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Flight Dynamics and Control of an eVTOL Concept Aircraft with a

Propeller-Driven Rotor

Umberto Saetti

Postdoctoral Fellow

School of Aerospace Engineering

Georgia Institute of Technology

Atlanta, GA 30313

Jacob Enciu

Assistant Research Professor

Joseph F. Horn

Professor

Department of Aerospace Engineering

The Pennsylvania State University

University Park, PA 16802

ABSTRACT

The objective of this investigation is three-fold. First, to assess the ﬂight dynamics of an electric Vertical Take-Off

and Landing (eVTOL) concept aircraft with a propeller-driven rotor. Second, to develop a Stability and Control

Augmentation System (SCAS) for this concept aircraft. Third, to verify the potential safety beneﬁts of the concept

aircraft by analyzing the autorotation performance following a total loss of power. The paper begins with a description

of the simulation model, including a detailed discussion on the inﬂow model of the propellers that drive the main rotor.

Next, the ﬂight dynamics are assessed at hover and in forward ﬂight. A SCAS based on Dynamic Inversion (DI) is

developed to provide stability and desired response characteristics about the roll, pitch, yaw, and heave axes for speeds

ranging from hover to 80 kts. Additionally, an RPM governor is implemented to hold the main rotor angular speed

constant at its nominal value. Finally, simulations that make use of the SCAS are performed to analyze the autorotation

performance following total loss of power.

INTRODUCTION

The use of tip-jets to provide the torque to power rotor blades

has long been considered as an alternate approach to more

traditional shaft-driven rotor systems. This is because the tip-

driven rotor approach leads to the elimination of transmission

and anti-torque rotor, the latter of which allows to shorten the

tail boom. The associated reduction in the overall mechan-

ical complexity, moving parts, and weight of the aircraft re-

sults in decreased power required (or equivalently, in more

payload capacity for the same weight) and decreased mainte-

nance costs. Additionally, because the angular dynamics of

the main rotor is no longer coupled with the directional dy-

namics of the rotorcraft, an increase or decrease in collective

pitch will not result in an off-axis response about the yaw axis.

This is a favorable response characteristic when comparing

the tip-driven rotorcraft to standard helicopters, for which the

heave-yaw dynamics are strongly coupled. In fact, helicopter

pilots typically need to counteract an increase or decrease in

collective pitch with pedals input to keep the desired heading.

A summary of the various approaches to tip-driven rotors that

were proposed and implemented over the years, including il-

lustrious rotorcraft examples, is provided below:

•Cold Tip Jets Compressed air is forced out of aft-facing

nozzles at the blade tips. An engine-driven air compres-

Presented at the VFS International 76th Annual Forum &

Technology Display, Virginia Beach, VA, October 6–8, 2020. Copy-

right c

2020 by the Vertical Flight Society. All rights reserved.

sor located in the fuselage pumps the air through a rotat-

ing seal and into hollow rotor blades. The FIAT 7002,

an Italian helicopter which ﬁrst ﬂew in 1961 adopted this

approach.

•Hot Tip Jets These jets burn fuel to heat the air for

greater thrust. Two types of these jets exist, the ﬁrst in

which the fuel is added to the air and burnt at the blade

tip, the second in which the exhaust gasses from a turbine

engine located in the fuselage are expelled from nozzles

at the blade tips. The Fairy Rotodyne, a British com-

pound gyroplane, used hot tip jets and completed its ﬁrst

ﬂight in 1957.

•Ramjets Ramjet engines are mounted on the blade tips.

These jet engines use the engine’s forward motion to

compress incoming air without axial or centrifugal com-

pressors to then add fuel and ignite it. This solution was

used on the Hiller YH-32 Hornet, an American ultralight

helicopter that ﬁrst ﬂew in 1950.

•Pulsejets Jet engines in which combustion occurs in

pulses are mounted on the blade tips. This method was

demonstrated on the American Helicopter XH-26 Jet

Jeep, an experimental helicopter developed in 1951.

Although the use of tip-jets is appealing because of the advan-

tages described above, it also comes with signiﬁcant draw-

backs that have prevented this approach to vertical ﬂight to

develop into a major commercial success. These drawbacks

1

Figure 1: F-Helix: an eVTOL concept aircraft with a

propeller-driven rotor.

include the pressure losses and sealing challenges associated

with transporting compressed air or exhaust gasses to the

blade tips, the high centrifugal loads acting on the jet en-

gines, and the relatively high noise levels produced by the

tip-jet when compared to those of the usual sources of rotat-

ing blade noise. Recently, an electric Vertical Take-Off and

Landing (eVTOL) concept aircraft was proposed in which the

main rotor is driven by rotor-mounted propellers rather than

by tip-jets (Ref. 1). While this design enjoys the same advan-

tages of tip-driven rotors, it eliminates the difﬁculties related

to transporting compressed air or exhaust gasses to the blade

tips. Additionally, it may relax the disadvantages associated

with the high noise levels of tip-jets. This eVTOL concept

aircraft is called “F-Helix” and is shown in Fig. 1. The torque

to power the main rotor blades is provided by two pairs of

counter-rotating co-axial propellers, called “eProps”, which

are powered by two electric engines each and mounted on a

beam, called “power mast”, that is rigidly connected to the ro-

tor hub. The eProps are placed at a radial location of roughly

half of the rotor radius. Lift is entirely generated by its two-

bladed rotor which, along with the fuselage, is based on that

of the Silvercraft SH-4. The SH-4 is an Italian light helicopter

and is shown in Fig. 2. The absence of torque exchanged be-

tween the fuselage and the main rotor greatly reduces the yaw

moment required to trim and control the aircraft, such that

small ducted fans replace the tail rotor. These ducted fans are

located at the original location of the tail rotor and are called

“yaw fans”. The yaw fans provide thrust in opposite direc-

tions. The desired yawing moment is produced by varying

the yaw fans angular speed independently. When not used,

the yaw fans rotate at an idle speed that overall creates zero

yaw moment to eliminate time delays in thrust production as-

sociated with rotor inﬂow development. The general charac-

teristics of the F-Helix eVTOL concept aicraft are reported in

Table 1. It is worth noting that the conﬁguration described

here differs from that in Ref. 1as it retains the fuselage of the

SH-4, including tail boom and empennage, and as it mounts

the yaw fans further back on the tail boom.

