Research on Lazy Theta* Route Planning Algorithm Based on Grid Point Optimization

Featured Application

This research can be applied to route planning in a complex threat environment.

Abstract

In recent years, the problem of route planning in complex battlefield environments has attracted significant attention. With the increasingly worrying international situation, safety and flyability in a continuously changing threat environment are critical factors in route planning research. Thus, this paper proposes an improved Lazy Theta* algorithm that adapts to a complex battlefield environment and finds the optimal route. Specifically, given the low computational efficiency and data redundancy of the existing environmental threat modeling, the developed scheme first employs an octree grid to divide the environment into a grid. Furthermore, based on a real environmental threat model and flight constraints, we design a Lazy Theta* algorithm based on octree grid points, which shortens the planning path and reduces the path cost. Finally, this paper proposes an equally spaced B-spline to smooth the route and improve its smoothness and flyability. Several simulated experiments verify that the smoothed route improves safety and flight ability while reducing the route’s distance. Overall, the simulation results prove that the proposed method significantly improves the planning efficiency and flyability compared with traditional methods.

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. For instance, Ou 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. Xiang 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. Zhou 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, Nasir 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. Gammell 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. Such a strategy affords an efficient, parallel, global search-based method. Shorakaei et al. [19] combined the parallel genetic algorithm with a probability map to realize the cooperative route planning of multiple UAVs. The parallel genetic algorithm overcomes the premature problem very well; however, it does not consider the UAV flight performance constraint. Duan et al. [20] developed a multi-frequency vibration genetic algorithm that uses two mutation operations acting on the entire population and elite individuals, respectively, providing global and local diversity for the population. Thereby, it effectively avoids the algorithm’s prematurity and improves its search accuracy.

Spurred by the shortcomings of the above research, this paper proposes a Lazy Theta* route planning method based on octree grid optimization. Based on 3D gridded scene modeling, this work develops a Lazy Theta* [21]–based algorithm that uses grid points instead to realize route planning. The complex terrain environment and flight constraints are considered during planning, and a route cost function that considers various threat costs is designed to improve the route’s safety and flyability during an actual flight. Finally, an equally spaced B-spline is proposed to smooth the flight path and further improve the path’s smoothness and flight ability.

The innovation of this paper is route planning based on a grid point subdivision environment. This method can adapt to environments of differing complexity. According to the complexity of the environment, different grid scales are set to achieve different step-size route planning, which can more effectively avoid various flight obstacles and improve flight safety and flight efficiency. It also provides a new idea for route planning.

2. Octree Meshing Modeling

Based on the meshing model, we establish a geographic environment model in which the static terrain is associated with a three-dimensional grid through terrain meshing to form a phasic mapping data model. This strategy models and retrieves the terrain environment by using the spatial and temporal properties of the grid to index the terrain environment [22].

2.1. Octree Meshing

Octree [23] is a tree-like data structure that describes a three-dimensional space. Figure 1 illustrates an Octree in which each node represents a cubical volume element and has eight child nodes. The sum of the eight child node volumes equals the parent node volume.

Octrees recursively divide space into eight squares, organized and stored in memory as octrees [24]. Figure 2 depicts the data organization of the environment illustrated in Figure 1. Specifically, the ball P in Figure 1 is stored within a child node, continuously divided into octaves to obtain a cube storing the ball P.

In the octree data storage setup, the idle nodes can be pruned, and only the parent and the target nodes are retained [25], saving memory and improving search efficiency. When planning a route, the octree grid divides the global environment, significantly improving planning accuracy, safety, and efficiency.

Compared with the standard grid, the octree grid is more convenient to identify and query objects. Figure 3 shows that the calculation methods of the standard grid and the octree grid subdivision model are different. The grid occupied by the red numbers (letters) in the figure indicates that it is also occupied by the model.

Figure 3a illustrates that the standard grid subdivision model is queried by traversing all grids to find the grids occupied by the model; this method is slow in computing large-scale models.

Figure 3b shows the query method of the octree grid subdivision model. It searches the grid occupied by the model through the continuous octree division of space, which has a higher search efficiency. At the same time, due to the characteristics of the octree structure, it is convenient to encode the grid. In addition, octree grids can be pruned to remove unoccupied grids, reducing the number of grids while further improving search efficiency.

2.2. Encoding Method

As illustrated in Figure 4, the cubic grid structure includes grid cells, grid points, grid edges, and grid faces, representing the cube’s eight vertices, 12 edges, and six faces, respectively [26].

The grid coding method is divided into cell coding, grid edge coding, and grid point coding according to the grid structure. A subspace of the same size is further divided into subspaces until the target level, or the highest level of the grid, is obtained. The grid coding is coded in a Z-curve manner, and the grid coordinates are coded in octal numbers in the order presented in Figure 1 [26,27,28].

The steps for converting grid coordinates $\left(X,Y,Z\right)$ at a level $N$ to octal encoding are as follows.

