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Dynamic Inversion Flight Control Laws for Autonomous Transition of Tilt-Rotor Aircraft

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This article describes the development, implementation, and demonstration of dynamic inversion (DI) flight control laws for autonomous transition of tilt-rotor aircraft from hover (helicopter mode) to cruise flight (air-plane mode). The DI control laws are based on a multi-loop architecture and do not require gain scheduling, although they still need to be scheduled with the linearized aircraft flight dynamics at discrete speed increments. A generic multi-rotor/wing simulation model is adapted to model the flight dynamics of two tilt-rotor configurations: one similar to a Bell XV-15 and one reminiscent of a Joby S4. The simulation models are trimmed at discrete speed increments from hover to cruise flight and linearized. Model-order reduction methods are lever-aged to reduce the order of these linearized models and make them suitable for flight control design. The DI flight control laws are demonstrated in batch simulations for both aircraft configurations to yield a successful autonomous transition from hover to cruise flight with excellent tracking of the commanded forward speed and minimal off-axis response.
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Dynamic Inversion Flight Control Laws for Autonomous Transition of
Tilt-Rotor Aircraft
Umberto Saetti
Assistant Professor
Department of Aerospace Engineering
University of Maryland
College Park, MD 20740
ABSTRACT
This article describes the development, implementation, and demonstration of dynamic inversion (DI) flight
control laws for autonomous transition of tilt-rotor aircraft from hover (helicopter mode) to cruise flight (air-
plane mode). The DI control laws are based on a multi-loop architecture and do not require gain scheduling,
although they still need to be scheduled with the linearized aircraft flight dynamics at discrete speed increments.
A generic multi-rotor/wing simulation model is adapted to model the flight dynamics of two tilt-rotor configu-
rations: one similar to a Bell XV-15 and one reminiscent of a Joby S4. The simulation models are trimmed at
discrete speed increments from hover to cruise flight and linearized. Model-order reduction methods are lever-
aged to reduce the order of these linearized models and make them suitable for flight control design. The DI
flight control laws are demonstrated in batch simulations for both aircraft configurations to yield a successful
autonomous transition from hover to cruise flight with excellent tracking of the commanded forward speed and
minimal off-axis response.
INTRODUCTION
Future Vertical Lift (FVL) vehicles and electric Vertical
Take-Off and Landing (eVTOL) vehicles intended for Ur-
ban Air Mobility (UAM) feature multiple rotors, rigid ro-
tor systems, and high levels of aerodynamic interactions
which pose significant modeling, simulation, and control
challenges. FVL configurations are include winged sin-
gle main rotor (wSMR), lift-offset coaxial (LOC), and
tilt-rotor configurations (Refs. 14). Attempts to cate-
gorize the various UAM eVTOL configurations are pro-
vided in Refs. 5, 6. In these studies, configurations
are differentiated between rotary-wing cruise and fixed-
wing cruise. The rotary-wing cruise category includes
rotary-wing and lift-fan aircraft, whereas the fixed-wing
category includes lift+cruise, tilt-wing/rotor, and tailsit-
ter aircraft. Tilt-rotor/wing configurations, which are
the only common configuration between FVL and UAM,
pose particular control challenges due to their hybrid de-
sign, which combines the features of both helicopters
and fixed-wing aircraft. Key challenges include: (i) the
transition from Vertical Take-Off and Landing (VTOL)
(or helicopter) mode and high-speed forward flight (or
airplane) mode, characterized by the tilting motion of
the rotors and/or wings tilt; (ii) significant differences in
the stability and response characteristics between VTOL,
high-speed forward flight, and transition modes; (iii) sig-
nificant rotor-on-rotor and rotor-on-wing interactions that
vary dramatically between operational modes and have
similarly dramatic effects on handling qualities; (iv) the
Presented at the 2nd International Conference on Advanced Air
Mobility Systems, Singapore, December 4–6, 2024.
presence of redundant control surfaces, such that the con-
trol signal needs to be reallocated intelligently across the
redundant control effectors; and (v) proneness to gust dis-
turbances and therefore need flight control systems able
to mitigate this deficiencies.
All of these control challenges require flexible, yet ro-
bust, control architectures to provide desired stability, re-
sponse, and handling quality characteristics across the
flight envelope. Moreover, it is important that the closed-
loop dynamics yield a simple, predictable response to
commanded output that is common across operational
modes to minimize pilot workload. Conversion control
laws have been investigated in the past, with examples
including total energy control system algorithm (Ref. 7),
nonlinear time-varying (NTV) control (Ref. 8), nonlinear
optimal control (Ref. 9), active adaptive model inversion
(Ref. 10), and hybrid DI and explicit model following
(EMF) (Ref. 11). While all of these studies achieved suc-
cessful conversion of more or less sophisticated models
of tilt-rotor aircraft, only the latter focused on handling-
quality evaluations. In fact, both DI and EMF are model-
following control methods that are particularly suited for
gust disturbance rejection and for stability, performance,
and handling quality evaluations as specified in the Aero-
nautical Design Standard-33 (ADS-33) specifications for
military rotorcraft (Ref. 12).