While previous investigations focused on the performance and

design optimization of the F-Helix eVTOL concept aircraft

(Ref. 1), this paper concentrates on the study of its ﬂight

dynamics and control characteristics. More speciﬁcally, the

objectives of the paper are to assess the open-loop ﬂight dy-

Figure 2: Silvercraft SH-4.

Table 1: General characteristics of the F-Helix eVTOL

concept aircraft.

Description Value

Main rotor number of blades 2

Main rotor radius 14.815 ft

Main rotor angular speed 410 RPM

Capacity 4

Maximum take-off weight 1900 lbs

Maximum speed 80 kts

Cruise speed 65 kts

eProp radial location 8 ft

eProp radius 1.4 ft

eProp angular speed 2510 RPM

Power required to hover 160 shp

namics of the aircraft, to develop a Stability and Control Aug-

mentation System (SCAS), and to verify the potential safety

beneﬁts of this concept aircraft by analyzing the autorotation

performance following a total loss of power. The architecture

chose for the SCAS in Dynamic Inversion (DI) an increas-

ingly popular model-following scheme among aircraft and ro-

torcraft manufacturers, and within the aerospace ﬂight con-

trols community in general (see, e.g., Ref. 2for ﬁxed-wing

aircraft and Ref. 3for rotorcraft). A key aspect of DI is the

feedback linearization loop, where the plant model is inverted

to simultaneously decouple the controlled axes and eliminate

the need for gain scheduling (even though the plant model

in the feedback linearization loop still requires being sched-

uled with the ﬂight condition). These aspects provide a con-

venient framework for the control of eVTOL vehicles due to

their wide range of operating conditions which spans hover,

forward ﬂight, and the transition between these two. In fact,

several recent studies investigated the use of DI for eVTOL

vehicles (Refs. 4–6).

The paper begins with a description of the simulation model,

including a detailed discussion on the inﬂow model of the pro-

pellers that drive the main rotor. Next, the ﬂight dynamics are

assessed at hover and in forward ﬂight. A SCAS based on DI

is developed to provide stability and desired response charac-

teristics about the roll, pitch, yaw, and heave axes for speeds

ranging from hover to 80 kts. Additionally, an RPM governor

is implemented to hold the main rotor angular speed constant

at its nominal value. To verify the potential safety beneﬁt of

2

the conﬁguration, simulations that make use of the SCAS are

performed to analyze the autorotation performance following

total loss of power. Final remarks summarize the overall ﬁnd-

ings of the study.

ROTORCRAFT SIMULATION MODEL

A simulation model is developed for the evaluation of the trim

conditions, assessment of the stability and control characteris-

tics of the aircraft, and for the design of a SCAS. The simula-

tion model is largely based on Ref. 7and is comprised of three

main modules representing the fuselage and empennage, main

rotor, and eProps. The dynamics of each module is described

below.

Fuselage and Empennage

The fuselage aerodynamic coefﬁcients are approximated by

using the aerodynamic model of the UH-60 Black Hawk fuse-

lage of Ref. 8as a representative conventional fuselage shape.

Since the propeller-drive rotor eVTOL concept aircraft retains

the SH-4 fuselage and empennage, the aerodynamic data is

adapted to the SH-4 fuselage by using the main rotor radius

and disk area as the reference length and area, respectively, for

the scaling of the nondimensional aerodynamic coefﬁcients.

Static aerodynamic models are developed for the prediction

of the aerodynamic loads produced by the fuselage. The hor-

izontal stabilizer and vertical tail’s lift, drag, and pitch mo-

ments are estimated by means of simple ﬁnite wing models.

Simpliﬁed wake models are incorporated to simulate the ef-

fect of the main rotor wake on the empennage components.

Main Rotor

The main rotor model is formulated using a quasi-steady tip

path plane model based on analytical integrations of the blade

element equations. A 3-state Pitt-Peters inﬂow model (Ref. 9)

is used for the prediction of the dynamic inﬂow components

of the main rotor. The rotor model neglects the lead-lag dy-

namics as its effect on the rotorcraft trim, stability, and control

characteristics are secondary and thus beyond the scope of the

present investigation. The resulting closed-form model allows

for an efﬁcient calculation of the quasi-steady ﬂapping angles

and the rotor forces and moments. A more detailed descrip-

tion of this rotor model can be found in Ref. 7.

eProp Dynamic Inﬂow Model

Each eProp is constituted of two counter-rotating coaxial pro-

pellers mounted on a beam attached rigidly to the main rotor.

Because the eProps are mounted on the main rotor, they expe-

rience an axial ﬂow which is a summation of their tangential

speed (i.e., the angular speed of the main rotor times the the

radial position of the eProps) and the projection of the veloc-

ity vector of the rotorcraft on the direction of rotation of the

eProp propellers. It is worth noting that in forward ﬂight the

incident and parallel velocities to each eProp are periodic with

respect to the rotor azimuth angle ψMR with a period of 2πor

one rotor revolution. Consider now a reference eProp. The

upper (or front) rotor is subject to both axial and tangential

ﬂows. The resulting climb (or axial ﬂow) and advance ratios

for the upper rotor are given by:

λcu=ΩreProp +usinψ

ΩePropReProp

(1a)

µu=ucosψ

ΩePropReProp

(1b)

where uis the longitudinal speed of the aircraft in body-ﬁxed

frame and ψis the azimuthal position of the reference eProp.

