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Preliminary control and stability analysis of a long-range eVTOL

aircraft

Miguel Cuadrat-Grzybowski ∗, Jakob J. Schoser †, and Saullo G. P. Castro ‡

Faculty of Aerospace Engineering, Delft University of Technology, 2629 HS Delft, Netherlands

This study proposes a strategy to incorporate control and stability aspects into the preliminary

design of a tandem-wing, long-range eVTOL aircraft concept. Four operational phases are

considered: cruise, transition, hover, and ground operation. For cruise, a method to design

for open-loop stability and size aerodynamic control surfaces is presented. Furthermore, a

controller is designed to improve handling qualities. For hover controllability by diﬀerential

thrust is considered, and for ground operation, the positioning of the landing gear is performed

according to clearance and tip-over requirements. A novel analytical model is derived for the

tandem wing aircraft in order to estimate during the preliminary design phase the stability

derivatives of the aerodynamic forces and moments. The transition manoeuvre between vertical

and horizontal ﬂight is only treated with qualitative considerations, due to the highly nonlinear

dynamics involved during this ﬂight phase.

I. Introduction

Control and stability of eVTOL vehicles is a challenging topic due to the great variety of ﬂight conditions encountered

between vertical take-oﬀ and landing, transition and cruise. This study presents a set of methods to be used in the

preliminary design stage of an eVTOL aircarft to design for stability and controllability.

The subject of this study is the Wigeon, an eVTOL concept that was developed by ten students at the Delft University

of Technology [

1

]. It is a long-range tandem tilt-wing eVTOL for four passengers designed to take oﬀ and land on

conventional helipads. The Wigeon is targeted towards European, North American and Southeast Asian markets for

comfortable inter-city travel with short door-to-door travel times [

2

]. The thrust for both vertical and horizontal ﬂight is

generated using open propellers [

3

] placed on the leading edges of the wings, which rotate during the transition phase.

A render of the eVTOL can be seen in Figure 1. The most important characteristics of the aircraft are summarised in

Table 1. These parameters were obtained through an iterative design process, which incorporated the methods presented

in this study [1].

The article begins with a section II on estimating the location of the centre of gravity, which is relevant for stability

and control in all ﬂight phases. The subsequent sections are structured according to the diﬀerent operational phases

of the Wigeon. In section III, a novel procedure to evaluate and design for open-loop stability and controllability of

tandem-wing aircraft in cruise is presented, along with sizing of aerodynamic control surfaces and the design of a

feedback controller to improve handling qualities. Following this, section IV contains an analysis of controllability in

hover, and the implications for the centre of mass and rotor placement. In section V, the intermediate stage between

cruise and hover is discussed: transition. In section VI, it is explained how considerations of ground stability enter the

design process. Section VII, ﬁnally, combines the aspects of the previous sections by suggesting a way to integrate

stability and control into an iterative, multi-disciplinary design process. The article concludes with an overview of

veriﬁcation procedures in Appendix A, as well as the most important results and recommendations for future work in

section VIII.

∗Corresponding author, BSc student, email: M.Cuadrat-Grzybowski@student.tudelft.nl

†Corresponding author, BSc student, ORCID: 0000-0002-5663-4921, email: jakob.schoser@gmail.com

‡Corresponding author, Assistant Professor, ORCID: 0000-0001-9711-0991, email: S.G.P.Castro@tudelft.nl

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 1 Render of the long-range eVTOL in cruise conﬁguration.

Table 1 Design parameters of Wigeon

Parameter Value

MTOM [kg] 2790.1

OEM [kg] 1428.9

Range [km] 400

Cruise speed [m/s] 72.2

Stall speed [m/s] 40

No. passengers and pilot [-] 5

Parameter Value

Wing span [m] 8.2

Total wing area [m2] 19.8

Fuselage length [m] 7.3

Lift to drag ratio [-] 16.3

No. of engines [-] 12

Maximum thrust per rotor [N] 3745

II. Centre of Gravity Location

As a ﬁrst step in each design iteration for stability and control, the centre of gravity (CG) range needs to be identiﬁed.

The location of the CG depends on the positioning of the aircraft components (which make up the operational empty

weight), as well as the loading state. Figure 2 shows an exemplary loading diagram, which illustrates the movement of

the CG location during the loading and boarding of the aircraft. The eVTOL must be stable and controllable on the

ground and in the air for any CG within this range, such that it can be ﬂown with diﬀerent loading conﬁgurations. With

only 7 cm, the CG range is very small. This is due to the passengers, which make up the largest portion of the payload,

being located close to the CG. Furthermore, the use of batteries means that neither the mass nor the CG location change

during refuelling.

III. Stability and Control in Cruise

Most of the ﬂight time of the Wigeon will be spent in cruise, so it is essential that the aircraft is easy and safe to ﬂy

in this conﬁguration. This section explores stability and control for a tandem wing aircraft with the following structure:

subsection III.A derives expressions for the CG limits due to static longitudinal stability and pitch controllability. These

limits are then used to perform an optimal sizing and positioning of the wings with respect to each other, as explained

in subsection III.B. In subsection III.C, the sizing process for the vertical tail and aerodynamic control surfaces is

detailed. This is followed by a derivation of analytical expressions for linearised stability and control derivatives in

subsection III.D, as well as simulation results for open-loop dynamics in subsection III.E. Finally, in subsection III.F, a

design for a controller is proposed to improve the handling qualities of the Wigeon.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 2 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 2 An exemplary loading diagram, showing the change of the CG x-position as a function of the loading

state (origin at the nose, x-axis pointing aft). The assumed order of loading in this case is ﬁrst luggage, then the

pilot, and ﬁnally the passengers. Two alternative boarding patterns are shown: back to front and front to back.

A. CG Limits due to Static Longitudinal Stability and Pitch Controllability

All control moments depend on the location of the centre of gravity, since that determines the moment arm that

the control force has. However, the pitch moment

𝑀

is especially aﬀected because the weight acts in the X-Z-plane.

Therefore, only the criterion for pitch controllability is be addressed here while roll and yaw criteria are discussed in

subsection III.C, where the control surfaces are sized.

The free-body diagram of the tandem wing conﬁguration representing straight, symmetric horizontal ﬂight including

aerodynamic forces and the weight can be seen in Figure 3.

Fig. 3 Free-body diagram showing all aerodynamic loads at horizontal ﬂight with associated distances for the

tandem wing conﬁguration.

The non-dimensional moment at the CG is as follows:

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 3 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

𝐶𝑚=𝐶𝑚𝑎𝑐 𝑓 𝑤𝑑 ·𝑆𝑓 𝑤𝑑 ¯𝑐𝑓 𝑤𝑑

𝑆¯𝑐−𝐶𝐷𝑓 𝑤𝑑 ·𝑧𝑐𝑔 𝑓 𝑤𝑑 𝑆𝑓 𝑤 𝑑

𝑆¯𝑐+𝐶𝑇𝑓 𝑤𝑑

𝑧𝑐𝑔 𝑓 𝑤𝑑 𝑆𝑓 𝑤 𝑑

𝑆¯𝑐+𝐶𝐿𝑓 𝑤𝑑 ·(𝑥𝑐𝑔 −𝑥𝑎𝑐 𝑓 𝑤𝑑 )𝑆𝑓 𝑤 𝑑

𝑆¯𝑐

+𝑉𝑟

𝑉2

· (𝐶𝐷𝑟𝑒 𝑎𝑟 ·𝑧𝑐𝑔𝑟𝑒𝑎𝑟 𝑆𝑟 𝑒𝑎𝑟

𝑆¯𝑐−𝐶𝐿𝑟 𝑒𝑎𝑟 ·(𝑥𝑎𝑐𝑟 𝑒𝑎𝑟 −𝑥𝑐 𝑔 )𝑆𝑟𝑒𝑎𝑟

𝑆¯𝑐+𝐶𝑚𝑎𝑐𝑟 𝑒𝑎𝑟

𝑆𝑟𝑒𝑎𝑟 ¯𝑐𝑟𝑒𝑎𝑟

𝑆¯𝑐−𝐶𝑇𝑟𝑒𝑎𝑟

𝑧𝑐𝑔𝑟𝑒 𝑎𝑟 𝑆𝑟𝑒𝑎𝑟

𝑆¯𝑐)(1)

The subscript

𝑓 𝑤𝑑

is for the forward wing and

𝑟𝑒𝑎𝑟

is related to the most aft wing.

𝐶𝑇𝑖

is the thrust coeﬃcient deﬁned

as the thrust normalised by the dynamic pressure force.

𝑉𝑟

is the velocity felt by the rear wing.

𝑆

and

¯𝑐

are the mean

aerodynamic chord and area of the entire aircraft. In this section the following are assumed to be as:

𝑆=𝑆𝑓 𝑤𝑑 +𝑆𝑟 𝑒𝑎𝑟 (2) ¯𝑐=𝑠𝑓 𝑤 𝑑 ·¯𝑐𝑓 𝑤 𝑑 +𝑠𝑟 𝑒 𝑎𝑟 ·¯𝑐𝑟 𝑒 𝑎𝑟 (3)

with

𝑠𝑖

being the ratio of the wing area

𝑆𝑖

by the total area

𝑆

.

𝐶𝑚𝑎𝑐

is the aerodynamic moment coeﬃcient at the

aerodynamic centre,

𝑥𝑎𝑐

is the horizontal location of the aerodynamic centre and

𝑧𝑐𝑔𝑟𝑒 𝑎𝑟

and

𝑧𝑐𝑔 𝑓 𝑤𝑑

are the vertical

distances between the aerodynamic centre of the rear wing and forward wing respectively and the centre of gravity. It

can be seen that the normal force components are neglected as they are known to be small and can be neglected when

the free stream is normal to the propeller area [4].

In order to evaluate the aircraft’s natural controllability without diﬀerential thrust or thrust vectoring, thrust is

neglected for further estimations. The further the centre of gravity moves forward, the more diﬃcult it becomes to pitch

the aircraft up. In order for it to be controllable, the aircraft must be able to attain

𝐶𝑚>

0even at its most forward

centre of gravity position.

𝑥𝑐𝑔 >1

𝐶𝐿𝑓 𝑤𝑑 +𝐶𝐿𝑟 𝑒𝑎𝑟

𝑆𝑟𝑒𝑎𝑟

𝑆𝑓 𝑤𝑑 𝑉𝑟

𝑉2(𝐶𝐿𝑓 𝑤𝑑 𝑥𝑎 𝑐 𝑓 𝑤𝑑 +𝐶𝐷𝑓 𝑤𝑑 𝑧𝑐𝑔 𝑓 𝑤 𝑑 −𝐶𝑚𝑎𝑐 𝑓 𝑤𝑑 ¯𝑐𝑓 𝑤𝑑

−𝑉𝑟

𝑉2𝑆𝑟𝑒𝑎𝑟

𝑆𝑓 𝑤𝑑 −𝐶𝐿𝑟 𝑒𝑎𝑟 𝑥𝑎𝑐𝑟𝑒 𝑎𝑟 −𝐶𝑚𝑎𝑐𝑟 𝑒𝑎𝑟 ¯𝑐𝑟 𝑒 𝑎𝑟 +𝐶𝐷𝑟 𝑒𝑎𝑟 𝑧𝑐𝑔𝑟 𝑒𝑎𝑟 )(4)

where the limit of controllability is the trim condition where

𝐶𝑚=

0;

𝐶𝐿𝑓 𝑤𝑑

can be inﬂuenced by installing mobile

surfaces on the trailing edge of the front wing. The distributed rotors would increase their eﬀectiveness and help to

achieve higher magnitudes of

𝐶𝐿

. These mobiles surfaces are elevators (with elevator deﬂection

𝛿𝑒

) which increase the

control authority over the aircraft.

The limiting factor for static open-loop stability is at high velocities. Hence, the aircraft must be statically stable at

cruise where the highest velocity is achieved.

In order to estimate the stability properties of the design, for a step disturbance in the angle of attack

𝛼

, the moment

equation seen in Equation 1 is diﬀerentiated w.r.t. to 𝛼leading to:

𝐶𝑚𝛼=𝜕𝐶𝑚/𝜕𝛼 =−𝐶𝐷𝛼𝑓 𝑤𝑑 ·𝑧𝑐𝑔 𝑓 𝑤𝑑 𝑆𝑓 𝑤𝑑

𝑆¯𝑐+𝐶𝐿𝛼𝑓 𝑤𝑑 ·(𝑥𝑐𝑔 −𝑥𝑎𝑐 𝑓 𝑤 𝑑 )𝑆𝑓 𝑤𝑑

𝑆¯𝑐

+𝐶𝐷𝛼𝑟𝑒𝑎𝑟 · (1−𝜕𝜖

𝜕𝛼 ) · 𝑧𝑐𝑔𝑟𝑒𝑎𝑟 𝑆𝑟 𝑒 𝑎𝑟

𝑆¯𝑐𝑉𝑟

𝑉2

−𝐶𝐿𝛼𝑟𝑒𝑎𝑟 · (1−𝜕𝜖

𝜕𝛼 ) · (𝑥𝑎𝑐𝑟𝑒 𝑎𝑟 −𝑥𝑐𝑔 )𝑆𝑟 𝑒𝑎 𝑟

𝑆¯𝑐𝑉𝑟

𝑉2

(5)

where

𝜕𝜖

𝜕𝛼

is the downwash eﬀect felt by the rear wing. The latter can be estimated using Equation 6 from [

5

] (with the

addition of 𝜂𝜖).

𝑑𝜖

𝑑𝛼 =𝜂𝜖

𝐾𝜖Λ

𝐾𝜖Λ=0

𝐶𝐿𝛼𝑓 𝑤𝑑

𝜋 𝐴𝑅 𝑓 𝑤𝑑 (𝑟

𝑟2+𝑚2

𝑡𝑣

0.4876

𝑟2+0.6319 +𝑚2

𝑡𝑣

+"1+𝑟2

𝑟2+0.7915 +5.0734𝑚2

𝑡𝑣 0.3113#·

1−𝑚2

𝑡𝑣

1+𝑚2

𝑡𝑣 )(6)

where

𝑚𝑡𝑣 =

2

·𝑣𝑡/𝑏

(where

𝑣𝑡

is the vertical distance between the rear wing aerodynamic centre and the forward wing

aerodynamic centre). An assumption is made based on the geometry of the aircraft that both wings are perfectly straight,

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 4 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 4 Variation of the downwash gradient of the front wing on the rear wing with changing total wing surface

area (where the ratio of wing areas is kept constant). Both results obtained using Equation 6 and lifting-line

theory are displayed.

resulting in

𝑣𝑡

being equal to the maximum height of the fuselage. The parameter

𝑟=

2

· (𝑥𝑎𝑐𝑟 𝑒𝑎𝑟 −𝑥𝑎 𝑐 𝑓 𝑤𝑑 )/𝑏𝑓 𝑤 𝑑

(with

𝑏𝑓 𝑤𝑑

being the span of the forward wing) and

𝐾𝜖Λ

is a function of the quarter chord sweep angle

Λ𝑐/4

of the

forward wing [6].

