2.1 Aircraft system modelling

The aircraft dynamics model consists of five parts: rotor system, propeller, fuselage, and two empennage surfaces. The rotor model’s aerodynamic forces and moments are obtained based on the blade element method. The flapping motion of the rigid blade is simulated by applying the equivalent flapping offset and flapping spring method [Reference Yuan, Thomson and Chen2]. The complex inflow interference between coaxial rotors generates strong nose-up moments during low-speed flight [Reference Weiand and Krenik22]. Here, an analytical model with the first harmonic inflow approximation [Reference Ruddell8, Reference Johnson23] is applied.

The uniform inflows of coaxial rotors are calculated by revised momentum theory, that is,

(1) \begin{equation}\begin{array}{*{20}{l}}{{f_{Gu}}{C_{Tu}} = 2{v_{iu}}\sqrt {{\mu ^2} + {\lambda _u}} }\\[5pt]{{f_{Gl}}{C_{Tl}} = 2{v_{il}}\sqrt {{\mu ^2} + {\lambda _l}} }\end{array}\end{equation}

where ${f_{Gu}}$ and ${f_{Gl}}$ are ground effect factors. The inflow ${\lambda _u}$ and ${\lambda _l}$ includes self-induced velocity and airflow interference approximated by rotor interference factors [Reference Valkov24, Reference Zhao, Yu and Chen25]. Detailed descriptions of inflow calculation are given in Ref. (Reference Zhao, Yu and Chen25).

Ground effect, increasing the rotor thrust when the helicopter is near the ground surface, is significant to the takeoff and landing performance. The “ground cushion” effect decreases the induced speed, i.e. decreases the induced power required. The influence of ground effect on power performance diminishes as the increase of flight altitude and flight. Cheeseman and Bennett [Reference Cheeseman and Bennett26] proposed the ground effect factor for the single rotor:

(2) \begin{equation}{f_G} = 1 - \frac{{\sigma {a_\infty }\lambda }}{{4{C_T}}}\frac{{{{(R/4{H_R})}^2}}}{{1 + {{(\mu /\lambda )}^2}}}\end{equation}

Coaxial rotors have different ground effect factors due to mutual interference and different ground distances [Reference Lim, McAlister and Johnson27]. The measured data from Ref. (Reference Lim, McAlister and Johnson27), as shown in Fig. 3, indicates that the lower rotor has more thrust increment in the hover state. In contrast, the upper rotor is less sensitive to ground space changes, with only about 8% increment.

Figure 3. Ground effect of coaxial rotors in hover.

However, there is no quantified data on ground effect of the coaxial rotors regarding forward flight. Therefore, it is assumed that the trend of the coaxial rotors with speed is similar to that of a single rotor. The ground effect of the coaxial rotors is simulated by modifying Equation (2) with coefficients ${K_{Gu}}$ and ${K_{Gl}}$ :

(3) \begin{align}{f_{Gu}} & = 1 - {K_{Gu}}\frac{{\sigma {a_\infty }{\lambda _u}}}{{4{C_{Tu}}}}\frac{{{{(R/4{H_{Ru}})}^2}}}{{1 + {{(\mu /{\lambda _u})}^2}}}\nonumber\\[4pt]{f_{Gl}} & = 1 - {K_{Gl}}\frac{{\sigma {a_\infty }{\lambda _l}}}{{4{C_{Tl}}}}\frac{{{{(R/4{H_{Rl}})}^2}}}{{1 + {{(\mu /{\lambda _l})}^2}}}\end{align}

where ${K_{Gu}}$ and ${K_{Gl}}$ are fitted to meet the thrust increment at ground height of around $0.25R$ (shown in Fig. 3). They are set as 0.163 and 0.838 in this research, respectively. The coefficients are less than one due to the additional interference increasing the inflow ratio ( ${\lambda _u}$ and ${\lambda _l}$ ).

