Conference PaperPDF Available

Downwash-aware Control Allocation for Over-actuated UAV Platforms

Authors:
  • Beijing Institute for General Artificial Intelligence

Abstract and Figures

Tracking position and orientation independently affords more agile maneuver for over-actuated multirotor Unmanned Aerial Vehicles (UAVs) while introducing unde- sired downwash effects; downwash flows generated by thrust generators may counteract others due to close proximity, which significantly threatens the stability of the platform. The complexity of modeling aerodynamic airflow challenges control algorithms from properly compensating for such a side effect. Leveraging the input redundancies in over-actuated UAVs, we tackle this issue with a novel control allocation framework that considers downwash effects and explores the entire allocation space for an optimal solution. This optimal solution avoids downwash effects while providing high thrust efficiency within the hardware constraints. To the best of our knowledge, ours is the first formal derivation to investigate the downwash effects on over-actuated UAVs. We verify our framework on different hardware configurations in both simulation and experiment
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Downwash-aware Control Allocation for Over-actuated UAV Platforms
Yao Su1˚, Chi Chu1,2˚, Meng Wang1, Jiarui Li1,3, Liu Yang1,4, Yixin Zhu5,6, Hangxin Liu1:
Project Website: https://marvel-uav.github.io
Abstract Tracking position and orientation independently
affords more agile maneuver for over-actuated multirotor
Unmanned Aerial Vehicles (UAVs) while introducing unde-
sired downwash effects; downwash flows generated by thrust
generators may counteract others due to close proximity,
which significantly threatens the stability of the platform. The
complexity of modeling aerodynamic airflow challenges control
algorithms from properly compensating for such a side effect.
Leveraging the input redundancies in over-actuated UAVs, we
tackle this issue with a novel control allocation framework that
considers downwash effects and explores the entire allocation
space for an optimal solution. This optimal solution avoids
downwash effects while providing high thrust efficiency within
the hardware constraints. To the best of our knowledge, ours is
the first formal derivation to investigate the downwash effects
on over-actuated UAVs. We verify our framework on different
hardware configurations in both simulation and experiment.
I. INTRO DUC TIO N
Over-actuated UAV platforms with independent position
and orientation tracking provide more agile maneuver com-
pared with traditional multirotors. A straightforward realiza-
tion is to tilt propellers [1–4] and generate thrust forces in
non-collinear directions. As a result, many platforms employ
actively tiltable thrust generators [5–7], achieving higher
thrust efficiency and enabling omnidirectional flights [3, 7].
Adopting tiltable thrust generators unfortunately also in-
troduces a common side effect—the downwash effect [8],
which has been rarely studied in the context of over-actuated
UAVs. This effect occurs when the airflow generated by
one thrust generator/propeller passes through and interacts
with the other(s), resulting in deteriorated trajectory track-
ing performance and lower trust efficiency; see Fig. 1 for
an illustration. In the literature, the downwash effects are
primarily treated by compensation [9–13] or as disturbances
to be slowly attenuated by adding integrators into trajectory
tracking controller [7, 14]. However, the former approach
needs numerous experimental data to learn the platform-
specific compensator, which cannot be generalized to other
platforms. The latter solution is slow in response and hence
has undesirable transitional behavior (e.g., obvious drop in
the flow direction). Critically, both approaches only handle
the downwash effect after it occurs and are inefficient in
terms of energy, requiring extra thrusts to compensate.
In this paper, we tackle the downwash effects from a novel
control allocation perspective for over-actuated UAVs with
actively tiltable thrust generators. Due to input redundancy,
there exists an infinite number of solutions to allocate desired
˚Y. Su and C. Chu contributed equally. :Corresponding author. 1Beijing In-
stitute for General Artificial Intelligence (BIGAI). Emails: {suyao, chuchi,
wangmeng, lijiarui, yangliu, liuhx}@bigai.ai. 2Department
of Automation, Tsinghua University. 3College of Engineering, Peking University.
4Academy of Arts & Design, Tsinghua University. 5Institute for Artificial In-
telligence, Peking University. Email: yixin.zhu@pku.edu.cn. 6School of
Artificial Intelligence, Peking University.
