# Coupling Dynamics and Three-Dimensional Trajectory Optimization of an Unmanned Aerial Vehicle Propelled by Electroaerodynamic Thrusters

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{1}–F

_{6}, can be controlled independently by adjusting the voltage. Through the combined control of the EAD thrusters, the desired control torque and propulsive force can be generated within a certain range, which is mainly limited by the maximum voltage. The EAD-UAV in this paper is designed by adding several groups of EAD thrusters for attitude control based on the UAV in [9]. In this paper, the influence of structural change on the aerodynamic characteristics of UAVs is not considered. The airframe parameters are the same as those in [9], the aerodynamic parameters only consider the lift coefficient and drag coefficient, and the lateral force coefficient and aerodynamic moment coefficients are set to 0. It is assumed that the EAD-UAV can track the aircraft deflection angle and track the inclination angle during flight, and the fuselage axis direction is always consistent with the flight speed direction, so the lift coefficient and drag coefficient remain unchanged. The lift of the EAD-UAV comes from the installation angle of the wing. The drag coefficient is estimated according to the drag and flight speed given in [9], and the lift coefficient is estimated according to the lift-drag ratio given in [9]. The six EAD thrusters of the EAD-UAV are assumed to have the same thrust at the same voltage, and the structural parameters of the EAD thrusters are consistent with those in [9]. The thrust of thrusters 3, 4, 5, and 6 has two directions, which are positive to the right and upward, and negative for the opposite. The + and − of F

_{3}, F

_{4}, F

_{5}, and F

_{6}represent the direction of thrust. The fuselage parameters of the EAD-UAV and the parameters of the EAD propeller are given in Table A1.

## 2. EAD-UAV Attitude–Path Coupling Dynamics

#### 2.1. Reference Frames

- (1)
- Ground coordinate system ${O}_{\mathrm{G}}{x}_{\mathrm{G}}{y}_{\mathrm{G}}{z}_{\mathrm{G}}$

- (2)
- Body coordinate system $O{x}_{\mathrm{t}}{y}_{\mathrm{t}}{z}_{\mathrm{t}}$

- (3)
- Speed coordinate system $O{x}_{\mathrm{V}}{y}_{\mathrm{V}}{z}_{\mathrm{V}}$

- (4)
- Track coordinate system $O{x}_{\mathrm{h}}{y}_{\mathrm{h}}{z}_{\mathrm{h}}$

#### 2.2. Forces and Torques Acting on EAD-UAVs

- (1)
- Aerodynamic forces

- (2)
- Aerodynamic torques

- (3)
- EAD forces and torques

#### 2.3. Dynamic Equations of EAD-UAVs

- (1)
- Dynamic equation of the motion of the center of mass of an EAD-UAV

- (2)
- Dynamic equation of an EAD-UAV rotating around the centroid

- (3)
- Kinematic equation of the motion of the center of mass of an EAD-UAV

- (4)
- Kinematic equation of an EAD-UAV rotating about its center of mass

- (5)
- Supplementary equations

- (6)
- Relationship between the thrust and voltage of an EAD thruster

- (7)
- Coupling dynamic equation of an EAD-UAV

## 3. Trajectory Optimization Using the Integrative Bezier Shaping Approach (IBSA)

#### 3.1. Optimization Problem

#### 3.2. Bezier State Approximation

^{·}represents the derivative of the actual time, t, from which one can obtain the Bezier representation of x and its first and second derivatives at the start time ${T}^{(k-1)}$ and end time ${T}^{(k)}$ of the k-th segment of the trajectory.

#### 3.3. Nonlinear Programming Problem (NLP)

## 4. Numerical Results

#### 4.1. Single-Target Optimal Flight Control

_{max}. Optimal single-target trajectories of the EAD-UAV with U

_{max}= 80 kV were designed by the BSA and GPM and are shown in Figure 6. The mass of the EAD-UAV was 2.6 kg, and when U

_{0}= 7.7 kV and U

_{max}= 80 kV, the maximum thrust of each EAD thruster, F

_{max}, was 14.4184 N. Except for the boundary constraints between the start and target, there were no constraints on the speed and attitude of the EAD-UAV, only on the U

