Impact of differential torsional rotor cant on the flight characteristics of a passenger-grade quadrotor

Abstract

At the Institute of Flight Systems at DLR, studies have been performed to understand the flight characteristics of novel eVTOL configurations. As part of these studies, previously conducted handling qualities assessments on a two-passenger generic quadrotor configuration had revealed major deficiencies about its yaw axis. Based on this result, the quadrotor model has been modified by differential torsional canting to improve its yaw characteristics. This paper analyzes the resulting impacts of such modification on the flight performance, dynamic stability and handling qualities. A piloted simulator test campaign was conducted to assess predicted and assigned handling qualities levels in compliance with the quantitative and qualitative performance standards of ADS-33E. The results show an improvement in midterm yaw response at the expense of increase in total required power. The pilot ratings and comments confirm the improvement on the yaw response upon the flown MTEs.

1 Introduction

In the previous study [1], an initial investigation on the flight dynamics and handling qualities of a passenger-grade virtual quadrotor model was conducted. The linear flight dynamics analyses revealed unstable phugoid modes about the pitch and roll axes. Furthermore, the criteria-based handling qualities predictions revealed poor yaw characteristics. Based on the literature study, the model was modified by the so-called differential torsional rotor cant to improve the yaw characteristics. Furthermore, an automatic control system was integrated to provide less oscillations during the simulator tests. This study aims to show the impacts of the differential torsional rotor canting on the flight characteristics of a passenger-grade quadrotor in terms of flight performance, linear dynamics, and handling qualities. The paper starts with the motivational background and continues with technical approach, introducing the process flow, as well as the baseline and the modified variants of the quadrotor model used. This is followed by the trim analysis comparison between the baseline and the modified variants. Thereafter, linear time-invariant models of these variants are compared at hover state. Next, the handling qualities of the baseline variant and one selected modified variant are compared by offline criteria and pilot ratings on the flown MTEs for the hover and low-speed regime in the flight simulator. Finally, conclusions of the study are given.

2 Motivation

The interest in eVTOLs has been growing since the last decade. Currently, there are more than 600 registered eVTOLs in the World eVTOL Aircraft Directory of the Vertical Flight Society [2], divided into multiple categories with a broad spectrum of use cases. Quadrotors are one of the well-established candidates among the eVTOL concepts categorized under multirotor configurations. Quadrotor configurations distribute the thrust within an array of four rotors, usually vectoring parallel to each other. These rotors mostly have an rpm-driven architecture, where each rotor is powered by an electric motor. The use of electric motors enables less complexity and lower mass by eliminating sophisticated parts, such as swashplates, bearings, hinges and transmission.

Quadrotor vehicles are controlled by varying the rotational speed and thereby the thrust in individual rotors using conventional helicopter inceptors. The maneuvers around the lateral and longitudinal axes (roll & pitch) are achieved by the anti-symmetric thrust along these axes, whereas the differential torque of the rotors is employed in order to initiate maneuver around the vertical axis (yaw).

The unique control concept of such quadrotor systems also brings some challenges. For one, generating upward lift through four independent rotors inherently tend to trigger low frequency unstable oscillations about the lateral and longitudinal axes in hover. Such vehicle behaviours are called phugoid modes, where the initial pitch/roll motion results with a speed increase in the direction of the motion, and thus leads to an interchanging oscillation between the transnational and angular kinetic energy. Another challenge emerges by the dynamic response characteristics of the vehicle about the yaw axis. In this case, the torque difference produced by the rotors may not be as effective as the thrust produced by a tail rotor to initiate the yaw motion. As this fact does not pose a control deficit for the small-scale quadrotor drones, the growing size of the vehicle for the passenger transport raises the inertia exponentially. Hence, the differential torque may no longer affect the vehicle yaw control adequately, which is why additional control modifications may be required to overcome these challenges.

The German Aerospace Center has been conducting studies to understand the general aspects of the flight characteristics of such passenger-size multirotor configurations. The virtual models¹ are put through quantitative and qualitative analysis and evaluated through the criteria and mission task elements given in the ADS-33E [3].

In the previous study [1], unstable flying characteristics of the quadrotor model reflected on the handling qualities ratings of the pilots falling mostly between Level 2 and 3 with respect to the Cooper-Harper Handling Qualities Rating Scale (HQR) [4]. The comments of the pilots concentrated on the unstable oscillations about the pitch and roll axis, as well as the tardy yaw response of the vehicle to the pedal inputs.

As a response to these feedback, some modifications and improvements have been applied on the virtual flight model from the previous study [1]. The model has been equipped with a stability augmentation system to cancel the phugoid oscillations. For further improvements in the yaw response, a constructive approach was required. In this regard, Niemiec et. al. [5] investigated the effects of multiple flapwise and torsional canting modes on the trim and flight dynamics of a small-scale quadrotor drone. They showed that a differential torsional canting mode can increase the yaw authority of the vehicle by up to 325%. In addition, the roll sensitivity is further increased, while the pitch sensitivity is reduced due to change in the rotor flow conditions resulting from the rotor cant. To investigate the effects of this approach on a larger scale quadrotor vehicle, the flight model has been modified by applying the differential torsional cant.

