Genetic Programming to Optimize 3D Trajectories

Abstract

Trajectory optimization is a method of finding the optimal route connecting a start and end point. The suitability of a trajectory depends on not intersecting any obstacles, as well as predefined performance metrics. In the context of unmanned aerial vehicles (UAVs), the goal is to minimize the route cost, in terms of energy or time, while avoiding restricted flight zones. Artificial intelligence techniques, including evolutionary computation, have been applied to trajectory optimization with varying degrees of success. This work explores the use of genetic programming (GP) for 3D trajectory optimization by developing a novel GP algorithm to optimize trajectories in a 3D space by encoding 3D geographic trajectories as function trees. The effects of parameterization are also explored and discussed, demonstrating the advantages and drawbacks of custom parameter settings along with additional evolutionary computational techniques. The results demonstrate the effectiveness of the proposed algorithm, which outperforms existing methods in terms of speed, automaticity, and robustness, highlighting the potential for GP-based algorithms to be applied to other complex optimization problems in science and engineering.

1. Introduction

1.1. Background and Problem Definition

Pathfinding essentially refers to search algorithms that connect the start and end points using a navigable path. Pathfinding in two-dimensional (2D) space or 2D routing is widely applied to navigational and logistical problems that may or may not be confined to a pathway network [1]. Increasingly complex problems are encountered in this field, especially with the introduction of additional considerations such as multiple points (Traveling Salesman Problem) [2], multiple agents (Vehicle Routing Problem) [3], or dynamism (transient obstacles, alternative routes, Dynamic Vehicle Routing Problem) [4]. Pathfinding in three-dimensional (3D) space expands upon 2D routing, with applications for autonomous or remotely operated vehicles, such as unmanned aerial vehicles (UAVs) [5,6] or unmanned underwater vehicles (UUVs) [7,8], where height and depth are, respectively, incorporated in a route or trajectory as a third spatial dimension. See Appendix A for background details.

The main advantage of 3D pathfinding is the ability to navigate in more complex environments, using routes and passages over or under no-go zones by varying altitude. An assumption in similar 2D problems is that danger zones extend to infinity in the third dimension and the results of route optimization are bound to a cost surface. However, this is not always the case [9]. With robotic vehicles becoming increasingly common and agile, the need to exploit concavities, holes, tunnels, and the spaces above and beneath structures becomes evident. Defining the extent of barriers in the third dimension allows many new routes to be considered in the optimization process.

Nevertheless, trajectory optimization in a space where transport infrastructure is irrelevant, such as aerial, submarine, subterranean, or extraterrestrial routes, is complicated by the fact that the number of possible routes is infinite [10]. The path must actively and automatically avoid physical obstacles and restricted areas, as they are implicitly avoided in pre-existing transport networks. Barrier zones around parks, hospitals, airports, or military installations may have legal, cost, or security significance. Other physical obstacles to avoid include buildings, terrain, vegetation or atmospheric phenomena. Therefore, for an exact optimization that sequentially evaluates all possible routes, it is unrealistic to find the optimum in finite time due to the immense computational resources required for “large search spaces with large numbers of potential solutions” [11] [p. 1]. Therefore, a different strategy is needed to search and sample within the solution space.

1.2. Literature Review

Evolutionary computation is an inexact, biologically inspired approach to optimization [11]. Unlike exact optimization, heuristic algorithms mimic natural evolution by testing the fitness of individuals within a population over multiple generations and applying selective pressures through a cost (or fitness) function. The procedure for finding a heuristic solution is called metaheuristics. Nature-inspired metaheuristic methods, such as evolutionary algorithms and genetic algorithms (GAs), are designed to find a good enough solution by subsampling the enormous and potentially infinite number of possible solutions in 3D space, rather than exploring them sequentially. This paper focuses on nature-inspired metaheuristic methods for spatial optimization of 3D trajectories. See Appendix B for methodological details on the literature review.

1.2.1. Trajectory Optimization Algorithms

Particle swarm optimization (PSO) [12] is the most popular technique for stochastic trajectory optimization in the related literature. For example, Mokhtari [13] generated 3D paths by mapping a B-spline to PSO-generated points and validated candidate solutions by measuring their distance from obstacles. Oultiligh et al. [14] compared Dijkstra, Voronoï, PSO, and fuzzy logic and concluded that PSO is superior because it offers the shortest possible path, while Dijkstra has difficulty in complex environments and Voronoï guarantees obstacle avoidance but cannot offer an optimal path. Other common techniques include deep learning, e.g., Qie et al. [15], to plan paths for multiple UAVs while avoiding threats.

