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Interactional Aerodynamics Modeling and Flight Control Design of
Multi-Rotor Aircraft
Umberto Saetti
Assistant Professor
Department of Aerospace Engineering
University of Maryland
College Park, MD 20740
Feyyaz Guner
Independent Scholar
38106 Braunschweig, Germany
ABSTRACT
This paper focuses on the integration of generalized rotor-on-rotor interactional aerodynamics models in state-variable
form within flight dynamics simulations of multi-rotor unmanned aircraft systems (UAS), and on their subsequent
linearization. The aircraft chosen for this investigation is the Malloy TRV-80 coaxial quadcopter. Upon trimming
the flight dynamics at hover, linearized models of said dynamics are compared to nonlinear simulations, and with
linear and nonlinear adopting dynamic inflow models that do not account for rotor-on-rotor inflow interference ef-
fects. Model-order reduction methods are investigated to guide the development of linearized models that are tractable
for flight control design while still predicting the effect of rotor-on-rotor interactions on the vehicle flight dynamics.
Model-following flight control laws based on these reduced-order linearized models that account for the rotor-on-rotor
interactions are developed and tested against similar control laws based on linearized models that do not account for
rotor-on-rotor interactions. It is found that, for the coaxial quadcopter configuration in consideration, rotor-on-rotor
interactions result in a lower magnitude roll/pitch rate response to lateral/longitudinal velocity and in a lower pitch/roll
damping. These changes in stability derivatives yield roll and pitch subsidence modes at a lower frequency and roll
and pitch oscillation modes with a lower natural frequency. Additionally, roll and pitch on-axis frequency responses
indicate that the interactional inflow dynamics results in higher gains and phases at low to medium frequencies (ap-
proximately 0.1 to 4 rad/s). No significant improvements are detected by designing model-following control laws
based on reduced-order models that account on rotor-on-rotor interactions.
INTRODUCTION
Vertical Take-Off and Landing (VTOL) Unmanned
Aerial Systems (UAS) typically feature multiple rotors,
high levels of aerodynamic interactions, high rotor RPM,
and variable speed rotors. These features 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
minimum blade sweep per time step, so computational
cost goes up with smaller, higher revolutions-per-minute
(RPM) rotors. Modeling aerodynamic interactions and
multiple rotors requires larger amounts of wake to be
computed, and rigid rotor systems require more costly
structural dynamics models of the blades. Thus, while
more advanced aeromechanics models are feasible, albeit
complex, it is likely that in many cases, execution speeds
required for real-time simulations or routine design ap-
plications will remain elusive.
At the same time, there is an increasing need for these ad-
vanced aeromechanic models to be capable of fast predic-
Presented at the 6th Decennial VFS Aeromechanics Special-
ists’ Conference, Santa Clara, CA, Feb 6–8, 2024. Copy-
right © 2024 by the Vertical Flight Society. All rights reserved.
Distribution Statement A. Approved for public release; distri-
bution is unlimited.
tion of the rotor-on-rotor interactional aerodynamics ro-
tor loads to determine flight characteristics and for rapid
evaluation of flight trajectories that are dynamically feasi-
ble. It is therefore critical that high fidelity aeromechan-
ics models are formulated in such a 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 most practical control design methodologies, but
linear model analysis provides many physical insights to
system dynamics. To this end, a first-order state vari-
able implementation of the rotor inflow that can be ef-
ficiently linearized is a highly desirable feature for fu-
ture advanced simulations. While state-space implemen-
tation of the free-vortex wake of the rotor exist, these im-
plementations are complex, yield a very high number of
states for typical flight dynamics simulations (i.e., thou-
sands or tens of thousands), and have not yet been ex-
tended to variable RPM rotors (Refs. 1–6). When the
rotor speed is constant, these high-order aeromechanics
models can be used to identify lower-order models from
frequency response data (Ref. 7).
For variable speed RPM rotors, low-order inflow models
that were recently developed in the rotorcraft community
and that can predict the rotor-on-rotor interactional aero-
dynamics could be exploited. These models are based
1
on the Combined Momentum Theory and Simple Vortex
Theory Inflow (CMTSVT) model (Refs. 8,9) which con-
stitutes the simplest inflow model for capturing the rotor-
on-rotor inflow interference effects of generic multirotor
configurations. However, these models are only valid for
rotors with parallel axes of rotation and with equal radius
and rotor speed. Thus, these models require a generaliza-
tion to the case of rotors arbitrarily oriented in space and
without necessarily the same radius and/or angular speed.
These formulations of the inflow dynamics allow the
interactional inflow dynamics to be time-marched, and
effectively used to augment the flight dynamics of the
multi-rotor UAS. As such, multi-rotor UAS simulations
become self-contained and inherently linearizable, thus
do not require external processing to generate approx-
imate linear models. Linearized models could then be
used to assess the handling qualities of the vehicle, and
for real-time simulations. A major challenge lies in mak-
ing the linearized models tractable for control design
while still capturing the rotor-on-rotor interactional aero-
dynamics. In fact, the order of the linearized models will
require being significantly reduced to remove the need
for estimating or measuring the states associated with the
rotor, inflow, and other high-order dynamics.
Another challenge is be the use of these reduced-order
models that account for complex, and potentially ad-
verse, aerodynamic interactions between rotors in model-
following control architectures like dynamic inversion
(DI) and explicit model following (EMF) to compen-
sate aforementioned interactions, and to guarantee satis-
factory stability, handling qualities, and performance re-
quirements.
The objectives of the present investigation are articulated
as follows: (i) generalization of the CMTSVT model to
rotors with arbitrary radius and angular speed; (ii) imple-
mentation and linearization of low-order dynamic inflow
models that account for the rotor-on-rotor interactional
aerodynamics of generic multi-rotor configurations; (iii)
assessment the effect of the rotor-on-rotor interactional
aerodynamics on the flight dynamics; (iv) investigation
of model-order reduction methods to guide the develop-
ment of linearized models that are tractable for flight con-
trol design and predict the effect of rotor-on-rotor inter-
actions on the vehicle flight dynamics; and (v) implemen-
tation and demonstration of model-following flight con-
trol laws based on the reduced-order linearized models in
nonlinear simulations.