The factor

𝜂𝜖

is a correction factor applied to the downwash gradient to better match the results obtained using the

lifting line theory [

7

]. Figure 4 shows an exemplary sensitivity analysis of the downwash gradient, which reveals that

Equation 6 overestimated the downwash gradient by roughly a factor of 2. Therefore, it is set that 𝜂𝜖=0.5.

Furthermore, for propeller aircraft, an additional downwash is created due to the propellers which has to be taken

into account. This is written as follows [6]:

𝑑𝜖

𝑑𝛼 𝑝

=6.5(𝑠𝑖𝑛(6𝜙))2.5· 𝜌·𝑃2

𝑏𝑟 ·𝑆3

𝑓 𝑤𝑑 ·𝐶3

𝐿𝑓 𝑤𝑑

𝑙4

ℎ·𝑊3!1/4

(7)

where

𝜙

is the angle between the wings deﬁned as

𝜙=𝑎𝑟𝑐𝑠𝑖𝑛 (𝑚𝑡 𝑣 /𝑟)

,

𝑃𝑏𝑟

is the shaft power per engine and

𝐶𝐿𝑓 𝑤𝑑

is

the lift coeﬃcient for the cruise condition.

It is now essential to estimate the drag derivatives 𝐶𝐷𝛼using the polar drag equation leading to:

𝐶𝐷𝛼𝑖

=2·𝐶𝐿𝛼𝑖

𝐶𝐿𝑖

𝜋𝐴𝑅𝑖𝑒𝑖

(8)

with 𝐶𝐿𝑖being the lift coeﬃcient of one of the wings in cruise condition.

For static longitudinal stability, it is required that

𝐶𝑚𝛼<

0, such that the aircraft restores its initial state after a

disturbance in angle of attack. This results in the neutral stability CG position curve as follows:

𝑥𝑐𝑔 =©«

𝐶𝐿𝛼𝑓 𝑤𝑑 𝑥𝑎𝑐 𝑓 𝑤 𝑑 +𝐶𝐷𝛼𝑓 𝑤𝑑 𝑧𝑐 𝑔 𝑓 𝑤𝑑 + (𝐶𝐿𝛼𝑟 𝑒𝑎𝑟

𝑥𝑎𝑐𝑟𝑒 𝑎𝑟 𝑆𝑟𝑒𝑎𝑟

𝑆𝑓 𝑤𝑑 −𝐶𝐷𝛼𝑟𝑒𝑎𝑟 𝑧𝑐 𝑔𝑟𝑒 𝑎𝑟 )(1−𝜕𝜖

𝜕𝛼 )𝑉𝑟

𝑉2

𝐶𝐿𝛼𝑓 𝑤𝑑 +𝐶𝐿𝛼𝑟𝑒𝑎𝑟

𝑆𝑟𝑒𝑎 𝑟

𝑆𝑓 𝑤𝑑 (1−𝜕𝜖

𝜕𝛼 )𝑉𝑟

𝑉2ª®®¬

(9)

At this point, it is essential to verify the sensitivity of the latter equation w.r.t to the aspect ratio design variable. As

it can be seen that

𝑥𝑐𝑔𝑚𝑎 𝑥

is a function of

𝐶𝐿𝛼

of both wings and that the latter is a function of the aspect ratio

𝐴𝑅

, it

must be seen how sensitive the maximum value is to a change in the aspect ratio. This is illustrated in Figure 5.

From Figure 5, it can be seen that the most aft allowable cg position does vary signiﬁcantly. An increase in aspect

ratio is hence favourable for the stability limit. This must be taken into consideration for future design phases.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 5 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 5 Sensitivity analysis on the aft CG position as a function of 𝐴𝑅.

B. Relative Wing Sizing and Placement

As Equation 9 shows, the location of the neutral point and thus static longitudinal stability in cruise depends strongly

on the geometry and positioning of the wings. More speciﬁcally, it depends on the ratio of wing surfaces

𝑆𝑟 𝑒𝑎𝑟 /𝑆𝑓 𝑤𝑑

,

𝑥𝑎𝑐 𝑓 𝑤𝑑

,

𝑥𝑎𝑐𝑟𝑒 𝑎𝑟

,

𝑧𝑐𝑔 𝑓 𝑤𝑑

, and

𝑧𝑐𝑔𝑟𝑒 𝑎𝑟

. Moreover, there are indirect geometric dependencies through other terms in

Equation 9.

𝐶𝐿𝛼𝑓 𝑤𝑑

and

𝐶𝐿𝛼𝑟𝑒𝑎𝑟

do not only depend on the

𝐶𝑙𝛼

of their respective aerofoils. Instead, they are given by Equation 10

[

8

]. From this, it can be seen that

𝐶𝐿𝛼

depends on wing aspect ratio, sweep and aerofoil lift slope. The last two are taken

as ﬁxed based on aerodynamic considerations [7]. Therefore, the aspect ratio 𝐴𝑅 remains as a free design variable.

𝑑𝐶𝐿

𝑑𝛼 =𝐶𝐿𝛼=𝐶𝑙𝛼𝐴𝑅

2+4+𝐴𝑅 𝛽

𝜂1+tan(Λ0.5𝐶)2

𝛽2(10)

Another implicit geometric dependency of the neutral point location stems from the downwash gradient

𝜕𝜖

𝜕𝛼

. As

shown in Equation 6 and Equation 7, this parameter depends on the aspect ratio

𝐴𝑅 𝑓 𝑤 𝑑

, span

𝑏𝑓 𝑤𝑑

and surface area

𝑆𝑓 𝑤𝑑

of the forward wing, and the horizontal and vertical distance between the wings (

Δ𝑥

and

Δ𝑧

, respectively). Note

that Equation 9 neglects the impact of the upwash of the rear wing on the front wing. This is deemed an acceptable

simpliﬁcation as an analysis using lifting line theory showed that the upwash gradient to be an order of magnitude lower

than the downwash gradient [7].

In addition to this, there are dependencies on the vertical wing positions, aspect ratios and cruise lift coeﬃcients

through the drag contributions. Finally, the term

(𝑉𝑟/𝑉)2

is taken as 1, which is the value suggested by [

5

] for a

high-mounted stabiliser. This value was chosen since the vertical distance between the wings is similar as between a

low-mounted wing and a high-mounted stabiliser, while the horizontal distance is even greater.

As for the forward CG limit for pitch controllability, similar geometric dependencies could be identiﬁed. Again,

𝑆𝑟 𝑒𝑎𝑟 /𝑆𝑓 𝑤𝑑

,

𝑥𝑎𝑐 𝑓 𝑤𝑑

,

𝑥𝑎𝑐𝑟𝑒 𝑎𝑟

,

𝑧𝑐𝑔 𝑓 𝑤𝑑

, and

𝑧𝑐𝑔𝑟𝑒 𝑎𝑟

are directly included in the equation (Equation 4). However, the only

inﬂuence of the aspect ratio is on the drag coeﬃcients of the wing, which is a small eﬀect that makes controllability less

dependent on aspect ratio. In addition to these, the length of the mean aerodynamic chords

¯𝑐𝑓 𝑤 𝑑

and

¯𝑐𝑟 𝑒𝑎𝑟

, as well as

the maximum increase in 𝐶𝐿𝑓 𝑤𝑑 that the elevators can oﬀer. This is discussed in more depth in subsection III.C.

A sensitivity study of the neutral point and controllability limit found that

𝑆𝑟 𝑒𝑎𝑟 /𝑆𝑓 𝑤𝑑

and

𝐴𝑅 𝑓 𝑤 𝑑

are the most

powerful parameters to aﬀect the stability and controllability limits of the aircraft (see [

1

,

2

,

9

]). The relative wing

size strongly aﬀects both stability and controllability, while the front wing aspect ratio mainly impacts stability.

𝐴𝑓

is

especially important since it not only impacts the lift slope of the front wing, but also the downwash which in turn

impacts the rear wing.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 6 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 6 Heat map showing

𝐶𝑚𝛼

as a function of the front wing aspect ratio and the ratio between wing surface

areas. The orange line indicates the limit for static longitudinal stability, while the blue line indicates the limit for

pitch controllability at stall. The blue line denotes the design point of the Wigeon.

Having identiﬁed these two key design variables allowed to plot the constraints aﬀecting wing placement and sizing

as contour lines in a 2D plot with

𝑆𝑟 𝑒𝑎𝑟 /𝑆𝑓 𝑤𝑑

on one axis and

𝐴𝑅 𝑓 𝑤 𝑑

on the other. Such a plot can be seen in Figure 6.

Note that the total wing area is kept constant as to not aﬀect the lift of the aircraft in cruise.

The plot shows that a small aspect ratio on the front wing is required for longitudinal stability, implying an increase

in induced drag. Also, since the root chord is limited in order to not interfere with other elements of the aircraft, the

small aspect ratio would require a short wingspan.

The consequent reduction in the space available for rotors on the front wing could mean that rotors would have to be

relocated from the front wing to the back wing, meaning that the front rotors would have to perform at a higher throttle

setting in hover than the rear engines. Reducing the number of engines on the front wing could also have a negative

impact on control redundancy in hover. The outcome of this analysis agrees with the results from [

10

], who found that

reducing the aspect ratio of the front wing in a tandem-wing eVTOL aircraft to 25% of the rear wing could allow it to be

longitudinally stable.

C. Vertical Tail and Control Surface Design

In horizontal ﬂight, there are nine state variables to be controlled [

11

]. As in conventional aircraft, the Wigeon

controls these states using the control surfaces that create rolling, pitching, and yawing moments.

The choice of aerodynamic control surfaces, as opposed to diﬀerential thrust and thrust vectoring, is based on the

ﬁndings of Chen [

12

], who concluded that conventional aerodynamic surfaces are much more eﬀective for steady level

ﬂight than thrust vectoring.

In addition to this, a vertical tail is designed since it was found that a tandem wing is generally unstable in the lateral

direction (see [1]).

1. Vertical tail and rudder sizing

This section presents the diﬀerent required steps to size of the vertical tail, in terms of its required surface area,

starting from an initial estimate obtained using a class I method. After sizing the vertical tail, the stability requirements

and ﬁnally controllability requirements are derived for a one engine inoperative (OEI) condition. The highest value

obtained from either the stability or controllability requirement is chosen as the ﬁnal design.

In order to initialise the analysis and sizing, a so-called class I method [

13

] is used. This method assumes a vertical

tail volume coeﬃcient ¯

𝑉𝑣which yields an equation for the tail area 𝑆𝑣being as follows:

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 7 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

𝑆𝑣=¯

𝑉𝑣·𝑆𝑏

𝑙𝑣

(11)

where

𝑙𝑣

is the vertical tail moment arm as

𝑙𝑣=(𝑥𝑣−𝑥𝑐𝑔 )

. A value of 0.04 for the volume coeﬃcient is chosen (as

initial estimate) using values found in [

13

]. Furthermore, it is assumed that the vertical tail is placed at the end of the

fuselage. Due to the assumed small size of the vertical tail w.r.t. to the fuselage length,

𝑙𝑓 𝑢𝑠

, this results in initial

estimate for the aerodynamic centre of the vertical tail to be

𝑥𝑣≈𝑙𝑓 𝑢𝑠

. Additionally, an estimate of the root chord,

𝑐𝑣𝑟

, must be performed from the surface area and taper ratio

𝜆𝑣

. Furthermore,

𝑏𝑣

is the vertical tail span which can

be obtained from

𝑏𝑣=√𝐴𝑅𝑣·𝑆𝑣

. The initial value for

𝐴𝑅𝑣

is assumed to be 1.25, chosen using [

13

] and later its

sensitivity to

𝑆𝑣

and

𝑏𝑣

is veriﬁed in order to understand the importance of this design parameter. The aspect ratio is

also treated as a design variable in the multi-disciplinary framework [

1

]. Furthermore, another design variable is the TE

(trailing edge) sweep angle,

Λ𝑣𝑇𝐸

which is also maximised or optimised (in order to increase the eﬀective moment arm

𝑙𝑣

and in order to have rudder outside the wake of the rear wing as much as possible). The ﬁnal design variable is the

taper ratio,

𝜆𝑣

, chosen to be initially 0.40 in order to obtain an approximated elliptical side force distribution. With

these design variables, it is possible to compute the required aerodynamic and geometric properties starting from the

MAC, ¯𝑐𝑣, and the root chord, 𝑐𝑣𝑟using:

¯𝑐𝑣=2/3·𝑐𝑣𝑟·(1+𝜆𝑣+𝜆2

𝑣)

(1+𝜆𝑣)(12) 𝑐𝑣𝑟=2

1+𝜆𝑣·𝑆𝑣

𝑏𝑣

(13)

Having the obtained the initial values, a more accurate estimate of the moment arm,

𝑙𝑣

, can be done by using the

x-and y-positions of the LEMAC of the vertical tail (which are a function of the TE sweep), assuming that the root

chord is entirely on the fuselage and ﬁnally that the aerodynamic centre is at quarter-chord of the MAC. The moment

arm becomes:

𝑙𝑣=𝑙𝑓 𝑢𝑠 −𝑐𝑣𝑟+𝑋𝐿𝐸 𝑀 𝐴𝐶𝑣+0.25 ·¯𝑐𝑣(14)

where

𝑋𝐿𝐸 𝑀 𝐴𝐶𝑣

is the LEMAC position in the x-direction (from the leading edge of the aerodynamic surface), computed

as follows:

𝑌𝑀 𝐴𝐶𝑣=𝑏𝑣

6·1+2·𝜆𝑣

1+𝜆𝑣

(15) 𝑋𝐿𝐸 𝑀 𝐴𝐶𝑣=𝑌𝑀 𝐴𝐶𝑣·𝑡𝑎𝑛 (Λ𝐿 𝐸𝑣)(16)

It is now possible to present the two diﬀerent requirements that the vertical tail must satisfy.

Having initialised the vertical tail design, the stability requirement must be speciﬁed. In fact, in order to have lateral

static stability it must hold that:

𝐶𝑛𝛽>

0. This stability derivative has multiple components: the wing terms, the

fuselage term and ﬁnally the vertical tail component.