A detailed definition of rotor canting is given in [5]. According to this definition, the torsional cant ($\chi$) is described as the tilt of the rotor about the axis extending radially from the geometric center of the vehicle towards the rotor hub called the boom axis. As is illustrated in Fig. 1, the rotor plane sets a new orientation along its canted axis. The resulting inclinations of the rotors are shown with red arrows indicating the canting direction with respect to the body-fixed xy-plane. Depending on the canting mode, a positive or negative $\chi$ is applied to the individual rotors. As there are four separate torsional canting modes introduced, Fig. 1 illustrates an example of the differential torsional canting mode, which is the focused configurational modification in this study.

Fig. 1

Torsional canting on quadrotor

3 Technical approach

The technical approach incorporates automated steps in a framework. The models are created using the CPACS [6] data format. The aeromechanical computations take place in HOST [7] provided by Airbus Helicopters. Three main types of computations are available in HOST: trim analysis, linear time-invariant modeling and non-linear time domain simulation. The piloted test campaign is conducted in the flight simulator AVES [8], a research simulation facility operated by the Institute of Flight Systems at DLR in Braunschweig.

Fig. 2

Framework process flow

Figure 2 shows the process flow of the framework. Initially, the model parameters are gathered in a table within Excel. Then, the CPACS file of the vehicle model is created through a Python script (CPACS MODELER) by reading these parameters. At this stage, the CPACS file contains all the necessary data for the computations. Moreover, the model geometry can be visualized using TIGLViewer [9] developed for CPACS. The HOST configuration is derived from CPACS file using the in-house developed tool-wrapper CPACS4HOST (C4H). The created model can be used both in offline analysis and simulator tests. In this work, the offline analysis comprises of trim analysis and the linearization of the flight models executed in HOST. Following the offline analysis computations, C4H updates the CPACS file with the result data delivered by HOST. Thus, the whole process data from parameterization to final analysis is contained in a single file. As for the simulator tests, the non-linear time domain simulation feature of HOST is connected to AVES (more details in Sect. 6), thus enabling a real-time simulation environment using the same vehicle configuration data.

Following sections give details of the design and modeling of the baseline and modified quadrotor configurations.

Fig. 3

3-View visualization of the baseline configuration (see Rotor in Table 1 for a, b and c)

Table 1 Design parameters of the baseline configuration (** identifies the complemented parameters)

3.1 Modeling of the quadrotor

The studied baseline configuration is taken from the parameterized model given in [10], which is a two-passenger rpm-controlled quadrotor configuration sized with respect to the urban air mobility mission requirements of NASA. The mission profile consists of two separate legs with 37.5 nmi each to be flown at a headwind of 10 kts with a reserve of 20 min [11,12,13]. Table 1 shows the parameters of the baseline configuration, while Fig. 3 depicts a representative 3-view visualization of this model. Here, parameters that have not been provided in the aforementioned studies are internally complemented with additional data.

The fuselage aerodynamics are modeled with a simple drag area of $D/q=0.6\,\hbox {m}^{2}$ ($C_{X,f}=1$) combined with a negative pitching moment (see Fig. 4). The drag area was selected based on the given data of the generic quadrotor models with different passenger sizes [11, 14], where D is the cruise drag and q is the dynamic pressure. Here, the assumed drag area contains the wetted area of the fuselage along with the undercarriage, faired rotor hubs and support arms. With the drag area and the rotor radius being the reference values, a linear function for the pitching moment coefficient of the fuselage $C_{M,f}$ represents a virtual moment resulting from the drag force acting on the fuselage nose at the level of CoG.

Fig. 4

Fuselage aerodynamic coefficients

The rotors are placed equidistantly in a cross arrangement from the geometric center in lateral and longitudinal directions by the factor $\text {a}=1.25$, multiplied with the rotor radius. Here, the geometric center is assumed as being the location of the center of gravity and the fuselage aerodynamic center. This approach neglects the moments that occur due to the offset positioning of the thrust elements, hence provides equal force distribution onto the rotors at hover. Moreover, the forward and aft rotors are separated in vertical direction. Although the rotor-rotor interaction is neglected in here, study shows that such separation in vertical direction can reduce the power up to 4.5% in forward flight [11]. First, the aft rotors are vertically positioned with respect to the center of gravity by a distance factor of b and the forward rotors are separated from the rear rotors by a distance determined by factor c, to be multiplied by the rotor radius. Figure 3 shows the rotor indexing and rotational directions along with the vehicle orientation in the body-fixed coordinate system. It should be noted that the positive rotor rotational direction for each $\Omega _i$ acts in the counter-clockwise (CCW) direction, whereas the positive yaw motion $\psi$ acts in the clockwise (CW) direction. In that respect, rotors 1 and 2 rotate in negative sense, and rotors 3 and 4 rotate in positive sense.

The rotor blade geometry is modeled rectangular with a constant NACA 23012 profile starting at the 20% of the rotor radius. The blade is twisted with a constant angle of $35^{\circ }$ at the root and a linear distribution of $-18.4^{\circ }/\hbox {m}$, such that the tip profile remains untwisted. The blades are numerically discretisized with 20 elements, employing the Pitt-Peters inflow model [7]. In HOST, the radial spacing of the blade elements are unequally distributed, such that the circle area of each blade element is identical. As for the azimuthal discretization of the rotor blades, a $5^{\circ }$ spacing is selected, which corresponds to 72 equally spaced radial stations per rotor revolution. For the simplicity of this study, the blades are considered rigid, neglecting the effects of blade elasticity.

