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Analytical Model and Control Solutions for Unmanned Aerial Vehicle Maneuvers in a Vertical Plane

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Abstract

In this study, we present a class of nonlinear analytical solutions for the dynamics of a fixed wing unmanned aircraft vehicle (UAV). These solutions are needed for the integration and fusion of sensor data for input to guidance and control algorithms. Derivation and integration of the 3-rd order vector differential equation of motion, and its applications to various dynamical models are presented. It is assumed that (a) acceleration due to aerodynamic lift, and the difference between the propulsive thrust and aerodynamic drag accelerations are not changed; (b) the bank angle is zero; (c) the sideslip angle is zero. The general integral and the corresponding analytical solutions for a class of flight trajectories consist of six independent integrals for heading angle, magnitude of velocity vector, time, altitude, and two components of the position vector. This explicit expression with respect to the governing parameters facilitates its direct incorporation into the development and design of trajectories, targeting, guidance and control schemes. It is shown that the first integrals which have been shown valid for a variety of aircraft platforms, re-entry vehicles and missiles, can specifically be applied to UAVs in which such control solutions are needed for sense and avoid situations. An illustrative example highlights the applicability of the general integral for range of trajectories and conditions pertinent to UAV flight patterns.

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Abbreviations

C L , C D :

Aerodynamic lift and drag coefficients

D, L :

Aerodynamic drag and lift

\(\textbf {T}_{F}^{E}\) :

Transformation matrix from F-frame to E-frame

T :

Thrust

W :

Weight

a T , a D , a L :

Propulsive thrust, aerodynamic drag and lift accelerations

H 1 :

Difference between thrust component and aerodynamic drag accelerations, [m/s 2]

H 2 :

Acceleration due to thrust component and aerodynamic lift, [m/s 2]

\(\textbf {e}, \textbf {e}_{i}^{F} ~(i=1,2,3)\) :

Unit vectors of F-frame

j :

Jerk vector

h :

Vertical coordinate or altitude, [m]

g, g 0 :

Gravitational acceleration vector and its magnitude, [m/s 2]

m :

Mass, [k g]

c i , i = 1, ... , 6:

Integration constants

r E :

Position vector in Earth centered inertial frame

t :

Time, [s]

v :

Magnitude of velocity vector, [m/s]

β, ϕ, ψ :

Sideslip, bank and heading (velocity yaw) angles respectively, [r a d]

β m :

Ballistic coefficient

ξ, η :

Horizontal coordinates, [m]

γ :

Flight path angle, [r a d]

ω :

Angular velocity vector

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Authors and Affiliations

  1. 2540 Dole Street, Holmes 202A, Honolulu, HI, 96822, USA

    Dilmurat Azimov

  2. 2540 Dole Street, Holmes 302, Honolulu, HI, 96822, USA

    John Allen

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  1. Dilmurat Azimov
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  2. John Allen
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Correspondence to Dilmurat Azimov.

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Azimov, D., Allen, J. Analytical Model and Control Solutions for Unmanned Aerial Vehicle Maneuvers in a Vertical Plane. J Intell Robot Syst 91, 725–733 (2018). https://doi.org/10.1007/s10846-017-0669-4

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  • DOI: https://doi.org/10.1007/s10846-017-0669-4

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