(a) Conventional control allocation framework
(b) Proposed downwash-aware control allocation framework
(c) Config. 1 (d) Config. 2 (e) Trust efficiency over the flight
Fig. 1: Comparison between the proposed control allocation
framework with conventional ones when tracking the reference
trajectory indicated by the light grey. (a) Conventional control
allocation framework fails to track stably as downwash effects ap-
pear twice, highlighted in green and blue. Exemplar configurations
solved by the conventional control allocation framework may lead
to (c) two and (d) one pair of downwash effects, where arrows
and cylinders stand for the thrust forces and downwash flows,
respectively. (b) The proposed framework avoids downwash effects
and thus maintains both stable tracking performance and high trust
efficiency over the challenging flight. (e) We further compare the
thrust efficiency of ideal (the reference trajectory), conventional,
and our proposed allocation frameworks.
force and torque commands to the low-level commands of
thrust generators. This observation makes it possible to find
a proper allocation such that no air flows would counteract
with other thrust generators through a flight, thus reducing
or even eliminating downwash effects beforehand.
We first incorporate the aerodynamics model for down-
wash effect analysis and investigate the relationship between
downwash effect avoidance and thrust efficiency. Next, we
extend our nullspace-based control allocation framework [14]
by adding downwash avoidance constraints and a thrust
efficiency index in the objective function. In simulation,
we verify the proposed downwash-aware control allocation
framework on different over-actuated UAV platforms. In
experiment, we build physical platforms that combine com-
mercial quadcopters with passive gimbal joints as 3-Degree
of Freedom (DoF) thrust generators and verify the proposed
framework. Collectively, we demonstrate that the proposed
framework can fully explore the entire allocation space and
find the optimal allocation solution that avoids downwash
effects and maintains a high thrust efficiency.
2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS 2022)
October 23-27, 2022, Kyoto, Japan
978-1-6654-7927-1/22/$31.00 ©2022 IEEE 10478
A. Related Work
Downwash effects have recently drawn an increas-
ing attention, primarily on computational models of two
UAVs [15–17] to achieve better motion cooperation [18, 19].
Downwash effects for multi-UAV systems are more chal-
lenging to handle. The most straightforward solution is to
keep enough safety distance among UAVs to avoid the
interference introduced by downwash effects [20]. Learning-
based method has also been proposed to compensate for
the downwash effects among multirobot swarm [21,22].
Different from the above work in multi-agent scenarios, we
study over-actuated UAV platforms, wherein several thrust
generators are physically connected to a common frame. By
developing a centralized control allocation framework, our
framework avoids the downwash effects by exploiting input
redundancy when generating low-level control commands of
thrust generators.
Commanding each actuator given the desired total wrench
of the platform, the control allocation of over-actuated UAV
platforms is a constrained nonlinear optimization problem
and is generally difficult to solve with high efficiency.
Prior work leverages gradient-descent [23], force decompo-
sition [24], iterative approach [25], separation method [26],
and linear approximation [27] to reduce the computational
complexity. However, none can incorporate input constraints
while providing exact solutions with satisfactory efficiency.
This limitation was first solved by Su et al. [14], who devised
anullspace-based control allocation framework; henceforth,
we referred to this framework as the conventional allocation
framework. This paper extends this framework by incorpo-
rating a downwash effect avoidance constraint and adding a
thrust efficiency index to the objective function. As a result,
various UAV platforms with 3-DoF thrust generators [1–
4, 7, 28] can achieve any arbitrary attitude without downwash
effects while maintaining high thrust efficiency along the
entire possible configuration space.
B. Overview
We organize the remainder of the paper as follows. Sec-
tion II presents the dynamics model of the UAV system with
downwash effect modeling. We analyze downwash effects
and study the relation between downwash effect avoidance
and thrust efficiency in Section III. Section IV describes the
hierarchical control structure and the proposed downwash-
aware allocation framework. Section V and Section VI show
the simulation and experiment results with comprehensive
evaluations. Finally, we conclude the paper in Section VII.
II. PLATF ORM MO DEL W ITH AE RODYNAMICS
The over-actuated UAV system discussed in this paper
adopts regular quadcopters with 2-DoF passive gimbal mech-
anism, serving as 3-DoF thrust generators [7]. This system
has demonstrated various configurations depending on the
number of thrust generators and mainframe design, and
its dynamics is mathematically equivalent to some seminal
platforms [2–4, 7].
Thrust Generator j
Passive Gimbal
Mechanism
Main
Frame
didi,j
dj
αj
βj
yj
zj
xj
oi,j
Thrust Generator i zw
yw
xw
zByB
xB
Downwash Flow
Proj(i,j)
Fig. 2: Coordinate systems of the over-actuated UAV platform.
Regular quadcopters are connected to the mainframe by 2-DoF
passive gimbal mechanism, serving as 3-DoF thrust generators.