Here it is assumed that the rotor mast is parallel to the air-

craft’s vertical axis. Because the eProps operate at high climb

ratios, the wake contraction of the upper rotor is neglected

(Ref. 10). Also, any skewing in the upper rotor wake is ne-

glected as well. This is justiﬁed by the fact that the maximum

skewing angle, which occurs at ψ=kπ, with k=0,1,2,... is

small:

χmax =tan−1umax

ΩreProp ≈21.4 deg (2)

when assuming a maximum longitudinal speed umax =80 kts.

It follows that the lower (or rear) rotor is assumed to fully

operate in the wake of the upper rotor so that the incident ve-

locity to the lower rotor is effectively the wake velocity of the

upper rotor. The resulting climb and advance ratios for the

lower rotor are:

λcL=λu+ΩreProp +usinψ

ΩePropReProp

(3a)

µl=µu(3b)

where λuis the inﬂow of the upper rotor. Each propeller is

modelled with a 1-state inﬂow model similar to the one de-

scribed in Ref. 9. The general form of the inﬂow model is:

˙

λ=−3

4πVλ+3

8πΩCT(4)

where:

λis the rotor inﬂow,

Ωis the rotor speed,

CTis the thrust coefﬁcient, and

Vis a function of climb ratio, advance ratio, and inﬂow.

Vis given by the following equation:

V=µ2+ (λc+λ)(λc+2λ)

pµ2+ (λc+λ)2(5)

Since the eProps are collocated opposite to each other, the

second eProp will operate at an azimuth angle (ψ+π). This

leads to a 4-state nonlinear system describing the inﬂow of the

upper and lower propellers on each eProp. The thrust coefﬁ-

cients are calculated by means of BEMT (Ref. 10).

3

Complete Model

The resulting equations of motion of the rotorcraft can be rep-

resented as a nonlinear system of the form:

˙

x=f(x,u)(6)

where xis the state vector uis the control input vector. The

state vector is:

xT=huvwpqrφ θ ψ (λ0λ1cλ1s)MR ΩMR ψMR

(λuλl)eProp1(λuλl)eProp2i(7)

where:

u,v,ware the velocities in the body-ﬁxed frame,

p,q,rare the angular rates,

φ,θ,ψare the Euler angles,

(λ0λ1cλ1s)MR are the main rotor inﬂow components,

ΩMR is the main rotor angular speed,

ψMR is the main rotor azimuth angle, and

(λuλl)ePropiare the induced velocities at the ith eProp’s

upper and lower rotor, respectively.

The input vector is:

uT=B1cA1cθTR θ0ΩeProp(8)

where:

B1cis the lateral stick,

A1cis the longitudinal stick,

θTR are the pedals,

θ0is the collective stick, and

ΩeProp is the eProp angular speed, assumed to be equal for

both eProps.

The lateral stick is used to control the roll attitude and indi-

rectly the lateral speed. The longitudinal stick is used to con-

trol the pitch attitude and indirectly the longitudinal speed.

The pedals are responsible for controlling the yaw fans’ an-

gular speed and are used to control the heading angle. The

collective stick is used to provide control along the vertical

axis. Lastly, the eProp angular speed is used to control the

main rotor RPM and is assumed to be equal for all propellers.

An iterative algorithm based on Newton-Rhapson is used to

trim the aircraft model at incremental speeds ranging from

hover to 80 kts (level ﬂight) at intervals of 10 kts. The weight

chosen for this analysis is the maximum take-off weight (i.e.,

1900 lb). The trim attitude across this range of speeds is

shown in Fig. 3. This ﬁgure shows that while the the ro-

torcraft is trimmed with zero bank angle, the resulting sideslip

angle is very small. In fact, the sideslip angle is approximately

zero at all speeds. This tendency in the trim attitude differs

greatly from standard helicopters in the sense that trimming

helicopters with zero sideslip results in a non-zero bank angle

and viceversa. This trim analysis indicates that because there

is no torque exchanged between the main rotor and fuselage,

the F-Helix eVTOL concept aircraft can simultaneously be

trimmed with zero sideslip and zero bank angle.

0 10 20 30 40 50 60 70 80

-1

0

1

(deg)

0 10 20 30 40 50 60 70 80

-4

-2

0

(deg)

0 10 20 30 40 50 60 70 80

True airspeed, VTAS (kts)

-5

0

5

10

(deg)

10-3

Figure 3: Trim attitude vs. true airspeed.

DYNAMIC STABILITY

To study the open-loop stability of the rotorcraft, linear time-

invariant models are obtained by linearizing the nonlinear dy-

namics at hover and 80 kts level ﬂight. The weight used for

this analysis is the maximum take-off weight (i.e., 1900 lb).