First the wing contribution, for unswept wings (at quarter-chord) is derived using [8] and is as follows:

(𝐶𝑛𝛽)𝑤=𝐶2

𝐿·1

4𝜋𝐴𝑅𝑤·𝑆𝑤𝑏𝑤

𝑆𝑏 (17)

where the subscript

𝑤

refers to one wing and

𝐶𝐿

is the lift coeﬃcient at cruise. The second required terms for the

computation of 𝐶𝑛𝛽is the fuselage term estimated using [14] with Equation 18 and Equation 19:

(𝐶𝑛𝛽)𝑓 𝑢𝑠 =−2𝜈

𝑆𝑏 (18) 𝜈=∫𝑙𝑓 𝑢𝑠

0

𝜋

4·𝑤(𝑥)2𝑑𝑥 (19)

where

𝜈

is the eﬀective volume of the fuselage and

𝑤(𝑥)

is the width as a function of the longitudinal position

𝑥

starting

from the nose. This are approximated with an elliptical shape resulting in 𝑤(𝑥)=(𝑤𝑚𝑎 𝑥 /2) · 1−𝑥

𝑙𝑓 𝑢𝑠 /22

.

The third term is related to the vertical tail as follows:

(𝐶𝑛𝛽)𝑣=−𝐶𝑌𝑣𝛼·1−𝑑𝜎

𝑑𝛽 ·𝑉𝑣

𝑉2

·𝑆𝑣𝑙𝑣

𝑆𝑏 (20)

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 8 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

where

(𝐶𝑌𝑣𝛼)𝑣

is the derivative of the side force coeﬃcient

𝐶𝑌

(of the vertical tail) w.r.t

𝛼

. This derivative is basically

negative

𝐶𝐿𝛼

of the vertical tail.

𝜎

is the side wash (assumed to be 0 for simplicity) and

𝑉𝑣

is the velocity of the airﬂow

at the vertical tail (assumed to be equal to the aircraft airspeed as the ﬂow would be undisturbed due to the height

diﬀerence between the wings).

The ﬁnal equation relating to the stability requirement for 𝑆𝑣can be derived and results in Equation 21.

𝑆𝑣=(𝐶𝑛𝛽− (𝐶𝑛𝛽)𝑓 𝑢𝑠 − (𝐶𝑛𝛽)𝑤 ,𝑟 𝑒𝑎𝑟 +𝑓 𝑤𝑑 )

𝐶𝐿𝑣𝛼·𝑆𝑏

𝑙𝑣

(21)

where

𝐶𝑛𝛽

is taken to be 0.0571 in order to provide a suﬃcient stability margin as found in [

15

] and to account for the

previously deﬁned assumptions.

For the controllability condition, the vertical tail should provide a suﬃcient counter-acting yaw moment for an

asymmetric thrust condition.

In order to obtain a reasonable estimate, several design variables must be identiﬁed being the maximum rudder

deﬂection

𝛿𝑟𝑚𝑎𝑥

, the span and chord ratio of the rudder and the vertical tail

𝑏𝑟

𝑏𝑣

and

¯𝑐𝑟

¯𝑐𝑣

respectively and the minimum

controllable speed 𝑉𝑀 𝐶 . These are estimated using [16] and can be summarised in Table 2.

Table 2 Lateral design variables.

Design variable Value/Range

𝛿𝑟𝑚𝑎𝑥 [deg] 25

𝑏𝑟/𝑏𝑣[-] 0.7-1.0

¯𝑐𝑟/¯𝑐𝑣[-] 0.15-0.4

𝑉𝑀𝐶 /𝑉𝑠𝑡 𝑎𝑙𝑙 [-] 1.2

For the geometric parameters such as

𝑏𝑟

𝑏𝑣

and

¯𝑐𝑟

¯𝑐𝑣

, a sensitivity analysis towards

𝑆𝑣

is performed in order to verify the

most optimal pair of values for the lowest area.

Additionally, the same is performed for a combination of Λ𝑣𝑇 𝐸 and 𝐴𝑅𝑣.

For the controllability requirement, the vertical tail and rudder must be sized in such a manner that an OEI condition

can be controlled, where the OEI is deﬁned in this section as losing all engines from one side of the aircraft. The created

yaw moment due to an asymmetric thrust condition can be computed using Equation 22 to Equation 24 [15]:

𝑁𝑎=𝑁𝐸+𝑁𝐷(22) 𝑁𝐸=2

𝑛

𝑖

𝑇

𝑛𝐸

𝑦𝑖(23) 𝑁𝐷≈0.25 ·𝑁𝐸(24)

where

𝑁𝐸

is the sum of the individual asymmetric yaw moments due to an asymmetric thrust per engine

𝑇/𝑛𝐸

, with a

moment arm

𝑦𝑖

and ﬁnally

𝑛

being the number of engines on one half-wing.

𝑁𝐷

is the yaw moment due to the drag of

the engine (which for variable pitch propellers is a quarter of 𝑁𝐸[15]) and 𝑛𝐸is the number of propellers.

It is now possible to show the yaw moment equilibrium equation which relates to the lateral trim condition obtained

with Equation 25 and Equation 26 [16]:

𝑁=𝑁0+𝑁𝑎+𝑁𝛿𝑟·𝛿𝑟+𝑁𝛽·𝛽+𝑁𝛿𝑎·𝛿𝑎=0(25) 𝐶𝑛𝛿𝑟=−𝐶𝐿𝑣𝛼·𝑆𝑣𝑙𝑣

𝑆𝑏 ·𝜏𝑟·𝑏𝑟

𝑏𝑣

(26)

with

𝑁0=

0as the vertical tail has a symmetric airfoil,

𝐶𝑛𝛿𝑟

is the yaw control derivative w.r.t rudder deﬂection and

𝜏𝑟

being the rudder eﬀectiveness which is as follows [16]:

𝜏𝑟=1.129 ·¯𝑐𝑟

¯𝑐𝑣0.4044

−0.1772 (27)

Assuming that the aircraft is not slipping (

𝛽=

0) and no aileron deﬂection is applied (

𝛿𝑎=

0), an equation for

𝑆𝑣

can be obtained. The aforementioned is as follows:

𝑆𝑣=𝑁𝑎

0.5𝜌𝑉2

𝑀𝐶 ·𝐶𝐿𝑣𝛼·𝑙𝑣·𝜏𝑟· (𝑏𝑟/𝑏𝑣) · 𝛿𝑟

(28)

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 9 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Having derived the stability and controllability limits for

𝑆𝑣

with Equation 21 and Equation 28, the limiting case

must be identiﬁed and as a result the highest value obtained from both equations is used for the ﬁnal design.

It is now possible to ﬁnd the sensitivity of the surface area

𝑆𝑣

and the span

𝑏𝑣

for the set of design variables and in

the same time ﬁnd an optimal value. First, the Λ𝑣𝑇𝐸 with 𝐴𝑅𝑣pair is selected as can be seen in Figure 7.

(a) Sensitivity of the vertical tail span,

𝑏𝑣

as a function of

Λ𝑣𝑇𝐸

with 𝐴𝑅𝑣.

(b) Sensitivity of the vertical tail span,

𝑆𝑣

as a function of

Λ𝑣𝑇𝐸

with 𝐴𝑅𝑣.

Fig. 7 Sensitivity analysis of both 𝑏𝑣and 𝑆𝑣parameters.

It can be easily seen, that

𝑆𝑣

has a very low sensitivity to the sweep angle, whereas it decreases with increases

𝐴𝑅𝑣

.

The span,

𝑏𝑣

, on the other hand has a localised minimum around

(𝐴𝑅𝑣=

1

.

05

,Λ𝑣𝑇𝐸 =

39

deg)

. It is therefore necessary

to ﬁnd a compromise between both the surface area and the span, and it must be noted that the larger the span and

sweep, the larger the required structure to support it, which increases the mass. It is hence decided that an

𝐴𝑅𝑣

of 1.4

and a TE sweep of 25

deg

is a good choice when taking into account all the aforementioned. Finally, it is possible to

select the required

𝑏𝑟

𝑏𝑣

and

¯𝑐𝑟

¯𝑐𝑣

in order to match the both the controllability and stability (represented as a black contour

line) requirements and can be seen in Figure 8.

(a) Sensitivity of the vertical tail span,

𝑏𝑣

as a function of

𝑐𝑏𝑟

𝑏𝑣

and ¯𝑐𝑟

¯𝑐𝑣

, for the selected aspect ratio and sweep.

(b) Sensitivity of the vertical tail area, 𝑆𝑣as a function of 𝑏𝑟

𝑏𝑣

and ¯𝑐𝑟

¯𝑐𝑣

, for the selected aspect ratio and sweep.

Fig. 8 Sensitivity analysis of both

𝑆𝑣

and

𝑏𝑣

for the controllability design variables and the stability requirement.

From Figure 8, it can clearly be seen that both variables are sensitive and aﬀected in the same manner by the design

variables. Therefore, to provide the most optimum values in terms of stability and to provide less stress to the vertical

tail, the values are taken to be: 𝑏𝑟

𝑏𝑣

=1.0and ¯𝑐𝑟

¯𝑐𝑣

=0.24.

Finally, following the

𝑆𝑣

estimation it is possible to estimate all geometric properties of the vertical tail and rudder.

These can be visualised in Figure 9 for the speciﬁc example.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 10 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 9 Visualisation of the vertical tail and rudder with geometrical properties

𝑏𝑣=

1

.

503 m, root chord

𝑐𝑣𝑟=1.534 m and tip chord 𝑐𝑣𝑡=0.613 m.

2. Elevator sizing

The elevator is an essential control surface for pitch control authority, especially it is vital at low speeds as it is the

limit of controllability. As it is done for previous control surfaces, it is required to assume a range for a set of design

variables in order to obtain the best elevator sizing possible (that can be found in [

17

]). The choice of designing a simple

elevator or an elevon is to be veriﬁed by checking what are the required geometric properties for pitch control at the

lowest speed and what would the value of the speciﬁc span dimensions have to be computed. For the speciﬁc case study,

upon further scrutiny of both the aileron and elevator sizing, the pitching control surface is deﬁned to be an elevon

placed on both the forward and rear wings, working in a similar manner to an aileron but for pitch control. This is

because it was found that high span ratio values are needed for the control surfaces.

It is noted that a certain increase in

𝐶𝐿𝑓 𝑤𝑑

and a decrease in

𝐶𝐿𝑟𝑒 𝑎𝑟

are needed in order to obtain a feasible CG

range (see [

9

]). This ensures not only that the aircraft can be trimmed at stall, but also that the elevators can be utilised

to control the aircraft in all other horizontal ﬂight conditions. The general lift coeﬃcient equation can be seen in

Equation 29. The previously described required increase in lift coeﬃcient, caused by the elevator deﬂection, can be

identiﬁed and re-written in Equation 30.

𝐶𝐿𝑖=𝐶𝐿𝛼𝑖·𝛼+𝐶𝐿𝛿𝑒𝑖·𝛿𝑒(29) Δ𝐶𝐿𝑖=±𝐶𝐿𝛿𝑒𝑖·𝛿𝑒(30)

where

𝐶𝐿𝛿𝑒𝑖

is the control derivative of one of the lift coeﬃcients w.r.t a deﬂection input. It can be observed that for the

rear wing, the required change in lift coeﬃcient is negative whereas for the forward a positive change is required to

obtain better pitching up capability.

The control derivative can estimated using Equation 31 [

17

] with an additional derived correction factor to account

for the fuselage width clearance:

𝐶𝐿𝛿𝑒𝑖≈𝜏𝑒·𝑏𝑒

𝑏𝑖·𝐶𝐿𝛼𝑖(31)

where

𝜏𝑒

is the elevator eﬀectiveness which can also be computed using Equation 37, where the ratio to be used is

𝑆𝑒/𝑆𝑖

. It must also be noted that the aircraft’s control derivative

𝐶𝐿𝛿𝑒

is diﬀerent from the above. However, this aircraft

derivative is not required as the elevator is designed for a speciﬁc increase and decrease in the forward and rear lift

coeﬃcient, respectively, and not for the entire aircraft. It can therefore be possible to optimise for the best set of

𝑆𝑒/𝑆𝑖

and

𝑏𝑒/𝑏𝑖

, where a special attention must be placed on the chord ratio as well in order to minimise the impact on the

wing box. As a last note, a clearance 𝑤𝑐𝑙𝑒 𝑎𝑟 of 0.5 m is taken in order to account for a local fuselage width of 1 m.

Finally, the elevators must be able to trim and allow for a pitching up moment at stall which can be translated to

𝐶𝑚>

0. Using Equation 1, it can be seen that the moment coeﬃcient is a function of both lift coeﬃcients and hence by

extension the elevator deﬂection

𝛿𝑒

and the respective wing control derivatives. In order to aﬀect the ﬂow over the rear

wing as little as possible, a maximum elevator deﬂection of 10

deg

is chosen (which is smaller than what can be found in

[

17

]). The sensitivity analysis of the pitching moment coeﬃcient w.r.t

𝑆𝑒/𝑆𝑖

,

¯𝑐𝑒/¯𝑐𝑖

and the elevator span ratio

𝑏𝑒/𝑏𝑖

can now be performed and visualised in Figure 10.

A similar pattern is observed for both the area and chord ratio design variables due to their geometric relationship.

However, it can be noticed that the moment coeﬃcient is slightly more sensitive to the chord ratio (although it has a

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 11 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

(a) Sensitivity of

𝐶𝑚

as a function of area ratio

𝑆𝑒

𝑆𝑖

and

𝑏𝑒

in

percentages of wing span 𝑏𝑖.

(b) Sensitivity of

𝐶𝑚

as a function of

¯𝑐𝑒

¯𝑐𝑖

and

𝑏𝑒

in percentages

of wing span 𝑏𝑖.

Fig. 10 Sensitivity of the pitching moment coeﬃcient w.r.t.

𝑆𝑒/𝑆𝑖

,

¯𝑐𝑒/¯𝑐𝑖

and

𝑏𝑒

for a maximum elevator

deﬂection of 𝛿𝑒=10 deg.

more restricted design space due to the presence of the wingbox). For the elevon wingspan ratio, a value of

𝑏𝑒

𝑏𝑖

=

0

.

868

is selected due to the fuselage clearance constraint. The outer limit of the elevator is placed at 99% of the wing’s span in

order to ensure good roll control when designing the ailerons. As previously mentioned, the limiting design variable is

the chord ratio which is selected to be

¯𝑐𝑒

¯𝑐𝑖

=

0

.