Each quadcopter generates downwash flow in thrust’s opposite
direction.
A. System Frames and Configuration
Fig. 2 outlines the system frames and configurations.
Let FWdenote the world coordinate frame and attach the
platform frame FBto the geometric center of the UAV
platform. We define the central position of the main frame as
ξ
ξ
ξ rx, y, zsT, the attitude in the roll-pitch-yaw convention
as η
η
η rφ, θ, ψsT, and the platform angular velocity in FB
as ν
ν
ν rp, q, rsT. Actuator frames Fis are attached to the
geometric center of the ith 3-DoF thrust generator.
B. Platform Dynamics
The dynamics model of this over-actuated UAV platform
can be described as in Yu et al. [7].
«mW:
ξ
:
ξ
:
ξ
BJ
J
JB9ν
9ν
9νW
BR
R
R0
0I3u
u
u`mgˆz
ˆz
ˆz
Bτ
τ
τg`extu
u
u, (1)
where the translational dynamics are expressed in the world
frame FW, whereas the rotational dynamics are described in
body-fix frame FB.mand J
J
Jare the total mass and inertia
matrix of the platform, respectively.
:
ξ
ξ
ξand 9
ν
ν
νare the linear
and angular acceleration of the central frame, respectively. g
is the acceleration due to gravity, Bτ
τ
τgis the gravity torque
due to the displacement of its center of mass (CoM) from
the geometric center [3], ˆ
z
z
z r0,0,1sT, and
u
u
uřN
i1
B
iR
R
R Tiˆz
ˆz
ˆz
řN
i1pd
d
diˆB
iR
R
R Tiˆz
ˆz
ˆzqJ
J
Jξpα
α
α, β
β
βq
J
J
Jνpα
α
α, β
β
βqT
T
T , (2)
where Ti,αi, and βidenote the magnitude of thrust, tilting,
and twisting angles of the ith thrust generator. Nis the
number of thrust generators, d
d
dithe distance vector from FB’s
center to each Fi, and extu
u
uthe external force/torque input,
assumed to be caused by downwash effects.
C. Downwash Effect Modeling
As elaborated by Khan et al. [15], for the zone of
flow establishment (ZFE), the velocity field of a quadcopter
follows a Gaussian distribution,
Vpz, rq VZFE,max pzqe´1
2´r´Rm0
0.5Rm0`0.075pz´z0´R0q{Kvisc ¯2
,(3)
10479
with
VZFE,maxpzq V0rc1´c2Kvisc pz´z0q{R0s,(4)
where zand rare the vertical and radial separations, re-
spectively. Kvisc is the viscosity constant. z0,R0, and V0are
the position, contracted radius, and induced velocity of the
efflux plane, respectively. Rm0is the radial location of the
maximum velocity at each cross-section. c1and c2are two
parameters, which can be experimentally determined.
With the model introduced in Jain et al. [17], the thrust
change caused by oncoming flows for every propeller is
estimated:
ti,j ´bv
N
ÿ
k1
Vpzi,j,k, ri,j,k qti,j ,@j1,¨ ¨ ¨ ,4(5)
where ti,j is the thrust generated by the jth propeller of ith
quadcopter module, defined by:
ti,j KTω2
i,j ,(6)
where ωi,j is the rotational speed, zi,j,k and ri,j,k are the
vertical and radial separations between ith quadcopter’s jth
propeller and kth quadcopter’s downwash flow, and bvis the
thrust decay coefficient, obtained experimentally.
We calculate the ith quadcopter’s thrust and torque distur-
bance caused by the downwash effects as in Ruan et al. [29]:
Ti
Mi»
1 1 1 1
b´b´b b
´b´b b b
´cτcτ´cτcτ
»
ti,1
ti,2
ti,3
ti,4
,(7)
where Miaffects the low-level attitude control of ith
quadcopter. Mi rMx
i, M y
i, M z
isTare the torque outputs
in Fi.bis a constant defined as ba{?2, where ais
the distance of each propeller to the quadcopter center. cτ
is a constant defined as cτKτ{KT, where Kτis the
propeller drag constant, and KTthe standard propeller thrust
constant. Timainly influences the high-level control as
external force, and we can have
extu
u
uW
BR
R
RpřN
i1
B
iR
R
RTiˆz
ˆz
ˆzq
řN
i1pd
d
diˆB
iR
R
RTiˆz
ˆz
ˆzq.(8)
Section VI adopts this downwash effect model for simulation
with the parameters acquired from experimental data.
