Flocking Behaviors of a Body Attitude Coordination Model with Velocity Alignment

Abstract

Body attitude coordination plays an important role in multi-airplane synchronization. In this paper, we study the flocking dynamics of a modified model for body attitude coordination. In contrast to the original body attitude alignment models in Degond et al. (Math. Models MethodsAppl. Sci., 27(6):1005-1049, 2017) and Ha et al. (Discrete Contin. Dyn. Syst., 40(4):2037-2060, 2020), we introduce the velocity alignment term and assume the velocity of each agent is variable. More precisely, the adjoint coefficient will vary with the linked individual changes. In this case, synchronization would include the body attitude alignment and velocity alignment. It will generate a new collective behaviour which is called body attitude flocking. As results, we present two sufficient frameworks leading to the body attitude flocking by technique estimates. Also, we show the finite-in-time stability of the system which is valid on any finite time interval. In addition, we formally derive a kinetic model of the model for body attitude coordination using the BBGKY hierarchy. We prove the well-posedness of the kinetic equation and show a rigorous justification for the meanfield limit of our model. Moreover, we present a sufficient condition for asymptotic flocking in the kinetic model. Finally, we also give the numerical simulations to verify our analysis results.

Keywords

Body attitude coordination / flocking / stability / measure valued solutions / mean-field limit

References

[1] G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato, and J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29(10):1901-2005, 2019.

[2] V. Arnold, Mathematical Methods of Classical Mechanics, Springer, 1978.

[3] J. Cañizo, J. Carrillo, and J. Rosado, A well-posedness theory in measures for some kinetic models of collective motion, Math. Models Methods Appl. Sci., 21(3):515-539, 2011.

[4] J. Carrillo, M. Fornasier, J. Rosado, and G. Toscani, Asymptotic flocking dynamics for the kinetic Cucker-Smale model, SIAM J. Math. Anal., 42(1):218-236, 2010.

[5] Z. L. Chen and X. X. Yin, The kinetic Cucker-Smale model: Well-posedness and asymptotic behavior, SIAM J. Math. Anal., 51(5):3819-3853, 2019.

[6] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Trans. Automat. Contr., 52(5):852862, 2007.

[7] P. Degond, A. Diez, and A. Frouvelle, Body-attitude coordination in arbitrary dimension, arXiv: 2111.05614, 2021.

[8] P. Degond, A. Diez, and M.-Y. Na, Bulk topological states in a new collective dynamics model, SIAM J. Appl. Dyn. Syst., 21(2):1455-1494, 2022.

[9] P. Degond, A. Frouvelle, and S. Merino-Aceituno, A new flocking model through body attitude coordination, Math. Models Methods Appl. Sci., 27(6):1005-1049, 2017.

[10] J. G. Dong and X. P. Xue, Synchronization analysis of Kuramoto oscillators, Commun. Math. Sci., 11(2):465-480, 2013.

[11] F. Dörfler, and F. Bullo, Synchronization in complex networks of phase oscillators: A survey, Automatica, 50(6):1539-1564, 2014.

[12] R. C. Fetecau, S. Y. Ha, and H. Park, An intrinsic aggregation model on the special orthogonal group SO(3): Well-posedness and collective behaviours, J. Nonlinear Sci., 31(5):74, 2021.

[13] R. C. Fetecau, S. Y. Ha, and H. Park, SIAM J. Appl. Dyn. Emergent behaviors of rotation matrix flocks, Syst., 21(2):1382-1425, 2022.

[14] F. Golse and S. Y. Ha, A mean-field limit of the Lohe matrix model and emergent dynamics, Arch. Ration. Mech. Anal., 234:1445-1491, 2019.

[15] S. Y. Ha, D. Kim, J. Lee, and N. S. Eun, Emergent dynamics of an orientation flocking model for multi-agent system, Discrete Contin. Dyn. Syst., 40(4):2037-2060, 2020.

[16] S. Y. Ha, J. Kim, and X. T. Zhang, Uniform stability of the Cucker-Smale model and its application to the mean-field limit, Kinet. Relat. Models, 11:1157-1181, 2018.

[17] S. Y. Ha, D. Ko, and S. W. Ryoo, Emergent dynamics of a generalized Lohe model on some class of Lie groups, J. Stat. Phys., 168:171-207, 2017.

[18] S. Y. Ha and J. G. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci., 7(2):297-325, 2009.

[19] S. Y. Ha and S. W. Ryoo, Asymptotic phase-locking dynamics and critical coupling strength for the Kuramoto model, Comm. Math. Phys., 377:811-857, 2020.

[20] S. Y. Ha and E. Tadmor, From particle to kinetic and hydrodynamic descriptions of flocking, Kinet. Relat. Models, 1(3):415-435, 2008.

[21] Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, 1984.

[22] J. R. Lawton and R. W. Beard, Synchronized multiple spacecraft rotations, Automatica, 38(8): 1359-1364, 2002.

[23] Z. C. Li and X. P. Xue, Cucker-Smale flocking under rooted leadership with fixed and switching topologies, SIAM J. Appl. Math., 70(8):3156-3174, 2010.

[24] M. A. Lohe, J. Phys. Non-Abelian Kuramoto models and synchronization, A: Math. Theor., 42(39):395101, 2009.

[25] S. Motsch and E. Tadmor, Heterophilious dynamics enhances consensus, SIAM Rev., 56(4):577621, 2014.

[26] Z. Y. Qiao, Y. C. Liu, and X. Wang, Multi-cluster flocking behavior analysis for a delayed CuckerSmale model with short-range communication weight, J. Syst. Sci. Complex., 35:137-158, 2022.

[27] H. Spohn, Large Scale Dynamics of Interacting Particles, Springer, 2012.

[28] R. Tron, R. Vidal, and A. Terzis, Distributed pose averaging in camera networks via consensus on SE(3), in: 2008 Second ACM/IEEE International Conference on Distributed Smart Cameras, 1-10, 2008.

[29] T. Vicsek, A. Czirók, E. B. Jacob, I. Cohen, and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75(6):1226, 1995.

[30] C. Villani, Optimal Transport: Old and New, Springer, 2009.

[31] X. Y. Wang and X. P. Xue, Formation behaviour of the kinetic Cucker-Smale model with noncompact support, P. Roy. Soc. Edinb. A, 153(4):1315-1346, 2022.