1.2.2. 3D Trajectory Optimization

Since 3D trajectories are not restricted to pre-existing transport infrastructure, there is an additional degree of freedom as to where paths can be placed [16]. Therefore, 3D trajectory optimization is a combination of pathfinding and cost minimization. Of the nine studies that discussed 3D trajectory optimization, only five considered obstacle avoidance (see

Table A1. Comparison of the literature on 3D path optimization with no-fly zones.

	Method	Performance Metrics	Overhead	Advantages	Disadvantages
Zhang and Duan [18]	Population-based meta-heuristic: Adapted pigeon-inspired optimization and path smoothing.	Composite metric including length, power consumption and constraint violation.	Problem size dependent. Not specified.	Applicable to moving no-fly areas.	Long and exponentially increasing with increasing number of nodes in solution representation.
Zhang et al. [20]	Network solving heuristic: Multistep search, Adapted A* algorithm.	Detection probability, travel distance.	Problem size dependent, observed: 6.13–18.48 s.	Applicable to dynamic environments.	Continuous 3D space is discretized into finite search space, problem size dependent computation time.
Hildemann and Verstegen [9]	Population-based meta-heuristic: Adapted NSGA 2 and path smoothing.	Travel time, noise emission, energy consumption.	Problem size dependent, observed: 2–20 h.	Applicable to highly constrained air space.	Long and exponentially increasing with increasing number of nodes in solution representation.
Hai-yan et al. [17]	Population-based meta-heuristic: Particle Swarm Optimization with improved K-means clustering.	Terrain obstacle avoidance, detection probability, maneuverability constraints.	Sufficiently efficient for pre-route planning in actual combat.	Applicable to multiple alternative route planning.	Rudimentary and inexact terrain obstacles. Performance only demonstrated with two alternative routes.
Yang et al. [19]	Population-based meta-heuristic: Genetic programming using binary trees to encode movement increments.	Problem size dependent. Not specified.	Travel distance and threat avoidance.	Improvement over GA applied to the same problem.	Minimal exploitation of the third dimension.

in Appendix C). Four of them used population-based metaheuristic methods, including two swarm- and two evolutionary-based approaches. Hai-yan et al. [17] succeeded in planning multiple alternative routes with PSO. Zhang and Duan [18] used Predator-Prey Pigeon-Inspired Optimization (PPPIO) in a dynamic 3D environment, effectively incorporating a fourth dimension. Their novel algorithm is not only an improvement over PIO, but also outperformed PSO and differential evolution (DE). Evolutionary algorithms were used in the form of genetic algorithms encoding path vertices (Hildemann and Verstegen [9]) and as genetic programming encoding sequential trajectory traversal steps (Yang et al. [19]). Also with a focus on dynamic environments, Zhang et al. [20] used a semidiscretized multistep search algorithm, an improved version of A*, to successfully plan UAV stealth paths while avoiding radar.

Ma et al. [21] and Sundaran [22] emphasized the importance of path smoothness to facilitate navigation, citing the turning ability of different vehicles. Both studies used Bézier and B-spline curve generation, respectively, to obtain smoothed paths, whereas Zhang and Duan [18] used a dynamically feasible path smoothing strategy called $\kappa$ -trajectories. Hildemann and Verstegen [9] also mentioned the importance of path smoothness, due to the energy loss from repeated braking and acceleration when traveling on an uneven route. In particular, in their study of trajectory optimization using genetic algorithms (GAs) for air taxis, Hildemann and Verstegen [9] (and earlier Hildemann [23]) used a directly translated chromosome to represent solutions, with individual genes consisting of the coordinates of points that compose a 3D linestring geometry. This approach, where chromosomes are lists of vertices (one point per gene), is extremely computationally demanding and has a length fixed to the number of output vertices.