The modelling of the propeller is similar to that of the rotor, such as the aerodynamic loads and uniform induced inflow. Considering the propeller is controlled by a collective pitch, no flapping motion is simulated. A constant control allocation between the rotor longitudinal cyclic and propeller pitch is applied. Define coefficient ${K_{A2pp}}$ as the ratio between their incremental stick input, that is,

(4) \begin{equation}{K_{A2pp}} = \frac{{\Delta {\delta _{pp}}}}{{\Delta {\delta _A}}}\end{equation}

Then their stick inputs are

(5) \begin{align}{\delta _A} & = {\delta _{A0}} + \Delta {\delta _A}\nonumber\\[4pt]{\delta _{pp}} & = {\delta _{pp0}} + {K_{A2pp}}\Delta {\delta _A}\end{align}

where ${\delta _{A0}}$ and ${\delta _{pp0}}$ are determined by initial hover trim state.

The empennage surface aerodynamics is modelled with a lift-curve slope prior to stall and a general curve fit for large angles of attack. The fuselage aerodynamic loads are calculated using the scaled aerodynamic load coefficients [Reference Tr28, Reference Pleasants29].

The flight dynamic model for the coaxial compound helicopter is expressed in the form of

(6) \begin{equation}\dot X = F(X,U,t)\end{equation}

where

\begin{equation*}\left\{ {\begin{array}{*{20}{l}}{X = {{\left[ {\begin{array}{*{20}{c}}u \quad v \quad w \quad p \quad q \quad r \quad \Phi \quad \Theta \quad \Psi \\[4pt]X \quad Y \quad Z \quad \delta \quad {}\quad {}\quad {} \quad {}\quad \quad \quad \quad {}\end{array}} \right]}^T}}\\[20pt]{U = \dot \delta = [{{\dot \delta }_c},{{\dot \delta }_A},{{\dot \delta }_B},{{\dot \delta }_r},{{\dot \delta }_{pp}}{]^T}}\end{array}} \right.\end{equation*}

where time derivatives of sticks $\dot \delta $ are introduced as control variables to avoid unreasonable discontinuities, also known as bang-bang control. In optimisation, numerical precision loss and error propagation might be caused by the multiple-magnitude variables. Hence, they are normalised and scaled to near unity to improve computational efficiency and rate of convergence, detailed can be found in Refs (Reference Zhao, Yu and Chen25, Reference Yan and Chen30).

A biodynamic model [Reference Bottasso, Maisano and Scorcelletti31] considering pilot response delay is integrated into the dynamic model. After passing through the pilot biodynamic model, the pilot intentions are translated into actual control displacements in the cockpit, presented as

(7) \begin{equation}{H_p}(s) = \frac{{{T_L}s + 1}}{{{T_I}s + 1}}\frac{{{e^{ - {\tau _p}s}}}}{{{T_N}s + 1}}\end{equation}

where ${e^{ - {\tau _p}s}}$ is a pure time delay for pilot’s cognitive responsiveness, $({T_L}s - 1)/({T_I}s + 1)$ reflects the lead/lag of pilot’s control input, ${T_N}$ is the neuromuscular lag time constant.

2.2 Takeoff path prediction

The takeoff path prediction is transformed into a trajectory optimisation problem, which calculates the best operations and paths subjected to certain constraints. Details of the algorithm are given in Refs (Reference Yan and Chen30, Reference Yu, Yan and Chen32). A short takeoff procedure consists of two stages: (1) level acceleration to liftoff velocity; (2) climb over a certain obstacle at height ${H_O}$ , as shown in Fig. 4. The first subfigure presents the ideal takeoff, which is widely used as the nominal path [Reference Bibik and Narkiewicz15, Reference Enciu and Rosen16, Reference Carlson20, Reference Sharma33]. The helicopter first accelerates at a typical rate $0.2g$ [Reference Carlson20] and holds constant height to accumulate kinetic energy for more excess power. After reaching liftoff speed, ${V_{LO}}$ , the helicopter performs a climb out at angle $\gamma $ (typically ${6^ \circ }$ [Reference Carlson20]) and constant airspeed. Among all liftoff speeds, there is a critical speed, ${V_{cL}}$ , for minimum takeoff distance to clear an obstacle.

Figure 4. Takeoff trajectory.