• Convert $\left(X,Y,Z\right)$ to binary number ${\left(X\prime ,Y\prime ,Z\prime \right)}_2$.
• According to the order $Z\prime \to Y\prime \to X\prime$, the bits are crossed to obtain the code $H_2=Z\prime \otimes Y\prime \otimes X\prime$.
• Shift $H_2$ to the left by three bits to obtain the encoded $H_2\prime =H_2<<3$.
• Convert the binary code $H_2\prime$ to the octal code $I_8$.

The above steps constitute the octal coding method of the grid cells, and on this basis, the grid points are encoded, with the relevant steps being presented below.

• Reverse the cross-bit of the binary cell code $H_2$ to obtain the binary coordinate $\left(X_2,Y_2,Z_2\right)$.
• The calculation formula of the first hexagram limit grid point code $H{d}_1$ is presented in Equation (1).

  $$\{\begin{array}{l}{X}_2\prime =X_2+1\\ Y_2\prime =Y_2+1\\ Z_2\prime =Z_2+1\\ H{d}_1=Z_2\prime \otimes Y_2\prime \otimes X_2\prime \end{array}$$

  (1)

The above coding method encodes according to the Z-shaped curve based on each level [29].

2.3. Building a Grid Point Map

Before route planning, a grid point map needs to be constructed, mainly including the following steps [30]:

• Determine the grid level within the environmental range required by the task; i.e., the grid scale is determined, and the minimum step size of the flight is limited. Let the grid side length of grid level $N$ be $l_N$. Then, the minimum step size $S$ satisfies Equation (2). According to the flight mission requirements, determine the flight range, reduce the unnecessary grids, and shorten the planning path.

  $$\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.l_N<S<\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.l_N$$

  (2)
• To determine the grid level and planning range, divide the grid into a point grid.
• Based on the gridded scene, assign a value to the grid point, where zero means the grid point is passable, and one means the grid point is occupied and not passable.

3. Route Planning Modeling

3.1. Flight Condition Constraints

During the flight mission, the aircraft is affected by the external environment and threats, and so various constraints must be added to the aircraft. This section mainly considers these constraints [31].

3.1.1. Maximum Range Constraint

Maximum range is the maximum flight path length of an aircraft. Let the maximum range be $L_{\max }$ and the actual flight route comprise $n$ segments. Then, the maximum range constraint is defined:

$${\displaystyle \sum _{i=1}^n{L}_i}<L_{\max }$$

(3)

3.1.2. Maximum Steering Angle Constraint

The maximum steering angle refers to the aircraft’s maximum steering angle in the horizontal direction in a unit of time. In a grid point map, maximum steering angle constraints can be converted into a horizontally connected domain problem. This paper utilizes the Lazy Theta* algorithm for route planning. Hence, the maximum steering angle is $\arctan 2$, and the steering angle constraint is:

$$\phi -\arctan 2\le 0$$

(4)

3.1.3. Maximum Climb/Dive Angle Constraints

The maximum climb/dive angle is the aircraft’s maximum allowed climb and dive angle. The maximum climb/dive angle constraint in the grid point map can be transformed into a vertically connected domain problem. The maximum climb/dive angle is $\arctan 2$, and the climb/dive angle constraint is:

$$\alpha -\arctan 2\le 0$$

(5)

3.1.4. Turning Radius Constraints

The minimum turning radius constraint is transformed into a limit on the number of nodes to be expanded. Hence, according to the heading of the current node, five of the eight nodes are selected to be expanded around the current node. As illustrated in Figure 5, $m$ is the current node and $n$ is its parent node. The route planning efficiency is improved by limiting the number of nodes to be expanded.

3.1.5. Flight Altitude Constraints

During route planning, the aircraft’s flight altitude must be limited. The minimum flight height is represented by $h_{\min }$, and the maximum by $h_{\max }$. The current height $h$ of the aircraft should satisfy Equation (6).

$$h_{\max }\le h\le h_{\max }$$

(6)

3.2. Threat Modeling

Threats are external factors that may adversely affect the aircraft’s flight safety, survival, and mission performance [32].

3.2.1. Terrain Threat

Terrain threat mainly refers to the threat posed to a flight by basic topography and a mountainous environment. The basic terrain model can be expressed as:

$$z_1(x,y)=\sin (y+a)+b\cdot \sin (x)+c\cdot \cos (d\cdot \sqrt{x^2+y^2})+e\cdot \cos (y)+f\cdot \sin (g\cdot \sqrt{x^2+y^2})$$

(7)

where $x$ and $y$ are the coordinates of the model points projected onto the horizontal plane, $z_1$ is the elevation value that corresponds to the horizontal plane, and $a,b,c,d,e,f,g$ are the constant factors that control the undulation of the reference terrain.

