integrating task pre-assignment and task re-assignment is proposed for the UAV formation CTAP, which is verified to work well in terms of mission reliability by extensive simulation results.
2. To reflect system resilience, a task pre-assignment model considering probabilistic constraints is established, and a multipopulation multi-objective evolutionary algorithm containing specially designed genetic and migration operators is proposed.
3. For dealing with emergencies, a real-time task re-assignment algorithm is designed based on CNP for three re-assignment mechanisms.
The rest of this paper is organized as follows. In Section 2, the problem description is given first, and then the models of task preassignment and re-assignment are given, respectively. In Section 3, the algorithm for solving the task pre-assignment problem is designed. And the algorithm for solving the task re-assignment problem is given in Section 4. In Section 5, several simulations are carried out to verify the effectiveness of the algorithms. Finally, the conclusion is given in Section 6.
2. UAV formation task pre-assignment and re-assignment problem definition
Cooperative task pre-assignment problem (CTPAP) and cooperative task re-assignment problem (CTRAP) are two sub-problems of UAV formation CTAP. These two types of task assignments occur at different stages and have different emphases. In this section, task pre-assignment and task re-assignment are first introduced specifically, and then the corresponding mathematical models are constructed separately.
2.1. Problem description
Before combat, the task pre-assignment process based on detected target information (i.e., location, value, etc.) and UAV formation information (i.e., ammunition, fuel, etc.) is required. That is, optimizing a task pre-assignment scheme for the UAV formation that makes the combat indicator optimal while satisfying the constraints. During combat, unexpected events can make the current scheme less optimal or infeasible, such as the UAV being destroyed or task execution failing. Therefore, the process of adjusting the current scheme in real-time, i.e. the task reassignment process, is needed to make the UAV formation adapt to the current battlefield situation. Three conditions that can trigger task re-assignment are as follows:
Condition 1. The UAV survives and successfully executes the current task, and there is at least one remaining task set containing the current task.
Condition 2. The UAV survives but fails to execute the task, while no remaining task set containing the current task exists.
Condition 3. The UAV is destroyed, while its remaining task set contains tasks that are not performed by other UAVs.
Different conditions will trigger different re-assignment processes. If condition 1 occurs, then cancel the remaining tasks on the current target and reroute the UAV. If condition 2 or 3 occurs, then reassign the task and reroute the UAV. A schematic diagram is given in Fig. 1.
The CTAP is studied based on the assumptions given below:
H1. Targets and UAVs are heterogeneous, ie. targets have different values and risks, and UAVs have different speeds, ammunitions, maximum ranges and values.
H2. Only attack tasks are performed on each target.
H3. The same target can be attacked multiple times within a given limit.
H4. The same UAV can attack the same target multiple times.
H5. UAVs have different flight heights to avoid a collision.
H6. All UAVs, including those staying at the airport, can receive information related to task re-assignment.
For ease of representation, define the symbols I, and I as lh : = (1,2, ..., n}, In : = {0, 1, ..., n}. Assume that the UAV formation contains Ny UAVs and Nr targets are detected. The set of UAVs and the set of targets are denoted as U = (U1, U2, ..., UNU} and T = (T1, T2, ..., TNT}, where U; is the ith UAV and Tj is the jth target. Destroying different targets does not have the same impact on the battlefield situation, i.e., the value of the targets is not the same. Meanwhile, the heterogeneous UAVs have different values. In addition, when the same UAV attacks different targets, the probability of UAV survival and the probability of task completion are different. The value of Ui and Tj are denoted as VUi, and VTj, respectively; Psuij and Psu,ij denote the probability of task completion and the probability of U; survival when Ui attacks Tj, respectively. Through task assignment, each UAV will possess a task set, in which the tasks have different priorities. Let My, denote the task set for Ui, MyUi = {Tmi1, Tmi2, ..., TmiNUi}, where Ny, denotes the number of tasks assigned to Uj, and T,,; denotes the number of the jth target in MUi.
