Nonlinear Dynamics

, Volume 88, Issue 4, pp 2651–2669 | Cite as

Geometric control formulation and nonlinear controllability of airplane flight dynamics

  • Ahmed M. Hassan
  • Haithem E. Taha
Original Paper


Linear controllability conditions for linearized systems are not necessary. That is, there exists a class of nonlinear systems that are linearly uncontrollable but nonlinearly controllable. Geometric control theory provides useful tools for analyzing nonlinear controllability of dynamical systems. In particular, it allows for identification of the ability to generate motions along unactuated (non-intuitive) directions through specific interactions between the system dynamics and control inputs. In this work, the six-degrees-of-freedom, rigid-airplane flight dynamics is considered and formulated in a geometric control framework. Then, nonlinear controllability analysis is performed. The analytical tools of geometric control theory allowed scrutiny of the system dynamics to assess the relation between airplane configuration and controllability. Moreover, new rolling and pitching mechanisms that can be exploited at high angles of attack (e.g., stall recovery) are identified. Finally, a thrust-only flight control system is analyzed in this framework.


Flight dynamics Geometric nonlinear control Nonlinear controllability Thrust-only flight control systems 

List of symbols


Velocity components in the body frame


True airspeed equals \(\sqrt{U^2{+}V^2{+}W^2}\)


Angular velocity components in the body frame

\(\phi , \theta , \psi \)

Euler angles defining body frame with respect to an inertial frame


Position coordinate in the east direction in the inertial frame of reference


Position coordinate in the north direction in the inertial frame of reference




Gravitational acceleration


Aircraft mass

\(F_X, F_Y, F_Z\)

Components of the aerodynamic force vector in the body frame

\({\mathcal {L}}, M, N\)

Components of the aerodynamic moment vector in the body frame


Mean aerodynamic chord


Wing span


Wing area

\(\delta _{\mathrm{e}}, \delta _{\mathrm{a}}, \delta _{\mathrm{r}}, \delta _{\mathrm{t}}\)

Elevator, aileron, rudder, and throttle deflections, respectively


Dynamic pressure

\(\alpha \)

Angle of attack

\(\beta \)

Sideslip angle

\(J_X, J_Y, J_Z, J_{XZ} \)