Using the same evolutionary algorithm, Hildemann [23] suggested an alternative, more efficient chromosome encoding format that uses genes to encode functions, a method known as genetic programming (GP). This type of evolutionary algorithm has been applied to similar optimization problems in two [24,25] and three [5,6,19] dimensions. It differs from GA in that a program or other executable structure, rather than a sequence of coordinates, is encoded genetically. These programs are modified as the constituent genes are modified, evaluated using a cost function, and a subset of the fittest results is selected to produce the next generation. As chromosomes reproduce, genes are exchanged, combined, and mutated, and successive generations are expected to produce better solutions and evolve toward an optimal trajectory.

1.3. Research Gap

The studies above show that a great deal of research has already been invested in trajectory optimization and that several methods using artificial intelligence, including evolutionary algorithms, are currently in use. However, these methods are overwhelmingly limited to two dimensions, discontinuous space, or both. Only a small fraction of the surveyed studies venture into the third dimension and continuous space, and they find limitations in terms of speed and automaticity. Evolutionary algorithms were used in four papers: GA applied to mobile robots in 2D [26] and to UAVs in 3D [9,19], and GP applied to USVs in 2D [27]. In GP, genes are considered terms within a function, where each path represents a function. That is, solutions can be represented by syntax trees instead of vertex lists as in GA. GP-based approaches thus take advantage of function trees, which are programmatic representations and can be manipulated as genetic programs. As GP-based approaches are severely underrepresented in the existing literature, this suggests that applying GP to the 3D trajectory optimization problem is a novel approach worth investigating.

1.4. Research Questions

Given the importance of trajectory optimization in 3D environments, the main objective here is to design, implement, and evaluate a GP-based solution for 3D trajectory optimization by encoding 3D geographic trajectories as function trees and considering 3D obstacles and barriers. With the focus on chromosome representation and efficient validation, the following research questions are raised:

• RQ1. Can functions encoding curves be effectively evolved to solve a minimization problem?
• RQ2. How does the geometry processing method affect optimization speed?

2. Materials and Methods

2.1. Study Area and Data Preparation

Two areas of New York City (NYC), U.S.A. (Figure 1), and one area of Porto, Portugal (Figure 2), were selected to test the route optimization solution. NYC has been previously studied for air taxi trajectory optimization [9] and Porto provides a more topographically interesting scenario. Both cities are rife with flight obstacles due to flight regulations (restricted airspace) and public disturbance (protected airspace). A fixed height above ground or sea level is the normal lower flight limit. In NYC, the Federal Aviation Administration (FAA) imposes a flight height limit due to plane air traffic of 213.36 m [28]. Preparation of the barrier dataset consisted of creating buffer zones around restricted land use areas and extruding them to the appropriate flight height clearance limits to produce 3-dimensional restricted flight zones (red polyhedrons in Figure 1 and Figure 2). Additionally, a global minimum and maximum flight altitude was specified.

The first scenario contained 17 barrier polygons around Clove Lakes Park in northern Staten Island and was prepared by manually selecting points to be used as start and end points, to provide various combinations of obstacles between pairs of points in order to assess the GP performance (Figure 1). A second scenario in NYC required navigating beyond the airport’s restricted zone near Cunningham Park, and the height limit was varied or removed to assess the optimization’s response to the changing z-dimension availability. The third scenario was central Porto and Gaia (Figure 2), in northern Portugal, and contained the deep valley of the Douro river, potentially aerially navigable.

Figure 3 illustrates the data preparation workflow for the two study areas. Data were obtained from open online sources, primarily OpenStreetMap [29] and OpenDataNYC [30], and filtered by type. Sensitive buildings and areas were then buffered laterally and vertically to create 3-dimensional restricted zones. Because intersection algorithms using simple geometries are significantly faster than algorithms using more complex meshes, barrier datasets were decomposed into their geometric primitives, such as faces, triangles, or convex hulls. For convex hulls, a barrier is intersected if any of the vertices of the 3D line are contained within the hull. The terrain representation of Porto consisted of a raster DEM, in TIFF format, and can theoretically be triangulated and sparsified for improved performance [31].

2.2. Research Methods

2.2.1. Evolutionary Computation Framework

Distributed Evolutionary Algorithms in Python (DEAP) is a Python-built framework built in Python for developing evolutionary computation techniques [32]. Function trees (syntax trees) are typified as individuals, consisting of operators and terminals (primitives) that comprise the primitive set [33]. The operators in the primitive set were selected to maximize the diversity achievable relative to the program size. Basic arithmetic functions such as addition, subtraction, multiplication, division, and negation were used in conjunction with the sine and cosine trigonometric functions to produce solution curves that can theoretically match any trajectory shape.