The ideal path neglects the hover transition to acceleration and final climb out. In order to smooth the trajectory and improve the feasibility of the flight path, a height variation margin of $ \pm 0.5{\rm{m}}$ is applied to Phase I. The acceleration rate is below $0.25g$ considering the extra propulsion. Though decelerating velocity for fewer the takeoff distance is theoretically feasible in climbing, it is prohibited for ease of implementation [Reference Schmitz and Vause18]. No other constraint for the climbing angle or speed is included. According to takeoff descriptions in Fig. 4(b), three flight parameters directly influence the takeoff: nose-down attitude, liftoff speed and obstacle height. The paths with varying parameters are considered for design a global optimum.

The cost function of the takeoff problem is described as

(8) \begin{equation}\min J = \frac{{{X_f}}}{{{X_g}}} + 0.1\frac{{{t_f}}}{{{t_g}}} + \int_{{t_0}}^{{t_f}} \left(\dot \delta _{\rm{c}}^2 + \dot \delta _{\rm{A}}^2 + \dot \delta _{\rm{B}}^2 + \dot \delta _{\rm{r}}^2\right){\rm{d}}t\end{equation}

including (a) terminal distance and time, and (b) sum of the quadratic control derivations throughout the path. The distance and time are scaled by their estimated distance and time yielded from the ideal takeoff path, that is

(9a) \begin{equation}{X_g} = \frac{{V_{LO}^2}}{{2{a_L}}} + \frac{{{H_O}}}{{\tan \gamma }}\end{equation}

(9b) \begin{equation}{t_g} = \frac{{{V_{LO}}}}{{{a_L}}} + \frac{{{H_O}}}{{{V_{LO}}\tan \gamma }}\end{equation}

which helps improve the convergence speed and increases weight on control derivations for better handling quality.

To solve the nonlinear boundary value problem, a collocation method called direct multiple shooting [Reference Yan and Chen30, Reference Chi, Yan and Chen34] is utilised. The problem is then transcribed into discrete nonlinear programming (NLP) by discretizing continuous states and controls into short time segments. Finally, the solution is obtained by solving NLP with the sequential quadratic programming method.

2.3 Trajectory assessment

The minimum distance and stick variation rate are achieved through optimisation with certain constraints. Here, we aim to develop a global optimum propeller control. Hence, the paths and relevant manoeuvres with varying flight parameters are evaluated. The general layout of takeoff prediction and evaluation is presented in Fig. 5.

Figure 5. Flow chart of takeoff path prediction and evaluation.

The takeoff distance, which is crucial for obstacle avoidance, is the most important indicator. It represents the capability of the aircraft. Besides, two quantities are applied to evaluate the ease of pilot implementation. One is pilot workload based on control inputs, and the other is path sensitivity to takeoff parameters.

2.3.1 Pilot workload evaluation

A helicopter’s short takeoff significantly depends upon the pilot’s control commands [Reference Enciu and Rosen21]. Although a minimum sum of the quadratic control derivations is achieved, the inherent workload of the takeoff procedure is not alleviated without a suitable control strategy. The coaxial compound helicopter largely relies on nose-down and propeller propulsion for acceleration, limited by insufficient propulsive force from the rotor. Besides, the auxiliary propeller and the intensive interference between rotors complicate the flight dynamics of the vehicle. These factors underline the importance and difficulty of longitudinal control.

Time-frequency representations (TFRs) [Reference Yan and Chen30, Reference Tritschler, O’Connor and Klyde35] is introduced to reflect the temporal variations in a manoeuvre. For a control input time history, $\delta (t)$ , the pilot time-varying cutoff frequency is given by ${\omega _{{\rm{co}}}}(t)$ , and the scalogram TFR of the stick power is given by ${G_{\delta \delta }}(\omega ,t)$ , which is relevant to the operation magnitude. There are periods with high time-varying cutoff frequencies, while the low stick activities have little or no practical consequence. Thus, time-varying power frequency, ${G_\omega }(t)$ is proposed as [Reference Tritschler, O’Connor and Klyde35]

(10) \begin{equation}{G_\omega }({t_i}) = {\omega _{co}}({t_i})\max {G_{\delta \delta }}(\omega ,{t_i})\end{equation}

which multiplies time-varying cutoff frequency by the maximum signal power at that instant ${t_i}$ . As such, the power frequency is reduced whenever the pilot stick activity is low. Conversely, the power frequency is magnified whenever the pilot stick activity is high.