Fig. 2 gives an example diagram of the task assignment with three UAVs and seven targets. The task pre-assignment ultimately results in a task assignment scheme containing the task set assigned to each UAV and the order in which the tasks are executed, as shown in Fig. 2(a), where the scheme is MUi = {T2,T1,T6}, MU2, {T1, T3, T4}, My, = (MU3 T3, T5, T7}. In this research, the distance between the targets is determined using the Euclidean distance, and the line connecting the two targets represents the flight path of the related UAV. For example, Us takes off from the airport, attacks targets Ta, 13, Ts, T7 in turn and eventually returns to the airport. Fig. 2(b) gives the case of task re-assignment, where U; successfully executes T2, the re-assignment triggered is to cancel the attack task of Us on T2; U, is destroyed while attacking Tg, then the reassignment on T6 is triggered, which results in Tg being assigned to Us and the remaining task set for Uz at this moment is {T7, T6}.
2.2. Multi-objective optimization model for UAV formation CTPAP
In practice, there is no guarantee that a target will be successfully attacked. Meanwhile, with limited resources, not limiting the number of tasks to ensure a successful attack on the target makes the combat inefficient. Therefore, it is necessary to consider mission reliability and limit the number of tasks assigned to each target. To ensure mission reliability, probability constraints are introduced, which set a threshold for each target that reflects the minimum attack success probability. Specifically, a set of attack tasks is assigned to each target such that the success probability is not less than the given minimum value and the number of attacks is not greater than the limit. In order to improve the combat efficiency of the UAV formation, we expect a task assignment scheme that allows the UAV formation to achieve a high attack benefit at a low cost. However, high gains are often accompanied by high costs, so multi-objective optimization is considered in the modelling to optimize these conflicting indicators simultaneously. The value of the targets successfully attacked is taken as the combat benefit, and the combat cost consists of two components, i.e., the value of the destroyed UAV and the range cost. Denote the location of T; as Lr, and let ... be the decision variable. ... means that Ui flies from Tji to Tj2, to perform the Ith attack task on Tj3 otherwise, ... The objectives are given as follows
(1) Maximizing the expectation of the value of the target being successfully attacked.
... (1)
where αj1 denotes the upper limit of the number of tasks assigned to Tj2 and the expression for ... is as follows.
...
The objective (1) is transformed into the following form, which minimizes the expectation of the value of the unsuccessfully performed targets.
... (2)
(2) Minimizing the combat cost of the UAV formation.
... (3)
where ... the distance between Tj1 and Tj2. If the same UAV attacks "the same target several times consecutively, then at this moment ....
Let ... Where ... Based on H1-H5 and the above analysis, the multi-objective optimization for UAV formation CTPAP (MOCTPAP) can be formulated as the following combinatorial optimization problem.
... (4)
... (5)
... (6)
... (7)
... (8)
... (9)
... (10)
where A; indicates the maximum ammunition load of Ui; Psj2 denotes the minimum execution success probability for Tj2; MR; indicates the maximum range of Ui; and timemi, is the execution time of task ... Considering the limited ammunition load, constraint (5) ensures that the number of tasks assigned to each UAV does not exceed the UAV's maximum ammunition load; constraint (6) ensures that the number of attack tasks assigned to each target does not exceed the given maximum; constraint (7) ensures the mission reliability, i.e., the success probability for each target cannot be lower than the given minimum value; constraint (8) ensures that the actual range of each UAV cannot exceed its maximum range; constraint (9) ensures that each UAV performs the tasks in the order in which the tasks are assigned; and constraint (10) is the range of values of decision variables.
2.3. Task re-assignment model for UAV formation CTRAP
Compared with the task pre-assignment phase, the task reassignment phase has higher requirements on the assignment time, i.e., it needs to achieve real-time task assignment. In order to save the time of judging non-dominated solutions, the multiobjective optimization is transformed into single-objective optimization. During combat, it is ensured that failed targets are assigned to the UAV if resource constraints are satisfied. Therefore, constraint (7) is not considered in the task re-assignment. For considering both the combat benefit and cost, the minimization of the weighted sum of J2 and J3 corresponding to the reassigned task performed by the UAV is taken as the objective function. Let ..." denote the position of Ui when the re-assignment process is triggered; RMUi denotes the current remaining tasks set of Ui, ie, the set of tasks that have not yet been executed by Ui, and ... where are the numbers of the remaining targets in RMUi; and Tj0 denotes the numbers of the currently reassigned target. Let ... (0, 1} be the decision variable. If Tj0 is reassigned to U; and assigned to the Ith position in RMy,, then ... = O. Based 0 H1-H6 and the above analysis, the UAV formation CTRAP for unexpected events is formulated as the following optimization problem.
