# Research on Lazy Theta+
## 1 Introduction  
In modern warfare, air defense systems continuously improve, prohibiting aircraft from reaching the target area in a complex environment. Therefore, how to reach the destination and complete the task while facing multiple threats is currently a significant problem [1]. To solve this problem, scholars have proposed various methods for improving a flight's efficiency and safety. Among these methods, route planning reduces the influence of a complex environment on the aircraft's power consumption and improves flight safety [2]. Therefore, route planning has attracted significant research attention. The literature currently presents several route planning algorithms, each with its advantages and disadvantages, and applicable scenarios. According to the planning method, the algorithms can be divided into traditional and intelligent planning algorithms [3]. The former algorithms include the Dijkstra algorithm [4], the artificial potential field method [5], and the simulated annealing algorithm [6]. Although the traditional algorithms ensure route optimality, their computation burden is long and thus unsuitable for real-time planning. Therefore, only a few route planning applications adopt this route planning type. In contrast, modern intelligent algorithms are widely used and closer to actual needs. The most commonly used are the A*, genetic, RRT, and ant colony [7] algorithms. The A* algorithm [8; 9; 10], the most commonly studied route planning algorithm, is inspired by the Dijkstra algorithm and introduces a heuristic function to realize fast searching. The A* algorithm has been extended to facilitate mission requirements. [PERSON], [PERSON] et al. [11] utilized the Hdl_graph_slam mapping algorithm to construct a two-dimensional grid map and introduced a path smoothing strategy and a safety protection mechanism in the A* algorithm to determine whether obstacles exist between the two path nodes. The latter strategy aimed to eliminate redundant and minimum angle points, improving path smoothness and search efficiency. [PERSON] et al. [12] proposed an A* algorithm that combined a multi-objective path planning strategy with a greedy algorithm, and improved the cost function to make the A* algorithm converge faster and have a shorter path length. [PERSON] et al. [13] decomposed the three-dimensional route planning into two parts and replaced the original three-dimensional planning with two-dimensional and height planning. In this scheme, first, the dynamic time warping distance was used as a heuristic function to conduct two-dimensional planning. Then, height information was obtained from the calculated two-dimensional route and combined with a digital elevation map. The three-dimensional route plan was generated. This method effectively remapped the problem from a three-dimensional space into a two-dimensional space. By reducing the dimension, the A* algorithm's efficacy was increased by drastically lowering the number of search nodes. The RRT algorithm [14; 15; 16] is a smart probabilistic roadmap method (PRM) with flexible search capabilities, and can be employed for path planning in various complex environments. However, RRT does not consider the path cost, and the obtained path is not optimal. Spurred by these defects, [PERSON] et al. [17] put forward the RRT*-Smart algorithm, which realized an intelligent sampling search through heuristic sampling. Although this method affords a faster convergence, it presents poor adaptability. [PERSON] et al. [18] suggested the Informed-RRT* algorithm, which limits the node sampling range by generating elliptical sampling subsets and increases the convergence velocity of the best route. Nevertheless, the overall time cost is still significant. Simulating the process of biological genetic evolution is the fundamental idea of a genetic algorithm. According to the principle of selection, crossover, mutation, and other operations, the problem is gradually approached from the initial to the optimal solution.