The paper begins with a detailed mathemetical descrip-
tion of the dynamic modeling of rotor-on-rotor interac-
tions. Next, the simulation model used in this study is
presented. Upon trimming the flight dynamics at hover,
linearized models of said dynamics are compared to non-
linear simulations, and with linear and nonlinear adopt-
ing dynamic inflow models that do not account for rotor-
on-rotor inflow interference effects. Model-order reduc-
tion methods are be investigated to guide the develop-
ment of linearized models that are tractable for flight con-
trol design while still predicting the effect of rotor-on-
rotor interactions on the vehicle flight dynamics. Model-
following flight control laws based on these reduced-
order linearized models that account for the rotor-on-
rotor interactions are developed and tested against similar
control laws based on linearized models that do not ac-
count for rotor-on-rotor interactions. Final remarks sum-
marize the study’s overall findings and identify future de-
velopments.
ROTOR-ON-ROTOR INTERACTIONAL
AERODYNAMICS
This section presents an extension of the state-space
CMTSVT model (Refs. 8, 9) to rotors arbitrarily oriented
in space and without necessarily the same radius and/or
angular speed. Consider the case where an arbitrary num-
ber Nof rotors are sufficiently close such that their inflow
dynamics is mutually interacting. For simplicity, con-
sider two mutually-interacting rotors that are arbitrarily
positioned and oriented in space and have radii that are
not necessarily equal. This setup is shown in Fig. 1.
Then, the inflow of the ith rotor can be expressed as the
summation between the self-induced inflow of that rotor
and the interference inflow from all other rotors:
λ
λ
λi=λ
λ
λsi+
N
∑
j=1,j=i
λ
λ
λinti j (1)
where:
λ
λ
λT
i=λi0λi1cλi1sis the total inflow of the ith rotor in
the local hub frame,
λ
λ
λT
si=hλsi0λsi1cλsi1siis the self-induced inflow of the
ith rotor in the local hub frame, and
λ
λ
λT
inti j =hλinti j0λinti j1cλinti j1siis the interference inflow
on the ith from the jth rotor in the ith rotor hub frame.
The self-induced inflow dynamics of the ith rotor are
given by:
˙
λ
λ
λsi=ΩT
T
TWi→HiM
M
M−1
ii −L
L
L−1
ii (T
T
THi→Wiλ
λ
λsi)+(T
T
THi→WiF
F
Fii)
(2)
where M
M
Mii is the apparent mass matrix and L
L
Lii is the static
gain inflow matrix. These matrices are given by:
M
M
M=
8
3π0 0
016
45π0
0 0 16
45π
(3a)
L
L
Lii =
0.5
VTi
0 0
15π
64VTi
tan χi
2
4 cos χi
Vi(1+cos χi)0
0 0 4
Vi(1+cos χ)
(3b)
Additionally, F
F
FT
ii = [CTi−CMi−CLi], where CTi,CMi,
and CLiare the thrust, pitching moment, and rolling mo-
ment coefficients of the ith rotor in the local hub frame.
2
Fig. 1: Two mutually-interacting rotors that are arbitrarily positioned and oriented in space.
The transformation from the ith rotor hub frame to the ith
rotor wind frame is:
k
k
kWi
j
j
jWi
i
i
iWi
=
1 0 0
0 cosψwi−sin ψwi
0 sinψwicos ψwi
| {z }
T
T
THi→Wi
k
k
kHi
j
j
jHi
i
i
izhi
(4)
where:
ψwi=
tan−1vHi
uHi,counterclockwise rotor
tan−1−vHi
uHi,clockwise rotor
(5)
The total inflow and advance ratios of the ith rotor are
given by:
µi=qu2
Hi+v2
Hi
ΩiRi
(6a)
µzi=wHi
ΩiRi
(6b)
VTi=qµ2
i+µzi−λi02(6c)
where uHi,vHi, and wHiare the longitudinal, lateral, and
vertical velocity components of the ith rotor hub. The
skew angle of the ith rotor is:
χi=tan−1µi
λi0−µzi(7)
The mass flow parameter is defined as:
Vi=µ2
i+λi0−µzi2λi0−µzi
VTi
(8)
The interference inflow on the ith from the jth rotor is
given by:
λ
λ
λinti j =ηi jL
L
Li jλ
λ
λsj(9)
where ηi j is a parameter that scales the non-dimensional
quantities of the jth rotor to non-dimensional quantities
of the ith rotor:
ηi j =ΩjRj
ΩiRi
(10)
Note that this inflow scaling parameter is introduced as
a novelty in this paper to generalize CMTSVT to rotors
with different radius and angular speed. The interference
matrix L
L
Li j is given by the multiplication of three matrices:
L
L
Li j =G
G
Gi jR
R
Rγi j L
L
Lj j (11)
where:
R
R
Rγi j =1
VTj1−1.5µ2
j
1 0 −3µj
0 31−1.5µ2
j0
−1.5µj0 3
(12)
G
G
Gi j =
g00i j g01si j g01ci j
g1s0i j g1s1si j g1s1ci j
g1c0i j g1c1sij g1c1ci j
(13)
and where L
L
Lj j is the static inflow gain matrix defined
above. The elements of the G
G
Gi j matrix are given by the
following integrals:
3
g00i j =−1
4π2Z2π−ψwi
−ψwiZRj/Ri
0Z2π−ψwj
−ψwj
Ki jdψjˆridˆridψi(14a)
g01ci j =−1
4π2Z2π−ψwi
−ψwiZRj/Ri
0Z2π−ψwj
−ψwj
Ki j cos ψjdψjˆridˆridψi(14b)
g01si j =−1
4π2Z2π−ψwi
−ψwiZRj/Ri
0Z2π−ψwj
−ψwj
Ki j sin ψjdψjˆridˆridψi(14c)
g1c0i j =−1
π2Z2π−ψwi
−ψwiZRj/Ri
0Z2π−ψwj
−ψwj
Ki jdψjˆr2
icosψidˆridψi(14d)
g1c1ci j =−1
π2Z2π−ψwi
−ψwiZRj/Ri
0Z2π−ψwj
−ψwj
Ki j cos ψjdψjˆr2
icosψidˆridψi(14e)
g1c1si j =−1
π2Z2π−ψwi
−ψwiZRj/Ri
0Z2π−ψwj
−ψwj
Ki j sin ψjdψjˆr2
icosψidˆridψi(14f)
g1s0i j =−1
π2Z2π−ψwi
−ψwiZRj/Ri
0Z2π−ψwj
−ψwj
Ki jdψjˆr2
isinψidˆridψi(14g)
g1s1ci j =−1
π2Z2π−ψwi
−ψwiZRj/Ri
0Z2π−ψwj
−ψwj
Ki j cos ψjdψjˆr2
isinψidˆridψi(14h)
g1s1si j =−1
π2Z2π−ψwi
−ψwiZRj/Ri
0Z2π−ψwj
−ψwj
Ki j sin ψjdψjˆr2
isinψidˆridψi(14i)
where jidentifies the acting rotor whereas irepresents
the receiving rotor. To find the integrand function Kij ,
one must first define the relative position of the rotors.