_{max}of the EAD thruster. It can be seen from Figure 7 that the attitude angles of the EAD-UAV are within reasonable ranges during flight. As can be seen from Figure 8, the angle of attack and sideslip angle of the EAD-UAV are within the limited range, which satisfies the assumption that the lift coefficient and drag coefficient are constant. Figure 9 and Figure 10 reveal that the thrust of the EAD thruster always satisfies the maximum voltage constraint during flight, indicating that the dynamic model of the EAD-UAV is reasonable and that the BSA can fully satisfy the dynamic constraints. It can be seen from Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 that the BSA and GPM optimization results are very similar. The flight time of the optimal trajectory based on the BSA is 192.5029 s, and the calculation time is 0.7824 s. Meanwhile, the flight time of the optimal trajectory obtained through GPM is 189.8874 s, and the calculation time is 49.3636 s. The difference between the BSA and GPM results is 1.38%, and the calculation time of the BSA is only 1.58% of that of GPM. Thus, the BSA has considerable advantages in trajectory planning. In the figures, the +/− of U and F represent the thrust direction, + indicates that the EAD thruster generates the right or upward thrust, and − indicates the opposite meaning.

_{max}was in the range of [50:2:80] kV. With increasing U

_{max}, the flight time decreases. A larger U

_{max}makes the EAD-UAV have a higher flight speed and greater maneuverability. This result is in agreement with the expectations. When the voltage is higher than 50 kV and the thrust is higher than 5.2735 N, the simulation can converge to get the optimal trajectory. This level of thrust is similar to that in [9], and we can optimize the EAD thruster to reach this level, which shows that it is feasible to use BSA to optimize the trajectory of an EAD-UAV. The average deviation between the flight times of the optimal trajectories obtained by the BSA and GPM is 1.14%, and the average calculation time of the BSA is only 1.95% of that of GPM, which shows that the BSA has obvious advantages in terms of calculation efficiency and that the calculation accuracy is not much different from that of GPM.

_{max}= 60 kV and U

_{max}= 80 kV designed by the BSA are shown in Figure 11. It can be seen from Figure 12 that the attitude angles of the EAD-UAV are within reasonable ranges during flight. As can be seen from Figure 13, the angle of attack and sideslip angle of the EAD-UAV are within the limited range. Figure 14 and Figure 15 reveal that the thrust of the EAD thruster always satisfies the maximum voltage constraint during flight. The maximum thrust during flight is only 4.2 N, which is similar to that in [9]. This level of thrust is expected to be achieved by improving the EAD thruster. It can be seen from Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 that the trajectory and other indexes under different voltages are very close, indicating that the voltage required for flight under the optimal energy consumption is relatively low. As shown in Table A5, the flight time, energy consumption, and average power of the flight trajectory with the optimal energy consumption were basically the same when U

_{max}was in the range of [60:2:80], indicating that applied voltage was much lower than U

_{max}to reduce energy consumption, leading to relatively lower battery performance requirements. The UAV power in [9] is 600 W, and the average power of the EAD-UAV under the optimal energy consumption is 1.4 kW. Considering that the EAD-UAV has more thrusters, the higher power is reasonable. Battery performance can be optimized based on the UAV in [9] to achieve this level of power. In the case of optimal time, the thrusters tend to increase the power as much as possible to reduce the flight time. The average power was 4~8 times that in the case of the optimal energy consumption, reducing the flight time by 18–28%. When U

_{max}was in the range of [60:2:80], the average power reached 5~11 kW, which is very demanding for the battery, meaning it may be difficult to find a qualified power supply. The energy consumption taking minimum energy consumption as the optimization objective is 106.6640 W·h, much less than that in the case of optimal time, which can be used as a reference for the selection of battery capacities of EAD-UAVs in the design of power supply systems. The energy consumption taking time as the optimization objective is 323.9982~589.3262 W·h, which is 3~5.5 times that in the case of the optimal energy consumption. Although higher energy consumption can achieve rapid transfer between targets, larger energy consumption requires greater battery capacity. This will lead to greater battery weight, which will pose a greater challenge to the design of the EAD-UAV, considering its low thrust. Therefore, balancing the energy consumption and flight time is very important when optimizing the flight trajectory of EAD-UAVs.

