Conference PaperPDF Available

Generic Tilt-Rotor Simulation Model with Coupled Flight Dynamics, State-Variable Aeromechanics, and Aeroacoustics

Authors:

Abstract and Figures

This paper describes the development, implementation, and validation of a generic tilt-rotor simulation model with coupled flight dynamics, state-variable aeromechanics, and aeroacoustic. A major novelty of this work lies in the integration of the flight dynamics with a state-space free-vortex wake code that adopts a near-wake vortex-lattice model. This way, the flight dynamics are augmented by the vortex wake dynamics so that the coupled flight and wake dynamics is self-contained and inherently linearizable. The model is implemented for a Bell XV-15 tiltrotor and validated against U.S. Army/NASA XV-15 flight-test data and other data in the literature. Flight control design is performed to provide desired stability, performance, handling-quality properties and to allow for a fully-autonomous transition between hover in helicopter mode and high-speed flight in aircraft mode. The simulation model has clear applications in the development and testing of advanced flight control laws, aeromechanics analysis, and in the prediction of aerodynamically-generated noise in generalized maneuvering flight.
Content may be subject to copyright.
Generic Tilt-Rotor Simulation Model with Coupled Flight Dynamics,
State-Variable Aeromechanics, and Aeroacoustics
Umberto Saetti
Assistant Professor
Department of Aerospace Engineering
University of Maryland
College Park, MD 20740
Batın Bu˘
gday
Graduate Research Assistant
Department of Aerospace Engineering
University of Maryland
College Park, MD 20740
ABSTRACT
This paper describes the development, implementation, and validation of a generic tilt-rotor simulation model with
coupled flight dynamics, state-variable aeromechanics, and aeroacoustic. A major novelty of this work lies in the inte-
gration of the flight dynamics with a state-space free-vortex wake code that adopts a near-wake vortex-lattice model.
This way, the flight dynamics are augmented by the vortex wake dynamics so that the coupled flight and wake dynam-
ics is self-contained and inherently linearizable. The model is implemented for a Bell XV-15 tiltrotor and validated
against U.S. Army/NASA XV-15 flight-test data and other data in the literature. Flight control design is performed
to provide desired stability, performance, handling-quality properties and to allow for a fully-autonomous transition
between hover in helicopter mode and high-speed flight in aircraft mode. The simulation model has clear applica-
tions in the development and testing of advanced flight control laws, aeromechanics analysis, and in the prediction of
aerodynamically-generated noise in generalized maneuvering flight.
INTRODUCTION
Simulations of rotorcraft flight dynamics have advanced sig-
nificantly over the past decade. To provide rapid simulations
of generalized maneuvering flight, flight dynamics models
were once restricted to relatively low fidelity aeromechan-
ics models, e.g., finite-state inflow, rigid blade element mod-
els. On the other hand, comprehensive aeromechanics simu-
lations historically used much higher fidelity aeromechanics,
e.g., free vortex-wake modeling or even computational fluid
dynamics (CFD) coupled with structural dynamics, at the cost
of longer run times with analysis restricted to trim or very sim-
ple maneuvers. In recent years, increasingly higher fidelity
aeromechanics are making their way into flight dynamics sim-
ulations, and even real-time piloted simulations. Real-time
time-accurate free wake modeling was implemented in the
General Helicopter (GenHel) (Ref. 1) simulation and demon-
strated using the CHARM Wake Module (Ref. 2) in the early
2000s. The CHARM Wake Module was subsequently imple-
mented in Navy training simulators coupled with ship airwake
effects (Ref. 3). The FLIGHTLAB simulation tool has the ca-
pability to use Viscous Vortex Particle Models (VVPM) (Ref.
4). Recently, state-space free vortex wake models were cou-
pled with a UH-60 rotor and linearized using time-periodic
systems tools to achieve real-time simulation (Refs. 5, 6).
Both free wake and VVPM models have the capability to pre-
dict rotor inflow and rotor wake interference effects in gen-
eralized maneuvering flight, moving beyond simplified mod-
Presented at the VFS International 79th Annual Forum &
Technology Display, West Palm Beach, FL, May 16–18, 2023.
Copyright © 2023 by the Vertical Flight Society. All rights reserved.
els and empirical models that have been historically used in
flight simulations. In 2017, Ref. 7 even investigated the fea-
sibility of using Euler-based Computational Fluid Dynamics
(CFD) methods in real-time simulation models. While those
results indicated that such an approach is theoretically possi-
ble, the CFD would need to be greatly simplified and rely on
massively parallelized computations, which would still be a
challenge with contemporary computing capabilities. In any
case, the gap between the fidelity of rotorcraft flight simula-
tion models and rotorcraft comprehensive models is steadily
closing.
In the meantime, new configurations have become more com-
plex. Future vertical lift (FVL) and urban air mobility (UAM)
configurations feature multiple rotors, high levels of aerody-
namic interactions, and in the case of UAM, high revolutions-
per-minute (RPM) and/or variable speed rotors. These fea-
tures drive the need for advanced aeromechanics models while
at the same time making real-time speeds (or even sufficiently
fast execution speeds for routine design) much more difficult.
For example, time steps in rotor models are driven by the min-
imum blade sweep per time step, so computational cost goes
up with smaller, higher RPM rotors. Modeling aerodynamic
interactions requires larger amounts of wake to be computed,
and rigid rotor systems require more costly structural dynam-
ics models of the blades. Thus, while more advanced aerome-
chanics models are feasible, it is likely that in many cases,
execution speeds required for real-time simulations or routine
design applications will remain elusive.
It is therefore critical that the coupled flight dynamics and
high-fidelity aeromechanics models are formulated in such a
1
way, that they can be readily linearized and/or simplified to
extract more tractable and less expensive models. Linearized
state space models are particularly attractive from the control
designer’s perspectives. Not only are linearized models used
in a majority of practical control design methodologies, but
linear model analysis provides many physical insights to sys-
tem dynamics. Practical examples include the effect of the
rotor wake aerodynamic interactions with fuselage, aerody-
namic appendices, and obstacles (e.g., the ship deck in ship-
board approach and landing operations, or surrounding build-
ings in UAM) on the handling qualities. To this end, a first-
order state variable implementation of the coupled flight dy-
namics and aeromechanics (or state-space implementation)
that can be efficiently linearized is a highly desirable feature
for future advanced simulations.
The objective of the present effort is to develop a flight dy-
namics simulation model of a generic tiltrotor aircraft for use
in flight dynamics and control, aeromechanics, and aeroa-
coustics analyses. To this end, the tiltrotor flight dynamics is
coupled with a free-vortex wake code in state-variable form,
and with an aeroacoustics solver. This model is intended
to be suitable for detailed aeromechanics analysis as well as
aeroacoustic predictions while making real-time speeds when
linearized. This enhanced simulation model, implemented
in MATLAB®/Simulink, draws inspiration from GTRSIM
(Ref.8) while incorporating several key improvements. Im-
provements include a blade-element model borrowed from
GenHel (Ref. 1) and adapted to centrally-hinged blades (Ref.
9), dynamic inflow modeling (Ref. 10) when not coupled with
the free-vortex wake, and a set of dynamic analysis algorithms
like trim, periodic trim, linearization, and model-order reduc-
tion.
The paper begins with an overview on the simulation model
and all of its components. This is followed by a detailed
mathematical description of the linearization, periodic trim,
and model-order reduction algorithms, as well as a discussion
on the aeroacoustic solver. Next, flight control design is dis-
cussed. Time- and frequency-domain validation is performed
against U.S. Army/NASA flight-test data and other data in the
literature for a Bell XV-15 tiltrotor implementation. Results
demonstrate the application of the trim, linearization, aeroa-
coustics, state-space free-vortex wake, and flight control al-
gorithms. Final remarks summarize the overall findings of the
study and future developments are identified.
SIMULATION MODEL
Overview
The general tilt-rotor (GenTR) simulation model is represen-
tative of a Bell XV-15 (Fig. 1) and loosely based on GTR-
SIM (Ref. 8) while introducing a number of significant im-
provements. More specifically, the model includes the rigid-
body dynamics, a blade element rotor model coupled with
flapping dynamics adapted from GenHel (Ref. 1) and Ref.
9, and a 3-state Pitt-Peters (Ref. 10). The model also in-
cludes nonlinear lookup tables for the fuselage, blade, and
rotor-on-rotor, rotor-on-wing, and rotor-on-empennage inter-
actional aerodynamics. Additional dynamics associated with
the landing gear, propulsion system, and flight control sys-
tem (FCS) can be switched on or off arbitrarily. The model is
coupled with a state-space free-vortex wake code with a near-
wake model (Ref. 11) which can also be used to simulate the
tip vortex dynamics only (Ref. 5, 6). The model is formu-
lated as a first-order dynamic system such that it is suitable
for linearization, order reduction, and flight control design.