DEAP allows extensive customization of its internal algorithms to improve the optimization method’s performance for specific use cases, for example, when higher accuracy or speed is required, or when complex scenarios require larger trees and more iterations. Several default parameter values were derived empirically or limited due to computational constraints. Koza [34] suggested a tree height limit of 17 and a maximum number of generations of 50. Poli et al. [33] made some general recommendations while emphasizing that optimal parameters are highly application-dependent. They suggested a duration of 10 to 50 generations, stating that the most productive search occurs at this early stage and, like Koza [34], posited a population size of 500, or “as high as possible”.

Table A2. Genetic programming parameters.

Parameter	Type	Default	Description
pop_size	int	1000	Population size
max_height	int	17	Size limit per individual, by number of levels
max_length	int	80	Size limit per individual, by length
init_min	int	1	Minimum length of new individuals
init_max	int	2	Maximum length of new individuals
mut_min	int	1	Minimum length of new mutations
mut_max	int	3	Maximum length of new mutations
ngen	int	250	Number of generations
patience	int	100	Number of generations to wait for improvements
cxpb	float	0.5	Probability of mating two individuals
mutpb	float	0.1	Probability of mutating an individual
start	float	0	Starting point on reference axis
end	float	1	Ending point on reference axis
nsegs	int	100	Number of line segments in the 3D path
hof_size	int	10	Number of elite individuals to preserve
elitism	Bool	True	Whether to implement elitism
dbl_tourn	Bool	False	Whether to use two-stage tournament selection
tournsize	int	3	Number of individuals per tournament
parsimony_size	float	1.7	Weight of individual size in double tournament,
			in the range $[1,2]$
fitness_first	Bool		Whether to consider fitness first during double
			tournament selection, as opposed to size first
adaptive_mode	Bool	True	Whether to quantify intersections

in Appendix D shows default values for all parameters and describes them in detail.

2.2.2. GP Optimization Workflow

Figure 4 shows the generalized GP optimization workflow developed on top of the DEAP framework. During initialization, a starting population is randomly generated according to user-set parameters. These candidate solutions are then validated by checking whether their intersection with the barriers is an empty set. Valid solutions are then evaluated by quantifying their fitness. A cost function measures the fitness of an individual as a solution, according to the optimization criteria, and the population is subsequently ranked by fitness. The cost function returns the navigational cost of the route, which is the object of minimization, so lower fitness values mean higher suitability. With all the fitnesses known, the best-performing individual is compared to the stopping criteria. If this solution is sufficiently apt, the routine terminates and the optimization is complete; otherwise, the population is varied by mating pairs of individuals or randomly mutating genes in an individual. Individuals with better fitness (lower cost) have a higher probability of mating, mutating, or passing on to the next generation unchanged. After this variation, the offspring generated becomes the starting population of the next generation, and the process repeats. The processes of selection, crossover, and mutation are collectively referred to as genetic operators. The data and code are publicly accessible on Zenodo [35]. See Appendix E for a detailed description of each step in Figure 4.

3. Results

3.1. RQ1—Function Tree Representation

The GP method in 2D is robust against complex obstacle environments and usable results are obtained in a short time. The results in 3D can be faster in some cases, if a simple route over nearby obstacles is available, although the time per evaluation is longer. Considering the number of generations, the duration of 2- and 3-dimensional optimization is mainly affected by population size, geometry size (number of vertices), tree size, and scenario complexity. The quality of optimization is mainly affected by the broadness of the search, which depends on the variation and selection settings (including elitism).

Figure 5 shows two trajectories evolved to connect points b and $bc$. The 2D trajectory is clamped to the ground and must navigate between and around barriers, whereas the 3D trajectory can pass over barriers whenever doing so would shorten the route. Due to the innate smoothness of the generated trajectories and the weight of height minimization, the 3D route does not go over the obstacles nearest to the start and end points. Instead, it circumvents them similarly to the 2D route and only directly passes above the obstacles near the center of the trajectory.