... (11)
... (12)
... (13)
... (14)
... (15)
where α1, α2 [0,1] are weights, α1 + α2 = 1; di ;, denotes the changed range of U; after assigning Tj, to the Ith position of RMUi,
...
LT0, represents the location of the airport; RA; and RVi denote the remaining ammunitions and the remaining range of Ui after performing all tasks in RMUi before Tj0 is assigned to Ui, respectively. Constraints (12) and (13) ensure that when Tj0 is assigned to a UAV, the UAV's range change and additional ammunition requirement do not exceed the UAV's remaining range and remaining ammunition, respectively; constraint (14) ensures that the order in which the remaining tasks are executed; and constraint (15) is the range of values of decision variables.
3. Improved multi-objective evolutionary algorithm for MOCTPAP
UAV formation CTAP as a combinatorial optimization problem is NP-hard. Traditional optimization methods are not effective in solving large-scale CTAP. Therefore, in this section, we proposed a multi-objective optimization algorithm based on the ideas of evolution and multiple populations. The main focus is on the construction of the chromosome encoding method, evolutionary operators, and the migration operator applicable to MOCTPAP.
3.1. Chromosome encoding
The MOCTPAP has the following two characteristics, 1) the number of targets assigned depends on the limited resources of the UAV formation (i.e, ammunition, range), and 2) the attack task assigned to each target is uncertain. Therefore, the dimension of chromosomes cannot be fixed. To reduce the chromosome dimension, the chromosome contains only of the assigned target and UAV information, as shown in Fig. 3. The first and second rows are the target number and UAV number, respectively. Both UAVs and targets are randomly selected to ensure randomness when encoding chromosomes on the basis of satisfying constraints (5)-(10), and the detailed process is shown in Algorithm 1. In order to show the task set of each UAV more clearly, the genes in the chromosome are sorted according to the UAV number, as shown in Fig. 3. The yellow part shows a UAV attacking the same target multiple times, in which case the UAV will attack the target continuously.
3.2. Crossover operator
Crossover operators that can increase population diversity play an important role in the algorithm. The constructed chromosomes need to satisfy given constraints and genes are coupled to each other, which leads to offsprings that do not satisfy the constraints, i.e., infeasible solutions, when using the traditional crossover methods. Therefore, it is necessary to construct a suitable crossover operator according to the setting of the MOCTPAP and the characteristics of chromosomes. The assignment of UAVs is an important factor in the task scheme, in addition, the task scheme needs to satisfy the given constraints. Therefore, under the premise of satisfying constraints (7)-(8), the crossover operator is constructed based on the idea of changing UAVs. Considering that the amount of ammunition can directly affect the task allocation, the UAVs are selected based on different rules to replace the UAVs at the crossover points under different ammunition surplus cases. The crossover parents Fi, F, are selected from the population using the roulette method. Let dim_F; denote the dimension of F;. First, two points py, P2 € min{dim_F,,dim_F,} (P1 <p2)>
The crossover operator is illustrated by the example of four UAVs attacking three targets. The information of UAVs and targets is given in Table 1 and Fig. 4. Let a; = 3 and Ps; = 0.8(j El;). Parents F;(i El) and a procedure of crossover operation on F; are given in Fig. 4, where the randomly chosen crossover points are py = 3, р» = 4. First, the crossover operation is performed on gene 3, at which time the intersection of the set of UAVs with remaining ammunition for F, and the set of UAVs in the gene fragment of F, is {2,3}, i.e, RMU1n GFU2 = {2,3}. Suppose that Us, which satisfied constraints (7)-(8), is randomly selected to replace U; in gene 3. Then, the crossover operation is performed on gene 4, at which time, there is still RMU1n GFU2 = {2,3}, and Us is selected to replace U> in gene 4. At this point, constraint (8) is satisfied. In addition, the success probability of attacking T3 is 0.853147, so constraint (7) is also satisfied. Finally, the offspring of F; obtained, as shown in Fig. 4.
