Nonlinear Dynamics

, Volume 88, Issue 4, pp 2651–2669

# Geometric control formulation and nonlinear controllability of airplane flight dynamics

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

## Abstract

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.

## Keywords

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

## List of symbols

UVW

Velocity components in the body frame

$$V_\mathrm{T}$$

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

PQR

Angular velocity components in the body frame

$$\phi , \theta , \psi$$

Euler angles defining body frame with respect to an inertial frame

$$P_\mathrm{E}$$

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

$$P_\mathrm{N}$$

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

h

Altitude

g

Gravitational acceleration

m

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

$${\bar{c}}$$

Mean aerodynamic chord

b

Wing span

S

Wing area

$$\delta _{\mathrm{e}}, \delta _{\mathrm{a}}, \delta _{\mathrm{r}}, \delta _{\mathrm{t}}$$

Elevator, aileron, rudder, and throttle deflections, respectively

q

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

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