Assume that the ith rotor hub has the following (dimen-
sional) coordinates expressed in the body frame:
r
r
r→Hi=xii
i
iB+yij
j
jB+zik
k
kB(15)
whereas the jth rotor body-frame (dimensional) coordi-
nates are:
r
r
r→Hj=xji
i
iB+yjj
j
jB+zjk
k
kB(16)
Then, the position of the ith rotor with respect to the jth
rotor is:
r
r
rHj→Hi=r
r
r→Hi−r
r
r→Hj
=T
T
TB→Hj
xi
yi
zi
−
xj
yj
zj
=xi ji
i
iHj+yi j j
j
jHj+zi jk
k
kHj
(17)
where the transformation T
T
TB→Hjwill be defined in the
next chapter. The aerodynamic calculation points of the
ith rotor expressed in the jth rotor hub coordinates are:
xACi
yACi
zACi
=
xi j
yi j
zi j
+T
T
TB→HjT
T
THi→B
| {z }
T
T
THi→Hj
−ricosψi
risinψi
0
(18)
where riis the dimensional radial coordinate 0 ≤
ri≤Ri. The aerodynamic calculation points are non-
dimensionalized with respect to the jth rotor radius, such
that:
ˆxACi
ˆyACi
ˆzACi
=1
Rj
xACi
yACi
zACi
(19)
Then, the interference of the jth rotor on the ith rotor can
be computed according to Heyson (Ref. 10), such that:
Ji j =1−(−ˆxACicosψj+ˆyACisin ψj) + Rcsin χjcos ψj
[Rc+ (cosψj+ˆxACi)sin χj−ˆzACicos χj]Rc+rc
(20)
where:
Rc=h1+ (−ˆxACi)2+ˆy2
ACi+ (−ˆzACi)2(21)
−2(−ˆxACicosψj+ˆyACisinψj)i1/2
(22)
The variable rcis the nondimentional core radius of the
tip vortex, which helps smoothen out large induced ve-
locites just to the right/left of the vortex tube (which are
mathematically correct according to this formulation, but
not necessarily physically accurate). A typical tip vor-
tex core size value is approximately 5% of the rotor ra-
dius (Ref. 11). Thus, in this implementation, rc≈0.05.
Finally, the integrand function Ki j can be found as Ki j =
Ji j k
k
kHj·k
k
kHior, equivalently, T
T
THi→Hj3,3. It is worth not-
ing that the total inflow harmonic coefficients λ
λ
λiare in
fact an output of the system of dynamic inflow equations
rather than states. Because this quantities are typically
used to calculate the thrust coefficient and other variables
before being computed in the inflow calculations, it is de-
sirable to express them as states. This avoids algebraic
loops in the rotor simulation. As such, the dynamics of
the total inflow is expressed as a first-order filter with the
following form:
˙
λ
λ
λi=1
τλ¯
λ
λ
λi−λ
λ
λi(23)
where ¯
λ
λ
λiis the total inflow calculated at each time step
using Eq. (1), and τλis the filter time constant which
4
should be chosen as at least one order of magnitude faster
than the fastest inflow dynamics. Based on this setup, the
inflow dynamics has the following state vector:
x
x
xT
inflow = [λ
λ
λs1··· λ
λ
λsNλ
λ
λ1··· λ
λ
λN](24)
FLIGHT DYNAMICS SIMULATION
MODEL
The aircraft of interest is the Malloy TRV-80 coaxial
quadcopter, which is shown in Fig. 2. This UAS has four
sets of coaxial rotors, each with only revolutions-per-
minute (RPM) degrees of freedom and can carry roughly
80 lbs payload. The TRV-80 has been tested operationally
as a package delivery drone with the Marines. A scaled
version of the TRV-80 with overlapped rotors has under-
gone wind-tunnel testing at the US Army 7 ×10 wind-
tunnel at NASA Ames research center (Ref. 12). Initial
modeling using ART’s FLIGHTLAB modeling tool was
performed by the Army Research Laboratory (ARL), and
the model input data is available. An ongoing effort by
U.S. Army Combat Capabilities Development Command
(CCDC) at NASA Ames is aimed at flight testing the
TRV-80 to obtain frequency response and trim data. This
flight data as well as system identified models (Ref. 13)
will soon be available and be used in future work. The
properties of the TRV-80 are reported in Table 1.
Fig. 2: Malloy TRV-80 coaxial quadcopter.
Table 1: General characteristics of the TRV-80-like
generic multi-rotor model (Ref. 13).