The model also incorporates useful features such as periodic
trim (Ref. 12), model-order reduction, and linearization, as
well as blade surface loads predictions. The latter allows for
acoustic predictions of the coupled flight and rotor dynam-
ics in generalized maneuvering flight when coupled with an
in-house aeroacoustic solver (Ref. 13). Another major differ-
ence with respect to GTRSIM is that GenTR is implemented
in MATLAB®/Simulink, an interpreted scientific computing
language, as opposed to the FORTRAN language (used for
GTRSIM) which is compiled.
Fig. 1: Bell XV-15.
Mathematical Formulation
The rotorcraft flight dynamics are formulated as a nonlinear
time-periodic system
˙
x
x
x=f
f
f(x
x
x,u
u
u,t)(1a)
y
y
y=g
g
g(x
x
x,u
u
u,t)(1b)
where x
x
xRnis the state vector, u
u
uRmis the control input
vector, y
y
yRlis the output vector, and tis the dimensional
time in seconds. It is convenient to note that dimensional time
can be related to the azimuth angle ψof a reference blade, also
known as non-dimensional time, via the following relation:
ψ=t, where is the rotor angular speed in rad/s. It fol-
lows that the fundamental period of the system is T= (2π)/
seconds, which corresponds to 2πradians or one rotor revolu-
tion. The nonlinear functions f
f
fand g
g
gare T-periodic in time
such that:
f
f
f(x
x
x,u
u
u,t) = f
f
f(x
x
x,u
u
u,t+T)(2a)
g
g
g(x
x
x,u
u
u,t) = g
g
g(x
x
x,u
u
u,t+T)(2b)
The state vector is:
x
x
xT=x
x
xT
Fx
x
xT
Rx
x
xT
Cx
x
xT
W(3)
where:
x
x
xFare the fuselage states,
2
x
x
xRare the rotor states,
x
x
xCare the control system states, and
x
x
xWare the free-vortex wake states.
Each of these set of states is described along with its associ-
ated dynamics below.
The pilot input vector is:
u
u
uT=hδlat δlon δcol δped XFL IDIFF IINACB βFTlon βFTlat βFTped i
(4)
where:
δlat and δlon are the lateral and longitudinal stick positions,
δcol is the collective stick position,
δped is the pedal position,
XFL is the flap setting,
IDIFF is the differential collective input,
IINACB is the nacelle beep switch position, and
βFTlon βFTlat βFTped are the longitudinal stick, lateral stick,
and pedal control force trim switches.
The pilot inputs are converted to control effector inputs via
the relations in Ref. 8. The control effector inputs are:
u
u
uT
C= [(θ0θ1sθ1c)R(θ0θ1sθ1cLδeδrδaδf(5)
where:
θ0θ1sθ1care the collective, longitudinal cyclic, and lateral
cyclic swashplate inputs,
δeis the elevator position,
δris the rudder position,
δais the aileron position, and
δfis the flaps position.
The subscripts ()Rand ()Lindicate the right and left rotors,
respectively.
Fuselage Dynamics
The fuselage states are:
x
x
xT
F= [uvwpqrφ θ ψ x y z](6)
where:
u,v,ware the longitudinal, lateral, and vertical velocities
in the body-fixed frame,
p,q,rare the roll, pitch, and yaw rates,
φ,θ,ψare the Euler angles, and
x,y,zare the positions in the North-East-Down (NED)
frame.
The dynamics of these states is governed by the equations of
motion for a rigid body.
Rotor Dynamics
The rotor state vector is:
x
x
xT
R=hβ0β1sβ1c˙
β0˙
β1s˙
β1cλ0λ1sλ1cψR
β0β1sβ1c˙
β0˙
β1s˙
β1cλ0λ1sλ1cψLwiRH wiRVi(7)
where:
β0,β1s,β1care the flapping angles in multi-blade coordi-
nates,
λ0,λ1s,λ1c,are the rotor induced inflow ratio harmonics,
is the rotor angular velocity
ψis the azimuth angle of a reference rotor blade, and
wiRH ,wiRV are the rotor induced velocities on the horizontal
and vertical stabilizers, respectively.
One of the major differences of the GenTR simulation model
with the original GTRSIM implementation (Ref. 8) is the use
of a blade-element model to calculate rotor forces and mo-
ments, as well as the flapping dynamics. The rotor blades
are modeled as centrally-hinged blades with a flapping spring
(Ref. 9), such that the flapping dynamics of the individual
blade is given by:
¨
βi+2λ2
ββi=sλ2
βkβ
Iβ22(pHcosψ+qHsinψ)
+˙pHsinψ+˙qHcosψ1
IβZR
eR
(reR)fzi(r)dr
(8)
where:
βiis the flapping angle of the ith blade,
λβ=r1+kβ
2Ibis the flapping frequency ratio,
kβis the flapping spring stiffness,
phand qhare the rotor hub angular rates in rotor hub axes,
Iβis the blade flapping moment of inertia,
ψi=ψ+k2π
Nb,k=1,...,Nbis the azimuth angle of the ith
blade,
fzinormal force on the ith blade,
Ris the rotor blade radius, and
ris the rotor blade radial coordinate.
In general, the angular rates in the rotor hub axes as well as
the normal force on the ith blade are functions of the states and
pilot control inputs such that ph=ph(x
x
x,u
u
u,t),qh=qh(x
x
x,u
u
u,t),
and Fzi=Fzi(x
x
x,u
u
u,t). The relationships between these vari-
ables and the states and pilot control inputs (i.e., the blade-
element model) is omitted for brevity. Nonetheless, this
blade-element model is based on that of Ref. 1 but adapted
for centrally-hinged rotor blades.
Another major difference with the original GTRSIM imple-
mentation (Ref. 8) is the use of dynamic inflow modeling as
opposed to quasi-static inflow modeling. The inflow dynam-
ics of each rotor is based on a 3-state Pitt-Peters inflow model
(Ref. 10):
1
M
M
M
˙
λ0
˙
λ1s
˙
λ1c
+L
L
L1
λ0
λ1s
λ1c
=
CT
Cl
CM
(9)
where CT,Cla, and CMaare the rotor thrust, roll moment, and
pitch moment coefficients, respectively, expressed in each ro-
tor wind frame. These quantities are calculated with the blade-
element model. The matrices L
L
L1and M
M
Mare found in Ref. 10.
Note that when the rotor is two-way coupled with the free-
vortex wake, the Pitt-Peters inflow states are bypassed and the
3
induced velocities at each blade element as predicted by the
free-vortex wake are used instead.
Control System Dynamics
The control system state vector is:
x
x
xT
C=βmθ0Xlon Xlat Xped (10)
where:
βmis the mast conversion angle,
θ0is the differential collective trim, and
Xlon Xlat Xped are longitudinal stick, lateral stick, and pedal
force-feel trim position.
The dynamics of these states are based on those found in Ref.
8.
State-Variable Aeromechanics
A state-space free-vortex wake model was implemented into
the GenTR flight simulation code following the model of Ref.
11. This free-vortex wake code incorporates a near-wake
vortex-lattice model for improved rotor blade and induced ve-
locity prediction over a tip-vortex only model, although the
tip-vortex only dynamics can still be simulated if needed (Ref.
5, 6). In PDE form, the trajectory of a wake node at non-
dimensional time ψand wake age ζis governed by:
r
r
r(ψ,ζ)
ψ +r
r
r(ψ,ζ)
ζ =1
V
V
V(r
r
r(ψ,ζ)) (11)
where the right-hand side is governed by the Biot-Savart law,
which is the most expensive part of the free-wake computation
(Refs. 14, 15). The wake model was implemented in the state-
variable form generally following the approach of (Ref. 16),
with some updates. The first update involves the inclusion of
states that define the vortex strength at each of the free wake
nodes, allowing for varying vortex strength along the filament.
A second update is the addition of a vortex-lattice model to
represent the shed vorticity and inboard vorticity distribution
near the rotor blade. A vortex-lattice model with a large core
size (0.05 R) is used to approximate vortex sheets trailing the
rotor for a fraction of a revolution. The fundamental equa-
tions for the vortex lattice are similar to the tip vortex but use
quadrilateral elements whose boundary condition is specified
by the bound circulation on the blade elements of the rotor.