#### 4.2. Multi-Target Continuous Optimal Flight Control

_{max}= [50:2:80] kV. The IBSA results are convergent within this range, which shows that the flight trajectory can be controlled by adjusting only the EAD-UAV voltage. With increasing U

_{max}, the flight time decreases. A larger U

_{max}makes the EAD-UAV have a higher flight speed and greater maneuverability. This result is in agreement with the expectations. The simulation using IBSA can converge to obtain the optimal trajectory with a U

_{max}higher than 50 kV and a thrust limit higher than 5.2735 N. This indicates that the trajectory optimization of the EAD-UAV using IBSA can be realized with a thrust level similar to that in [9], which shows the feasibility of IBSA. As the BSA does not consider the subsequent optimization process when optimizing the trajectory of the current segment, it will leave a relatively poor initial value for the subsequent trajectory optimization problem, and the dynamic model of the EAD-UAV is relatively complex. Therefore, when using the BSA for trajectory optimization, only the first segment or first two segments of the three-segment flight trajectory converge, and the third segment flight trajectory diverges.

## 5. Conclusions

_{max}higher than 50 kV and a thrust limit higher than 5.2735 N. This level of thrust is similar to that in [9], and we can optimize the EAD thruster to reach this level, which indicates the feasibility of the BSA in single-target scenarios and IBSA for trajectory optimization of the EAD-UAV. The simulation showed that using the BSA to optimize the 3D trajectory of an EAD-UAV yielded results 1.14% different from the optimized performance index of GPM and a calculation time that was only 1.95% of that of GPM. Using the minimum energy consumption as the optimization goal, the average power was 1.4 kW, which is achievable. In the case of optimal time, the average power was four to eight times that in the case of the optimal energy consumption, leading to very high requirements for the battery. Therefore, balancing the energy consumption and flight time is very important when optimizing the flight trajectory of an EAD-UAV. Hence, the IBSA can overcome the poor convergence issue of the BSA under the continuous acceleration constraint for multi-target flight trajectories. For the EAD-UAV with the coupled dynamics, the IBSA can rapidly produce 3D trajectory optimization results.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Airframe parameters | Total mass (kg) | 2.6 |

Wingspan (m) | 5.14 | |

Characteristic area (m^{2}) | 4.8 | |

Lift coefficient | 0.24 | |

Drag coefficient | 0.03 | |

Moment of inertia (kg·m^{2}) | J_{x} = 2.8 | |

J_{y} = 0.4 | ||

J_{z} = 1.6 | ||

J_{xy} = 0.17 | ||

EAD thruster parameters | Radius of emitting electrode(mm) | 0.1 |

Airfoil of collecting electrode | NACA0010 | |

Gap between electrodes(mm) | 60 | |

Span of electrode(m) | 3 | |

Dimensionless constant C_{0} | 0.7 | |

Ion mobility μ (m^{2}·V^{−1}·s^{−1}) | 3 × 10^{−4}(cited from [51]) | |

Number of electrode pairs in each thruster | 8 | |

Thrust center distance (m) | l_{1} = 0.1 | |

l_{2} = 0.1 | ||

l_{3} = 0.2 |

Objective | $\mathit{x}$ (m) | $\mathit{y}$ (m) | $\mathit{z}$ (m) | $\mathit{V}$ (m/s) | $\mathit{\theta}$ (°) | ${\mathit{\psi}}_{\mathit{V}}$ (°) | $\mathit{\vartheta}$ (°) | $\mathit{\psi}$ (°) | $\mathit{\gamma}$ (°) | ${\mathit{\omega}}_{\mathit{x}}$ (°/s) | ${\mathit{\omega}}_{\mathit{y}}$ (°/s) | ${\mathit{\omega}}_{\mathit{z}}$ (°/s) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Start | 0 | 20 | 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Target | 1500 | 220 | 200 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Objective | $\mathit{x}$ (m) | $\mathit{y}$ (m) | $\mathit{z}$ (m) | $\mathit{V}$ (m/s) | $\mathit{\theta}$ (°) | ${\mathit{\psi}}_{\mathit{V}}$ (°) | $\mathit{\vartheta}$ (°) | $\mathit{\psi}$ (°) | $\mathit{\gamma}$ (°) | ${\mathit{\omega}}_{\mathit{x}}$ (°/s) | ${\mathit{\omega}}_{\mathit{y}}$ (°/s) | ${\mathit{\omega}}_{\mathit{z}}$ (°/s) |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Start | 0 | 20 | 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

Target 1 | 500 | 120 | 50 | |||||||||

Target 2 | 1000 | 120 | 150 | |||||||||

Target 3 | 1500 | 220 | 200 | 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