Varying the maximum tree height shows a clear advantage for higher values in terms of fitness (Figure 6a). Note that fitness is a measure of the cost of the route and the objective is to minimize the cost. Therefore, lower fitness translates into better trajectories. In the search for local minima, there continues to be significant variability in the 95% confidence interval in subsequent generations, and the propensity for this is directly correlated with the height limit of the tree. Although evaluation time increases proportionally to tree size, the increased diversity of larger trees allows them to outperform more restricted trees. In the initial 100 generations, the median value is higher (orange line in Figure 6a), before being mostly superseded by the smallest trees until 190 generations, with the largest trees finally dominating the optimization. This suggests that parameters have distinct advantages at different stages of the evolutionary process. Lower height limits may yield quicker solutions but do so inconsistently. Higher limits ensure finer convergence, at the expense of solution size (Figure 6b) and optimization speed (Figure 6c).

Figure 7 shows the trajectories generated after 1000 generations, with quantified intersections, compared to the results of Hildemann [23] and Hildemann and Verstegen [9]. Our results are congruent with theirs with regard to the overall path. The most surprising aspect is the difference in length and smoothness: Under identical constraints (height limitation 213 m), the GP line is 12,469 m in length, compared to 12,513 m (from [9]). Although this is a minuscule reduction in length, the gradual curvature in the GP solution presents a significant reduction in flight time and energy cost due to the increased acceleration and deceleration required to navigate sharp changes in heading. In addition to the reduced computational load of the GP method, this inherent smoothness is another significant improvement over Hildemann and Verstegen [9].

Very complicated trajectories can have a huge computational cost, due to the tendency for bloating and the limited feedback available to the selection algorithm. In these cases, the optimization is nothing more than a random search. Hildemann and Verstegen [9] used a seeding method to create a progenitor trajectory that was then refined incrementally using GA and repair. Although this method works well, the seeding process requires prior knowledge of the environment. Quantifying intersection solves this problem in the scenarios tested, showing that our novel implementation outperforms Hildemann and Verstegen [9] in terms of automaticity. Although it requires significantly more time to solve, this is offset by the overall reduction in computational demand. The processing time compares extremely favorably to Hildemann [23], as shown in

Table 1. Processing time per generation (in seconds) for population size 30.

Strategy	Generations	Duration (seconds)
GA [23]	15	16,440
GP (this paper)	15	11.7

, where a drastic reduction in processing time is observed from hours to a few seconds over 15 generations.

Varying the global height limit also shows how the algorithm can adapt to different scenarios. In the real world, there is a flight height limit of 213 m, which forces the optimization to navigate around restricted areas of 300 m height. The 500 m limit represents a situation where the airport zone is completely impassable, but the 300 m high zone over the park can be flown over. With no global height limit, the optimal route goes directly over the airport zone (blue line in Figure 7). In the Porto–Gaia scenario, trajectories can navigate between and over restricted areas without intersecting the DEM (Figure 8).

Table 2. Effects of parameterization on optimization performance and results.

Scenario	dem	ngen	end	pop_size	cxpb	mutpb	hof_size	Optimum	Duration (s)
Clove Lakes i to e	no	400	44	1000	0.5	0.02	20	4555.25	10.09
Clove Lakes i to e	no	400	85	1000	0.5	0.1	20	4505.02	25.74
Clove Lakes i to e	no	400	400	1000	0.5	0.5	20	4445.51	321.29
Clove Lakes i to e	no	400	57	1000	0.1	0.1	20	4746.49	6.21
Clove Lakes i to e	no	400	134	1000	0.9	0.1	20	4488.21	89.27
Cunningham Park	no	1000	52	100	0.5	0.1	2	15,502.89	29.33
Cunningham Park	no	1000	646	1000	0.5	0.1	20	12,346.46	478.06
Cunningham Park	no	1000	905	10,000	0.5	0.1	200	6768.44	736.87
Cunningham Park	no	1000	332	100	0.75	0.25	10	12,507.67	442.04
Cunningham Park	no	1000	650	1000	0.75	0.25	100	12,749.04	790.16
Cunningham Park	no	1000	1000	10,000	0.75	0.25	1000	12,453.76	620.23
Porto G to R	no	250	76	500	0.9	0.5	10	1222.9	27.77
Porto G to R	yes	250	250	500	0.9	0.5	10	1197.17	739.96

shows a comparison of the results of several runs, and the generations required to achieve a heuristically optimal solution with 30 generations of patience. Varied parameters are highlighted in blue.