The near wake model is implemented in state-variable form,
similar to those presented in Ref. 5. The extension of the near
wake behind the blade is user specified. The results shown
in this paper adopt a short distance of 1/6 of a revolution, af-
ter which it is assumed the vortex sheet rolls into a single-tip
vortex which initiates at the end of the near-wake model. To
avoid algebraic feedback loops between the wake model and
the rotor model, the bound circulations at the rotor blades are
filtered and added to the system of ODE as states. Their dy-
namics are governed by a system of first-order filters with a
very small time constant τΓb=0.02(2π)/. The resulting
system of ODE expressed in dimensional time is:
˙
r
r
rNW =A
A
Aζr
r
rNW +V
V
V(r
r
rNW (t,ζ)) (12a)
˙
r
r
rTV =A
A
Aζr
r
rTV +V
V
V(r
r
rTV (t,ζ)) (12b)
˙
Γ
Γ
ΓNW =A
A
AζΓ
Γ
ΓNW (12c)
˙
Γ
Γ
ΓTV =A
A
AζΓ
Γ
ΓTV (12d)
˙
Γ
Γ
Γb=1
τΓbΓ
Γ
Γ(r
r
rB(t)) Γ
Γ
Γb(12e)
where r
r
rNW ,Γ
Γ
ΓNW ,r
r
rTV , and Γ
Γ
ΓTV represent the node positions
and vortex strengths of the near wake and trailing vortex wake
elements respectively. Matrix A
A
Aζis the finite difference ma-
trix corresponding to the 5PBU4 scheme from Ref. 16. The
filtered bound circulation on the blades is represented by Γ
Γ
Γb,
and r
r
rBis the position of the rotor blade elements. The bound-
ary conditions for the near wake elements are the locations
and filtered bound circulation on the blade elements, and the
boundary conditions for the trailing vortex elements are the
position of the last outboard near wake element and the max-
imum vortex strength in the last row of near wake elements.
The free-vortex wake can be coupled loosely or tightly. Loose
coupling does not feed back the induced velocities at the rotor
blade elements as predicted by the vortex wake, such that the
flight dynamics is invariant with respect to the free-wake dy-
namics. Conversely, tight coupling feeds back the induced ve-
locities at the rotor blade elements computed with the vortex
wake and so that the flight and wake dynamics are coupled.
AEROACOUSTICS
Blade Geometry and Loads
While for flight dynamics predictions it is sufficient to calcu-
late the blade loads based on a discretization of the blade in
the sole spanwise direction, aeroacoustic calculations require
increased resolution of the blade loads distribution over the
blade surface. In addition, a detailed blade geometry is re-
quired to calculate the unit vectors perpendicular to the blade
surface, as well as the velocity vector of each blade surface
panel. As such, the upper and lower blade surfaces are mod-
eled with a discrete number of panels both in the spanwise and
chordwise directions. Note that this approach is used solely
for aeroacoustic calculations and does not directly affect any
flight dynamics calculation.
The rotor blade airfoil adopted in GenTR is a NACA 0012.
The thickness distribution is described by the following equa-
tion (Ref. 17):
zt(xc) = ¯
t
0.20.2969xc0.3516x2
c+0.2843x3
c0.1015x4
c
(13)
where ¯
tis the maximum thickness of the blade section
expressed as a fraction of the chord, and xcis the non-
dimensional chordwise coordinate. Note that xc=0 corre-
sponds to the leading edge and xc=1 corresponds to the trail-
ing edge. The maximum thickness of the blade section is
4
assumed to be 12% of the blade chord. Since the airfoil in
consideration is symmetric, Eq. (13) is used for describing
the sectional geometry of both the upper and lower blade sur-
faces. The chordwise panels are distributed unevenly across
the blade chord. In fact, the chordwise panels distribution is
proportional to the square of the non-dimensional chordwise
coordinate, such that the chordwise panels are concentrated
toward the leading edge of the section. Consider a twist an-
gle θ(y,t)function of the spanwise location yand of time t,
given by the summation of inherent blade twist distribution,
blade pitch input, and dynamic twist. Then, the longitudinal
and vertical position of an arbitrary blade surface element in
blade frame axes is given by the following equation:
x
z=cosθsin θ
sinθcos θ xc
±tzt(xc)1/4
0c(14)
where t=0.2076 ft is the maximum thickness of the blade
section and cis the blade chord. In the implementation, the
blade chord is assumed constant. The blade frame axes are
defined such that their origin lies at the intersection of the
quarter-chord line, the mean line, and the blade root. The x
axis is along the chordwise direction, pointing from the lead-
ing to the trailing edge; the yaxis is along the spanwise di-
rection, pointing from the blade root to the blade tip; and the
zaxis is perpendicular to xand y, pointing from the lower
to the upper surface (for zero twist). Figure 2 shows the dis-
cretized geometry of the Bell XV-15 rotor blade with 10 span-
wise panels and 10 chordwise panels. Based on this geometry,
the unit vectors normal to each blade element are calculated
and stored for the undeformed blade geometry, and rotated
according to the transformation matrix in Eq. (14) at each
time step based on the current blade pitch at each spanwise
location. For aeroacoustic calculations, the blade surface lo-
cations, velocities, and unit vectors normal to the surface are
transformed from the blade to the inertial frame via the trans-
formations defined in Ref. 8.
The surface pressure distribution for each blade spanwise seg-
ment is calculated based on the local lift coefficient, and on
NACA 0012 lookup tables based on thin airfoil theory. The
pressure coefficient chordwise distribution on each spanwise
segment is given by the following equation:
CP=
1v
V+vα
VCL2
upper surface
1v
Vvα
VCL2
lower surface
(15)
where CLis the lift coefficient of the spanwise segment, v/V
is the velocity ratio, and vα/Vis the velocity increment ra-
tio. The latter two quantities are functions of the normalized
chordwise coordinate and the corresponding lookup tables can
be found in from Ref. 17. The blade gauge pressure chord-
wise distribution at each spanwise segment is then found by:
˜p=1
2ρu2
T+u2
PCP(16)
where uTand uPare respectively the tangential and perpen-
dicular velocities of the oncoming flow to the blade spanwise
Fig. 2: Discretized geometry of the undeformed generic
tiltrotor blade with 10 spanwise panels and 10 chordwise
panels.
segment in consideration. As an example, Figure 3 shows
the blade geometry and surface pressure distribution for the
XV-15 trimmed at hover, 120 kts forward flight, and 170 kts
forward flight.
Aeroacoustic Solver
The noise generated aerodynamically by the rotors is com-
puted in terms of acoustic pressure at a desired observer lo-
cation. The acoustic pressure may correspond to points fixed
in space (e.g., observers or microphones on the ground) or to
a points moving with the aircraft (e.g., observers or micro-
phones in the cockpit). The acoustic pressure at each location
can be described according to the impermeable emission sur-
face formulation (Ref. 18) as follows:
4πp(x
x
x,t) = 1
c0
tZΣρ0c0un+˜pˆ
n
n
n·ˆ
r
r
r
rΛret
dΣ+
ZΣ˜pˆ
n
n
n·ˆ
r
r
r
r2Λret
dΣ+p
Q(x
x
x,t)(17)
where: x
x
xis the observer location,
Σis the emission surface (i.e., the surface in space-time that
emitted sound that reached the observer x
x
xat time t),
ρ0is the density of the undisturbed air,
c0is the speed of sound of the undisturbed air,
unis the rotor blade velocity normal to the blade surface,
˜p=pp0is the gauge pressure on the blade surface,
ˆ
n
n
nis the unit vector normal to the blade surface,
ˆ
r
r
r=x
x
xy
y
y
||x
x
xy
y
y|| is the emission direction,
y
y
yis the source location,
Λ=hp12Mnˆ
n
n
n·ˆ
r
r
r+M2
niret,
Mnis the local Mach number normal to the blade surface,
and
p
Q(x
x
x,t)is the quadrupole term.
5
(a) Hover.
(b) 120 kts.
(c) 170 kts.
Fig. 3: Blade geometry and surface pressure distribution for a
Bell XV-15 tiltrotor in trimmed flight.
The integrands in Eq. (17) are evaluated at the retarded time,
which is defined as:
τ=tr
c0
(18)
where r=||x
x
xy
y
y|| is the distance between the observer loca-
tion x
x
xand the source location y
y
ywhen the sound was emitted.
It is worth noting that the quadrupole term p
Q(x
x
x,t)was in-
cluded in Eq. (17) for the sake of generality but will not be
considered in this preliminary investigation.
The aeroacoustic solver that is used to compute the noise gen-
erated aerodynamically by the main rotor blades is that of
Ref. 13. This code is a MATLAB®implementation of the
algorithm in Ref. 19 and adopts a marching cubes strategy
to find the impermeable Ffowcs Williams-Hawkings surface
(Refs. 13, 20). The acoustic pressure is then calculated based
on an emission surface formulation of the Ffowcs Williams-
Hawkings equations shown in Eq. (17). It is worth noting
that the aeroacoustic code does not currently solve for the
quadrupole term, i.e., broadband noise.
