Cooperative Drone Transportation of a Cable-Suspended Load: Dynamics and Control
Abstract
:1. Introduction
2. Mathematical Model
2.1. Reference Frames
- Two Body–fixed frames, , . The longitudinal axis is positive out of the nose of the i-th platform in its selected plane of symmetry, aims in the direction of the fuselage/frame bottom, and completes a right-handed triad.
- Two Local–Horizontal–Local–Vertical frames, , . The frame is attached to the aircraft’s center of gravity and, under the assumption of a flat and non–rotating Earth, is always parallel to .
- A vehicle structural reference frame, , used to locate the center of gravity and all platform components: the axes are parallel to the Body–fixed frame axes such that , , and . Stationlines (ST) are measured positively aft along the longitudinal axis. Buttlines (BL) are lateral distances, positive to the right, and waterlines (WL) are measured vertically, positive upward. In what follows, without loss of generality, a sample multirotor configuration is considered. In such a case, lies on the top surface of the frame and is aligned vertically with the drone’s geometric center over the plane.
2.2. Equations of Motion
3. Problem Statement and Solution
3.1. Controller Design
3.2. Stability and Control Analysis
4. Results
4.1. Simulation Platform
4.2. Performance Indicators
- Maneuver time, . The simulation is assumed to be concluded at time when two conditions are simultaneously satisfied: (1) the formation control residual error falls below , where and are expressed in m and , respectively, and (2) the absolute value of remains below degrees for a predetermined time of 5 s.
- Average swing angle, . This is calculated as , where the integral
- Propulsive energy, . This measures the total energy delivered by the motors to the propellers and is calculated through the integral
4.3. Test Cases
4.3.1. Simulation Case 1
4.3.2. Simulation Case 2
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
BL | Buttline |
FC | Formation Controller |
H | Hook point |
IMU | Inertial Measurement Unit |
LQR | Linear Quadratic Regulator |
MPC | Model Predictive Control |
PC | Payload Controller |
PID | Proportional–Integral–Derivative |
ST | Stationline |
UAV | Unmanned Aerial Vehicle |
VC | Velocity Controller |
WL | Waterline |
Appendix A. Elements of Matrices A and B
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Parameter | Symbol | Value | Units |
---|---|---|---|
Multirotor | |||
Mass | m | 70 | kg |
Center of gravity position | 0 | m | |
–0.15 | m | ||
Moments of inertia | 10.61 | kg m2 | |
10.31 | kg m2 | ||
19.74 | kg m2 | ||
0.037 | kg m2 | ||
–0.043 | kg m2 | ||
–0.003 | kg m2 | ||
Center of pressure position | 0 | m | |
–0.125 | m | ||
Frame drag areas | 0.22 | m2 | |
1.03 | m2 | ||
Propeller | |||
Number of blades | 2 | ||
Radius | R | 0.5 | m |
Mean aerodynamic chord | 0.086 | m | |
Chord @ | 0.103 | m | |
Lift-curve slope | a | 5.9 | rad−1 |
Pre-cone angle | 0 | rad | |
Root pitch angle | 0.7854 | rad | |
Total twist | –0.6981 | rad | |
Load | |||
Mass | 100 | kg | |
Reference area | 0.785 | m2 | |
Drag coefficient (sphere) | 0.5 | ||
Cable | |||
Nominal length | L | 15 | m |
Hooke’s constant | K | 90,950 | N m−1 |
Damping coefficient | C | 215 | N m−1 s |
Hook point position | 0 | m | |
–0.2 | m |
Subsystem | FC | FC+PC |
---|---|---|
Index | FC | FC + PC | Variation [%] |
---|---|---|---|
[s] | 23.5 | 17.9 | –23.8 |
[deg s] | 22.96 | 15.95 | –30.5 |
[deg] | 0.98 | 0.89 | –9.2 |
[kJ] | 327.05 | 249.31 | –23.8 |
Index | FC | FC + PC | Variation [%] |
---|---|---|---|
[s] | 38.1 | 31.0 | –18.6 |
[deg s] | 26.90 | 15.62 | –41.9 |
[deg] | 0.71 | 0.50 | –28.7 |
[kJ] | 530.53 | 431.77 | –18.61 |
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Costantini, E.; de Angelis, E.L.; Giulietti, F. Cooperative Drone Transportation of a Cable-Suspended Load: Dynamics and Control. Drones 2024, 8, 434. https://doi.org/10.3390/drones8090434
Costantini E, de Angelis EL, Giulietti F. Cooperative Drone Transportation of a Cable-Suspended Load: Dynamics and Control. Drones. 2024; 8(9):434. https://doi.org/10.3390/drones8090434
Chicago/Turabian StyleCostantini, Elia, Emanuele Luigi de Angelis, and Fabrizio Giulietti. 2024. "Cooperative Drone Transportation of a Cable-Suspended Load: Dynamics and Control" Drones 8, no. 9: 434. https://doi.org/10.3390/drones8090434