Parameter Value Units
Mass and inertia
Gross weight, W61.7 lb
Roll-axis moment of inertia, Ixx 1.303 sl-ft2
Pitch-axis moment of inertia, Iyy 3.738 sl-ft2
Yaw-axis moment of inertia, Izz 4.370 sl-ft2
Roll/yaw-axes product of inertia, Ixz 0 sl-ft2
Fuselage
Frontal Drag Area, D10.36 ft2
Vertical Drag Area, D22.30 ft2
Sideward Drag Area, D30.86 ft2
Rotors
Number of blades, Nb2 -
Radius, R1.1 ft
Blade chord at root, croot 0.181 ft
Blade chord at tip, ctip 0.095 ft
Geometric pitch, θ019.7 deg
Blade twist, θtw −12.35 deg
Longitudinal location (body axes) ±2.094 ft
Lateral location (body axes) ±1.313 ft
Vertical location (body axes) ±0.088 ft
Flight dynamics modeling is performed using a generic
multi-rotor flight dynamics simulation code developed
at University of Maryland (UMD). This code is imple-
mented in MATLAB®/Simulink and contains a 6-DoF
rigid-body dynamic model of the fuselage, nonlinear
aerodynamic lookup tables for the fuselage, rotor blades,
and empennage (if any), rigid flap rotor blade dynamics,
a three-state Pitt-Peters inflow model (Ref. 14 for each
rotor, and simple RPM dynamics. This software can be
used to model rotorcraft configurations with any number
of rotors/wings. Note that to simulate the TRV-80, the
flapping dynamics was disabled such that the rotors are
assumed to be rigid. The RPM dynamics is modeled as a
first-order filter with a time constant of 0.05 seconds. The
3-state Pitt-Peters inflow model for each rotor is trans-
formed into a 6-state model according to the generalized
CMTSVT theory presented in the section above. A visual
representation of the TRV-80 geometry modeled with this
code is shown in Fig. 3.
The rotorcraft flight dynamics are formulated as a nonlin-
ear time-invariant (NLTI) system in first-order form such
that they are suitable for linearization, order reduction,
and flight control design:
˙
x
x
x=f
f
f(x
x
x,u
u
u)(25a)
y
y
y=g
g
g(x
x
x,u
u
u)(25b)
where x
x
x∈Rnis the state vector, u
u
u∈Rmis the control
input vector, and y
y
y∈Rlis the output vector. The state
vector is:
x
x
xT=x
x
xT
RB x
x
xT
R1··· x
x
xT
R8(26)
where x
x
xRB are the rigid-body states and x
x
xRiare the states
5
Fig. 3: TRV-80 geometry as modeled with the UMD generic multi-rotor flight dynamics simulation code.
of the ith rotor. The rigid-body (or fuselage) states are:
x
x
xT
RB =huvwpqrφ θ ψi(27)
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 state vector of the ith rotor is:
x
x
xT
Ri=λ
λ
λT
siλ
λ
λT
iΩiψi(28)
where Ωiis the angular speed and azimuth angle of the
ith rotor and ψiis the azimuth angle of a reference blade
of the ith rotor. The pilot input vector is:
u
u
uT=δlat δlon δcol δped(29)
where:
δlat and δlon are the lateral and longitudinal stick posi-
tions,
δcol is the collective stick position, and
δped is the pedal position.
The pilot inputs are converted to control effector inputs,
i.e., the rotor commanded angular velocities Ωcmdi, via
the following mixing matrix:
Ωcmd1
Ωcmd2
Ωcmd3
Ωcmd4
Ωcmd5
Ωcmd6
Ωcmd7
Ωcmd8
=
1−1 1 1
1−1 1 −1
−1−1 1 −1
−1−1 1 1
1 1 1 −1
1 1 1 1
−1 1 1 1
−1 1 1 −1
δlat
δlon
δcol
δped
(30)
The rotor numbering convention is shown in Fig. 4.
Fig. 4: Rotor numbering convention and verse of
rotation (blue: clockwise, orange: counterclockwise).
LINEARIZED MODELS
Trim Algorithm
For an NLTI system, like the coupled rigid-body and rotor
dynamics in Eq. (25), an equilibrium condition coincides
with a fixed point in the state space x
x
x∗,u
u
u∗. Equilibria can
be stable, neutrally stable, or unstable. In the case of a
stable system about a particular equilibrium point, equi-
librium can be reached by time marching the dynamics
of the system until convergence to steady state. That is,
if the initial conditions are close enough to that equilib-
rium point. For a system that is either neutrally stable
or unstable, trim solution methods are required. While
the inflow dynamics is generally stable, if it is coupled
with the rigid-body dynamics of a hovering vehicle which
are typically unstable (Ref. 15), then the dynamics of the
coupled system will be unstable. As such, a trim solution
method suitable for unstable equilibria is implemented.
The trim problem can be stated as follows. Given a pre-
scribed function in time ˙
x
x
x∗, the goal is to solve for a sub-
6
set of the state vector ˆ
x
x
x∈Rp, with p≤nand for a subset
of the control input vector ˆ
u
u
u∈Rq, with q≤msubject to:
˙
x
x
x∗=f
f
f(x
x
x∗,u
u
u∗)(31)
. The trim variables for this problem are thus given by
augmenting the subset of the state vector with the control
input and output vectors:
Θ
Θ
ΘT=ˆ
x
x
xTˆ
u
u
uT(32)
which leads to p+qtrim variables. The constraints are
given by the ntrim targets ˙
x
x
x∗. Then, the trim problem to
be solved is given by:
e
e
e(Θ) = ˙
x
x
x∗−f
f
f(ˆ
x
x
x,ˆ
u
u
u) = 0
0
0 (33)
where e
e
e(Θ)is the error vector. It is clear that, to make
the problem square such that a unique solution exists, the
number of trim variables must be equal to that of the
constraints (i.e.,p+q=n). It follows that n−p−q
conditions still need to be specified. Note that if the m
control inputs are given and the corresponding equilib-
rium solution is required, then the problem in consider-
ation becomes a closed system as q=0 and p=n. On
the other hand, in the case where one or more (possibly
all) of the mcontrol inputs are unknown, then each in-
put is used to ensure some desired condition. For typical
aerospace vehicles, the flight dynamics are invariant with
respect to position and heading (Ref. 16) such that the
position and heading can be arbitrarily assigned and re-
moved from the vector of unknowns. Since these vehicles
typically employ control about four axes (i.e., roll, pitch,
yaw, and heave) leading to four control inputs, fixing the
three components of the position (x,y,z) and heading (ψ)
at equilibrium leads to a square problem.
