Nomenclature

$D,Y,L$

Aerodynamic forces (drag, side force, lift)

${\bf{F}}$

force vector

${\bf{g}}$

gravity vector

${\bf{H}}$

angular momentum vector

h

altitude

$\tilde{I}$

inertia tensor

k

control conversion constant

K

aerodynamic surface-body interference factor

$m,M$

mass of part and total mass

$\bar L,M,N$

aerodynamic moments (in roll, pitch, yaw)

O

reference frame origin

$P,Q,R$

angular velocities (in roll, pitch, yaw)

P

power

R

rotation matrix

$\tilde R$

inertia tensor translation matrix

${\bf{r}}$

position vector

t

time

${\bf{T}}$

torque vector

$T,Q$

propeller thrust and torque

${\bf{U}}$

control input

$U,V,W$

linear velocities

${\bf{v}}$

velocity vector

${V_T}$

flight speed

$x,y,z$

position coordinates

Greek Symbol

$\alpha $

aircraft main body angle-of-attack

$\beta $

aircraft main body side-slip angle

$\delta $

deflection or tilt angle

$\Delta {V_T}$

increment in flight speed at the wing due to canard wake

$\epsilon $

downwash angle

$\varepsilon $

stability cost function

$\lambda $

propeller rotation direction (1 counterclockwise or −1 clockwise)

$\rho $

air density

$\phi ,\theta ,\psi $

Euler angles (roll, pitch, yaw)

$\omega $

angular velocity vector

$\Omega $

angular velocity cross product in matrix form

Subscripts

$aL,aR,w,c$

aerodynamic actuators (left aileron, right aileron, wing, canard)

B

main body reference frame

CB

canard-body interference

e

exposed aerodynamic surface, or equilibrium condition

E

Earth fixed inertial reference frame

$f,e,r$

aerodynamic actuators (flap, elevator, rudder)

i

aerodynamic surface moving part index

j

rotor moving part index

lat

lateral

long

longitudina

m

x or z reference coordinate for pitch moment calculation

pivot

wing or canard pivot point

$RC,LC$

right and left canard

$RW,LW$

right and left wing

$R1,R2,R3,R4$

rotor 1, 2, 3, 4

rel

relative velocity or acceleration

sp

setpoint

th

threshold

WB

wing-body interference

2.0 Aircraft concept

The concept of aircraft studied is a VTOL UAV with the capacity to tilt the wing and the canard, being that the two front propellers are fixed, and the rear propellers tilt along with the wing, is shown in Fig. 1, the hover configuration, and in Fig. 2, the cruise configuration. The aircraft has aerodynamic controls such as flaps, elevator, aileron and rudder. This configuration is somewhat of a hybrid, or combination, between the tilt-wing and lift-plus-cruise configurations, where the stabilising frontal surface (canard) is decoupled from the frontal rotors. The aircraft total weight is 6.44 kg.

Figure 1. Aircraft concept, hover configuration.

Figure 2. Aircraft concept, cruise configuration.

At take-off, landing and hovering, the aircraft should perform like a quadcopter, with the four rotors pointing upwards, and no use of aerodynamic controls to stabilise vehicle attitude. The transition strategy from hover to horizontal flight is that the wing would tilt gradually from the vertical position to the horizontal position so that the rotors in the wing would accelerate the vehicle forward. During hovering, the canard should be at a vertical position to reduce the planform area in the wake of the front rotors. At the start of the transition manoeuvre, it should tilt into a low-drag horizontal position so that it progressively acquires aerodynamic loading to stabilise horizontal flight. In addition, it can be actuated to correct aircraft attitude. Therefore, at horizontal flight, it should perform like a fixed-wing aircraft with canard and attitude and trajectory control given primarily by the aerodynamic control surfaces. Thereby, the transition manoeuvre from horizontal flight to hovering would be just the opposite sequence.

With this configuration, there is the possibility for stable and controllable flight at any flight speed from hovering to horizontal flight at maximum speed. In other words, it should not have stall speed provided the combination of wing, and canard tilt angle is well mapped for trimming the aircraft.

To have reasonable hovering control and performance, the centre of gravity must be at the longitudinal midpoint between the rotors in vertical position so that, in this condition, the four propellers are producing the same thrust. Also, having a canard configuration while in horizontal flight, it is possible to have a positive static margin even with the centre of gravity so far from the aircraft nose.

3.0 Dynamics modelling

The aircraft dynamics model shall be divided into parts of constant mass as shown in Fig. 3, where the origins of the reference frames are at the centre of mass of each component [Reference Daud Filho9, Reference Daud Filho and Belo27]. The wing and canard tilts with respect to the fixed pivot points ${P_W}$ and ${P_C}$ , which are positioned on the one-quarter chord of their exposed root chords, whereas the front rotors have their coordinate frames $({O_{R3}},{O_{R4}})$ fixed with respect to the main body coordinate frame.

Figure 3. Aircraft multi-body reference frames and dynamics model.

3.1 Linear motion

The linear motion equation derivation first step is to define the aircraft dynamic total linear momentum in the Earth fixed inertial reference frame and then derive it with respect to time, which leads to the force equation,

(1) \begin{align}{{\bf{F}}_E} = {m_B}{{\dot{\bf v}}_{{E_B}}} + {m_{LW}}{{\dot{\bf v}}_{{E_{LW}}}} + {m_{RW}}{{\dot{\bf v}}_{{E_{RW}}}} + {m_{LC}}{{\dot{\bf v}}_{{E_{LC}}}} + {m_{RC}}{{\dot{\bf v}}_{{E_{RC}}}} + \sum\limits_{j = 1}^4 {m_{{R_j}}}{{\dot{\bf v}}_{{E_{{R_j}}}}}\end{align}

Now, expanding the equation in terms of the body coordinate frame B fixed at the aircraft main body centre of mass, whose position is not affected by the wing or canard tilt. Thus, the acceleration vector of the main body is defined,

(2) \begin{align}{{\dot{\bf v}}_E} = {{\dot{\bf v}}_B} + {\omega _B} \times {{\bf{v}}_B}\end{align}

Additionally, applying the definition of acceleration of a moving point A with respect to a moving point B [Reference Meriam and Kraige28] to compute the acceleration of the remaining parts,

(3) \begin{align}{{\dot{\bf v}}_{{E_i}}} = {{\dot{\bf v}}_B} + {\omega _B} \times {{\bf{v}}_B} + {\dot \omega _B} \times {{\bf{r}}_{i/B}} + {\omega _B} \times ({\omega _B} \times {{\bf{r}}_{i/B}}) + 2{\omega _B} \times {{\bf{v}}_{re{l_{i/B}}}} + {{\bf{a}}_{re{l_{i/B}}}}\end{align}

Moreover, the position vector of each part with respect to the main body origin is defined by the sum of the position of the pivot point plus the rotating vector from the pivot point to the part,

(4) \begin{align}{{\bf{r}}_{i/B}} = {{\bf{r}}_{pivo{t_i}/B}} + R_B^i{{\bf{r}}_{i/pivo{t_i}}}\end{align}

Thus, taking the time derivative of Equation (4), the relative velocity vector ( ${{\bf{v}}_{re{l_{i/B}}}}$ ) is obtained in Equation (5), and the second time derivative leads to the relative acceleration vector in Equation (6).

