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doi: 10.3934/jgm.2021005
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Multi-agent systems for quadcopters
Richard Carney 1, , Monique Chyba 1,, , Chris Gray 1, , George Wilkens 1, and Corey Shanbrom 2,
Department of Mathematics, University of Hawaii at Manoa, 2565 McCarthy Mall, Honolulu, Hawaii 96822, USA
Department of Mathematics and Statistics, California State University, Sacramento, 6000 J St., Sacramento, CA 95819, USA
* Corresponding author: M. Chyba
Received December 2020 Revised March 2021 Early access April 2021
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Unmanned Aerial Vehicles (UAVs) have been increasingly used in the context of remote sensing missions such as target search and tracking, mapping, or surveillance monitoring. In the first part of our paper we consider agent dynamics, network topologies, and collective behaviors. The objective is to enable multiple UAVs to collaborate toward a common goal, as one would find in a remote sensing setting. An agreement protocol is carried out by the multi-agents using local information, and without external user input. The second part of the paper focuses on the equations of motion for a specific type of UAV, the quadcopter, and expresses them as an affine nonlinear control system. Finally, we illustrate our work with a simulation of an agreement protocol for dynamically sound quadcopters augmenting the particle graph theoretic approach with orientation and a proper dynamics for quadcopters.
Keywords: Quadcopter, rigid body dynamics, graph Laplacian, multi-agent coordination, network consensus, weighted network, decentralized control.
Mathematics Subject Classification: Primary: 70E60; 93A14; Secondary: 70E55.
Citation: Richard Carney, Monique Chyba, Chris Gray, George Wilkens, Corey Shanbrom. Multi-agent systems for quadcopters. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021005
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Figure 1. Rendezvous Missions with Unweighted Network. Displays two different communication network scenarios for a 4-agent rendezvous mission. Agreement positions coincide but trajectories differ
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Fig. 1 corresponding to rendezvous missions with Unweighted Network">Figure 2. Comparison of the $ x, y, z $-motions for agent 1 for the scenarios of Fig. 1 corresponding to rendezvous missions with Unweighted Network
Figure 3. Comparison between trajectories on rendezvous missions with unweighted and weighted networks. The solid curves represents the trajectories for the unweighted network and the dashed ones for the weighted network. The two scenarios converge to the same agreement value
Figure 3. Observe that each component converges much more rapidly for the weighted network, corresponding to the difference in their $ \lambda_2 $ values: $ 13.060>1 $">Figure 4. Comparison of the $ x, y, z $-motions for agent 1 for the rendezvous missions displayed in Figure 3. Observe that each component converges much more rapidly for the weighted network, corresponding to the difference in their $ \lambda_2 $ values: $ 13.060>1 $
Figure 5. Rendezvous missions comparing and unweighted network (solid curves) to a time-varying weighted one (dashed curves). They both agree on the consensus joint value. The points where the dashed curves diverge from the solid ones correspond to the addition of edges
Figure 5. The components of the trajectory corresponding to the time-varying network clearly converge more rapidly">Figure 6. Comparison of the $ x, y, z $-motions for agent 1 for the rendezvous mission of Figure 5. The components of the trajectory corresponding to the time-varying network clearly converge more rapidly
Figure 7. A quadcopter rising straight up, then yawing while hovering, then flying straight along the body $ x $-axis. Top: position in space. Bottom: orientation angles over time
Figure 7. Top: angular velocities of each of the four motors over time. Bottom: total thrust over time">Figure 8. The controls used to produce the motion in Figure 7. Top: angular velocities of each of the four motors over time. Bottom: total thrust over time
Figure 9. Three drones start at position $ (0, 0, 0) $, $ (0, 9, 0) $ and $ (15, 9, 0) $ with initial yaw angles $ 0 $, $ -\pi/4 $, and $ \pi/2 $. Their trajectories to the rendezvous position are shown
Figure 10. The yaw $ \psi $ over time for each drone. Total flight times differ for each drone
Figures 9 and 10. We show the four motor speeds and total thrust for each drone as functions of time">Figure 11. The controls used to produce the motions in Figures 9 and 10. We show the four motor speeds and total thrust for each drone as functions of time
Figure 12. Comparison of two methods of motion planning. Dashed curves include dynamics; solid curves do not. Left: $ x $-coordinate over time. Right: $ y $-coordinate over time
Table 1. Parameters used in simulations
Constant Symbol Value
drone mass $ m $ 0.468
drone inertia $ J $ diag($ (3.8278, 3.8288, 7.6566)\cdot10^{-3} $)
rotor inertia $ J_r $ diag$ (0, 0, 2.8385\cdot 10^{-5}) $
distance to rotor $ d $ 0.25
thrust coefficient $ K_r $ $ 2.9842\cdot 10^{-5} $
translational drag $ C_D $ $ (5.5670, 5.5670, 6.3540)\cdot10^{-4} $
rotational drag $ C_{\tau} $ $ (5.5670, 5.5670, 6.3540)\cdot 10^{-4} $
propeller drag $ K_d $ $ 3.2320\cdot 10^{-7} $
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Richard Carney Monique Chyba Chris Gray George Wilkens Corey Shanbrom
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