3.2 Differential torsional canting

Differential torsional canting is one of the four introduced torsional rotor canting modes by [5]. As schematized in Fig. 5, this mode alternates the torsional canting of the individual rotors in a positive or negative sense in a way that the resulting thrust symmetry in hover is not violated. In addition, the in-plane thrust component of each rotor can contribute to yaw acceleration without including any lateral or longitudinal acceleration. Unlike in the original definition, CW-rotating rotors are canted negatively and the CCW-rotating rotors are canted positively in the configuration studied. This canting mode is denoted by the canting angle vector

$$\begin{aligned} \varvec{\chi } = \bigl (-\chi _d, -\chi _d, \chi _d, \chi _d\bigr ), \end{aligned}$$

(1)

where the positive angle $\chi _d$ is the same for each rotor.

Since the rotors are canted relative to the rotor boom axis, a transformation is applied to obtain the rotor orientation relative to the aircraft frame. The orientation of the i-th rotor with respect to the aircraft roll axis (x) is defined by the angle $\phi _i$, while the orientation about the pitch axis (y) is defined by $\theta _i$ as shown in Fig. 5. Given the rotor azimuth angle ($\Psi _i$), the orientation of each rotor is calculated by the following relation

$$\begin{aligned} \begin{aligned} \phi _{i}&= -\varvec{\chi }_i \; \cos \Psi _i\\ \theta _{i}&= \varvec{\chi }_i \; \sin \Psi _i \qquad , \end{aligned} \end{aligned}$$

(2)

for $i=1, 2, 3, 4$. In this regard, the parameterization of $\phi _i$ and $\theta _i$ with respect to their $\psi _i$ is given in Table 2.

Fig. 5

Canted quadrotor configuration schema

Table 2 Differential torsional canting parameterization of the quadrotor configuration

4 Trim analysis

The baseline model ($\chi _d=0^{\circ }$) is trimmed from hover state to $150\,\hbox {km/h}$ forward flight at the standard sea-level conditions. The trim characteristics of the modified models canted by $2.5^{\circ }, 5^{\circ }$ and $7.5^{\circ }$ are compared with respect to the baseline trim characteristics. The results are shown in Figs. 6, 7, 9 and 10, respectively.

Fig. 6

Trim characteristics of the baseline model ($\chi _d=0^{\circ }$) in forward flight

In Fig. 6, the left plot illustrates the thrust of the front and rear rotors specified by the rotor area, the middle plot shows the pitch attitude of the vehicle, and the right plot shows the total power required for the baseline model. At the hover condition, all rotors create identical thrust, corresponding to a disc loading of $143.65\,\hbox {N/m}^{2}$ as stated in Table 1. The forward flight is achieved through pitch-down attitude of the vehicle, where the rotors are vectored towards the flight direction, in order to create forward thrust. To maintain this attitude, the rear rotors produce higher thrust, while front rotors produce less thrust. This tendency can be traced at an increasing rate up to ${65}\,\hbox {km/h}$ horizontal speed. From this point on, the tendency starts to decrease, as the front rotors spin faster to compensate the increasing pitch-down moment of the fuselage. The pitch-down attitude increases with a linear rate and reaches $-10^{\circ }$ at $150\,\hbox {km/h}$. The total power is put together through induced drag resulting from the air inflow passing through the rotors, blade profile drag and parasite drag of the fuselage. The maximum total power in low-speed region is located at hover due to the dominance of the induced power. With increasing tangential flow of the rotor blades in forward flight, the induced power declines, which leads to a drop in the total power. The minimum required power is reached at about ${65}\,\hbox {km/h}$, which is the speed, where the rotors partially start to work against fuselage pitch-down moment. Likewise, the increasing parasite drag of the fuselage becomes significant in the total power from that point on. Since the blade pitch is not adjustable, vectoring the rotors towards the flight direction also causes higher profile drag that contributes to the increment of the total power in high-speed region. Therefore, it is essential to consider the aspects of the blade design for rpm-controlled rotors separately from pitch-controlled rotors.

Fig. 7

Thrust comparison relative to the baseline model

Figure 7 shows the thrust of the modified variants relative to the baseline model. It can be seen that the thrust values of the front and rear rotors separate asymmetrically as the canting angle grows. There are two main constraints causing this separation. To understand the problem, the longitudinal and vertical force components acting on the vehicle at hover are illustrated in Fig. 8. The first constraint is the decrement of the vertical thrust component ($F_{z,front}$ and $F_{z,rear}$) in the rotors. To compensate the loss of thrust in the vertical direction resulting from the canting mode, the rotors should produce higher thrust, which triggers the second constraint: combined with the vertical rotor separation, torsional differential canted rotors create a nose-up pitching moment $M_{\chi _d}$ about the center of gravity resulting from the longitudinal opposite force component of the canted rotors ($F_{x,front}$ and $F_{x,rear}$). To satisfy these two constraints while maintaining the hover trim, the rear rotors carry out the bigger portion of the vertical thrust, the while front rotors set in a relatively lower thrust in order to cancel out the pitching moment. It has to be noted that the thrust separation between the front and rear rotors is not distributed evenly. At hover, the rear rotors of the $\chi _d=2.5\,^{\circ }$ variant produce 1.2% higher thrust, where the front rotors produce 1% less thrust relative to the baseline. For $\chi _d=5^{\circ }$, this separation is 2.6% to -2%, and for $\chi _d=7.5^{\circ }$, 4.2% to $-$2.5%, respectively.