3.3. Mutation operator
The mutation operator plays the role of avoiding the algorithm from falling into a local optimum, which is another important part of making the population evolution. Crossover offsprings are mutated according to the mutation probability. Both gains and battle losses of UAVs are related to the tasks performed, while the range is not only related to the tasks but also to the execution order of the tasks. Therefore, the mutation operator is constructed based on the idea of exchanging genetic information at two mutation points. According to the different tasking scenarios of UAVs, the following three mutations are designed: 1) swap the positions of two genes; 2) swap the UAVs of two genes; 3) swap the targets of two genes. First, two mutation points 41, d2 (41 <q)>
An example diagram of the mutation operation for offspring 1 in Fig. 4 is given in Fig. 5. Let the randomly selected mutation points be dı = 3 and gq, = 6. As can be seen in Fig. 5, the UAVs in both gene 3 and gene 6 are assigned no more than 2 targets. Therefore, the lines 7-16 of Algorithm 3 are executed. Swap the UAVs in gene 3 and gene 6, at this point, Us and U, satisfy constraint (8). Furthermore, the probability of successful attacks on T; and T, are 0.8404 and 0.94, respectively, so constraint (7) is satisfied. Therefore, the mutation operation on offspring 1 is completed. The obtained mutation offspring is shown in Fig. 5.
3.4. Multi-subpopulation multi-objective evolutionary algorithm
Based on the constructed chromosome encoding method and the evolutionary operators, i.e. Algorithms 1-3, the multipopulation multi-objective evolutionary algorithm (MPMOEA) is constructed. Let G denote the maximum number of iterations; NS denote the number of subpopulations; the size of Ith subpopulation is denoted as Sı; Pl and Pl, denote the crossover and mutation probabilities of the Ith subpopulations (Ilys). The subpopulations are set to migrate individuals every IG generations according to the migration operator given in Fig. 6. The steps of the migration operator are as follows. Step 1: Let i = 1. Step 2: Calculate the rank of chromosomes in the ith and (i+1)th subpopulation. Step 3: Chromosomes with high rank in the ith subpopulation are replaced with chromosomes in the Pareto-optimal front of the (i+1)th subpopulation. Step 4: Let i = i+ 1, if i< NS- 1, then turn to Step 2, otherwise, turn to the next step. Step 5: Calculate the rank of chromosomes in the NSth and first subpopulation. Step 6: Chromosomes with high rank in the NSth subpopulation are replaced with chromosomes in the Pareto-optimal front of the first subpopulation.
The constructed algorithm applicable to solving MOCTPAP is shown in Algorithm 4. MPMOEA mainly consists of 9 steps, as follows. Step 1: Input the information of targets, UAVs, and algorithm parameters. Step 2: Initialize NS subpopulations based on the chromosome encoding method in Algorithm 1. Step 3: Calculate the fitness value of each chromosome in each subpopulation separately. Step4: Based on the roulette method and Algorithms 2-3, generate a population SSP! (VI Ens) of size n, for the subpopulation SP, of size nj. Step 5: Combine populations SP, and SSP! (YI Elns) to obtain a new population QL. Step 6: Calculate the rank of chromosomes in population a (Vl EIns) using the fast nondominated sorting approach, and select n; chromosomes with low rank from QL to construct a new subpopulation SPL. Step 7: Letg = q,andq = q+ 1. Step 8: If the given migration condition is satisfied, chromosome migration is performed, otherwise, go to the next step. Step 9: If the termination condition is satisfied, then combine all subpopulations and output the Pareto-optimal set of the obtained population, otherwise, repeat Step 1-Step 8. Fig. 7 gives the flow chart of MPMOEA.
The UAV formation can only perform one task assignment scheme, so a solution needs to be selected from the Pareto-optimal set. Let № is the number of objectives; 6, € [0, 1](1 €ly,) denotes the weight, which satisfies >" Bı = 1; and fl represents the value of the ith objective in the jth non-dominated solution. First, the decision maker sets the value of the weights based on the degree of preference for the objective; then calculate F = =, в for the jth non-dominated solution of the Pareto-optimal front; and finally, select the Pareto solution corresponding to the smallest Fİ.
4. An real-time CNP-based algorithm for CTRAP
According to the analysis in Section 1, three trigger conditions that may be encountered by UAVs during combat will trigger task re-assignment. For the rapidly changing combat situation, the current task assignment scheme needs to be adjusted in real-time to suit the battlefield situation. Although evolutionary algorithms can provide high-quality solutions to the CTPAP, they cannot achieve real-time task assignment. Therefore, a real-time algorithm for CTRAP needs to be developed.