In practice this problem is solved iteratively, in that a can-
didate solution is refined over a series of computational
steps until a convergence criteria is reached. Consider the
candidate solution at iteration kof the algorithm, Θ
Θ
Θk. One
iteration of the algorithm begins with evaluating the cost
function in Eq. (33). If the infinity norm of the cost func-
tion is less than an arbitrary tolerance (i.e.,∥e
e
ek∥∞), then
a solution is found. If not, the algorithm proceeds with
linearizing the system about the candidate solution. The
resulting state-space matrices (see Eq. (37a)) are used to
define the Jacobian matrix of the trimming algorithm:
J
J
Jk= [A
A
AkB
B
Bk](34)
where J
J
Jk∈Rn×(n+m). The mcolumns corresponding
to those states and/or controls that are specified are trun-
cated from the Jacobian matrix, leading to a modified Ja-
cobian matrix ˆ
J
J
Jk∈Rn×n. Next, the solution is updated
using the Newton-Raphson method (Ref. 17):
ˆ
Θ
Θ
Θk+1=Θ
Θ
Θk−ˆ
J
J
J−1
ke
e
ek(35)
The next iteration of the algorithm then proceeds with
this new candidate solution until the stopping criteria is
met.
When applied to the flight dynamics in consideration, the
trim variables are chosen as:
Θ
Θ
ΘT=uvwpqrφ θ x
x
xT
R1··· x
x
xT
R8(36)
Thus, the interactional inflow dynamics is trimmed to-
gether with the rigid-body dynamics.
Model Order Reduction
Once a trim solution is found, the coupled rigid-body and
inflow dynamics are linearized about that trim solution
using perturbation methods (Ref. 18) to yield a linear
time-invariant (LTI) system in first-order form:
˙
x
x
x=A
A
Ax
x
x+B
B
Bu
u
u(37a)
y
y
y=C
C
Cx
x
x+D
D
Du
u
u(37b)
where A
A
A∈Rn×nis the system matrix, B
B
B∈Rn×mis the
control matrix, C
C
C∈Rl×nis the output matrix, and D
D
D∈
Rl×mis the feed-through matrix. Linearization is per-
formed via perturbation methods (Ref. 18). A major chal-
lenge lies in making the linearized models tractable for
control design while still capturing the rotor-on-rotor in-
teractional aerodynamics. Because the rotor (e.g., inflow)
dynamics are typically stable and significantly faster than
that of the rigid-body dynamics, the order of the cou-
pled rigid-body and rotor dynamics can be reduced by
means of singular perturbation theory, and in particular
via residualization (Ref. 19). This removes the need to
measure or estimate higher-order dynamics states asso-
ciated with the rotors and their inflow dynamics while
retaining information on the influence of the residualized
dynamics onto the slower rigid-body dynamics.
Assuming one or more states to have stable dynamics
which are faster than that of the remaining states, the state
vector in Eq. (26) is partitioned into slow and fast com-
ponents:
x
x
xT=x
x
xT
sx
x
xT
f(38)
Then, the system in Eq. (37a) 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(39)
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(40)
where:
ˆ
A
A
A=A
A
As−A
A
AsfA
A
Af−1A
A
Afs (41a)
ˆ
B
B
B=B
B
Bs−A
A
AsfA
A
Af−1B
B
Bf(41b)
Note that A
A
Afmust be invertible. This is guaranteed if Af
is asymptotically stable, i.e., all eigenvalues have their
real part that is strictly negative. The slow states are
chosen as the rigid-body states with the exception of the
7
position and heading states which are truncated as the
rotorcraft dynamics are invariant with respect to these
states (Ref. 16), whereas the fast states are taken as the
rotor states:
x
x
xs=x
x
xRB (42a)
x
x
xT
f=x
x
xT
R1··· x
x
xT
R8(42b)
This way, an 8-state residualized system is obtained that
still accounts for the higher-order dynamics. For the
residualized model to retain information about the influ-
ence of the residualized dynamics on the outputs, con-
sider partitioning the output equations in Eq. (37b) as:
y
y
y=C
C
CsC
C
Cfx
x
xs
x
x
xf+D
D
Du
u
u(43)
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(44)
where:
ˆ
C
C
C=C
C
Cs−C
C
CfA
A
Af−1A
A
Afs (45a)
ˆ
D
D
D=D
D
D−C
C
CfA
A
Af−1B
B
Bf(45b)
These residualized models are used for flight control de-
sign.
FLIGHT CONTROL DESIGN
The flight control architecture chosen for this study is dy-
namic inversion (DI). Application of DI control laws to
rotorcraft can be found in, e.g., Refs. 5, 20–29. A key
aspect of DI is the reliance on model inversion to cancel
the plant dynamics and track a desired reference model.
One convenient feature of DI is that it inverts the plant
model in its feedback linearization loop, which, com-
pared to other more conventional model-following con-
trol strategies such as explicit model following (EMF),
eliminates the need for gain scheduling. However, the
plant model used for feedback linearization still needs to
be scheduled with the flight condition. A generic DI con-
troller as applied to a linear system is shown in Fig. 5.
The key components are a command model (also known
as command filter or reference model) that specifies de-
sired response to pilot commands, a feedback compensa-
tion on the tracking error, and an inner feedback loop that
achieves model inversion (i.e., the feedback linearization
loop).
A multi-loop DI control law largely based on Refs.
5, 20, 24, 26 is designed to enable autonomous flight.
The schematic of the closed-loop multi-rotor dynamics
is shown in Fig. 6. 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 Translational Rate
Command (TRC). The inner loop 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,
Rate Command / Attitude Hold (RCAH) is used for the
yaw axis, and a TRC response is used for the heave axis.
RESULTS
Trim
The generalized CMTSVT model is implemented within
the UMD generic multi-rotor flight dynamics simulation
code for a configuration similar to that of the TRV-80.