**Table A4.**Flight and calculation times for single-target trajectory optimization using the BSA and GPM.

U_{max} (kV) | F_{max} (N) | BSA | GPM | ||
---|---|---|---|---|---|

T_{total} (s) | Calculation Time (s) | T_{total} (s) | Calculation Time (s) | ||

50 | 5.2735 | 231.2105 | 0.6726 | 229.0850 | 32.8628 |

52 | 5.7436 | 228.5899 | 0.7488 | 226.3433 | 40.4004 |

54 | 6.2337 | 225.9591 | 0.6525 | 224.1937 | 32.7720 |

56 | 6.7437 | 223.3179 | 0.6856 | 221.0971 | 37.7848 |

58 | 7.2736 | 220.6755 | 0.6389 | 219.0316 | 31.0693 |

60 | 7.8234 | 218.0338 | 0.7345 | 215.6894 | 31.2783 |

62 | 8.3932 | 215.3979 | 0.6979 | 212.9133 | 31.8652 |

64 | 8.9830 | 212.7728 | 0.6976 | 210.3341 | 36.7595 |

66 | 9.5926 | 210.1608 | 0.7502 | 207.6127 | 38.3360 |

68 | 10.2222 | 207.5660 | 0.7256 | 205.0823 | 34.9859 |

70 | 10.8717 | 204.9927 | 0.6866 | 202.4180 | 41.3066 |

72 | 11.5412 | 202.4239 | 0.7748 | 199.8713 | 37.7527 |

74 | 12.2306 | 199.8926 | 0.7923 | 197.3255 | 40.7273 |

76 | 12.9400 | 197.4132 | 0.8059 | 194.8418 | 38.5275 |

78 | 13.6692 | 194.9420 | 0.7460 | 192.2157 | 45.8105 |

80 | 14.4184 | 192.5029 | 0.7824 | 189.8874 | 49.3636 |

**Table A5.**Flight time, energy consumption, and average power of single-target trajectory optimization for J

_{T}and J

_{Energy}.

U_{max} (kV) | F_{max} (N) | J_{T} | J_{Energy} | ||||
---|---|---|---|---|---|---|---|

T_{total} (s) | Energy Consumption (W·h) | Average Power (W) | T_{total} (s) | Energy Consumption (W·h) | Average Power (W) | ||

60 | 7.8234 | 218.0338 | 323.9982 | 5.3496 × 10^{3} | 267.8697 | 106.6642 | 1.4335 × 10^{3} |

62 | 8.3932 | 215.3979 | 345.6179 | 5.7764 × 10^{3} | 267.8685 | 106.6637 | 1.4335 × 10^{3} |

64 | 8.9830 | 212.7728 | 368.5284 | 6.2353 × 10^{3} | 267.8684 | 106.6637 | 1.4335 × 10^{3} |

66 | 9.5926 | 210.1608 | 391.7572 | 6.7107 × 10^{3} | 267.8683 | 106.6637 | 1.4335 × 10^{3} |

68 | 10.2222 | 207.5660 | 416.7118 | 7.2274 × 10^{3} | 267.8686 | 106.6638 | 1.4335 × 10^{3} |

70 | 10.8717 | 204.9927 | 442.7956 | 7.7762 × 10^{3} | 267.8681 | 106.6636 | 1.4335 × 10^{3} |

72 | 11.5412 | 202.4239 | 470.2307 | 8.3628 × 10^{3} | 267.8712 | 106.6648 | 1.4335 × 10^{3} |

74 | 12.2306 | 199.8926 | 498.1934 | 8.9723 × 10^{3} | 267.8683 | 106.6637 | 1.4335 × 10^{3} |

76 | 12.9400 | 197.4132 | 527.2413 | 9.6147 × 10^{3} | 267.8700 | 106.6643 | 1.4335 × 10^{3} |

78 | 13.6692 | 194.9420 | 557.5341 | 1.0296 × 10^{4} | 267.8678 | 106.6635 | 1.4335 × 10^{3} |

80 | 14.4184 | 192.5029 | 589.3262 | 1.1021 × 10^{4} | 267.8711 | 106.6648 | 1.4335 × 10^{3} |

**Table A6.**Flight and calculation times for multi-target trajectory optimization using the IBSA and BSA.

U_{max} (kV) | IBSA | BSA | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

ΔT^{(1)}(s) | ΔT^{(2)}(s) | ΔT^{(3)}(s) | T_{total}(s) | Calculation Time (s) | ΔT^{(1)}(s) | ΔT^{(2)}(s) | ΔT^{(3)}(s) | T_{total}(s) | Calculation Time (s) | |

