# Research on Lazy Theta+
## 7 Conclusions  
This paper proposes a Lazy Theta* route planning algorithm based on grid point optimization, which calculates the optimal route by setting the grid size according to the threat environment and actual mission requirements, and the proposed approach is verified by three groups of experiments. In experiment 1, octree grids of different levels are used to divide the model. The changes in the number of meshes, model fit, and calculation time required to mesh the model at different mesh levels are analyzed. The results show that as the grid level increases, the number of grids required for the subdivision model and the model fit will also increase. The number of grids increases exponentially after the eighth level, and the model fit continues to increase at a stable rate. In experiment 2, grid A*, grid point A*, grid Lazy Theta*, grid point Lazy Theta*, RRT * are used for route planning simulations. At the same time, the data of path length, path cost, maximum turning angle, maximum pitch angle, and calculation time are obtained. By analyzing and comparing these data, it is found that: (1) With the increase in the grid level, the path length, path cost, maximum turning angle, and maximum pitch angle of A*, Lazy Theta*, and RRT will be optimized, but the running time will also increase. (2) In the same level of grid, the algorithm based on the grid point has a shorter path length, a smaller path cost, maximum pitch angle, maximum turning angle, and less running time than that based on the grid. (3) The path planned by the grid point Lazy Theta* algorithm has the lowest path length, path cost, maximum turn angle, and maximum pitch angle, and the grid point A* has the smallest calculation time. Grid point Lazy Theta* can improve the flyability of the route and the flight safety in complex environments while ensuring computational efficiency. (4) At Level 7, the grid point Lazy Theta* is able to plan a major route with as short a running time as possible. In experiment 3, the smoothing results of the equally spaced B-spline proposed in this paper and the uniform B-spline are compared, the findings demonstrate that the proposed smoothing approach increases safety and flyability. In the future, the optimized Lazy Theta* algorithm can be further combined with dynamic obstacle avoidance, which has a broad application value. Future works will consider more environmental factors and actual mission requirements for more realistic applications in the field of route planning. The effectiveness of the proposed approach in route planning is verified by the above experiments. This approach can determine the subdivision fineness of the model and the step size of route planning by setting the grid level, and has more path choices in the face of complex environments. However, it also has some shortcomings. The following are plans to improve these deficiencies: (1) The increase in grid level will lead to a multiplication of the running time, and a better mesh generation method is needed to reduce the impact on running time. (2) This paper only considers the route planning of a single aircraft; the next step will be to try multi-aircraft cooperative route planning. The approach proposed in this paper may be able to inspire some research in other fields. For example, for the spatial control of a city, buildings can be identified by gridding the city, and buildings can be regarded as obstacles to plan the city's transportation network or communication site network to improve the utilization of space and the order of planning. Conceptualization, Z.G. and X.X. ; methodology, Z.G. ; chart data extraction, L.W. ; validation, Z.G. ; customary route plan design, M.C. ; algorithm programming, Z.G. ; data curation, L.W. ; writing--original draft preparation, L.W. ; writing--review and editing, Z.G. and X.X. ; supervision, L.W. ; funding acquisition, M.C. All authors have read and agreed to the published version of the manuscript.