# Evaluation of Hunting-Based Optimizers for a Quadrotor Sliding Mode Flight Controller

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- First evaluation of SMC parameter optimization for quadrotor flight system with hunting-based algorithms ALO and GWO
- Parameters obtained by ALO and GWO provided more confidence and repeatability during optimization process
- Parameters obtained by ALO and GWO provided lower tracking error
- Novel extension of such optimization approaches to SMC controller tuning, usually applied to PID control

## 2. Quadrotor Dynamics

## 3. Altitude and Attitude Control—Sliding Mode Flight System

## 4. Hunting-Based Optimizers

#### 4.1. Ant Lion Optimizer

#### 4.1.1. Random Walk of Ants

#### 4.1.2. Ant Lions Building Traps

#### 4.1.3. The Entrapment of Ants in Traps

#### 4.1.4. Ant Lions Catching Ants and Re-Building Traps

Algorithm 1: ALO algorithm pseudocode. |

#### 4.2. Grey Wolf Optimizer

- Alpha ($\alpha $) are dominant wolves and thus followed by the rest of the pack.
- Beta ($\beta $) are second in command helping alphas in the decision process and establish a bridge between alphas and the lower levels.
- Delta ($\delta $) are third in the pack hierarchy; while submitted to alphas and betas, they submit the lowest rank, which is called omega. Deltas represent wolves such as scouts, sentinels, elders, hunters, and caretakers.
- Omega ($\omega $) represent the rest of population solutions.

Algorithm 2: Pseudocode for the GWO algorithm. |

## 5. Simulations and Discussion

^{®}R-2014b and represent a time of 300 s. A sampling frequency of ${f}_{s}=30$ Hz was selected, within the feasible sample time limit for a small quadrotor with a diameter around 50 cm [37].

#### 5.1. Fitness Function and Optimization Methodology

^{®}. The fitness of each potential solution (particle, ant, or wolf) were evaluated using Equation (25) considering a full quadrotor flight (see Figure 3) which explores all common VTOL movements (vertical take-off and landing and curves) during 300 s (see Table 5). Although no trajectory benchmark is available, most works consider rectangular, helical, elliptical, or mixed paths [9,13], in such a way that the main VTOL movements be explored. It is noteworthy and required that the same reference trajectory be used for all optimization algorithms. The proposed general fitness function J is defined as:

- statistical Best is the minimum fitness function value (${F}_{i}$) (or best value) obtained in ${N}_{runs}$.$$Best=mi{n}_{i=1}^{{N}_{runs}}{F}_{i}$$
- statistical Worst is the maximum fitness function value (${F}_{i}$) (or worst value) obtained in ${N}_{runs}$.$$Worst=ma{x}_{i=1}^{{N}_{runs}}{F}_{i}$$
- statistical Median is the middle fitness function value (${F}_{i}$) in a sorted list (or median value) obtained in ${N}_{runs}$. If there are two middle numbers (${N}_{runs}$ is even), the median is their average.
- statistical Mean is the average performance of a stochastic algorithm applied ${N}_{runs}$ times, where ${F}_{i}^{*}$ is the optimal solution at the ith run.$$Mean=\frac{1}{{N}_{runs}}\sum _{i=1}^{{N}_{runs}}{F}_{i}^{*}$$
- statistical Standard deviation (Std) indicates the optimizer stability and robustness, preferably as small as possible.$$Std=\sqrt{\frac{1}{{N}_{runs}-1}\sum {({F}_{i}^{*}-Mean)}^{2}}$$

#### 5.2. Flight Simulation

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Subsystems | |
---|---|

Translational | Rotational |

$X}_{\Delta}={\left(\right)}^{x}T\in {\mathbb{R}}^{6$ | $X}_{\Theta}={\left(\right)}^{\varphi}T\in {\mathbb{R}}^{6$ |

${X}_{\Delta}={\left(\right)}^{{x}_{1}}T$ | ${X}_{\Theta}={\left(\right)}^{{x}_{7}}T$ |

$U}_{\Delta}={U}_{1$ | ${U}_{\Theta}={\left(\right)}^{{U}_{2}}T$ |

$\dot{{X}_{\Delta}}=f\left(\right)open="("\; close=")">{X}_{\Delta},{U}_{\Delta}={f}_{\Delta}$ | $\dot{{X}_{\Theta}}=f\left(\right)open="("\; close=")">{X}_{\Theta},{U}_{\Theta}={f}_{\Theta}$ |