(5) \begin{align}{{\bf{v}}_{re{l_{i/B}}}} = \frac{d}{{dt}}({{\bf{r}}_{i/B}}) = \frac{d}{{dt}}\left({{\bf{r}}_{pivo{t_i}/B}} + R_B^i{{\bf{r}}_{i/pivo{t_i}}}\right) = \dot R_B^i{{\bf{r}}_{i/pivo{t_i}}}\end{align}

(6) \begin{align}{{\bf{a}}_{re{l_{i/B}}}} = \frac{d}{{dt}}({{\bf{v}}_{re{l_{i/B}}}}) = \frac{d}{{dt}}\big(\dot R_B^i{{\bf{r}}_{i/pivo{t_i}}}\big) = \ddot R_B^i{{\bf{r}}_{i/pivo{t_i}}}\end{align}

As the rotation of the wing and tail are aligned with the y-axis of the main body coordinate frame, the rotation matrix ( $R_B^i$ ) is defined as shown in the Equation (7), where ${\delta _{w,c}}$ is the wing or canard tilt angle.

(7) \begin{align}R_B^i = \left[ {\begin{array}{c@{\quad}c@{\quad}c}{\cos {\delta _{w,c}}} &0 &{\sin {\delta _{w,c}}}\\[5pt] 0 &1& 0\\[5pt] { -\!\sin {\delta _{w,c}}} &0 &{\cos {\delta _{w,c}}}\end{array}} \right]\end{align}

Furthermore, the applied net force vector in Equation (8) is the sum of the aerodynamic and propulsive forces ( ${{\bf{F}}_B}$ ), the weight of the aircraft main body ( ${m_B}R_B^E{{\bf{g}}_E}$ ), the weight of the left and right-wing and canard $\big(\!\sum\nolimits_{i = 1}^4 {m_i}R_B^E{{\bf{g}}_E}\big)$ and the weight of each rotor $\big(\!\sum\nolimits_{j = 1}^4 {m_j}R_B^E{{\bf{g}}_E}\big)$ .

(8) \begin{align}{{\bf{F}}_E} = {{\bf{F}}_B} + {m_B}R_B^E{{\bf{g}}_E} + \sum\limits_{i = 1}^4 {m_i}R_B^E{{\bf{g}}_E} + \sum\limits_{j = 1}^4 {m_j}R_B^E{{\bf{g}}_E}\end{align}

Now substituting Equations (2), (3), (5), (6) and (8) into Equation (1), making use of the vector cross-product transformation into skew-symmetric matrix representation $({\Omega _B} = {\omega _B} \times )$ and $({\dot \Omega _B} = {\dot \omega _B} \times)$ , summing the individual masses parts $\big( {M = {m_B} + \sum\nolimits_{i = 1}^4 {m_i} + \sum\nolimits_{j = 1}^4 {m_j}} \big)$ and rearranging terms gives the resulting Equations (9) and (10).

(9) \begin{align}{{\dot{\bf v}}_B} = - {\Omega _B}{{\bf{v}}_B} + \frac{{{{\bf{F}}_B}}}{M} + R_B^E{{\bf{g}}_E} - {\bf{f}}\end{align}

(10) \begin{align}{\bf{f}}& = \frac{1}{M}\sum\limits_{i = 1}^4 \big\{ {m_i}\big[\big({{\dot \Omega }_B} + {\Omega _B}{\Omega _B}\big){{\bf{r}}_{i/B}} + \big(2{\Omega _B}\dot R_B^i + \ddot R_B^i\big){{\bf{r}}_{i/pivo{t_i}}}\big]\big\} \nonumber \\[3pt] {}& \quad + \frac{1}{M}\sum\limits_{j = 1}^4 \big\{ {m_j}\big[\big({{\dot \Omega }_B} + {\Omega _B}{\Omega _B}\big){{\bf{r}}_{j/B}} + \big(2{\Omega _B}\dot R_B^j + \ddot R_B^j\big){{\bf{r}}_{j/pivo{t_j}}}\big]\big\} \end{align}

3.2 Angular motion

Derivation of the angular motion equation begins by defining the aircraft total angular momentum,

(11) \begin{align}{{\bf{H}}_B} = {\tilde I_B}{\omega _B} + \sum\limits_{i = 1}^4 \big\{ {\tilde I_{{B_i}}}{\omega _{{B_i}}} + {{\bf{r}}_{i/B}} \times \big({m_i}{{\bf{v}}_{{B_i}}}\big)\big\} + \sum\limits_{j = 1}^4 \big\{ {\tilde I_{{B_j}}}{\omega _{{B_j}}} + {{\bf{r}}_{j/B}} \times \big({m_j}{{\bf{v}}_{{B_j}}}\big)\big\} \end{align}

So, the velocity vector of each part in the main body coordinate frame is defined,

(12) \begin{align}{{\bf{v}}_{{B_i}}} = {{\bf{v}}_B} + {\omega _B} \times {{\bf{r}}_{i/B}} + {{\bf{v}}_{re{l_{i/B}}}} = {{\bf{v}}_B} + {\omega _B} \times {{\bf{r}}_{i/B}} + \dot R_B^i{{\bf{r}}_{i/pivo{t_i}}}\end{align}

The time derivative of the aircraft total angular momentum is the sum of the net torque acting at the aircraft main body centre of mass ( ${{\bf{T}}_B}$ ) and the moments due to each part weight by the distance to the main body centre of mass,

(13) \begin{align}\frac{d}{{dt}}({{\bf{H}}_B}) = {{\bf{T}}_B} + \sum\limits_{i = 1}^4 \big\{ {{\bf{r}}_{i/B}} \times {m_i}R_B^E{{\bf{g}}_E}\big\} + \sum\limits_{j = 1}^4 \big\{ {{\bf{r}}_{j/B}} \times {m_j}R_B^E{{\bf{g}}_E}\big\} \end{align}