Figure 4shows the eigenvalues of the linearized full-order

system. Speciﬁcally, Fig. 4a shows all the eigenvalues of

the system, where the eigenvalues to the far left of the com-

plex plane are relative to the inﬂow dynamics of the eProps,

whereas those closer to the origin concern the rigid-body, and

main rotor inﬂow and angular dynamics. From this ﬁgure, it

is clear that the eProp inﬂow dynamics is stable and consider-

ably faster than the rigid-body and main rotor dynamics both

at hover and at 80 kts level ﬂight. The low-frequency eigen-

values are shown in greater detail in Fig. 4b. The eigenval-

ues at hover are summarized in Table 2, which identiﬁes the

modes based on analysis of the eigenvectors. An effort was

done to classify the rigid-body modes according to the termi-

nology that is used to classify modes that are often seen on

typical aircraft. However, it should be kept in mind that this

is indeed not a typical aircraft. Most notably, there are two

unstable modes at hover: a slower mode involving the longi-

tudinal speed uand the pitch attitude θ(i.e., phugoid), and a

faster mode involving a coupled roll-pitch oscillation. This is

behavior is typical of helicopter at hover. Additionally, it is

worth noting that the main rotor angular speed mode is stable

at this ﬂight condition. This may be explained physically by

observing that if the main rotor angular speed increases, then

the inﬂow at the eProp rotor disk increases as well. However,

an increase in inﬂow coincides with a decrease in induced in-

ﬂow and a resulting decrease in thrust, which in terms causes

the rotor to slow down. The opposite is also true. The eigen-

values and modes at 80 kts forward ﬂight are summarized in

Table 3. This table shows that the dutch roll mode is unstable

with characteristics that, per ADS-33E handling qualities re-

quirements for rotorcraft (Ref. 11) does not achieve Level 3 in

the predicted lateral-directional oscillatory handling qualities

4

(i.e., the minimal level of handling qualities for safe ﬂight; see

Fig. 23 of Ref. 11). This result can be explained by a lack of

yaw damping because of the absence of the tail rotor. This

issue can be alleviated by providing additional yaw damping

via feedback control to the yaw fans and/or by increasing the

vertical tail size. Additionally, the main rotor angular speed

mode is stable with a frequency roughly double than that at

hover. This result indicates that forward speed has a beneﬁcial

effect to the main rotor angular speed mode in the sense that it

moves the eigenvalue associated to the mode further to the left

of the complex plane. Although the main rotor angular speed

mode is stable both at hover and in forward ﬂight, the mode

has relatively low frequency, which leads to a relatively slow

response to disturbances. For this reason, an RPM governor

should be implemented to enhance the response characteris-

tics of the main rotor angular speed mode.

-120 -100 -80 -60 -40 -20 0 20

Real

-8

-6

-4

-2

0

2

4

6

8

Imag

Hover

80 kts

(a) All eigenvalues.

-2.5 -2 -1.5 -1 -0.5 0 0.5

Real

-1.5

-1

-0.5

0

0.5

1

1.5

Imag

2.5 2 1.5 1 0.5

0.985

0.94

0.86 0.76 0.64 0.5 0.34 0.16

0.985

0.94

0.86 0.76 0.64 0.5 0.34 0.16

Hover

80 kts

(b) Rigid-body eigenvalues.

Figure 4: Eigenvalues of the linearized full-order system at

hover and high-speed level ﬂight.

Table 2: Summary of the eigenvalues at hover.

Mode Eigenvalue

Roll-Pitch Oscillation −1.3387 ±0.2397i

Roll-Pitch Oscillation 0.2697 ±0.5620i

Phugoid 0.0276 ±1.0171i

Heave Subsidence −0.1053

Yaw Subsidence −0.0045

Main Rotor Angular Speed −0.1416

Main Rotor Coning Inﬂow −1.8445

Main Rotor Cyclic Inﬂow (2×)−10.4566

eProp Inﬂow −101.6494

eProp Inﬂow −101.1208

eProp Inﬂow −101.1476

eProp Inﬂow −101.6137

Table 3: Summary of the eigenvalues at 80 kts level ﬂight.

Mode Eigenvalue

Short Period −1.3377 ±1.1845i

Phugoid −0.0386 ±0.2476i

Dutch Roll 0.4672 ±0.2049i

Coupled Subsidence/Spiral Mode −2.1776

Coupled Subsidence/Spiral Mode −1.7793

Main Rotor Angular Speed −0.2209

Main Rotor Coning Inﬂow −16.4377

Main Rotor Cyclic Inﬂow −24.9357 ±7.7208i

eProp Inﬂow −94.7127

eProp Inﬂow −94.2907

eProp Inﬂow −94.6667

eProp Inﬂow −94.3279

Another important aspect of the propeller-driven rotor conﬁg-

uration that does not arise from analyzing its modes of motion

is that the dynamics of the heave and yaw motions are decou-

pled. Because there is no torque exchanged between the main

rotor and fuselage, a collective stick input will not result in

an off-axis response about the yaw axis. This is a favorable

response characteristic when comparing the propeller-driven

rotor concept aircraft to standard helicopters, for which the

heave-yaw dynamics are strongly coupled. In fact, helicopter

pilots typically need to counteract an increase or decrease in

collective pitch with input to the pedals to keep the desired

heading. In many helicopters this is done through mechanical

mixing between the collective and pedals.

FLIGHT CONTROL DESIGN

The architecture chosen for the SCAS is Dynamic Inversion

(DI), a popular model-following scheme among aircraft and

rotorcraft manufacturers, and within the aerospace ﬂight con-

trols community in general (e.g., see Refs. 3,12–16). A

generic DI controller as applied to a linear system is shown

in Fig. 5. The key components are a command model (also

known as commend model or reference model) that speciﬁes

desired response to pilot commands, a feedback compensation

on the tracking error, and an inner feedback loop that achieves

model inversion (also known as feedback linearization loop).

5

Dynamic inversion is used to provide stability, disturbance re-

jection, and Rate Command - Attitude Hold (RCAH) response

about the roll, pitch, yaw, and heave axes. Additionally, a DI-

based governor is implemented to hold the main rotor angular

speed constant. Because the system is designed with RCAH

response type it is expected that it might be implemented as

a partial authority ﬂight control system integrated with the

existing mechanical controls to the main rotor on the SH-4.