25. In the same manner as for the rudder, the selected design ratios can be

multiplied by the wing geometric properties in order to obtain the elevator size.

3. Aileron sizing

In order to design and size the ailerons, the roll rate requirement for small aircraft is needed. The aircraft must be able

to roll faster or at the same rate as demanded by regulations. This involves a combination of the airfoil aerodynamics,

wing geometry and ﬁnally a control derivative estimation.

As a ﬁrst step, some design variables must be identiﬁed and deﬁned. These are: the aileron-wing surface and span

ratio,

𝑆𝑎/𝑆𝑖

and

𝑏𝑎/𝑏

respectively and the maximum aileron deﬂection

𝛿𝑎𝑚𝑎𝑥

. The aileron span is found by assuming

the inner and outer positions, 𝑏1and 𝑏2respectively, leading to 𝑏𝑎=𝑏2−𝑏1as can be seen in Figure 11. The chosen

values and ranges of the geometric parameters are summarised in Table 3 as seen in literature [18].

Fig. 11 Aileron geometry, position w.r.t the wing and

coordinate y used.

Table 3 Aileron design variables.

Design variable Value/Range

𝛿𝑎𝑚𝑎𝑥 [deg]±30

𝑏2/(𝑏𝑖/2)[-] 0.70-0.95

𝑆𝑎/𝑆𝑖[-] 0.05-0.2

Due to the elevon wingspan and relative position on both wings,

𝑏2=

0

.

99

·𝑏𝑖/

2. The outer position of the

aileron is hence slightly higher than as it can be found in Table 3. A maximum deﬂection of

𝛿𝑎𝑚𝑎𝑥 =±

30

deg

is

assumed. Furthermore, a particular attention must be noted on the range of

𝑆𝑎/𝑆𝑖

(which can be seen in Table 3), as for

conventional aircraft the typical range is 0.05-0.1 [

18

]. This diﬀerence is taking into account the tandem wing nature of

the eVTOL, hence

𝑆𝑖

can reach values that are less than half the value of the total area

𝑆

. This hence explains the higher

maximum limit set for the surface ratio and the lower limit for the inner limit.

Having deﬁned the necessary geometric properties, the physical problem can be explained. Due to its relatively

small mass, the aircraft must be able to roll 60

deg

in 1.3 s [

19

]. This is further conﬁrmed for V/STOL aircraft in [

11

],

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 12 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

where the requirement is slightly lower. For this the following equilibrium equation for steady roll is used:

L=L𝛽·𝛽+ L𝑝·𝑝𝑏

2𝑉+ L𝛿𝑟·𝛿𝑟+ L𝛿𝑎·𝛿𝑎=0(32)

where

𝑝

is the roll rate and for a pure roll

𝛽=

0and no deﬂection in rudder is used

𝛿𝑟=

0. The latter with the regulation

requirement yield the following:

𝑝=−2𝑉

𝑏·𝐶𝑙𝛿𝑎

𝐶𝑙𝑝·𝛿𝑎𝑚𝑎𝑥 ≥ ±60 ·𝜋/180

1.3(33)

with 𝑉being the slowest speed at which a controlled roll manoeuvre can be performed which is assumed to be 𝑉𝑀𝐶 .

It is further assumed that the wing is straight, and this assumption is supported by the fact that the sweep at

quarter-chord is 0 and due to that the wing is approximately straight. It is now possible to estimate two required

derivatives,

𝐶𝑙𝛿𝑎=𝑑𝐶𝑙

𝑑 𝛿𝑎

and

𝐶𝑙𝑝=𝑑𝐶𝑙

𝑑𝑝𝑏

2𝑉

, obtained using simple strip theory[

14

,

20

]. These can be found using

Equation 34 [18] and Equation 35 [20].

(𝐶𝑙𝛿𝑎)𝑖=−𝐶𝐿𝛼𝑖𝜏𝑎𝑐𝑟𝑖

𝑆𝑖𝑏𝑖𝑦2

2+2

3

𝜆𝑖−1

𝑏𝑖

𝑦3𝑏1

𝑏2

(34) (𝐶𝑙𝑝)𝑖=−(𝐶𝑙𝛼𝑖+𝐶𝑑0𝑖)𝑐𝑟𝑖𝑏𝑖

24 ·𝑆𝑖(1+3𝜆𝑖)(35)

where

𝐶𝑙𝛼

and

𝐶𝑑0

are lift curve slope and zero lift drag coeﬃcient of the wing airfoil and

𝑖

refers to either the forward or

rear wing. It must be noted that in order to obtain the aircraft’s

𝐶𝑙𝛿𝑎

and

𝐶𝑙𝑝

, a correction factor which accounts for the

diﬀerent wing sizes has to be implemented. This is due to the deﬁnition of the aircraft’s roll moment coeﬃcient

𝐶𝑙

as:

𝐶𝑙=L𝑓 𝑤𝑑 + L𝑟 𝑒𝑎𝑟

0.5𝜌𝑉2·𝑆𝑏

=𝐶𝑙𝑓 𝑤𝑑 ·𝑆𝑓 𝑤 𝑑 𝑏𝑓 𝑤𝑑

𝑆𝑏 +𝐶𝑙𝑟 𝑒𝑎𝑟 ·𝑆𝑟 𝑒𝑎𝑟 𝑏𝑟 𝑒𝑎𝑟

𝑆𝑏 𝑉𝑟

𝑉2

(36)

where

𝑏

is the span of the entire aircraft. Finally,

𝜏𝑎

is the aileron eﬀectiveness that can be estimated using Equation 37

[18].

𝜏𝑎=−6.624 ·𝑆𝑎

𝑆𝑖4

+12.07 ·𝑆𝑎

𝑆𝑖3

−8.292 ·𝑆𝑎

𝑆𝑖2

+3.295 ·𝑆𝑎

𝑆𝑖+0.004942 (37)

With all the aforementioned, it is now possible to proceed with the sizing procedure. This must ensure that

Equation 33 is satisﬁed and with an assumed

𝑏2

value, optimal values for

𝑆𝑎/𝑆𝑖

and

𝑏1

can be obtained through a

sensitivity analysis. Additionally, the aileron is constrained within the geometry of the elevon and this is evaluated as

follows: 𝑆𝑎

𝑆𝑖(𝑏1)=1

𝑆𝑖

𝑐𝑎𝑡+𝑐𝑎𝑟(𝑏1)

2·2·𝑏𝑎(𝑏1)(38)

with 𝑐𝑎𝑡and 𝑐𝑎𝑟being the tip and root chords respectively. All the aforementioned can be visualised with Figure 12.

From Figure 12, it can be seen that both variables aﬀect the roll rate of the aircraft in a similar manner. The

intersection of the geometric constraint from Equation 38 and the roll requirement is the most optimum design for the

aileron. This is found to be: 𝑆𝑎

𝑆𝑖

=0.115 and 𝑏1=0.4703 ·𝑏𝑓 𝑤𝑑 /2.

The ﬁnal elevon design, for the speciﬁc example, can be be visualised (for the forward wing) in Figure 13.

D. Stability and Control Derivatives

In the following, a novel analytical model is derived for the tandem wing aircraft in order to estimate during the

preliminary design phase the stability derivatives of the aerodynamic forces and moments

𝑋

(forward force),

𝑍

(down

force) and

𝑀

(pitch moment) for longitudinal motions and

𝑌

(side force),

L

(roll moment) and

𝑁

(yaw moment) for

lateral motion. The state variables for longitudinal motion are: the dimensionless velocity perturbation

ˆ𝑢

, the angle of

attack

𝛼

, the pitch angle

𝜃

, and the dimensionless pitch rate

𝑞¯𝑐

𝑉0

. In the case of lateral motion the state variables are:

the side-slip angle

𝛽

, the bank angle

𝜙

, the dimensionless roll and yaw rates

𝑝𝑏

2𝑉0

and

𝑟 𝑏

2𝑉0

respectively. This method

combines both known semi-empirical methods (that are adapted to account for a two-winged aircraft) and new physical

derivations. The preliminary model is veriﬁed using stability derivatives obtained for other aircraft from [11].

1. Longitudinal aerodynamic forces

The corresponding longitudinal aerodynamic force coeﬃcients 𝐶𝑋and 𝐶𝑍are as follows:

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 13 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 12 Sensitivity analysis of the roll rate,

𝑝

, as a func-

tion of the surface ratio

𝑆𝑎/𝑆𝑖

and the inner dimension

𝑏1.

Fig. 13 Elevon geometry visualisation for the forward

wing.

𝐶𝑋=𝐶𝐿𝑠𝑖𝑛(𝛼) − 𝐶𝐷𝑐𝑜𝑠 (𝛼) + 𝐶𝑇(39) 𝐶𝑍=−𝐶𝐿𝑐𝑜𝑠 (𝛼) − 𝐶𝐷𝑠𝑖𝑛(𝛼) + 𝐶𝑇𝑖𝑇(40)

with

𝛼

being the angle of attack,

𝐶𝑇

being the thrust coeﬃcient deﬁned previously and

𝑖𝑇

being the eﬀective incidence

angle of the propeller total thrust force w.r.t to the stability axis system.

When estimating the dynamic stability behaviour of the aircraft, the main focus is on small disturbances that deviate

the aircraft from its trim (equilibrium) condition. Due to the aforementioned, the small angle approximation can be used

for the angle of attack, resulting in:

𝐶𝑋≈𝐶𝐿𝛼−𝐶𝐷+𝐶𝑇(41) 𝐶𝑍≈ −𝐶𝐿−𝐶𝐷𝛼(42)

The aerodynamic pitching moment coeﬃcient, 𝐶𝑚, has already been derived and can be found in Equation 1.

2. Velocity stability derivatives

The ﬁrst stability derivatives to be discussed in this section are the derivatives w.r.t

ˆ𝑢=Δ𝑉

𝑉0

, the change in initial

velocity normalised by the initial velocity

𝑉0

(in trim condition). The derivatives are hence

𝐶𝑋𝑢

,

𝐶𝑍𝑢

and

𝐶𝑚𝑢

. Using

Equation 41, Equation 42 and Equation 1 and the transformation 𝑑

𝑑ˆ𝑢=𝑀𝑑

𝑑𝑀 , the equations are as follows:

𝐶𝑋𝑢=𝑀2

0

1−𝑀2

0

𝐶𝐿,0𝛼0−3𝐶𝐷,0−3𝐶𝐿 ,0𝑡𝑎𝑛 (𝛾0) − 𝑀0𝐶𝐷𝑀(43)

𝐶𝑍𝑢=−𝑀2

0

1−𝑀2

0

𝐶𝐿,0−𝑀0𝐶𝐷𝑀𝛼0(44)

𝐶𝑚𝑢=𝑀0·"𝐶𝐿𝑀𝑓 𝑤𝑑 · (𝑥𝑐𝑔 −𝑥𝑎𝑐 𝑓 𝑤 𝑑 ) · 𝑆𝑓 𝑤 𝑑

𝑆¯𝑐−𝐶𝐿𝑀𝑟𝑒 𝑎𝑟 · (𝑥𝑎 𝑐𝑟𝑒 𝑎𝑟 −𝑥𝑐𝑔 ) · 𝑆𝑟 𝑒 𝑎𝑟

𝑆¯𝑐𝑉𝑟

𝑉2#+

𝐶𝑇𝑢𝑓𝑤 𝑑

𝑧𝑐𝑔 𝑓 𝑤𝑑 𝑆𝑓 𝑤 𝑑

𝑆¯𝑐−𝐶𝑇𝑢𝑟𝑒𝑎𝑟

𝑧𝑐𝑔𝑟𝑒 𝑎𝑟 𝑆𝑟𝑒𝑎𝑟

𝑆¯𝑐𝑉𝑟

𝑉2

(45)

where the subscript 0relates to the initial equilibrium condition being the cruise condition,

𝑀0

is the initial mach

number,

𝛾0

is the initial ﬂight path angle and ﬁnally

𝐶𝐿𝑖𝑀

and

𝐶𝐷𝑀

are the lift and drag derivatives w.r.t mach number

which account for compressibility eﬀects. The latter drag term terms can be approximated to 0 compared to the lift

term as the aircraft will ﬂy in the subsonic incompressible regime. This also is already done for Equation 45 (which is

derived by diﬀerentiating Equation 1 w.r.t

ˆ𝑢

), where the drag terms are neglected. The aforementioned

𝐶𝐿𝑀

derivative

and 𝐶𝑇𝑢were found in [14], where for the latter the constant power case is taken.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 14 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 14 Simpliﬁed representation of an idealised pull-up manoeuvre with velocity

𝑉

, radius

𝑅

and pitch rate

𝑞

for a generic aircraft.

3. Angle of attack stability derivatives

The derivatives can be found in Equation 46 and Equation 47.

𝐶𝑋𝛼=𝐶𝐿𝛼𝛼0+𝐶𝐿,0−𝐶𝐷𝛼+𝐶𝑇𝛼(46) 𝐶𝑍𝛼=−𝐶𝐿𝛼−𝐶𝐷𝛼𝛼0−𝐶𝐷,0(47)

where it is assumed that thrust is not a function of the angle of attack leading to

𝐶𝑇𝛼=

0. For

𝐶𝑚𝛼

, the equation is

already derived and can be found in Equation 5.

4. Pitch rate stability derivatives

A general estimate for the change in geometric angle of attack must be ﬁrst done in order to estimate the required

stability derivatives

𝐶𝑍𝑞

and

𝐶𝑚𝑞

whose eﬀects are dominant during a pull-up manoeuvre. It is also essential to mention

that the forward force term 𝐶𝑋𝑞is usually neglected as seen in both [11, 14], leading to 𝐶𝑋𝑞≈0.

For an idealised pull-up manoeuvre several aspects are assumed. First, the velocity

𝑉

and the load factor

𝑛

is

assumed to be constant. Secondly, it is assumed that the aircraft motion follows a perfect circle with a radius

𝑅

, assumed

to be signiﬁcantly larger than the size of the aircraft. The general situation can be portrayed in Figure 14.

The change in geometric angle of attack can be estimated by Equation 48, which again assumes that

𝑅

is signiﬁcantly

larger than the overall length of the aircraft and uses the small angle approximation. Additionally, the radius

𝑅

can be

expressed as a function of the pitch rate

𝑞

and velocity

𝑉

with

𝑅=𝑉/𝑞

. From the latter, Equation 48 can be rewritten

into Equation 49.

