# A Two-Stage Co-Evolution Multi-Objective Evolutionary Algorithm for UAV Trajectory Planning
## 2 Related Research Background
### At-Cmoea  
The constraints of UAVs cause the search space to generate multiple infeasible domains that partition the objective space into multiple unevenly distributed subspaces. Most CMOEAs focus on the diversity, convergence, and feasibility of the solutions (planning solutions), ignoring the role of the infeasible solutions (the planning solutions of \(CV\
eq 0\)), which leads to insufficient selection pressure on the nondominated solutions. An archive-based two-stage evolutionary algorithm with constraints (AT-CMOEA) provides an effective global search strategy that fully considers the balanced relationship among objectives and between constraints and objectives, and this algorithm utilizes information from both feasible and infeasible solutions to improve the diversity and convergence of the solutions [12]. In this section, the AT-CMOEA is briefly introduced. Figure 1: Multi-objective optimization problems with feasible solutions. (**a**) denotes the decision space; (**b**) denotes the objective space. The AT-CMOEA divides the evolutionary process into two stages--exploration and exploitation. The exploration stage explores the search space extensively through two populations, one biased towards constraints and one biased towards the objective, so that the search resources are concentrated in the promising regions as much as possible. The exploitation stage obtains a set of well-distributed Pareto-optimal solutions based on the useful individual information found in the exploration stage, and the two populations with different characteristics cooperate to converge to the constrained Pareto front (CPF). This is organized as follows:  
1. The exploration stage consists of two populations: the main population, which does not consider any constraints and aims to explore the search space quickly and efficiently, and the assistant population, which takes all the constraints as an additional objective and retains some infeasible solutions close to the feasible domain, with the aim of jumping out of the local optimum and finding a promising feasible domain. Finally, all the solution information is archived for further use in the exploitation phase. 2. The exploitation stage is more concerned with fast convergence to CPF, where the main population consists of the populations from the exploitation stage and the assistant population considers all constraints in order to obtain a set of well-distributed Pareto-optimal feasible solutions. In addition, by exchanging the offspring produced by the two populations, the assistant population continuously helps the main population to converge from the infeasible domain to the CPF by using the promising information it carries. The AT-CMOEA uses different populations to determine the CPF from different stages and fully considers the effects of constraints and infeasible solutions on the objective space. The main and auxiliary populations co-evolve schematically, as shown in Figure 2.  
The red dots in Figure 2a,b denote the main population, the blue dots denote the assistant population, the grey area denotes the feasible domain, the red solid line denotes the Pareto front while considering the constraints, the blue solid line denotes the Pareto front without considering the constraints, and the dashed arrows denote the direction of the run of the populations. Figure 2c shows that the main and assistant populations were simultaneously clustered in the feasible domain constraining the Pareto front. There are many constraints coupled in MOUTP; if the constraints of the UAV are ignored, then the generated key trajectory points will ignore the actual flight situation of the UAV, which will have an impact on the stability and safety of the UAV. The constraint-handling technique of the two-stage co-evolution of the AT-CMOEA is of high significance. Figure 2: Schematic representation of the co-evolution of the main and assistant populations.