3.2. RQ2—Geometry Processing Method

3.2.1. Initialization and Convergence

Population size is one of the two major numerical parameters of GP [34]. Poli et al. [33] recommend using the highest value that can be gracefully handled by the system in use. Figure 9 shows the outcomes of four different values for population size with all other parameters constant. As expected, it shows a clear improvement in optimization efficiency but a direct cost to evaluation time. Over 10 optimizations, larger populations converge more finely in terms of minimum fitness, but diverge more in terms of evaluation time.

Considering the stochastic nature of the initialization, variation, and selection operations, it is not surprising that optimizations with the same inputs can vary significantly. To visualize the convergence of the evolutionary process, multiple optimizations for the same problem were computed, varying the random seed state. The results show a relatively high diversity in fitness values in earlier generations, which eventually converges as the objective is reached. The 95% confidence intervals in Figure 10 show that optimizations with different starting states converge to a near-optimal solution after 50 generations for the scenario shown in Figure 10a, and 20 generations for the scenario in Figure 10b. In the third scenario, Figure 10c, the optimization fails to converge after 400 generations. Due to the complexity of this scenario, the search is vulnerable to local optima, and the initial state has a significant effect on the outcome. The minimum fitness after 400 generations ranges from 11,979 m to 25,434 m for a route with a reference theoretical minimum distance of 11,813 m.

3.2.2. Variation and Selection

Crossover and mutation directly affect the diversity of the population, which is effectively the search radius. Additionally, crossover can produce more specialized offspring by combining the beneficial traits of both parents. Figure 11 shows how optimization performance increases with a higher probability of both crossover and mutation. This illustrates how larger values allow for a quicker and broader search and may aid in avoiding or escaping local minima. Increasing these values also increases the processing time because more evaluations are required for the higher number of altered offspring.

Considering size as an additional tournament stage in the selection process means that smaller solutions are preferred if fitness is similar. Figure 12 shows the reduced average size in the double tournament experiment and a slight improvement in the optimization result.

3.2.3. Elitism

Elitism can improve the speed of convergence because it preserves the best individuals. Therefore, the population’s best fitness is protected from degradation with continued evolution. The optimization results show that the minor loss of diversity is offset by the increase in speed and quality, depending on the size of the elite fraction (see Figure 13). However, the effect is neither proportional nor consistent: after 100 generations, elitism with a 3% elite fraction outperforms 10% in terms of fitness, although 10% performs best in terms of optimization speed and solution size.

Effective bloat control using elitism is well documented [36,37], and the effect of elitism on solution size, especially in later generations, is also shown in Figure 13. After 120 generations, tree size starts to proliferate and this bloating is constrained proportionally to the elite fraction, albeit inconsistently.

3.2.4. Quantifying Validity

The first generations tend to contain more invalid solutions, and once the first valid trajectories are found, an evolutionary process of refinement centered on these trajectories is triggered. In very hard-to-navigate scenarios, with densely packed barriers or narrow passages, it is unlikely that the algorithm randomly arrives at a valid solution that can then be refined. Thus, the rough-stage optimization fails and the binary validation method is ineffective. Hildemann and Verstegen [9] circumvents this problem by “seeding” vertices within a navigable channel, and then using GA and a repair mechanism to refine this rough progenitor trajectory. This requires some foreknowledge of the barrier layout, and a more automated and potentially generalizable solution is for the algorithm to recognize such navigable channels itself. Thus, to adapt the trajectory to a highly complex airspace environment, it is necessary to drive the optimization in the right direction by providing explicit feedback. Solutions that intersect the barrier geometry to different degrees would have to be penalized differently, i.e., the validation criterion must be quantified.

Towards this end, an additional parameter was introduced to “allow” intersection. Enabling the adaptive_mode parameter means that the fitness of invalid/intersecting trajectories is not identically calculated, but depends on the amount of intersection. Trajectories that intersect barriers over most of their length will, thus, be less fit than those that intersect only in a small part. By discriminating between solutions based on the size of the intersecting set, the refining evolutionary process can begin before a valid solution is found. This refinement is then driven by incremental improvements in fitness over incremental reductions in the size of the intersecting set. The algorithm can then find more intricate solution trajectories, as it can know from the size of the intersection when it is evolving in the right direction.

