Path Planning of Mobile Robots Based on an Improved Particle Swarm Optimization Algorithm
Abstract
:1. Introduction
2. Related Work
3. Principles of PSO and DE
3.1. Particle Swarm Optimization (PSO)
3.2. Differential Evolution Algorithm (DE)
- Establish the initial population and initialize the parameters:
- 2.
- Mutation operation:
- 3.
- Cross operation:
- 4.
- Select operation.
4. Algorithm Improvement
4.1. Improved Particle Swarm Optimization Based on Corporate Governance Idea (IPSO)
- 1.
- ${\omega}^{*}\cdot {V}_{i}^{t}$
- 2.
- ${c}_{1}\cdot {r}_{1}\cdot ({P}_{i}{}^{t}-{X}_{i}^{t})$
- 3.
- ${c}_{2}\cdot {r}_{2}\cdot (Lbes{t}_{i}{}^{t}-{X}_{i}^{t})$
- 4.
- $\beta \cdot {c}_{3}\cdot {r}_{3}\cdot \psi \cdot (Ad{m}^{t}-{X}_{i}^{t})$
- 5.
- The improved mathematical formula of pos.ition update formula
4.2. Improved Differential Evolution Algorithm Based on Adaptive Parameters
- Adaptive optimization of scaling factor F
- 2.
- Adaptive optimization of cross probability factor CR
4.3. Hybrid IPSO with IDE (IPSO-IDE)
4.3.1. The Principle of IPSO-IDE Algorithm
- step1:
- Initialization parameters, including acceleration factor, number of support votes Opvote, and number of negative votes Owvote, etc.
- step2:
- Initialize the particle swarm randomly, including dimension D, number of population particle N, position X, velocity V, etc.
- step3:
- Calculate the particle fitness value Fit according to the set objective function.
- step4:
- Based on the fitness value Fit, calculate the individual best position P_{i}^{t}, the local best position Lbest_{i}^{t}, and elect Adm according to (12) to obtain the global best position.
- step5:
- Update position X and velocity V according to the improved (9) (10) to generate the elite population with high quality.
- step6:
- Use (13) to process the boundary.
- step7:
- Use the elite population as the initial population of IDE, combine the adaptive parameters (14) (15), and use (6) (7) (8) to perform “high intensity” iterative optimization.
- step8:
- Apply the optimized result of IDE algorithm to Leader of the updated particle swarm.
- step9:
- If the termination condition is met, stop the algorithm and output the optimal results. Otherwise, go to Step2.
4.3.2. Experimental Verification of IPSO-IDE Model
5. Path Planning Based on IPSO-IDE Algorithm
5.1. Design of Fitness Function
- Path length function
- 2.
- Penalty function
5.2. Path Smoothing
- Basic principle of cubic spline interpolation
- In n subintervals [x_{i}, x_{i}_{+1}] (i = 0, 1, …, n − 1), S(x) is a cubic polynomial.
- S(x_{i}) = y_{i} (i = 0, 1, 2, …, n − 1).
- The first derivative and the second derivative of S(x) in the interval [x_{min}, x_{max}] are continuous.
- 2.
- Determine the equation of Spline Interpolation
- According to S(x), which is composed of n cubic polynomials, the polynomial expression can be obtained as:
- According to $S({x}_{i})={y}_{i}(i=0,1,2,\dots ,n-1)$, it can be concluded that:
- According to the continuity of the differential, it can be concluded that:
- 3.