The mountain environment model mainly refers to higher natural mountains in the flight route, represented by an exponential function. The mathematical model is described as:

$$z_2(x,y)={\displaystyle {\sum }_{i=1}^n{h}_i\exp [-{(\frac{x-x_i}{x_{st}})}^2-{(\frac{y-y_i}{y_{st}})}^2]}$$

(8)

where $z_2(x,y)$ is the peak elevation value, $(x_i,y_i)$ is the center coordinate of the $i$ peak, $h_i$ is the terrain parameter that controls the peak’s height, and $x_{si}$,$y_{si}$ are the attenuation of the peak along the $x$ and $y$ axes, respectively, to control the slope. $n$ represents the total number of peaks.

The safety threshold requirement should be met between the flight height and the terrain height, and the terrain constraint can be expressed as:

$$\min (z_i-h_{p_i})-h_{safe}\ge 0$$

(9)

where $z_i$ is the height of the current waypoint, $h_{p_i}$ is the terrain height at the current waypoint, and $h_{safe}$ is the safe height threshold.

The terrain threat probability is presented below:

$$P_G=\{\begin{array}{ll}0& \min (z_i-h_{p_i})-h_{safe}\le 0\\ 1& \min (z_i-h_{p_i})-h_{safe}>0\end{array}$$

(10)

3.2.2. Detection Threat

The detection threat mainly refers to the threat caused when the aircraft is detected by a radar, which is related to the radar’s maximum detection distance. The radar signal-to-noise ratio formula is given by:

$$\raisebox{1ex}{$S$}\!\left/ \!\raisebox{-1ex}{$N$}\right.=\frac{P_t{G}_t{G}_{r}p\sigma {\lambda }^2}{{(4\pi )}^3{K}_s{L}_m{B}_n{T}_s{d}_R{}^4}$$

(11)

where $\frac{P_t{G}_t{G}_{r}p\sigma {\lambda }^2}{{(4\pi )}^3{K}_s{L}_m{B}_n{T}_s}$ is a parameter related to the performance of the radar, regarded as a constant. After transforming the above formula, the following equation is obtained:

$$\raisebox{1ex}{$S$}\!\left/ \!\raisebox{-1ex}{$N$}\right.=K_a\frac{1}{d_R{}^4}$$

(12)

where $d_R$ represents the straight-line distance between the aircraft and the radar, and $K_a$ is the radar performance parameter.

The radar threat range can be simplified as an inverted hemisphere, with its horizontal section illustrated in Figure 6.

The radar detection threat probability is formulated as:

$$P_R=\{\begin{array}{ll}0& d_R>R_{\max }\vee h_R<H_b\\ \frac{R_{\max }-d_R}{R_{\max }+d_R}·\frac{R^4{}_{\max }}{R^4{}_{\max }+d_R{}^4}& d_R<R_{\max }\wedge h_R>H_b\end{array}$$

(13)

where $R_{\max }$ is the radar’s maximum detection range and $H_b$ is the radar detection height boundary, given by:

$$H_b=K_a{L}^2$$

(14)

where $K_a$ is the radar detection coefficient, and $L$ is the distance from the aircraft’s horizontal projection to the radar’s center, which is less than the radar’s blind area detection $H_b$.

3.2.3. Fire Threat

Fire threats include surface-to-air missiles and anti-aircraft artillery threats [33].

• Surface-to-air missile threat model

The threat range of surface-to-air missiles can be approximated as a drum, with its horizontal section being a sector (Figure 7). Since the missile launch direction is uncertain, the horizontal section can be approximated as a circle. Due to the missile’s maneuverability, the target within the shortest distance cannot be attacked, and so the cross-section of the missile threat range can be simplified as a ring.

In Figure 7, $\overline{AB}$ is the highest limit of the strike area and the maximum height is $h_m{}^{\max }$, $\overline{DC}$ is the lowest limit of the strike area and the lowest height is $h_m{}^{\min }$, and $\overline{BC}$ is the farthest limit of the strike area and the farthest distance is $R_m{}^{\max }$. $\overline{AED}$ is the closest limit of the strike area and the closest distance is $R_m{}^{\min }$.

The aircraft’s flying altitude is typically lower than the striking height of the surface-to-air missile. Therefore, the influence of the height on the threat is ignored, with the corresponding overall missile threat probability being formulated as:

$$P_m=\{\begin{array}{ll}0& d_m\le R_m{}^{\min }\\ \frac{4(R_m{}^{\max }-d_m)(d_m-R_m{}^{\min })}{{(R_m{}^{\max }-R_m{}^{\min })}^2}& R_m{}^{\min }\le d_m\le R_m{}^{\max }\\ 0& d_m\ge R_m{}^{\max }\end{array}$$

(15)

where $d_m$ is the slant distance between the aircraft and the launch point of the missile, and $R_m{}^{\min }$ and $R_m{}^{\max }$ are the shortest and the farthest distances the missile can strike, respectively.

• Anti-aircraft gun threat model

The threat range of the anti-aircraft gun is approximately a sphere, with its horizontal section depicted in Figure 8.