Examples of interference matrices (i.e., the matrix G
G
G)
computed based on the TRV-80 geometry are provided
below:
G
G
G12 =
0.3502 0 0
0 0.0752 0
0 0 0.0752
(46a)
G
G
G21 =
0.5899 0 0
0 0.3162 0
0 0 0.3162
(46b)
G
G
G13 =
0 0 0.1053
0 0.0444 0
0 0 0.0444
(46c)
G
G
G14 =
0 0 0
0 0.0403 0
0 0 0.0322
(46d)
The matrix G
G
G12 shows the interference of rotor 2 on rotor
1, whereas the opposite is true for G
G
G21. As expected, the
influence of rotor 2 on rotor 1 is of a lesser entity given
that rotor 2 acts in the wake of rotor 1. Because rotors 1
and 2 are coaxial, only on-axis terms are present in the
G
G
G12 and G
G
G21 matrices. On the other hand, because, for
instance, rotors 1 and 3 do not share the axis of rotation,
the terms in the G
G
G13 matrix are not only on-axis terms. In
fact, the influence of rotor 3 on rotor’s 1 average inflow
is zero as there is no overlap between the rotors, and be-
cause the rotors are on the same plane. The same applies
for rotors 1 and 4.
kint =2√2Tu
Tl3/21+Tu
Tl−3/2
=1.201 (47)
where Tuand Tlare the thrusts of the upper and lower ro-
tor, respectively. This value is similar to that reported by
Leishman (Ref. 30), i.e.,kint =1.266 for coaxial rotors
operated at equal rotor torque and with the lower rotor
operating in the far wake wake of the upper rotor. Differ-
ences in these results may be attributed to the fact that, in
the TRV-80 configuration, the lower rotor does not act in
the fully-developed wake of the upper rotor. Moreover,
the CMTSVT model used does not model wake decay
and/or contraction. Wake decay and contraction effects
could possibly be implemented based on Ref. 31.
8
Fig. 5: Block diagram for dynamic inversion as applied to a linear system.
Fig. 6: Schematic of the closed-loop multi-rotor dynamics.
Table 2: Performance of one pair of counter-rotating
coaxial rotors trimmed at hover.
Variable Units Rotor 1 Rotor 2
(Upper) (Lower)
λ0- 0.082 0.100
λ1s- 0 0
λ1c- 0.0 0.0
λS0- 0.056 0.035
λS1s- 0 0
λS1c- 0 0
Tlb 8.7 6.7
CT- 0.0092 0.0071
Php 0.39 0.38
CP- 0.00075 0.00072
Ωrad/s 261.3 261.3
Open-Loop Dynamics
The flight dynamics trimmed at hover are linearized
based on perturbation methods (Ref. 18), and subse-
quently residualized, to assess the effect of the interac-
tional dynamics on the rigid-body dynamics. Figure 7
shows the eigenvalues of the residualized dynamics with
and without rotor-on-rotor interactions. The latter case is
achieved by conveniently setting the interference matri-
ces Gto zero. Numerical values for the eigenvalues, as
well as the modal properties of the corresponding modes,
are reported quantitatively in Table 3. These results
show that rotor-on-rotor interactions significantly affect
the hovering cubic (Ref. 15) eigenvalues. More specifi-
-3 -2 -1 0 1
Real
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Imag
Interactions Off
Interactions On
Roll
Oscillation
Roll
Subsidence
Pitch
Subsidence
Pitch
Oscillation
Heave
Subsidence
Yaw
Subsidence
Fig. 7: Comparison between the residualized dynamics
eigenvalues at hover with and without rotor-on-rotor
interactions.
cally, roll and pitch oscillation modes result in lower nat-
ural frequencies and the same applies to the roll and pitch
subsidence modes. On the other hand, the vertical (heave)
and directional (yaw) dynamics appear to be largely un-
affected. Thus, it is concluded that, for this particular
coaxial quadcopter configuration, the rotor-on-rotor in-
teractional aerodynamics results in roll and pitch oscil-
lation modes with a lower natural frequencies, and sub-
sidence modes at lower frequencies. On the other hand,
the heave and yaw modes are not significantly affected.
To better explain these difference in stability properties,
the stability derivatives of the residualized dynamics are
9
Table 3: Hover modal characteristics with and without rotor-on-rotor interactions.
Mode Interactional Eigenvalues Natural Damping
Aerodynamics [rad/s] Frequency, Ratio, ζ
ωn[rad/s]
Roll Subsidence Off −2.848 - 1
On −2.270 - 1
Pitch Subsidence Off −2.034 - 1
On −1.451 - 1
Roll Oscillation Off 0.580 ±1.735i1.829 -
On 0.292 ±1.122i1.159 -
Pitch Oscillation Off 0.394 ±1.216i1.278 -
On 0.058 ±0.456i0.460 -
Heave Subsidence Off −0.470 - 1
On −0.429 - 1
Yaw Subsidence Off −0.078 - 1
On −0.089 - 1
Table 4: Lateral stability and control derivatives.
Parameter Interactional Value Units
Aerodynamics
Yv
Off −0.025 1/s
On −0.033
Lv
Off −0.296 rad/(ft-s)
On −0.077
Lp
Off −1.662 1/s
On −1.728
Lδlat
Off 0.0011 rad/(s2-%)
On 0.0010
investigated. Consider the decoupled, reduced-order, lat-
eral dynamics:
˙v
˙p
˙
φ
=
Yv0g
LvLp0
0 1 0
v
p
φ
+
0
Lδlat
0
δlat (48)
Note that, at hover and for a symmetric configuration like
the TRV-80, the lateral dynamics is decoupled from the
directional dynamics. The stability and control deriva-
tives corresponding to the lateral dynamics are reported
in Table 4. Rotor-on-rotor interactions result in a roll rate
response due to lateral velocity (i.e.,Lv) that is signifi-
cantly lower in magnitude and in a pitch damping (i.e.,
Lp) that is also significantly lower in magnitude. The
lower magnitude of Lpexplains the lower frequency of
the roll subsidence mode when compared to the case of
no rotor-on-rotor interactions, whereas the combination
of Lvand Lpresults in a lower-frequency roll oscillation
mode.
