# American Institute of Mathematical Sciences

doi: 10.3934/jgm.2021005
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 1 Department of Mathematics, University of Hawaii at Manoa, 2565 McCarthy Mall, Honolulu, Hawaii 96822, USA 2 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

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.

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
##### References:
 [1] W. Anderson Jr. and T. Morley, Eigenvalues of the Laplacian of a graph, Linear and Multilinear Algebra, 18 (1985), 141-145.  doi: 10.1080/03081088508817681.  Google Scholar [2] AUVs on-board capability for image analysis for real time situation assessment, https://www.youtube.com/watch?v=qo0VWWvTgdU Google Scholar [3] A. Bloch, Nonholonomic Mechanics and Control, $2^{nd}$ edition, Springer-Verlag, New York, 2015. doi: 10.1007/978-1-4939-3017-3.  Google Scholar [4] A. Bloch, I. Hussein, M. Leok and A. Sanyal, Geometric structure-preserving optimal control of the rigid body, Journal of Dynamical and Control Systems, 15 (2009), 307-330.  doi: 10.1007/s10883-009-9071-2.  Google Scholar [5] A. Bloch, P. Krishnaprasad, J. Marsden and G. Sanchez De Alvarez, Stabilization of rigid body dynamics by internal and external torques, Automatica, 28 (1992), 745-756.  doi: 10.1016/0005-1098(92)90034-D.  Google Scholar [6] A. Bloch, N. Leonard and J. Marsden, Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. Automat. Control, 45 (2000), 2253-2270.  doi: 10.1109/9.895562.  Google Scholar [7] H. Bouadi, M. Bouchoucha and and M. Tadjine, Sliding mode control based on backstepping approach for an UAV type-quadrotor, International Journal of Mechanical and Mechatronics Engineering, 1 (2007), 39-44.   Google Scholar [8] H. Bouadi and M. Tadjine, Nonlinear observer design and sliding mode control of four rotors helicopter, International Journal of Aerospace and Mechanical Engineering, 1 (2007), 354-359.   Google Scholar [9] F. Bullo, J. Cortes and and S. Martinez, Distributed Control of Robotic Networks, Princeton University Press, 2009.  doi: 10.1515/9781400831470.  Google Scholar [10] F. Bullo and A. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar [11] Y. Cao, W. Yu, W. Ren and G. Chen, An overview of recent progress in the study of distributed multi-agent coordination, IEEE Transactions on Industrial Informatics, 9 (2013), 427-438.  doi: 10.1109/TII.2012.2219061.  Google Scholar [12] R. Carney, M. Chyba et al., Multi-agents path planning for a swarm of unmanned aerial vehicles, 2020 IEEE International Geoscience and Remote Sensing Symposium, (2020). Google Scholar [13] R. Chandrasekaran, L. Colombo, M. Camarinha, R. Banavar and A. Bloch, Variational collision and obstacle avoidance of multi-agent systems on Riemannian manifolds, 2020 European Control Conference, (2020), 1689–1694. Google Scholar [14] S. Chung, A. Paranjape, P. Dames, S. Shen and V. Kumar, A survey on aerial swarm robotics, IEEE Transactions on Robotics, 34 (2018), 837-855.  doi: 10.1109/TRO.2018.2857475.  Google Scholar [15] M. Chyba, R. Carney, C. Gray and Z. Trimble, Increasing small unmanned aerial system real-time autonomy, 2020 IEEE International Geoscience and Remote Sensing Symposium, (2020). Google Scholar [16] M. Chyba, T. Haberkorn, R. Smith and G. Wilkens, A geometric analysis of trajectory design for underwater vehicles, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 233-262.  doi: 10.3934/dcdsb.2009.11.233.  Google Scholar [17] F. Costa, J. Ueyama, T. Braun, G. Pessin, F. Osório and P. Vargas, The use of unmanned aerial vehicles and wireless sensor network in agricultural applications, IEEE International Symposium on Geoscience and Remote Sensing, (2012). doi: 10.1109/IGARSS.2012.6352477.  