The center of the sphere is the deployment position of the anti-aircraft gun, and the radius is the effective range. The threat probability of the anti-aircraft gun can be simplified to:

$$P_a=\{\begin{array}{ll}1& d_a<R_a\\ \frac{1}{d_a}& {\mathrm{R}}_a\le d_a\le R_a{}^{\max }\\ 0& d_a>R_a{}^{\max }\end{array}$$

(16)

where $d_a$ is the slant distance from the aircraft to the anti-aircraft gun, $R_a{}^{\max }$ is the farthest strike distance of the anti-aircraft gun, and $R_a$ is the effective range.

3.2.4. No-Fly Zone Threat

A cylinder represents the no-fly zone, and the threat likelihood associated with the no-fly zone is given as:

$$P_N=\{\begin{array}{ll}0& l_i\cap S_{NFZ}=\varnothing \\ 1& l_i\cap S_{NFZ}\ne \varnothing \end{array}$$

(17)

where $l_i$ represents the $i$ route and $S_{NFZ}$ is the no-fly zone.

4. Lazy Theta* Algorithm Based on Grid Point Optimization

4.1. Map Grid Point

Current smart algorithms focus their planning process on the grid’s center, with the grid itself operating as a mobile unit. Nevertheless, the actual motion trajectory varies, and the aircraft’s location is not always at the center of the grid. Furthermore, flight conflicts are more likely in the actual flight location [34].

Using the grid points illustrated in Figure 9 allows the aircraft to select among more routes and simultaneously choose a planned flight route closer to the real flight route.

In Figure 10a, for a grid with an edge length $a$, the aircraft flies within the unit grid in the left, right, and top-right directions dependent on the grid’s center.

$l$ is the flight length per unit of time, as defined by Equation (18).

$$a<l<\sqrt{2}a$$

(18)

The aircraft illustrated in Figure 10b is located at a grid point map with eight movable directions. $l$ is defined by:

$$1/2a<l<\raisebox{1ex}{$\sqrt{2}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.a$$

(19)

4.2. Theta* Algorithm

In order to search for solutions, A* uses a grid-based heuristic method. Researchers have made several tweaks to the A* algorithm to increase its search efficiency, but the current variants still suffer from many issues. Theta* is an improvement of the A* algorithm that takes into account a visual inspection to produce a more accurate route prediction [35].

As can be seen in Figure 11a, the paths generated by the A* algorithm can only advance in grid units, hence the route that has been planned does not meet the genuine path condition depicted in Figure 11b.

Equation (20) shows the cost function of the A* algorithm.

$$f(s)=g(s)+h(s)$$

(20)

In this case, where $h(s)$ corresponds to the cost estimated from the current node to the target, and $g(s)$ refers to the path cost from start node $s_{start}$ to the current one $s$, the path cost includes the path length cost and threat cost in some threat areas. There are two lists in algorithm A*: $open list$ and $close list$, one to hold the current points and the other to rank the order in which nodes should be extended. The A* algorithm updates $g(s)$ and the current node $s$ to the un-extended neighbor $s\prime$, and updates the parent nodes $parent(s\prime )$ and $g(s)$ of $s\prime$ by comparing the new path while the shortest present one is in effect.

Similarly, the Theta* algorithm’s cost function is conveyed via Equation (20). In contrast to the A* technique, the parent node of the current node in Theta* can be any vertex. The specific calculation steps are described below [36]:

• Initialize $open list$ and $close list$; put $s_{start}$ to $open list$.
• When the $open list$ is empty, the search cannot determine the shortest route.
• Choose the smallest node $f(s)$ in the $open list$ and insert it into the $close list$ and make it the current node.
• Given a current node $s$ with a target node $s_{goal}$, determine the route from the original point to the final destination using $s$’s parent node $parent(s)$, and the search terminates. If s is not the destination node, step 5 is executed.
• Search the $close list$ with each nearby node $s\prime$ to $s$. If $s\prime$ is in the lists, disregard $s\prime$. If not, go to step 6.
• Compute the cost of $s\prime$, then check the visibility of the $parent(s)$ and $s\prime$.
• When there are two visible points: the route cost of $s_{start}\to \cdots \to parent(s)\to s\to s\prime$ is recorded as $g(s\prime )$, and the path cost of $s_{start}\to \cdots \to parent(s)\to s\prime$ is recorded as $g(parent(s))+g\prime (s\prime )$, where $g\prime (s\prime )$ is the path cost of $parent(s)\to s\prime$. If $g(parent(s))+g\prime (s\prime )<g(s\prime )$, update $g(s\prime )=g(parent(s))+g\prime (s\prime )$. If $s\prime$ already appears on the $open list$, update $s\prime$. If not, add $s\prime$ to $open list$.
• There are two points that are not visible. Check whether $s\prime$ is present already in the $open list$; if so, write down the cost of $s_{start}\to \cdots \to s\to s\prime$ as $g(s)+g\prime (s\prime )$ and the cost of $s_{start}\to \cdots \to s$ as $g(s\prime )$, if $g(s)+g\prime (s\prime )<g(s\prime )$, $g(s\prime )=g(s)+g\prime (s\prime )$. Otherwise, add $s\prime$ to the $open list$.
• Sort nodes in the $open list$ based on the path cost, and go to step 2.

