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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.
REFERENCES
1. Saetti, U., Enciu, J., and Horn, J. F., “Performance and
Design Optimization of the F-Helix eVTOL Concept,”
Proceedings of the 75th Annual Forum of the Vertical
Flight Society, May 13-16, 2019.
2. Stevens, B. L., and Lewis, F. L., Aircraft Control
and Simulation: Dynamics, Controls Design, and Au-
tonomous Systems, Third Edition, John Wiley and Sons,
Inc., New York, 2015.
3. Horn, J. F., “Non-Linear Dynamic Inversion Control
Design for Rotorcraft,” Aerospace, Vol. 6, (3), 2019.
DOI: 10.3390/aerospace6030038
4. Lombaerts, T., Kaneshige, J., Schuet, S., Hardy, G.,
Aponso, B., and Shish, K., “Nonlinear Dynamic Inver-
sion based Attitude Control for a Hovering Quad Tiltro-
tor eVTOL Vehicle,” Paper AIAA 2019-0134, Proceed-
ings of the AIAA Scitech Forum, Jan 7-11, 2019.
5. Lombaerts, T., Kaneshige, J., Schuet, S., Hardy, G.,
Aponso, B., and Shish, K., “Dynamic Inversion based
Full Envelope Flight Control for an eVTOL Vehicle
using a Unified Framework,” Paper AIAA 2020-1619,
Proceedings of the AIAA Scitech Forum, Jan 6-10,
2020.
6. Lombaerts, T., Kaneshige, J., and Feary, M., “Con-
trol Concepts for Simplified Vehicle Operations of a
Quadrotor eVTOL Vehicle,” Paper AIAA 2020-3189,
Proceedings of the AIAA Aviation Forum, June 15-19,
2020.
12
7. Enciu, J., and Rosen, A., “Simulation of Coupled
Helicopter-Slung Load-Pilot Dynamics,” Journal of the
American Helicopter Society, Vol. 62, (1), 2017, pp. 1–
13(13). DOI: 10.4050/JAHS.62.012007
8. Howlett, J. J., “UH-60A Black Hawk Engineering Simu-
lation Program: Volume I - Mathematical Model,” Tech-
nical report, NASA-CR-166309, 1981.
9. Pitt, D. M., and Peters, D. A., “Theoretical Prediction of
Dynamic-Inflow Derivatives,” Vertica, Vol. 5, (1), 1883,
pp. 21–34.
10. Leishman, J. G., Principles of Helicopter Aerodynam-
ics, Cambridge University Press, New York, New York,
2006.
11. Anon, Aeronautical Design Standard Performance
Specification, Handling Qualities Requirements for Mil-
itary Rotorcraft, ADS-33-PRF, USAAMCOM, 2000.
12. Saetti, U., Horn, J. F., Lakhmani, S., Lagoa, C., and
Berger, T., “Design of Dynamic Inversion and Explicit
Model Following Control Laws for Quadrotor Inner and
Outer Loops,” Proceedings of the 74th Annual Forum of
the Vertical Flight Society, May 14-17, 2018.
13. Saetti, U., Horn, J. F., Villafana, W., Sharma, K., and
Brentner, K. S., “Rotorcraft Simulations with Coupled
Flight Dynamics, Free Wake, and Acoustics,” Proceed-
ings of the 72nd Annual Forum of the American Heli-
copter Society, May 16-19, 2016.
14. Saetti, J., U., Horn, J. F., Lakhmani, S., Lagoa, C., and
Berger, T., “Design of Dynamic Inversion and Explicit
Model Following Control Laws for Quadrotor UAS,”
Journal of the American Helicopter Society, Vol. 65, (3),
2020, pp. 1–16(16). DOI: 10.4050/JAHS.65.032006
15. Caudle, D. B., DAMAGE MITIGATION FOR ROTOR-
CRAFT THROUGH LOAD ALLEVIATING CONTROL,
Master’s thesis, The Pennsylvania State University, Uni-
versity Park, PA, 2014.
16. Spires, J. M., and Horn, J. F., “Multi-Input Multi-Output
Model-Following Control Design Methods for Rotor-
craft,” Proceedings of the 71st Annual Forum of the
American Helicopter Society, May 5-7, 2015.
17. Kokotovic, P. V., O’Malley, R. E., and Sannuti, P.,
“Singular Perturbations and Order Reduction in Control
Theory, an Overview,” Automatica, Vol. 12, (2), 1976,
pp. 123–132. DOI: 10.1016/0005-1098(76)90076-5
18. Saetti, U., and Horn, J. F., “Use of Harmonic Decompo-
sition Models in Rotorcraft Flight Control Design with
Alleviation of Vibratory Loads,” Proceedings of the 72nd
Annual Forum of the American Helicopter Society, May
16-19, 2016.
19. Blakelock, J. H., Automatic Control of Aircraft and
Missiles, John Wiley & Sons, New York, 1965.
DOI: 10.1017/S0001924000057912
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