Google Scholar [18] P. Doherty and P. Rudol, A UAV search and rescue scenario with human body detection and geolocalization, Australian Conference on Artificial Intelligence, (2007), 1–13. doi: 10.1007/978-3-540-76928-6_1.  Google Scholar [19] M. Erdelj and E. Natalizio, UAV-assisted disaster management: Applications and open issues, 2016 International Conference on Computing, Networking and Communications (ICNC), (2016). doi: 10.1109/ICCNC.2016.7440563.  Google Scholar [20] J. Kim, S. Kim, C. Ju and H. Son, Unmanned aerial vehicles in agriculture: a review of perspective of platform, control, and applications, IEEE Access, 7 (2019). doi: 10.1109/ACCESS.2019.2932119.  Google Scholar [21] S. Knorn, Z. Chen and R. Middleton, Collective control of multiagent systems, IEEE Transactions on Control of Network Systems, 3 (2016), 334-347.  doi: 10.1109/TCNS.2015.2468991.  Google Scholar [22] G. Lafferriere, A. Williams, J. Caughman and J. Veerman, Decentralized control of vehicle formations, Systems & Control Letters, 54 (2005), 899-910.  doi: 10.1016/j.sysconle.2005.02.004.  Google Scholar [23] A. Lewis and R. Murray, Configuration controllability of simple mechanical control systems, SIAM Journal on Control and Optimization, 35 (1997), 766-790.  doi: 10.1137/S0363012995287155.  Google Scholar [24] T. Luukkonen, Modelling and Control of Quadcopter, Independent research project, Aalto University in Espoo, Finland, 2011. Google Scholar [25] M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, 2010.  doi: 10.1515/9781400835355.  Google Scholar [26] M. Mueller and T. D'Andrea, Stability and control of a quadrocopter despite the complete loss of one, two, or three propellers, 2014 IEEE International Conference on Robotics and Automation (ICRA), (2014), 45–52. doi: 10.1109/ICRA.2014.6906588.  Google Scholar [27] I. Okoloko, Path planning for multiple spacecraft using consensus with LMI avoidance constraints, IEEE Aerospace Conference, (2012), 1–8. doi: 10.1109/AERO.2012.6187118.  Google Scholar [28] R. Olfati-Saber, J. Fax and R. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233.  doi: 10.1109/JPROC.2006.887293.  Google Scholar [29] P. Rooney, A. Bloch and C. Rangan, Trees, forests, and stationary states of quantum Lindblad systems, arXiv: 1810.11144, (2018). Google Scholar [30] A. Sanyal, A. Bloch and N. McClamroch, Dynamics of multibody systems in planar motion in a central gravitational field, Dynamical Systems, 19 (2004), 303-343.  doi: 10.1080/14689360412331309160.  Google Scholar [31] R. Smith, M. Chyba, G. Wilkens and C. Catone, A geometrical approach to the motion planning problem for a submerged rigid body, International Journal of Control, 82 (2009), 1641-1656.  doi: 10.1080/00207170802654410.  Google Scholar [32] V. Stepanyan and K. Krishnakumar, Estimation, navigation and control of multi-rotor drones in an urban wind field, AIAA Information Systems-AIAA Infotech @ Aerospace, (2017). doi: 10.2514/6.2017-0670.  Google Scholar [33] S. Waharte and N. Trigoni, Supporting search and rescue operations with UAVs, 2010 International Conference on Emerging Security Technologies, (2010). doi: 10.1109/EST.2010.31.  Google Scholar [34] S. Wich and L. Koh, Conservation Drones: Mapping and Monitoring Biodiversity, Oxford University Press, New York, 2018.   Google Scholar

show all references