According to Figure 12, path 2 uses the A* algorithm from $s$ to $s\prime$ and path 1 uses the Theta* algorithm, checking the visibility of $s\prime$ and $parent(s)$. It can be observed that path 1 has a lower path cost.

4.3. Grid Point-Based Lazy Theta* Algorithm

Based upon the Theta* algorithm, the Lazy Theta* algorithm is enhanced, reducing unnecessary visibility calculations; i.e., it directly judges that $s\prime$ is visible to $parent(s)$, and puts it directly in $open list$. When $s\prime$ is removed from the $open list$ for expansion, check the visibility of $s\prime$ with the $parent(s)$. If it is visible, the next neighboring point will be located. Otherwise, it looks for a neighboring parent node in the $close list$ [37].

A* can swiftly compute a route associated to a grid edge; in many cases, the planned route is constrained by the form of the grid. However, the Theta* algorithm can design a route at any angle [38]. However, there is a cap on the possible routes between nodes because of the number of vertices in the square grid. Therefore, grid point-based route planning may not only enhance the accuracy of planned routes and enable better danger avoidance, but it can also expand the range of aircrafts and give a more practical method for planning better routes.

Lazy Theta* grid path planning is shown in Figure 13a. If the aircraft makes a turn when it hits an obstruction while it is still within the red zone, it will crash into the black circle. Figure 13b depicts the grid environment at level $N=2$ and then employs the Lazy Theta* algorithm to plan the path. The pilot completes the steering of the aircraft before striking the red barrier. This avoids obstacles and collisions. There are also more paths to choose from that are closer to the true flight path.

After a grid point Lazy Theta* has been designed, the previous grid’s vertex are replaced with grid points, and the visibility of the $parent(p)$ and the neighboring grid point $p\prime$ is checked if the present grid point $p$ is expanded. If they are visible to each other, the route keeps growing using $p\prime$ as the present grid point until it reaches the desired grid point [39,40].

It is common practice to use the distance between the starting and ending grid points as a proxy for the projected cost involved in moving from one grid point to another. When the moving distance of the aircraft is short, combined with the characteristics of the grid point, we use the heuristic function given by Euclid [41], which is written as:

$$h(s)=\sqrt{{(x_{goal}-x_i)}^2+{(y_{goal}-y_i)}^2+{(z_{goal}-z_i)}^2}$$

(21)

where $(x_{goal},y_{goal},z_{goal})$ are the destination point coordinates and $(x_i,y_i,z_i)$ are the current point coordinates.

When the aircraft is far away from the target point, affected by the curvature of the earth, the calculation of the straight-line distance between the two points is no longer applicable. At this time, the Haversive distance is used as the heuristic function, which is written as:

$$hav(\frac{d}{R})=hav({\phi }_2-{\phi }_1)+\cos ({\phi }_1)\cos ({\phi }_2)hav({\lambda }_2-{\lambda }_1)$$

(22)

In Equation (22), $d$ represents the distance between two places, $R$ is the radius of the earth, and $({\phi }_1,{\lambda }_1)$ and $({\phi }_2,{\lambda }_2)$ are the latitude and longitude coordinates of two points, respectively. The Haversive distance allows you to calculate the distance between two latitude and longitude coordinates, suitable for long flight distances.

5. B-Spline Route Smoothing

The B-spline curve is a parametric curve appropriate for three-dimensional space, and a smooth continuous curve can be obtained by specifying the control points for calculation. The route planned by the intelligent algorithm is usually a polyline, which must be smoothed to be close to the real flight trajectory [42].

5.1. B-Spline Uniform Sampling

The control point sequence of the route is ${\left\{P_i\right(x_i,y_i,z_i\}}_{i=1}^N$. By entering the order $K$ of the B-spline curve and setting the curve parameter $t$, the corresponding coordinates on the curve are obtained [43,44]:

$$\{\begin{array}{l}x(t)={\displaystyle \sum _{i=1}^N{x}_i·B_{i,k}(t)}\\ y(t)={\displaystyle \sum _{i=1}^N{y}_i·B_{i,k}(t)}\\ z(t)={\displaystyle \sum _{i=1}^N{z}_i·B_{i,k}(t)}\end{array}$$

(23)

where $B_{i,k}(t)$ is the mixing function of the B-spline:

$$B_{i,K}(t)=\frac{(t-Knot(i))B_{i,K-1}(t)}{Knot(i+K-1)-Knot(i)}+\frac{(Knot(i+K)-t)B_{i+1,K-1}(t)}{Knot(i+k)-Knot(i+1)}$$

(24)

with

$$Knot(i)=\{\begin{array}{ll}0& i<K\\ i-K+1& K\le i\le N\\ N-K+2& i>N\end{array}$$

(25)

$$B_{i,1}(t)=\{\begin{array}{ll}1& Knot(i)\le t\le Knot(i+1)\\ 1& Knot(i)\le t\le Knot(i+1)\wedge t=N-K+2\\ 0& \mathrm{otherwise}\end{array}$$

(26)

The uniform B-spline performs uniform sampling on the definition domain of the parameter $t$, as shown in Figure 14, and the red square in the figure refers to the control point, Black dots represent sampling points, where the spatial distribution of the sampling point sequence is not uniform.