However, the details of implementation of the SCAS system

is a matter of future research.

Reduced-Order Models

The very ﬁrst step to developing a ﬂight control law is to ob-

tain linear models representative of the rotorcraft dynamics

across the ﬂight conditions of interest. For this reason, linear

models are derived at incremental forward ﬂight speeds Vby

means of linearization.

˙

x=A(V)x+B(V)u(9)

where the state and control vectors are given in Eqs. (7) and

(8), and where the coefﬁcient matrices are functions of the

total speed of the aircraft V=√u2+v2+w2. Since DI re-

quires full-state feedback, to make the controller design more

tractable, reduced-order models are utilized for the synthe-

sis of the controller rather than the full-order model. The

reduced-order model is obtained by means of residualization,

a methodology deriving from Singular Perturbation Theory

(Ref. 17). The key assumption is that the dynamics of a sub-

set of states are substantially faster than that of the remaining

states. It follows that the dynamics of the fast subset of states

reach steady-state more quickly than that of the slow subset.

By this principle, an approximate reduced-order model that

represents the slow and neglects fast phenomena of the sys-

tem is derived. Speciﬁcally, the state vector of the full-order

model is partitioned into slow and fast components:

xT=xT

sxT

f(10)

where:

ˆ

xT

s=uvwpqrφ θ ΩMR(11a)

ˆ

xT

f=(λ0λ1cλ1s)MR (λuλl)eProp1(λuλl)eProp2(11b)

It is worth noting that the heading angle ψis truncated from

the state vector as it does not affect the dynamics of the air-

craft. Similarly, the main rotor azimuth angle ψMR is trun-

cated from the state vector as it does not affect the averaged

dynamics (over one rotor revolution) of the aircraft. The full-

order model is re-written as:

˙

xs

˙

xf=AsAsf

Afs Afxs

xf+Bs

Bfu(12)

Following the derivation in Ref. 18, the following 9-state

reduced-order models are obtained:

˙

xs=ˆ

Axs+ˆ

Bu (13)

where:

ˆ

A=As−AsfAf−1Afs (14a)

ˆ

B=Bs−AsfAf−1Bf(14b)

To demonstrate the approach, the reduced-order model eigen-

values are compared to those of the full-order model at hover,

as shown in Fig. 6. Note that the reduced-order model

eigenvalues nearly overlay corresponding full-order model

eigenvalues at low frequency. The analysis shows that the

reduced-order model is indeed a good approximation of the

low-frequency dynamics of the aircraft.

Dynamic Inversion Flight Control Law

Deﬁne the following output vector, corresponding to the con-

trolled variables of the nonlinear system (i.e., the aircraft dy-

namics):

yT=p q r VzΩMR(15)

where Vzis the vertical speed in the inertial frame (positive

up). The output matrix that relates the state vector to the out-

put vector:

C(V) =

0 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 −10000V0

0 0 0 0 0 0 0 0 1

(16)

It is worth noting that the output matrix is a function of the

forward speed Vand therefore requires scheduling. The ob-

jective of the DI control law is that the output ytracks a refer-

ence trajectory ycmd(t)given by:

yT

cmd =pcmd qcmd rcmd Vzcmd ΩMRcmd(17)

with desired response characteristics. For this reason the ref-

erence trajectory is fed through ﬁrst-order command models

(also known as command ﬁlters or ideal responses) which dic-

tate the desired response of the system. The command models

are also used to extract the ﬁrst derivative of the ﬁltered refer-

ence trajectory for use in the proportional-integral (PI) com-

pensator described below. The command models are of the

following form:

Gideal(s) = 1

τs+1(18)

where τis the command model time constant, which is the in-

verse of the command model break frequency (i.e.,τ=1/ωn).

Table 4shows the values used for the parameters of the com-

mand models.

In order for the controls to appear explicitly in the output

equations, the output equations are differentiated once, yield-

ing:

˙p

˙q

˙r

˙

Vz

˙

ΩMR

=CAx +CBu (19)

6

Figure 5: DI controller as applied to a linear system.

-1.5 -1 -0.5 0 0.5

Real

-1.5

-1

-0.5

0

0.5

1

1.5

Imag

Full-Order Model

Reduced-Order Model

Figure 6: Comparison between the eigenvalues of the

full-order and reduced-order models at hover.

Table 4: Command models break frequencies.

Command ωn[rad/s]

Roll Rate 4.5

Pitch Rate 4.5

Yaw Rate 2.0

Vertical Position 1.0

Main Rotor Angular Speed 3.0

PI compensation is used to reject external disturbances and to

compensate for discrepancies between the approximate model

used in this derivation and the actual bare-airframe dynamics

of the aircraft. The resulting DI control law is found by solv-

ing for the control vector, leading to:

u= (Cˆ

B)−1(ν−Cˆ

Axs)(20)

where νis the pseudo-command vector and eis the error as

deﬁned respectively in Eqs. (21) and (22).