##### References:
 [1] W. Anderson Jr. and T. Morley, Eigenvalues of the Laplacian of a graph, Linear and Multilinear Algebra, 18 (1985), 141-145.  doi: 10.1080/03081088508817681.  Google Scholar [2] AUVs on-board capability for image analysis for real time situation assessment, https://www.youtube.com/watch?v=qo0VWWvTgdU Google Scholar [3] A. Bloch, Nonholonomic Mechanics and Control, $2^{nd}$ edition, Springer-Verlag, New York, 2015. doi: 10.1007/978-1-4939-3017-3.  Google Scholar [4] A. Bloch, I. Hussein, M. Leok and A. Sanyal, Geometric structure-preserving optimal control of the rigid body, Journal of Dynamical and Control Systems, 15 (2009), 307-330.  doi: 10.1007/s10883-009-9071-2.  Google Scholar [5] A. Bloch, P. Krishnaprasad, J. Marsden and G. Sanchez De Alvarez, Stabilization of rigid body dynamics by internal and external torques, Automatica, 28 (1992), 745-756.  doi: 10.1016/0005-1098(92)90034-D.  Google Scholar [6] A. Bloch, N. Leonard and J. Marsden, Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. Automat. Control, 45 (2000), 2253-2270.  doi: 10.1109/9.895562.  Google Scholar [7] H. Bouadi, M. Bouchoucha and and M. Tadjine, Sliding mode control based on backstepping approach for an UAV type-quadrotor, International Journal of Mechanical and Mechatronics Engineering, 1 (2007), 39-44.   Google Scholar [8] H. Bouadi and M. Tadjine, Nonlinear observer design and sliding mode control of four rotors helicopter, International Journal of Aerospace and Mechanical Engineering, 1 (2007), 354-359.   Google Scholar [9] F. Bullo, J. Cortes and and S. Martinez, Distributed Control of Robotic Networks, Princeton University Press, 2009.  doi: 10.1515/9781400831470.  Google Scholar [10] F. Bullo and A. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar [11] Y. Cao, W. Yu, W. Ren and G. Chen, An overview of recent progress in the study of distributed multi-agent coordination, IEEE Transactions on Industrial Informatics, 9 (2013), 427-438.  doi: 10.1109/TII.2012.2219061.  Google Scholar [12] R. Carney, M. Chyba et al., Multi-agents path planning for a swarm of unmanned aerial vehicles, 2020 IEEE International Geoscience and Remote Sensing Symposium, (2020). Google Scholar [13] R. Chandrasekaran, L. Colombo, M. Camarinha, R. Banavar and A. Bloch, Variational collision and obstacle avoidance of multi-agent systems on Riemannian manifolds, 2020 European Control Conference, (2020), 1689–1694. Google Scholar [14] S. Chung, A. Paranjape, P. Dames, S. Shen and V. Kumar, A survey on aerial swarm robotics, IEEE Transactions on Robotics, 34 (2018), 837-855.  doi: 10.1109/TRO.2018.2857475.  Google Scholar [15] M. Chyba, R. Carney, C. Gray and Z. Trimble, Increasing small unmanned aerial system real-time autonomy, 2020 IEEE International Geoscience and Remote Sensing Symposium, (2020). Google Scholar [16] M. Chyba, T. Haberkorn, R. Smith and G. Wilkens, A geometric analysis of trajectory design for underwater vehicles, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 233-262.  doi: 10.3934/dcdsb.2009.11.233.  Google Scholar [17] F. Costa, J. Ueyama, T. Braun, G. Pessin, F. Osório and P. Vargas, The use of unmanned aerial vehicles and wireless sensor network in agricultural applications, IEEE International Symposium on Geoscience and Remote Sensing, (2012). doi: 10.1109/IGARSS.2012.6352477.  Google Scholar [18] P. Doherty and P. Rudol, A UAV search and rescue scenario with human body detection and geolocalization, Australian Conference on Artificial Intelligence, (2007), 1–13. doi: 10.1007/978-3-540-76928-6_1.  Google Scholar [19] M. Erdelj and E. Natalizio, UAV-assisted disaster management: Applications and open issues, 2016 International Conference on Computing, Networking and Communications (ICNC), (2016). doi: 10.1109/ICCNC.2016.7440563.  Google Scholar [20] J. Kim, S. Kim, C. Ju and H. Son, Unmanned aerial vehicles in agriculture: a review of perspective of platform, control, and applications, IEEE Access, 7 (2019). doi: 10.1109/ACCESS.2019.2932119.  