Components of the inertia matrix in the body frame


  1. 1.
    Bianchini, R.M., Stefani, G.: Graded approximations and controllability along a trajectory. SIAM J. Control Optim. 28(4), 903–924 (1990)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Blajer, W., Kołodziejczyk, K.: A geometric approach to solving problems of control constraints: theory and a DAE framework. Multibody Syst. Dyn. 11(4), 343–364 (2004)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blajer, W., Seifried, R., Kołodziejczyk, K.: Servo-constraint realization for underactuated mechanical systems. Arch. Appl. Mech. 85(9–10), 1191–1207 (2015)CrossRefGoogle Scholar
  4. 4.
    Brunovsky, P.: Local controllability of odd systems. Math. Control Theory 1, 39–45 (1974)Google Scholar
  5. 5.
    Burcham, F.W., Burken, J., Maine, T.A.: Flight testing a propulsion-controlled aircraft emergency flight control system on an F-15 airplane. National Aeronautics and Space Administration, Office of Management, Scientific and Technical Information Program (1994)Google Scholar
  6. 6.
    Burcham, F.W., Maine, T.A., Burken, J.J.: Development and flight test of an augmented thrust-only flight control system on an MD-11 transport airplane. National Aeronautics and Space Administration, Office of Management, Scientific and Technical Information Program (1996)Google Scholar
  7. 7.
    Crouch, P.E.: Spacecraft attitude control and stabilization: applications of geometric control theory to rigid body models. IEEE Trans. Autom. Control 29(4), 321–331 (1984)CrossRefMATHGoogle Scholar
  8. 8.
    Crouch, P.E., Byrnes, C.I.: Local accessibility, local reachability, and representations of compact groups. Math. Syst. Theory 19(1), 43–65 (1986)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fumagalli, A., Masarati, P., Morandini, M., Mantegazza, P.: Control constraint realization for multibody systems. J. Comput. Nonlinear Dyn. 6(1), 011,002 (2011)CrossRefGoogle Scholar
  10. 10.
    Hassan, A.M., Taha, H.E.: Airplane loss of control problem: linear controllability analysis. Aerosp. Sci. Technol. 55, 264–271 (2016)CrossRefGoogle Scholar
  11. 11.
    Hermes, H.: On local controllability. SIAM J. Control Optim. 20(2), 211–220 (1982)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Jurdjevic, V.: Polynomial control systems. In: Proceedings of 22nd IEEE CDC, vol. 2, pp. 904–906 (1983)Google Scholar
  13. 13.
    Kalman, R.E., Ho, Y.C., N, K.: Controllability of linear dynamical systems. Contrib. Differ. Equ. 1, 189–213 (1963)MathSciNetGoogle Scholar
  14. 14.
    Kwatny, H.G., Dongmo, J.E.T., Chang, B.C., Bajpai, G., Yasar, M., Belcastro, C.: Nonlinear analysis of aircraft loss of control. J. Guid. Control Dyn. 36(1), 149–162 (2012)CrossRefGoogle Scholar
  15. 15.
    Lemaignan, B.: Flying with no flight controls: handling qualities analyses of the baghdad event. In: AIAA Atmospheric Flight Mechanics Conference, AIAA-2005-5907 (2005)Google Scholar
  16. 16.
    Masarati, P., Morandini, M., Fumagalli, A.: Control constraint of underactuated aerospace systems. J. Comput. Nonlinear Dyn. 9(2), 021,014 (2014)CrossRefGoogle Scholar
  17. 17.
    McLean, D.: Automatic Flight Control Systems. Prentice Hall, Englewood Cliffs (1990)MATHGoogle Scholar
  18. 18.
    Munthe-Kaas, H., Owren, B.: Computations in a free lie algebra. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 357(1754), 957–981 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Nelson, R.C.: Flight Stability and Automatic Control, 2nd edn. McGraw-Hill, New York (1998)Google Scholar
  20. 20.
    Nelson, R.C., Pelletier, A.: The unsteady aerodynamics of slender wings and aircraft undergoing large amplitude maneuvers. Prog. Aerosp. Sci. 39(2), 185–248 (2003)CrossRefGoogle Scholar
  21. 21.
    Nijmeijer, H., Van der Schaft, A.: Nonlinear Dynamical Control Systems. Springer, Berlin (1990)CrossRefMATHGoogle Scholar
  22. 22.
    Polhamus, E.C.: A concept of the vortex lift of sharp-edge delta wings based on a leading-edge-suction analogy. Technical Report NASA TN D-3767, Langley Research Center, Langley Station, Hampton (1966)Google Scholar
  23. 23.
    Sontag, E.D., Sussmann, H.J.: Nonlinear output feedback design for linear systems with saturating controls. In: Proceedings of the 29th IEEE Conference on Decision and Control, pp. 3414–3416. IEEE (1990)Google Scholar
  24. 24.
    Stevens, B.L., Lewis, F.L.: Aircraft Control and Simulation. Wiley-Interscience, New York (2003)Google Scholar
  25. 25.
    Sussmann, H.J.: A general theorem on local controllability. SIAM J. Control Optim. 25(1), 158–194 (1987)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Taha, H.E.: Aircraft Guidance Along a Prescribed Trajectory Using Inverse Simulation and Model Reference Adaptive Control. Cairo University, Giza (2008)Google Scholar
  27. 27.
    Taha, H.E., Hajj, M.R., Beran, P.S.: State-space representation of the unsteady aerodynamics of flapping flight. Aerosp. Sci. Technol. 34, 1–11 (2014)CrossRefGoogle Scholar
  28. 28.
    Tucker, T.: Touchdown: the development of propulsion controlled aircraft at NASA Dryden. NASA History Office, Office of Policy and Plans, NASA Headquarters (1999)Google Scholar
  29. 29.
    Yamasaki, K., Ichihara, Y., Yasui, H., Hiranuma, S.: A thrust-only flight-control system as a backup for loss of primary flight controls. Mitsubishi Heavy Ind. Tech. Rev. 48(4), 1 (2011)Google Scholar
  30. 30.
    Yan, Z., Taha, H.E., Hajj, M.R.: Geometrically-exact unsteady model for airfoils undergoing large amplitude maneuvers. Aerosp. Sci. Technol. 39, 293–306 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Mechanical and Aerospace EngineeringUniversity of CaliforniaIrvineUSA

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