Parameter | Description | Value |
---|---|---|

m | total quadrotor mass | $0.52$ kg |

g | gravity acceleration | $9.8$ ms${}^{-2}$ |

L | quadrature arm length | $0.235$ m |

d | propeller resistance coefficient (drag factor) | $7.5\times {10}^{-7}$ |

b | propeller lift coefficient (thrust factor) | $3.13\times {10}^{-5}$ |

${J}_{r}$ | moment of propeller inertia around Z axis | $8.66\times {10}^{-7}$ |

${I}_{x}$ | moment of inertia around X axis | $6.228\times {10}^{-3}$ kgm${}^{2}$ |

${I}_{y}$ | moment of inertia around Y axis | $6.228\times {10}^{-3}$ kgm${}^{2}$ |

${I}_{z}$ | moment of inertia around Z axis | $1.121\times {10}^{-2}$ kgm${}^{2}$ |

Roll |

$\begin{array}{c}\hfill {\displaystyle {e}_{\varphi}={\varphi}_{R}-\varphi ={\varphi}_{R}-{x}_{7}}\\ \hfill {\displaystyle {S}_{\varphi}=\dot{{e}_{\varphi}}+{\lambda}_{2}{e}_{\varphi}=\dot{{\varphi}_{R}}-{x}_{8}+{\lambda}_{2}\left(\right)open="("\; close=")">{\varphi}_{R}-{x}_{7}}\end{array}$ |

Pitch |

$\begin{array}{c}\hfill {\displaystyle {e}_{\theta}={\theta}_{R}-\theta ={\theta}_{R}-{x}_{9}}\\ \hfill {\displaystyle {S}_{\theta}=\dot{{e}_{\theta}}+{\lambda}_{3}{e}_{\theta}=\dot{{\theta}_{R}}-{x}_{10}+{\lambda}_{3}\left(\right)open="("\; close=")">{\theta}_{R}-{x}_{9}}\end{array}$ |

Yaw |

$\begin{array}{c}\hfill {\displaystyle {e}_{\psi}={\psi}_{R}-\psi ={\psi}_{R}-{x}_{11}}\\ \hfill {\displaystyle {S}_{\psi}=\dot{{e}_{\psi}}+{\lambda}_{4}{e}_{\psi}=\dot{{\psi}_{R}}-{x}_{12}+{\lambda}_{4}\left(\right)open="("\; close=")">{\psi}_{R}-{x}_{11}}\end{array}$ |

Control Component | Controller Parameters | |||
---|---|---|---|---|

Altitude z | Roll $\mathit{\varphi}$ | Pitch $\mathit{\theta}$ | Yaw $\mathit{\psi}$ | |

${U}_{eq}$ | ${\lambda}_{1}$ | ${\lambda}_{2}$ | ${\lambda}_{3}$ | ${\lambda}_{4}$ |

${U}_{sm}$ | ${k}_{1},{\delta}_{1}$ | ${k}_{2},{\delta}_{2}$ | ${k}_{3},{\delta}_{3}$ | ${k}_{4},{\delta}_{4}$ |

**Table 5.**Trajectory used in the optimization with initial condition $\psi \left(0\right)=-\frac{\pi}{10}$.

Time Interval | ${\mathit{z}}_{\mathit{R}}$ | ${\mathit{\varphi}}_{\mathit{R}}$ | ${\mathit{\theta}}_{\mathit{R}}$ | ${\mathit{\psi}}_{\mathit{R}}$ |
---|---|---|---|---|

$0<t\le 50s$ | 5 | 0 | 0 | 0 |

$50s<t\le 80s$ | 3 | $\frac{\pi}{5}$ | 0 | 0 |

$80s<t\le 140s$ | 1 | 0 | $-\frac{\pi}{5}$ | 0 |

$140s<t\le 180s$ | 4 | $-\frac{\pi}{3}$ | 0 | 0 |

$t>180s$ | 0 | 0 | 0 | 0 |

Measure Fitness Value | Method | ||
---|---|---|---|

PSO | ALO | GWO | |

Best | $822,840$ | $823,860$ | $822,850$ |

Worst | $925,520$ | $864,350$ | $833,150$ |

Median | $825,060$ | $836,580$ | $823,350$ |

Mean | $838,060$ | $841,590$ | $823,510$ |

Std | $0.0305$ | $0.0153$ | $0.0018$ |

Controller Parameters | Mean Values (30 runs) | Best Values (30 runs) | |||||
---|---|---|---|---|---|---|---|

