Genetic Programming to Optimize 3D Trajectories
Abstract
:1. Introduction
1.1. Background and Problem Definition
1.2. Literature Review
1.2.1. Trajectory Optimization Algorithms
1.2.2. 3D Trajectory Optimization
1.3. Research Gap
1.4. Research Questions
- 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
2.2. Research Methods
2.2.1. Evolutionary Computation Framework
2.2.2. GP Optimization Workflow
3. Results
3.1. RQ1—Function Tree Representation
3.2. RQ2—Geometry Processing Method
3.2.1. Initialization and Convergence
3.2.2. Variation and Selection
3.2.3. Elitism
3.2.4. Quantifying Validity
4. Discussion
4.1. RQ1—Function Tree Representation
4.2. RQ2—Geometry Processing Method
4.3. Applicability and Scalability
4.4. Comparison with Similar Methods
4.5. Limitations and Potential Solutions
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Appendix A. Background Concepts
- t: trajectory duration,
- : initial condition,
- : initial trajectory for finite interval t,
- : input trajectory for finite interval t,
- : long term cost of following trajectory,
- : minimum and maximum trajectory height limits,
- d: minimum distance of trajectory to obstacles.
Appendix B. PRISMA Literature Review Methods
Appendix C. Comparison with Related Literature
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. |
Appendix D. 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 | |||
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 |
- g: number of generations;
- n: number of vertices in the output trajectory;
- m: number of barrier geometries;
- k: average number of vertices in each barrier geometry;
- i: number of iterations in the optimization;
- f: complexity of the intersection test.
Appendix E. GP Optimization Workflow
- Translate to origin;
- Normalize;
- Rotate about origin;
- Scale to geographic size;
- Translate to destination.
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Strategy | Generations | Duration (seconds) |
---|---|---|
GA [23] | 15 | 16,440 |
GP (this paper) | 15 | 11.7 |
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 |
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Kotze, A.; Hildemann, M.J.; Santos, V.; Granell, C. Genetic Programming to Optimize 3D Trajectories. ISPRS Int. J. Geo-Inf. 2024, 13, 295. https://doi.org/10.3390/ijgi13080295
Kotze A, Hildemann MJ, Santos V, Granell C. Genetic Programming to Optimize 3D Trajectories. ISPRS International Journal of Geo-Information. 2024; 13(8):295. https://doi.org/10.3390/ijgi13080295
Chicago/Turabian StyleKotze, André, Moritz Jan Hildemann, Vítor Santos, and Carlos Granell. 2024. "Genetic Programming to Optimize 3D Trajectories" ISPRS International Journal of Geo-Information 13, no. 8: 295. https://doi.org/10.3390/ijgi13080295