III. DOWN WAS H EFFE CT ANALYSI S
A. Downwash Constraint Derivation
As shown in Fig. 2, the radial distance between ith
quadcopter’s downwash flow and jth quadcopter’s center is
defined as Oi,j , which be calculated by
Oi,j b}di,j }2´ }projpi, jq}2,(9)
d
d
di,j d
d
dj´d
d
di,(10)
projpi, jq dotpd
d
di,j ,B
iR
R
Rˆz
ˆz
ˆzq,(11)
where dot refers to the dot product of two vectors. By having
B
iR
R
R,Oi,j is a function of αiand βi. If we build a vector
O
O
Opα, βq rO2
1,2;...;O2
N,N ´1s P RNpN´11by stacking
O2
i,j , we can calculate a minimum distance vector O
O
Omin to
Algorithm 1: Downwash Constraint Calculation
Data: di,B
iR
R
R, N, omin constant
Result: O
O
Omin
iÐ1, j Ð2, k Ð0;
O
O
Omin ÐzerospNpN´1q,1q;
for i1¨ ¨ ¨ Ndo
for j1¨ ¨ ¨ Ndo
if ijthen
kÐk`1;
di,j Ðdj´di;
projpi, jq Ð dotpdi,j ,B
iR
R
Rˆz
ˆz
ˆzq;
if projpi, jq ď 0then
O
O
Ominpkq Ð 0
else
O
O
Ominpkq Ð o2
min
end
end
end
end
constraint O
O
O. As a result, the downwash effect avoidance can
be achieved by requiring
O
O
Opα, βq ě O
O
Omin.(12)
Of note, as shown in Algorithm 1, we need only this
constraint when the downwash flows go through other quad-
copters in the positive direction. As this inequality constraint
is highly nonlinear, we approximately include this constraint
into the nullspace-based allocation framework by first-order
linearization, to be detailed in Section IV-B.
B. Downwash Effect Avoidance and Thrust Efficiency
The “thrust efficiency index” was defined by Ryll et
al. [30] to quantify wasted internal forces in over-actuated
multirotor systems. Formally, it is defined as
ηf}řN
i1
B
iR
R
R Tiˆz
ˆz
ˆz}
řN
i1Ti
}J
J
Jξpα
α
α, β
β
βqT
T
T}
řN
i1Ti
}u
u
up1 : 3,1q}
řN
i1Ti
, ηfP r0,1s
(13)
where ηfis a configuration-dependent ratio between the sum
of vectored thrusts and the sum of total thrust magnitudes.
We study the relation between downwash avoidance and
thrust efficiency for three different over-actuated UAV plat-
forms with four, five, and six 3-DoF thrust generators; Fig. 3
summarizes the results. When the platforms fly vertically
(see Figs. 3a, 3d and 3g), downwash effects still appear as
most prior allocation frameworks [7, 14] if we only try to
maintain maximum thrust efficiency (ηf1). By exploring
the entire configuration space, other feasible configurations
might both avoid downwash effects and maintain high thrust
efficiency (see Figs. 3c, 3f and 3i). This finding motivates us
to propose a new allocation framework that efficiently finds
such a configuration for the over-actuated UAV platforms.
In Eq. (13), the numerator of ηfis provided by wrench
command u
u
uof tracking controller, which can be treated as
a constant value in allocation. To include thrust efficiency
index into the objective function of the nullspace-based allo-
cation framework, we choose to minimize the denominator of
Eq. (13) (řN
i1Ti); please refer to Section IV-B for details.
10480
(a) Four: Config. 1 (b) Four: Config. 2 (c) Four: Config. 3
(d) Five: Config. 1 (e) Five: Config. 2 (f) Five: Config. 3
(g) Six: Config. 1 (h) Six: Config. 2 (i) Six: Config. 3
Fig. 3: Thrust efficiency and downwash effect avoidance for
different over-actuated UAV platforms. Infinite number of thrust
force configurations can generate the same required wrench com-
mand with different thrust efficiencies. For each platform, three
configurations are provided as examples. Four, ve, six refer to the
platform with 4, 5, or 6 3-DoF thrust generators, respectively. Same
notations are applied for the rest of this paper.
IV. DOW NWAS H-AWARE CO NT RO LLE R DES I GN
The overall controller has a hierarchical architecture,
shown in Fig. 4. The high-level trajectory tracking controller
(see Fig. 4a) (i) calculates the desired force/torque command
(6-DoF wrench command) for the entire platform, and (ii)
allocates the force/torque command to tilting angle αi, twist-
ing angle βi, and thrust Tiof each 3-DoF thrust generator.