The idea of CNP is widely used in the construction of dynamic task assignment algorithms, which can satisfy the requirement of time. Therefore, the task re-assignment algorithm is given based on the CNP idea. The flow of re-assignment triggered by different trigger conditions is different. For condition 1, first, information about the currently successfully attacked target is broadcast; then, each UAV deletes the task for that target from the task set; and finally, the path of the UAV that deleted the task is replanned. For conditions 2-3, the task re-assignment process is shown in Fig. 8. It is 1) initialize the current information; 2) bid on the task; 3) each UAV submitting a bid; and 4) the UAV winning the bid.
The task re-assignment triggered by all conditions affect the flight path of the UAV. In addition, for conditions 2-3, when the UAV tenders for a task, the location where the new task is placed in the task set directly affects the cost of the UAV. Therefore, the following information needs to be included in the contract: 1) information about the bidding UAV (number, current position); 2) the ranking of the assigned task in the task set (conditions 2-3); 3) the reassigned target number (conditions 2-3); 4) the objective function value (conditions 2-3); and 5) the path that needs to be replanned. Suppose the currently attacked target is Tj. The following two types of bids are designed according to different trigger events.
1) T; is successfully executed. If the remaining task sets contain attack tasks on Tj, then the bid takes the following form.
...
where NiRMb, represents the number of tasks in RMUi; k ∈NiRM, indicates the rank of Tj in ... represent the path between ..., and ..., is replanned. If ... then ..., and ..., are the positions of the target corresponding to the (k-1)th and (k+1)th tasks in RMy,, respectively. If k = 1, then Lik-1 4 = LnowUi, and ... is the position of the target corresponding to the (k+1)th task in RMUi. If k = NiRM, then Li+4, = LT0, and Lik-i, is the position of the target corresponding to the (k-1)th task in RMUi.
2) T; is failed executed. The remaining task sets do not contain attack tasks on Tj, then the bid takes the following form.
...
where ... indicates that Tj is ranked at the Ith position in the new ... represent that the path between Lit-1, and LTj, and the path between LTj and Li are replanned. If l = 1, then Lit-1 = LnowUi and Lil is the position of the target corresponding to the first task in the original RMUi. If .... then Lil-1 and Lil are the positions of the target corresponding to the (1-1)th and Ith tasks in the original RMUi respectively. If l| = NiRM + 1, then Lil = LT0, and ... is the position of the target corresponding to the last task in the original RMUi.
The real-time CNP-based algorithm constructed for CTRAP (RTCNP-CTRAP) is shown in Algorithm 5. First, initialize the target and UAV information. Then, the type of trigger event is judged. Next, the re-assignment process is performed according to the different trigger conditions. Finally, update the remaining resources of the UAV formation. If trigger condition 1 occurs, then 1) delete the task for the current target in RMy, (i Eln,); 2) replan the UAV path. If trigger condition 2 occurs, then 1) broadcast task information; 2) each UAV with remaining ammunition puts the bidding task at any position in its remaining task set, and selects the scheme that minimizes (11) and satisfies constraints (12)-(15) as the bid; 3) select the best bid and assign the task to the winning UAV; 4) replan the UAV path. If trigger condition 3 occurs, then execute the process corresponding to condition 2 for each task that needs to be reassigned in the remaining task set of the dead UAV. The flowchart of RTCNP-CTRAP is given in Fig. 9.
5. Numerical results
Three scenarios with different scales are simulated, including a small-scale scenario giving clear details of task assignment and reassignment and two larger-scale scenarios testing the performance of the algorithms. The parameters in MPMOEA are taken as follows. Three subpopulations are considered with п; = 60(l €l3). The crossover probabilities are Pİ = 0.8, R2 = 0.85, P1t = 0.9, and mutation probabilities are Pl, = 0.1, P2, = 0.15, P3M, = 0.2. Let the maximum number of population iterations G equal to 200, and migrate chromosomes between subpopulations every 10 iterations. The location of the airport is set to Ly, = (0,0). To avoid the impact on the optimization due to the different magnitudes of the two terms in objective function J3, the two terms are made to have the same magnitude by converting the unit of the range in the examples. Numerical experiments are implemented by using Python 3.8 on a computer with Intel Core i5-10210U CPU @ 1.60 GHz 2.11 GHZ and 4.00 GB RAM.