Model following control architectures such as Dynamic
Inversion (DI) and Explicit Model Following (EMF) are
especially well-suited for providing gust disturbance re-
jection. This is because their 2-degree-of-freedom (DoF)
structure, with separate feed forward and feedback paths,
1
allows to tune the rotorcraft response to gusts inde-
pendently from responses to outer guidance/navigation
loops. In contrast, in 1-DoF classic control methodolo-
gies, the responses to gusts and outer guidance/navigation
loops cannot be independently tuned. A key aspect of
both architectures is the reliance on model inversion to
cancel the plant dynamics and track a desired reference
model. EMF is essentially a linear design method that
uses a simplified model inversion in the feed-forward
path to to follow the desired reference model, while
feedback design is applied to the high-order linear plant
models that are either identified from flight-test data
or obtained from physics-based models via linearization
schemes. Generally, both feed-forward inversion and
feedback compensation needs to be scheduled with with
the flight condition to account variations in the flight dy-
namics. DI, on the other hand, inverts the plant model in
its feedback linearization loop, which eliminates the need
for gain scheduling. However, the plant model used for
feedback linearization still needs to be scheduled with the
flight condition. But if the plant model is known at each
flight condition, either from flight test or from simula-
tion, DI becomes particularly attractive from the control
system designer’s perspective due to its ease of imple-
mentation.
With regards to past work on DI for tilt-rotor conversion,
(Ref. 11) only developed outer-velocity DI control loops,
whereas the inner-attitude loop was based on DI. More-
over, these control laws were not demonstrated for auto-
matic transition, rather for piloted transition, and were
only demonstrated for quasi-Linear Parameter-Varying
Model (qLPV) Models (Ref. 13).
As such, the objectives of the present investigation are
two fold. The first objective is to develop DI flight con-
trol laws for tilt-rotor aircraft that provide desired stabil-
ity, performance, and handling quality characteristics at
hover, in high-forward flight, and during conversion be-
tween the two. The second objective is to demonstrate
said flight control laws for two different tilt-rotor con-
figurations: a tilt-rotor similar to an XV-15 and a tilt-
multirotor similar to a Joby S4.
The paper begins with an overview of the simulation
model and its adaptation to two tilt-rotor configura-
tions: one similar to a Bell XV-15, and one reminis-
cent of a Joby S4 eVTOL. This is followed by a de-
tailed mathematical description of the linearization, trim,
and model-order reduction algorithms. Next, DI control
laws are developed based on a multi-loop architecture
and parametrized with flight speed to enable autonomous
transition from hover to cruise flight. These flight control
laws are demonstrated for both configurations through a
simulation involving the transition from hover to 160 kts
forward flight. Final remarks summarize the overall find-
ings of the study and future developments are identified.
SIMULATION MODELS
Overview
Simulation models are based on an in-house
MATLAB®/Simulink generic multi-rotor/wing flight
dynamics and control code with the following character-
istics:
The aircraft fuselage is modeled as a rigid-body.
Fuselage aerodynamic forces are calculated based
on equivalent flat-plate frontal, lateral, and vertical
areas (Ref. 14).
The user can specify any number of rotors and/or
wings at arbitrary (and time-varying) orientations on
the aircraft body.
Accuracy of the rotor aeromechanics is selectable,
with modeling methods being as simple as static in-
flow and no blade flapping, or as comlpex as dy-
namic inflow (Ref. 15) and flexible blades based on
finite element methods (FEM). Rotor aerodynam-
ics can also account for rotor-on-rotor interactions
(Refs. 1618).
Wings accuracy level is also selectable. The most
complex model consists in lifting-line theory. Rotor-
on-wing aerodynamics inreractions are calculated
based on the circulation at each wing element/panel
using Biot-Savart and Hyeson (Ref. 19).
The model is implemented in MATLAB®to ease the
design and testing of flight control laws.
The model is integrated with a baseline control law
based on Dynamic Inversion (Refs. 2022) with
inner-attitude and outer-velocity control loops.
Simulation models are developed for two different tilt-
rotor aircraft configurations: a tilt-rotor similar to an XV-
15 (Refs. 21,23) and a tilt-multirotor similar to a Joby S4.