Δ𝛼≈𝑠𝑖𝑛(Δ𝛼)=𝑥−𝑥𝑐𝑔

𝑅(48)

Δ𝛼=(𝑥−𝑥𝑐𝑔 )

¯𝑐·𝑞¯𝑐

𝑉0

(49)

Having derived the general equation for the change in angle of attack, it is now possible to estimate the stability

derivatives of the down normal force and pitching moment deﬁned as 𝐶𝑍𝑞and 𝐶𝑚𝑞respectively.

For the latter, the approximation

𝐶𝑍≈ −𝐶𝐿

and the change in lift due to the pitch rate

𝑞

can be used as seen in

Equation 50.

Δ𝐶𝐿=−𝐶𝐿𝛼𝑓 𝑤𝑑 ·(𝑥𝑐𝑔 −𝑥𝑎𝑐 𝑓 𝑤𝑑 )

¯𝑐·𝑆𝑓 𝑤 𝑑

𝑆·𝑞¯𝑐

𝑉0+𝐶𝐿𝛼𝑟𝑒𝑎𝑟 ·𝑆𝑟 𝑒𝑎𝑟 (𝑥𝑎𝑐𝑟𝑒 𝑎𝑟 −𝑥𝑐𝑔 )

𝑆¯𝑐𝑉𝑟

𝑉2

·𝑞¯𝑐

𝑉0

(50)

The derivative can hence be identiﬁed which leads to:

𝐶𝑍𝑞≈𝐶𝐿𝛼𝑓 𝑤𝑑 ·(𝑥𝑐𝑔 −𝑥𝑎𝑐 𝑓 𝑤 𝑑 )

¯𝑐·𝑆𝑓 𝑤 𝑑

𝑆−𝐶𝐿𝛼𝑟𝑒𝑎𝑟 ·(𝑥𝑎 𝑐𝑟𝑒 𝑎𝑟 −𝑥𝑐𝑔 )

¯𝑐·𝑆𝑟 𝑒𝑎𝑟

𝑆𝑉𝑟

𝑉2

(51)

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 15 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

From the latter, the moment coeﬃcient derivative is as follows:

𝐶𝑚𝑞≈ − 𝐶𝐿𝛼𝑓 𝑤𝑑 ·𝑆𝑓 𝑤 𝑑 (𝑥𝑐𝑔 −𝑥𝑎𝑐 𝑓 𝑤𝑑 )2

𝑆¯𝑐2+𝐶𝐿𝛼𝑟𝑒 𝑎𝑟 ·𝑆𝑟 𝑒𝑎𝑟 (𝑥𝑎 𝑐𝑟𝑒 𝑎𝑟 −𝑥𝑐𝑔 )2

𝑆¯𝑐2𝑉𝑟

𝑉2!(52)

5. Angle of attack rate stability derivatives

These stability derivatives are due to the time diﬀerence associated to the front wing downwash which aﬀects the

rear wing. The latter alters the lift force on the rear wing and hence also the pitching moment. These derivatives are

deﬁned with the dimensionless change of angle of attack

¤𝛼¯𝑐

𝑉0

. The eﬀect on the vehicle drag can be neglected [

14

],

which leads to 𝐶𝑋¤𝛼≈0.

In order to ﬁnd an analytical estimate of the rest of the derivatives, ﬁrst the time diﬀerence that the ﬂow takes

between both wings can be approximated in Equation 53 and the downwash can hence be linearised and computed with

Equation 54 [14].

Δ𝑡≈(𝑥𝑎𝑐𝑟𝑒 𝑎𝑟 −𝑥𝑎𝑐 𝑓 𝑤 𝑑 )

𝑉0

=𝑙𝑤

𝑉0

(53) 𝜖(𝑡) ≈ 𝜖0+𝑑𝜖

𝑑𝛼 · (𝛼− ¤𝛼Δ𝑡)(54)

The additional lag term can be identiﬁed and accounted for with the aid of the product rule

𝑑 𝛼

𝑑¤𝛼¯𝑐/𝑉0

=𝑑 𝜖

𝑑 𝛼

𝑙𝑤

¯𝑐

which

leads to the following:

𝐶𝑍¤𝛼=−𝐶𝐿𝛼𝑟𝑒 𝑎𝑟 ·𝑆𝑟 𝑒𝑎𝑟

𝑆·𝑉𝑟

𝑉2

·𝑑𝜖

𝑑𝛼

𝑙𝑤

¯𝑐(55)

𝐶𝑚¤𝛼=−𝐶𝐿𝛼𝑟𝑒 𝑎𝑟 ·𝑆𝑟 𝑒𝑎𝑟

𝑆·𝑉𝑟

𝑉2

·𝑑𝜖

𝑑𝛼

𝑙𝑤(𝑥𝑎𝑐𝑟𝑒 𝑎𝑟 −𝑥𝑐𝑔 )

¯𝑐2(56)

6. Side-slip stability derivatives

For the lateral motion, the derivatives of the side force

𝑌

, yaw moment

𝑁

and roll moment

L

must be estimated for

a small disturbance in side-slip angle 𝛽.

First, the dominant term to 𝐶𝑌𝛽is from the vertical tail and can be estimated as follows:

𝐶𝑌𝛽≈ −𝐶𝑌𝑣𝛼·1−𝑑𝜎

𝑑𝛽 ·𝑉𝑣

𝑉2

·𝑆𝑣

𝑆(57)

with the diﬀerent lateral parameters being already deﬁned in subsubsection III.C.1. The yaw moment derivative,

𝐶𝑛𝛽

, is

also presented in the same section. The last stability derivative,

𝐶𝑙𝛽

has multiple terms that depend on lift distribution,

vertical tail position and wing characteristics (dihedral, quarter-chord sweep and lift curve slope) and position. These

are obtained by combining a semi-empirical method from [

8

] for the wing contribution (corrected by a required factor,

already derived in the previous section) and an approximate analytical estimate due to the vertical tail. The result is:

𝐶𝑙𝛽=

2

𝑤=1"−𝐶𝐿𝛼𝑤Γ𝑤

4·2/31+2𝜆𝑤

1+𝜆𝑤−1.2√𝐴𝑅𝑤𝑍𝑤 𝑓 (𝑙𝑓 𝑢 𝑠 +𝑤𝑓 𝑢𝑠 )

𝑏2

𝑤#𝑆𝑤𝑏𝑤

𝑆𝑏 𝑉𝑤

𝑉2

+𝐶𝑌𝛽𝑣·𝑧𝑣

𝑏(58)

where the ﬁrst term is the component for both wings and accounts for the wing and wing-fuselage interference, obtained

from [

8

]. Due to the dual-wing nature of the aircraft, this interference is averaged per wing with the term

𝑆𝑤𝑏𝑤

𝑆𝑏

.

Γ𝑤

is

the dihedral angle of the wing and

𝑍𝑤 𝑓

is the distance above the centre line of the wing. Indeed, a high wing has a

negative contribution to the derivative, which hence is stabilising. The ﬁnal component is due to the vertical moment

arm,

𝑧𝑣

, from the aerodynamic centre of the vertical tail to the CG of the aircraft. The latter assumes a small initial

angle of attack, 𝛼0.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 16 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

7. Roll rate stability derivatives

The dimensionless roll rate

𝑝𝑏

2𝑉0

stability derivatives are

𝐶𝑌𝑝

,

𝐶𝑙𝑝

and

𝐶𝑛𝑝

. In order to understand why all the

lateral aerodynamic forces and moments change due to a roll rate, it can be understood by a change in the geometric

angle of attack (as it is seen for the pitch rate). This change in angle of attack for the vertical tail can be estimated as

Δ𝛼𝑣≈𝑝𝑧

𝑉0

=𝑧

𝑏·𝑝𝑏

2𝑉0and for the wing it is Δ𝛼=2𝑦

𝑏·𝑝𝑏

2𝑉0.

First, for the side force derivative the dominant term is due to the vertical tail. Due to the 0.4 vertical tail taper ratio,

the side force distribution can be approximated to be elliptical on the vertical tail. Hence, using [

14

], the equation is as

follows:

𝐶𝑌𝑝≈ (𝐶𝑌𝑝)𝑣=−8

3𝜋𝑉𝑣

𝑉2

·𝑏𝑣𝑆𝑣

𝑏𝑆 ·𝐶𝐿𝛼𝑣(59)

The second derivative is

𝐶𝑙𝑝

and it is due to a span-wise change in the sectional lift distribution. It has already been

estimated previously when sizing the aileron (see subsubsection III.C.3).

Finally, the yaw moment also changes with the roll rate and can be estimated with:

𝐶𝑛𝑝≈ −𝑙𝑣

𝑏· (𝐶𝑌𝑝)𝑣−1

8 𝐶𝐿𝑓 𝑤𝑑, 0

𝑆𝑓 𝑤 𝑑 𝑏𝑓 𝑤𝑑

𝑆𝑏 +𝐶𝐿𝑟 𝑒𝑎𝑟 ,0

𝑆𝑟 𝑒𝑎𝑟 𝑏𝑟𝑒 𝑎𝑟

𝑆𝑏 𝑉𝑟

𝑉2!(60)

where the ﬁrst contribution is due to the vertical tail and the second one is due to the wings. Due to the angle of attack

change, the sectional drag varies along the wing as

Δ𝐶𝑑=−(𝐶𝑙 ,0+𝐶𝑑𝛼) · Δ𝛼≈ −𝐶𝑙,0𝑝 𝑦

𝑏

𝑝𝑏

2𝑉0

. Hence this diﬀerence in

the

𝑋−

force along the wing, when integrated over the whole span results in an induced yaw moment (using simple strip

theory). An approximation of the integral associated to the tandem wing correction can be seen as the second term of

Equation 60.

8. Yaw rate stability derivatives

The last set of stability derivatives are due to a yaw rate

𝑟 𝑏

2𝑉0

, and are the following:

𝐶𝑌𝑟

,

𝐶𝑙𝑟

and

𝐶𝑛𝑟

. In the same

manner as for a pitch rate, a yaw rate induces a change in the geometric angle of attack for all the aircraft’s lifting

surfaces. This change in angle of attack, is Δ𝛼𝑟=(𝑥−𝑥𝑐𝑔)𝑟

𝑉0.

With the aforementioned explained, it is now possible to ﬁnd analytical equations for the three stability derivatives.

In a similar manner than for the roll rate derivative,

𝐶𝑌𝑟

represents the change in side force due to an induced change in

angle of attack, and its main contribution is due to the vertical tail. This can be written as:

𝐶𝑌𝑟=2·𝐶𝑌𝑣𝛼·𝑆𝑣𝑙𝑣

𝑆𝑏 ·𝑉𝑣

𝑉2

(61)

The roll moment derivative follows from the previous equations and can be written as follows:

𝐶𝑙𝑟=𝑧𝑣

𝑏·𝐶𝑌𝑟+1

4· 𝐶𝐿𝑓 𝑤𝑑, 0

𝑆𝑓 𝑤 𝑑 𝑏𝑓 𝑤𝑑

𝑆𝑏 +𝐶𝐿𝑟 𝑒𝑎𝑟 ,0

𝑆𝑟 𝑒𝑎𝑟 𝑏𝑟𝑒 𝑎𝑟

𝑆𝑏 𝑉𝑟

𝑉2!(62)

where the ﬁrst term is due to the vertical tail (assuming a small initial angle of attack) and the second term is related to

the induced change in lift due to a yaw rate (equivalent to a change in angle of attack) which consequently creates a roll

moment. The latter equation is an approximation of the integral from simple strip theory by assuming an elliptical

distribution of lift over the wing.

The ﬁnal stability derivative can be estimated using:

𝐶𝑛𝑟=−𝑙𝑣

𝑏·𝐶𝑌𝑟−1

4· 𝐶𝐷𝑓 𝑤𝑑, 0

𝑆𝑓 𝑤 𝑑 𝑏𝑓 𝑤𝑑

𝑆𝑏 +𝐶𝐷𝑟 𝑒𝑎𝑟 ,0

𝑆𝑟 𝑒𝑎𝑟 𝑏𝑟𝑒 𝑎𝑟

𝑆𝑏 𝑉𝑟

𝑉2!(63)

where a similar pattern emerges with the ﬁrst term being due to the vertical tail and the second being an approximation

using simple strip theory of the wing contributions.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 17 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Table 4 Deﬁnitions and derived equations of the control derivatives.

Control Derivative Equation

𝐶𝑋𝛿𝑒=𝑑𝐶𝑋

𝑑 𝛿𝑒𝐶𝑋𝛿𝑒=0[11]

𝐶𝑍𝛿𝑒=𝑑𝐶𝑍

𝑑 𝛿𝑒𝐶𝑍𝛿𝑒=Í2

𝑖=1(−1)𝑖𝜏𝑒·𝑆𝑖

𝑆

𝑏𝑒

𝑏𝑖·𝐶𝐿𝛼𝑖·𝑉𝑖

𝑉2

𝐶𝑚𝛿𝑒=𝑑𝐶𝑚

𝑑 𝛿𝑒𝐶𝑚𝛿𝑒=−Í2

𝑖=1𝜏𝑒·𝑆𝑖

𝑆

𝑏𝑒

𝑏𝑖·𝐶𝐿𝛼𝑖·|𝑥𝑐𝑔−𝑥𝑎 𝑐𝑖|

¯𝑐

𝐶𝑌𝛿𝑎=𝑑𝐶𝑌

𝑑 𝛿𝑎𝐶𝑌𝛿𝑎=0[11]

𝐶𝑙𝛿𝑎=𝑑𝐶𝑙

𝑑 𝛿𝑎𝐶𝑙𝛿𝑎=Í2

𝑖=1−𝐶𝐿𝛼𝑖𝜏𝑎𝑐𝑟𝑖

𝑆𝑖𝑏𝑖·h𝑦2

2+2

3

𝜆𝑖−1

𝑏𝑖𝑦3i𝑏1

𝑏2

𝑆𝑖𝑏𝑖

𝑆𝑏 𝑉𝑖

𝑉2

𝐶𝑛𝛿𝑎=𝑑𝐶𝑛

𝑑 𝛿𝑎𝐶𝑛𝛿𝑎=−0.2·𝐶𝐿, 0·𝐶𝑙𝛿𝑎[8]

𝐶𝑌𝛿𝑟=𝑑𝐶𝑌

𝑑 𝛿𝑟𝐶𝑌𝛿𝑟=𝐶𝐿𝑣𝛼·𝑆𝑣

𝑆·𝜏𝑟·𝑏𝑟

𝑏𝑣

𝐶𝑙𝛿𝑟=𝑑𝐶𝑙

𝑑 𝛿𝑟𝐶𝑙𝛿𝑟=𝑧𝑣

𝑏·𝐶𝐿𝑣𝛼·𝑆𝑣

𝑆·𝜏𝑟·𝑏𝑟

𝑏𝑣

𝐶𝑛𝛿𝑟=𝑑𝐶𝑛

𝑑 𝛿𝑟𝐶𝑛𝛿𝑟=−𝐶𝐿𝑣𝛼·𝑆𝑣𝑙𝑣

𝑆𝑏 ·𝜏𝑟·𝑏𝑟

𝑏𝑣[16]

9. Control derivatives

Having designed the aerodynamic control surfaces for cruise, the aircraft’s control properties are described by the

aid of control derivatives. These are the changes in the aerodynamic loadings due to deﬂections in elevator

𝛿𝑒

(for

longitudinal control), aileron 𝛿𝑎and rudder 𝛿𝑟(for lateral and directional control). These are summarised in Table 4.