Furthermore, the time derivative of the Equation (11) leads to,

(14) \begin{align} \frac{d}{{dt}}({{\bf{H}}_B}) &= {\tilde I_B}{\dot \omega _B} + {\omega _B} \times \big({\tilde I_B}{\omega _B}\big) + \sum\limits_{i = 1}^4 \bigg\{ \frac{d}{{dt}}\big({\tilde I_{{B_i}}}{\omega _{{B_i}}}\big) + {\omega _B} \times \big({\tilde I_{{B_i}}}{\omega _{{B_i}}}\big) + {m_i}\bigg[\frac{d}{{dt}}({{\bf{r}}_{i/B}}) \times {{\bf{v}}_{{B_i}}} + {{\bf{r}}_{i/B}} \times \frac{d}{{dt}}({{\bf{v}}_{{B_i}}})\bigg]\bigg\}\nonumber \\ & \quad + \sum\limits_{j = 1}^4 \bigg\{ \frac{d}{{dt}}\big({\tilde I_{{B_j}}}{\omega _{{B_j}}}\big) + {\omega _B} \times \big({\tilde I_{{B_j}}}{\omega _{{B_j}}}\big) + {m_j}\bigg[\frac{d}{{dt}}({{\bf{r}}_{j/B}}) \times {{\bf{v}}_{{B_j}}} + {{\bf{r}}_{j/B}} \times \frac{d}{{dt}}({{\bf{v}}_{{B_j}}})\bigg]\bigg\} \end{align}

It is also necessary to define the inertia tensor in Equation (15), being the sum of a rotation and translation of the inertia tensor from its reference frame to B through the parallel axis theorem in Equation (15). Moreover, the inertia tensor time derivative is defined in Equation (16),

(15) \begin{align}{\tilde I_{{B_i}}} = R_B^i{\tilde I_i}R_B^{{i^T}} + {m_i}\tilde R_B^i\end{align}

(16) \begin{align}\frac{d}{{dt}}\big({\tilde I_{{B_i}}}\big) = \dot R_B^i{\tilde I_i}R_B^{{i^T}} + R_B^i{\tilde I_i}\dot R{_B^{i^{T}}} + {m_i}\frac{d}{{dt}}\big(\tilde R_B^i\big)\end{align}

Similarly to the inertia tensor, the aircraft surface moving parts angular velocity is the sum of the main body angular velocity and the rotating aerodynamic surface angular velocity in Equation (17). In Equation (18) the time derivative is taken to define the angular acceleration.

(17) \begin{align}{\omega _{{B_i}}} = R_B^i{\omega _i} + {\omega _B} = {\left[ {\begin{array}{c@{\quad}c@{\quad}c}0 &{{{\dot \delta }_i}} &0\end{array}} \right]^T} + {\omega _B}\end{align}

(18) \begin{align}{\dot \omega _{{B_i}}} = \frac{d}{{dt}}(R_B^i{\omega _i} + {\omega _B}) = {\left[ {\begin{array}{c@{\quad}c@{\quad}c}0 &{{{\ddot \delta }_i}}& 0\end{array}} \right]^T} + {\dot \omega _B}\end{align}

Now, for the angular velocity of the rotors in Equation (19) it is necessary to take into account the angular velocity of the rotors themselves ( ${\omega _{{R_j}}}$ ) and the direction of rotation ( ${\lambda _j} = 1$ or $ - 1$ ). Its time derivative is shown in Equation (20).

(19) \begin{align}{\omega _{{B_j}}} = R_B^j{\omega _{{R_j}}} + R_B^j{\omega _j} + {\omega _B} = R_B^j{\left[ {\begin{array}{c@{\quad}c@{\quad}c}{{\lambda _j}{\omega _{{R_j}}}}& 0 &0\end{array}} \right]^T} + {\left[ {\begin{array}{c@{\quad}c@{\quad}c}0 &{{{\dot \delta }_j}} &0\end{array}} \right]^T} + {\omega _B}\end{align}

(20) \begin{align}{\dot \omega _{{B_j}}} = \frac{d}{{dt}}\left( {R_B^j{\omega _{{R_j}}} + R_B^j{\omega _j} + {\omega _B}} \right) = \dot R_B^j{\left[ {\begin{array}{c@{\quad}c@{\quad}c}{{\lambda _j}{\omega _{{R_j}}}} &0 &0\end{array}} \right]^T} + R_B^j{\left[ {\begin{array}{c@{\quad}c@{\quad}c}{{\lambda _j}{{\dot \omega }_{{R_j}}}}& 0 & 0\end{array}} \right]^T} + {\left[ {\begin{array}{c@{\quad}c@{\quad}c}0 &{{{\ddot \delta }_j}} &0\end{array}} \right]^T} + {\dot \omega _B}\end{align}

Finally, substituting Equations (14), (17)–(20) into Equation (13) and rearranging it, the following equation is obtained.

(21) \begin{align}{\dot \omega _B} = {A^{ - 1}}(\!-\!B{\omega _B} - C{{\dot{\bf v}}_B} - D{{\bf{v}}_B} - {\bf{E}} + {\bf{F}} + {{\bf{T}}_B})\end{align}

The coefficients A, B, C, D, ${\bf{E}}$ and ${\bf{F}}$ in Equation (21) are defined, respectively, in Equations (22)–(27).

(22) \begin{align}A = {\tilde I_B} + \sum\limits_{i = 1}^4 {\tilde I_{{B_i}}} + \sum\limits_{j = 1}^4 {\tilde I_{{B_j}}}\end{align}

(23) \begin{align}B = {\Omega _B}{\tilde I_B} + \sum\limits_{i = 1}^4 \bigg\{ \frac{d}{{dt}}\big({\tilde I_{{B_i}}}\big) + {\Omega _B}{\tilde I_{{B_i}}}\bigg\} + \sum\limits_{j = 1}^4 \bigg\{ \frac{d}{{dt}}\big({\tilde I_{{B_j}}}\big) + {\Omega _B}{\tilde I_{{B_j}}}\bigg\} \end{align}

(24) \begin{align}C = \sum\limits_{i = 1}^4 \{ {m_i}{{\bf{r}}_{i/B}} \times \} + \sum\limits_{j = 1}^4 \{ {m_j}{{\bf{r}}_{j/B}} \times \} \end{align}