The low-level controller (see Fig. 4b) of each quadcopter (i)
regulates the individual attitude to the desired values and (ii)
provides the required thrust force.
A. High-level Control
Without downwash effects, the dynamics equation (i.e.,
Eq. (1)) can be rewritten following Su et al. [31]
«W:
ξ
:
ξ
:
ξ
B9ν
9ν
9ν1
m
W
BR
R
R0
0BJ
J
J´1u
u
u`gˆz
ˆz
ˆz
BJ
J
J´1Bτ
τ
τg.(14)
We design the feedback-linearization controller as
u
u
udmW
BR
R
RT0
0
0
0
0
0BJ
J
Jˆ„u
u
uξ
u
u
uν´gˆz
ˆz
ˆz
Bτ
τ
τg˙,(15)
where the superscript dindicates the desired values. Our
above controller design transfers platform dynamics ex-
pressed by Eq. (14) into a simple double integrator [32],
«W:
ξ
:
ξ
:
ξ
B9
ν
9
ν
9
νu
u
uξ
u
u
uν.(16)
Position
Controller
Attitude
Controller
Downwash
Aware
Control
Allocation
Quadcopter
Plants
Trajectory
Position
Feedback
Attitude
Feedback
Feedback
Linearization
Controller
(a) High-level platform trajectory tracking controller (100 Hz)
Quadcopter
PID Onboard
Controller
PWM
Mapper Quadcopter 

Propeller
Velocity
Mapper
(b) Low-level 3-DoF thrust generator controller (500 Hz)
Fig. 4: Hierarchical control architecture. (a) The high-level
position and attitude tracking controller gives desired 6-DoF wrench
command u
u
udto the downwash-aware control allocation through
feedback linearization. u
u
udis then allocated as the desired thrusts and
joint angles for each 3-DoF thrust generator to maintain high thrust
efficiency and avoid downwash effects. (b) In low-level control,
each quadcopter module regulates its joint angles and thrust with
an onboard PID controller. The angular velocity commands are
converted to PWM signals for motor actuation.
Two virtual inputs u
u
uξand u
u
uνcan be designed with trans-
lational and rotational errors to track predefined reference
position and attitude trajectory. We close this control loop
by an LQR controller that considers communication delay
and improves system robustness [32, 33].
B. Downwash-aware Control Allocation
The nullspace-based control allocation framework of over-
actuated UAVs has been proposed in Su et al. [14] to solve
α
α
α,β
β
β, and T
T
Tfrom u
u
udwhile maintaining defined input con-
straints. To avoid downwash effects and maintain high thrust
efficiency, we modify the framework and reformulate the
Quadratic Programming (QP) problem as described below.
An intermediate variable F
F
Fis defined as
F
F
Fpα
α
α, β
β
β, T
T
Tq F
F
FT
1¨ ¨ ¨ F
F
FT
NTPR3Nˆ1,(17)
where
F
F
Fipαi, βi, Tiq Ti»
sin βi
´sin αicos βi
cos αicos βi
.(18)
With F
F
F, we can transform the nonlinear allocation problem
to a linear one,
u
u
udJ
J
Jξpα
α
α, β
β
βq
J
J
Jνpα
α
α, β
β
βqT
T
TW F
W F
WF, (19)
where W
W
WPR6ˆ3Nis a constant allocation matrix with full
row rank. Therefore, F
F
Fcan be solved from u
u
udwith a general
solution form,
F
F
Fpα
α
α, β
β
β, T
T
Tq W
W
W:u
u
ud`N
N
NWZ
Z
Z, (20)
where N
N
NWPR3Nˆp3N´6qis the nullspace of W
W
W, and Z
Z
ZP
Rp3N´61is an arbitrary vector.
As discussed in Su et al. [14], Eq. (20) is linearized
with the first-order Taylor expansion and relaxed with slack
variable s
s
sPR3Nˆ1,
s
s
s`F
F
FpX
X
X0q ` BF
F
F
BX
X
XˇˇˇˇX
X
XX
X
X0
X
X
XW
W
W:u
u
ud`N
N
NWZ
Z
Z, (21)
10481
where X
X
Xis defined as X
X
X rα
α
αT, β
β
βT, T
T
TTsT,r¨s0is the value
of a variable at last time step, and r¨s is the difference w.r.t.
the previous time step of a variable.