νp

νq

νr

νVz

νΩMR

=

¨pcmd

¨qcmd

˙rcmd

˙

Vzcmd

˙

ΩMRcmd

+KP

ep

eq

er

eVz

eΩMR

KI

Repdt

Reqdt

Rerdt

ReVzdt

ReΩMR dt

(21)

e=ycmd −y; (22)

The 5-by-5 diagonal matrices KPand KIidentify the pro-

portional and integral gain matrices, respectively. Note that

the coefﬁcient matrices Cˆ

B)−1(V)and Cˆ

A(V)are functions

of the total speed of the aircraft V. For this reason, from a

practical standpoint, these matrices are computed ofﬂine at

incremental speeds from hover to 80 kts at 10 kts intervals

and stored. When the linearized DI controller is implemented

on the nonlinear aircraft dynamics, the coefﬁcient matrices

Cˆ

B)−1and Cˆ

Aare computed at each time step via interpola-

tion based on the current total airspeed V(t)and on the tabled

stored ofﬂine. It is important to note that what is implemented

on the nonlinear aircraft dynamics is linearized DI. However,

because the coefﬁcient matrices are scheduled with total air-

speed, where scheduling effectively introduces a nonlinear

relation between the aircraft states and the feedback control

input, the controller implemented is effectively nonlinear DI

(NLDI) (Ref. 3). A block diagram of the linearized DI ﬂight

control law is shown in Fig. 7. Note that turn coordination and

turn compensation laws are incorporated in the feed-forward

path of the control system (Ref. 19).

Error Dynamics

Feedback compensation is needed to ensure the system tracks

the command models. It can be demonstrated (Ref. 2) that

for a DI control law the output equation must be differenti-

ated ntimes for the controls to appear explicitly in the output

equation:

e(n)=y(n)

cmd −ν(23)

Since in the present study the output equation requires being

differentiated only once for the controls to appear explicitly

7

Figure 7: DI ﬂight control law.

in the output equation, a PI control strategy is applied to the

pseudo-command vector:

ν=˙ycmd (t) + KPe(t) + KIZt

0

e(τ)dτ(24)

Substituting Eq. (24) into Eq. (23) leads to the closed-loop

error dynamics:

˙e(t) + KPe(t) + KIZt

0

e(τ)dτ=0 (25)

The gains are chosen such that the frequencies of the error

dynamics are of the same order as the command ﬁlters (i.e.,

ﬁrst order), ensuring that the bandwidth of the response to

disturbances is comparable to that of an input given by a pilot

or outer loop. By taking the Laplace transform, and there-

fore switching to the frequency domain, the error dynamics

becomes

e(s)s2+sKP+KI=0 (26)

To obtain the gains that guarantee the desired response, the er-

ror dynamics of Eq. (26) is set equal to the following second-

order system:

s2+2ζ ωns+ωn2=0 (27)

yielding the following proportional and integral gains:

KP=2ζ ωn(28a)

KI=ωn2(28b)

Table 5shows the natural frequencies and damping ratios used

for disturbance rejection. It is worth noting that because the

Table 5: Disturbance rejection natural frequencies and

damping ratios.

ωn[rad/s]ζ

pcmd 4.5 0.7

qcmd 4.5 0.7

rcmd 2 0.7

Vzcmd 1 0.7

ΩMRcmd 3 0.7

Table 6: DI inner loop compensation gains.

KPKI

qcmd 6.3 20.25

qcmd 6.3 20.25

rcmd 2.8 4

Vzcmd 1.4 1

ΩMRcmd 4.2 9

plant is inverted in the feedback linearization loop such that

the system being controlled is effectively a set of integrators,

there is no need for gain scheduling. However, the plant model

used for feedback linearization still requires to be scheduled

with the ﬂight condition. Table 6shows the PI gains.

Attitude Hold and Airspeed Hold Modes

Attitude hold and airspeed hold modes were added to the con-

troller for the autorotation simulations, as these were extended

8

simulations that required more precise control of the aircraft

ﬂight path and speed. The attitude hold mode requires some

small modiﬁcation to the DI control law. First, the roll and

pitch attitude command signals are constructed from the com-

manded body axis roll and pitch rates.

For the pitch axis, the pitch attitude command is constructed

using the exact Euler angle kinematics:

˙

θcmd =qcmd cosφ+rsinφ(29a)

θcmd =Z˙

θcmddt (29b)

For the roll axis, it is sufﬁciently accurate to compute the com-

manded roll attitude as an integration of the body axis roll rate.

The error dynamics are then modiﬁed such that the rate inte-

grator is replaced by proportional attitude compensation, and

an integrated attitude error is added. This can be thought of

as proportion plus integrator plus double integrator compen-

sation on the angular rate. For example, the psuodo-control

for for pitch is:

νθ=˙qcmd +KP(qcmd −q)+ KI(θcmd −θ)+KII Z(θcmd −θ)dt

(30)

If we deﬁne the tracking error in terms of angular rate, this

feedback controller linearizes to third order error dynamics:

˙e(t) + KPe(t) + KIZe(τ)dτ+KII ZZ e(τ)dτ=0 (31)

The Laplace transform of the error dynamics yields

e(s)(s3+s2KP+KIs+KII ) = 0 (32a)

(s2+2ζ ωns+ωn2)(s+p) = 0 (32b)

The gains can now be deﬁned by the natural frequency, damp-

ing ratio, and a real axis pole such that:

KP=2ζ ωn+p(33a)

KI=ωn2+2ζ ωnp(33b)

KII =ωn2p(33c)

For the attitude hold modes, the natural frequency and damp-

ing ratio are exactly the same as used in the rate command

mode as shown in Table 5. The real axis pole is selected as

1/5 of the natural frequency (thus p=0.9 for both roll and

pitch).