Fig. 8

Rotor forces and resulting moment about the center of gravity in hover condition

In forward flight, the trend for more thrust in the rear rotors and less thrust in the front rotors with increasing canting angle remains unchanged. Still, some significant patterns regarding the behaviour of the rotors can be observed in Fig. 7. With increasing horizontal speed from hover, the relative thrust produced in the canted rear rotors initially decrease before reaching a minimum, where the $\theta _i$ orientation of these rotors are compensated by the pitch angle - $\theta$ (see Fig. 9). The more the nose pitches down in this phase, the more the rear rotors face upward, thus gaining more vertical thrust component. Since the rear rotors do not contribute to the forward thrust before eventually turning into the flight direction, they adjust the thrust merely to compensate the weight. Once the rear rotors start to turn towards the flight direction, the vertical component of the rear rotor thrust starts to decrease again. At this speed,  the relative thrust curves start to grow after passing a local minimum. It is worthy to mention that the selected canting mode also brings a $\phi _i$ rotation to the rotors, which constantly degrades the vertical component of the rotor thrust and can only be compensated by increasing the overall thrust in the rear rotors. This why the relative thrust grows non-linear at higher speeds for canted rear rotors. The front rotors produce less thrust in the same correlation in order to maintain the pitch attitude of the fuselage.

Figure 9 shows the pitch attitude of all compared models. It can be seen that the canted models can maintain the hover at $\theta =0^{\circ }$, since the rear rotors cancel out the pitching moment as mentioned previously. However, the differential torsional cant leads to a slightly more nose-down tendency in forward flight. Since these nose-down attitude deviations are considerably small, a noticeable difference can only be observed at models with higher cant angle, e.g., starting from $\chi _d={5}^{\circ }$. This can be considered as a small trade-off due to the relaxation of the front rotors in forward flight, so that the rear rotors producing the bigger portion of the overall thrust have to be further vectored towards the flight direction in order to compensate the drag.

Fig. 9

Pitch angles of the baseline and modified models

Fig. 10

Total required power comparison relative to the baseline model

Figure 10 shows the total required power of the canted models relative to the baseline model. The alteration of the total power for $\chi _d=2.5^{\circ }$ variant is negligibly low with a 0.4% deviation at its maximum. The $\chi _d=5^{\circ }$ variant has a maximum deviation of 1.7%, while this value lies at 5.1% for $\chi _d=7.5^{\circ }$. At hover as well as in the low-speed regions, the offset between the curves occurs mainly due to the overall increment at the induced drag and profile drag. As these parameters highly correlate with the generated thrust, the induced power and profile power of the rear rotors are greater than those of the front rotors regardless the canting. Furthermore, these power components lessen for the front rotors with the growing canting angle and vice versa for the rear rotors. With the increasing horizontal speed, the induced power starts to decay, while the profile power component becomes more dominant for each rotor. Here, the rear rotors experience significantly higher growth in the profile than the front rotors. This is due to the fact that the rear rotors expose to higher blade loading compared to the front rotors resulting from the asymmetric thrust distribution. Apart from induced and profile power, the parasite power does not significantly contribute to the change in the total power relative to the baseline model, since the maximum difference of the pitch attitude between the model variants occur by about $1.5^{\circ }$ at $150\,\hbox {km/h}$ (see Fig. 9).

5 Linear flight dynamics

A linear quadrotor flight model is needed to assess the influence of differential torsional canting on the flight dynamics in terms of stability derivatives. Therefore, non-linear models of the baseline ($\chi _d={0}^{\circ }$) and modified variants ($\chi _d=2.5^{\circ }$, $5^{\circ }$ and $7.5^{\circ }$) are numerically linearized at the hover trim condition. It is worth rementioning that the vehicle axes are completely decoupled in quadrotor configurations, unlike in the standard helicopter configurations with main rotor-tail rotor arrangement. Hence, the dynamic properties of each axis can be studied separately.

The linearized bare-airframe model structure is given in state-space form by

$$\begin{aligned} \mathbf {\dot{x}} = \textbf{A}\textbf{x} + \textbf{B}\textbf{u}, \end{aligned}$$

(3)

with the state and control vectors given as

$$\begin{aligned} \begin{aligned} \textbf{x}&= \begin{bmatrix} u&v&w&p&q&r&\phi&\theta&\psi \end{bmatrix}^T\\ \textbf{u}&= \begin{bmatrix} \delta _0&\delta _x&\delta _y&\delta _p \end{bmatrix}^T. \end{aligned} \end{aligned}$$

(4)

The system matrix $\textbf{A}$ and the control matrix $\textbf{B}$ are obtained following the numerical linearization. For the initial trim state at hover conditions, the $\textbf{A}$ and $\textbf{B}$ matrices are

$$\begin{aligned} \textbf{A} = \begin{bmatrix} X_u &{} 0 &{} 0 &{} 0 &{} X_q &{} 0 &{} 0 &{} -g &{} 0 \\ 0 &{} Y_v &{} 0 &{} Y_p &{} 0 &{} 0 &{} g &{} 0 &{} 0 \\ 0 &{} 0 &{} Z_w &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} L_v &{} 0 &{} L_p &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ M_u &{} 0 &{} 0 &{} 0 &{} M_q &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} N_r &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ \end{bmatrix} \end{aligned}$$