- Smoothing by spline interpolation
6. Simulation Experiments and Analysis
6.1. The Number of Path Nodes Experiment
6.2. Path Planning Experiment
6.2.1. First Experiment: Compared with Different Traditional Heuristic Algorithms
6.2.2. Second Experiment: Compared with Different Improved Algorithms
6.2.3. Third Experiment: Verification of Big Map
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
Abbreviation | meaning |
PSO | particle swarm optimization |
IPSO | improved particle swarm optimization |
DE | differential evolution |
IDE | optimized differential evolution |
IPSO- IDE | improved particle swarm optimization based on differential evolution |
GA | genetic algorithm |
CR | crossover probability factor |
DPG-PSO | democracy-inspired particle swarm optimizer with the concept of peer groups |
ABC | artificial bee colony |
PSO-ABC | hybrid algorithm based on PSO and ABC |
Algorithm 1: Code for manager selects |
//Initialize operator, Owner, operator vote, manager vote, administrator particle. |
$\mathrm{Initialize}:\mathit{Opvote}:=\mathsf{\varphi},\mathit{ovote}:=1-\mathsf{\varphi},\mathrm{Operator},\mathit{Owner},\mathit{Adm},{\mathrm{Adm}}_{i}^{\mathrm{d}}$,vote_{i}^{d}: [0,1]. |
1.While (fit > fit_{min}) 2. For t = 1 to T //T is the number of iterations. 3. For i = 1 to N //N is the number of particles in the population. 4. For d = 1 to D //D for dimension. 5. If (vote_{i}^{d} < Opvote) 6. ${\mathrm{Adm}}_{i}^{\mathrm{d}}$$\leftarrow Operato{r}^{d}$ //If the particles vote for the operator. 7. Opvote ← Opvote + 1/(M.N) //Add up the votes. 8. Opvote ← Opvote/(Opvote+Owvote) //Standardize voting. 9. Else 10. ${\mathrm{Adm}}_{i}^{\mathrm{d}}$$\leftarrow Owne{r}^{d}$ //If the particle votes for the owner. 11. Owvote ← Owvote + 1/(M.N) //Add up the votes. 12. Owvote←Owvote/(Opvote+Owvote) //Standardize voting. 13. End if 14. $//\mathrm{The}{e}^{\left(1-vot{e}_{Adm}\right)}$ expressed as follows. 15. $\mathrm{If}({\mathrm{Adm}}_{i}^{\mathrm{d}}$$=Operato{r}^{d}$) 16. ${e}^{\left(1-vot{e}_{Adm}\right)}$ ←1 17. End if 18. $\mathrm{If}({\mathrm{Adm}}_{i}^{\mathrm{d}}$$=Owne{r}^{d}$) 19. ${e}^{\left(1-vot{e}_{Adm}\right)}$ ←2.3 20. End if 21. $\mathrm{Update}Operato{r}^{d}$ and V_{t} 22. End for 23. End for 24.End while |
Algorithm2: Code for IPSO-IDE |
Initialize: pro = 1/N, ϕ ← 0.7, Opvote: = ϕ, ovote: = 1− ϕ, |
c_{1} ← 0.5, c_{2} ← 0.5, c_{3} ← 1.2, c_{4} ← 1, X, V, Operator, Owner, Leader, Lbest, P. |
1. While (fit > fit_{min}) 2. For t = 1 to T //T is the number of iterations. 3. For i = 1 to N //N is the number of particles in the population. 4. For d = 1 to D //D for dimension. |
//The following is the calculation of the optimal position of an individual based on the fitness value fit. |