A number of observations can be noted in the expressions of the control derivatives. First, the

𝑋−

and

𝑌−

control

derivatives to elevator and aileron deﬂection, respectively, are zero. This is approximation found in literature and can

be safely assumed as a preliminary estimate. The second concept which is recurrent in the expression of the control

derivatives is 𝜏which refers to the control surface eﬀectiveness and is already deﬁned previously. This term allows to

see how eﬀective the aerodynamic control surface (for a change in deﬂection) are when translated to a local increase in

lift (or side-force). Thirdly, it must be understood that the elevator and aileron are placed on both wings of the aircraft

(analogous to an aircraft with both a canard and a tail for the elevator), leading to the summing nature of the elevator

control derivative derived from Equation 31 and roll control derivative equations. Finally, for

𝐶𝑙𝛿𝑎

, the equation is

derived and corrected with a combined method using strip theory [20] and [18], as explained in subsubsection III.C.3.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 18 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

10. Results and speciﬁc consideration for lateral open-loop stability

The stability and control derivatives obtained in this section are summarised in Table 5.

Table 5 Summarised stability and control derivatives for both longitudinal and lateral motion for clean cruise

conﬁguration.

Longitudinal Force Derivatives Normal Force DerivativesPitch Moment Derivatives

𝐶𝑋𝑢=-0.16374 𝐶𝑍𝑢=−0.024899 𝐶𝑚𝑢=0.0061871

𝐶𝑋𝛼=0.2487 𝐶𝑍𝛼=−3.6501 𝐶𝑚𝛼=−0.1320

𝐶𝑋¤𝛼=0𝐶𝑍¤𝛼=−3.6320 𝐶𝑚¤𝛼=−10.0364

𝐶𝑋𝑞=0𝐶𝑍𝑞=−2.4294 𝐶𝑚𝑞=−22.9669

𝐶𝑋𝛿𝑒=0𝐶𝑍𝛿𝑒=0𝐶𝑚𝛿𝑒=−3.8147

Lateral Force Derivatives Roll Moment Derivatives Yaw Moment Derivatives

𝐶𝑌𝛽=−0.1443 𝐶𝑙𝛽=−0.05192 𝐶𝑛𝛽=0.05404

𝐶𝑌¤

𝛽=0𝐶𝑛¤

𝛽=0

𝐶𝑌𝑝=−0.02243 𝐶𝑙𝑝=−0.7462 𝐶𝑛𝑝=−0.02082

𝐶𝑌𝑟=0.1469 𝐶𝑙𝑟=0.1585 𝐶𝑛𝑟=−0.08167

𝐶𝑌𝛿𝑎=0𝐶𝑙𝛿𝑎=−0.09817 𝐶𝑛𝛿𝑎=0.01013

𝐶𝑌𝛿𝑟=0.8693 𝐶𝑙𝛿𝑟=0.1746 𝐶𝑛𝛿𝑟=−0.4423

For lateral motions, the stability derivatives

𝐶𝑛𝛽

and

𝐶𝑙𝛽

deﬁne the lateral stability behaviour in terms of Dutch Roll

convergence and spiral stability. The spiral stability limit is determined by the equation:

𝐸=𝐶𝐿,0· (𝐶𝑙𝛽·𝐶𝑛𝑟−𝐶𝑛𝛽·𝐶𝑙𝑟)=0(64)

For Dutch Roll stability is determined by the Routh’s Discriminant which should be positive:

𝑅(𝐶𝑙𝛽, 𝐶𝑛𝛽)=𝐵·𝐶(𝐶𝑙𝛽, 𝐶𝑛𝛽) · 𝐷(𝐶𝑙𝛽, 𝐶𝑛𝛽) − 𝐴· (𝐷(𝐶𝑙𝛽, 𝐶𝑛𝛽))2−𝐵2·𝐸(𝐶𝑙𝛽, 𝐶𝑛𝛽)>0(65)

where the relations for 𝐴,𝐵,𝐷(𝐶𝑙𝛽, 𝐶𝑛𝛽)and 𝐸(𝐶𝑙𝛽, 𝐶𝑛𝛽)can be found in [11].

Both can now be plotted in the (−𝐶𝑙𝛽, 𝐶𝑛𝛽)plane and can be seen in Figure 15.

Fig. 15 Dutch roll and spiral stability limits plotted in the (

−𝐶𝑙𝛽, 𝐶𝑛𝛽

) plane. The aircraft is represented by a dot

in the plane with its initial properties, and diﬀerent changes to the dihedral lead to diﬀerent positions in the space.

From Figure 15, it can be seen that the speciﬁc eVTOL has both a divergent Dutch Roll and an unstable spiral.

Usually, an unstable spiral can easily be dealt with by the pilot as it is a very slow motion. Dutch Roll, however, should

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 19 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

be stable as it can be detrimental to passenger comfort. To address the latter, the dihedral of the aircraft was changed, as

it was previously explained that

𝐶𝑙𝛽

is a function of the dihedral angle for both wings (see Equation 58). It was observed

that if the dihedrals are set to

Γ𝑓 𝑤𝑑 =−

0

.

5

deg

and

Γ𝑟𝑒𝑎𝑟 =−

4

.

0

deg

, the aircraft has a convergent Dutch Roll (with a

suﬃcient margin to account for assumptions). The smaller forward dihedral is chosen in order to minimise the risk of

the forward wing touching ground during hover or vertical ﬂight, whereas the rear wing dihedral was limited to -4.0

deg

so that it does not aﬀect the propellers eﬃcacy and aforementioned computations for yaw control.

E. Simulation of Open-Loop Dynamics

The non-dimensional and linearised equations of motion of an aircraft for both symmetric and asymmetric motions

have been derived in [

11

]. A brief overview of the derivation process applied to obtain those equations is provided

in this section. The equations of motion expressed in matrix form are transformed in a state-space form in order to

simulate the aircraft responses to speciﬁc disturbances and control inputs. Those equations of motion were linearised

for a steady, straight, symmetric ﬂight in the stability reference frame. This is valid for small disturbances from an

initial equilibrium condition of steady, straight, symmetric ﬂight (i.e. being the reference condition). Since the aircraft

motion studied in this report was described by a set of linear equations of motion, the stability of a speciﬁed equilibrium

condition is independent of the input or disturbance.

For the equations of motion the stability reference frame is adopted, where the x-axis lies in the symmetry plane and

its direction is situated along the longitudinal component of the velocity of the centre of gravity. The z-axis also lies in

the symmetry plane and points downwards perpendicular to the x-axis. The y-axis points out of the x-z plane as to

complete the right-handed coordinate system.

The equations of motion can be rewritten into the following form:

C1¤

®𝑥+C2®𝑥+C3®𝑢=®

0(66)

where C

1

,C

2

and C

3

are stability matrices,

®𝑥

is the state vector containing the required responses and

®𝑢

is the input

vector. First, the required matrices are derived for the symmetric case, resulting in:

C1=

−2𝜇𝑐·¯𝑐/𝑉00 0 0

0(𝐶𝑍¤𝛼−2𝜇𝑐) · ¯𝑐/𝑉00 0

0 0 −¯𝑐/𝑉00

0𝐶𝑚¤𝛼·¯𝑐/𝑉00−2𝜇𝑐·𝐾𝑦 𝑦 2·¯𝑐/𝑉0

(67)

C2=

𝐶𝑋𝑢𝐶𝑋𝛼𝐶𝑍0𝐶𝑋𝑞

𝐶𝑍𝑢𝐶𝑍𝛼−𝐶𝑋0(𝐶𝑍𝑞+2𝜇𝑐)

0 0 0 1

𝐶𝑚𝑢𝐶𝑚𝛼0𝐶𝑚𝑞

(68) C3=

𝐶𝑋𝛿𝑒

𝐶𝑍𝛿𝑒

0

𝐶𝑚𝛿𝑒

(69)

where in the symmetric case, the state vector, ®𝑥=[ˆ𝑢 𝛼 𝜃 𝑞 ¯𝑐/𝑉0]𝑇and ®𝑢is equal to the elevator deﬂection 𝛿𝑒.

Second, the same procedure is performed for the asymmetric case resulting in the following:

C1=

(𝐶𝑌¤

𝛽−2𝜇𝑏) · 𝑏/𝑉00 0 0

0−1

2·𝑏/𝑉00 0

0 0 −4𝜇𝑏𝐾2

𝑥𝑥 ·𝑏/𝑉04𝜇𝑏·𝐾𝑥𝑧 ·𝑏/𝑉0

𝐶𝑛¤

𝛽·𝑏/𝑉00 4𝜇𝑏·𝐾𝑥𝑧 ·𝑏/𝑉0−4𝜇𝑏𝐾2

𝑧𝑧 ·𝑏/𝑉0

(70)

C2=

𝐶𝑌𝛽𝐶𝐿,0𝐶𝑌𝑝(𝐶𝑌𝑟−4𝜇𝑏)

0 0 1 0

𝐶𝑙𝛽0𝐶𝑙𝑝𝐶𝑙𝑟

𝐶𝑛𝛽0𝐶𝑛𝑝𝐶𝑛𝑟

(71) C3=

𝐶𝑌𝛿𝑎𝐶𝑌𝛿𝑟

0 0

𝐶𝑙𝛿𝑎𝐶𝑙𝛿𝑟

𝐶𝑛𝛿𝑎𝐶𝑛𝛿𝑟

(72)

where for the asymmetric case

®𝑥=

[

𝛽 𝜙 𝑝𝑏/(

2

𝑉0)𝑟𝑏/(

2

𝑉0)

]

𝑇

and

®𝑢=[𝛿𝑎𝛿𝑟]𝑇

, where

𝛿𝑎

and

𝛿𝑟

are functions of time

or are input to the model as arrays.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 20 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

The ﬁnal step requires transforming the matrices C

1

,C

2

and C

3

into the state-space matrices A,B,Cand Din the

following:

¤

®𝑥=A®𝑥+B®𝑢&®𝑦=C®𝑥+D®𝑢(73)

where

®𝑦

is the output vector which is chosen to be equal to

®𝑥

, resulting in Cbeing the identity matrix and Dbeing a

matrix containing only zeroes.

The computation of Aand Bis implemented using Equation 74.

A=−C1−1C2&B=−C1−1C3(74)

The latter results in A𝑠for the symmetric case and A𝑎for the asymmetric case.

It can be seen that a set of additional inputs are required. These inputs are the aircraft’s non-dimensional inertia are

as follows:

𝐾2

𝑥𝑥 =𝐼𝑥𝑥

𝑚𝑏2

,

𝐾2

𝑦𝑦 =𝐼𝑦 𝑦

𝑚¯𝑐2

,

𝐾2

𝑧𝑧 =𝐼𝑧 𝑧

𝑚𝑏2

and

𝐾𝑥𝑧 =𝐼𝑥 𝑧

𝑚𝑏2

. The ﬁnal aerodynamic inputs to the state-space system

are 𝐶𝐿,0,𝐶𝑋0and 𝐶𝑍0computed as follows:

𝐶𝑋0=𝑊 𝑠𝑖𝑛 (𝜃0)

1/2𝜌𝑉2

0𝑆(75) 𝐶𝑍0=−𝑊𝑐𝑜𝑠(𝜃0)

1/2𝜌𝑉2

0𝑆(76)

which require the weight

𝑊

, the airspeed

𝑉0

, the air density

𝜌

and ﬁnally

𝜃0

which is the initial pitch angle. Finally, the

non-dimensional mass 𝜇𝑐and 𝜇𝑏must be computed using:

𝜇𝑐=𝑊

𝑔𝜌𝑆 ¯𝑐(77) 𝜇𝑏=𝑊

𝑔𝜌𝑆 𝑏 (78)

Using the stability and control derivatives given in Table 5, the values for the state-space matrices can be computed.

Based on this, the poles and zeroes of both the symmetric and asymmetric state-space system are found. They are

displayed in Figure 16.

In the symmetric system, all poles have negative real parts, meaning that they are open-loop stable. There is

one periodic eigenmode (pair of complex poles) and two aperiodic eigenmodes (real poles). This is in contrast to

conventional aircraft, which have two periodic symmetric eigenmodes. In the asymmetric system, there is again one

periodic eigenmode and two aperiodic modes. This is the same as for conventional aircraft, where the Dutch roll, the

aperiodic roll and spiral modes are observed. The spiral mode is unstable for the Wigeon (as for many conventional

aircraft), but the other eigenmodes are stable.

This behaviour is favourable in the sense that the aircraft is stable in all modes except the spiral, which can be

deemed acceptable due to it being very slow. However, while stability is an essential criterion for controlling an aircraft,

it is not the only one. As Figure 17 shows, a small step input to the elevator (a typical input given by a pilot to change

the pitch angle) results in a very large change in

𝑉

and

𝜃

. Furthermore, the response is very slow to settle on its ﬁnal

value, with a large overshoot in all state variables. This needs to be addressed with a closed-loop ﬂy-by-wire system.

Figure 18 shows the response of the asymmetric states to a pulse-shaped rudder input. The Dutch Roll mode causes

high-frequency oscillations in all states, while the unstable spiral mode causes a slow divergence that is especially

visible in the roll angle and yaw rate. Since the Dutch roll is very unpleasant for the occupants of an aircraft and can

cause nausea, it is essential that the oscillations are reduced.

Finally, Figure 19 shows the response of the aircraft states to a pulse-shaped aileron input. The responses are

qualitatively similar as for the rudder, except for an initial peak in

𝑝

, which is the primary function of the aileron.

Furthermore, the magnitude of the response is smaller.

It has therefore been established that a controller is required to achieve good handling qualities of the aircraft in

cruise. It needs to decrease the response time and overshoot for the elevator response, reduce the oscillations of the

Dutch roll and potentially eliminate the instability due to the spiral mode.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 21 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 16 Map of the open-loop poles and zeroes of

the aircraft in cruise. Crosses indicate poles, circles

indicate zeroes.