The emission surface is approximated at each time step by a
mesh of triangular faces where ∆Σiis the surface area of the
ith triangle. Based on this approach, Eq. (17) is re-written as:
4πp(x
x
x,t) = 1
c0
tI1+I2(19)
where:
I1=ZΣρ0c0vn+˜pˆ
n
n
n·ˆ
r
r
r
rΛret
dΣ=ZΣQ1(y
y
y,tr/c0)
rΛret
dΣ
(20a)
I2=ZΣ˜pˆ
n
n
n·ˆ
r
r
r
r2Λret
dΣ=ZΣQ2(y
y
y,tr/c0)
r2Λret
dΣ(20b)
Then, the integrals contained in the terms I1and I2can be
approximated numerically by direct summation over all trian-
gles that compose the emission surface as:
I1
Ntri
i=1Q1(y
y
y,tr/c0)
rΛret
∆Σi(21a)
I2
Ntri
i=1Q2(y
y
y,tr/c0)
r2Λret
∆Σi(21b)
(21c)
where Ntri is the total number of triangles. Note that the con-
trol surface is described as the sum of twelve total surfaces,
i.e., the upper and lower surfaces of each of the three rotor
blades of the XV-15. Note that no end caps at the tip and root
of the blade are used. The time derivative
tI1is obtained
via a centered finite difference scheme over the observer time
evaluations:
tI1(ti)I1(ti+1)I1(ti1)
2t(22)
LINEARIZATION, TRIM, AND MODEL
ORDER REDUCTION
Approximate Linearized Time-Invariant Models
Consider the NLTP dynamics of Eq. (1). Let x
x
x(t)and u
u
u(t)
represent a periodic solution of the system such that x
x
x(t) =
6
x
x
x(t+T)and u
u
u(t) = u
u
u(t+T). Then, the NLTP dynamics
can be linearized about the periodic solution. Consider the
case of small disturbances:
x
x
x=x
x
x+x
x
x(23a)
u
u
u=u
u
u+u
u
u(23b)
where x
x
xand u
u
uare the state and control perturbation vectors
from the candidate periodic solution. A Taylor series expan-
sion is performed on the state derivative and output vectors.
Neglecting terms higher than first order results in the follow-
ing equations:
f
f
f(x
x
x+x
x
x,u
u
u+u
u
u,t) = f
f
f(x
x
x,u
u
u,t) + F
F
F(t)x
x
x+G
G
G(t)u
u
u
(24a)
g
g
g(x
x
x+x
x
x,u
u
u+u
u
u,t) = g
g
g(x
x
x,u
u
u,t) + P
P
P(t)x
x
x+Q
Q
Q(t)u
u
u
(24b)
where:
F
F
F(t) = f
f
f(x
x
x,u
u
u)
x
x
xx
x
x,u
u
u,G
G
G(t) = f
f
f(x
x
x,u
u
u)
u
u
ux
x
x,u
u
u(25a-b)
P
P
P(t) = g
g
g(x
x
x,u
u
u)
x
x
xx
x
x,u
u
u,Q
Q
Q(t) = g
g
g(x
x
x,u
u
u)
u
u
ux
x
x,u
u
u(25c-d)
Note that the state-space matrices in Eq. (25) have T-periodic
coefficients such that:
F
F
F(t) = F
F
F(t+T),G
G
G(t) = G
G
G(t+T)(26a-b)
P
P
P(t) = P
P
P(t+T),Q
Q
Q(t) = Q
Q
Q(t+T)(26c-d)
Equations (24a) and (24b) yield a linear time-periodic (LTP)
approximation of the NLTP system of Eq. (1) as follows:
˙
x
x
x=F
F
F(t)x
x
x+G
G
G(t)u
u
u(27a)
y
y
y=P
P
P(t)x
x
x+Q
Q
Q(t)u
u
u(27b)
Hereafter, the notation is simplified by dropping the in
front of the linearized perturbation state and control vectors
while keeping in mind that these vectors represent perturba-
tions from a periodic equilibrium. Next, the state, input, and
output vectors of the LTP systems are decomposed into a finite
number of harmonics of the fundamental period via Fourier
analysis:
x
x
x=x
x
x0+
N
i=1
x
x
xic cos2πit
T+x
x
xis sin2πit
T(28a)
u
u
u=u
u
u0+
M
j=1
u
u
ujc cos 2πjt
T+u
u
ujs sin 2πjt
T(28b)
y
y
y=y
y
y0+
L
k=1
y
y
ykc cos 2πkt
T+y
y
yks sin 2πkt
T(28c)
where N,M, and Lare the number of harmonics retained in the
state, control input, and output vector, respectively. Should
the interest be in capturing number-of-blades-per-revolution
(Nb/rev) phenomena, then one could set N=M=L=kNb,
with k=1,2,.. . depending on the number of super harmonics
of interest. On the other hand, if the interest is to capture the
averaged dynamics only, then N=M=L=0. As shown
in Ref. 21, the harmonic decomposition methodology can be
used to transform the LTP model into an approximate higher-
order linear time-invariant (LTI) model in first-order form:
˙
X
X
X=A
A
AX
X
X+B
B
BU
U
U(29a)
Y
Y
Y=C
C
CX
X
X+D
D
DU
U
U(29b)
where the augmented state, control, and output vectors X
X
X
Rn(2N+1),U
U
URm(2M+1), and Y
Y
YRl(2L+1), respectively, are
given by:
X
X
XT=x
x
xT
0x
x
xT
1cx
x
xT
1s... x
x
xT
Nc x
x
xT
Ns (30a)
U
U
UT=u
u
uT
0u
u
uT
1cu
u
uT
1s... u
u
uT
Mc u
u
uT
Ms(30b)
Y
Y
YT=y
y
yT
0y
y
yT
1cy
y
yT
1s... y
y
yT
Lc y
y
yT
Ls(30c)
with A
A
ARn(2N+1)×n(2N+1),B
B
BRn(2N+1)×m(2M+1),C
C
C
Rl(2L+1)×n(2N+1), and D
D
DRl(2L+1)×m(2M+1). Closed-form
expressions for these matrices can be found in Ref. 21. It
is worth noting that harmonic decomposition does not rely
on state transition matrices, which makes the methodology
more computationally efficient and less numerically sensitive
than other approaches such as the Lyapounov-Floquet method
(Ref. 22) and frequency lifting methods (Ref. 23).
Periodic Trim Algorithm
A necessary step towards the approximation of the NLTP dy-
namics with harmonic decomposition models is determination
of the periodic orbit about which the NLTP system is lin-
earized, which involves computing the states and controls that
result in a periodic equilibrium (i.e., trimming a vehicle about
a periodic orbit). Formally, the problem consists of determin-
ing the periodic state and control input x
x
x(t)and u
u
u(t)such
that:
˙
x
x
x=f
f
f(x
x
x,u
u
u,t)(31)
where ˙
x
x
x(t) = ˙
x
x
x(t+T).
Several methods exist for trimming a vehicle about a peri-
odic orbit, such as: averaged approximate trim, time march-
ing trim, autopilot trim (Ref. 24), periodic shooting (Ref. 25),
harmonic balance (Ref. 26), and modified harmonic balance
(Ref. 27). The latter is used in GenTR as it offers three major
advantages when compared to other techniques: it is based on
harmonic decomposition and thus does not rely on state tran-
sition matrices, it simultaneously solves for the approximate
higher-order LTI dynamics about the periodic solution, and
it can be used to compute open-loop higher-harmonic control
(HHC) inputs that attenuate arbitrary state harmonics. The
choice of modified harmonic balance is further justified by the
fact it can calculate unstable periodic orbits, unlike methods
such as average approximate trim or time marching trim. Be-
cause the flight dynamics of rotorcraft may be stable or unsta-
ble (Ref. 28), it is important to select a periodic trim solution
method capable of solving for unstable periodic orbits.
The modified harmonic balance solution strategy is an itera-
tive algorithm, in that a candidate solution is refined over a
7
series of computational steps until a convergence criteria is
reached. A key assumption is that the fundamental period
Tof the nonlinear time-periodic system is known. Given a
candidate periodic solution x
x
x
k(t)and u
u
u
k(t)at iteration k, one
iteration of the algorithm is articulated in the following steps:
1. The candidate periodic solution is approximated using a
Fourier series with a finite number of harmonics and re-
written in terms of its respective Fourier coefficients:
X
X
XT
k=x
x
xT
k0x
x
xT
k1cx
x
xT
k1s... x
x
xT
kNc x
x
xT
kNs (32a)
U
U
UT
k=u
u
uT
k0u
u
uT
k1cu
u
uT
k1s... u
u
uT
kMc u
u
uT
kMs (32b)
Note that the harmonics retained for the state and control
input are Nand M, respectively. Because the algorithm
simultaneously solves for the periodic solution and the
necessary control inputs that ensure it, the harmonic re-
alizations of the candidate periodic solution and control
inputs are combined into a single vector of unknowns at
iteration k:
Θ
Θ
ΘT
k=hX
X
XT
kU
U
UT
ki(33)
where Θ
Θ
ΘkRn(2N+1)+m(2M+1).