The geometry of these two tilt-rotor vehicles is shown
in trimmed flight at hover and high-speed forward flight
in Fig. 1. Note that for illustration purposes, the Joby
S4-like model features a scaled XV-15 fuselage. This
is because it was not possible to reproduce the Joby S4
fuselage based on publicly available data. The general
characteristics of both aircraft are reported in Tables 1
and 2. The XV-15 properties are largely based on Ref.
23, whereas the S4 mass properties are scaled from the
those of the XV-15 based on relative mass and dimen-
sions. Additionally, geometry parameters were estimated
from publicly available pictures / drawings found online.
Nonlinear Dynamics
The rotorcraft flight dynamics are formulated as a nonlin-
ear 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 di-
mensional time in seconds. It is convenient to note that
2
Table 1: General characteristics of the XV-15-like model (Ref. 23).
Parameter Value Units
Mass and inertia
Gross weight, W13000 lb
Roll-axis moment of inertia, Ixx 52795 sl-ft2
Pitch-axis moment of inertia, Iyy 21360 sl-ft2
Yaw-axis moment of inertia, Izz 66335 sl-ft2
Roll/yaw-axes product of inertia, Ixz 1234 sl-ft2
CG fuselage station 25 ft
CG butt line 0 ft
CG water line 6.8 ft
Fuselage
Frontal drag area 23.11 ft2
Sideward drag area 131.83 ft2
Vertical drag area 184.11 ft2
Fuselage station (center of pressure) 24.42 ft
Butt line (center of pressure) 0 ft
Water line (center of pressure) 5.42 ft
Rotors
Number of blades 3 -
Radius 12.5 ft
Mean blade chord 1.19 ft
Blade twist 40.9 deg
Blade weight 213.14 lb
Blade flapping inertia 102.5 sl-ft2
Hub flapping spring 225 (sl-ft)/deg
Angular speed 61.68 rad/s
Shaft length 4.67 ft
Fuselage station (shaft pivot point) 25 ft
Butt line (shaft pivot point) ±16.08 ft
Water line (shaft pivot point) 8.33 ft
Wing
Span 32.17 ft
Mean chord 5.26 ft
Twist 3 deg
Sweep 6.5 deg
Fuselage station (aerodynamic center) 24.31 ft
Butt line (aerodynamic center) 0 ft
Water line (aerodynamic center) 7.99 ft
Horizontal Stabilizer
Span 20 ft
Mean chord 3.92 ft
Fuselage station (aerodynamic center) 46.67 ft
Butt line (aerodynamic center) 0 ft
Water line (aerodynamic center) 8.58 ft
Vertical Stabilizers (Center)
Span 7.68 ft
Mean chord 3.73 ft
Fuselage station (aerodynamic center) 47.5 ft
Butt line (aerodynamic center) ±6.41 ft
Water line (aerodynamic center) 9.64 ft
3
Table 2: General characteristics of the Joby S4-like model.
Parameter Value Units
Mass and inertia
Gross weight, W4800 lb
Roll-axis moment of inertia, Ixx 28650 sl-ft2
Pitch-axis moment of inertia, Iyy 1962 sl-ft2
Yaw-axis moment of inertia, Izz 17118 sl-ft2
Roll/yaw-axes product of inertia, Ixz 0 sl-ft2
CG fuselage station 10.2 ft
CG butt line 0 ft
CG water line 3.86 ft
Fuselage
Frontal drag area 26.69 ft2
Sideward drag area 135.08 ft2
Vertical drag area 135.08 ft2
Fuselage station (center of pressure) 10.15 ft
Butt line (center of pressure) 0 ft
Water line (center of pressure) 2.66 ft
Rotors
Number of blades 5 -
Radius 6.04 ft
Mean blade chord 0.9 ft
Blade twist 30 deg
Shaft length {1.65,3.58,1.93}ft
Fuselage station (shaft pivot point) {1.77,10.2,20.77}ft
Butt line (shaft pivot point) 8.29,±19.5,±8.29}ft
Water line (shaft pivot point) {5.89,5.23,8.8}ft
Wing
Span 39 ft
Mean chord 4.11 ft
Twist 0 deg
Sweep 2 deg
Fuselage station (aerodynamic center) 10.77 ft
Butt line (aerodynamic center) 0 ft
Water line (aerodynamic center) 5.5 ft
V-Tail
Span 16.08 ft
Mean chord 3.8 ft
Twist 0 deg
Sweep 20 deg
Dihedral 35 deg
Fuselage station (aerodynamic center) 27.55 ft
Butt line (aerodynamic center) 0 ft
Water line (aerodynamic center) 5.5 ft
4
(a) XV-15 at hover. (b) XV-15 in high-speed forward flight.
(c) Generic tilt-rotor aircraft at hover. (d) Generic tilt-rotor in high-speed forward flight.
