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A controllability perspective of dynamic soaring

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Abstract

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.

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Abbreviations

AR:

Aspect ratio of the wing

b :

Wing span

C :

Controllability matrix

\(C_{\mathrm{L}}\) :

Lift coefficient

\(C_{\mathrm{D}}\) :

Drag coefficient

\(C_{\mathrm{D}_{0}}\) :

Zero lift drag coefficient

e :

Oswald efficiency factor

\(f({\varvec{x}})\) :

Drift vector

g :

Acceleration due to gravity

\(g_j({\varvec{x}})\) :

Control input vector field

K :

Aerodynamic coefficient

LARC:

Lie algebraic rank condition

m :

Mass of the vehicle

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

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

S :

Wing planform area

V :

True air speed

\(V_\mathrm{w}\) :

Wind velocity

x :

Position vector along east direction

y :

Position vector along north direction

z :

Altitude

\(\alpha \) :

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

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Author information

Authors and Affiliations

  1. Research Center for Modeling and Simulation, National University of Sciences and Technology, H-12 Campus, Islamabad, Pakistan

    Imran Mir & Adnan Maqsood

  2. Mechanical and Aerospace Engineering Department, University of California, Irvine, USA

    Haitham Taha & Sameh A. Eisa

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  1. Imran Mir
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Correspondence to Haitham Taha.

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Appendix A: Lie brackets

Appendix A: Lie brackets

In this appendix, we show some of the Lie brackets that completes the LARC (specified by Eq. (30)) between the vector fields \({\varvec{f}}\) and \({\varvec{g}}_p\) and \({\varvec{g}}_q\) under linear wind shear conditions.

$$\begin{aligned}&{[}{\varvec{g}}_p,{\varvec{g}}_q{]}= \begin{bmatrix} 0\\ 0\\ \frac{\cos \phi }{\cos \theta }\\ 0\\ 0\\ 0\\ \cos \phi \tan \theta \\ -\sin \phi \\ \end{bmatrix}, \;\;\\&{[}{\varvec{f}},{\varvec{g}}_p{]}= \begin{bmatrix} 0\\ \frac{C_{\mathrm{L}_{a}} \rho S V \sin \phi (\alpha _{0L} + \gamma - \theta )}{2m}\\ \frac{C_{\mathrm{L}_{a}} \rho S V \cos \phi (\alpha _{0L} + \gamma -\theta )}{2m\cos \gamma }\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix}, \;\;\\&{[}{\varvec{g}}_p,[{\varvec{f}},{\varvec{g}}_p]{]}= \begin{bmatrix} 0\\ -\frac{C_{\mathrm{L}_a} \rho S V \cos \phi (\alpha _{0L} + \gamma - \theta )}{2m}\\ -\frac{(C_{\mathrm{L}_a} \rho S V \sin \phi (\alpha _{0L} + \gamma - \theta )}{2m\cos \gamma }\\ 0\\ 0\\ 0\\ 0\\ 0\\ \end{bmatrix},\\ \end{aligned}$$
$$\begin{aligned}&{[}{\varvec{f}},{\varvec{g}}_q{]} \begin{bmatrix} \frac{\beta V \cos \gamma \cos \psi \sin \gamma \sin \phi }{\cos \theta }- \frac{C_{\mathrm{L}_{a}}^2 K \rho S V^2 \cos \phi (2 \alpha _{0L} + 2 \gamma - 2\theta )}{2m}\\ -\frac{\beta \cos \psi \sin ^2\gamma \sin \phi }{\cos \theta }- \frac{C_{\mathrm{L}_{a}}\rho S V \cos ^2\phi }{2m} -\frac{C_{\mathrm{L}_{a}} \rho S V \sin ^2\phi \tan \theta (\alpha _{0L} + \gamma - \theta )}{2m} \\ \frac{C_{\mathrm{L}_{a}}\rho S V \cos \phi \sin \phi \tan \theta (\alpha _{0L} + \gamma - \theta )}{2 m \cos \gamma }-\frac{C_{\mathrm{L}_{a}}\rho S V \cos \phi \sin \phi }{2 m\cos \gamma }- \frac{\beta \sin \gamma \sin \phi \sin \psi }{\cos \gamma \cos \theta }\\ \frac{V \cos \gamma \cos \psi \sin \phi }{\cos \theta }\\ \frac{V \cos \gamma \sin \phi \sin \psi }{\cos \theta }\\ 0\\ 0\\ 0 \end{bmatrix}. \end{aligned}$$

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Mir, I., Taha, H., Eisa, S.A. et al. A controllability perspective of dynamic soaring. Nonlinear Dyn 94, 2347–2362 (2018). https://doi.org/10.1007/s11071-018-4493-6

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