Abstract :

In this study, we develop a shape formation controller based on regular distance-based control law. As distance rigidity theory may not be successful in achieving desired shape when the control system defined under minimally rigid graph. As a matter of fact, the formation may converge to ambiguous framework when the group of mobile robots did not initialize in proper way. This is a well-known issue with distance-based formation control while there are multiple equilibrium points in the dynamics. We introduce a new controller by adding virtual constraints to the system. The performance of control system is exemplified through numerical simulations and the convergence to the desired formation for all initial conditions validated with stability analysis.

---

Keywords :

Formation control, Multi-agent systems, Networked systems.

---

References :

1. Anderson, B.D.O., Yu, C., Fidan, B., and Hendrickx, J.M. (2008). Rigid graph control architectures for autonomous formations. IEEE Control Syst. Mag., vol. 28, no. 6, pp. 48-63.
2. Anderson, B. D. O., Sun, Z., Sugie, T., Azuma, S., and Sakurama, K. (2017). Formation shape control with distance and area constraints. IFAC J. Syst. Control, vol. 1, pp. 2-12.
3. Aryankia, K. and Selmic, R.R., 2022. Formation Control of Nonlinear Multi-Agent Systems Using Three-Layer Neural Networks. arXiv preprint arXiv:2203.04381.
4. Asimow, L. and Roth, B. (1979) The rigidity of graphs II. J. Math. Anal. Appl., vol. 68, no. 1, pp. 171-190.
5. Babazadeh, R. and Selmic, R. (in press). “Distributed Directed Distance-Based Formation Control With Almost Global Asymptotic Stability That Prevents Reflections in 3-D Space”, Automatica.
6. Bereg, S. (2005). Certifying and constructing minimally rigid graphs in the plane,” Proc. Ann. Symp. Comp. Geom., pp. 73-80, Pisa, Italy
7. Cao, Y., Sun, Z., Anderson, B., and Sugie, T. (2019). Almost global convergence for distance-and area constrained hierarchical formations without reflection. Proc. IEEE Intl. Conf. Control Autom., pp. 1534-1539.
8. de Queiroz, M., X. Cai, and Feemster, M. (2019). Formation Control of Multi-Agent Aystems: A Graph Rigidity Approach. Hoboken, NJ: Wiley.
9. Ferreira-Vazquez, E.D., Hernandez-Martinez, E.G., Flores-Godoy, J. J., Fernandez-Anaya, G., and Paniagua-Contro, P. (2016). Distance-based formation control using angular information between robots. J. Intell. & Robotic Syst., vol. 83, no. 3-4, pp. 543-560.
10. Ferreira-Vazquez, E.D., Flores-Godoy, J.J., HernandezMartinez, E.G. and Fernandez-Anaya, G. (2016). Adaptive control of distance-based spatial formations with planar and volume restrictions. Proc. IEEE Conf. Control Appl., pp. 905-910, Buenos Aires, Argentina.
11. Gazi, V. and Passino, K.M. (2011). Swarm Stability and Optimization, Berlin: Springer-Verlag.
12. Habibian S. and Losey, D. P. (2022) “Encouraging Human Interaction With Robot Teams: Legible and Fair Subtask Allocations,” in IEEE Robotics and Automation Letters, vol. 7, no. 3, pp. 6685-6692, doi: 10.1109/LRA.2022.3174264.
13. Izmestiev, I. (2009). Infinitesimal rigidity of frameworks and surfaces. Lectures on Infinitesimal Rigidity, Kyushu University, Japan.
14. Jackson, B. (2007). “Notes on the rigidity of graphs,” Notes of the Levico Conference, Levico Terme, Italy
15. Jing, G. and Wang, L. (2020). Multiagent flocking with angle-based formation shape control. IEEE Trans. Autom. Control, vol. 65, no. 2, pp. 817-823.
16. Krick, L., Broucke, M.E. and Francis, B.A. (2009). Stabilisation of infinitesimally rigid formations of multi-robot networks. Intl. J. Control, vol. 82, no. 3, pp. 423-439.
17. Lan, X., Xu, W., and Wei, Y. (2018). Adaptive 3D distance-based formation control of multiagent systems with unknown leader velocity and coplanar initial positions. Complexity, vol. 2018, article ID 1814653.
18. Liu, T., Fernandez-Kim, V., and de Queiroz, M. (2020). Switching formation shape control with distance+area/angle feedback. Syst. & Control Lett., vol. 135, Article 104598.
19. Liu, L., de Queiroz, M., Zhang, P., and Khaledyan, M. (2021). Further results on the distance and area control of planar formations. Intl. J. Control, vol. 94, no. 3, pp. 767-783.
20. Liu, T. and de Queiroz, M. (2021) “Distance + Angle-Based Control of 2-D Rigid Formations,” in IEEE Transactions on Cybernetics, vol. 51, no. 12, pp. 5969-5978, doi: 10.1109/TCYB.2020.2973592.
21. Liu, T., de Queiroz, M. and Sahebsara, F. (2020). Distance-based planar formation control using orthogonal variables. Proc. IEEE Conf. Control Techn. Appl., pp. 64-69, Montreal, Canada
22. Liu, T. and de Queiroz, M. (2021). An orthogonal basis approach to formation shape control. Automatica, Vol. 129, ISSN 0005-1098.
23. Mehdifar, F., Bechlioulis, C. P., Hendrickx, J. M., & Dimarogonas, D. V. (2021). 2-D Directed Formation Control Based on Bipolar Coordinates. arXiv preprint arXiv:2108.00916.
24. Mesbahi, M. and Egerstedt, M. (2010). Graph Theoretic Methods in Multiagent Networks, Princeton, NJ: Princeton University Press.
25. Oh, K.-K., Park, M.-C., and Ahn, H.-S. (2015). A survey of multi-agent formation control. Automatica, vol. 53, pp. 424-440.
26. Olfati-Saber, R. and Murray, R. M. (2002). Graph rigidity and distributed formation stabilization of multi-vehicle systems. Proc. IEEE Conf. Dec. Control, vol. 3, pp. 2965-2971, Las Vegas, NV.
27. Park, M., Kim H., and Ahn, H. (2017). Rigidity of distance-based formations with additional subtendedangle constraints. Proc. Intl. Conf. Control, Autom., and Syst., pp. 111-116, Jeju, Korea.
28. Sahebsara, F. and de Queiroz, M., Avoiding Flip Ambiguities During Sequentially Grown Formations, IFAC-PapersOnLine, Volume 54, Issue 20, 2021, Pages 657-662, ISSN 2405-8963.
29. Sugie, T., Anderson, B.D., Sun, Z., and Dong, H. (2018). On a hierarchical control strategy for multi-agent formation without reflection. Proc. IEEE Conf. Dec. Control, pp. 2023-2028, Miami Beach, FL.