(25) \begin{align}D = \sum\limits_{i = 1}^4 \big\{ {m_i}\big[\big(\dot R_B^i{{\bf{r}}_{i/pivo{t_i}}} \times \big) + {{\bf{r}}_{i/B}} \times {\Omega _B}]\big\} + \sum\limits_{j = 1}^4 \big\{ {m_j}\big[\big(\dot R_B^j{{\bf{r}}_{j/pivo{t_j}}} \times \big) + {{\bf{r}}_{j/B}} \times {\Omega _B}\big]\big\} \end{align}

\begin{align*}{\bf{E}} = \sum\limits_{i = 1}^4 \bigg\{ \bigg[\frac{d}{{dt}}\big({\tilde I_{{B_i}}}\big) + {\Omega _B}{\tilde I_{{B_i}}}\big]R_B^i{\omega _i} + {\tilde I_{{B_i}}}\frac{d}{{dt}}\big(R_B^i{\omega _i}\big) + {m_i}\big[\dot R_B^i{{\bf{r}}_{i/pivo{t_i}}} \times \big({\Omega _B}{{\bf{r}}_{i/B}} + \dot R_B^i{{\bf{r}}_{i/pivo{t_i}}}\big)\end{align*}

\begin{align*} + {{\bf{r}}_{i/B}} \times \big(\big({\dot \Omega _B} + {\Omega _B}{\Omega _B}\big){{\bf{r}}_{i/B}} + \big(2{\Omega _B}\dot R_B^i + \ddot R_B^i\big){{\bf{r}}_{i/pivo{t_i}}}\big)\bigg]\bigg\} \end{align*}

\begin{align*} + \sum\limits_{j = 1}^4 \bigg\{ \bigg[\frac{d}{{dt}}\big({\tilde I_{{B_j}}}\big) + {\Omega _B}{\tilde I_{{B_j}}}\big]\big(R_B^j{\omega _{{R_j}}} + R_B^j{\omega _j}\big) + {\tilde I_{{B_j}}}\frac{d}{{dt}}\big(R_B^j{\omega _{{R_j}}} + R_B^j{\omega _j}\big) + {m_j}\big[\dot R_B^j{{\bf{r}}_{j/pivo{t_j}}} \times \big({\Omega _B}{{\bf{r}}_{j/B}} + \dot R_B^j{{\bf{r}}_{j/pivo{t_j}}}\big)\end{align*}

(26) \begin{align} + {{\bf{r}}_{j/B}} \times \big(\big({\dot \Omega _B} + {\Omega _B}{\Omega _B}\big){{\bf{r}}_{j/B}} + \big(2{\Omega _B}\dot R_B^j + \ddot R_B^j\big){{\bf{r}}_{j/pivo{t_j}}}\big)\bigg]\bigg\} \end{align}

(27) \begin{align}{\bf{F}} = \sum\limits_{i = 1}^4 \big\{ {{\bf{r}}_{i/B}} \times {m_i}R_B^E{{\bf{g}}_E}\big\} + \sum\limits_{j = 1}^4 \big\{ {{\bf{r}}_{j/B}} \times {m_j}R_B^E{{\bf{g}}_E}\big\} \end{align}

3.3 Forces and torques

The net force vector is the sum of the aircraft aerodynamic forces and the propellers propulsive forces, where S is the main body-to-wind-axes rotation matrix,

(28) \begin{align}{{\bf{F}}_B} = {S^T}\left[ {\begin{array}{c}{ - D}\\[5pt] Y\\[5pt] { - L}\end{array}} \right] + \sum\limits_{j = 1}^4 \left( {R_B^j\left[ {\begin{array}{c}{{T_j}}\\[5pt] 0\\[5pt] 0\end{array}} \right]} \right)\end{align}

Furthermore, the net torque vector is also the sum of the aerodynamic moments and propeller torques,

(29) \begin{align}{{\bf{T}}_B} = {S^T}\left[ {\begin{array}{c}{\bar L}\\[5pt] M\\[5pt] N\end{array}} \right] + \sum\limits_{j = 1}^4 \left( {R_B^j\left[ {\begin{array}{c}{{\lambda _j}{Q_j}}\\[5pt] 0\\[5pt] 0\end{array}} \right] + {{\bf{r}}_{j/B}} \times R_B^j\left[ {\begin{array}{c}{{T_j}}\\[5pt] 0\\[5pt] 0\end{array}} \right]} \right)\end{align}

The propeller data used in the simulations is the APC 12 × 5 propeller performance data provided by the manufacturer website [Reference Propellers29].

It is important to note that the front propellers of the tilt-wing UAV configuration under study are expected to operate under a high angle-of-attack conditions, which would affect the resulting propeller thrust, torque and moments, particularly at high flight speeds. So, a blade element model of a helicopter rotor would be more precise, however, it would make the modelling and simulation much more complex.

3.4 Attitude propagation equation

The aircraft angular velocity and Euler angle rates are not the same, they relate through the attitude propagation equation of Equation (30), whose demonstration can be found in [Reference Stevens, Lewis and Johnson8].

(30) \begin{align}\left[ {\begin{array}{c}{\dot \phi }\\[5pt] {\dot \theta }\\[5pt] {\dot \psi }\end{array}} \right] = \left[ {\begin{array}{c@{\quad}c@{\quad}c}1 &{\tan \theta \sin \phi } &{\tan \theta \cos \phi }\\[5pt] 0 &{\cos \phi } &{ - \sin \phi }\\[5pt] 0 &{\sin \phi \sec \theta } &{\cos \phi \sec \theta }\end{array}} \right]\left[ {\begin{array}{c}P\\[5pt] Q\\[5pt] R\end{array}} \right]\end{align}

3.5 Navigation equation

Additionally, the velocity in Earth’s fixed inertial frame is defined,

(31) \begin{align}{{\bf{v}}_E} = {\left[ {\begin{array}{c@{\quad}c@{\quad}c}{{{\dot x}_E}} &{{{\dot y}_E}} &{{{\dot z}_E}}\end{array}} \right]^T} = R_E^B{{\bf{v}}_B}\end{align}

Furthermore, the aircraft altitude h is defined as $h = - {z_E}$ since in the Earth’s fixed inertial reference frame, the ${z_E}$ coordinate points downwards.

3.6 Controls dynamic equation

Every aerodynamic control surface actuation and rotor angular velocity are modelled as first-order dynamic systems,

(32) \begin{align}\dot \delta = \frac{1}{{{\tau _\delta }}}({k_\delta }{U_\delta } - \delta )\end{align}

(33) \begin{align}{\dot \omega _R} = \frac{1}{{{\tau _R}}}({k_{{\omega _R}}}{U_{{\omega _R}}} - {\omega _R})\end{align}

3.7 Aerodynamic modelling

The aerodynamic forces and moments are modelled according to the following equations of this section. Also, the equations, aerodynamic coefficients and dynamic derivatives were obtained using the extensive methods of [Reference Hoak30–Reference Houghton and Carpenter32]. The wing and canard have constant chord Naca 0012 aerofoils.