Th airspeed hold mode is implemented as an outer loop con-

troller that computes the pitch rate command fed to the pitch

axis rate command / attitude hold controller. The design is

based on a simple dynamic inverse model of the longitudinal

velocity dynamics. We assume acceleration is proportional to

perturbations in pitch attitude, and the pitch attitude follows

directly from the pitch rate commands:

˙u=−g∆θ(34)

∆˙

θ=qcmd (35)

If we can deﬁne the control law:

qcmd =−(1/g)( ¨ucmd + (KP+KDs)(ucmd −u)) (36)

Then the velocity error dynamics in Laplace domain are spec-

iﬁed by:

(ucmd(s)−u(s))(s2+KDs+KP) = 0 (37)

The gains can be selected to have desired natural frequency

and damping ratio as discussed in Eq. (28b). However, for

our assumptions to hold there needs to be frequency separa-

tion between the inner and outer loop controller. Thus, the

natural frequency of the velocity dynamics was chosen as 1/5

of the pitch axis natural frequency (0.9 rad/s), while the damp-

ing ratio was also selected as 0.7. In practice, if the primary

function of the controller is too hold velocity and allow for lin-

ear changes in velocity command, the second derivative term

in the control law is not necessary. For the autorotation study,

the velocity commands were ramp inputs, and thus the second

derivative is 0.

Flight Control Law Validation

The DI ﬂight control law is implemented on the nonlinear air-

craft dynamics and tested in batch simulations. The ﬂight

condition in consideration is 65 kts level ﬂight, which cor-

responds to the speed for maximum range. Figure 8shows

the response of the controlled variables to a 20% longitudinal

stick doublet. This ﬁgure shows how the closed-loop system

tracks the commanded pitch rate while providing stability on

the remaining axes. More speciﬁcally, the off-axis response is

very well-contained and the main rotor angular speed is held

approximately constant by the RPM governor. Figure 9shows

the eProp inﬂow response to the 20% longitudinal stick dou-

blet. The amplitude of the oscillation of the inﬂow ratio for

the eProps upper (or front) rotors is substantially higher than

that of the eProps lower (or rear) rotors. This suggests that

the upper (or front) rotor of each eProp acts as a ﬁlter for the

inﬂow of the lower (or rear) rotor. Although further validation

on the other controlled axes is omitted for brevity, it is con-

cluded that the DI ﬂight control law is successful in stabilizing

the aircraft while providing desired RCAH response about the

roll, pitch, yaw, heave, and main rotor angular speed axes.

AUTOROTATION SIMULATION

The F-Helix design results in low disk loading and high ro-

tor inertia. This has the potential beneﬁt of providing good

autorotation performance, which is a signiﬁcant safety advan-

tage when compared to other eVTOL conﬁgurations. Autoro-

tation performance is often measured in terms of the Autoro-

tation Index (AI), which represents the ratio of total kinetic

9

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-2

0

2

p (rad/s)

10-3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-0.1

0

0.1

q (rad/s)

Response Command

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-2

0

2

r (rad/s)

10-3

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

-0.03

-0.02

-0.01

0

Vz (m/s)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (s)

43.76

43.77

43.78

MR (rad/s)

Figure 8: Response of the controlled variables to a

longitudinal stick doublet at 65 kts level ﬂight.

energy in the rotor in nominal conditions per unit gross weight

to the rotorcrtaft disk loading. Figure 10 shows trends of AI

values for legacy rotorcraft (as documented in Ref. 10), and

includes both the SH-4 and the F-Helix for reference. The AI

value of F-Helix is 170, which is clearly well above the typ-

ical value of ﬁelded rotorcraft. The AI value was calculated

by approximating the eProps as point masses weighting 50 lb

each, and by assuming the power mast to be a uniform beam

weighting 40 lb.

To verify this potential safety beneﬁt, simulations were per-

formed to analyze autorotation performance following total

loss of power. To analyze this maneuver, it was necessary to

enhance the basic DI controller by adding the attitude and air-

speed hold modes as discussed above. In addition to these

modiﬁcation, an autorotation mode was implemented using

some basic logic and simple modiﬁcations to the DI con-

troller:

1. Following power loss, the collective pitch is no longer

a viable mechanism for altitude control. Therefore, the

vertical axis DI controller is disabled, and the collective

is moved to a ﬁxed position deemed suitable for autoro-

tation. For most piloted rotorcraft, this usually corre-

sponds to the bottom position of the collective lever. In

this study, trim sweeps were performed for descending

ﬂight at various forward airspeeds, and the value of col-

lective corresponding to zero torque at the nominal RPM

was determined. This value was consistently around 0.07

rad (4.0 deg). Following detection of power loss, the col-

lective is ramped to this value at a rate of 3 deg/s.

2. The RPM governor is also disabled. The inversion pre-

sented in Eq. 20 then becomes a 3 x 3 inversion.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (s)

0.72

0.73

0.74

0.75

0.76

0.77

0.78

0.79

0.8

0.81

eProp inflow ratio,

Upper Rotor (eProp1)

Lower Rotor (eProp1)

Upper Rotor (eProp2)

Lower Rotor (eProp2)

Figure 9: Response of the eProp inﬂow ratio to a longitudinal

stick doublet at 65 kts level ﬂight.

3. The trim analysis discussed above was also used to ﬁnd

the forward airspeed at which the F-Helix can autorotate

with minimal descent rate. The minimum descent rate

was found to be 4.5 m/s when the forward airspeed was

41 knots. Thus, following a power failure, the airspeed

controller commands a 41 kts forward speed, ramping in

this command at 1 m/s/s (resulting in an acceleration or

deceleration of approximately 0.1 g).

At this time it was not possible to perform a successful au-

torotation simulation for speeds below 10 m/s. This was due

to a numerical instability that seemed to occur when the rotor

was near a vortex ring condition. It is suspected that this is a

numerical issue with the Pitt-Peters inﬂow model implemen-

tation and not a limitation of the conﬁguration.

The simulation analysis consisted of the following steps and

assumptions:

1. Power failure was simulated by ramping down the eProp

RPM to 0 over the span of 1 second.