2.3.2 Sensitivity evaluation

The optimum distance varies with flight parameters. Hence, path sensitivity is proposed for aircraft safety, aiming at evaluating the possible deviation from the expected path due to flight parameter variation. Although an optimal path is available in theoretical prediction, the actual trajectory might deviate from it due to pilot manoeuvre skills or environmental disturbance. In this case, the final distance might be much higher than the expected, resulting in obstacle contact. Therefore, a robust flight path is of vital importance for aircraft safety. In order to evaluate the robustness, the sensitivities of takeoff distance to liftoff speed, flight attitude, and obstacle height are analysed.

The takeoff distance could be expressed as a function of flight parameters according to the trajectory descriptions in Fig. 4(b). Based on the form of Equation (9a), the distance to a certain height is expressed as

(11) \begin{align} X & = \underbrace {({S_a} + {S_{a2h}}{H_O})}_{{\rm{related}}\,\,{\rm{to}}\,\,{\rm{acceleration}}}\left( ({V_{LO}} - \underbrace {({S_v} + {S_{v2h}}{H_O})}_{{\rm{related}}\,\,{\rm{to}}\,\,{\rm{critical}}\,\,{\rm{speed}}} \right)^2\nonumber\\& + {S_x} + \underbrace {{S_\gamma }{H_O}}_{{\rm{related}}\,\,{\rm{to}}\,\,{\rm{climb}}\,\,{\rm{angle}}}\end{align}

where ${S_{( \bullet )}}$ are sensitivity indices. The schematic diagram (Fig. 6) shows the relationship between the six sensitivity indices, where two calculated paths (solid lines), denoted as ${X_1}$ and ${X_2}$ , and an imaginary path (dotted line) at zero height are presented.

Figure 6. Schematic diagram of sensitivity indices.

$({S_a} + {S_{a2h}}{H_O})$ represents the concavity of a quadratic curve, and refers to the distance variation to square of speed, i.e. is relevant to acceleration rate. It usually increases with height due to lift reduction as out of ground effect. $({S_v} + {S_{v2h}}{H_O})$ is related to the critical speed, ${V_{cL}}$ , representing the speed for the minimum distance, $({S_x} + {S_\gamma }{H_O})$ . Generally speaking, the critical speed increases with ${H_O}$ [Reference Schmitz and Vause18] since high translational velocity benefits climbing. The last coefficient, ${S_\gamma }$ , refers to the distance increment with obstacle height, and is relevant to the climbing angle.

The polynomial is a possible relationship between the distance and takeoff parameters, where high fitting coefficients indicates unpleasant deviation from the desired path. According to Fig. 6 and Equation (11), the sensitivities could be classified into three groups:

1. 1. baseline imaginary takeoff performance, including ${S_a}$ , ${S_v}$ and ${S_x}$

2. 2. performance increment with obstacle height, including ${S_{a2h}}$ , ${S_{v2h}}$ and ${S_\gamma }$

3. 3. total sensitivity at certain height, such as ${S_a} + {S_{a2h}}{H_O}$

3.1 Model validation

In the coaxial compound helicopter, several key features influence the takeoff, such as the interference between rotors, rigid blades, and the propeller. In order to simulate these characteristics, a validated model is needed. Therefore, a rotorcraft with rich data published [Reference Ruddell8, Reference Valkov24, Reference Pleasants29], XH-59A, is chosen as the prototype helicopter. Meanwhile, a propeller instead of the original propulsion is applied. The calculated stick movements, aircraft attitude and power required are validated against the flight test of the pure coaxial helicopter [Reference Ruddell8], as shown in Fig. 7. The large longitudinal stick movement at low speed indicates well-modelled interference between the coaxial rotors.

Figure 7. Level flight trim validation against flight test.

A propeller is applied to the coaxial compound helicopter, its parameters are the same as that in Ref. (Reference Yuan, Thomson and Chen2). The low-speed performance of the propeller is verified against wind-tunnel data [Reference Dunham, Gentry and Coe36]. The detail of the experimental eight-blade propeller settings and parameters can be found in Ref. (Reference Dunham, Gentry and Coe36). In propeller validation, the advancing ratio, lift and power coefficients are normalised by rotor rotation (rpm) and diameter. According to Fig. 8, the calculated results of the aircraft and propeller show good agreement with the flight test data.