Cases have been observed where optimization could not generate a valid solution due to the minimum required complexity of the solution. Figure 14 shows one example of how intersection quantification can “push” evolution towards a valid solution. This method is significantly slower because all intersections between the trajectory and barriers are measured and summed, instead of invalidating the trajectory after the first intersection is found. Since these solutions are not completely invalidated, the final solution may be invalid, having only minimized but not eliminated the amount of intersection. To avoid this, the intersection_cost parameter may be adjusted to impose a sufficient penalty to evolve away from even small intersections, towards a fully valid trajectory.

Three validation methods were developed for 3D intersections, each with their respective strengths and weaknesses. They were compared by applying the same simple trajectories within the Clove Lakes scenario. The Shapley-based2.5D method was the fastest, ranging from 20 to 50 s for a 100-generation optimization. The first SciPy-based method using Delaunay simplices was the slowest, requiring 2 to 5 hours for the same task. The second SciPy method using convex hulls required between 40 and 100 min.

4. Discussion

4.1. RQ1—Function Tree Representation

In this work, loosely typed GP is used, which allows any primitive or terminal to be passed as an argument to a primitive. Using strongly typed GP allows additional Boolean-type primitives to be incorporated. In such an implementation, conditionals such as if_then_else, operators such as xor, or Boolean terminals could also be used in the syntax tree. The efficacy of strongly typed GP applied to trajectory optimization remains to be investigated.

An alternative approach for extracting two distinct outputs from the same input to a function tree is to initialize individuals as pairs of function trees that evolve in tandem, with a single fitness value calculated for the pair. The crossover and mutation functions then need to be similarly modified to process the function pairs. This approach is more difficult to implement and tends to larger solutions, and its efficacy has not been investigated.

4.2. RQ2—Geometry Processing Method

Ideally, a population of valid trajectories would be initialized before the evolutionary process begins. Subsequent generations would then refine these individuals by adapting them more precisely to the problem. However, on highly obstructed routes it may be nearly impossible to randomly generate a valid geometry due to the complexity required. In these cases, invalid solutions are kept and their fitness is evaluated, applying a penalty proportional to the “magnitude” of their invalidity. Doing so allows the evolution to find niches or incrementally solve a very complex optimization that would not be possible without informative feedback. Another option is to seed waypoints within the navigable airspace by generating random points in the study area and using them as checkpoints for optimization.

Parameter influence was discernible to various extents and it was possible to broadly calibrate the model for different use cases by prioritizing speed, accuracy, or search radius. Certain parameters proved to be disproportionately significant, especially population size and parameters that directly improve diversity. Taking into account the test results of varying parameter values, it is possible to identify configurations that are more or less suitable for the 3D trajectory optimization problem. Population size should be as large as possible, preferably 1000 or more, and no less than 300. Size limits and variation probabilities depend on the complexity of the scenario but can be safely increased to 17 for tree height, 90% for crossover, and 25% for mutation. Elitism should be implemented with an elite fraction of 5–10%.

4.3. Applicability and Scalability

The case studies presented vary in trajectory length from 5 to 25 km and represent a common application for low-altitude routing over moderately large distances in a highly obstructed space. In the areas studied, the universal height limit has been ignored in some cases to demonstrate the ability of the method to find trajectories over restricted areas. Topographic models that incorporate terrain as an additional restriction allow for more precise pathfinding but significantly increase solving time. In two dimensions, the method can be applied to maritime routing, robot path planning, or autonomous vehicles. In three dimensions, the method can work for UAVs and submarines.

Scaling the method to much larger areas should have a very limited effect on speed, provided that the geometrical complexity is not excessive. Performance depends on the number of features and vertices within features, as well as the granularity of the trajectory and the desired coordinate precision. Geometries that are geographically larger or smaller will require the same amount of computational effort if the number of features and vertices are similar.

4.4. Comparison with Similar Methods

Compared to Hildemann [23], the optimization improved the computational requirements and allowed results to be obtained in a much shorter time. The optimization process was successful in all presented use cases and showed reliable convergence in numerous experiments. In addition, the solutions are comparatively lightweight in terms of storage and processing requirements, and the bloat-control methods implemented were able to minimize their size and maximize their processing speed.

The main improvement over Hai-yan et al. [17] and Yang et al. [19] is the complexity and density of obstacles that the new method is able to navigate through. The intrinsic trajectory smoothness also yields shorter and more efficient routes than Hai-yan et al. [17], Zhang and Duan [18], and Zhang et al. [20].

