The dual-system VTOL UAV can be modelled as a multi-rotor airframe mounted on a fixed-wing airframe to add the hover and VTOL capabilities to a normal fixed-wing airplane. Assuming rigidity of the airplane, the equations of motion for small air vehicles presented in [11] can be used to model the airplane’s motion in the 3-dimensional space. The motion van be fully described by 12 states, 3 angular velocity components and 3 linear velocity components in the body axis, 3 Euler angles defining the orientation of the body axis w.r.t the earth fixed axis, and 3 components of the position of the airplane w.r.t the earth fixed axis. The rigid body 6 degrees of freedom (6-DOF) set of equations of motion is made up of two groups of equations, the Dynamics equations, and the Kinematics equations.
The Dynamics equations relating applied forces and moments to the time rate of change of linear and angular velocities in the body axis.
\(\dot{V}=\frac{1}{m}F-\omega \times V\) \(\dot{\omega }={I}^{-1}\left(H-\omega \times I\omega \right)\) (1)
The \(F=[{f}_{x},{f}_{y},{f}_{z}{]}^{T}\&H=[L,M,N{]}^{T}\) are the vectors representing the total forces and moments acting on the airplane due to aerodynamic, thrust, and gravity effects which should be calculated from another set of equations then substituted in Eq. (1). \(V=[u,v,w{]}^{T} \& \omega =[p,q,r{]}^{T}\) are the linear and angular velocity vectors in the body axis, \(m\) is the airplane's mass and \(I\) is the inertia tensor.
The Kinematics equations, Eq. (2), relate the velocity and angular rates vectors \(V \& \omega\) in the body axis to the velocity and Euler angles' rates in the inertial frame of reference which describes the orientation of the airplane in the 3-dimensional space. Integrating them yields the position and attitude of the airplane w.r.t the inertial axis.
\(\dot{P}={R}_{1}V\) \(\dot{A}={R}_{2}\omega\) (2)
Where \(P=[x,y,z{]}^{T}\)& \({\rm A}=[\varphi ,\theta ,\psi {]}^{T}\) are the position vector w.r.t the inertial frame of reference and the attitude of the airplane described by the Euler angles, \({R}_{1} \& {R}_{2}\) are the transformation matrices used in Eq. (2) which are functions of the Euler angles as expressed in Eq. (3)
$$\begin{array}{c}{R}_{1}=\left[\begin{array}{ccc}{C}_{\theta }{C}_{\psi }& {S}_{\varphi }{S}_{\theta }{C}_{\psi }-{C}_{\varphi }{S}_{\psi }& {C}_{\varphi }{S}_{\theta }{C}_{\psi }-{S}_{\varphi }{S}_{\psi }\\ {C}_{\theta }{S}_{\psi }& {S}_{\varphi }{S}_{\theta }{S}_{\psi }+{C}_{\varphi }{C}_{\psi }& {C}_{\varphi }{S}_{\theta }{S}_{\psi }-{S}_{\varphi }{C}_{\psi }\\ -{S}_{\theta }& {S}_{\varphi }{C}_{\theta }& {C}_{\varphi }{C}_{\theta }\end{array}\right]\\ {R}_{2}=\left[\begin{array}{ccc}1& \text{s}\text{i}\text{n}\varphi \text{t}\text{a}\text{n}\theta & \text{c}\text{o}\text{s}\varphi \text{t}\text{a}\text{n}\theta \\ 0& \text{c}\text{o}\text{s}\varphi & -\text{s}\text{i}\text{n}\varphi \\ 0& \text{s}\text{i}\text{n}\varphi \text{s}\text{e}\text{c}\theta & \text{c}\text{o}\text{s}\varphi \text{s}\text{e}\text{c}\theta \end{array}\right]\end{array}$$
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Where \({C}_{\theta },{S}_{\varphi }\) are short-hand appreviations of \({cos}\theta ,{sin}\varphi\) and so on.