Figure 8. Low-speed propeller validation against wind tunnel test.

3.2 Method feasibility evaluation

For validation of the optimisation technique, the results generated by the theoretical model (solid curves) are compared against the flight test data of UH-1B [Reference Schmitz and Vause18]. Figure 9 shows the distance to clear a 50-ft obstacle as a function of liftoff speed for three excess power conditions. Detailed flight test operating parameters, such as gross weight, density altitude, and ambient temperature, are varied as per each power condition [Reference Schmitz and Vause18].

Figure 9. Distance to clear a 50-ft obstacle, as a function of excess power and liftoff velocity.

In Fig. 9, the optimal path model results show satisfying agreement with flight test data. In the two cases with relatively high excess power, the aircraft’s takeoff procedure is similar to the ideal flight illustrated in Fig. 4. In other words, the acceleration and climbing ability is not degraded. By contrast, the theoretical predictions in the low-power case slightly underestimate the test results due to two factors. Firstly, the accuracy of the flight test data was influenced by integrated errors over a long takeoff time, especially in the heavily-loaded condition. The other reason is that the optimisation algorithm intends to search for the minimum distance to clear a given obstacle height, too ideal for realistic performance. After all, the theoretical model should be sufficient for predicting takeoff path and analysing relevant procedures.

3.3 Trim analysis

In the takeoff of a heavily loaded aircraft, the additional in-ground-effect (IGE) thrust benefits the acceleration. The influence of ground effect to level-flight trim power is presented in Fig. 10, with varying skid height and airspeed. The environmental air density altitude is assumed as 4,500m, atmospheric temperature as ISA + 15deg for simulating the heavily loaded condition. The takeoff weight is determined by the IGE hover with excess power for 0.5m/s climbing rate [37].

Figure 10(a) shows the total power required ${P_T}$ as a function of skid height and forward flight speed with pure rotors without the propeller. The power gain from ground effect decreases as velocity or skid height increases, in line with the trend of ground effect. In the level flight trim of coaxial compound helicopter with turning propeller, the airframe attitude is preset to determine the propeller pitch [Reference Yuan, Thomson and Chen3, Reference Ferguson7]. The power required at different pitch attitudes $\Theta $ and relevant gross weight are shown in Fig. 10(b). With similar excess power, there’s no significant difference between their total power required. However, high attitude results in high propeller power ( ${P_{pp}}$ ), leaving less power for rotor lift. As a result, the aircraft’s maximum gross weight is degraded.

4.1 Initial propeller thrust design

The influence of airframe attitude on takeoff distance is presented in Figs 11 and 12. Different flight parameters are considered for an overall evaluation. The slices at certain attitude and liftoff speeds are also given. The coefficient ${K_{A2pp}}$ is assumed as an optimisation parameter to exclude its influence, and the optimised results are given in Fig. 13.

Figure 11. Takeoff distances under different airframe pitch angles ${\Theta _{\min }} = - {5^ \circ }$ .

Figure 12. Takeoff distances under different airframe pitch angles ${\Theta _{\min }} = - {10^ \circ }$ .

Figure 13. Optimal allocation variation with takeoff parameters under different initial pitch angles.

According to the three figures, the initial pitch angle for takeoff is set as ${1^ \circ }$ for three reasons. Firstly, as discussed in the trim analysis (Fig. 10(b)), the aircraft is of heavier gross weight with a lower pitch angle. In this case, the controller designed for a severe condition is compatible with a lighter aircraft. The second factor is relevant to the distance. According to Figs 11(b) and 12(b), the distances show a near-linear trend with attitude. It indicates that the distance reduction is caused by the light weight other than the initial propulsion. In other words, despite the heavier weight, the takeoff distance is hardly degraded by the attitude, especially as liftoff speed increases. Although there is a sharp rise at low speeds, it is avoidable by performing a short takeoff. Thus, it is feasible to introduce the propeller while maintaining the vehicle’s maximum gross weight. Last but not least, low initial propulsion benefits the following operation, as indicated by Fig. 13. The case of ${\Theta _{{\rm{ini}}}}= 1^\circ $ has the broadest positive range and smallest variation of the ${K_{A2pp}}$ . Thus, the designed controller would not be too far from the optimum. On the contrary, a high attitude limits the propeller pitch increment, resulting in reverse propulsion increment from the rotors and propeller.