5. If (fit (X_{i}(t)) ≤ fit (P_{i}(t))) 6. P_{i}(t) ← X_{i}(t) 7. Else 8. P_{i}(t) ← X_{i}(t) 9. End if |
//The following is the calculation of the local optimal position based on the fitness value fit. |
10. If (fit(X_{i}(t)) ≤ fit(Lbest_{i}(t))) 11. Lbest_{i}(t) ← X_{i}(t) 12. Else 13. Lbest_{i}(t) ←Lbest_{i}(t − 1) 14. End if |
//The following is the calculation of the global optimal position based on the fitness value fit. |
15. If (fit(X_{i}(t)) ≤ fit(Lbest_{i}_{±1}(t)) or (fit(X_{i}(t)) ≤ fit(Lbest_{i}_{±2}(t))) 16. Lbest_{i}_{±1} (t) ← Lbest_{i}(t) |
//Elect an Adm according to the expression (3-15) |
17. Leader, Leader_fit, Opvote, Owvote ← Choose_Leader (Operator, Owner) |
//Update position X and speed V to generate a better elite group, and take the elite group as the initial group of IDE algorithm. |
18. DE_list ← X 19. h(t) ← Mutation (DE_list(t), fit(DE_list(t))) //variation. 20. v(t) ← Crossover (DE_list(t), h(t), fit(DE_list(t))) //cross. |
//Selection. |
21. DE_list(t) ← Selection (v(t), DE_list(t)) (Mutation, Crossover, and Selection: |
respectively refer to the mutation, crossover, and selection operations in the DE algorithm) |
//Apply the optimized result of IDE algorithm to Leader of the updated particle swarm. |
22. DE_fitness ← fit (DE_list(t)) 23. min_f ← Minimum DE_fitness 24. min_position ← Minimum DE_fitness position 25. If (min_f ← Leader_fit) 26. Leader ← min_position 27. End if 28. Normalize Opvote and Owvote 29. If (Leader^{d}(t) = Opvote_{i}^{d}(t)) 30. ${e}^{(1-vot{e}_{leader})}$← 1 31. Else 32. ${e}^{(1-vot{e}_{leader})}$← 2.3 33. End if 34. Update X_{t} and V_{t} //Update the position and velocity of the particle. 35. End for 36. End for 37. End while |
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Test Function | Expression | Value Range | Min | Dim |
---|---|---|---|---|
Sphere | ${f}_{1}={\displaystyle \sum _{i=1}^{D}{x}_{i}^{2}}$ | [−100,100] | 0 | 30 |
Step | ${f}_{2}={\displaystyle \sum _{i=1}^{D}{(floor({x}_{i}+0.5))}^{2}}$ | [−10,10] | 0 | 30 |
H14 | ${f}_{3}=\mathrm{exp}(0.5\cdot {\displaystyle \sum _{i=1}^{D}{x}_{i}^{2}})-1$ | [−1.28,1.28] | 0 | 30 |
Schwefel’s P2.22 | ${f}_{4}={\displaystyle \sum _{i=1}^{D}\left|{x}_{i}\right|}+{\displaystyle \prod _{i=1}^{D}\left|{x}_{i}\right|}$ | [−10,10] | 0 | 30 |
Alpine | ${f}_{5}={\displaystyle \sum _{i=1}^{D}\left|{x}_{i}\cdot \mathrm{sin}{x}_{i}\right|}+0.1\cdot \left|{x}_{i}\right|$ | [−10,10] | 0 | 30 |
Quadric | ${f}_{6}={\displaystyle \sum _{j=1}^{D}({\displaystyle \sum _{i}^{j}{x}_{i}}}{)}^{2}$ | [−100,100] | 0 | 30 |
Rastrigin | ${f}_{7}={\displaystyle \sum _{i=1}^{D}[}{x}_{i}^{2}-10\cdot \mathrm{cos}2\pi {x}_{i}+10]$ | [−5.12,5.12] | 0 | 30 |