5.2. B-Spline Equally Spaced Sampling

The equidistant sampling point sequence of the B-spline curve is more suitable for aircraft route planning, as it is a polyline according to the connection of the uniform sampling points of B-spline about $t$, and accumulates the length of the line segment along the polyline. When the length is close to the sampling interval, the equidistant sampling method uses the binary method to search for a point so that the point reaches the previous sampling point. The distance is equal to the sampling spacing [45].

The calculation steps are as follows.

• According to B-spline uniform sampling, a uniform sampling point sequence about parameter $t$ is obtained, denoted as ${\{p(t_m)\}}_{m=1}^M={\{p(x(t_m),y(t_m),z(t_m))\}}_{m=1}^M$, where $p(t_m)\triangleq p(x(t_m),y(t_m),z(t_m))$ is the $m$-th sampling point about the parameter $t$, $t_m=\frac{(m-1)·(N-K+1)}{M}$, and $M\gg N-K$.
• Let point $P_d\triangleq p(t_1)=p(x(t_1),y(t_1),z(t_1))$, and put $p_d$ into $S_{d,s}$, where $p_d$ is the first sampling point. Let $m_a=1$ be used to mark the sampling point $p(t_{m_a})$ in $S_t$, and $m_a$ is used for the iterative process of the dichotomy.
• In $S_t$, starting from point $p(t_{m_a})$, find the next sampling point $p(t_{m{\prime }_a})$, which is required to satisfy Equation (27).

  $$m_a^{\prime }\triangleq \arg {\min }_m\left\{m\right|d_{\sum }(p_d,p(t_m))\ge d,m_a\le m\le M\}$$

  (27)

  where $d_{\sum }(,)$ represents the cumulative distance and $d_{\sum }(p_d,p(t_m))$ is the path length from point $p_d$ to point $p(t_m)$ along the polyline specified by $S_t$.

(1)

If any point $p(t_m)$ has $d_{\sum }(p_d,p(t_m))<d(m_a\le m\le M)$, then: let $m_a^{\prime }=M$, $p_d=p(t_M)$; put point $p_d$ into $S_{d,\epsilon }$; point $p_d$ is the last sampling point of $S_{d,\epsilon }$; and move to step 5.

(2)

If $m_a^{\prime }=M$, the last point of $S_t$ has been searched. Let $p_d=p(t_M)$, and place point $p_d$ into $S_{d,\epsilon }$, where point $p_d$ is the last sampling point of $S_{d,\epsilon }$. Skip to step 5.

(3)

If $m_a^{\prime }<M$, then $d_{\sum }(p_d,p(t_{m_a^{\prime }-1}))<d$, meaning that the point whose cumulative distance to point $p_d$ is $d$ is located on $p(t_{m_a^{\prime }})p(t_{m_a^{\prime }+1})$. Move to step 4 to search for this point by dichotomy.

(4)

On the line segment $p(t_{m_a^{\prime }})p(t_{m_a^{\prime }+1})$, use the dichotomy to find the point $p(\overline{t})$, such that $d_{\sum }(p_d,p(\overline{t}))\approx d$ or $\left|d_{\sum }(p_d,p(\overline{t}))-d\right|\le \epsilon$, where $\epsilon$ is the sampling spacing error, which is used to control the search accuracy of the dichotomy. Let $p_d=p(\overline{t})$, put $p_d$ into $S_{d,s}$ as a new sampling point, and let $m_a=m_a^{\prime }$. Move to step 3 to continue execution.

(5)

Obtain $S_{d,s}$ by calculation; that is, the sequence of the equally spaced sampling points of the B-spline route.

Figure 15 is a schematic diagram of the accumulated distance, and Figure 16 is a schematic diagram of a smooth route with equal spacing sampling, where the red squares represent the control points and the black dots represent the sampling points

6. Simulation Experiment and Analysis

6.1. Meshing Experiment

• Simulation

In the same environment, the same model is divided into a single-scale octree grid with increasing levels ranging over 1~10, and the number of divisions is 1000 times. Dividing the model into different levels requires calculating and analyzing the number of meshes to fit the model expressed by:

$$O=\frac{V_M}{V_{\mathrm{n}}}$$

(28)

where $V_M$ is the model volume and $V_{\mathrm{n}}$ is the mesh volume required to enclose the model.

Figure 17 illustrates the effect of the octree meshing model, which is the seventh-level meshing.

The experimental results of the other levels of octree meshing are shown in Figure A1 of Appendix A. The following is a statistical analysis of the relevant data obtained from the experiment.