Google Scholar [21] S. Knorn, Z. Chen and R. Middleton, Collective control of multiagent systems, IEEE Transactions on Control of Network Systems, 3 (2016), 334-347.  doi: 10.1109/TCNS.2015.2468991.  Google Scholar [22] G. Lafferriere, A. Williams, J. Caughman and J. Veerman, Decentralized control of vehicle formations, Systems & Control Letters, 54 (2005), 899-910.  doi: 10.1016/j.sysconle.2005.02.004.  Google Scholar [23] A. Lewis and R. Murray, Configuration controllability of simple mechanical control systems, SIAM Journal on Control and Optimization, 35 (1997), 766-790.  doi: 10.1137/S0363012995287155.  Google Scholar [24] T. Luukkonen, Modelling and Control of Quadcopter, Independent research project, Aalto University in Espoo, Finland, 2011. Google Scholar [25] M. Mesbahi and M. Egerstedt, Graph Theoretic Methods in Multiagent Networks, Princeton University Press, 2010.  doi: 10.1515/9781400835355.  Google Scholar [26] M. Mueller and T. D'Andrea, Stability and control of a quadrocopter despite the complete loss of one, two, or three propellers, 2014 IEEE International Conference on Robotics and Automation (ICRA), (2014), 45–52. doi: 10.1109/ICRA.2014.6906588.  Google Scholar [27] I. Okoloko, Path planning for multiple spacecraft using consensus with LMI avoidance constraints, IEEE Aerospace Conference, (2012), 1–8. doi: 10.1109/AERO.2012.6187118.  Google Scholar [28] R. Olfati-Saber, J. Fax and R. Murray, Consensus and cooperation in networked multi-agent systems, Proceedings of the IEEE, 95 (2007), 215-233.  doi: 10.1109/JPROC.2006.887293.  Google Scholar [29] P. Rooney, A. Bloch and C. Rangan, Trees, forests, and stationary states of quantum Lindblad systems, arXiv: 1810.11144, (2018). Google Scholar [30] A. Sanyal, A. Bloch and N. McClamroch, Dynamics of multibody systems in planar motion in a central gravitational field, Dynamical Systems, 19 (2004), 303-343.  doi: 10.1080/14689360412331309160.  Google Scholar [31] R. Smith, M. Chyba, G. Wilkens and C. Catone, A geometrical approach to the motion planning problem for a submerged rigid body, International Journal of Control, 82 (2009), 1641-1656.  doi: 10.1080/00207170802654410.  Google Scholar [32] V. Stepanyan and K. Krishnakumar, Estimation, navigation and control of multi-rotor drones in an urban wind field, AIAA Information Systems-AIAA Infotech @ Aerospace, (2017). doi: 10.2514/6.2017-0670.  Google Scholar [33] S. Waharte and N. Trigoni, Supporting search and rescue operations with UAVs, 2010 International Conference on Emerging Security Technologies, (2010). doi: 10.1109/EST.2010.31.  Google Scholar [34] S. Wich and L. Koh, Conservation Drones: Mapping and Monitoring Biodiversity, Oxford University Press, New York, 2018.   Google Scholar
Rendezvous Missions with Unweighted Network. Displays two different communication network scenarios for a 4-agent rendezvous mission. Agreement positions coincide but trajectories differ
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
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
. 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: $13.060>1$
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
. 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
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
. 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
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
The yaw $\psi$ over time for each drone. Total flight times differ for each drone
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
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
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 $K_d$ $3.2320\cdot 10^{-7}$
 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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