Similarly, the downwash avoidance constraint (see
Eq. (12)) can be approximated by another linear equation
as a linear inequality constraint,
O
O
OpX
X
X0q ` BO
O
O
BX
X
XˇˇˇˇX
X
XX
X
X0
X
X
XěO
O
Omin.(22)
The physical constraints of the platform are designed as
X
X
Xmin ´X
X
XoďX
X
XďX
X
Xmax ´X
X
Xo,(23)
X
X
Xmin ďX
X
XďX
X
Xmax.(24)
The objective function is designed as
X
X
XTQ
Q
Q1X
X
X`s
s
sTQ
Q
Q2s
s
s`Z
Z
ZTQ
Q
Q3Z
Z
Z`P
P
PTX
X
X, (25)
where Q
Q
Q1´3are three positive semi-definite weighting ma-
trices. As introduced in Section III-B, the thrust efficiency
index is included as P
P
PTpX
X
Xo`X
X
Xq, with
P
P
PT0
0
01ˆ2Nγ1
1
11ˆNPR1ˆ3N.(26)
Then we have
P
P
PTpX
X
Xo`X
X
Xq γ
N
ÿ
i1
Ti,(27)
where γis the scaling factor. Of note, P
P
PTX
X
Xois a constant,
thus removed from the objective function.
After solving this optimization problem Eqs. (21) to (25),
we can approximately calculate the desired X
X
Xfor next step
with discrete integration,
X
X
XX
X
Xo`X
X
X. (28)
To eliminate the approximation errors, we utilize nullspace
projection with
Z
Z
Z˚N
N
N:
WpF
F
FpX
X
Xq ´ W
W
W:u
u
udq,(29)
F
F
F˚W
W
W:u
u
ud`N
N
NWZ
Z
Z˚.(30)
Finally, with exact solution F
F
F˚, low-level commands α
α
αd,β
β
βd,
and T
T
Tdcan be recovered with inverse kinematics:
Td
ibF2
ix `F2
iy `F2
iz,(31)
αd
iatan2Fiy, Fiz q,(32)
βiasinpFix
Tiq.(33)
C. Low-level Control
The joint angles of each quadcopter module are controlled
by separate PID controllers based on the error dynamics:
:αd
ikP αeα`kI α żeαdt `k
9eα,
:
βd
ikP β eβ`k żeβdt `kD β
9
eβ,
(34)
where kr¨sαand kr¨sβare constant PID gains, and
eααd
i´αe
i,
eββd
i´βe
i,(35)
are error terms with joint angle feedback αe
i,βe
ifrom
onboard IMU. The related torque commands are determined
by
Mx
iBJx
i
:αd
icos βi,
My
iBJy
i
:
βd
i,
Mz
iBJx
i
:
αd
isin βi.
(36)
For each quadcopter module, with Eqs. (6) and (7), the
angular velocity ωi,j of each propeller can be calculated,
later converted to the PWM signal to drive the motor.
V. SIM ULATI ON AN D EXPE RI M EN T SE TU PS
A. Simulation Setup
Before conducting physical experiments, we develop a
simulation platform in Matlab Simulink/Simscape to eval-
uate and characterize the proposed downwash-aware control
allocation framework. In addition to the UAV’s physical
parameters obtained from system identification, the dynam-
ics of propeller motors and saturation, control frequencies,
measurement noise, and communication noise and delays,
the simulator also incorporates the downwash aerodynamics
model introduced in Section II-C based on experimental data.
The proposed allocation framework was verified on two
over-actuated platforms with four and six 3-DoF thrust
generators, respectively. Table I summarizes the physical and
software properties acquired from the physical system used
in simulation, where m0and I0refer to the mass and inertia
matrix of the mainframe, and miand Iirefer to the mass
and inertia matrix of each 3-DoF thrust generator.
TABLE I: Physical and Software Properties in Simulation
Parameter Four Six
m0{kg0.020 0.030
mi{kg 0.050 0.036
diagpI0q{kg ¨cm2r3.20 3.20 4.70s r4.50 4.50 6.20s
diagpIiq{kg ¨cm2r0.35 0.35 0.55s r0.16 0.16 0.29s
l{m0.21 0.18
a{m0.068 0.032
tmax{N0.30 0.15
Communication delay/sec 0.02 0.02
Remote PC control rate/Hz 100 100
Onboard control rate/Hz 500 500
B. Experiment Setup
As shown in Fig. 5, the quadcopters are connected to
the central frame by 2-DoF passive gimbal mechanism,
which have no rotation-angle limitations, thus can be utilized
as 3-DoF thrust generators. We use Crazyflie 2.1 as the
quadcopter module. The weight of Crazyflie 2.1 is 27gwith
a maximum 60gtotal payload. For the platform with four 3-
DoF thrust generators, we upgraded the motors, propellers,
and batteries of the Crazyflie for larger thrust force.