2

Analysis of results

Once the model is divided 1000 times, the average data values are obtained to define the number of grids. Then, a model-fitting process is required for each level, as reported in

Table 1. Single-scale octree gridded data.

Grid Level	Number of Grids	Model Fit
1	1	40.2%
2	8	40.2%
3	41	65.40%
4	264	73.76%
5	1245	78.65%
6	6457	83.70%
7	30,994	87.24%
8	147,925	91.08%
9	711,874	93.16%
10	3,405,644	94.56%

.

By analyzing the data in Table 1, the changing trend illustrated in Figure 18 is obtained, highlighting that as the mesh level increases, the number of meshes required increases exponentially, and the corresponding model fit also increases significantly.

In this experiment, the number of meshes required for one to ten-level mesh generation models and the degree of model fit are analyzed. Through the analysis of the data, it is concluded that with the increase in the grid level, the number of grids required is also increasing, and it increases exponentially after the eighth level, while the growth rate of the model fit is relatively stable. Therefore, in this experiment, we preliminarily judged that the eighth level is a more suitable calculation level.

6.2. Route Planning Algorithm Comparison Experiment

We compare the proposed route planning algorithm through two experiments and verify its application value. The first set of experiments compares the A*, Lazy Theta*, and RRT* algorithms based on the grid and grid points in a three-layer grid environment. The evaluation is based on comparing their path length, path cost, maximum pitch angle, and maximum turning angle. The route planning is set in a cuboid environment of size $S=100\times 100\times 10$, where $P_{\mathrm{s}}=(20,20,0)$ are the starting point coordinates and $P_{\mathrm{e}}=(90,90,0)$ are the target point coordinates.

The coordinates of the radar threat center are (20, 10, 0), the detection height boundary is $H_b=1$, and the detection range is $R_{\max }=15$. The surface-to-air missile threat coordinates are (70, 70, 0), the minimum strike distance is $R_m{}^{\min }=4$, and the maximum distance is $R_m{}^{\max }=6$. The coordinates of the anti-aircraft gun are (50, 60, 0), the effective strike distance is $R_a=2$, and the maximum strike distance is $R_a{}^{\max }=3$. There is a no-fly zone with the central coordinates of (58, 78, 0), and the radius is $R_{NFZ}=5$.

We set the aircraft’s minimum flight altitude $h_{safe}=0.1$ and the threat weight $(\alpha ,\beta ,\gamma ,\sigma ,\epsilon ,\zeta )=(0.5,0.1,0.1,0.1,0.1,0.1)$.

6.2.1. Route Planning Algorithm Comparison Experiment

• Simulation

This simulation compares and analyzes the route planning results of A*, Lazy Theta* and RRT* at three to five grid levels, and compares them with the algorithms under the grid points.

The route planning simulation at the fifth grid level is shown in Figure 19, Figure 19a is the result of the grid Lazy Theta* and its top view, and Figure 19b is the result of grid point Lazy Theta* proposed in this paper.

Other levels as well as the A* and RRT* simulation results are shown in Figure A2, Figure A3 and Figure A4 in Appendix B. Next, the experimental data of A*, Lazy Theta*, and RRT* route planning based on the grid and grid points at three to five levels are analyzed and compared.

2

Analysis of results

Table 2. Multi-scale grid comparison data.

Algorithm	Grid Level	Path Length	Path Cost	Max Pitch Angle	Max Turn Angle	Running Time
Grid A*	3	120.89	103.00	4.04	45.14	0.91
	4	113.48	93.93	4.04	45.14	1.53
	5	100.93	82.35	4.04	45.14	1.95
Grid point A*	3	113.57	96.64	5.71	45.14	0.98
	4	95.71	82.60	3.02	45.14	1.02
	5	91.93	76.35	2.02	45.14	1.72
Grid Lazy Theta*	3	116.40	97.35	4.04	63.46	1.34
	4	110.63	89.86	3.03	56.32	2.36
	5	98.15	79.37	1.59	22.83	2.54
Grid point Lazy Theta*	3	105.02	86.39	2.86	63.64	1.25
	4	91.26	74.01	1.99	36.87	2.02
	5	87.90	68.01	1.46	16.58	2.38
RRT*	step size = 25	164.97	132.02	14.33	62.41	1.91
	step size = 12.5	145.73	116.61	14.37	58.43	3.44
	step size = 6.25	105.10	84.14	10.58	58.59	5.23

displays the experimental results, and the data change trend is illustrated in Figure 20.

The data change diagram in Figure 20 can be obtained through the simulation experiment. As the grid level increases, the path length and path cost obtained by A*, Lazy Theta*, and RRT* planning are reduced to varying degrees.

Under the same grid level, the path length, path cost, and running time based on the grid points are significantly smaller than those based on the grid. A closer look at Figure 20 shows that the grid point-based Lazy Theta* algorithm has the lowest values in four aspects: path length, path cost, maximum pitch angle, and maximum turning angle, and the grid-based A* algorithm has the shortest running time. This indicates that the grid point-based Lazy Theta* algorithm can better control the aircraft’s flight and turning angles than the competing algorithms, preserving the route to be as short as possible. In this case, the flight operation difficulty is reduced, and the flight safety and efficiency are improved. Similarly, we can find that as the grid level increases, the running time of the three algorithms increases in varying degrees.