Eqs. (1) & (2) are the 12 ordinary differential equations describing the change of the 12 states of motion \(x\left(t\right)=[V,\omega ,A,P{]}^{T}\) in response to applied forces and moments on the airplane. The system of equations of motion is solved numerically for the values of the states vector \(x\) at each time step, Runge-Kutta 4th order numerical integration method [12] was used in our simulations.
The total force acting on the airplane is the summation of the aerodynamic, gravity, and thrust forces as given in Eq. (4). The total moments are due to aerodynamics and thrust only assuming the body axis origin is located at the center of gravity CG.
$$\begin{array}{c}F={f}_{a}+{f}_{t}+{f}_{g}\\ H={h}_{a}+{h}_{t}\end{array}$$
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The gravity forces acting on the airplane are function of the airplanes’ mass \(m\), gravitational acceleration g, pitch angle \(\theta\), roll angle \(\varphi\) and is given in Eq. (5)
$${F}_{g}=mg\left[\begin{array}{c}-{sin}\theta \\ {cos}\theta {sin}\varphi \\ {cos}\theta {cos}\varphi \end{array}\right]$$
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The aerodynamic and thrust forces and moments can be calculated as the summation of the contributions of the fixed-wing and the multi-rotor airframes. The aerodynamic and thrust forces and moments of the fixed-wing and the quadrotor are calculated separately and summed together to find the total forces and moments acting on the VTOL UAV, the aerodynamic interaction between the rotors and the fixed-wing airframe is neglected in this study, similar to other earlier studies in the literature [3–4, 10]. For dual-system VTOL UAVs there were no data or mathematical models yet available that allows for calculating the aerodynamic forces and moments of the whole airplane accounting for the aerodynamic effect of the operation of the vertical rotor on the fixed-wing airframe. The dual-system airplane considered in our simulations is the (Aerosonde HQ VTOL UAV built by Textron systems) which is created by augmenting the fixed-wing Aerosonde UAV with 4 vertical rotors to gain the VTOL capabilities.
2.1 Fixed-wing Airframe Model
Porchazka et al [7] performed computational fluid dynamics CFD simulations to estimate the aerodynamic derivatives of the Aerosonde HQ UAV at a certain flight speed and certain running speed of the vertical rotors, however, the derivatives calculated cannot be used through the whole flight envelope with different rotational speeds of the vertical rotors. The mass and inertia parameters in of the Aerosonde HQ UAV calculated in [7] were used in our study with the aerodynamic derivatives of the fixed-wing Aerosonde UAV supplied in [11].
The thrust force \({f}_{{t}_{fw}}\) from the fixed-wing airframe is given in Eq. (6) as follows
$${f}_{{t}_{fw}}=\frac{1}{2}\rho {S}_{prop}{C}_{prop}\left[\begin{array}{c}{\left({K}_{motor}{\delta }_{t}\right)}^{2}-{{V}_{t}}^{2}\\ 0\\ 0\end{array}\right]$$
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Where \(\rho\) is the ambient air density, \({\delta }_{t}\in \left[\text{0,1}\right]\) is the throttle input signal to the forward thrust propeller. Other parameters describing the motor and the propeller are supplied in [11]. The aerodynamic forces and moments from the fixed-wing airframe are given as a function of the airplane's flow conditions, derivatives, and geometric parameters in addition to the control surfaces inputs as in Eq. (7).