(34) \begin{align}D = \frac{1}{2}\rho {({V_T} + \Delta {V_T})^2}{C_{{D_{{W_e}}}}}{S_{{W_e}}} + \frac{1}{2}\rho V_T^2\left( {{C_{{D_B}}}{S_B} + {C_{{D_{{C_e}}}}}{S_{{C_e}}}} \right) + \frac{1}{4}\rho {V_T}{S_W}{\bar c_W}{C_{{D_q}}}Q\end{align}

(35) \begin{align}Y = \frac{1}{2}\rho V_T^2{S_W}\left( {{C_{{Y_\beta }}}\beta + {C_{{Y_{{\delta _r}}}}}{\delta _r}} \right) + \frac{1}{4}\rho {V_T}{S_W}{b_W}\left( {{C_{{Y_p}}}P + {C_{{Y_r}}}R + {C_{{Y_{\dot \beta }}}}\dot \beta } \right)\end{align}

(36) \begin{align}L = \frac{1}{2}\rho V_T^2{K_{CB}}\left( {{C_{{L_B}}}{S_B} + {C_{{L_{{C_e}}}}}{S_{{C_e}}}} \right) + \frac{1}{2}\rho {({V_T} + \Delta {V_T})^2}{K_{WB}}{C_{{L_{{W_e}}}}}{S_{{W_e}}} + \frac{1}{4}\rho {V_T}{S_W}{\bar c_W}\left( {{C_{{L_q}}}Q + {C_{{L_{\dot \alpha }}}}\dot \alpha } \right)\end{align}

(37) \begin{align}\bar L = \frac{1}{4}\rho {V_T}{S_W}b_W^2\left( {{C_{{l_p}}}P + {C_{{l_r}}}R + {C_{{l_{{{\dot \beta }_{WBT}}}}}}\dot \beta } \right) + \frac{1}{2}\rho V_T^2{S_W}{b_W}\left( {{C_{{l_\beta }}}\beta + {C_{{l_{{\delta _{{a_L}}}}}}}{\delta _{{a_L}}} - {C_{{l_{{\delta _{{a_R}}}}}}}{\delta _{{a_R}}} + {C_{{l_{{\delta _r}}}}}{\delta _r}} \right)\end{align}

\begin{align*}M &= \frac{1}{2}\rho V_T^2\Big(({x_B} - {x_{{m_B}}}){S_B}{K_{CB}}({C_{{L_B}}}\cos \alpha + {C_{{D_B}}}\sin \alpha ) - ({z_B} - {z_{{m_B}}}){S_B}{K_{CB}}({C_{{L_B}}}\sin \alpha - {C_{{D_B}}}\cos \alpha ) \\[5pt] & + {C_{{m_B}}}{\bar{c}_W}{S_B} + ({x_B} - x^{\prime}_{{C_{e}}}){S_{{C_{e}}}}{K_{CB}}({C_{{L_{{C_{e}}}}}}\cos \alpha + {C_{{D_{{C_{e}}}}}}\sin \alpha )\\[5pt] &- ({z_B} - z^{\prime}_{{C_{e}}}){S_{{C_{e}}}}{K_{CB}}({C_{{L_{{C_{e}}}}}}\sin \alpha - {C_{{D_{{C_{e}}}}}}\cos \alpha ) + {C_{{m_{{C_{e}}}}}}{\bar{c}_C}{S_{{C_{e}}}}\Big)\end{align*}

\begin{align*} + {1 \over 2}\rho {({V_T} + \Delta {V_T})^2}{S_{{W_e}}}(({x_B} - x^{\prime}_{{W_e}}){K_{WB}}({C_{{L_{{W_e}}}}}\cos (\alpha - \varepsilon ) + {C_{{D_{{W_e}}}}}\sin (\alpha - \varepsilon ))\end{align*}

(38) \begin{align} - ({z_B} - z^{\prime}_{{W_e}}){K_{WB}}({C_{{L_{{W_e}}}}}\sin (\alpha - \varepsilon ) - {C_{{D_{{W_e}}}}}\cos (\alpha - \varepsilon )) + {C_{{m_{{W_e}}}}}{\bar c_W}) + {1 \over 4}\rho {V_T}{S_W}\bar c_W^2({C_{{m_q}}}Q + {C_{{m_{\dot \alpha }}}}\dot \alpha )\end{align}

(39) \begin{align}N = \frac{1}{4}\rho {V_T}{S_W}b_W^2\Big( {{C_{{n_p}}}P + {C_{{n_r}}}R + {C_{{n_{\dot \beta }}}}\dot \beta } \Big) + \frac{1}{2}\rho V_T^2{S_W}{b_W}\Big({C_{{n_\beta }}}\beta + {C_{{n_{{\delta _{{a_L}}}}}}}{\delta _{{a_L}}} - {C_{{n_{{\delta _{{a_R}}}}}}}{\delta _{{a_R}}} + {C_{{n_{{\delta _r}}}}}{\delta _r}\Big)\end{align}

3.8 Energy consumed during flight

The propellers require power to produce the thrust and torque, which is provided by the onboard batteries, which have a total energy capacity of 10400 mmAh. Thus, the aircraft range and autonomy must be computed concerning the batteries energy capacity.

The energy is the integral of the power over time, as defined in Equation (40). However, the energy capacity of the batteries is defined in mmAh, so, in Equation (40), the power must be divided by the battery nominal voltage (considered Lipo 4S = 14.8V) to compute the current, and divide by 3.6 to convert to mmAh.

(40) \begin{align}E(mmAh) = \int_{{T_0}}^{{T_{end}}} P{\kern 1pt} dt = \int_{{T_0}}^{{T_{end}}} \frac{P}{{3.6V}}{\kern 1pt} dt\end{align}

4.0 Aircraft transition trajectory

Aircraft transition trajectory is the sequence of equilibrium points, or trim conditions, which combines state variables that make the flight state derivatives equal to zero. Moreover, it is also essential because it provides an initial condition for flight simulation and a flight condition to linearise the aircraft dynamics equations of motion [Reference Stevens, Lewis and Johnson8].

So, the trim conditions are defined by the combination of variables of the state vector of Equation (41), with a corresponding control vector of Equation (42), that results in the scalar cost function of Equation (43) equal to zero. Note that this scalar is the sum of the time derivatives squared of the states related to linear and angular motion, which can be computed using Equations (9) and (21) for a given input state vector ${\bf{X}}$ .