Fig. 17 Open-loop response of the airspeed

𝑉

, the

angle of attack

𝛼

, the pitch angle

𝜃

, and the pitch

rate 𝑞to a step elevator input of -0.005 rad.

Fig. 18 Open-loop response of the sideslip angle

𝛽,

the roll angle

𝜙

, the roll rate

𝑝

, and the yaw rate

𝑟

to a pulse rudder input of 0.025 rad (for 1 second).

Fig. 19 Open-loop response of the sideslip angle

𝛽

, the roll angle

𝜙

, the roll rate

𝑝

, and the yaw rate

𝑟

to a pulse rudder input of 0.025 rad (for 1 second).

F. Controller Design

In this section, the design of a closed-loop controller for the Wigeon is described. The purpose of this controller

is to improve on the open-loop dynamics of the eVTOL in terms of stability and handling quality. Since the Wigeon

spends most of its mission time in cruise, the present study focuses on a controller to make the aircraft easy to ﬂy in

cruise. The design of controllers for VTOL operation and the transition phase are beyond the scope of this preliminary

design method.

1. Control allocation

In order to introduce the controller design, it is essential to qualitatively mention the required control allocation.

It is important to know what the pilot commands are and how these commands can be translated to deﬂections of the

aerodynamic surfaces or varying angular speed of the propellers. For cruise, a control stick for pitch, a side stick for roll

and pedals for yaw are connected to a ﬂight control system that directly controls the deﬂections of the control surfaces

(through the use of actuators) and corrects accordingly for any instabilities. The pilot can therefore set a certain attitude

angle for pitch and a target heading angle for yaw control which automatically sets a roll rate for roll control when a

certain turn manoeuvre must be performed, where for the roll rate a certain maximum bank angle is allowed within the

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 22 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

ﬂight envelope. Finally, to control the aircraft during hover, a collective lever will be used by the pilot which modulates

the speed of the propellers

∗

. Additionally, pedals can be used by the pilot to control the rudder, as in for cruise, and

during ground operations it can be used as diﬀerential braking.

2. Controller architecture and closed-loop dynamics

A high-level overview of the controller architecture (created in Simulink) can be seen in Figure 20. While the pitch

controller only consists of one feedback loop with a PI (proportional integral) controller, the lateral controller is more

sophisticated. It is inspired by a lecture by How [21] on a controller for coordinated turns.

Fig. 20 Architecture of the controller for cruise.

In order to improve the Wigeon’s longitudinal dynamics a feedback loop from the pitch angle

𝜃

is used. The pilot

sets a target pitch angle

𝜃𝑡 𝑎𝑟 𝑔𝑒𝑡

, which is compared to the current pitch angle (measured by an inertial measurement

unit) and then fed through a PI controller. The dynamics of the elevators are modelled using the transfer function

1

0.15𝑠+1

,

which is the transfer function proposed by [

21

] for the aileron. Modelling the elevator as a transfer function takes into

account that its response speed is limited and occurs with a delay.

After tuning the controller gains with Simulink’s PID tuner app, the resulting gain and phase margins are 22.3 dB

and 147 degrees, as seen in Figure 21. The step responses of the longitudinal states to a step input to

𝜃𝑡 𝑎𝑟 𝑔𝑒𝑡

is shown in

Figure 22. It can be seen that the aircraft is not only stable, but also responds quickly with minimal overshoot.

Fig. 21 Bode plot of the closed-loop response of

𝜃

to

𝜃𝑡 𝑎𝑟 𝑔𝑒𝑡 set by the pilot.

Fig. 22 Closed-loop response of the airspeed

𝑉

, the

angle of attack

𝛼

, the pitch angle

𝜃

, and the pitch rate

𝑞

to a step input of 0.1 rad to 𝜃𝑡 𝑎𝑟 𝑔𝑒𝑡 .

∗URL https://evtol.com/features/behind-the-controls-of-an-evtol-aircraft-a-test-pilots-perspective/ [cited 15 June 2021]

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Preliminary control and stability analysis of a long-range eVTOL aircraft

The lateral controller is structured as described by How [

21

] for a lateral controller that takes a heading input and

performs coordinated turns. A gain K calculates the appropriate roll angle

𝜙

based on the error in the yaw angle

𝜓

. The

error in the yaw angle is then fed through a controller block, whose output is compared to the roll rate which has a gain

(proportional controller) applied. The output of this comparison is fed as a command to the aileron actuator. At the

same time, another feedback loop with a washout ﬁlter on the yaw rate attempts to bring the yaw rate to zero. This is the

yaw damper designed to reduce low-frequency oscillations (Dutch roll).

The aileron and rudder actuators are modelled as

1

0.15𝑠+1

and

3.33

𝑠+3.33

, respectively. In order to reduce oscillations in

the response, a PI controller was used for CTRL 4 in Figure 20 instead of the proportional controller in the original work.

In Figure 23, the Bode plot showing the response of the heading angle

𝜓

to the pilot input

𝜓𝑡 𝑎𝑟 𝑔𝑒𝑡

is shown. The

system is closed-loop stable with a gain margin of 12.4 dB and a phase margin of -180 degrees. These margins are not

as good as for the longitudinal case, so there is further room for optimisation. This is conﬁrmed by the step responses

shown in Figure 24 are also slower and more oscillatory than for the longitudinal controller.

In Figure 25, the poles and zeroes of the open-loop system can be seen. All poles are now stable (with a negative

real part), which is an improvement over the open-loop system in Figure 16. However, the asymmetric system now has

two zeroes in the right half-plane. This can lead to the system’s initial response being in the opposite direction of its

ﬁnal value. This can, in fact, be observed in the evolution of

𝑟

in Figure 24. This conﬁrms that in future iterations of the

design, the lateral controller may have to be reviewed to improve handling qualities.

Fig. 23 Bode plot of the closed-loop response of

𝜓

to

𝜓𝑡 𝑎𝑟 𝑔𝑒𝑡 set by the pilot.

Fig. 24 Closed-loop response of the sideslip angle

𝛽

, the

roll angle

𝜙

, the roll rate

𝑝

, the yaw rate

𝑟

, and the yaw

angle 𝜓to a step input of 0.5 rad to 𝜓𝑡𝑎 𝑟𝑔 𝑒𝑡 .

IV. Controllability in Hover

In hover mode, the oncoming airspeed experienced by the vehicle is very low. Therefore, aerodynamic control

surfaces are not an eﬀective means of control and thrust vectoring and diﬀerential thrust must be used. In order to

quantify the controllability of the eVTOL in hover, the Available Control Authority Index (ACAI) developed by Du et al.

[

22

] is used. The ACAI is designed to evaluate available control authority of hovering multirotor vehicles with ﬁxed

rotors. While the Wigeon can tilt its rotors (by tilting the wings), neglecting this possibility for hover control simpliﬁes

the analysis considerably while also being conservative.

Du et al. [

22

] model the dynamics of a hovering multicopter using a state-space system of the form given in

Equation 79. 8 states are considered, which are given in Equation 80. These include the altitude

ℎ

, the roll angle

𝜙

, the

pitch angle

𝜃

, the yaw angle

𝜓

, the vertical speed

𝑣ℎ

, the roll rate

𝑝

, the pitch rate

𝑞

, and the yaw rate

𝑟

. Equation 81

shows the control variables, which are the total thrust force

𝑇

, the roll moment

𝑁

, the pitch moment

𝑀

, and the yaw

moment 𝐿. The weight 𝑚𝑎·𝑔is also included in this vector for the sake of convenience.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 24 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 25 Map of the closed-loop poles and zeroes of the aircraft in cruise. Crosses indicate poles, circles indicate

zeroes.

¤𝑥

¤𝑥

¤𝑥=𝐴𝑥

𝑥

𝑥+𝐵𝑢

𝑢

𝑢(79) 𝑥

𝑥

𝑥=hℎ 𝜙 𝜃 𝜓 𝑣 ℎ𝑝 𝑞 𝑟i𝑇

(80)

𝑢

𝑢

𝑢=𝐹

𝐹

𝐹−𝐺

𝐺

𝐺=h𝑇 𝐿 𝑀 𝑁 i𝑇−h𝑚𝑎·𝑔000i𝑇

(81)

According to Du et al. [

22

], there are two necessary and suﬃcient conditions for multirotor controllability in hover:

1) rank C(𝐴, 𝐵)=8

2) ACAI > 0

C(𝐴, 𝐵)

is deﬁned by Equation 82, where

𝐴

and

𝐵

are the state and input matrices of the state-space system in

Equation 79. However, this condition was not found to be limiting for the eVTOL under any condition.

C(𝐴, 𝐵)=h𝐵 𝐴𝐵 𝐴2𝐵 . . . 𝐴7𝐵i(82)

The procedure for calculating the ACAI is described in [

22

] in detail. The calculation is implemented in Python

using the Matlab Toolbox [

23

] developed by Du et al. [

22

] as an example and means of veriﬁcation. At this point, only

the inputs required to perform the calculation are listed in Table 6. Note that since the origin of the coordinate system is

the centre of mass, this is also implicitly an input to the calculation.

Table 6 Input parameters to the calculation of the ACAI [22].

Parameter Description

𝑥1, 𝑥2, . . . , 𝑥𝑚x-position of each rotor w.r.t the centre of mass

𝑦1, 𝑦2, . . . , 𝑦𝑚y-position of each rotor w.r.t the centre of mass

𝐾1, 𝐾2, . . . , 𝐾𝑚Maximum thrust of each rotor

𝑘𝜇Ratio between reactive torque and thrust of a rotor

𝑤1, 𝑤2, . . . , 𝑤𝑚Direction of rotation of each rotor

𝜂1, 𝜂2, . . . , 𝜂𝑚Eﬃciency parameter of each rotor

𝑚𝑎Vehicle mass

In order to calculate the ACAI of the eVTOL, it is assumed that the rotors are located at 0.5 m and 6.1 m from the

aircraft nose (for the front and rear wing, respectively), and evenly spaced between 1.0 m and 4.1 m outwards from the

symmetry plane. The parameter

𝑘𝜇

is assumed to be 0.1 based on the values used by Du et. al [

22

]. As for the direction

of rotation of the propellers, they are all taken to be rotating inboard.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 25 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Using these values, along with the MTOM and the maximum thrust per rotor, an ACAI of 350.0 is obtained. Since

this is positive, the aircraft is controllable. However, the design is not the most eﬃcient design possible for hover, since

the forward CG location causes the front rotors to be more loaded in hover.

This was highlighted in a sensitivity study conducted to understand the inﬂucence of diﬀerent parameters. Figure 26

shows the dependency of the ACAI is on the centre of gravity location

𝑥𝑐𝑔

. The best controllability is achieved when

the centre of gravity lies in the middle between the wings. The Wigeon’s centre of gravity lies in front of this, leading to

suboptimal performance. Due to this, the ACAI of the Wigeon increases when the front wing is moved forward, away

from the centre of gravity (which gives them a larger moment arm).

Fig. 26 Variation of the ACAI with 𝑥𝑐𝑔 .

Lateral placement of rotors also plays a role: spreading the rotors out over a larger range increases the ACAI. In

other words, placing the rotors closer to the symmetry plane on one side and further towards the wingtips on the other

side improves performance. Other parameters behave as expected, where a larger mass reduces the ACAI and a higher

thrust increases it.

Finally, the eﬀect of the parameter

𝑘𝜇

was investigated (see Figure 27). The contribution of this coeﬃcient is crucial

for yaw control, and estimating it accurately at a preliminary stage is challenging. However, variations in

𝑘𝜇

do not lead

to uncontrollable behaviour unless

𝑘𝜇=

0. Its value also does not create more restrictive limits on other parameters, so

ﬁnding a precise value is not necessary at a preliminary stage.

The ACAI also allows to evaluate performance after rotor failures. To model this, the eﬃciency factors

𝜂

are set to

zero for the corresponding rotors. This method can be used to ensure that hover control is failsafe.

V. Transition

Having established how the eVTOL can be made controllable both in hover and in cruise ﬂight, it remains to be

shown that the available control inputs can achieve the transition between these two ﬂight modes. This phase of the

ﬂight is very diﬃcult to model, so only a qualitative statement will be made in this article.

The dominant aspect of transition is the acceleration/deceleration between stall speed and zero airspeed. This is

achieved by tilting the rotors between their vertical orientation in hover to a horizontal orientation in cruise ﬂight. Due

to the low airspeed, most of the control authority will be obtained from the rotors rather than aerodynamic surfaces.

Therefore, it is assumed that control of other state variables than horizontal speed is achieved in the same way as in

hover.

In order to design a controller for the transition phase, certain aerodynamic aspects must be taken into account.

These mostly relate to the nature of the ﬂow during transition which becomes very diﬃcult to predict due to hysteresis.

In fact, during transition, there is a sudden change from fully separated to attached ﬂow which in turn translates to a

sudden change in lift over the wings. Additionally, the modelling unpredictability is enhanced due to the phenomenon of

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 26 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 27 Variation of the ACAI with 𝑘𝜇.

hysteresis where the stalling characteristics of the aircraft depend on the ﬂow’s history. Furthermore, an unpredictable

atmosphere can increase the complexity of the model, which e.g. the case of a sudden change in wind gust (whether it is

a change in magnitude or direction), the ﬂow over wings may re-attach or separate sooner/later than predicted.

For the pure transition phase itself, the controller must also be able to react accordingly for a range of diﬀerent

situations such as if the ﬂow separates/(re-)attaches ﬁrst at the forward wing, or if the ﬂow ﬁrst (re-)attaches/separates

at the rear wing or for the rarest case that the nature of the ﬂow changes at the same time for both wings. It can be

understood that all these diﬀerent situations create diﬀerent and unpredictable pitching moments, that can be aggravated

by an unpredictable atmosphere as explained previously. The latter introduces a dangerous transient response, which

shows the need for a robust controller.

There is yet another complication: due to the high angles of attack attained during transition, there is an increased

risk that the rear rotors could enter the wake of the front rotors or front wing, thereby leading to a sudden loss in thrust.

The aircraft has its entire wing tilting which will create a larger wake, which is a risk that will certainly need attention.

Because of this, a quantitative analysis of transition dynamics is beyond the scope of this paper.

VI. Ground Stability

The placement of the landing gear is restricted in the design of the Wigeon aircraft due to crashworthiness

considerations that dictate that no stiﬀ structure shall be located directly below human occupants [

1

]. Therefore, the rear

landing gear is placed far aft behind the passenger cabin (at 4.76 m from the nose) and the traditional single nose gear is

replaced by two gears (at 1.36 m from the nose) to be able to place them next to the pilot rather than below.