Fig. 1: Tilt-rotor aircraft geometry at hover and high-speed forward flight.
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 an-
gular speed, in rad/s, of the slowest rotor. It follows that
the fundamental period of the system is T= (2π)/sec-
onds, which corresponds to 2πradians or one revolution
of the slowest rotor. 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
RB x
x
xT
R1··· x
x
xT
RN(3)
where x
x
xRB are the rigid-body states and x
x
xRiare the states
of the ith of Nrotors. The rigid-body state vector is com-
mon to all aircraft and is given by:
x
x
xT
RB = [uvwpqrφ θ ψ x y z](4)
where:
u,v,ware the longitudinal, lateral, and vertical veloc-
ities 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 general form of the ith rotor state vector is:
x
x
xT
Ri=hβ
β
βT
M˙
β
β
βT
Mλ
λ
λTψRi(5)
where:
β
β
βMare the flapping angles in multi-blade coordinated,
λ
λ
λT= [λ0λ1cλ1s]is a vector containing the dynamic
inflow components,
is the rotor angular speed, and
ψRis the azimuth angle of a reference blade.
It is worth noting that the state vector differs between the
two aircraft models as the S4-like configuration does not
feature blade flapping. Because the rotors are considered
to be rigid, the flapping states are removed from the S4-
like configuration state vector.
The pilot input vector is:
u
u
uT=δlat δlon δcol δped δaux(6)
where:
δlat is the lateral and longitudinal stick position,
5
δlat and δlon is the longitudinal stick position,
δcol is the collective stick position,
δped is the pedals position, and
δaux is an auxiliary input that controls the rotor/wing
tilt angle.
The pilot inputs are converted to actuator inputs via a
mixing matrices scheduled with aircraft absolute speed.
Note that, in this study, the XV-15-like model does not
adopt the control mixing relations in Ref. 23. The con-
trol effector inputs are:
u
u
uT
C=u
u
uT
Ru
u
uT
W(7)
where u
u
uRare the rotor actuator inputs and u
u
uWare the
wing actuator inputs. The actuator inputs for the ith ro-
tor are:
u
u
uT
Ri= [θ0θ1cθ1scmd βR](8)
where:
θ0θ1sθ1care the collective, longitudinal cyclic, and
lateral cyclic swashplate inputs,
cmd is the commanded rotor speed, and
βRis the rotor tilt angle.
Note that, in this study, the Joby-S4 rotors are assumed
to be pitch-only controlled and their angular speed to be
prescribed. The actuator inputs for the ith wing are:
u
u
uT
Wi= [βWδTE ··· δTEN](9)
where βWiis the wing tilt angle and δTE jis the deflection
of the jth trailing edge.
Trim, Linearization, and Model-Order Reduction
Linearized, time-invariant models are obtained by trim-
ming the rotorcraft flight dynamics at a desired flight
conditions via the periodic trim algorithm of Ref. 24,
25. Subsequently, the rotorcraft flight dynamics are lin-
earized about each trim point via perturbation methods
by only retaining the averaged (or zeroth) dynamics. 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 dy-
namics, 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.
This can be achieved through residualization, a portion
of singular perturbation theory that pertains to LTI sys-
tems (Ref. 26). Application of residualization to the ro-
torcraft flight dynamics can be found in several published
research studies, e.g., Refs. 20, 21, 2729. Residualiza-
tion begins with the assumption that one or more states
have stable dynamics that are faster than that of the re-
maining states. Then, the state vector in Eq. (3) is parti-
tioned into slow and fast components:
x
x
xT=x
x
xT
sx
x
xT
f(10)
Then, the linearized dynamics 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(11)
By neglecting the dynamics of the fast states (i.e.,˙
x
x
xf=0)
and performing a few algebraic manipulations, the equa-
tions 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(12)
where:
ˆ
A
A
A=A
A
AsA
A
AsfA
A
Af1A
A
Afs (13a)
ˆ
B
B
B=B
B
BsA
A
AsfA
A
Af1B
B
Bf(13b)
Note that A
A
Afmust be invertible. This is guaranteed if A
A
Afis
asymptotically stable, i.e., all eigenvalues have their real
part that is strictly negative. This condition is satisfied by
choosing the rotor dynamics as the fast dynamics, since
the rotor dynamics are typically stable with eigenvalues
on the far left of the complex plane. The slow states, on
the other hand, are chosen as the rigid-body states with
the exception of the position and heading states which
are truncated as the rotorcraft dynamics are invariant with
respect to these states (Ref. 30):
x
x
xs=x
x
xRB (14a)
x
x
xT
f=x
x
xT
R1x
x
xT
R2(14b)
This way, an 8-state residualized system is obtained that
still accounts for the higher-order dynamics. A similar
procedure is applied to the output equations of the lin-
earized dynamics, which are re-formulated as:
y
y
y=C
C
CsC
C
Cfx
x
xs
x
x
xf+D
D
Du
u
u(15)
Then, it can be shown that the residualized output equa-
tions are:
˙
Y
Y
Y=ˆ
C
C
Cx
x
xs+ˆ
D
D
Du
u
u(16)
where:
ˆ
C
C
C=C
C
CsC
C
CfA
A
Af1A
A
Afs (17a)
ˆ
D
D
D=D
D
DC
C
CfA
A
Af1B
B
Bf(17b)
These reduced-order output equations capture the influ-
ence of the residualized dynamics on the output of the
system.