2. The state derivative vector computed along the candidate
periodic solution over a single fundamental period is also
decomposed into a finite number of harmonics up to the
Nth via Fourier analysis. As such, the error vector at it-
eration kis defined using the n(2N+1)integral relations
that ensure the periodicity of the state dynamics (Ref.
27):
e
e
eT
k=W
W
W"˙
x
x
x
k0T˙
x
x
x
kic 2πi
Tx
x
x
kis T˙
x
x
x
kis +2πi
Tx
x
x
kic T#
(34)
where e
e
ekRn(2N+1)and W
W
WRn(2N+1)×n(2N+1)is a di-
agonal scaling matrix that is used to convert errors corre-
sponding to Euler angles and angular rates from radians
and rad/s to deg and deg/s. This way, errors in the Euler
angles and angular rates have comparable units to posi-
tion and speed states, which are expressed in meters and
m/s.
3. The NLTP dynamics are linearized at incremental time
steps along the candidate periodic solution, yielding the
following LTP system:
˙
x
x
x=F
F
Fk(t)x
x
x+G
G
Gk(t)u
u
u(35)
4. The LTP model is approximated with a higher-order LTI
using the harmonic decomposition methodology:
˙
X
X
X=A
A
AkX
X
X+B
B
BkU
U
U(36)
where the augmented state and control vectors are given
in Eqs. (30a) and (30b).
5. The Jacobian matrix of the harmonic balancing algo-
rithm at iteration kis formed using the LTI system co-
efficient matrices:
J
J
Jk= [A
A
AkB
B
Bk](37)
where J
J
JkRn(2N+1)×[n(2N+1)+m(2M+1)]. It is clear that,
in its current form, the Jacobian matrix is not square and
thus not invertible. To make the Jacobian matrix square
and invertible, m(2M+1)trim conditions must be spec-
ified. A detailed discussion of these conditions is pro-
vided in Ref. 27 but is omitted here for brevity. En-
forcing these trim conditions results in a modified square
Jacobian ˆ
J
J
Jk. Newton-Raphson (Ref. 29) is then used to
find a candidate periodic solution update (in harmonic
form) according to:
ˆ
Θ
Θ
Θk+1=ˆ
Θ
Θ
Θkˆ
J
J
J1
ke
e
ek(38)
where ˆ
Θ
Θ
Θkis the vector of unknowns.
6. The candidate periodic solution update is reconstructed
in the time domain using Eqs. (28a) and (28b).
The algorithm is stopped when ekbecomes less than an
arbitrary tolerance. The high-order LTI system computed as
part of the last iteration constitutes an approximation of the
NLTP dynamics about the periodic solution thus found, and
can readily be used for stability analysis or feedback control
design. A flowchart of the algorithm is shown in Fig. 4.
Model Order Reduction
To eliminate the need to measure or estimate states associ-
ated with the higher-order dynamics, where the higher-order
dynamics include rotor and higher harmonic dynamics, it is
desirable to reduce the order of the linearized dynamics. This
is a necessary step to make linearized models tractable for
practical control design purposes. Ideally, these reduced-order
models are the same order as the rigid-body dynamics while
still retaining information on the higher-order dynamics. This
can be achieved through residualization, a portion of singu-
lar perturbation theory that pertains to LTI systems (Ref. 30).
Assuming one or more states to have stable dynamics which
are faster than that of the remaining states, the state vector in
Eq. (30a) is be partitioned into fast and slow components:
X
X
XT=X
X
XT
sX
X
XT
f(39)
Then, the system in Eq. (29a) can be re-written as:
˙
X
X
Xs
˙
X
X
Xf=A
A
AsA
A
Asf
A
A
Afs A
A
AfX
X
Xs
X
X
Xf+B
B
Bs
B
B
BfU
U
U(40)
By neglecting the dynamics of the fast states (i.e.,˙
X
X
Xf=0) and
performing a few algebraic manipulations, the equations for
a reduced-order system with the state vector composed of the
slow states may be found:
˙
X
X
Xs=ˆ
A
A
AX
X
Xs+ˆ
B
B
BU
U
U(41)
8
Fig. 4: Periodic trim solution algorithm flowchart (Ref. 27).
where:
ˆ
A
A
A=A
A
AsA
A
AsfA
A
Af1A
A
Afs (42a)
ˆ
B
B
B=B
B
BsA
A
AsfA
A
Af1B
B
Bf(42b)
Note that Afmust be invertible. This is guaranteed if Afis
asymptotically stable, i.e., all eigenvalues have their real part
that is strictly negative. The slow states are chosen as the
zeroth harmonic fuselage states with the exception of the po-
sition and heading states which are truncated, whereas the fast
states are taken as the remaining states, including the higher
harmonics:
X
X
XT
s= [u0v0w0p0q0r0φ0θ0](43a)
X
X
XT
f=x
x
xR0x
x
xC0x
x
xW0x
x
xT
1cx
x
xT
1s... x
x
xT
Nc x
x
xT
Ns (43b)
This way, an 8-state residualized system was obtained that still
accounts for the higher-order dynamics. For the residualized
model to retain information of the influence or the residual-
ized dynamics not only on the zeroth harmonics of the output,
but also on its higher-output harmonics, consider partitioning
the output equations in Eq. (29b) as:
Y
Y
Y=C
C
CsC
C
CfX
X
Xs
X
X
Xf+D
D
DU
U
U(44)
Then, it can be shown that the residualized output equations
are:
˙
Y
Y
Y=ˆ
C
C
CX
X
Xs+ˆ
D
D
DU
U
U(45)
where:
ˆ
C
C
C=C
C
CsC
C
CfA
A
Af1A
A
Afs (46a)
ˆ
D
D
D=D
D
DC
C
CfA
A
Af1B
B
Bf(46b)
If now the augmented output vector is selected to coincide
with the augmented state vector, such that:
Y
Y
YT=x
x
xT
0x
x
xT
1cx
x
xT
1s... x
x
xT
Nc x
x
xT
Ns (47)
then, the residualized model will be able to predict the influ-
ence of the residualized dynamics on the zeroth and higher-
harmonic states.
FLIGHT CONTROL DESIGN
The flight control architecture chosen for this study is Non-
linear Dynamic Inversion (NDI). Application of NDI control
laws to rotorcraft can be found in, e.g., Refs. 3140. A key
aspect of DI is the reliance on model inversion to cancel the
plant dynamics and track a desired reference model. One con-
venient feature of NDI is that it inverts the plant model in its
feedback linearization loop, which, compared to other more
conventional model-following control strategies such as ex-
plicit model following (EMF), eliminates the need for gain
scheduling. However, the plant model used for feedback lin-
earization still needs to be scheduled with the flight condition.
A multi-loop NDI control law largely based on Refs. 31,35,37
is designed to enable fully autonomous flight, including a
fully-automatic transition from helicopter to aircraft mode.
The schematic of the closed-loop tiltrotor dynamics is shown
in Fig. 5. The outer loop controller tracks longitudinal and
lateral ground velocities commands in the heading frame and
calculates the desired pitch and roll attitudes for the inner loop
to track. The desired response type for the outer loop is Trans-
lational Rate Command (TRC). The inner loop achieves sta-
bility, disturbance rejection, and desired response characteris-
tics about the roll, pitch, yaw, and heave axes. When coupled
with the outer loop, an Attitude Command / Attitude Hold
(ACAH) response is used for the roll and pitch axes, Rate
9
Command / Attitude Hold (RCAH) is used for the yaw axis,
and a TRC response is used for the heave axis. A generic
DI controller as applied to a linear system is shown in Fig.
6. The key components are a command model (also known
as command filter 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 (i.e., the feedback linearization loop).
Inner Loop
The inner-loop NDI controller is based on the 8-state reduced-
order model of Eq. (42) with a state vector given in Eq.