5.1. Scenario 1: 6 UAVs, 11 targets assignment scenario
The detected target information (location Lt, value Vr.) is shown in Table 2, and information of the UAV formation (value VUi, range MR;, speed S;, ammunition A;) is shown in Table 3. It is worth clarifying that the value, ammunition, range and speed of different UAVs are different. Table 4 gives the values of PE and Pi Elk, Jj El). The minimum success probability and maximum number of tasks assigned to each target are Ps, = 0.8, а; = 3.
The Pareto-optimal front of MOCTPAP obtained by MPMOEA is shown in Fig. 10. The circular dots in Fig. 10 are the obtained nondominated solutions, and it is clear that the distribution of the non-dominated solutions is relatively uniform. Let the weights 61, 6, in the solution selection strategy are all set to 0.5. In the obtained Pareto-optimal front, the green dot is the non-dominated solution selected for actual combat, and its specific task assignment scheme is shown in Table 5. To verify the effectiveness and advantages of MPMOEA, it is compared with the single population multiobjective evolutionary algorithm (SPMOEA) with population size N = 180, crossover probability P! = 0.8 and mutation probability Pi, = 0.2. The red triangles in Fig. 10 represent the Pareto-optimal front calculated by SPMOEA. It can be seen that the non-dominated solution of MPMOEA dominates the non-dominated solution of SPMOEA, i.e. the result of MPMOEA is better than that of SPMOEA. In addition, the convergence of the algorithms is compared using the same initial population, as shown in Fig. 11, which shows a better convergence trend for MPMOEA.
The CPU running times for 20 simulations using the two algorithms separately are shown in Fig. 12, which shows that MPMOEA takes less time than SPMOEA. The process of determining the nondominated solution is the main time-consuming part of the algorithm. In each iteration, the complexity of this process for MPMOEA and SPMOEA is O(M - ES?) and O(M «(> S;)"), respectively, where M denotes the number of objectives. In this scenario, the number of computations required in each iteration to determine the non-dominated solution for MPMOEA and SPMOEA are 86400 and 259200, respectively. It is obvious that MPMOEA takes much less time than SPMOEA in this scenario.
The process of the UAV formation executing the task assignment scheme in Table 5 is simulated, and the task execution process obtained is shown in the appendix. The blue solid line and the red dashed line indicate the paths flown by the UAVs and the paths not yet flown, respectively. Appendix A (1) gives the initial allocation scheme and the paths of the UAVs. The details of the task execution and re-assignment processes are as follows.
1) U1 reaches Lr, and attacks Ts. U, survives and successfully attacks Ts, at which point the re-assignment is not triggered, as shown in appendix A (2).
2) U6 reaches Ly, and attacks Tg. Us survives and successfully attacks Tg, at which point the re-assignment process to cancel Tg from My, is triggered, as shown in appendix A (3).
3) U3 reaches Lr, and attacks Te. Us survives but the attack mission fails, at which point the re-assignment is not triggered, as shown in appendix A (4).
4) U4y, U2, U6, U4 and U1 attack T6, T10, T1, T2 and T7 respectively. The attack missions are all successful and the UAVs all survive. The re-assignment is not triggered, and the above processes are shown in appendix A (5)-(9).
5) U6 reaches Lt, and attacks T9. U6 survives but the attack mission fails, at which point То needed to be re-assigned. After bidding, T9 is assigned to Us and the result of the reassignment is shown in appendix A (10).
6) U5 and U6 attack T11 and T3 in sequence. The attack missions are successful and the UAVs survive. The re-assignment is not triggered, as shown in appendix A (11)-(12).
7) U4 and U5 attack T9 and T4 in sequence. Both UAVs are destroyed, at which point T3 and T4 need to be reallocated. After bidding, T9 and T4 are assigned to U1 and U2 respectively, as shown in appendix A (13)-(14).
8) U1 and U2 attack T9 and T4 in turn. The UAVs survive but the attack tasks fail. After bidding, T9 and T4 are reassigned to U2, U3 respectively, as shown in appendix A (15)-(16).
9) U3 reaches LT, and attacks T4. U3 survives but the attack task fails, at which point T4 needs to be reallocated. After bidding, T4 is reassigned to U3, as shown in appendix ? (17).
10) U3 and U2 attack T4 and T9 in turn. The attack missions are successful and the UAVs survive, as shown in appendix A (18)-(19). Since then, all missions have been completed and the UAVs have returned to the airport.