(5)

$$\begin{aligned} \textbf{B} = \begin{bmatrix} X_{\delta _x} &{} 0 &{} 0 &{} 0 \\ 0 &{} Y_{\delta _y} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} Z_{\delta _0} \\ 0 &{} L_{\delta _y} &{} 0 &{} 0 \\ M_{\delta _x} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} N_{\delta _p} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 \\ \end{bmatrix} \end{aligned}$$

(6)

where g is the only non-derivative parameter, representing the gravitational constant as ${9.806}\,\hbox {m/s}^{2}$. Calculating the eigenvalues of the system without any control input ($\textbf{u} = \textbf{0}$) yields the poles of the system, which reveals information about the dynamic modes. The poles of the linear quadrotor models at hover for all four canting variants are shown in Fig. 11, while Fig. 12 focuses on these poles in three regions revealing their dependence on differential torsional canting. In total, there are four stable poles located on the real axis for each model variant. These modes represent the critically damped ($\zeta = 1$) subsidences following a pitch, roll, yaw or heave perturbation.

Fig. 11

Pole locations at hover state

Fig. 12

Pole behaviours due to $\chi _d$ at hover state

As the cant angle increases, the pitch and roll subsidences decrease in frequency, and therefore move towards the origin on the real axis as shown in Fig. 12a. Furthermore, the separation between the frequencies of the roll and pitch subsidences increases, as the roll subsidence moves further towards the origin than the pitch subsidence. This asymmetric separation behaviour occurs due to the pitch-up moment ($M_{\chi _d}$) resulting from the combination of the vertical rotor separation and differential torsional cant, as depicted in Fig. 8 and discussed in Sect. 4. On the one hand, a positive canting modification ($\chi _d>0$) reduces $M_u$, as the front rotors experience favourable free-stream components with forward flight speed, whereas the rear rotors are exposed to less favourable free-stream components due to backward tilt about the y-axis (see $\theta _i$ in Fig. 5). On the other hand, this reduction is mitigated by counteracting $M_{\chi _d}$.

The yaw subsidence shows a notable pole movement with respect to increasing cant angle, as shown in Fig. 12b. Initially, the pole moves towards the unstable region for $\chi _d = {2.5}^{\circ }$. This is followed by a direction change for $\chi _d = {5}^{\circ }$ and $\chi _d = {7.5}^{\circ }$, where the pole starts to move away from the unstable region. It appears that there is a local minimum for the yaw damping derivative out of all the investigated cases at $\chi _d = {2.5}^{\circ }$ (see Table 3), which is broadly discussed in following dedicated subsection.

The open-loop dynamics exhibit oscillatory unstable modes for perturbations in roll and pitch, which are referred to as phugoid motion (Fig. 12c). With increasing $\chi _d$, the unstable roll phugoid increases in frequency, whereas the unstable pitch phugoid decreases. This means that canting results in a quicker roll and a slightly slower pitch phugoid motion, confirming Niemiec et. al. [5] regarding the changes in the roll and pitch responses mentioned in Sect. 1. It is also worthy to note that the roll phugoid poles move along an almost constant negative damping ratio ($\zeta = -Re(\lambda)/Im(\lambda) <0$), whereas the pitch phugoid poles tend to decrease in $\zeta$ with increasing $\chi _d$, by moving little less away from the baseline pole towards the origin. Similar to the roll/pitch subsidences, this asymmetric behaviour in the pitch phugoid pole movement occurs due to emerging $M_{\chi _d}$ from vertical rotor separation.

5.1 Yaw deficiencies

Results from the previous piloted simulator campaign based on non-linear time simulation [1] showed deficits in the handling qualities ratings regarding the yaw maneuverability of the vehicle. The overall achievable yaw bandwidth showed poor figures compared to vehicles with a dedicated tail rotor. However, a predicted Level 3 yaw bandwidth axis was not found to influence the assigned ratings as expected [1]. The lowest awarded ratings were at Level 2 for all tested MTEs, which were expected to be affected the most by the bandwidth criteria, namely the precision tasks Hover, Hover Turn and Pirouette [1]. This indicates that the influence of poor predicted levels in yaw axis will likely not correlate with the assigned levels for multirotor configurations as it does for helicopters. The assumption is that this is mainly due to the weak coupling between the heave and yaw axes.

For heave/yaw-coupled aircraft in a precision height hold task, the pilot has to compensate each and every small collective input with a corresponding pedal input in order to counteract the coupled yaw response. Therefore, the yaw bandwidth is closely related to the assigned ratings of such systems.

For heave/yaw-uncoupled aircraft, the yaw bandwidth would only be tested in closed-loop precision heading tracking requirements, which could be argued as not part of the commonly tested low-speed MTEs, with the possible exception of the Hover Turn MTE. The second anomaly was specifically the assigned ratings for the Hover Turn, which was also located in the Level 2 region. Here, the bandwidth was expected to have an even higher impact on the rating due to the precise heading capture requirement of the task. However, the pilots stated that the main aspect that affected the awarded ratings was the sluggish response of the yaw axis, requiring aggressive pedal inputs and compensation afterwards. This upshot indicates that an improvement in the moderate to large amplitude/low to moderate frequency attitude changes (attitude quickness) might have a bigger impact on the handling qualities, rather than the small amplitude/moderate to high frequency response as previously assumed (bandwidth/phase delay).