(43a). Residualized systems are obtained at discrete speed
intervals from 0 to 300 kts such that, effectively, ˆ
A
A
A=ˆ
A
A
A(V)
and ˆ
B
B
B=ˆ
B
B
B(V). Here, V=u2+v2+w2is the total speed
of the rotorcraft. Note that the nacelle angle is a function of
the flight speed and, as such, ˆ
A
A
Aand ˆ
B
B
Baccount for changes in
nacelle angle setting. However, this approach assumes that
the each speed is associated with a single nacelle angle (taken
as the mean of the conversion corridor bounds), which is not
necessarily true. In addition, the following output vector is
defined, corresponding to the controlled variables of the non-
linear system (i.e., the aircraft dynamics):
y
y
yT= [φ θ r Vz](48)
where Vzis the vertical speed in the heading frame (positive
up). The output matrix that relates the state vector to the out-
put vector:
C
C
C=C
C
C1
C
C
C2(49)
where:
C
C
C1=00000010
00000001(50a)
C
C
C2=0 0 0 0 0 1 0 0
0 0 10000V(50b)
C
C
C1corresponds to the roll and pitch attitudes whereas C
C
C2is
related to the yaw rate and vertical speed. The matrix C
C
C2is a
function of the total speed Vand therefore requires schedul-
ing. This partitioning is due to the fact that the output equa-
tions for φand θmust be differentiated twice to have the con-
trol inputs appear explicitly in the output equation, while the
same procedure requires being performed once for rand Vz:
¨
φ
¨
θ
˙r
˙
Vz
="C
C
C1ˆ
A
A
A2ˆ
x
x
x+C
C
C1ˆ
A
A
Aˆ
B
B
Bu
u
u
C
C
C2ˆ
A
A
Aˆ
x
x
x+C
C
C2ˆ
B
B
Bu
u
u#(51)
The objective of the DI control law is that the output y
y
ytracks
a reference trajectory y
y
ycmd(t)given by:
y
y
yT
cmd = [φcmd θcmd rcmd Vzcmd](52)
with desired response characteristics. For this reason, the ref-
erence trajectory is fed through first- or second-order com-
mand models which dictate the desired response of the sys-
tem. More specifically, φcmd and θcmd are fed through a
Table 1: Inner loop command models parameters.
Command ωn[rad/s]ζ
Roll Attitude, φ4.5 0.7
Pitch Attitude, θ4.5 0.7
Yaw Rate, r2.0 -
Vertical Position, Vz1.0 -
second-order system, whereas rcmd and Vzcmd are fed through
a first-order system. The command models are also used
to extract the first and second derivatives of the filtered ref-
erence trajectory for use in the proportional-integral (PI)
and proportional-integral-derivative (PID) compensators de-
scribed below. The command models are of the following
form:
G(1)
ideal(s) = 1
τs+1(53a)
G(2)
ideal(s) = ω2
n
s2+2ωnζ+ω2
n
(53b)
where τis the first-order command model time constant,
which is the inverse of the command model break frequency
(i.e.,τ=1/ωn). Additionally, ωnand ζare, respectively, the
natural frequency and damping ratio of the second-order com-
mand model. Table 1 shows the values used for the parameters
of the command models of the inner loop.
PI and PID compensation are used to reject external distur-
bances and to compensate for discrepancies between the ap-
proximate model used in this derivation and the actual bare-
airframe dynamics of the aircraft. The resulting DI control
law is found by solving for the control vector in Eq. (51),
leading to:
u
u
u=C
C
C1ˆ
A
A
Aˆ
B
B
B
C
C
C2ˆ
B
B
B1 ν
ν
ν"C
C
C1ˆ
A
A
A2
C
C
C2ˆ
A
A
A#ˆ
x
x
x!(54)
where ν
ν
νis the pseudo-command vector and e
e
eis the error as
defined respectively in Eqs. (55) and (56).
νφ
νθ
νr
νVz
=
¨
φcmd
¨
θcmd
˙rcmd
˙
Vzcmd
+K
K
KP
eφ
eθ
er
eVz
+K
K
KD
˙eφ
˙eθ
0
0
+K
K
KI
Reφdt
Reθdt
Rerdt
ReVzdt
(55)
e
e
e=y
y
ycmd y
y
y; (56)
The 4-by-4 diagonal matrices K
K
KP,K
K
KI, and K
K
KDidentify the pro-
portional, integral, and derivative gain matrices, respectively.
Note that the coefficient matrices (C
C
C1ˆ
A
A
Aˆ
B
B
B)1,C
C
C1ˆ
A
A
A2,(C
C
C2ˆ
B
B
B)1,
and C
C
C2ˆ
A
A
Aare functions of the longitudinal speed of the aircraft
Vx. For this reason, from a practical standpoint, these matrices
are computed offline at incremental longitudinal speeds from
0 to 300 kts at 20 kts intervals and stored. When the linearized
DI controller is implemented on the nonlinear aircraft dynam-
ics, the coefficient matrices (C
C
C1ˆ
A
A
Aˆ
B
B
B)1,C
C
C1ˆ
A
A
A2,(C
C
C2ˆ
B
B
B)1, and
C
C
C2ˆ
A
A
Aare computed at each time step via interpolation based
10
Fig. 5: Schematic of the closed-loop tiltrotor dynamics.
Fig. 6: DI controller as applied to a linear system.
on the current airspeed V(t)and on the lookup tables stored
offline. It is important to note that what is implemented on
the nonlinear aircraft dynamics is linearized DI. However, be-
cause the coefficient matrices are scheduled with the longitu-
dinal speed, and scheduling effectively introduces a nonlinear
relation between the aircraft states and the feedback control
input, the controller implemented is effectively nonlinear DI
(NDI) (Ref. 31). A block diagram of the linearized DI flight
control law is shown in Fig. 7.
Outer Loop
The objective of the outer loop is to track longitudinal and
lateral velocities in the heading frame, such that the reference
trajectory is given by:
y
y
yT
cmd =Vxcmd Vycmd (57)
The heading frame is a vehicle-carried frame where the x-axis
is aligned with the current aircraft heading, the z-axis is posi-
tive up in the inertial frame, and the y-axis points to the right,
forming a left-handed orthogonal coordinate system. The fol-
lowing equation shows the rotation from body to the heading
frame:
T
T
Th/b=
cosθsin φsin θcos φsin θ
0 cosφsin φ
sinθsin φcos θcos φcos θ
(58)
such that the velocities in the heading frame are given by:
Vx
Vy
Vz
=T
T
Th/b
u
v
w
(59)
The following approximate model of the longitudinal and lat-
eral dynamics of the helicopter is used to derive the outer loop
control law:
˙
Vx
˙
Vy
|{z}
˙
ˆ
x
x
x
=Xu0
0Yv
| {z }
ˆ
A
A
A
Vx
Vy
|{z}
ˆ
x
x
x
+g0
0g
| {z }
ˆ
B
B
B
θ
φ
|{z}
u
u
u
(60a)
x
y
|{z}
y
y
y
=0100
0001
| {z }
C
C
C
Vx
x
Vy
y
|{z}
ˆ
x
x
x
(60b)
where ˆ
x
x
xis the modified state vector, and ˆ
A
A
A(V), and ˆ
B
B
Bare the
modified system and control matrices. Note that these modi-
fied quantities are different from those used in the inner loop
control design. The stability derivatives in the system ma-
trix are scheduled with flight speed. The control matrix is
not scheduled with speed as it is only composed of zeros and
gravitational acceleration (i.e., g). The output matrix C
C
Cis also
not scheduled with speed as it is composed solely of ones and
11
Fig. 7: NDI inner loop.
Table 2: Outer loop command models parameters.
Command ωn[rad/s]ζ
Longitudinal Speed, Vx1 0.7
Lateral Speed, Vy1 0.7
zeros. The command models for the longitudinal and lateral
speed are first order. The natural frequencies and damping ra-
tios are given in Table 2. Following a similar procedure to the
inner loop yields an outer control law of the form:
u
u
u=C
C
Cˆ
A
A
Aˆ
B
B
B1ν
ν
νC
C
Cˆ
A
A
A2ˆ
x
x
x(61)
The reference trajectory is subtracted from the output to find
the error, which is compensated by a PI controller. The feed-
forward signal is subsequently added, leading to the pseudo-
control vector for the outer loop:
νx
νy=˙
Vxcmd
˙
Vycmd+K
K
KPex
ey+K
K
KIRexdt
Reydt(62)
The DI outer loop block diagram is shown in Fig. 8.
Because the tiltrotor flight envelope includes low-speed flight
(i.e., lower than 40 kts) as well as high-speed flight (i.e.,
greater than 60 kts), different control strategies are needed to
control the yaw rate for these two flight condition. Above 60
kts, turn coordination is used; below 40 kts no turn coordina-
tion (Ref. 41) is used; between 40 and 60 kts a blend between
the two is used. These three control strategies are summarized
as follows:
r
cmd =
rcmd V<VLS
rcmd +g
VsinφVVLS
VHS VLS VLS V<VHS
rcmd +g
VsinφVVHS
(63)
where VLS =40 kts, and VHS =60 kts.
Error Dynamics
Feedback compensation is needed to ensure the system tracks
the command models. It can be demonstrated (Ref. 42) 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(64)
For the output equations that require to be differentiated only
once, a PI control strategy is applied to the pseudo-command
vector:
ν=˙ycmd(t) + KPe(t) + KIZt
0
e(τ)dτ(65)
Substituting Eq. (65) into Eq. (64) leads to the closed-loop
error dynamics:
˙e(t) + KPe(t) + KIZt
0
e(τ)dτ=0 (66)
The gains are chosen such that the frequencies of the error
dynamics are of the same order as the command filters (i.e.,
12
Fig. 8: NDI outer loop.
