Nonlinear Dynamics

, Volume 94, Issue 4, pp 2347–2362 | Cite as

A controllability perspective of dynamic soaring

  • Imran Mir
  • Haitham TahaEmail author
  • Sameh A. Eisa
  • Adnan Maqsood
Original Paper


Dynamic soaring is an exquisite flying technique to acquire energy from the atmospheric wind shear. In this study, a geometric nonlinear controllability analysis of an unmanned aerial vehicle (UAV) under dynamic soaring conditions is performed. To achieve such an objective, the state-of-the-art mathematical tools of nonlinear controllability are summarized and presented to an aeronautical engineering audience. The dynamic soaring optimal control problem is then formulated and solved numerically. The controllability of the UAV along the optimal soaring trajectory is analyzed. More importantly, the geometric nonlinear controllability characteristics of generic flight dynamics are analyzed in the presence and absence of wind shear to provide a controllability explanation for the role of wind shear in the physics of dynamic soaring flight. It is found that the wind shear is instrumental in ensuring controllability as it allows the UAV attitude controls (pitch and roll) to play the role of thrust in controlling the flight path angle. The presented analysis represents a controllability-based mathematical proof for the energetics of flight physics.


Dynamic soaring Flight dynamics Optimal control Linear control Nonlinear controllability Geometric nonlinear control 

List of symbols


Aspect ratio of the wing


Wing span


Controllability matrix


Lift coefficient


Drag coefficient


Zero lift drag coefficient


Oswald efficiency factor


Drift vector


Acceleration due to gravity


Control input vector field


Aerodynamic coefficient


Lie algebraic rank condition


Mass of the vehicle

\(R_\Sigma ({\varvec{x}}_0, T)\)

Reachable set from \(x_0\) in exactly time T


Wing planform area


True air speed


Wind velocity


Position vector along east direction


Position vector along north direction



\(\alpha \)

Angle of attack

\(\gamma \)

Flight path angle

\(\Delta \)

Accessibility distribution

\(\theta \)

Pitch angle

\(\rho \)

Density of the air

\(\phi \)

Bank angle

\(\Phi (t,\tau )\)

State transition matrix

\(\psi \)

Azimuth measured clockwise from the y-axis


Compliance with ethical standards

Conflict of interest

The authors of this paper have no conflict of interest to declare.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Imran Mir
    • 1
  • Haitham Taha
    • 2
    Email author
  • Sameh A. Eisa
    • 2
  • Adnan Maqsood
    • 1
  1. 1.Research Center for Modeling and SimulationNational University of Sciences and TechnologyIslamabadPakistan
  2. 2.Mechanical and Aerospace Engineering DepartmentUniversity of CaliforniaIrvineUSA

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