When looking at the linear state-space representation of the quadrotor model, given by Eqs. (3) and (5), the two derivatives influencing the overall yaw performance are the yaw damping ($N_r$) and the pedal control sensitivity ($N_{\delta _p}$). The numerical values for the increase in the cant angle are given by Table 3. For helicopters in hover, $N_r$ is almost entirely due to the tail rotor, with a numerical value between $-0.25$ and $-0.4$, depending on the tail rotor design parameters [15]. The baseline quadrotor model shows a significantly lower yaw damping derivative compared to those of conventional helicopters. As $\chi _d$ increases, so does $N_r$ in negative sense. However, even at high canting angles, such as $\chi _d = {7.5}\,^{\circ }$, $N_r$ does not reach $-0.25$. Therefore, substantial increase in yaw damping has to be achieved through response quickening control augmentation, and cannot be achieved by the canting of the rotor alone. Based on this conclusion, the piloted simulations were conducted with the involvement of an ACAH flight control system, which is described in Sect. 6.2.

Table 3 Influence of $\chi _d$ on the yaw damping derivative and pedal control sensitivity derivative

The pedal control derivative ($N_{\delta _p}$) is defined as the initial angular acceleration per unit of control input (in this case percent of total stick displacement), and is recognized as a primary parameter affecting the pilot’s opinion of aircraft handling [15]. Here, significant improvements for increase in cant angle can be seen. For $\chi _d={7.5}\,^{\circ }$, the control sensitivity is increased by 244% accompanied by a maximum of 5% increase in the total power. An increase of 77% in the control sensitivity can be obtained with $\chi _d={2.5}^{\circ }$.

Overall, an increase in cant angle results in higher control sensitivity in the yaw axis. The $\chi _d={2.5}^{\circ }$ variant is of special interest: the yaw damping remains the lowest while still assuring increase in control sensitivity by a factor of 1.77. In addition, the decrease in roll phugoid stability is the smallest of the evaluated cant angle variants (see Fig. 12c). The reduction of the yaw damping results in a lower bare airframe yaw bandwidth. Nevertheless, the initial results showed that bandwidth has little impact on assigned ratings due to the fact that small amplitude high-frequency pedal corrections due to collective inputs are rarely needed in the quadrotor configuration during low-speed MTEs. With the goal in mind to increase moderate to large amplitude or low to moderate-frequency attitude response, the $\chi _d={2.5}^{\circ }$ configuration was selected for the piloted handling qualities evaluations.

6 Handling qualities assessments

The baseline variant ($\chi _d = 0^{\circ }$) and a modified quadrotor variant ($\chi _d = 2.5^{\circ }$) were assessed regarding their handling qualities using the two methods stated in ADS-33E [3]: predicted and assigned levels. The former is a criteria-based objective assessment method examining the flying qualities parameters obtained from the vehicle response to different input types (step input, frequency-sweep, etc.). The latter is a subjective assessment method awarded by the test pilots with respect to the Cooper-Harper Handling Qualities Rating Scale (HQR, see Fig. 17) [4] upon performing certain flight tasks called Mission Task Elements (MTEs). Here, the main focus of HQR is to determine the workload regarding performed pilot control compensation to complete the MTE. Eventually, both methods return a level index outlining the adequacy of the model for each obtained flying qualities parameter, as well as for each assigned HQR: Level 1, Level 2, and Level 3, representing minimal-, moderate-, and extensive pilot workload, respectively [15].

The simulator test campaign took place in the flight simulator facility AVES. In contrast to the analyses in the previous two sections, both quadrotor variants were augmented with simple flight controllers for the handling qualities assessments. The predicted and assigned levels were obtained using selected criteria and MTEs from the ADS-33E [3]. In addition to HQR, the model variants were also rated on the flown MTEs using Bedford Workload Scale (BWS, see Fig. 18) [16, 17], which determines the workload in terms of the potentially available spare capacity of the pilot for a secondary task, such as monitoring crew members, looking outside, listening and responding to the radio calls. The results were compared with DLR’s ACT/FHS² to evaluate the differences between a conventional helicopter and the generic quadrotor models.

Following sections elaborate on the selected criteria and MTEs, the setup of the simulation and the conduction of the simulator tests for both methods. Finally, the assessment results are given.

6.1 Selected criteria and MTEs

Unlike in the previous study [1], the main focus of the criteria and MTE selections lies on assessing the effect of the canting modification on the response of the vehicle to control inputs with different dynamic amplitudes and frequencies about the three axis. Three types of dynamic inputs have been targeted as classified in ADS-33E [3]:

• Small amplitude/moderate to high frequency,

• Small amplitude/low to moderate frequency,

• Moderate amplitude/low to moderate frequency.

In that regard, six ADS-33E criteria have been selected for the assessment of the predicted levels about the three axes:

• Bandwidth in Pitch, Roll and Yaw Axes,

• Dynamic Stability in Pitch and Roll Axes,

• Attitude Quickness in Yaw Axis.