50 | 76.4132 | 73.9339 | 76.5223 | 226.8695 | 6.8735 | 75.5783 | invalid | invalid | invalid | invalid |

52 | 75.4377 | 72.9035 | 75.4856 | 223.8268 | 5.8291 | 74.9793 | invalid | invalid | invalid | invalid |

54 | 74.6172 | 71.8952 | 74.5277 | 221.0401 | 5.2695 | 74.0551 | invalid | invalid | invalid | invalid |

56 | 73.5338 | 70.9669 | 73.5909 | 218.0917 | 6.7434 | 73.1152 | invalid | invalid | invalid | invalid |

58 | 72.8124 | 70.0394 | 72.6541 | 215.5058 | 5.7007 | 72.1822 | invalid | invalid | invalid | invalid |

60 | 71.8200 | 69.0879 | 71.7320 | 212.6399 | 5.9556 | 71.2523 | invalid | invalid | invalid | invalid |

62 | 70.7714 | 68.1487 | 70.7778 | 209.6978 | 6.7505 | 70.3290 | invalid | invalid | invalid | invalid |

64 | 69.8145 | 67.2227 | 69.8679 | 206.9051 | 7.0418 | 69.4099 | Invalid | invalid | invalid | invalid |

66 | 68.9297 | 66.2969 | 68.9316 | 204.1582 | 7.1343 | 68.4966 | invalid | invalid | invalid | invalid |

68 | 68.1025 | 65.4028 | 68.0002 | 201.5056 | 6.6272 | 67.5913 | invalid | invalid | invalid | invalid |

70 | 67.2949 | 64.5305 | 67.1137 | 198.9391 | 5.6748 | 66.6964 | invalid | invalid | invalid | invalid |

72 | 66.3007 | 63.6260 | 66.2121 | 196.1387 | 6.7939 | 66.8113 | invalid | invalid | invalid | invalid |

74 | 65.3456 | 62.7322 | 65.3454 | 193.4232 | 7.0412 | 64.9317 | invalid | invalid | invalid | invalid |

76 | 64.6345 | 61.8891 | 64.4766 | 191.0002 | 6.1002 | 64.0640 | invalid | invalid | invalid | invalid |

78 | 63.7714 | 61.0379 | 63.6160 | 188.4254 | 6.2392 | 63.2145 | invalid | invalid | invalid | invalid |

80 | 62.8919 | 60.1779 | 62.7654 | 185.8352 | 6.3757 | 62.3733 | invalid | invalid | invalid | invalid |

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**Figure 3.**Relationship between $O{x}_{\mathrm{g}}{y}_{\mathrm{g}}{z}_{\mathrm{g}}$ and $O{x}_{\mathrm{t}}{y}_{\mathrm{t}}{z}_{\mathrm{t}}$.

**Figure 4.**Relationship between $O{x}_{\mathrm{g}}{y}_{\mathrm{g}}{z}_{\mathrm{g}}$ and $O{x}_{\mathrm{h}}{y}_{\mathrm{h}}{z}_{\mathrm{h}}$.

**Figure 5.**Relationship between $O{x}_{\mathrm{V}}{y}_{\mathrm{V}}{z}_{\mathrm{V}}$ and $O{x}_{\mathrm{t}}{y}_{\mathrm{t}}{z}_{\mathrm{t}}$.

**Figure 7.**Attitude of the EAD-UAV in the single-target optimal trajectory for J = J

_{T}with U

_{max}= 80 kV.

**Figure 8.**Angle of attack and sideslip angle of EAD-UAV in the single-target optimal trajectory for J = J

_{T}with U

_{max}= 80 kV. (

**a**) Angle of attack, (

**b**) sideslip angle.

**Figure 9.**Voltages of EAD-UAV thrusters with single-target optimal trajectories for J = J

_{T}obtained using U

_{max}= 80 kV. (

**a**) Voltage of thruster 1, (

**b**) voltage of thruster 2, (

**c**) voltage of thruster 3, (

**d**) voltage of thruster 4, (

**e**) voltage of thruster 5, (

**f**) voltage of thruster 6.

**Figure 10.**Thrusts of EAD-UAV thrusters with single-target optimal trajectories for J = J

_{T}obtained using U

_{max}= 80 kV. (

**a**) Thrust of thruster 1, (

**b**) thrust of thruster 2, (

**c**) thrust of thruster 3, (

**d**) thrust of thruster 4, (

**e**) thrust of thruster 5, (

**f**) thrust of thruster 6.

**Figure 11.**Optimal single-target trajectory of an EAD-UAV for J

**=**J

_{Energy}with U

_{max}= 60 kV and U

_{max}= 80 kV.