FLIGHT CONTROL DESIGN
General Architecture
The flight control architecture chosen for this study
is Nonlinear Dynamic Inversion (NDI). Application of
NDI control laws to rotorcraft can be found in Refs.
11, 2022, 27, 29, 31, 32. A key aspect of DI is the re-
liance on model inversion to cancel the plant dynamics
and track a desired reference model. One convenient fea-
ture of NDI is that it inverts the plant model in its feed-
back linearization loop, which, compared to other more
conventional model-following control strategies such as
explicit model following (EMF), eliminates the need for
6
gain scheduling. However, the plant model used for feed-
back linearization still needs to be scheduled with the
flight condition. A generic DI controller as applied to
a linear system is shown in Fig. 2. 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 er-
ror, and an inner feedback loop that achieves model in-
version (i.e., the feedback linearization loop).
A multi-loop NDI control law based on Refs. 11, 20, 21,
31 is designed to enable autonomous flight in low-speed
flight (helicopter mode), cruise flight (airplane mode),
and transition between the two. The schematic of the
closed-loop rotorcraft dynamics is shown in Fig. 3. The
outer loop controller, shown in Fig. 4b tracks longitudi-
nal, lateral, and vertical ground velocities commands in
the heading frame and calculates the desired pitch and
roll attitudes for the inner loop to track, in addition to the
collective control input setting. The desired response type
for the outer loop is Translational Rate Command (TRC).
The inner loop, shown in Fig. 4a achieves stability, dis-
turbance rejection, and desired response characteristics
about the roll, pitch, yaw, and heave axes. When cou-
pled with the outer loop, an Attitude Command / Attitude
Hold (ACAH) response is used for the roll and pitch axes,
whereas Rate Command / Attitude Hold (RCAH) is used
for the yaw axis.
Inner-Attitude Loop
The modified state vector used for the inner-attitude loop
design is:
x
x
xT= [pqrφ θ](18)
The system and control matrices of the corresponding
modified system (i.e.,A
A
Aand B
B
B) are obtained by truncat-
ing the rows and columns corresponding with the body-
axes linear velocity states. Note that the stability and
control derivatives are a function of the total speed V=
u2+v2+w2. The controlled variables are:
y
y
yT= [φ θ r](19)
The output matrix that relates the state vector to the out-
put vector:
C
C
CT=C
C
CT
1C
C
CT
2(20)
where:
C
C
C1=00010
00001(21a)
C
C
C2=00100(21b)
In the equation above,C
C
C1corresponds to the roll and pitch
attitudes whereas C
C
C2corresponds to the yaw rate. This
partitioning is due to the fact that the output equations
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 the
yaw rate r:
¨
φ
¨
θ
˙r
=C
C
C1A
A
A2x
x
x+C
C
C1A
A
AB
B
Bu
u
u
C
C
C2A
A
Ax
x
x+C
C
C2ˆ
B
B
Bu
u
u(22)
The objective of the DI control law is that the output y
y
y
tracks a reference trajectory y
y
ycmd(t)given by:
y
y
yT
cmd = [φcmd θcmd rcmd](23)
with desired response characteristics. For this reason, the
reference trajectory is fed through first- or second-order
command models which dictate the desired response
of the system. More specifically, φcmd and θcmd are
fed through a second-order system, whereas rcmd is fed
through a first-order system. The command models are
also used to extract the first and second derivatives of the
filtered reference trajectory for use in the proportional-
integral (PI) and proportional-integral-derivative (PID)
compensators. 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
described below. The command models are of the follow-
ing form:
G(1)
ideal(s) = 1
τs+1(24a)
G(2)
ideal(s) = ω2
n
s2+2ωnζ+ω2
n
(24b)
where τis the first-order command model time constant,
which is the inverse of the command model break fre-
quency (i.e.,τ=1/ωn). Additionally, ωnand ζare, re-
spectively, the natural frequency and damping ratio of the
second-order command model. PI and PID compensation
are used to reject external disturbances and to compensate
for discrepancies between the approximate model used
in this derivation and the actual bare-airframe dynamics
of the aircraft. The resulting DI control law is found by
solving for the control vector in Eq. (22), leading to:
u
u
u=C
C
C1A
A
AB
B
B
C
C
C2B
B
B1ν
ν
νC
C
C1A
A
A2
C
C
C2A
A
Ax
x
x(25)
where ν
ν
νis the pseudo-command vector and e
e
eis the error
as defined in Eqs. (26) and (27), respectively.