Ackley | ${f}_{8}=-20\cdot \mathrm{exp}(-0.2\cdot \sqrt{\frac{1}{D}\cdot {\displaystyle \sum _{i=1}^{D}{x}_{i}^{2}}})-\mathrm{exp}(\frac{1}{D}\cdot {\displaystyle \sum _{i=1}^{D}\mathrm{cos}2\pi {x}_{i}})+20+e$ | [−32,32] | 0 | 30 |
Griewank | ${f}_{9}=\frac{1}{4000}\cdot {\displaystyle \sum _{i=1}^{D}{x}_{i}^{D}}-{\displaystyle \prod _{i=1}^{D}\mathrm{cos}(\frac{{x}_{i}}{\sqrt{i}})}+1$ | [−600,600] | 0 | 30 |
Algorithm | Parameter |
---|---|
IPSO-IDE | c_{1} = 0.5, c_{2} = 0.5, c_{3} = 1.2, c_{4} = 1; ω = 0.4–0.2; ϕ = 0.7; β = 1–1.5; Vmax = 0.6 × Range; F = 0.6; CR = 0.9–0.1 |
PSO | c_{1} = 2; c_{2} = 2; ω = 1; Vmax = 0.5 × Range; Vmax = 0.1 × Range |
DPG-PSO | c_{1} = 2, c_{2} = 1.5, c_{3} = 0.5, c_{4} = 0.8; ω = 0.2; ϕ = 0.7; Vmax = 0.5 × Range |
PSO-ABC | c_{1} = 2, c_{2} = 2; ω = 0.95–0.4; Vmax = Range |
IDE | F = 0.6; CR = 0.9–0.1; Vmax = Range |
Algorithm | Evaluation Index | IPSO-IDE | PSO | DPG-PSO | PSO-ABC | IDE |
---|---|---|---|---|---|---|
f_{1} | Mean | 0 | 1.539 | 0.075 | 2.9 × 10^{−5} | 1.7 × 10^{−27} |
Best | 0 | 1.330 | 0.002 | 2.83 × 10^{−5} | 1.12 × 10^{−30} | |
Worst | 0 | 1.86 | 0.236 | 3.0 × 10^{−5} | 1.6 × 10^{−26} | |
Std | 0 | 0.1809 | 0.0573 | 4.29 × 10^{−7} | 3.93 × 10^{−27} | |
M-iters | 332 | 1180 | 214 | 425 | 1993 | |
f_{2} | Mean | 0 | 0 | 0 | 0 | 0 |
Best | 0 | 0 | 0 | 0 | 0 | |
Worst | 0 | 0 | 0 | 0 | 0 | |
Std | 0 | 0 | 0 | 0 | 0 | |
M-iters | 15 | 1118 | 21 | 100 | 384 | |
f_{3} | Mean | 0 | 1.1045 | 2.79 × 10^{−6} | 5.36 × 10^{−6} | 0 |
Best | 0 | 0.88 | 1.36 × 10^{−8} | 3.82 × 10^{−6} | 0 | |
Worst | 0 | 1.15 | 1.21 × 10^{−5} | 6.84 × 10^{−6} | 0 | |
Std | 0 | 0.1067 | 4.06 × 10^{−6} | 9.82 × 10^{−7} | 0 | |
M-iters | 16 | 286 | 279 | 1065 | 1173 | |
f_{4} | Mean | 0 | 5.36 | 0.362 | 0.029 | 1.01 × 10^{−21} |
Best | 0 | 5.04 | 0.19 | 0.029 | 1.94 × 10^{−22} | |
Worst | 0 | 5.83 | 0.7 | 0.029 | 2.04 × 10^{−21} | |
Std | 0 | 0.2892 | 0.1968 | 3.88 × 10^{−18} | 7.3 × 10^{−22} | |
M-iters | 19 | 1248 | 135 | 186 | 1997 | |
f_{5} | Mean | 0 | 3.6045 | 0.0116 | 1.7435 | 2.88 × 10^{-19} |
Best | 0 | 1.99 | 0.002 | 0.039 | 8.01 × 10^{−28} | |
Worst | 0 | 4.8 | 0.04 | 2.41 | 6.73 × 10^{−19} | |
Std | 0 | 1.1845 | 0.0123 | 1.1265 | 6.73 × 10^{−19} | |
M-iters | 18 | 335 | 126 | 2000 | 2000 | |
f_{6} | Mean | 0 | 9.582 | 0.923 | 4.11 × 10^{−4} | 3.98 × 10^{−29} |
Best | 0 | 7.5 | 0.105 | 4 × 10^{−4} | 3.14 × 10^{−29} | |
Worst | 0 | 10.27 | 2.56 | 4.2 × 10^{−4} | 4.76 × 10^{−29} | |
Std | 0 | 1.0278 | 1.2421 | 1.15 × 10^{−5} | 1.13 × 10^{−29} | |
M-iters | 400 | 1432 | 463 | 765 | 2000 | |
f_{7} | Mean | 0 | 38.066 | 26.629 | 0.0051 | 15.17 |
Best | 0 | 27.98 | 19.14 | 0.004 | 10.94 | |
Worst | 0 | 59.34 | 35.63 | 0.0058 | 19.9 | |
Std | 0 | 12.1852 | 7.5275 | 0.001 | 4.4048 | |
M-iters | 25 | 503 | 208 | 665 | 738 | |
f_{8} | Mean | 3.09 × 10^{−16} | 8.432 | 9 × 10^{−4} | 0.004 | 0.4575 |