Based on the expected impact of canting modification on the pilot rating and anticipated relevancy for quadrotor configurations in low-speed regime, following ADS-33E MTEs have been selected for the piloted study:

• Hover to test the impact of canting modification on the dynamic stability in roll and pitch,

• Hover Turn to identify the changes in yaw sensitivity due to rotor canting during the heading change section.

• Pirouette to test the impact of canting modification on the ability to accomplish precision control of the vehicle simultaneously in all axes.

For all selected MTEs, the performance limitations refer to the Cargo/Utility Aircraft category with conditions for GVE (clear daylight & adequate visual cues) and no-wind.

6.2 Simulation setup

AVES features a replica of the ACT/FHS helicopter cockpit with conventional helicopter flight controls. The helicopter cockpit was considered adequate for the simulation campaign, although it does not directly represent the cockpit of quadrotor eVTOLs. To represent the studied configuration in the simulator tests, an eVTOL external visual model with a high-aft rotor arrangement was chosen (Fig. 13a). With the forward rotors being visible within the cockpit sight (Fig. 13b), it was intended to give the pilots a higher perception of flying in a quadrotor rather than sitting in a contemporary helicopter. A flight test scenario consists of a generic MTE test course including visual cues (hover boards, ground markings, etc.), designed in compliance with the ADS-33E MTEs for Utility / Cargo Aircraft category and GVE. The generic quadrotor flight models were run in HOST, whereas the flight model of the ACT/FHS was run in DLR’s in-house flight dynamics code (developed and validated specifically to replicate the ACT/FHS in real-time simulation).

Fig. 13

Virtual quadrotor model in AVES

The stability augmentation features an ACAH response type in pitch and roll. In terms of directional and heave control, a simple rate damping feedback in yaw and a RC response type in heave were respectively used. Although this work is not focused on flight control system augmentation, it was deemed necessary to include a control augmentation system, especially in the lateral and longitudinal axis, based on prior results [1]. The tuning of the control system is based on Level 1/Level 2 optimization for the ACT/FHS helicopter model. The control tuning was then evaluated using the baseline quadrotor model without canting and was found to yield boundary Level 1/Level 2 conditions. Thus, it was refrained from retuning the control parameters. Although a dedicated tuning for each canting variant would be desirable to improve the overall handling qualities even further, such modification would potentially also wash-out any direct impact of each canting on the ratings assigned by the pilots. Therefore, the selected control tuning remained unchanged during the handling qualities evaluations of all quadrotor variants, as well as for the ACT/FHS.

6.3 Conduction of the simulator tests

For the assessment of the predicted levels, sweep, step and pulse responses of the augmented nonlinear real-time quadrotor and ACT/FHS models at hover were recorded. The results were evaluated with respect to the aforementioned selected criteria.

The simulator test campaign was conducted with a single helicopter experimental test pilot, highly experienced in light utility helicopters.³ Due to the unusual flight characteristics of the quadrotor, a relatively long familiarization time was needed for the pilot to adopt his control strategy and to minimize training effects. After familiarization with each new test case, the pilot was asked to perform 2–4 repetitions and give a representative rating for the final run. In particular for the HQR discussions, a task performance display specified for each MTE was shown to the pilot after each run to support the subjective rating with objective performance data, such as ground track, attitude, altitude and time.

6.4 Results

The results for the predicted and assigned levels are shown in Figs. 14 and 15 respectively. The former depicts the level classification of the baseline quadrotor ($\chi _d={0}^{\circ }$), modified quadrotor ($\chi _d={2.5}^{\circ }$) and the ACT/FHS based on the results obtained from the previously mentioned criteria, while the latter shows the results of the subjective pilot ratings for each MTE with respect to HQR. The assigned workload ratings with respect to BWS are shown in Fig. 16.

Fig. 14

Predicted handling qualities levels (refer to ADS-33E [3] for the used symbols, acronyms and descriptions given in parentheses)

6.4.1 Predicted levels

The bandwidths for pitch and roll are located in the Level 1 region for all examined configurations (see Figs. 14a and 14b). Except for the roll bandwidth, the canted model exhibits higher bandwidth values compared to the uncanted baseline model. Increasing the cant angle to $\chi _d={2.5}^{\circ }$ moves the dynamic roll stability metric along the Level 1 / Level 2 border region (Fig. 14e), effectively moving the poles upwards along the constant damping ratio line $\zeta = 0.35$, whilst increasing in natural frequency $\omega _n$. This is congruent with the observations made in the linear analysis, where the roll phugoid mode exhibits a similar trend for increase in cant angle (see Fig. 12c), albeit in the unstable region due to a lack of stabilizing flight control feedback. The same holds for the dynamic pitch stability (Fig. 14d). Based on the linear model analysis, the expectation would have been a decrease in natural frequency whilst moving along the $\zeta = 0.35$ line. This behavior could be explained by the introduction of the FCS feedback providing the ACAH response type, but needs further investigation to be conclusive.

The bandwidths for the yaw axis (Fig. 14c) are located in the Level 1 region for all examined configurations. The modified variant shows higher bandwidth values compared to the baseline variant. Yaw quickness (Fig. 14f) is improved from Level 2 to Level 1 with the introduction of the rotor cant angle. This again is reflected by a significant increase in pedal control sensitivity observed in the linear analysis. Compared to the quickness value of a helicopter setup (ACT/FHS), one can see that the ACT/FHS model still shows quicker response relative to the modified quadrotor model. Overall, the most significant change in predictive handling qualities criteria between the baseline model and $\chi _d={2.5}^{\circ}$ is the increase in yaw attitude quickness.