νφ
νθ
νr
=
¨
φcmd
¨
θcmd
˙rcmd
+K
K
KP
eφ
eθ
er
+K
K
KD
˙eφ
˙eθ
0
+K
K
KI
Reφdt
Reθdt
Rerdt
(26)
e
e
e=y
y
ycmd y
y
y; (27)
The 3-by-3 diagonal matrices K
K
KP,K
K
KI, and K
K
KDidentify
the proportional, integral, and derivative gain matrices,
respectively. Command model parameters are chosen to
be the same as Ref. 21. Note that the coefficient matri-
ces (C
C
C1A
A
AB
B
B)1,C
C
C1A
A
A2,(C
C
C2B
B
B)1, and C
C
C2A
A
Aare functions of
the total speed of the aircraft V. For this reason, from
7
Fig. 2: DI controller as applied to a linear system.
Fig. 3: Schematic of the closed-loop tilt-rotor dynamics.
a practical standpoint, these matrices are computed of-
fline at incremental longitudinal speeds from hover to the
maximum speed of the aircraft at 20 kts intervals and
stored. When the linearized DI controller is implemented
on the nonlinear aircraft dynamics, the coefficient matri-
ces (C
C
C1ˆ
A
A
Aˆ
B
B
B)1,C
C
C1A
A
A2,(C
C
C2B
B
B)1, and C
C
C2A
A
Aare computed at
each time step via interpolation based on the current air-
speed V(t)and on the lookup tables stored offline. It is
important to note that what is implemented on the nonlin-
ear aircraft dynamics is linearized DI. However, because
the coefficient matrices are scheduled with the longitudi-
nal speed, and scheduling effectively introduces a nonlin-
ear relation between the aircraft states and the feedback
control input, the controller implemented is effectively
nonlinear DI (NDI) (Ref. 20).
Outer-Velocity Loop
The objective of the outer-velocity loop is to track longi-
tudinal and lateral velocities in the heading frame, such
that the reference trajectory is given by:
y
y
yT
cmd =Vxcmd Vycmd Vzcmd (28)
The heading frame is a vehicle-carried frame where the x-
axis is aligned with the current aircraft heading, the z-axis
is positive up in the inertial frame, and the y-axis points
to the right, forming a left-handed orthogonal coordinate
system. The following equation shows the rotation from
body to the heading frame:
T
T
TBh=
cosθsinφsin θcos φsin θ
0 cosφsinφ
sinθsinφcos θcos φcos θ
(29)
such that the velocities in the heading frame are given by:
Vx
Vy
Vz
=T
T
TBh
u
v
w
(30)
The outer-velocity loop is modified with respect to Ref.
11 not to include the auxiliary control input (i.e., the na-
celle/wing angle in the case of tilt-rotor) as a control vari-
able. Rather the nacelle/wing angle is prescribed with
flight speed, but can still be changed via open-loop pilot
inputs. The outer loop dynamics are designed based on
the following reduced-order dynamics:
˙u
˙v
˙w
=
Xu0 0
0Yz0
0 0 Zw
u
v
w
+
0XθXδcol
Yδφ0 0
0ZθZδcol
φcmd
θcmd
δcol
=A
A
Ax
x
x+B
B
Bu
u
u
(31a)
Vx
Vy
Vz
=
cosθ00 sinθ0
0 1 0
sinθ00 cosθ0
u
v
w
=C
C
Cx
x
x
(31b)
8
(a) Inner-attitude control loop.
(b) Outer-velocity control loop.
Fig. 4: Dynamic inversion inner-attitude and outer-velocity control loops.
where:
x
x
x,u
u
uare the reduced-order state, control input, and out-
put vectors used for outer-loop control design,
A
A
A,B
B
B,C
C
Care the system, control, and output matrices
used for outer-loop control design,
u,v,ware the longitudinal, lateral, and vertical veloc-
ities in the body-fixed frame,
Vx,Vy,Vzare the longitudinal, lateral, and vertical ve-
locities in the heading frame,
φcmd,θcmd are the roll and pitch attitudes commanded
to the inner loop,
Xu,Yv,ZW,Xθ,Yδφ,Zθare stability derivatives,
Xδcol,Zδcol are stability derivatives, and
θ0is the trim pitch attitude.