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
become:
e(s)s2+sKP+KI=0 (67)
To obtain the gains that guarantee the desired response, the er-
ror dynamics of Eq. (67) are set equal to the following second-
order system:
s2+2ζ ωns+ωn2=0 (68)
yielding the following proportional and integral gains:
KP=2ζ ωn(69a)
KI=ωn2(69b)
Similarly, for those outputs that require to be differenti-
ated twice, a PID control strategy is applied to the pseudo-
command vector:
ν=¨ycmd(t) + KD˙e(t) + KPe(t) + KIZt
0
e(τ)dτ(70)
Substituting Eq. (70) into Eq. (64) leads to the following
closed-loop error dynamics:
¨e(t) + KD˙e(t) + KPe(t) + KIZt
0
e(τ)dτ=0 (71)
and, therefore, to:
e(s)s3+KDs2+KPs+KI=0 (72)
Again, the gains are chosen such that the frequencies of the
error dynamics are of the same order as the command filters
(i.e., second order), ensuring that the bandwidth of the re-
sponse to disturbances is comparable to that of an input given
by a pilot or outer loop. To obtain the gains that guarantee the
desired response, the error dynamics of Eq. (72) are set equal
to the following third-order system:
(s2+2ζ ωns+ωn2)(s+p) = 0 (73)
yielding the following proportional, integral, and derivative
gains:
KD=2ζ ωn+p(74a)
KP=2ζ ωnp+ωn2(74b)
KI=ωn2p(74c)
This compensation strategy is used for ensuring trajectory
tracking in both the inner and outer loops. Tables 3 and 4 show
the natural frequencies, damping ratios, time constants, and
the integrator pole values, respectively, for the inner and the
outer loop. Note that the integrator pole pis usually chosen to
be one-fifth of the natural frequency, corresponding to about
one-fifth of the loop crossover frequency (Ref. 43). Further,
the outer loop error dynamics natural frequency must be 1/10
to 1/5 of the inner loop error dynamics natural frequency to
ensure sufficient frequency separation (Ref. 43). Addition-
ally, because the 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 must be
scheduled with the flight condition (i.e., with the aircraft speed
Vin this case). Tables 5 and 6 show the compensation gains
for the inner and outer loops.
RESULTS
Validation
The GenTR model is initially implemented for a Bell XV-15
tiltrotor. This choice is motivated by the availability of air-
craft data in the literature (see, e.g., Ref. 8) as well as US
13
Table 3: Inner loop disturbance rejection natural frequencies,
damping ratios, and integrator poles.
ωn[rad/s]ζp
φcmd 4.5 0.7 0.75
θcmd 4.5 0.7 0.75
rcmd 2 0.7 -
Vzcmd 1 0.7 -
Table 4: Outer loop disturbance rejection natural frequencies
and damping ratios.
ωn[rad/s]ζ
Vxcmd 1 0.7
Vycmd 1 0.7
Army/NASA flight test data, which makes the XV-15 plat-
form particularly convenient for model validation. The rotor-
craft is trimmed at hover in helicopter mode (i.e., with the na-
celles at βm=0 deg according to the definition in Ref. 8) and
at 170 kts in aircraft mode (i.e., with the nacelles at βm=90
deg) for validation. Figure 9 shows a comparison of the lat-
eral dynamics eigenvalues for each of these conditions with
those from US Army/NASA flight test data and from Ref. 9.
Figure 10 shows some sample frequency responses at hover
as compared to US Army/NASA XV-15 flight data and simu-
lation data from Ref. 9. While the general agreement is good,
it is worth noting that the available flight-test data does not
provide the exact information on CG location, moments of in-
ertia, weight, and flap setting. As such, it is difficult to draw
definitive conclusions.
Figure 10 shows the GenTR time response to lateral-
directional pilot inputs at hover as compared with US
Army/NASA XV-15 flight test data. This figure shows a gen-
erally good match between the GenTR and flight test data
time histories. The runtime performance is assessed with a
MATLAB®R2022b implementation and fourth-order Runge-
Kutta solver on a 2021 MacBook Pro computer equipped with
an Apple M1 Max processor. Simulation of the nonlinear
model with a non-dimensional time step of ψ=10 deg runs
approximately 40×faster than real-time.
Periodic Trim
As an example, the periodic trim algorithm is used to com-
pute the periodic solution for the state vector in level forward
Table 5: Inner loop compensation gains.
KPKIKD
φcmd 24.975 15.1875 7.05
θcmd 24.975 15.1875 7.05
rcmd 4 4 4
Vzcmd 2 1 -
Table 6: Outer loop compensation gains.
KPKI
Vxcmd 1.5 0.5625
Vycmd 1.5 0.5625
-0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2
Real
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Imag
GenTR
Padfield
Flight Test
(a) Hover.
-1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0
Real
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Imag
GenTR
Padfield
Flight Test
(b) 170 kts.
Fig. 9: Comparison of the lateral dynamics eigenvalues with
those from US Army/NASA flight test data and from Ref. 9.
flight at a forward speed of ˙x=120 kts with constant con-
trol setting. In this example, the free-vortex wake dynam-
ics is ignored, yielding a state vector with dimension n=41.
Additionally, only the pilot sticks and pedals are treated as
unknowns such that the dimension of the control input vec-
tor is m=4. Since the XV-15 has three rotor blades, the
first three state harmonics are retained in the numerical so-
lution of the periodic motion (N=3). Only the zeroth har-
monic is retained for the control input (M=0). The vector of
n(2N+1) + m(2M+1) = 291 unknowns is:
Θ
Θ
ΘT=x
x
xT
0x
x
xT
1cx
x
xT
1s... x
x
xT
Nc x
x
xT
Ns u
u
uT
0(75)
14
10010 1
-40
-20
0
Mag [dB]
p/ lat
GenTR
Flight Data
ID Model
Padfield
10010 1
-400
-200
0
Phase [deg]
10010 1
Frequency [rad/s]
0.4
0.6
0.8
1
Coherence
(a) p/δlat.
10010 1
-60
-40
-20
0
Mag [dB]
r/ ped
GenTR
Flight Data
ID Model
Padfield
10010 1
-400
-200
0
Phase [deg]
10010 1
Frequency [rad/s]
0
0.5
1
Coherence
(b) r/δped.
Fig. 10: Bare-airframe GenTR frequency responses at hover
compared with US Army/NASA XV-15 flight data and
simulation data from Ref. 9.
The n(2N+1) = 287 constraints are given by the integral re-
lations, i.e., those conditions that satisfy Eq. (34):
˙
x
x
x
0=0
0
0 (76a)
˙
x
x
x
ic =2πi
Tx
x
x
is (76b)
˙
x
x
x
is =2πi
Tx
x
x
ic (76c)
with the exception of the zeroth harmonic of the derivative of
the xposition state which is set to the desired forward speed
(i.e. ˙x0=120 kts) (Ref. 27). Because there are m(2M+1) = 4
unknowns more than there are constraints, the zeroth harmon-
ics of the position states (x,y,z)and heading ψ, denoted as
x0,y0,z0, and ψ0, are removed from the problem and set to
arbitrary values. This choice is justified by the fact that the
rotorcraft dynamics is invariant with respect to the zeroth har-
monic of the position and heading (Ref. 44) (that is, if it is
012345678910
20
30
40
50
60
70
lat [%]
012345678910
Time [s]
40
45
50
55
60
65
ped [%]
(a) Pilot inputs.
012345678910
-0.2
0
0.2
p [rad/s]
GenTR LTI Flight Test Padfield
012345678910
-0.2
0
0.2
0.4
0.6
r [rad/s]
012345678910
0
20
40
[rad]
012345678910
Time [s]
-0.1
-0.05
0
ay [ft/s2]
(b) Lateral dynamics states.
Fig. 11: GenTR time response to pilot inputs at hover as
compared with US Army/NASA XV-15 flight test data.
assumed that the air density does not depend on vertical po-
sition). This way, the number of unknowns decreases to 287
such that the problem is square and thus a unique solution is
guaranteed. The modified vector of unknowns is denoted as,
ˆ
Θ
Θ
ΘT=ˆ
x
x
xT
0x
x
xT
1cx
x
xT
1s... x
x
xT
Nc x
x
xT
Ns u
u
uT
0(77)
where ˆ
x
x
x0is the zeroth-harmonic state vector without the posi-
tion and yaw angle states included. Figure 12 shows the pe-
riodic angular rates obtained with the modified harmonic bal-
ance algorithm (solid line) using an error tolerance of 1e7.