Now consider the the decoupled, reduced-order, longitu-
dinal dynamics system matrix:
˙u
˙q
˙
θ
=
Xu0−g
MuMq0
0 1 0
u
q
θ
+
0
Mδlon
0
δlon (49)
Table 5: Longitudinal stability and control derivatives.
Parameter Interactional Value Units
Aerodynamics
Xu
Off −0.025 1/s
On −0.031
Mu
Off 0.103 rad/(ft-s)
On 0.026
Mq
Off −0.470 1/s
On −0.429
Mδlon
Off −0.0006 rad/(s2-%)
On −0.0005
Note that, at hover and for a symmetric configuration
like the TRV-80, the longitudinal dynamics is decoupled
from the vertical dynamics. The stability and control
derivatives corresponding to the longitudinal dynamics
are reported in Table 5. Similarly to the lateral dynam-
ics case, rotor-on-rotor interactions result in a pitch rate
response due to longitudinal velocity (i.e.,Mu) that is sig-
nificantly lower in magnitude and in a pitch damping (i.e.,
Mq) that is also significantly lower in magnitude. The
lower magnitude of Mqexplains the lower frequency of
the pitch subsidence mode when compared to the case of
no rotor-on-rotor interactions, whereas the combination
of Muand Mqresults in a lower-frequency pitch oscil-
lation mode. It is concluded that rotor-on-rotor interac-
tions result in a lower magnitude roll/pitch rate response
to lateral/longitudinal velocity and in a lower pitch/roll
damping. In turn, the combination of these effects yield
roll and pitch subsidence modes at lower frequency, and
roll and pitch oscillation modes with a lower natural fre-
quency.
To gain more insight on how the rotor-on-rotor interac-
tional aerodynamics affects the hover flight dynamics,
on-axis frequency responses are computed based on the
full-order dynamics with and without interactions. Figure
8 shows the roll and pitch on-axis frequency responses
10
with and without rotor-on-rotor interactions. Both roll
and pitch frequency responses show higher magnitude
and phase at low to medium frequencies of interest to
flight dynamics. An approximate categorization of these
frequencies is as follows: ω⪅1 rad/s corresponds to low
frequencies, 1 ⪅ω⪅10 coincides with medium frequen-
cies, and ω⪆10 rad/s signifies high frequencies. Magni-
tude differences are apparent up to approximately 1 rad/s,
whereas phase differences are accentuated for the lon-
gitudinal stick to roll rate response up to about 4 rad/s.
This is somewhat consistent with the results shown in
Ref. 7, where the interactional inflow dynamics affects
frequency responses mostly in the low to medium range.
10-1 100101
-120
-100
-80
-60
Mag [dB]
Interactions Off
Interactions On
10-1 100101
Frequency, [rad/s]
-200
-150
-100
-50
Phase [deg]
(a) p
δlat
(s).
10-1 100101
-120
-100
-80
-60
Mag [dB]
Interactions Off
Interactions On
10-1 100101
Frequency, [rad/s]
-400
-200
0
Phase [deg]
(b) q
δlon
(s).
Fig. 8: Roll and pitch on-axis frequency responses with
and without rotor-on-rotor interactions.
Figure 9 shows the yaw and heave on-axis frequency re-
sponses with and without rotor-on-rotor interactions. As
for the eigenvalues, both pedals to yaw rate and collective
to vertical speed response are not significantly affected
by rotor-on-rotor interactions. It is concluded that, for a
coaxial quadcopter configuration, rotor-on-rotor interac-
tions result in higher gain and phase of the on-axis roll
and pitch frequency responses at low to medium ranges
of the frequencies of interest to flight dynamics (i.e.,
0.1≤ω≤30 rad/s) (Ref. 32). On the other hand, the
heave and yaw on-axis frequency responses are largely
unaffected.
10-1 100101
-120
-100
-80
Mag [dB]
Interactions Off
Interactions On
10-1 100101
Frequency, [rad/s]
-350
-300
-250
Phase [deg]
(a) r
δped
(s).
10-1 100101
-100
-80
-60
Mag [dB]
Interactions Off
Interactions On
10-1 100101
Frequency, [rad/s]
-300
-250
-200
Phase [deg]
(b) w
δcol
(s).
Fig. 9: Yaw and heave on-axis frequency responses with
and without rotor-on-rotor interactions.
Closed-Loop Dynamics
The controller performance is assessed in both time and
frequency domains for controllers based on the residu-
alized dynamics with and without rotor-on-rotor interac-
tions. Time-domain analysis involves closed-loop simu-
lation time histories whereas frequency-domain analysis
focuses on: (i) stability margins, (ii) disturbance rejection
bandwidth and peak (DRB and DRP) (Ref. 33), and (iii)
closed-loop response.
Batch simulations are run to compare the closed-loop
performance of the flight control laws design based on
reduced-order models not accounting and accounting for
rotor-on-rotor interactions. The simulation consists in a
5 ft/s doublet in the lateral speed. These simulations are
11
shown in Fig. 10. Figure 10a shows a comparison be-
tween the velocities in the heading frame. It appears that
the on-axis responses almost overlap while the off-axis
responses are small in magnitude and do not present par-
ticular differences in tracking performance. Figure 10b
shows the closed-loop attitude response. Again, the on-
axis responses nearly overlap whereas the off-axis re-
sponses are small in magnitude and do not differ signifi-
cantly. From this time-domain analysis it appears that the
closed-loop response is similar for DI control laws based
on linearized models that do / do not account for the in-
teractional aerodynamics.
0246810
0
0.05
0.1
Vx [ft/s]
Interactions Off
Interactions On
0246810
-5
0
5
Vy [ft/s]
0246810
Time [s]
-2
-1
0
Vz [ft/s]
(a) Heading frame velocities.
0246810
-20
0
20
40
[deg]
Interactions Off
Interactions On
0246810
-0.5
0
0.5
[deg]
0246810
time [s]
-0.2
0
0.2
0.4
[deg]
(b) Euler angles.
Fig. 10: Closed-loop response for flight control laws
accounting and not accounting for rotor-on-rotor
interactions.
Next, frequency-domain specifications are analyzed. The
first specifications addressed are the gain and phase mar-
gins obtained from the broken loop response (Fig. 11).