Then, the DI outer-loop control law will be of the form:
u
u
u= (C
C
CB
B
B)1(ν
ν
νC
C
CA
A
Ax
x
x)(32)
where ν
ν
νis the pseudo-control vector. Like for the inner-
velocity loop, the stability and control derivatives are a
function of the total speed V, such that the coefficient
matrices (C
C
CB
B
B)1and C
C
CA
A
Aare also function of total speed.
Thus, these matrices are computed at discrete speed in-
crements and stored offline along with those of the inner
loop. The reference trajectory is subtracted from the out-
put to find the error, which is compensated by a PI con-
troller. The feed-forward signal is subsequently added,
leading to the pseudo-control vector for the outer loop:
νVx
νVy
νVz
=
˙
VVxcmd
˙
VVycmd
˙
VVzcmd
+K
K
KP
eVx
eVy
eVz
+K
K
KI
ReVxdt
ReVydt
ReVzdt
(33)
where K
K
KPand K
K
KIare 3-by-3 proportional and integral
gain matrices, respectively. The command models for the
9
longitudinal, lateral, and vertical speed are first order and
adopt the same values as in Ref. 21.
Error Dynamics
Feedback compensation gains that act on the error dy-
namics are designed according to the methods in 11, 20–
22, 27, 29, 31, 32.
Turn Coordination
Because the tilt-rotor 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 condi-
tion. Above 60 kts, turn coordination is used; below 40
kts no turn coordination (Ref. 33) 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
(34)
where VLS =40 kts, and VHS =60 kts.
RESULTS
The XV-15-like simulation model is validated both in
the frequency and time domains against US Army/NASA
flight-test (Ref. 34) data and other models in the litera-
ture (Ref. 35). The rotorcraft is trimmed at hover in heli-
copter mode (i.e., with the nacelles at βR=0 deg accord-
ing to the definition in Ref. 23) and at 170 kts in aircraft
mode (i.e., with the nacelles at βR=90 deg) for valida-
tion. Figure 5 shows a comparison of the lateral dynam-
ics eigenvalues for each of these conditions with those
from US Army/NASA flight test data (Ref. 34) and from
Ref. 35. Notably, the linearized dynamics eigenvalues
are close to those of both the system identified dynamics
from Ref. 34, and those of the FLIGHTLAB®simulation
model from Ref. 35. Moreover, the residualize dynamics
eivencalues nearly overlap those of the full-order dynam-
ics, indicating the accuracy of the model-order reduction
method used. Figure 6 shows some sample frequency re-
sponses at hover as compared to US Army/NASA XV-
15 flight data (Ref. 34) and simulation data from Ref.
35. Again, the linearized dynamics are very close to
those from flight-test data and the model in the litera-
ture, and the residualized model overlaps the full-order
frequency response. While for the cases presented the
general agreement is good, the available flight-test data
does not provide the exact information on CG location,
moments of inertia, weight, and flap setting. As such, it
is difficult to draw definitive conclusions.
Trim
As a first step, the simulation models are trimmed at
incremental flight speed from 0 to 280 kts for the XV-
15-like configuration, and from 0 to 200 kts for the S4-
like configuration. Note that, for the XV-15, the auxil-
iary control input (i.e., nacelle tilt angle) is prescribed
since the XV-15 conversion corridor is known (Ref. 36).
The prescribed tilt angle approximately corresponds to
the center of the conversion corridor. On the other hand,
because the S4-like configuration conversion corridor is
unknown, the auxiliary control input is used as a control
variable in place of the pitch attitude. More specifically,
the pitch attitude is prescribed to zero while the auxil-
iary control input is an unknown variable in the trim pro-
cess. The trim attitude and pilot control inputs are shown
in Fig. 7 for both aircraft. More specifically, Fig. 7a
shows the trim Euler angles for increasing flight speed.
While trim roll and yaw attitudes are zero across the flight
speed considered, which stems from the tilt-rotors being
symmetric about the xzplane, the trim pitch attitude
behaves differently, at least for the XV-15 varies with
flight speed. In fact, the trim pitch attitude is shown to
increase with increasing speed up to 100 kts, and subse-
quently steadily decrease. Above 100 kts, the pitch at-
titude decreases with increasing speed, as it would for
a fixed-wing aircraft (Ref. 37). Since the S4-like con-
figuration is trimmed with zero pitch attitude, and given
the symmetry of the configuration about the xzplane,
the trim roll, pitc, and yaw attitudes are all zero for this
configuration. Figure 7b shows the trim control inputs
for increasing speed. In this figure, lateral stick and ped-
als are shown to remain at the neutral position due to the
symmetry of the aircraft about the xzplane. The lon-
gitudinal stick follows the trend of the pitch attitude for
the XV-15, while it remains approximately neutral for the
S4-like model. Trim collective stick inputs first decrease
due to the lower power required in low-speed forward
flight compared to hover, and subsequently steadily in-
crease with flight speed. Nacelle control inputs for the
S4-like aircraft turn out to be similar to the XV-15, as the
nacelles are shown to tilt from 0 to 90 deg between 40
and 150 kts.