PSO | ALO | GWO | PSO | ALO | GWO | ||

${\lambda}_{1}$ | $0.24$ | $0.26$ | $0.27$ | $0.27$ | $0.26$ | $0.27$ | |

${k}_{1}$ | $8.00$ | $7.61$ | $8.43$ | $8.42$ | $8.83$ | $8.42$ | |

Height z | ${\delta}_{1}$ | $10.00$ | $8.91$ | $10.00$ | $10.00$ | $9.99$ | $10.00$ |

${\lambda}_{2}$ | $2.40$ | $3.67$ | $0.82$ | $0.5$ | $0.50$ | $0.50$ | |

${k}_{2}$ | $0.10$ | $0.10$ | $0.10$ | $0.10$ | $0.10$ | $0.10$ | |

Roll $\varphi $ | ${\delta}_{2}$ | $10.00$ | $10.00$ | $10.00$ | $10.00$ | $10.00$ | $10.00$ |

${\lambda}_{3}$ | $3.08$ | $6.03$ | $1.11$ | $0.44$ | $2.89$ | $0.44$ | |

${k}_{3}$ | $1.16$ | $1.29$ | $0.38$ | $0.10$ | $0.13$ | $0.10$ | |

Pitch $\theta $ | ${\delta}_{3}$ | $6.12$ | $7.55$ | $7.62$ | $1.54$ | $8.80$ | $1.68$ |

${\lambda}_{4}$ | $3.97$ | $5.61$ | $3.34$ | $2.50$ | $9.43$ | $0.25$ | |

${k}_{4}$ | $0.14$ | $1.13$ | $0.73$ | $0.10$ | $0.54$ | $1.05$ | |

Yaw $\psi $ | ${\delta}_{4}$ | $9.08$ | $7.67$ | $5.64$ | $10.00$ | $7.18$ | $8.68$ |

Flight Plan 1 |

$\psi \left(0\right)=-\frac{\pi}{5}$ |

$0<t\le 70s$, ${z}_{R}=3,{\varphi}_{R}={\theta}_{R}={\psi}_{R}=0$ |

$70s<t\le 120s$, ${z}_{R}=3,{\varphi}_{R}=\frac{\pi}{3},{\theta}_{R}={\psi}_{R}=0$ |

$120s<t\le 180s$, ${z}_{R}=3,{\varphi}_{R}=0,{\theta}_{R}=-\frac{\pi}{10},{\psi}_{R}=0$ |

$180s<t\le 250s$, ${z}_{R}=3,{\varphi}_{R}=-\frac{\pi}{10},{\theta}_{R}=0,{\psi}_{R}=0$ |

$t>250s$, ${z}_{R}={\varphi}_{R}={\theta}_{R}={\psi}_{R}=0$ |

Flight Plan 2—Parameter variation |

$t>0$, ${z}_{R}=6$ |

$80s\le t\le 130s$, Parameter variation of $+20\%$ in $m,{J}_{r},{I}_{x},{I}_{y},{I}_{z},d,b$ |

Flight Plan 3—Input disturbance and temporary motor failure |

the same FP 1 with constant input disturbance of $+0.1$ in ${U}_{1}$ |

$80s\le t\le 130s$, ${U}_{2}=0$ |

Flight Plan 4—Measurement noise |

the same FP 1 with added noise in $z,\varphi ,\theta ,\psi $ |

Flight Plan | Method | ${\mathit{ISE}}_{\mathit{T}}$ | ${\mathit{U}}_{\mathit{T}}$ | $|{\mathit{TV}}_{\mathit{U}}|$ | ${\mathit{std}}_{\mathit{T}}$ |
---|---|---|---|---|---|

