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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

x∈Rnis the state vector, u

u

u∈Rmis the control input

vector, y

y

y∈Rlis 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=Ωt, where Ωis the rotor angular speed in rad/s. It fol-

lows that the fundamental period of the system is T= (2π)/Ω

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˙

β1cλ0λ1sλ1cΩψR

β0β1sβ1c˙

β0˙

β1s˙

β1cλ0λ1sλ1cΩψLwiRH wiRVi(7)

where:

β0,β1s,β1care the ﬂapping angles in multi-blade coordi-

nates,

λ0,λ1s,λ1c,are the rotor induced inﬂow ratio harmonics,

Ωis 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+Ω2λ2

ββi=sλ2

β−kβ

IβΩ22Ω(pHcosψ+qHsinψ)

+˙pHsinψ+˙qHcosψ−1

IβZR

eR

(r−eR)fzi(r)dr

(8)

where:

βiis the ﬂapping angle of the ith blade,

λβ=r1+kβ

Ω2Ibis 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

ΩM

M

M

˙

λ0

˙

λ1s

˙

λ1c

+L

L

L−1

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

L−1and 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=βm∆θ0Xlon 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

ΩV

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)/Ω. The resulting

system of ODE expressed in dimensional time is:

˙

r

r

rNW =−ΩA

A

Aζr

r

rNW +V

V

V(r

r

rNW (t,ζ)) (12a)

˙

r

r

rTV =−ΩA

A

Aζr

r

rTV +V

V

V(r

r

rTV (t,ζ)) (12b)

˙

Γ

Γ

ΓNW =−ΩA

A

AζΓ

Γ

ΓNW (12c)

˙

Γ

Γ

ΓTV =−ΩA

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

0.20.2969√xc−0.3516x2

c+0.2843x3

c−0.1015x4

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

0c(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=

1−v

V+∆vα

VCL2

upper surface

1−v

V−∆vα

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=p−p0is 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

x−y

y

y

||x

x

x−y

y

y|| is the emission direction,

y

y

yis the source location,

Λ=hp1−2Mnˆ

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=t−r

c0

(18)

where r=||x

x

x−y

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,t−r/c0)

rΛret

dΣ

(20a)

I2=ZΣ˜pˆ

n

n

n·ˆ

r

r

r

r2Λret

dΣ=ZΣQ2(y

y

y,t−r/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,t−r/c0)

rΛret

∆Σi(21a)

I2≈

Ntri

∑

i=1Q2(y

y

y,t−r/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(ti−1)

2∆t(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

U∈Rm(2M+1), and Y

Y

Y∈Rl(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

A∈Rn(2N+1)×n(2N+1),B

B

B∈Rn(2N+1)×m(2M+1),C

C

C∈

Rl(2L+1)×n(2N+1), and D

D

D∈Rl(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

X∗T

k=x

x

x∗T

k0x

x

x∗T

k1cx

x

x∗T

k1s... x

x

x∗T

kNc x

x

x∗T

kNs (32a)

U

U

U∗T

k=u

u

u∗T

k0u

u

u∗T

k1cu

u

u∗T

k1s... u

u

u∗T

kMc u

u

u∗T

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

X∗T

kU

U

U∗T

ki(33)

where Θ

Θ

Θk∈Rn(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

ek∈Rn(2N+1)and W

W

W∈Rn(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

Jk∈Rn(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

Θk−ˆ

J

J

J−1

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 ∥ek∥∞becomes 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

As−A

A

AsfA

A

Af−1A

A

Afs (42a)

ˆ

B

B

B=B

B

Bs−A

A

AsfA

A

Af−1B

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

Cs−C

C

CfA

A

Af−1A

A

Afs (46a)

ˆ

D

D

D=D

D

D−C

C

CfA

A

Af−1B

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. 31–40. 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

B−1 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 cosφ−sin φ

sinθ−sin φcos θ−cos 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

B−1ν

ν

ν−C

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

VsinφV−VLS

VHS −VLS VLS ≤V<VHS

rcmd +g

VsinφV≥VHS

(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 1e−7.

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.

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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

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θ [deg]

012345678910

time [s]

-0.1

0

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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.

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