**Figure 12.**Attitude of the EAD-UAV in the single-target optimal trajectory for J = J

_{Energy}with U

_{max}= 60 kV and U

_{max}= 80 kV.

**Figure 13.**Angle of attack and sideslip angle of EAD-UAV in the single-target optimal trajectory for J = J

_{Energy}with U

_{max}= 60 kV and U

_{max}= 80 kV. (

**a**) Angle of attack, (

**b**) sideslip angle.

**Figure 14.**Voltages of EAD-UAV thrusters with single-target optimal trajectories for J = J

_{Energy}obtained using U

_{max}= 60 kV and U

_{max}= 80 kV. (

**a**) Voltage of thruster 1, (

**b**) voltage of thruster 2, (

**c**) voltage of thruster 3, (

**d**) voltage of thruster 4, (

**e**) voltage of thruster 5, (

**f**) voltage of thruster 6.

**Figure 15.**Thrusts of EAD-UAV thrusters with single-target optimal trajectories for J = J

_{Energy}obtained using U

_{max}= 60 kV and U

_{max}= 80 kV. (

**a**) Thrust of thruster 1, (

**b**) thrust of thruster 2, (

**c**) thrust of thruster 3, (

**d**) thrust of thruster 4, (

**e**) thrust of thruster 5, (

**f**) thrust of thruster 6.

**Figure 16.**Thrusts of EAD-UAV thrusters with multi-target optimal trajectory obtained using U

_{max}= 80 kV.

**Figure 17.**Velocity of an EAD-UAV with the multi-target optimal trajectory obtained using U

_{max}= 80 kV.

**Figure 18.**Acceleration of an EAD-UAV with the multi-target optimal trajectory obtained using U

_{max}= 80 kV.

**Figure 19.**Attitude of an EAD-UAV with the multi-target optimal trajectory obtained using U

_{max}= 80 kV.

**Figure 20.**Angle of attack and sideslip angle of EAD-UAV in the multi-target optimal trajectory with U

_{max}= 80 kV. (

**a**) Angle of attack, (

**b**) sideslip angle.

**Figure 21.**Voltages of EAD-UAV thrusters with multi-target optimal trajectory obtained using U

_{max}= 80 kV. (

**a**) Voltage of thruster 1, (

**b**) voltage of thruster 2, (

**c**) voltage of thruster 3, (

**d**) voltage of thruster 4, (

**e**) voltage of thruster 5, (

**f**) voltage of thruster 6.

**Figure 22.**Thrusts of EAD-UAV thrusters for multi-target optimal trajectory with U

_{max}= 80 kV. (

**a**) Thrust of thruster 1, (

**b**) thrust of thruster 2, (

**c**) thrust of thruster 3, (

**d**) thrust of thruster 4, (

**e**) thrust of thruster 5, (

**f**) thrust of thruster 6.

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## Share and Cite

**MDPI and ACS Style**

Lin, T.; Huo, M.; Qi, N.; Wang, J.; Wang, T.; Gu, H.; Zhang, Y.
Coupling Dynamics and Three-Dimensional Trajectory Optimization of an Unmanned Aerial Vehicle Propelled by Electroaerodynamic Thrusters. *Aerospace* **2023**, *10*, 950.
https://doi.org/10.3390/aerospace10110950

**AMA Style**

Lin T, Huo M, Qi N, Wang J, Wang T, Gu H, Zhang Y.
Coupling Dynamics and Three-Dimensional Trajectory Optimization of an Unmanned Aerial Vehicle Propelled by Electroaerodynamic Thrusters. *Aerospace*. 2023; 10(11):950.
https://doi.org/10.3390/aerospace10110950

**Chicago/Turabian Style**

Lin, Tong, Mingying Huo, Naiming Qi, Jianfeng Wang, Tianchen Wang, Haopeng Gu, and Yiming Zhang.
2023. "Coupling Dynamics and Three-Dimensional Trajectory Optimization of an Unmanned Aerial Vehicle Propelled by Electroaerodynamic Thrusters" *Aerospace* 10, no. 11: 950.
https://doi.org/10.3390/aerospace10110950