(41) \begin{align}{\bf{X}} = \left( {U,V,W,P,Q,R,\phi ,\theta ,\psi ,{x_E},{y_E},{z_E},{\delta _f},{\delta _e},{\delta _r},{\delta _{{a_L}}},{\delta _{{a_R}}},{\delta _w},{\delta _c},{\omega _{{R_1}}},{\omega _{{R_2}}},{\omega _{{R_3}}},{\omega _{{R_4}}}} \right)\end{align}

(42) \begin{align}{\bf{U}} = ({U_{{\delta _f}}},{U_{{\delta _e}}},{U_{{\delta _r}}},{U_{{\delta _{{a_L}}}}},{U_{{\delta _{{a_R}}}}},{U_{{\delta _w}}},{U_{{\delta _c}}},{U_{{\omega _{{R_1}}}}},{U_{{\omega _{{R_2}}}}},{U_{{\omega _{{R_3}}}}},{U_{{\omega _{{R_4}}}}})\end{align}

(43) \begin{align}j = {\dot U^2} + {\dot V^2} + {\dot W^2} + {\dot P^2} + {\dot Q^2} + {\dot R^2}\end{align}

Therefore, the transition trajectory was computed using the Sequential Simplex algorithm, described in [Reference Nelder and Mead33, Reference Walters, Morgan, Lloyd, Parker and Deming34], for the minimisation of the scalar cost function of Equation (43), given inputs of the state vector ${\bf{X}}$ with the constraints of longitudinal flight condition (input reference flight speed ${U_e}$ , lateral states imposed as zero and no angular velocities). The algorithm starting procedure implemented was the Corner Initial Method, described in [Reference Walters, Morgan, Lloyd, Parker and Deming34], and the stopping criterion used was scalar cost function value less than $1e - 10$ .

The equilibrium trajectory is computed in steps of ${U_e} = 0.1{\kern 1pt} m/s$ , from hover (0 m/s) to 20 m/s. And from 20 m/s to 30 m/s, the steps are 1.0 m/s. Therefore the equilibrium points correspond to the markers in Figs. 4–7. The reference altitude is 100 m.

Figure 4. Aircraft trim data as function of reference flight speed ${U_e}$ .

Figure 5. Aircraft aerodynamic coefficients and angle-of-attack trim data as a function of reference flight speed ${U_e}$ .

Figure 6. Total vertical force, propulsive or aerodynamic.

Figure 7. Required power from the propellers.

The algorithm was applied to the conceptual VTOL aircraft under study, generating the data presented in the following Fig. 4 for the propellers RPM at wing $({\omega _{Wing}})$ and at the front of the aircraft $({\omega _{Front}})$ and wing and canard tilt angle $({\delta _w},{\delta _c})$ , as a function of reference flight speed ${U_e}$ , being that the flap and elevator are not used to trim the aircraft. Note that at flight speed zero, hovering condition, both wing and canard are at tilt angle ${90^ \circ }$ having the propellers pointing upwards, and as the flight speed increases, the wing gradually leans forward until it is close to a horizontal position at higher speeds. At low flight speeds, the at the wing and front propellers have similar angular velocities, up until the wing is close to the horizontal position, where the angular velocity of the wing propellers has to increase to keep track of the horizontal velocity of the aircraft, whilst the front propellers reduce its angular velocity to reduce thrust.

Moreover, the transition strategy for the canard tilt angle is to move it entirely in one step from the vertical position to a more horizontal position at the beginning of the transition so that it would have low drag during acceleration. So, it tilts to 5° at the beginning of the manoeuvre. At flight speed close to ${U_e} = 16.7{\kern 1pt} m/s$ , the wing is at the maximum lift coefficient condition, so the canard must increase its tilt angle and angle-of-attack to trim the vehicle, which allows the continued acceleration and gradual wing tilting to a more horizontal position while maintaining negative main body angle-of-attack, which is essential to keep the front propellers producing thrust upwards and forward.

Furthermore, it is noticeable that there are two distinct regions of equilibrium conditions, that is, from hover until ${U_e} = 16.7{\kern 1pt} m/s$ where the wing is in the aerodynamically stalled region, while the main body is kept at 0° angle-of-attack. Therefore, most of the vehicle lifting force comes from the propellers, and the total power required is progressively reduced. The aerodynamic coefficients and angle-of-attack of the main body, wing and canard are presented in Fig. 5, where the wing reaches its maximum lift coefficient close to the frontier between regions of equilibrium conditions, and at higher flight speeds the drag coefficients are significantly reduced.

During the transition manoeuvre from hover to maximum speed, the total vertical force that sustains the vehicle switches between mostly propulsive to mostly aerodynamic, as shown in Fig. 6. The contribution of each vehicle component is also shown in this figure.

As most of the vertical force is provided by the propellers at low flight speed, the power required is also more significant, and as the vehicle flight speed gradually increases, simultaneously with the increase of the portion of aerodynamic vertical force, the power required for flight reduces, which is depicted in Fig. 7.

5.0 Transition flight simulation algorithm

The strategy to control the flight is shown in Fig. 8 with the control architecture diagram used for simulation, which has the following main blocks: aircraft dynamics, controller and reference scheduler. First, the aircraft dynamics block is the grouping of the dynamic equations previously described, being Equations (9), (21), (30), (31)–(33), which can be numerically integrated to obtain the state variables over time.

Figure 8. Control architecture diagram for simulation.

Furthermore, the controller block purpose is to compute the control vector ${\bf{U}}$ to stabilise the aircraft at the current steady state reference ${{\bf{X}}_e}$ , remembering that in the trimmed condition there is a control reference ${{\bf{U}}_e}$ that must be complemented by ${\bf{u}}$ to control the aircraft, given some state variables disturbances $({\bf{U}} = {{\bf{U}}_e} + {\bf{u}})$ . Thus the control system must compute the disturbance ${\bf{x}}$ from reference condition $({\bf{x}} = {\bf{X}} - {{\bf{X}}_e})$ .

The state vector is separated in longitudinal ${{\bf{X}}_{long}} = (U,W,Q,\theta ,h)$ and lateral states ${{\bf{X}}_{lat}} = (U,V,P,R,\phi ,\psi )$ , to have a decoupled system with longitudinal and lateral control. Lastly, the disturbances in the longitudinal and lateral motion are multiplied by longitudinal and lateral gain matrices to compute the controls: ${{\bf{u}}_{long}} = - {K_{long}}{{\bf{x}}_{long}}$ and ${{\bf{u}}_{lat}} = - {K_{lat}}{{\bf{x}}_{lat}}$ . In this way, every aerodynamic actuator and propeller command is computed and combined in the resulting ${\bf{u}}$ and final complete control vector ${\bf{U}}$ , which must then be saturated to avoid exceeding the maximum possible values.