Best | 4.4 × 10^{−17} | 7.18 | 6.58 × 10^{−5} | 0.004 | 7.55 × 10^{−15} | |
Worst | 4.44 × 10^{−16} | 9.51 | 1.8 × 10^{−3} | 0.004 | 1.34 | |
Std | 2.31 × 10^{−16} | 0.8422 | 8.26 × 10^{−4} | 0 | 0.6338 | |
M-iters | 51 | 430 | 182 | 548 | 1831 | |
Grade | N | - | - | - | - | |
f_{9} | Mean | 0 | 35.96 | 0.0664 | 2.01 × 10^{−6} | 0.0147 |
Best | 0 | 31.05 | 0.037 | 2.01 × 10^{−6} | 0 | |
Worst | 0 | 41.02 | 0.147 | 2.01 × 10^{−6} | 0.0172 | |
Std | 0 | 3.97 | 0.0495 | 0 | 0.0218 | |
M-iters | 21 | 2000 | 341 | 112 | 1222 |
Algorithm | Parameter |
---|---|
IPSO-IDE | c_{1} = 0.5, c_{2} = 0.5, c_{3} = 1.2, c_{4} = 1; ω = 0.4–0.2; ϕ = 0.7; β = 1–1.5; Vmax = 0.6 × Range; F = 0.9–0.1; CR = 0.9–0.1 |
DE | F = 0.6; CR = 0.7; Vmax = Range |
PSO | c_{1} = 2, c_{2}= 2; ω = 1; Vmax = 0.5 × Range;Vmax = 0.1 × Range |
ABC | n_{OnLooker} = 10; φ = 1.2; P = 0.5; Vmax = Range |
PSO-ABC | c_{1} = 2, c_{2} = 2; ω = 0.95–0.4; Vmax = Range |
DPG-PSO | c_{1} = 2, c_{2} = 1.5, c_{3} = 0.5, c_{4} = 0.8; ω = 0.2; ϕ = 0.7; Vmax = 0.5 × Range |
PSO-DE | c_{1} = 2, c_{2} = 2; F = 0.7; CR = 0.7; Vmax = 0.6 × Range |
IDE | F = 0.9–0.1; CR = 0.9–0.1; Vmax = Range |
Algorithm | Mean | Best | Worst | Std | Time |
---|---|---|---|---|---|
IPSO-IDE | 15.273 | 14.562 | 18.311 | 1.255 | 5.83 |
DE | 16.461 | 15.514 | 18.317 | 1.359 | 7.52 |
PSO | 22.735 | 18.919 | 33.489 | 4.833 | 6.53 |
ABC | 16.865 | 15.557 | 18.356 | 1.330 | 7.90 |
Algorithm | Mean | Best | Worst | Std | Time |
---|---|---|---|---|---|
IPSO-IDE | 14.656 | 14.374 | 15.487 | 0.376 | 7.08 |
PSO-ABC | 15.896 | 15.257 | 16.364 | 0.463 | 9.13 |
DPG-PSO | 17.636 | 16.325 | 18.689 | 0.971 | 7.52 |
PSO-DE | 15.147 | 14.572 | 15.793 | 0.547 | 11.13 |
IDE | 14.843 | 14.534 | 15.635 | 0.426 | 9.74 |
Algorithm | Mean | Best | Worst | Std | Time |
---|---|---|---|---|---|
IPSO-IDE | 143.982 | 143.362 | 144.527 | 0.738 | 16.51 |
PSO-ABC | 158.504 | 157.924 | 151.457 | 1.120 | 43.02 |
DPG-PSO | 157.771 | 157.158 | 159.090 | 0.819 | 35.21 |
PSO-DE | 145.137 | 144.443 | 146.481 | 0.781 | 43.12 |
IDE | 145.703 | 145.180 | 147.504 | 1.021 | 45.34 |
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Yuan, Q.; Sun, R.; Du, X. Path Planning of Mobile Robots Based on an Improved Particle Swarm Optimization Algorithm. Processes 2023, 11, 26. https://doi.org/10.3390/pr11010026
Yuan Q, Sun R, Du X. Path Planning of Mobile Robots Based on an Improved Particle Swarm Optimization Algorithm. Processes. 2023; 11(1):26. https://doi.org/10.3390/pr11010026
Chicago/Turabian StyleYuan, Qingni, Ruitong Sun, and Xiaoying Du. 2023. "Path Planning of Mobile Robots Based on an Improved Particle Swarm Optimization Algorithm" Processes 11, no. 1: 26. https://doi.org/10.3390/pr11010026