To summarize the above analysis: (1) Compared with the grid, the grid points are optimized in the path length, path cost, maximum turning angle, maximum pitch angle, and running time of the algorithm. (2) The running time increases as the grid level increases, but the path length, path cost, maximum turn angle, and maximum pitch angle are also optimized. (3) Among the five algorithms, the path planned by the grid point of Lazy Theta* has the highest flyability, and the grid point A* has the fastest planning time.

6.2.2. Grid Point-Based Lazy Theta* Route Planning at Different Grid Levels

This section carries out simulation tests and analyzes the outcomes of grid point-based Lazy Theta* algorithm planning at various grid levels. The route planning utilizes the grid levels ranging from N = 3 to N = 9, and compares the path length, path cost, maximum pitch angle, and maximum turning angle of the planned route. The grid level that is most suitable for route planning is selected.

• Simulation

Figure 21 illustrates the route planning results for grid point Lazy Theta* at grid levels 3 and 7.

The route planning simulation results at other grid levels are shown in Figure A5 of Appendix B. Next, the data obtained from this simulation experiment are analyzed and compared.

2

Analysis of results

The experimental data are reported in

Table 3. Lazy Theta* route planning data of different grid levels.

Algorithm	Grid Level	Path Length	Path Cost	Max Pitch Angle	Max Turn Angle	Running Time
Grid point Lazy Theta*	3	105.02	86.39	2.86	63.64	1.25
	4	91.26	74.01	1.99	36.87	2.02
	5	87.90	68.01	1.46	16.58	2.38
	6	83.67	66.58	4.04	4.62	4.45
	7	79.45	64.75	2.70	2.29	6.85
	8	78.36	63.82	8.05	7.44	13.45
	9	78.94	63.25	5.11	18.82	58.64

, and the corresponding trend of the data change is depicted in Figure 22.

By analyzing the above experimental results, it can be seen that with an increase in the grid level, the path length and path cost of the planning algorithm show a downtrend. After level 7, the decrease is minimal, and the maximum pitch angle is the smallest at level 5, which then gradually increases. The maximum turn angle is the smallest at level 7 and then gradually increases. The running time always increases; it grows slowly at levels 3–7 and grows very quickly after level 7. The algorithm has lost its timeliness when at level 9. Since the increase in the grid level will lead to an exponential increase in the number of calculations, according to the data analysis, the path length, path cost, maximum pitch angle, and maximum turning angle can be guaranteed to be as small as possible at the seventh level, while ensuring the calculation efficiency. Therefore, grid level 7 is the most appealing grid level for route planning, this conclusion is also similar to that of the eighth level grid obtained in Experiment 1.

6.2.3. Route Smoothing Experiment

• Simulation

For a grid level of $N=3$, the flight direction of the planned route obviously changes. It is convenient to observe the effect of the B-spline smoothing route, so the route planning result of the grid-based Lazy Theta* algorithm under the grid level $N=3$ is used as the smoothing experiment object. The competitor smoothing methods are the uniform quadratic B-spline, uniform cubic B-spline, equally spaced quadratic B-spline, and equally spaced cubic B-spline. By comparing the maximum altitude and path length after smoothing, a method suitable for smoothing the route is obtained. The equally spaced B-spline results for route smoothing are depicted in Figure 23, in which a green dotted line represents the original path, and the smoothed path is represented by a red curve.

Figure A6 in Appendix B complements the smooth simulation results of the uniform B-spline. The following is an analysis and comparison of the data obtained from the smooth simulation experiment.

2

Analysis of results

The experimental data of the smooth route are presented in

Table 4. Route smoothing experimental data.

Route Planning Algorithm	Smoothing Method	Path Length after Smoothing	Maximum Height after Smoothing
Grid point Lazy Theta*	Original route	105.02	5.00
	Uniform quadratic B-Splines	100.59	4.38
	Uniform cubic B-splines	98.61	4.17
	Equally spaced quadratic B-splines	96.89	4.17
	Equally spaced cubic B-splines	90.50	3.61

.

The data in Table 4 show that the uniform B-spline cannot connect to the route’s starting point, while the equally spaced B-spline is a smooth curve connecting to the route’s starting point. Therefore, the equally spaced B-spline is more suitable for the smoothing method of the route. Regarding path length, the path length of the equally spaced cubic B-spline is the shortest, which further improves flight efficiency and makes the planning result closer to the real flight path. From the perspective of maximum altitude, three B-splines with equal spacing can reduce altitude changes and make the flight more stable and controllable.

Overall, the above experimental analysis verifies the equidistant B-spline’s effectiveness, proving that the equidistant cubic B-spline is more suitable for route smoothing in complex terrains.

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.