$$\begin{array}{c}{F}_{lift}=\frac{1}{2}\rho {V}_{t}^{2}S{C}_{L}\left(\alpha ,q,{\delta }_{e}\right)\\ {{F}_{drag}}_{\text{ }}=\frac{1}{2}\rho {V}_{t}^{2}S{C}_{D}\left(\alpha ,q,{\delta }_{e}\right)\\ {F}_{side}=\frac{1}{2}\rho {V}_{t}^{2}S{C}_{Y}\left(\beta ,p,r,{\delta }_{a},{\delta }_{r}\right)\\ L=\frac{1}{2}\rho {V}_{t}^{2}Sb{C}_{l}\left(\beta ,p,r,{\delta }_{a},{\delta }_{r}\right)\\ M=\frac{1}{2}\rho {V}_{t}^{2}S\stackrel{-}{c}{C}_{m}\left(\alpha ,q,{\delta }_{e}\right)\\ N=\frac{1}{2}\rho {V}_{t}^{2}Sb{C}_{n}\left(\beta ,p,r,{\delta }_{a},{\delta }_{r}\right)\end{array}$$
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Where \({\delta }_{a},{\delta }_{e},{\delta }_{r}\) are the aileron, elevator, and rudder deflections respectively and \(S,c,b\) are the wing's area, chord, and span, respectively. All control surfaces deflections are under the limits \(-2{5}^{\circ }\le {\delta }_{i}\le 2{5}^{\circ }\). [11] includes values of all required coefficients and the parameters needed to calculate the lift and drag coefficients at wide range of angles of attack including pre and post stall angles of attack. The aerodynamic lift and drag forces are transformed to the body axis as in Eq. (8)
$${f}_{{a}_{fw}}=\left[\begin{array}{ccc}{cos}\alpha & 0& -{sin}\alpha \\ 0& 1& 0\\ {sin}\alpha & 0& {cos}\alpha \end{array}\right]\left[\begin{array}{c}\begin{array}{c}-{F}_{drag}\\ {F}_{side}\end{array}\\ -{F}_{lift}\end{array}\right]$$
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The fixed-wing airframe contribution to the moments due to thrust is zero assuming thrust line coincides with the airplane's x-axis, thus the aerodynamic moments is as in Eq. (9)
\({h}_{{a}_{fw}}={\left[\begin{array}{c}L\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}M\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}N\end{array}\right]}^{T}\) \({h}_{{t}_{fw}}={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^{T}\) (9)
2.2 Quadcopter Airframe Model
Quadcopter airframe contributes to the forces and moments exerted on the airplane by the thrust forces of each of the vertical rotors and the moment generated by them due to the rotors’ position relative to the center of gravity (CG) [13]. The thrust \(T\) and torque \(\tau\) created by the \({i}^{th}\) rotor is given by Eq. (10)
\({T}_{i}={K}_{T}{{\omega }_{i}}^{2}\) \({\tau }_{i}={K}_{D}{{\omega }_{i}}^{2}\) (10)
Where \({K}_{T}\&{K}_{D}\) are the coefficients relating the thrust and torque of a rotor, to its propellers rotational speed \(\omega\), as they vary according to the brushless motor used and the propeller, those coefficients are determined experimentally.
Each two opposite rotors have the same rotation directions opposite to the other pair of rotors, the attitude of the vehicle (roll, pitch, and yaw) can be changed by changing the rotation speeds of the rotors. The total thrust and moments exerted by the rotors are given as a function of the rotors’ rotational speeds as in Eq. (11) [13]
$$\left[\begin{array}{c}T\\ {\tau }_{\varphi }\\ {\tau }_{\theta }\\ {\tau }_{\psi }\end{array}\right]=\left[\begin{array}{cccc}{K}_{T}& {K}_{T}& {K}_{T}& {K}_{T}\\ -{K}_{T}{d}_{y}& -{K}_{T}{d}_{y}& {K}_{T}{d}_{y}& {K}_{T}{d}_{y}\\ {K}_{T}{d}_{x}& -{K}_{T}{d}_{x}& -{K}_{T}{d}_{x}& {K}_{T}{d}_{x}\\ {K}_{D}& -{K}_{D}& {K}_{D}& -{K}_{D}\end{array}\right]\left[\begin{array}{c}{{\omega }_{1}}^{2}\\ {{\omega }_{2}}^{2}\\ {{\omega }_{3}}^{2}\\ {{\omega }_{4}}^{2}\end{array}\right]$$