PSO | $1.58\times {10}^{3}$ | $47,240$ | $6.17\times {10}^{3}$ | $0.13$ | |

ALO | $1.48\times {10}^{3}$ | $47,243$ | $7.53\times {10}^{3}$ | $0.14$ | |

FP1 | GWO | $1.46\times {10}^{3}$ | $47,229$ | $7.73\times {10}^{3}$ | $0.13$ |

PSO | $3.11\times {10}^{3}$ | $47,398$ | $9.77\times {10}^{3}$ | $0.38$ | |

ALO | $2.95\times {10}^{3}$ | $47,398$ | $1.07\times {10}^{4}$ | $0.38$ | |

FP2 | GWO | $2.87\times {10}^{3}$ | $47,398$ | $1.11\times {10}^{4}$ | $0.38$ |

PSO | $4.26\times {10}^{3}$ | $46,690$ | $6.68\times {10}^{3}$ | $0.20$ | |

ALO | $3.63\times {10}^{3}$ | $46,684$ | $8.04\times {10}^{3}$ | $0.21$ | |

FP3 | GWO | $3.54\times {10}^{3}$ | $46,852$ | $8.05\times {10}^{3}$ | $0.24$ |

PSO | $1.57\times {10}^{3}$ | $47,265$ | $6.22\times {10}^{3}$ | $0.13$ | |

ALO | $1.48\times {10}^{3}$ | $47,316$ | $7.60\times {10}^{3}$ | $0.14$ | |

FP4 | GWO | $1.47\times {10}^{3}$ | $47,367$ | $7.62\times {10}^{3}$ | $0.15$ |

Flight Plan | Method | ${\mathit{ISE}}_{\mathit{T}}$ | ${\mathit{U}}_{\mathit{T}}$ | $|{\mathit{TV}}_{\mathit{U}}|$ | ${\mathit{std}}_{\mathit{T}}$ |
---|---|---|---|---|---|

PSO | $1.48\times {10}^{3}$ | $47,210$ | $6.85\times {10}^{3}$ | $0.13$ | |

ALO | $1.46\times {10}^{3}$ | $47,215$ | $8.06\times {10}^{3}$ | $0.13$ | |

FP1 | GWO | $1.49\times {10}^{3}$ | $47,210$ | $6.91\times {10}^{3}$ | $0.13$ |

PSO | $2.87\times {10}^{3}$ | $47,398$ | $1.11\times {10}^{4}$ | $0.38$ | |

ALO | $2.85\times {10}^{3}$ | $47,398$ | $1.16\times {10}^{4}$ | $0.38$ | |

FP2 | GWO | $2.87\times {10}^{3}$ | $47,398$ | $1.11\times {10}^{4}$ | $0.38$ |

PSO | $3.66\times {10}^{3}$ | $47,206$ | $6.81\times {10}^{3}$ | $0.34$ | |

ALO | $3.48\times {10}^{3}$ | $47,212$ | $8.01\times {10}^{3}$ | $0.35$ | |

FP3 | GWO | $3.66\times {10}^{3}$ | $47,206$ | $6.87\times {10}^{3}$ | $0.34$ |

PSO | $1.52\times {10}^{3}$ | $47,392$ | $6.68\times {10}^{3}$ | $0.15$ | |

ALO | $1.49\times {10}^{3}$ | $47,433$ | $7.87\times {10}^{3}$ | $0.15$ | |

FP4 | GWO | $1.51\times {10}^{3}$ | $47,283$ | $6.84\times {10}^{3}$ | $0.14$ |

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## Share and Cite

**MDPI and ACS Style**

Oliveira, J.; Oliveira, P.M.; Boaventura-Cunha, J.; Pinho, T.
Evaluation of Hunting-Based Optimizers for a Quadrotor Sliding Mode Flight Controller. *Robotics* **2020**, *9*, 22.
https://doi.org/10.3390/robotics9020022

**AMA Style**

Oliveira J, Oliveira PM, Boaventura-Cunha J, Pinho T.
Evaluation of Hunting-Based Optimizers for a Quadrotor Sliding Mode Flight Controller. *Robotics*. 2020; 9(2):22.
https://doi.org/10.3390/robotics9020022

**Chicago/Turabian Style**

Oliveira, Josenalde, Paulo Moura Oliveira, José Boaventura-Cunha, and Tatiana Pinho.
2020. "Evaluation of Hunting-Based Optimizers for a Quadrotor Sliding Mode Flight Controller" *Robotics* 9, no. 2: 22.
https://doi.org/10.3390/robotics9020022