6.4.2 Assigned levels

As depicted in Fig. 15, for the Hover MTE, the baseline quadrotor was awarded HQR-5. The pilot stated that the amount of aggressiveness during the transition from diagonal approach to hover stabilization had to be reduced compared to the ACT/FHS to avoid roll oscillations. Consequently, the initial control inputs to decelerate were decisive, because the ability to apply corrections during the transition was limited. As a result, only adequate task performance was achieved. For the modified quadrotor, the rating improves to HQR-4, since the controllability is improved according to the pilot. The task performance improved to desired levels, although the pilot still noticed deficiencies about the roll axis. For the ACT/FHS, a rating of HQR-3 was awarded because the pilot was able to apply the aggressiveness needed to achieve the desired task performance and no oscillations occurred. The BWS-ratings given in Fig. 16 are BWS-5 for both quadrotor variants and BWS-4 for the ACT/FHS, because the pilot had to compensate the deficiencies in the flying characteristics of the quadrotors. In general, the workload for this MTE is driven by the transition phase, in which only a reduced available spare capacity for a secondary task was available. The pilot stated that the workload for the hover phase was much lower.

Fig. 15

Assigned HQR-ratings

Fig. 16

Assigned BWS-ratings

For the Hover Turn MTE, the baseline quadrotor received a rating of HQR-5 and the modified quadrotor a rating of HQR-3. The task performance was mainly driven by holding the position by small corrections during the yaw phase. The transition to hover at the end of this MTE was less demanding, simply because there is no strict time limit for this phase. Therefore, less aggressive control inputs are sufficient, which reduce the risk of experiencing oscillations. The pilot stated that the controllability of the canted quadrotor is improved, thus resulting in a higher HQR. The ACT/FHS received a rating of HQR-3 as well, because pilot compensation was still a factor to achieve the desired task performance. However, according to the pilot, there was still a potential to fly this task more aggressively due to the better yaw characteristics. The BWS-ratings are BWS-4 for all cases, because neither the yaw phase nor the transition to hover phase caused a major reduction in the available spare capacity.

As all variants received a HQR-3 rating for the Pirouette MTE, the pilot stated that the ACT/FHS was the variant requiring the least amount of control compensations by minimal differences among all, followed by the modified quadrotor variant. It is worth to mention that the pilot was not limited by roll oscillations during the transition to hover, completing this phase within the required time limits. This can be associated with different demands on control inputs (mainly on roll axis instead of a combination of pitch and roll axes) and training effects during the simulation campaign. The BWS-ratings are BWS-4 for all cases, as no major reduction in the available spare capacity was perceived between the variants.

7 Conclusions

The impacts of the differential torsional canting on the flight characteristics of a passenger-grade virtual quadrotor configuration were investigated. Model variants with different cant angles were analyzed regarding trim performance, linear flight dynamics and handling qualities. The trim analyses for hover and level flight and linear bare-airframe flight dynamics at hover state were compared between the baseline variant ($\chi _d={0}^{\circ }$) and three modified variants ($\chi _d={2.5}^{\circ }$, ${5}^{\circ }$ and ${7.5}^{\circ }$). Finally, handling qualities assessments for the baseline variant and the $\chi _d={2.5}^{\circ }$ variant were made with an experimental helicopter pilot in a flight simulator campaign using selected criteria and MTEs from ADS-33E. Based on the results and discussions, following conclusions are noted:

• Differential torsional canting exhibits improving effects for the control sensitivity about the yaw axis, which was found as a deficit for the previously evaluated passenger-grade virtual quadrotor configurations,

• The assigned levels and pilot comments upon the simulator tests confirmed the observable reduction of the required control compensation due to differential torsional canting during performing the MTEs.

• Modifying the quadrotor model by differential torsional canting increases the total power required, mainly due to the increase in the rotor thrust to compensate the weight and rear rotors being partially facing against the flight direction,

• Moreover, combining differential torsional canting with vertical rotor separation (rear rotors higher than front rotors) leads to additional pitch-up moment, requiring greater thrust in the rear rotors than in the front rotors to maintain the moment equilibrium both in hover and in forward flight,

• The stabilizing effect of the differential torsional canting on the pitch phugoid motion is mitigated due to vertical rotor separation,

• To improve the rotor performance, individual blade design for the front and the rear rotor pairs is recommended.

Future works will mainly focus on two aspects in terms of improving the modeling capabilities for simulation-mature eVTOL configurations based on the knowledge gained during this study. The first aspect is the development of the design and sizing rules for eVTOL multirotor configurations. The second aspect is the task-tailored design of an automatic control system and investigation of the interactions between control tuning parameters, rotorcraft design parameters and handling qualities requirements for eVTOL configurations.

Appendix A Rating scales

See Figs. 17, 18.

Fig. 17

Cooper-Harper Handling Qualities Rating Scale (HQR) [4] - level categorization according to ADS-33E [3]

Fig. 18

Bedford Workload Scale (BWS) [16] - workload categorization according to NASA-STD-3001 [17]