The longitudinal and lateral gain matrices $({K_{long}},{K_{lat}})$ , to apply in the simulations, were computed using the linear quadratic regulator theory (LQR) [Reference Stevens, Lewis and Johnson8, Reference Ogata35] applied to linearised equations of motion at the equilibrium points, for each pair of longitudinal and lateral equilibrium vectors of the set of equilibrium points. The reference scheduler block must select the gain matrices, whose purpose is to choose the proper gain matrices and state reference vectors from the input flight speed setpoint $({U_{sp}})$ , altitude $({h_{sp}})$ and turn rate $({\psi _{sp}})$ ; however, in this paper only flight speed control is simulated.

The resulting gain matrices $({K_{long}},{K_{lat}})$ for the range of reference flight speeds are shown in the Appendix section 9.0.

The flight speed setpoint ${U_{sp}}$ is the aircraft desired speed. So, to accelerate or decelerate, the chosen flight transition concept is to move from equilibrium points gradually, that is, accelerate towards the next equilibrium point ${{\bf{X}}_e}$ slightly tilting the aerodynamic surfaces and wait until the control system diminishes the state disturbances ${\bf{x}}$ from the current equilibrium reference ${{\bf{X}}_e}$ , and allowing only the acceleration to continue until the next equilibrium vector if the dynamic system is sufficiently stabilised. The stability criterion to permit the change in state reference are the following cost functions: ${\varepsilon _1} = \sqrt {{u^2} + {v^2} + {w^2}} $ (velocity disturbance from reference), ${\varepsilon _2} = \sqrt {{P^2} + {Q^2} + {R^2}} $ (angular velocity disturbance from reference), ${\varepsilon _3} = \sqrt {{\phi ^2} + {\theta ^2} + {\psi ^2}} $ (attitude disturbance from reference), ${\varepsilon _4} = \sqrt {{h^2}} $ (altitude disturbance from reference), ${\varepsilon _5} = \sqrt {{{\dot U}^2} + {{\dot V}^2} + {{\dot W}^2}} $ (acceleration disturbance from reference) and ${\varepsilon _6} = \Delta t$ (time since last transition), which must be less than thresholds ${\varepsilon _{{1_{th}}}}$ , ${\varepsilon _{{2_{th}}}}$ , ${\varepsilon _{{3_{th}}}}$ , ${\varepsilon _{{4_{th}}}}$ , ${\varepsilon _{{5_{th}}}}$ and greater than ${\varepsilon _{{6_{th}}}}$ .

Additionally, the reference vectors are grouped with respect to the equilibrium flight velocity so that each equilibrium vector is related to an auxiliary variable named ${U_{index}}$ . Therefore, for an accelerated flight, the ${U_{index}}$ must increase gradually, whereas for a decelerated flight, the ${U_{index}}$ must decrease. This concept is better illustrated in Fig. 9, where it is shown that each ${U_{index}}$ is associated with flight speed and longitudinal and lateral states and controls. Thus, the flight speed setpoint aims the objective ${U_{index}}$ , and the control system switches between ${U_{index}}$ , which updates the current controller parameters $({{\bf{X}}_e},{{\bf{U}}_e},{K_{long}},{K_{lat}})$ gradually as the aircraft becomes sufficiently stable.

Figure 9. Reference scheduler parameters selection for simulation.

6.0 Transition flight simulation results

The control architecture diagram of Fig. 8 was implemented in the MATLAB/Simulink software environment [36] to simulate the transition flight from hover to cruise condition, and from cruise to hover condition, where the trim conditions are previously computed and stored in matrix form to be selected by the transition algorithm. The gain matrices $({K_{long}},{K_{lat}})$ as a function of reference flight speeds are shown in Appendix section 9.0. Furthermore, results are presented only for transition flight simulation, where the integration time steps were 0.0001 seconds, the reference altitude was 100 m, and the transition stability thresholds applied are listed in Table 1.

Table 1. Transition stability thresholds

First, in Fig. 10, one can see the results for the flight states for transition simulation of 0 to 30 m/s, and in Fig. 11 the controls. There is also the simulation for the transition from cruise to hover (30 m/s to 0 m/s), with the flight states in Fig. 12 and the controls in Fig. 13. The cost functions that govern the transition between trim points are shown in Fig. 14 and Fig. 15. Moreover, these results are followed by graphs with a closer look at the most critical transition phase for the accelerated flight focused on the simulation time from 85 to 95 seconds and for the decelerated flight for the simulation time from 20 to 30 seconds.

Figure 10. Simulation flight states, the transition from 0 to 30 m/s.

Figure 11. Simulation controls, the transition from 0 to 30 m/s.

Figure 12. Simulation flight states, the transition from 30 to 0 m/s.

Figure 13. Simulation controls, the transition from 30 to 0 m/s.

Figure 14. Simulation transition cost functions, the transition from 0 to 30 m/s.

Figure 15. Simulation transition cost functions, the transition from 30 to 0 m/s.

The complete transition from take-off (0 m/s) to maximum speed (30 m/s) takes 125 seconds and 1530 m of horizontal distance, and from maximum speed to landing (0 m/s) also takes 125 seconds and 1469 m. The results are summarised in Table 2.

Table 2. Longitudinal flight simulation results summary

There are two main control stages to complete the entire transition flight from hover (0 m/s) to maximum flight speed (30 m/s), which are defined respective to the reference flight speed ${U_e}$ so that at each stage there are some active controls and some remain neutral. They are defined in Table 3. For reference flight speed ${U_e}$ , from 0 to 1.9 m/s, the active controls are the four propellers angular velocities, and no aerodynamic actuator is active, therefore remaining in the reference position. That is because the aerodynamic dynamic pressure is too small to effectively control. For ${U_e}$ from 2.0 to 30.0 m/s, the active controls are the four propellers angular velocities, the deflections of flaps, elevators, rudder, ailerons and canard, being that the canard can only tilt $ - {10^ \circ }$ to $ + {10^ \circ }$ from the reference tilt angle to control the vehicle.

Table 3. Longitudinal flight active controls as function of reference flight speed

Moreover, there is a critical phase which is the transition from inclined wing aerodynamic stalled to more horizontal and aerodynamic loaded. This can be noticed in the trim data of Fig. 4 where it is shown that the wing position goes from ${90^ \circ }$ tilt angle at hover (0 m/s), gradually decreases the tilt angle until ${24^ \circ }$ at ${U_e} = 16.7m/s$ , where the canard must increase its tilt angle from ${5^ \circ }$ to ${20^ \circ }$ to trim the vehicle. In Fig. 10, one can see that the vehicle keeps a continuous acceleration up until ${U_e} = 20m/s$ , whilst having oscillations in attitude and altitude given that the wing is in the nonlinear range of angle-of-attack (close to the maximum lift coefficient). From ${U_e} = 20m/s$ to maximum flight speed at ${U_e} = 30m/s$ , there is an even higher acceleration rate, where the wing is at the linear range of angle-of-attack. Moreover, to keep the continuous acceleration and transition, the aircraft main body pitch angle is maintained at negative values so that the front propellers thrust point upwards and forward.