The time for 20 task re-assignments is counted in this scenario and is shown in Fig. 13. The average CPU runtime is 0.10157 ms, which shows that RTCNP-CTRAP can achieve real-time task reassignment.
5.2. Scenario 2: 13 UAVs, 24 targets assignment scenario
Tables 6 and 7 gives information of 13 UAVs and 24 targets. The values of PR and PE are given in https://github.com/gaoxh-github/ Parameter-values-for-scenarios-2-3. And the minimum success probability and maximum number of attack tasks assigned to each target are Ps; = 0.8, ?; = 3(j Ela). The comparison of the Paretooptimal fronts obtained by two algorithms is given in Fig. 14, which show that MPMOEA outperforms SPMOEA. And the CPU runtime of two algorithms for 20 experiments is given in Fig. 15. It shows that MPMOEA is more advantageous in terms of time. The red dot in Fig. 14 is the selected non-dominated solution, and the details of which are shown in Table 8. This assignment scheme is used as the initial task assignment to simulate the combat process of the UAV formation. Fig. 16 shows the task execution process and the re-assignment process obtained by the simulation.
In Fig. 16, IT and IR denote the Ith event and the re-assignment event triggered by it, respectively; the survival and death of Ui are denoted as i-S and i-D; j-C and j-W denote the cancellation and reassignment of Tj and the completion and non-completion of Tj are denoted as j-A and j-F, respectively. For convenience of description, three categories of events are selected from Fig. 16 for explanation. The part marked in green is the first type of event. 2T: 2-S,15-A indicates the second event, i.e. U2 successfully executed T15 and survived, at which point RMU2 = {T6, T0}. The re-assignment is triggered because T15 ∈MU4. 2R: 15-C indicates the cancellation of T15. And the remaining task set of Uy after the re-assignment is {T23, T14,T0}. The second type of event is the part marked yellow. 6T: 12S, 8-F indicates the 6th event, i.e. U,2 survived but did not successfully attack Tg. No re-assignment is triggered because T8 ∈ MUi. The part marked in pink is the third type of event. 21T: 7-D, 3-F indicates that in the 21st event, U7 died while attacking T3. Because T3 is included in RMUi and T12 and T24 are not attacked by other UAVs anymore, a re-assignment process is triggered. After the re-assignment, T12, is assigned to U3 ie. 21R: 12-W, and RMU3 = {T12, T0} after the assignment; U11 wins the T24, i.e. 21R: 24-W, and RMUi = {T24, T9, T3, T0} after the assignment. In Fig. 16, Li = (+170.943, -185.886) and L2 = (8.155, - 204.196), which are the positions of U3 and U11 when the corresponding re-assignments are triggered.
In order to reflect the survival of the UAVs and the completion of targets, 100 simulations of UAVs performing tasks are conducted. Table 9 gives the percentage of experiments corresponding to different UAV destruction rates, where ND, RD (Rp = ND/NU) and PD denote the number of UAVs destroyed, the UAV destruction rate and the percentage of experiments with different UAV destruction rates, respectively. From Tables 9 and it can be seen that 90% of the experiments have a UAV destruction rate below 0.384, and the UAV destruction rate is below 0.23 in more than half of the experiments. The percentages of experiments corresponding to different task completion rates are given in Table 10, where Nc, Rc, (Rc = Nc/NT) and Pc denote the number of completed targets, the target completion rate and the percentage of experiments with different target completion rates, respectively. According to the information in Table 10, more than half of the experiments have a task completion rate of 1, and 93% of the experiments have a target completion rate higher than 0.916, which verifies that the proposed framework can ensure high mission reliability. The time taken for 20 task re-assignments is counted in this scenario, which is shown in Fig. 17. And the average CPU runtime is 0.14386 ms.
5.3. Scenario 3: 25 UAVs, 45 targets assignment scenario
The information of 25 UAVs and 45 targets and the values of PR and Pr are given in https://github.com/gaoxh-github/Parametervalues-for-scenarios-2-3. And the minimum success probability and maximum number of attack tasks assigned to each target are Ps; = 0.8, aj = 3(j Elas). Fig. 18 gives the Pareto-optimal fronts of MOCTPAP obtained by MPMOEA and SPMOEA. The statistical CPU running times of the two algorithms for 20 experiments are shown in Fig. 19. Figs. 18 and 19 show that MPMOEA is better than SPMOEA in terms of both solution quality and computation time. Let 6, = 0.5, 62 = 0.5, then the selected non-dominated solution is the red dot in Fig. 18, whose details are shown in Table 11.