The distribution of ${K_{A2pp}}$ , shown in Fig. 13, reflects the best allocation variation with takeoff parameters. These curves could be classified into two groups by the attitude constraint. The group of ${\Theta _{\min }} = - {5^ \circ }$ is higher, where high ${K_{A2pp}}$ makes up the level acceleration. In each group, ${K_{A2pp}}$ increases with liftoff speed for better acceleration, decreasing with obstacle height for more climbing power. To sum up, the allocation shows a regular trend with flight parameters, with minor variations. In addition, there is a high coincidence at low speed, indicating the feasibility of a constant allocation.

4.2 Control allocation design

In this subsection, a constant allocation for global takeoff performance is designed. In order to avoid reverse propulsion from the propeller and rotor that significantly damages the rotorcraft’s handling quality, a positive ${K_{A2pp}}$ is preferred. Its influence on paths is calculated and evaluated, with ${K_{A2pp}}$ from 0.1 to 0.8 at an interval of 0.1.

4.2.1 Takeoff distance

The distances under different allocations are presented in Figs 14 and 15, with varying takeoff parameters. There are infeasible solutions and steep rises when the allocation is below 0.3 or above 0.6. Too low ${K_{A2pp}}$ results in a low propeller pitch increment as translational velocity increases, and a high ${K_{A2pp}}$ leads to insufficient power for climbing. In the cases between 0.3 and 0.6, low allocation usually achieves significantly shorter distances, especially at low liftoff speeds. According to Fig. 13, a high allocation is preferred at high velocity. By contrast, the length is slightly shortened or even longer according to the Figs 14(b), (c) and 15(b), (c). Therefore, a low coefficient is preferred from aspects of distance. Nevertheless, the range of 0.3–0.6 is chosen for an overall path evaluation on pilot workload and sensitivities.

Figure 14. Takeoff distances under different airframe pitch angles ${\Theta _{\min }} = - {5^ \circ }$ .

Figure 15. Takeoff distances under different airframe pitch angles ${\Theta _{\min }} = - {10^ \circ }$ .

4.2.2 Pilot workload analysis

The power frequencies of the stick movements under different ${K_{A2pp}}$ are shown in Figs 16 and 17. The obstacle height and relevant speeds are given in each subfigure, with speed increment of 2m/s. There is hardly any collective movement, resulting from two reasons. Firstly, pulling up the collective stick with little excess power is not practical. On the other hand, lowering the stick increases the pilot operation and degrades climbing. Thus, the collective is held constant. The longitudinal power frequency varies with fight paths. It rises with height or as pitch constraint from $ - {5^ \circ }$ to $ - {10^ \circ }$ , where more longitudinal operations are needed. Nevertheless, the pilot workload hardly changes with the allocation. In other words, although the allocation influences flight paths and operations with different propulsion, the range of power frequencies (multiplication of frequency by stick movement) are similar. Thus, the recommended allocation range is still between 0.3 and 0.6.

Figure 16. Pilot workload under different allocations ${\Theta _{\min }} = - {5^ \circ }$ .

Figure 17. Pilot workload under different allocations ${\Theta _{\min }} = - {10^ \circ }$ .

4.2.3 Sensitivity analysis

Table 2 presents the distance sensitivity indices under different allocations when ${\Theta _{\min }} = - {5^ \circ }$ , which are numerical representations of the distance variation in Fig. 14. The indices, ${S_a}$ and ${S_{a2h}}$ , represent the overall trend of paths against speed. Their rapid increases with allocation are in line with the steep rises at high ${K_{A2pp}}$ and high height. Thus, the low allocation has the lowest acceleration sensitivities for all heights. ${S_v}$ refers to the imaginary critical speed at zero obstacle height. It is the lowest when ${K_{A2pp}} = 0.3$ because of more excess power from low propulsion. The critical speed increment with height is presented by ${S_{v2h}}$ . Despite the lowest ${S_{v2h}}$ of high allocation (0.6), its critical speed ( ${V_{cL}} = {S_v} + {S_{v2h}}{H_O}$ ) is the highest for all heights. As a result, the level acceleration path, represented by ${S_x}$ , is longer. Although the lower ${S_\gamma }$ partially offsets its distance disadvantage as height increases, the total distance remains the longest. Therefore, the high allocation has high sensitivities despite the minor alleviation as height increases.