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The parameters \({d}_{x}\&{d}_{y}\) are the relative positions of the vertical rotors w.r.t the airplane CG and are supplied in [7]. The coefficients \({K}_{T}\&{K}_{D}\) of the quad-rotor system of the Aerosonde HQ UAV are not available, however, experimental data of the multi-rotor system of a VTOL UAV with similar mass and size [14] were used in our simulations. The aerodynamic contribution of the quadcopter system is neglected, the contributions of the quad are given in Eq. (12)
\({f}_{{t}_{quad}}={\left[\begin{array}{ccc}0& 0& -T\end{array}\right]}^{T}\) \({h}_{{{t}_{quad}}_{\text{ }}}={\left[\begin{array}{ccc}{\tau }_{\varphi }& {\tau }_{\theta }& {\tau }_{\psi }\end{array}\right]}^{T}\) \({f}_{{{a}_{quad}}_{\text{ }}}={h}_{{a}_{quad}}={\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^{T}\) (12)
The values of total forces & moments acting on the airplane are obtained by adding the contribution of the fixed-wing airframe, calculated from Eq. (6), (8), and (9), and the quadcopter airframe's contribution, calculated from Eq. (12), in addition to the gravity forces of Eq. (5).
2.3 Dual-system VTOL UAV Model
Adding the fixed-wing and quadcopter contributions to find the total forces and moments acting on the airplane we obtain a multi-input multi-output (MIMO) dynamic system describing the UAVs flight motion. The inputs are \(\left[\begin{array}{cc}\begin{array}{ccc}{\delta }_{a}& {\delta }_{e}& {\delta }_{r}\end{array}& \begin{array}{cccc}\begin{array}{cc}{\delta }_{t}& {\varOmega }_{1}\end{array}& {\varOmega }_{2}& {\varOmega }_{3}& {\varOmega }_{4}\end{array}\end{array}\right]\) where \({{\Omega }}_{i}={{\omega }_{i}}^{2}\), the 12 states are \(\left[\begin{array}{cccc}V& \omega & A& P\end{array}\right]\), and the outputs are selected to be the total velocity, angles of attack and sideslip, and climb angle \(\left[\begin{array}{cccc}{V}_{t}& \alpha & \beta & \gamma \end{array}\right]\) .
The redundancy/over-actuation property of the model appears clearly at this point, for example, a pitching motion can be obtained through the fixed-wing airframe contribution by deflecting the elevator or through the quadcopter airframe contribution by changing the rotational speeds of the forward or backward rotors. The same principle applies to other degrees of freedom like roll, pitch, and acceleration along the z-axis.
A simulation model was created in the Simulink/MATLAB environment to simulate the open-loop response of the dual-system VTOL UAV to its inputs. The model was trimmed at different flight speeds using the Linearization toolbox in Simulink to find the equilibrium values for all the inputs and states for a steady-level flight at the stall and above-stall speeds. Table 1 contains the trim points of the VTOL UAV including the trim control actions of elevator and thrust at different flight speeds and angles of attack. Those trim points are used for validation of the performance of the closed-loop system after applying the controller and the control allocation module. It was observed that the control allocation module designed commands the exact trim values of the inputs according to the desired flight speed at the steady-state condition as will be illustrated in the results section.
Table 1
Trim points of steady-level cruising flight at different flight speeds
Vt (m/sec) | α (rad) | δe (rad) | δt |
18 | 0.27605 | -0.25656 | 0.24879 |
20 | 0.20948 | -0.20596 | 0.27225 |
25 | 0.10629 | -0.12754 | 0.33450 |
30 | 0.04987 | -0.08466 | 0.39891 |
33 | 0.02757 | -0.06771 | 0.43800 |
35 | 0.01577 | -0.05874 | 0.46417 |
40 | -0.00638 | -0.041908 | 0.5298 |