Robust stochastic stabilization of attitude motion

  • Ehsan SamieiEmail author
  • Morad Nazari
  • Eric A. Butcher
  • Amit K. Sanyal


This study proposes robust stochastic stabilization of rigid body attitude motion within the framework of geometric mechanics, which can globally represent the attitude dynamics model. The system is subject to a stochastic input torque with an unknown variance parameter and an unknown nonlinear diffusion coefficient matrix. Our development starts with introducing a general notion of the stochastic stability in probability within the framework of geometric mechanics. Then, the Morse–Lyapunov (M–L) technique is employed to design a nonlinear continuous stochastic feedback control law. Finally, the asymptotic stability of the system is guaranteed in probability and the control gain parameters are obtained via solving a linear matrix inequality feasibility problem. An estimate of the region of attraction of the system is calculated to provide a better insight for tuning the control gain parameters. Two illustrative examples are performed based on the discretized model of the closed-loop system to demonstrate the effectiveness of the proposed control scheme.


Attitude motion Stochastic stabilization Morse–Lyapunov function Region of attraction Geometric mechanics 



Financial support from the National Science Foundation under Grant No. CMMI-1131646 is gratefully acknowledged.


  1. 1.
    Haas E (2002) Aeronautical channel modeling. IEEE Trans Veh Technol 51(2):254–264MathSciNetCrossRefGoogle Scholar
  2. 2.
    Li M, Zhou X, Rouphail N (2011) Quantifying benefits of traffic information provision under stochastic demand and capacity conditions: a multi-day traffic equilibrium approach. In: 2011 14th international IEEE conference on intelligent transportation systems (ITSC), pp 2118–2123Google Scholar
  3. 3.
    Primak S, Kontorovitch V, Lyandres V (2005) Stochastic methods and their applications to communications: stochastic differential equations approach. Wiley, ChichesterGoogle Scholar
  4. 4.
    Carden EP, Mita A (2011) Challenges in developing confidence intervals on modal parameters estimated for large civil infrastructure with stochastic subspace identification. Struct Control Health Monit 18(1):53–78Google Scholar
  5. 5.
    Jokipii JR, Parker EN (1969) Stochastic aspects of magnetic lines of force with application to cosmic-ray propagation. Astrophys J 155:777–798CrossRefGoogle Scholar
  6. 6.
    Sanyal A, Chaturvedi NA (2008) Almost global robust attitude tracking control of spacecraft in gravity. In: AIAA guidance, navigation, and control conferenceGoogle Scholar
  7. 7.
    Zhang R, Quan Q, Cai K-Y (2011) Attitude control of a quadrotor aircraft subject to a class of time-varying disturbances. IET Control Theory Appl 5(9):1140–1146MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chen Z, Huang J (2009) Attitude tracking and disturbance rejection of rigid spacecraft by adaptive control. IEEE Trans Autom Control 54:600–605MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sidi M (1997) Spacecraft dynamics and control: a practical engineering approach. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  10. 10.
    Samiei E, Izadi M, Sanyal AK, Butcher EA (2015) Robust stabilization of rigid body attitude motion in the presence of a stochastic input torque. In: 2015 IEEE international conference on robotics and automation, May 26–30, Seattle, WAGoogle Scholar
  11. 11.
    Samiei E, Sanyal AK, Butcher EA (2016) Almost global stochastic stabilization of attitude motion with unknown multiplicative diffusion coefficient. American Institute of Aeronautics and Astronautics Inc, AIAA, RestonCrossRefGoogle Scholar
  12. 12.
    Milnor J W (1963) Morse theory, No. 51. Princeton University Press, PrincetonGoogle Scholar
  13. 13.
    Moulay E (2011) Morse theory and Lyapunov stability on manifolds. J Math Sci 177(3):419–425MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Shucter MD (1993) A survey of attitude representations. J Astronaut Sci 41(4):439–517MathSciNetGoogle Scholar
  15. 15.
    Schaub H, Junkins JL (1996) Stereographic orientation parameters for attitude dynamics: a generalization of the Rodrigues parameters. J Astronaut Sci 44(1):1–19MathSciNetGoogle Scholar
  16. 16.
    Koditschek DE (1989) The application of total energy as a Lyapunov function for mechanical control systems. Contemp Math 97:131MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chaturvedi NA, McClamroch NH (2006) Almost global attitude stabilization of an orbiting satellite including gravity gradient and control saturation effects. In: 2006 American control conference, IEEE, pp 6Google Scholar
  18. 18.
    Khalil H K (2002) Nonlinear Systems, 3rd edn. Prentice Hall, Upper Saddle RiverzbMATHGoogle Scholar
  19. 19.
    Chaturvedi N, Sanyal A, McClamroch N (2011) Rigid-body attitude control. IEEE Control Syst 31(3):30–51MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Deng H, Krstic M, Williams RJ (2001) Stabilization of stochastic nonlinear systems driven by noise of unknown covariance. IEEE Trans Autom Control 46(8):1237–1253MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jazwinski AH (2007) Stochastic processes and filtering theory. Dover Publications, New YorkzbMATHGoogle Scholar
  22. 22.
    Samiei E, Torkamani S, Butcher EA (2013) On Lyapunov stability of scalar stochastic time-delayed systems. Int J Dyn Control 1(1):64–80CrossRefGoogle Scholar
  23. 23.
    Ibrahim RA (2007) Parametric random vibration. Dover Publication Inc, New YorkzbMATHGoogle Scholar
  24. 24.
    Lee T (2011) Geometric tracking control of the attitude dynamics of a rigid body on SO (3). In: American control conference (ACC), 2011, IEEE, pp 1200–1205Google Scholar
  25. 25.
    Nordkvist N, Sanyal AK (2010) A Lie group variational integrator for rigid body motion in SE(3) with applications to underwater vehicle dynamics. In: 2010 49th IEEE conference on decision and control (CDC), IEEE, pp 5414–5419Google Scholar
  26. 26.
    Bou-Rabee N, Owhadi H (2009) Stochastic variational integrators. IMA J Numer Anal 29(2):421–443MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schaub H, Junkins JL (2009) Analytical mechanics of space systems. AIAA, RestonzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ehsan Samiei
    • 1
    Email author
  • Morad Nazari
    • 2
  • Eric A. Butcher
    • 3
  • Amit K. Sanyal
    • 4
  1. 1.Department of Mechanical and Aerospace EngineeringNew Mexico State UniversityLas CrucesUSA
  2. 2.Department of Aerospace EngineeringEmbry-Riddle Aeronautical UniversityDaytona BeachUSA
  3. 3.Aerospace and Mechanical Engineering DepartmentUniversity of ArizonaTucsonUSA
  4. 4.Department of Mechanical and Aerospace EngineeringSyracuse UniversitySyracuseUSA

Personalised recommendations