Autnonomous Transition
The rotorcraft dynamics are linearized and residualized
at each discreet speed increment to obtain the DI con-
trol law coefficient matrices. Dynamic inversion control
laws are demonstrated for a closed-loop simulations of an
acceleration from hover to cruise flight, which includes a
conversion from helicopter to airplane mode. The simula-
tion consists in an acceleration from hover to 160 kts over
60 seconds. This simulation makes use of the nonlinear
rotorcraft dynamics. Figure 8a shows the heading-frame
velocities. In this figure, the longitudinal speed is shown
to track accurately the reference trajectory while the off-
axis responses in lateral and vertical speed are minimal.
Figure 8b shows how the inner-attitude loop is able to
10
-1 -0.5 0 0.5
Real
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Imag
Full-Order
Residualized
Padfield
Flight Data
(a) Hover.
-2 -1.5 -1 -0.5 0 0.5
Real
-6
-4
-2
0
2
4
6
Imag
Full-Order
Residualized
Padfield
Flight Data
(b) 170 kts.
Fig. 5: Comparison of the XV-15 lateral dynamics eigenvalues with those from US Army/NASA flight test
data (Ref. 34) and simulation data from Ref. 35.
10-1 100101
-40
-20
0
20
Mag [dB]
Full Order
Reduced Order
Padfield
Flight Data
10-1 100101
Frequency, [rad/s]
-300
-200
-100
0
Phase [deg]
(a) p/δlat.
10-1 100101
-60
-40
-20
0
Mag [dB]
Full Order
Reduced Order
Padfield
Flight Data
10-1 100101
Frequency, [rad/s]
-150
-100
-50
Phase [deg]
(b) r/δped.
Fig. 6: Bare-airframe frequency responses at hover compared with US Army/NASA XV-15 flight data (Ref. 34) and
simulation data from Ref. 35.
track the desired pitch attitude while maintaining zero roll
and yaw attitudes. The pitch attitude first decreases to di-
rect the rotor thrust forward and thus accelerate, and then
gradually increases as the aircraft reaches a steady accel-
eration. Finally, Fig. 8c shows the closed-loop control
inputs.
CONCLUSIONS
A generic multi-rotor/wing simulation model was
adapted to model the flight dynamics of two tilt-rotor
configurations: one similar to a Bell XV-15 and one rem-
iniscent of a Joby S4. The rotorcraft flight dynamics
were trimmed at discrete speed increments from hover to
cruise flight and linearized. Model-order reduction meth-
ods were leveraged to reduce the order of these linearized
models and make them suitable for flight control design.
Nonlinear dynamic inversion (NDI) control laws were
developed and implemented to enable autonomous tran-
sition given a prescribed profile of rotors tilt angle with
aircraft speed. These flight control laws were demon-
strated in batch simulations for both aircraft configura-
tions to yield a successful autonomous transition from
hover (helicopter mode) to cruise (airplane mode) flight
with excellent tracking of the commanded forward speed
and minimal off-axis response. Based on this work, it is
concluded that the proposed NDI flight control laws are
suitable for transitioning flight of tilt-rotor aircraft.
Future work will focus on demonstrating the method-
ology for other rotorcraft configurations, including
lift+cruise and tilt-wing aircraft, and on testing the pro-
posed flight control laws in piloted flight simulations.
11
0 50 100 150 200 250
-1
0
1
[deg]
XV-15
S4
0 50 100 150 200 250
0
2
4
6
8
[deg]
0 50 100 150 200 250
V [kts]
-1
0
1
[deg]
(a) Euler angles.
0 50 100 150 200 250
49
50
51
lat [%]
XV-15
S4
0 50 100 150 200 250
60
80
lon [%]
0 50 100 150 200 250
50
100
col [%]
0 50 100 150 200 250
49
50
51
ped [%]
0 50 100 150 200 250
V [kts]
0
50
100
aux [%]
(b) Control inputs.
Fig. 7: Trim variables for increasing speed.
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... The flight control architecture chosen for this study is Nonlinear Dynamic Inversion (NDI). Application of NDI control laws to rotorcraft can be found in [17,[21][22][23][24][25][26][27][28]. A key aspect of DI is the reliance on model inversion to cancel the plant dynamics and track a desired reference model. ...
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