As expected, since the tiltrotor in consideration has three ro-
tor blades with equal mass and since the state harmonics re-
tained in the solution are three, the higher-harmonic content
of the periodic solution is limited to the third sine and co-
sine harmonics. Additional harmonics which are multiples of
four (i.e., 6/rev, 9/rev, 12/rev, etc.) can readily be captured by
increasing the number of state harmonics retained in the so-
15
lution (i.e. N=6, N=9, N=12, etc.). It should be noted
that the algorithm is able to capture harmonics in the periodic
solution that are not only multiples of the Nb/rev harmonics.
For instance, if one rotor blade had a different mass with re-
spect to the others such that the rotor was imbalanced, then
the algorithm would capture the 1/rev component in the re-
sulting periodic solution, provided that the first harmonic is
retained in the solution. It is also worth noting that the al-
gorithm can be used to compute open-loop Higher-Harmonic
Control (HHC) inputs that attenuate arbitrary state harmonics
like vibrations at the center of gravity, as demonstrated in Ref.
27.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-2
-1
0
1
2
p [deg/s]
×10 -16 Initial guess Solution
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-4
-2
0
2
4
q [deg/s]
×10 -6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Nondimensional time, t/T
-2
-1
0
1
2
r [deg/s]
×10 -17
Fig. 12: Angular rates periodic equilibrium for an XV-15 at
120 kts level flight.
Free-Vortex Wake
A three-dimensional visualizations of the state-space free-
vortex wake with a near-wake vortex-lattice model are shown
in Fig. 13 for different flight conditions. These results cor-
respond to the tightly coupled flight and free-wake dynamics
trimmed at hover (Fig. 13a), at 120 kts during conversion be-
tween helicopter and airplane mode (Fig. 13b), and at 170 kts
in airplane mode (Fig. 13c). Figure 13a shows the contraction
of the wake below the rotor, indicative of the effectiveness of
the implementation. The interested reader is invited to con-
sult Ref. 11 for more detailed results on the validation of the
vortex wake model.
Aeroacoustics
The aeroacoustics solver is used to compute the acoustic pres-
sure corresponding to a single observer location moving with
the helicopter. The observer is fixed in the aircraft body frame.
In this condition, the nacelles are tilted vertically at βm=0
deg. The observer is located three rotor radii in front on the
main rotor and lies in the plane of rotation of the rotor (i.e.,
(a) Hover.
(b) 120 kts.
(c) 170 kts.
Fig. 13: Free-vortex wake with a near-wake vortex-lattice
model for a Bell XV-15 tiltrotor in trimmed flight.
approximately 6.92 ft along the water line). The observer po-
sition is shown qualitatively in Fig. 14. The chosen flight
condition is hover. The azimuthal resolution used for the sim-
ulation is ψ=1 deg, such that the number of time steps per
revolution is nψ=360. The spatial resolution used to dis-
cretize each rotor blade surface is 10 chordwise panels and 10
spanwise panels, as suggested in Ref. 45. Note that the blade
upper and lower surfaces are considered as two different sur-
faces. As such, a total 1200 panels are used to characterize the
surface of all rotor blades. Figure 15 shows the total acous-
tic pressure generated by the rotors, as well as the acoustic
pressure components, i.e., thickness, near-field loading, and
far-field loading for a hovering condition.
Closed-Loop Simulations
The closed-loop performance of the flight control law dis-
cussed above is demonstrated for a transition from hover to
16
Fig. 14: Location of an observer fixed in the helicopter body
frame.
0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13
Observer time [s]
-10
-8
-6
-4
-2
0
2
4
Acoustic pressure [Pa]
Total Thickness Loading 1Loading 2
Fig. 15: Noise generated aerodynamically by the rotors at
hover.
10 kts forward flight. Figure 16 shows the closed-loop re-
sponse to a 10 kts step input at hover. Figure 16a shows that
the velocity command is tracked by the closed-loop dynam-
ics while off-axis velocity responses are contained and stable.
Figure 16b shows how the tilrotor initially pitches forward to
accelerate and slowly settles to a small, positive pitch attitude
once the target velocity is reached. Like for the velocity re-
sponse, the off-axis responses in the roll and yaw angles are
minimal and stable. Figure 16c shows the closed-loop con-
trols corresponding to the maneuver in consideration. The
control system applies forward stick to achieve the desired
nose-down pitch rate to tilt the rotor thrust forward and sub-
sequently eases off the longitudinal stick once the target lon-
gitudinal speed is achieved. To compensate for the tilt of the
thrust vector forward and thus for the loss of thrust, the con-
troller applies positive collective stick to maintain altitude. As
such, the flight control law appears to track the commanded
input with desired response characteristics while guarantee-
ing stability and minimizing off-axis response.
CONCLUSIONS
A generic tiltrotor simulation model with coupled flight dy-
namics, state-variable aeromechanics, and aeroacoustic was
developed, implemented, and validated. The model was im-
plemented for a Bell XV-15 tiltrotor and validated both in the
frequency and time domains against U.S. Army/NASA XV-
15 flight-test data and other data available in the literature for
multiple operating conditions including hover, transition, and
forward flight. State-of-the-art time-periodic systems analy-
sis tools are used to compute the periodic equilibrium of the
rotorcraft at a desired flight condition to obtain higher-order
linear time-invariant models of the aircraft flight dynamics
that account for the higher harmonics. Model-order reduc-
tion methods are leveraged to reduce the order of these lin-
earized models and make them tractable for flight control de-
sign. Model-following flight control laws that are capable of
autonomous flight, including transition from hover to high-
speed forward flight were developed. State-space free-vortex
wake modeling was shown to provide qualitatively reason-
able predictions of the wake geometry in different flight con-
ditions. The model was demonstrated to predict noise for a
hovering condition and for an observer fixed in the aircraft
frame. Based on this work, it is concluded that the simu-
lation model developed has clear applications in flight dy-
namics and controls, aeromechanics, and in the prediction of
aerodynamically-generated noise in generalized maneuvering
flight.
Future work will focus on generating linearized models of
the coupled flight dynamics, state-space aeromechanics, and
acoustics. Model-order reduction methods will be investi-
gated to guide the development of linearized models that are
tractable for real-time simulations and flight control design
while still capturing the underlying physics. Among other ap-
plications, this simulation model will be used to investigate
complex shipboard interactions that caused fatal mishaps in
the past (Ref. 6, 46).
ACKNOWLEDGMENTS
This research was partially funded by the U.S. Govern-
ment under agreements no. W911W62120003 and no.
N000142312067. The views and conclusions contained in
this document are those of the authors and should not be inter-
preted as representing the official policies, either expressed or
implied, of the Aviation Development Directorate or the U.S.
Government.
REFERENCES
1Howlett, J. J., “UH-60A Black Hawk Engineering Simu-
lation Program. Volume 1: Mathematical Model, Technical
report, NASA-CR-166309, 1980.
17
012345678910
0
5
10
Vx [ft/s]
CL
Cmd
012345678910
-0.1
0
0.1
0.2
0.3
Vy [ft/s]
012345678910
Time [s]
-4
-2
0
2
Vz [ft/s]
(a) Longitudinal, lateral, and vertical velocities in the heading
frame.
012345678910
-1
-0.5
0
0.5
φ [deg]
012345678910
-6
-4
-2
0
2
θ [deg]
012345678910
time [s]
-0.1
0
0.1
0.2
ψ [deg]
(b) Roll, pitch, and yaw angles.
012345678910
4
5
6
δ
lat [in]
012345678910
0
10
20
30
δ
lon [in]
012345678910
7
8
9
δ
col [in]
012345678910
time [s]
1.5
2
2.5
3
δ
ped [in]
(c) Closed-loop control inputs.
Fig. 16: Closed-loop response to a 10 kts step input in the
longitudinal speed at hover.
2Horn, J. F., Bridges, D. O., Wachspress, D. A., and Rani,
S. L., “Implementation of a Free-Vortex Wake Model in Real-
Time Simulation of Rotorcraft, AIAA Journal of Comput-
ing, Information, and Communications, Vol. 3, (3), Mar 2006,
pp. 93–107.
doi: https://doi.org/10.2514/1.18273
3Keller, J. D., Wachspress, D. A., and Hoffler, J. C., “Real
Time Free Wake and Ship Airwake Model for Rotorcraft
Flight Training Applications, Proceedings of the 71st Annual
Forum of the American Helicopter Society, Virginia Beach,
VA, May 5-7, 2015.
4Zhao, J. and He, C., “A Viscous Vortex Particle Model for
Rotor Wake and Interference Analysis,” Journal of the Amer-
ican Helicopter Society, Vol. 55, (1), Jan 2010.
doi: https://doi.org/10.4050/JAHS.55.012007
5Saetti, U. and Horn, J. F., “Implementation and Lineariza-
tion of a Rotor Simulation with a State-Space Free-Vortex</