2. A 2.0 seconds delay was applied before turning on the

autorotation mode of the controller.

3. The simulation was performed starting in level ﬂight at

500 m altitude for initial airspeeds ranging from 10 to 70

kts and carried out for 50 seconds following the power

failure. This study focuses only on entry to autorotation,

and no attempt was made to simulate ﬂare and landing at

this time.

Figure 11 shows sample trajectories for the autorotation ma-

neuver simulated for seven different initial airspeeds (10 to 70

kts). Figure 12 shows the airspeed, vertical speed, and pitch

attitude response, while Fig. 13 show the collective pitch and

the main rotor and eProp RPM response. The results show that

in all cases the rotorcraft reaches a steady autorotative descent

characterized by a constant airspeed and descent rate and a

10

constant rotor RPM. As dictated by the autorotation mode of

the controller, the rotorcraft ends up in the same steady ﬂight

condition: 41 kts forward speed with a 4.5 m/s descent rate

and rotor speed near its nominal value. In the case of the

higher initial airspeeds (50 to 70 knots), the rotorcraft per-

forms a deceleration and thus its initial descent rate is less in

the initial phase of the maneuver, while for the slower airspeed

(10 to 40 knots) the rotorcraft must accelerate and loses more

altitude at the start of the maneuver. Because of the 2 sec-

ond delay imposed on power failure detection, Fig. 13 shows

that the vertical axis controller initially tries to increase col-

lective to main altitude, which delays descent but increases

the main rotor speed droop. Still, due to the high rotor iner-

tia, the highest rotor speed droop was only about 11.5 percent.

The 2 second delay is a conservative assumption, and clearly

performance could be improved with quicker detection.

Using this type of analyses, one can start to make the ﬁrst

steps towards prediction of the height-velocity envelope for

safe autorotation. At this stage, we only predict an entry phase

where the rotorcraft reaches a steady descent rate. However,

in emergency autorotations at low altitude the helicopter may

not reach a steady descent rate or have time to reach the for-

ward airspeed for minimal descent rate, especially when start-

ing from a low forward speed. In these cases, a more complex

dynamic maneuver would need to be simulated to test autoro-

tation capability. In any case, the results show the rotorcraft

can reach a steady controllable descent with a small transient

droop in rotor RPM. The advantage of the high inertia rotor

would also provide beneﬁt for the ﬁnal ﬂare and collective

breaking maneuvers to land the rotorcraft.

Figure 10: Autorotation index (recreated from Ref. 10).

CONCLUSIONS

The open-loop ﬂight dynamics of an eVTOL concept aircraft

with a propeller-driven rotor were assessed at hover and in

forward ﬂight. A Stability and Control Augmentation sys-

tem (SCAS) based on Dynamic Inversion (DI) was devel-

oped to provide stability and desired response characteristics

about the roll, pitch, yaw, and heave axes. Additionally, an

RPM governor was implemented to hold the main rotor angu-

lar speed constant at its nominal value with desired response

Figure 11: Flight trajectories for the autorotation simulations.

Figure 12: Airspeed, vertical velocity, and pitch attitude for

the autorotation simulations.

characteristics. To verify the potential safety beneﬁt of the

conﬁguration, simulations that make use of the SCAS were

performed to analyze the autorotation performance following

total loss of power. This work results in the following conclu-

sions:

1. The trim analysis indicates that, because there is no

torque exchanged between the main rotor and fuselage,

the F-Helix eVTOL concept aircraft can simultaneously

be trimmed with zero sideslip and zero bank angle. This

11

Figure 13: Collective pitch, rotor RPM, and eProp RPM for

the autorotation simulations.

tendency in the trim attitude differs greatly from stan-

dard helicopters in the sense that trimming helicopters

with zero sideslip results in a non-zero bank angle and

viceversa.

2. At high-speed forward ﬂight the Dutch roll mode is un-

stable with characteristics that, per ADS-33E handling

qualities requirements for rotorcraft, does not achieve

Level 3 in the predicted lateral-directional oscillatory

handling qualities. This result can be explained by a lack

of yaw damping because of the absence of the tail rotor.

This issue can be alleviated by providing additional yaw

damping via feedback control to the yaw fans and/or by

increasing the vertical tail size.

3. The main rotor angular speed mode is stable both at

hover and high-speed ﬂight. Forward speed has a beneﬁ-

cial effect on this mode as it increases its response band-

width. However, an RPM governor should be considered

for enhancing the response of the mode since its response

is relatively slow.

4. Because there is no torque exchanged between the main

rotor and fuselage, the dynamics of the heave and yaw

axes are decoupled. This is a favorable response char-

acteristic when comparing the propeller-driven rotor eV-

TOL concept aircraft to standard helicopters, for which

the heave-yaw dynamics are strongly coupled.

5. The amplitude of the oscillation of the inﬂow ratio for the

eProps upper (or front) rotors is substantially higher than

that of the eProps lower (or rear) rotors. This suggests

that the upper (or front) rotor of each eProp acts as a

ﬁlter for the inﬂow of the lower (or rear) rotor.

6. The autorotation results show that the rotorcraft can

reach a steady controllable descent with a small transient

droop in rotor RPM following total loss of power. The

advantage of the high-inertia rotor would also provide

beneﬁt for the ﬁnal ﬂare and collective breaking maneu-

vers to land the rotorcraft.

ACKNOWLEDGEMENTS

This research was partially funded by Vinati Srl under a spon-

sored research contract. The views and conclusions contained

in this document are those of the authors and should not be in-

terpreted as representing the policies, either expressed or im-

plied, of Vinati Srl.

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