In the experiment, we use the Noitom motion capture
system to measure the position and attitude of the central
frame. The main controller runs on a remote PC, which
communicates with the motion capture system through Eth-
ernet. The main controller calculates the desired thrust T
T
Td,
tilting angles α
α
αd, and twisting angles β
β
βdfor all quadcopter
10482
(a) Four (b) Six
Fig. 5: Hardware prototypes of the over-actuated UAV plat-
forms. The central frame is a rigid body made by carbon-fiber
tubes and 3-D printed parts. Commercial quadcopter Crazyflie 2.1
from Bitcraze is combined with 3-D printed 2-DoF passive gimbal
mechanism as the 3-DoF thrust generator. The platforms have (a)
four and (b) six thrust generators, respectively. Motor, propellers,
and batteries are upgraded to generate larger thrust forces.
modules. The communication between the remote PC and
each quadcopter is achieved by Crazy Radio PA antennas
(2.4G Hz). Each quadcopter is embedded with an onboard
IMU module, estimating the rotation angle given the attitude
of central frame η
η
η. Meanwhile, the onboard controller reg-
ulates the tilting and twisting angles to desired values and
provides the required thrust. The measurement rate of the
motion capture system, the remote PC controller, and the data
communication with each quadcopter are all set to 100 Hz.
The quadcopter’s onboard controller is set to 500 Hz for fast
low-level response. Fig. 6 shows the software architecture.
VI. SI M UL ATI ON A N D EXP E RI MEN T RESU LTS
A. Simulation Results
Fig. 7 summarizes the simulation results of two over-
actuated UAV platforms with the proposed downwash effect
model introduced in Section II-C. For the platform that has
four 3-DoF thrust generators, a reference attitude trajectory is
designed where the downwash effects occur twice (Fig. 7a).
As we can see, the downwash effect first appears at about
9s, when the platform is rotated at 90 degree along the axis
?2
2,?2
2,0s;T4and T1, as well as T3and T2, aligned
vertically (two pairs of downwash effect). With conventional
allocation framework, the downwash flows significantly in-
fluence the control of the platform; we noticed a drop in
Z axis with about 0.15m, and the control performance of
other 5-DoF is also deteriorated (Fig. 7b). Later, another
downwash effect appears at about 16s(T4and T2aligned
vertically), which finally makes the platform unstable. Using
the proposed the downwash-aware allocation framework,
the platform tracks the reference trajectory stably (Figs. 7f
and 7g) and maintain a high thrust efficiency (Fig. 7h). Please
see also Fig. 1 for better visualization.
For the platform that has six 3-DoF thrust generators, a
90 degree pitch reference trajectory (Fig. 7k) is utilized, and
three pairs of downwash effects happen at the final attitude.
With the conventional allocation framework, although the
platform is still stable, we noticed a 0.3mdrop in Z-axis with
more than 5sto compensate for position control (Fig. 7l).
Further, as we can see in Fig. 7m, this framework needs
Fig. 6: Platform communication setup in experiment. The remote
PC takes position and attitude feedback from motion capture sys-
tem, runs the high-level controller at 100 Hz, and sends commands
to each quadcopter through radio communication. Each quadcopter
runs low-level controller at 500 Hz with onboard IMU feedback.
more thrusts to compensate for the downwash aerodynamics,
inefficient in terms of energy. With our proposed downwash-
aware allocation framework, the control in the Z-axis is
maintained, and the thrust is not increased by much for
downwash avoidance. In summary, by exploring the entire
allocation space, the downwash effects are avoided, and the
high thrust efficiency is maintained.
B. Experiment Results
We conducted experiments on the over-actuated UAV
platform that has four 3-DoF thrust generators to compare
the conventional allocation framework and our proposed
downwash-aware allocation framework; see Fig. 8. Using the
conventional allocation framework, the platform is controlled
to track a 90 degree pitch reference trajectory (Fig. 8a),
where a pair of downwash effects appear, and an obvious
drop in the Z-axis is noticed (Fig. 8b). Although the uni-
formly high thrust efficiency is maintained by deploying all
the thrusts in the same direction, this framework requires
more thrust forces to slowly compensate downwash effects
with the integrator of position controller (Fig. 8c). Moreover,
the stability of the platform is influenced by more oscilla-
tions.