For the decelerated flight in Fig. 12, there are also two regions with an almost constant rate of deceleration, from ${U_e} = 30m/s$ to ${U_e} = 20m/s$ , and from ${U_e} = 20m/s$ to ${U_e} = 0m/s$ , where the aircraft pitch angle is kept at positive values for the most of the manoeuvre so that the front propellers would point upwards and backward. The altitude oscillates from roughly −1.0 to +1.0 m for the entire transition.

The wing tilting at the low flight speeds causes oscillations in attitude, and even in the lateral directional flight states such as roll and yaw Euler angles and angular velocities, causing sideslip, which is a result of the gyroscopic moments of the propellers tilting at high angular speeds.

To have a better understanding of the aircraft dynamics, its oscillatory movement, and control actions, the states, controls and cost functions were highlighted at the critical phase of the transition in Figs. 16, 17 and in Fig. 18 for the accelerated flight for simulation time 85–95 seconds and in Figs. 19, 20 and 21 for the decelerated flight for the simulation time of 20–30 seconds. In this critical phase, the aircraft performs the manoeuvre of the wing at high lift coefficient, and the canard is tilted at a high angle-of-attack. This manoeuvre causes a change in the dynamic response noticeable by the amplitude of the states and controls oscillations.

Figure 16. Simulation flight states, the transition from 0 to 30 m/s (simulation time 85 to 95 seconds).

Figure 17. Simulation controls, the transition from 0 to 30 m/s (simulation time 85–95 seconds).

Figure 18. Simulation transition cost functions, the transition from 0 to 30 m/s (simulation time 85–95 seconds).

Figure 19. Simulation flight states, the transition from 30 to 0 m/s (simulation time 20–30 seconds).

Figure 20. Simulation controls, the transition from 30 to 0 m/s (simulation time 20–30 seconds).

Figure 21. Simulation transition cost functions, the transition from 30 to 0 m/s (simulation time 20–30 seconds).

The switching between reference states and controls occurs when the transition cost functions reach their thresholds, being their time series depicted in Fig. 14 for the accelerated flight and in Fig. 15 for the decelerated flight. It is worth mentioning that the cost functions that primarily govern the transition are related to attitude and acceleration instabilities, that is, cost functions ${\varepsilon _2}$ , ${\varepsilon _3}$ and ${\varepsilon _5}$ . The cost function ${\varepsilon _6}$ (minimum time between transitions switching) is vital to impose a time constraint to wait for some stabilisation before switching to the following equilibrium point, otherwise, the switching would occur before the dynamics of the system can develop.

6.1 Full transition flight

The aircraft concept model has two Lipo 4S batteries for the propulsion system. Each battery has a 5200 mmAh energy capacity, resulting in a total of 10400 mmAh.

From Table 2 the transition from hover to cruise takes 125 seconds (2.08 minutes), 1530 m, and 1635 mmAh of energy, and the transition for landing 125 seconds (2.08 minutes), 1469 m, and 1644 mmAh, so there is left 7121 mmAh for the cruise at maximum speed $({U_e} = 30m/s)$ . So the two transition phases added together consumes 31.5 % of the available energy.

The power required at cruise speed is 263.7 W, as can be seen in Fig. 7. Dividing the total required power by the batteries rated voltage (14.8 V) gives the estimated required current of 17.81 A. Therefore, the available time for a cruise flight would be,

(44) \begin{align}{t_{cruise}} = \frac{{7.121{\kern 1pt} Ah}}{{17.81{\kern 1pt} A}} = 0.40{\kern 1pt} h = 23.98{\kern 1pt} minutes\end{align}

So, the estimated vehicle autonomy would be,

(45) \begin{align}{t_{autonomy}} = {t_{0to30}} + {t_{cruise}} + {t_{30to0}} = 2.08 + 23.98 + 2.08 = 28.14{\kern 1pt} minutes\end{align}

Also, the estimated vehicle range would be,

(46) \begin{align}\Delta {x_{total}} = \Delta {x_{0to30}} + \Delta {x_{cruise}} + \Delta {x_{30to0}} = 1530 + 30*{t_{cruise}} + 1469 = 46163{\kern 1pt} m\end{align}

7.0 Conclusion

This paper presented a tilt-wing VTOL UAV configuration with a canard and wing with two attached propellers capable of tilting and two fixed front propellers kept pointing upwards. To study the transition from hover to cruise flight condition, a dynamics model for this configuration was derived using multi-body equations of motion, which were used to compute the transition trajectory, that is, the combination of states and controls that keep the aircraft in steady condition for the range of desired flight speeds (0 to 30 m/s). Also, a transition manoeuvre algorithm using a gain-scheduled linear quadratic regulator (LQR) controller was developed to numerically simulate the transition flight from hover to cruise condition and from cruise condition to hover. The strategy is to switch between equilibrium states as the control system diminishes the state disturbances from the current equilibrium vector, as the transition stability criteria thresholds are satisfied. The numerical simulations showed that the designed model aircraft should have a smooth transition between flight speeds, except for the situation where the wing is at its maximum lift coefficient. This movement, from the stalled wing at a high angle-of-attack to tilt to a more horizontal and in the linear angle-of-attack range, is the most critical of the transition manoeuvre and makes the aircraft oscillate in attitude, altitude and even in the lateral directional flight states such as roll and yaw Euler angles and angular velocities, causing sideslip, which is a result of the gyroscopic moments. To have an accelerated transition, the aircraft should remain with a negative main body angle-of-attack to keep the front propellers thrust pointing upwards and forwards, whereas, for the decelerated transition flight, the aircraft main body should conversely maintain a positive angle-of-attack so that the front propellers points upwards and backwards, slowing down the vehicle.

The aircraft concept model full weight is 6.44 kg with a total battery capacity of 10400 mmAh. The numerical simulation showed that the vehicle transition flight from hover (at 0 m/s) to cruise at 30 m/s would take 125 seconds while travelling 1530 m of horizontal distance, consuming 1635 mmAh of battery energy. The transition flight from cruise at 30 m/s to hover condition would also take 125 seconds with 1469 m and 1644 mmAh. Therefore, both transition phases added together consumes 31.5 % of the available energy. Moreover, the estimated vehicle autonomy would be 28.14 minutes and a range of 46.16 km.

A possible future improvement of the mathematical model would include the blade element model similar to the modelling of helicopter rotors, applied mainly to the front propellers that are subject to situations of high angle-of-attack and horizontal flight speed. Modifying the front propellers to helicopter rotors, with cyclic and collective control, employing a swashplate, would be a substantial improvement.