4.5. Limitations and Potential Solutions

The trend towards local optima can be effectively offset by larger population sizes and increased diversity. This directly translates into slower optimization but guarantees more consistent results despite the random nature of the initialization and variation routines. The broadness of the search and the quality of the final result rely more on the starting parameters than on the length of the optimization.

Quantifying intersections in complicated scenarios has a significant processing cost. A semiautomatic method for solving problems that require a very wide lateral or vertical deviation is to iteratively search for the nearest nonintersecting point in the plane that contains the midpoint of the start and end points of the trajectory and is normal to the line that connects them. Using the distance from this point and the displacement between the start and midpoint, a stretching factor can be calculated and used to exaggerate the route geometry in a certain direction. In this way, the algorithm could yield trajectories with very wide turns early on, allowing the rest of the optimization (refinement) to be triggered.

Where the length of intersection is measured in the evaluation routine, with adaptive_mode enabled, the algorithm is currently susceptible to errors if there are overlapping polygons. Parts of the trajectory that intersect multiple polygons will contribute to the total intersection length for each polygon. Care must be taken that no barrier geometries overlap if this setting is enabled.

A possible performance increase is the spatial indexing of barrier geometries. With this implemented, nonintersecting barriers can be identified without calling an intersection algorithm if a barrier is sufficiently distant. This can potentially speed up the validation process, which accounts for 85–95% of the processing time.

PostGIS is a popular geospatial extension for PostgreSQL that supports 3D geometries and 3D geometric predicates. Barriers can be represented as 3D polyhedral surfaces or triangulated irregular networks, and trajectories as 3D line segments [38]. It should be possible to recreate this research using geographic objects stored in a database and substituting the 3D intersection algorithm with the equivalent function in PostGIS.

Another promising prospect is meta-evolutionary algorithms, in which a GA is used to optimize the GP parameters. The calibration performed during this research could theoretically be performed more exhaustively using an evolutionary algorithm operating over repeated runs of the same optimization. Similar to this are coevolutionary algorithms that take into account multiple UAVs and have the additional condition of no collision with other UAVs while traveling along the optimized route.

The cost function can incorporate four objectives with varying weights, although the total trajectory length is the dominant fitness criterion. For improved multiobjective optimization, it is necessary to alter the method of selection to obtain a Pareto front. For example, NSGA-III is a novel nondominated sorting approach based on NSGA-II (used successfully by Hildemann and Verstegen [9] for 3D routing) that has shown its effectiveness with up to 15 objectives [39], and presents a promising research opportunity for 3D trajectory optimization in the future.

5. Conclusions

In this work, a novel approach to the trajectory optimization problem in 3D was proposed, using genetic programs to encode geometries and applying selective pressures to minimize the cost of the route while avoiding intersections with obstacles. The method was applied to case studies in New York and Porto with different levels of navigational difficulty and with different constraints imposed to assess the adaptability of the algorithm. After devising a method of evolving functions and transforming them into geographic trajectories, it was possible to evolve geographic trajectories in place and subsequently achieve the trajectory optimization goal. The syntax tree representation of functions encoding curves proved capable of representing highly variable shapes to adapt to complicated and obstacle-rich environments. These trees are also easily processed and stored, and amenable to manipulation by evolutionary processes.

In complex use cases, it was necessary to guide the evolution towards validity, since the first-generation initialization of a valid geometry becomes exceedingly unlikely in the presence of too many barriers. An adaptive method, which incorporates the intersection length into the cost function, was able to find heuristically optimal trajectories in very confined spaces. An alternative may be to construct a seed function that operates on the initialization process, to increase diversity in first-generation individual creation.

Three methods of 3D geometry intersection were compared, with a clear inverse relationship between processing speed and geometry complexity. The Shapley-based intersection evaluation method is faster but restricts the resolvable details of the third dimension. Fully 3D methods come with a significant performance cost and, as a result, may be impractical in some use cases. The 3D representation of barrier geometries is limited to convex shapes, although this only means that additional preprocessing is required to accommodate concave polyhedrons. Full support for mesh geometry is an opportunity for future work, as is the investigation into the possible benefit of GPU processing.

This research not only showcases the versatility of GP but also demonstrates the possibility of solving spatial optimization problems where the form of the solution found differs widely from the final form. The only requirement is that the solution, or an encoded form of the solution, is derivable from a function.