Broken loop responses are obtained for both the inner at-
titude (roll attitude) and outer velocity (lateral velocity)
loops, as shown in Figs. 11a and 11b, respectively. Both
loops have similar crossover frequencies gain and phase
margins, as reported quantitatively in Table 6. The in-
ner loop gain margins are both well within typical design
requirements, i.e., GM >=6 dB whereas the phase mar-
gins appear to be significantly lower than typical design
requirements, i.e., PM >=45 deg.
10-1 100101102
-50
0
50
Magnitude [dB]
Interactions Off
Interactions On
10-1 100101102
[rad/s]
-400
-200
0
Phase [deg]
(a) BLφ
10-1 100101102
-100
-50
0
50
Magnitude [dB]
Interactions Off
Interactions On
10-1 100101102
[rad/s]
-400
-200
0
Phase [deg]
(b) BLVy
Fig. 11: Inner attitude and outer velocity broken loop
responses.
Figure 12 shows roll attitude (Fig. 12a) and lateral veloc-
ity (Fig. (12b) disturbance responses for both controllers.
Table 6 lists the attitude disturbance rejection bandwidth
(DRB) and peak (DRP) (Ref. 33) values for the two con-
trollers. Both controller show similar DRB and DRP.
Finally, Fig. 13 shows the closed-loop roll attitude (Fig.
13a) and lateral velocity (Fig. 13b) responses of the two
controllers as compared to their common command mod-
els. The closed-loop response is very similar across con-
trollers for both the inner and outer loops. The inner loop
design has good model tracking performance up to ap-
proximately 6 rad/s whereas the outer-loop tracking per-
formance is poorer, especially for frequencies exceed-
ing 2 rad/s. In conclusion, no significant improvements
were detected by designing a model-following control
law based on reduced-order models that account on rotor-
on-rotor interactions. While rotor-on-rotor interactions
result in significant differences in the flight dynamics at
low to medium frequencies, these frequencies might not
affect the control synthesis.
12
Table 6: Hover modal characteristics with and without rotor-on-rotor interactions.
Specification Loop Interactional Value Units
Aerodynamics
Gain Margin (GM)
GMφInner Off 17.55 dB
On 16.52
GMVyOuter Off 4.90 dB
On 5.83
Phase Margin (GM)
PMφInner Off 25.16 deg
On 24.41
PMVyOuter Off 42.50 deg
On 43.82
Disturbance Rejection Bandwidth (DRB)
DRBφInner Off 4.61 rad/s
On 4.80
DRBVyOuter Off 6.93 rad/s
On 6.89
Disturbance Rejection Peak (DRP)
DRPφInner Off 2.36 dB
On 2.34
DRPVyOuter Off 7.59 dB
On 6.51
CONCLUSIONS
The paper begins with a detailed mathemetical descrip-
tion of the dynamic modeling of rotor-on-rotor interac-
tions. Next, the simulation model used in this study is
presented. Upon trimming the flight dynamics at hover,
linearized models of said dynamics are compared to non-
linear simulations, and with linear and nonlinear adopting
dynamic inflow models that do not account for rotor-on-
rotor inflow interference effects. Model-order reduction
methods are investigated to guide the development of lin-
earized models that are tractable for flight control design
while still predicting the effect of rotor-on-rotor interac-
tions on the vehicle flight dynamics. Model-following
flight control laws based on these reduced-order lin-
earized models that account for the rotor-on-rotor inter-
actions are developed and tested against similar control
laws based on linearized models that do not account for
rotor-on-rotor interactions. Based on this work, the fol-
lowing conclusions can be reached:
1. Thus, it is concluded that, for this particular coax-
ial quadcopter configuration, the rotor-on-rotor in-
teractional aerodynamics results in roll and pitch
oscillation modes with a lower natural frequencies,
and subsidence modes at lower frequencies. On the
other hand, the heave and yaw modes are not signif-
icantly affected.
2. It is concluded that rotor-on-rotor interactions result
in a lower magnitude roll/pitch rate response to lat-
eral/longitudinal velocity and in a lower pitch/roll
damping. In turn, the combination of these effects
yield roll and pitch subsidence modes at lower fre-
quency, and roll and pitch oscillation modes with a
lower natural frequency.
3. It is concluded that, for a coaxial quadcopter config-
uration, rotor-on-rotor interactions result in higher
gain and phase of the on-axis roll and pitch fre-
quency responses at low to medium ranges of the
frequencies of interest to flight dynamics (i.e., 0.3≤
ω≤30 rad/s) (Ref. 32). On the other hand, the
heave and yaw on-axis frequency responses are
largely unaffected.
4. No significant improvements were detected by de-
signing model-following control laws based on
reduced-order models that account on rotor-on-rotor
interactions. While rotor-on-rotor interactions result
in significant differences in the flight dynamics at
low to medium frequencies, these frequencies might
not affect control design.
Future work will involve the validation of the simulation
model against TRV-80 flight test data.
ACKNOWLEDGMENTS
The authors extend our sincere gratitude to the U.S. Army
Combat Capabilities Development Command at NASA
Ames for providing the TRV-80 properties necessary to
configure the flight dynamics simulation model.
13
10-1 100101102
[rad/s]
-60
-50
-40
-30
-20
-10
0
10
Magnitude [dB]
Interactions Off
Interactions On
(a) φ′/φd
10-1 100101102
[rad/s]
-60
-50
-40
-30
-20
-10
0
10
Magnitude [dB]
Interactions Off
Interactions On
(b) V′
y/Vyd
Fig. 12: Inner attitude and outer velocity disturbance
response.
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10-1 100101102
-40
-20
0
Magnitude [dB]
Interactions Off
Interactions On
Command Model
10-1 100101102
[rad/s]
-400
-200
0
Phase [deg]
(a) φ/φcmd
10-1 100101102
-100
-50
0
Magnitude [dB]
Interactions Off
Interactions On
Command Model
10-1 100101102
[rad/s]
-400
-200
0
Phase [deg]
(b) Vy/Vycmd
Fig. 13: Inner attitude and outer velocity closed-loop
response.
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