With this restriction in mind, a total of ﬁve criteria are considered. The two ﬁrst among these are the maximum

turn-over angle and the minimum load on the steering wheels. The former requires that the centre of gravity of the

aircraft must be located at an angle of 55 degrees above the line connecting the nose wheels and the rear wheels. This

criterion is originally proposed for tricycle landing gears in [

5

], but is extended to quadricycle landing gears, as shown

in Figure 28.

The purpose of this requirement is to avoid the the eVTOL tipping over to the side while taxiing. The load criterion

on the steering wheels (which are the nose wheels in the case of the Wigeon) is related to controllability [

5

]. It requires

that at least 8% of the total aircraft weight must rest on the steering wheels for them to achieve their function.

Three additional requirements are the clearance of the wings when they are vertical, clearance of the rotors when the

wings are horizontal, and the maximum tip-back angle. The last two requirements are common in conventional aircraft,

but would only be relevant for emergency situations for the Wigeon. This is because in a conventional mission, it would

land vertically with the rotors oriented upwards. However, it is assumed that the landing gear can tolerate a landing with

wings in horizontal conﬁguration on a regular airﬁeld [

1

]. This could allow for a safe landing if the rotation mechanism

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 27 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Fig. 28 Deﬁnition of the turn-over angle for a quadricycle landing gear.

of one or both wings should fail during cruise.

The critical requirement for clearance is found to be the root chord of the wing when in vertical position, dictating

the required height of the landing gear. The track width of the landing gear is determined by the turn-over requirement.

The corresponding equations are Equation 83 and Equation 84. Note that it is chosen to use the same track width for the

front and rear wheels. This is because reducing the track width in one of them would have required an increase in the

other in order to satisfy the turn-over requirement. Therefore, this design minimises the maximum track width.

tan(𝜓𝑡 𝑜 )=𝑧𝑐𝑔 +ℎ𝑙 𝑔

𝑡𝑤 /2(83) tan(𝜙𝑐𝑙 )=tan(Γ)𝑦𝑟 𝑜 𝑡 +ℎ𝑙𝑔 +𝑧𝑓−𝑟𝑟 𝑜𝑡 /2

𝑦𝑟𝑜𝑡 −𝑡𝑤/2(84)

Here,

𝜓𝑡𝑜

is the turn-over angle,

𝑧𝑐𝑔

is the z-location of the aircraft’s centre of gravity,

ℎ𝑙𝑔

is the height of the

landing gear,

𝑡𝑤

is the track width of the landing gear,

𝜙𝑐𝑙

is the clearance angle,

Γ

is the front wing’s dihedral,

𝑦𝑟𝑜𝑡

is

the spanwise location where the wing rotates,

𝑧𝑓

is the z-location of the front wing root chord, and

𝑟𝑟𝑜𝑡

is the maximum

length of the chord behind the rotation point (i.e., radius of the circle traced by the trailing edge of the front wing when

rotating). In order to increase the clearance for the rotated wing,

𝑧𝑓

is decreased by 10 cm compared to its real value,

leading to a 10 cm higher landing gear.

Solving Equations 83 and 84 for

ℎ𝑙𝑔

and

𝑡𝑤

, with the recommended values

𝜓𝑡𝑜

= 55 degrees and

𝜙𝑐𝑙

= 5 degrees

from [

5

], and applying the additional 10 cm clearance, yields a landing gear height of 0.9278 m and a track width of

2.220 m. The track width is therefore 0.8400 m wider than the fuselage, meaning that the landing gear must be deployed

0.4200 m outboard of the fuselage.

VII. Integration of Stability and Control into the Design Process

So far, the design process for stability and control of the Wigeon has been presented in isolation, assuming that all

other aircraft parameters are known. However, this is rather diﬀerent from the approach taken to design the aircraft. To

obtain a convergent design, the calculations from diﬀerent disciplines are integrated in a single design code that allowed

for a high degree of automation. In this code, an initial set of parameters is iteratively altered until all requirements

are met. First, the wings are sized according to the procedure explained in subsection III.B, then hover controllability

is checked to see that any one engine could fail while maintaining controllability. If this is not initially the case, the

thrust-to-weight ratio is increased until the requirement is met. Then, the CG excursion is calculated and used to place

the landing gear optimally, and the process monitors whether the track width does not become too high. Sizing the

vertical tail and control surfaces is one of the last steps in one iteration, since they are not found to be an issue in most

cases, which limits their inﬂuence on other design aspects. The simulation and controller design for cruise is done

manually after a converged design had been found.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 28 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

VIII. Conclusion and Recommendations

In this article, a set of methods has been proposed to evaluate and design for stability and control of a tandem

tilt-wing eVTOL aircraft during the preliminary design phase. The procedure is presented using the Wigeon long-range

eVTOL concept as an example, but could be used for any other vehicle in the same category. Future work should

include veriﬁcation of the linearised dynamics model for cruise using CFD software. A number of CFD simulations

could be performed at diﬀerent angles with respect to the incoming ﬂow to characterise the behaviour of the aircraft.

Furthermore, the stability and control eﬀects in transition and the governing aerodynamics should be studied further.

Credit authorship contribution statement

M. Cuadrat-Grzybowski, J.J. Schoser: Methodology, investigation, formal analysis, validation, software, writing -

original draft preparation, writing - reviewing and editing. S.G.P. Castro: Conceptualization, supervision, writing -

reviewing and editing.

Acknowledgments

This work has been developed as part of the Design Synthesis and Exercise "Multi-Disciplinary Design and

Optimisation of a Long-Range eVTOL Aircraft", given during the Spring quarter in 2021. We would like to thank our

fellow companions Javier Alba Maestre, Egon Beyne, Michael Buszek, Alejandro Montoya Santamaria, Nikita Poliakov,

Koen Prud’homme van Reine, Noah Salvador López and Kaizad Wadia who are part of the team conducting this design.

Additionally, we would like to thank Davide Biagini and Dr. Ali Nokhbatolfoghahai for their generous help, advice

and supervision in composing this project. We would also like to thank Dr. Erik-Jan van Kampen for providing us with

guidance in the early phase of this project.

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https://arc-aiaa- org.tudelft.idm.oclc.org/doi/abs/10.2514/1.G000731.

[23]

Du, G.-X., and Quan, Q., “A Matlab Toolbox for Calculating an Available Control Authority Index of Multicopters,” , 3 2016.

URL http://rﬂy.buaa.edu.cn/resources.

A. Veriﬁcation Procedures

A. Veriﬁcation of Stability Derivatives Model

The veriﬁcation of the analytical model is performed in a series of steps. First, a number of small unit tests are

done in order to verify that the expected change in the stability derivatives is correct. Finally, the model’s outputs are

compared to already existing aircraft, in terms of magnitude and sign, where the latter is of utmost importance and

should be similar for a speciﬁc number of stability and control derivatives.

1. Unit test - CG shift

The unit test performed in this subsection is a shift in the CG of the aircraft. In order to perform a complete and

concise unit test veriﬁcation, only a small number of stability derivatives for both longitudinal (

𝐶𝑋𝛼

,

𝐶𝑚𝛼

and

𝐶𝑚𝑞

) and

lateral (𝐶𝑛𝛽and 𝐶𝑛𝛿𝑟) motions are shown due to their high number.

Speciﬁc simpliﬁed equations of the change of longitudinal stability derivatives can be found in [

11

] and are compared

to the ones obtained by the code. For the lateral stability derivatives, the qualitative shift is veriﬁed. The results of the

latter can be summarised in Table 7.

From Table 7, it can be seen that the unit tests are successful. Even though minor diﬀerences in

𝐶𝑚𝛼

and large

diﬀerences in

𝐶𝑚𝑞

can be seen, these can be simply explained by the fact that the equations used from [

11

] are simpliﬁed

and assume a conventional aircraft conﬁguration, which does not have the same analytical equations. Even with these

diﬀerences, the trend is still the same and hence with the aforementioned the analytical modules have been independently

veriﬁed.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 30 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Table 7 Summarised Results of the CG Unit Test, the expected outcome of those tests and the actual result.

Veriﬁcation Test Expected Outcome Obtained output

Shift in 𝑥𝑐𝑔 by factor of 2 Δ𝐶𝑋𝛼=0(no dependency)

Δ𝐶𝑚𝛼=11.973 (destabilising)

Δ𝐶𝑚𝑞=−41.1467

Decrease in 𝐶𝑛𝛽

Decrease in magnitude of 𝐶𝑛𝛿𝑟

Δ𝐶𝑋𝛼𝑚𝑜𝑑𝑒𝑙

=0

Δ𝐶𝑚𝛼𝑚𝑜𝑑𝑒𝑙

=11.756

Δ𝐶𝑚𝑞𝑚𝑜𝑑𝑒𝑙

=−33.21419

Δ𝐶𝑛𝛽𝑚𝑜𝑑𝑒𝑙

=−0.062499

Δ𝐶𝑛𝛿𝑟𝑚𝑜𝑑𝑒𝑙

=−0.35528

Shift in 𝑥𝑐𝑔 by factor of 0.5 Δ𝐶𝑋𝛼=0(no dependency)

Δ𝐶𝑚𝛼=−5.986615 (stabilising)

Δ𝐶𝑚𝑞=−5.54833

Increase in 𝐶𝑛𝛽

Increase in magnitude of 𝐶𝑛𝛿𝑟

Δ𝐶𝑋𝛼𝑚𝑜𝑑𝑒𝑙

=0

Δ𝐶𝑚𝛼𝑚𝑜𝑑𝑒𝑙

=−5.878075

Δ𝐶𝑚𝑞𝑚𝑜𝑑𝑒𝑙

=−12.9931

Δ𝐶𝑛𝛽𝑚𝑜𝑑𝑒𝑙

=0.028641

Δ𝐶𝑛𝛿𝑟𝑚𝑜𝑑𝑒𝑙

=0.162813

2. Comparison with existing aircraft

This subsection brieﬂy presents the strategy to verify the order of magnitude and sign of the stability derivatives

using the reference values for diﬀerent aircraft in clean cruise conﬁguration found in the appendices of [

11

]. These

values are generated using the vortex lattice method.

First, it is essential to mention the diﬀerences. The main diﬀerence that is observed is the magnitude of the

derivatives:

𝐶𝑚𝑞

,

𝐶𝑍¤𝛼

,

𝐶𝑚¤𝛼

,

𝐶𝑚𝛿𝑒

,

𝐶𝑍𝛿𝑒

and for certain aircraft the down-force

ˆ𝑢

-derivative

𝐶𝑍𝑢

. For

𝐶𝑚𝑞

and

𝐶𝑍𝛿𝑒

, it

can be explained by the fact that both wings are far from the centre of gravity and hence acting as a canard and a tail

simultaneously.

𝐶𝑚𝑞

is hence approximately doubled, whereas

𝐶𝑍𝛿𝑒

is zero due to the way the elevator is used. In

fact, the elevator is used in the same manner as an aileron and hence explains that both down-forces for both wings

cancel out. This further explains the fact that

𝐶𝑚𝛿𝑒

is approximately twice as large, as both elevators (instead of one for

conventional aircraft) allow for a higher pitching down moment. The

¤𝛼

-derivatives of the Wigeon are mostly 1.5 to

twice as large as reference values, which can be explained by their high sensitivity to the speciﬁc conﬁguration and

atmospheric conditions. Lastly,

𝐶𝑍𝑢

is signiﬁcantly smaller than for certain aircraft. This is mainly due to the fact that

the eVTOL is ﬂying at subsonic speeds which relates to very low compressibility eﬀects. Additionally, aero-elastic

eﬀects which also aﬀect the derivative are neglected.

In terms of similarities, it can be observed that the rest of derivatives have identical sign, especially for the dominant

stability derivatives w.r.t to their respective angle rates (

𝑞

,

𝑝

and

𝑟

) and the control derivatives which shows that the

model uses the same conventions. Last but not least, the assumption of the derivatives

𝐶𝑋¤𝛼

,

𝐶𝑋𝑞

,

𝐶𝑋𝛿𝑒

and

𝐶𝑌𝛿𝑎

being

zero is also the case of a wide range of diﬀerent and hence verifying the validity of the assumption.

With the latter, it can be concluded that the model shows results that are very similar to other aircraft and its

discrepancies can be easily explained by the dual-wing nature of eVTOL. Hence this conﬁrms that the model can be

used as an early preliminary tool to obtain stability and control derivatives of tandem wing (or large canard) aircraft.

Computational methods however must still be applied in order to obtain more accurate estimates of the stability behaviour

of the aircraft.

B. Veriﬁcation of Cruise Dynamics Using Numerical Model

In this section the system test to verify the dynamic model is presented. The veriﬁcation procedure consists of

verifying the value of the eigenvalues of the stability matrix A. These deﬁne the stability behaviour of the diﬀerent

eigenmotions of the aircraft. Before starting the procedure it is essential to mention that the veriﬁcation model outputs

the eigenvalues in normalised form, deﬁned as

𝜆𝑐=¯𝑐

𝑉0·𝜆

for longitudinal motions and

𝜆𝑏=𝑏

𝑉0·𝜆

for lateral and hence

must be transformed to their non-normalised form. The results are summarised in Table 8. From Table 8, it can be seen

that there is no diﬀerence between both models in terms of stability behaviour of the aircraft. The system test is hence

successful and the model implementation of the state-space matrices is conﬁrmed to be correct.

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 31 of 32

Preliminary control and stability analysis of a long-range eVTOL aircraft

Table 8 Eigenvalues computed by the model

𝜆

, the numerical model for veriﬁcation normalised

𝜆𝑐

or

𝜆𝑏

eigenvalues and the error

Eigenmotion Model eigenvalue 𝜆Numerical model eigenvalue 𝜆cor 𝜆bError:Re(𝜆),Im(𝜆)(%)

Short period −2.47; −0.89 −0.04332; −0.01557 0, 0

Phugoid −0.0226 ±0.0361 𝑗−0.0003960 ±0.0006318 𝑗0, 0

Aperiodic roll −2.52 ±0𝑗−0.28690 ±0𝑗0, 0

Dutch roll −0.00699 ±0.97 𝑗−0.0007948 ±0.1103 𝑗0, 0

Spiral 0.0125 ±0𝑗0.0014239 ±0𝑗0, 0

M Cuadrat-Grzybowski, JJ Schoser, SGP Castro Proceedings paper AIAA Scitech 2022 Forum Page 32 of 32