The simulated combat process using the assignment scheme in Table 11 as the initial task assignment is shown in Table 12, where L1 = (- 69.858, - 149.781), L5 = (215.208, - 116.130), L3 = (- 282, 219), L4 = (- 70.518, 54.764), and L5 = (- 150.919, - 124.106). Fig. 20 shows the CPU runtime for 20 re-assignment process at this scale, with an average time of 0.21178 ms. Tables 13 and 14 gives information on UAV survival and task completion for 100 simulation experiments of task execution processes. It can be seen that 91% of the experiments with a UAV destruction rate below 0.4, and 90% of the experiments with a target completion rate higher than 0.911. It is further verified that the proposed framework can still ensure high mission reliability at a large scale.
The ammunition load of the UAV formation is an important factor that affects the target completion rate. In order to verify the effect of ammunition load, the percentage of experiments corresponding to different UAV destruction and target completion rates for multiple ammunition load cases is given. 100 simulations are conducted for the cases of all UAVs with ammunition loads of 3, 4, 5 and 6, respectively. Fig. 21 gives the trend of the percentage of experiments with different target completion and UAV destruction rates. It is clear from Fig. 21 that the percentage of experiments with a target completion rate of 1 gradually increases as the number of ammunition increases, which indicates that the number of ammunition has a direct effect on the target completion rate. However, when the number of ammunition per UAV is increased to 6, the percentage of experiments with a target completion rate of 1 does not change much compared to the case of the number of ammunition of 5, which is due to the limitation of the total range of the UAV formation. Furthermore, it can be seen that the UAV destruction rate is hardly affected by the number of ammunition. The reason is that the destruction of the UAV is affected by its survival probability when performing the task.
5.4. Scalability test
To test the effect of problem size on the computational efficiency of MPMOEA, eight scale problems are tested, with the number of UAVs ranging from 6 to 55 and the number of targets ranging from 11 to 102. And 25 simulations are performed for each scale problem. The statistical CPU runtime is shown in Fig. 22. It is evident that the CPU runtime tends to scale linearly as the problem size increases, with average runtimes of 18.592 s, 33.870 s, 45.523 s, 63.336 s, 69.081 s, 94.702 s, 104.375 s and 119.947 s, respectively.
6. Conclusions
This paper focuses on the CTAP of UAV formations with the purpose of improving mission reliability and combat effectiveness. We develop a resilient mission planning framework that includes task pre-assignment and re-assignment modules. In the task preassignment module, probabilistic constraints that reflect the mission reliability are introduced, and a multi-population multiobjective evolutionary algorithm with specially designed evolutionary and migration operators is constructed. Meanwhile, a realtime CNP-based distributed algorithm capable of handling multiple emergencies is proposed in the task re-assignment module. The results of three simulation experiments of different scales verify the effectiveness of the proposed framework for task assignment before and during the combat. In addition, statistical results on numerous simulation experiments demonstrate the advantage of the proposed framework in ensuring mission reliability. In future research, the CTRAP for multiple types of tasks and UAV trajectory planning in environments containing obstacles can be further studied. Moreover, in order to further improve mission reliability, it makes sense to study real-time cooperative task assignment with comprehensive consideration of UAV combat capabilities, system health status, mission information and battlefield environment.
CRediT authorship contribution statement
Xinwei Wang: Funding acquisition, Methodology, Supervision, Writing - review & editing. Xiaohua Gao: Conceptualization, Methodology, Writing - original draft. Lei Wang: Funding acquisition, Supervision, Writing - review & editing. Xichao Su: Formal analysis, Supervision. Junhong Jin: Data curation, Methodology. Xuanbo Liu: Validation. Zhilong Deng: Supervision.
Acknowledgements
This work was supported by the National Key Research and Development Plan (Grant No. 2021 YFB3302501); and the National Natural Science Foundation of China (Grant Nos. 12102077, 12161076, U2241263).
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* Corresponding author.
E-mail address: [email protected] (X. Gao).
Peer review under responsibility of China Ordnance Society
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