Table 2. Distance sensitivity under different allocations ${\Theta _{\min }} = - {5^ \circ }$

The sensitivities when ${\Theta _{\min }} = - {10^ \circ }$ are shown in Table 3. Their trends with ${K_{A2pp}}$ are similar to the corresponding cases in Table 2, except the rapid decrease of acceleration indicator ${S_a}$ . The negative ${S_a}$ in the high-allocation case indicates the narrowed gap of acceleration sensitivities with the others. However, it remains the highest at all heights. From ${\Theta _{\min }} = - {5^ \circ }$ to $ - {10^ \circ }$ , there are no significantly numerical changes in height-related sensitivities, such as ${S_{a2h}}$ and ${S_{v2h}}$ . Because climbing ability largely depends upon liftoff speed for climbing power other than nose-down attitude. Hence, high allocation still has the highest sensitivities.

Table 3. Distance sensitivity under different allocations ${\Theta _{\min }} = - {10^ \circ }$

Finally, the proposed ${K_{A2pp}}$ for the coaxial compound helicopter is set as 0.3. Its takeoff distance is relatively the shortest among all allocations, as demonstrated in the distance discussion and relevant sensitivity indices ( ${S_x}$ , ${S_\gamma }$ ). In addition, its acceleration- and speed-related sensitivities at all heights are the lowest. Although the increments with height, ${S_{v2h}}$ and ${S_\gamma }$ , are slightly higher, the minor increases are acceptable considering the magnitude of speed and distance.

4.3 Assessment of the proposed controller

From the research above, the combination of low attitude ( ${1^ \circ }$ ) and moderate allocation ( ${K_{A2pp}}$ ) achieves the best takeoff performance among all cases. Here, a baseline strategy is introduced for assessing the proposed controller. Only the rotor is operated in the baseline, and the propeller is not engaged. The takeoff paths under the two strategies are calculated with the aircraft under the same gross weight and atmospheric conditions.

The distances under the two propeller control strategies are given in Fig. 18. According to the imaginary distances at ${H_O} = 0$ , the propeller control benefits the level acceleration. As a result, the proposed strategy (triangle symbols) requires fewer distances to the same height in all cases, with the average length shortened by 12.6%.

Figure 18. Takeoff distance comparison between the proposed and baseline strategies.

From Fig. 19, the longitudinal pilot workload is lower than the baseline. The mean power frequency is reduced by 9.8% on average, and the maximum by 7.3%. Higher alleviation is obtained in the cases of higher liftoff speed, height or nose-down, where longer acceleration is required. It is because the propulsion alleviates stick movement for acceleration.

Figure 19. Longitudinal pilot workload comparison between the proposed and baseline strategies.

The sensitivity indices under different strategies are presented in Table 4, with varying attitudes. Almost all indicators are significantly alleviated by applying the proposed method, indicating the path is less sensitive to liftoff speed or height deviation. Also, the distance is shorter, in line with results from Fig. 18. In addition, the path deviation caused by different attitudes is reduced, as shown by the narrowed gaps between cases of $ - {5^ \circ }$ and $ - {10^ \circ }$ . Therefore, the takeoff path sensitivities to all three flight parameters are fully alleviated.

Table 4. Distance sensitivity comparison between the proposed and baseline strategies

In practice, the non-rotating propeller would produce additional drag, which is neglected in this paper. Thus, it can be foreseen that the performance in real scenarios should be further worse than the baseline calculation results. On the other hand, the propeller effects are considered in this propeller-embedded optimisation. Meanwhile, the results indicate that its performance, including takeoff distance, pilot workload, and sensitivity, is significantly improved even compared to the baseline calculation results. This outcome demonstrates that it is feasible and helpful to integrate the propeller into the low-speed takeoff.