Using the proposed downwash-aware allocation frame-
work, the platform avoids the downwash effects by deploying
the proper thrust forces and maintains a high thrust efficiency
(Figs. 8h to 8j). Therefore, the position and attitude tracking
control performance is guaranteed along whole trajectory
(Figs. 8f and 8g). Fig. 8k shows keyframes of the experiment.
C. Discussion
The minimum downwash avoid distance omin in Algo-
rithm 1 has to be experimentally decided for different
platforms. omin 0means the downwash avoidance is not
activated. Large omin may result in no feasible solution to the
downwash-aware allocation problem. Despite that small omin
cannot fully avoid downwash flow, it can still improve the
control performance to some extent. We chose omin 7cm
in the experiment.
10483
0 5 10 15 20
Time (s)
-2
-1
0
1
2
3
Orientation (rad)
rollref
roll
pitchref
pitch
yawref
yaw
(a) Four (C): Attitude.
0 5 10 15 20
Time (s)
-0.2
-0.1
0
0.1
0.2
Position(m)
x y z
(b) Four (C): Position.
0 5 10 15 20
Time (s)
0
0.2
0.4
0.6
0.8
1
Thrust Forces(N)
0.8
0.9
1
1.1
1.2
Thrust Efficiency
T1T2T3T4
(c) Four (C): Thrust.
0 5 10 15 20
Time (s)
-4
-2
0
2
4
Tilting Angles (rad)
1 2 3 4
(d) Four (C): Tilting Angles.
0 5 10 15 20
Time (s)
-4
-2
0
2
4
Twisting Angles (rad)
1 2 3 4
(e) Four (C): Twisting Angles.
0 5 10 15 20
Time (s)
-2
-1
0
1
2
3
Orientation (rad)
rollref
roll
pitchref
pitch
yawref
yaw
(f) Four (D): Attitude.
0 5 10 15 20
Time (s)
-0.2
-0.1
0
0.1
0.2
Position(m)
x y z
(g) Four (D): Position.
0 5 10 15 20
Time (s)
0
0.2
0.4
0.6
0.8
1
Thrust Forces(N)
0.8
0.9
1
1.1
1.2
Thrust Efficiency
T1T2T3T4
(h) Four (D): Thrust.
0 5 10 15 20
Time (s)
-4
-2
0
2
4
Tilting Angles (rad)
1 2 3 4
(i) Four (D): Tilting Angles.
0 5 10 15 20
Time (s)
-4
-2
0
2
4
Twisting Angles (rad)
1 2 3 4
(j) Four (D): Twisting Angles.
0 5 10 15
Time (s)
0
0.5
1
1.5
2
Orientation (rad)
pitchref pitch
(k) Six (C): Attitude.
0 5 10 15
Time (s)
-0.2
-0.1
0
0.1
0.2
Position(m)
x y z
(l) Six (C): Position.
0 5 10 15
Time (s)
0
0.2
0.4
0.6
Thrust Forces(N)
0.8
0.9
1
1.1
1.2
Thrust Efficiency
T1
T4
T2
T5
T3
T6
(m) Six (C): Thrust.
0 5 10 15
Time (s)
-4
-2
0
2
4
Tilting Angles (rad)
1
4
2
5
3
6
(n) Six (C): Tilting Angles.
0 5 10 15
Time (s)
-4
-2
0
2
4
Twisting Angles (rad)
1
4
2
5
3
6
(o) Six (C): Twisting Angles.
0 5 10 15
Time (s)
0
0.5
1
1.5
2
Orientation (rad)
pitchref pitch
(p) Six (D): Attitude.
0 5 10 15
Time (s)
-0.2
-0.1
0
0.1
0.2
Position(m)
xyz
(q) Six (D): Position.
0 5 10 15
Time (s)
0
0.2
0.4
0.6
Thrust Forces(N)
0.8
0.9
1
1.1
1.2
Thrust Efficiency
T1
T4
T2
T5
T3
T6
(r) Six (D): Thrust.
0 5 10 15
Time (s)
-4
-2
0
2
4
Tilting Angles (rad)
1
4
2
5
3
6
(s) Six (D): Tilting Angles.
0 5 10 15
Time (s)
-4
-2
0
2
4
Twisting Angles (rad)
1
4
2
5
3
6
(t) Six (D): Twisting Angles.
Fig. 7: Simulation: Comparison of conventional and downwash-aware control allocation on two over-actuated UAV platforms. C
and D denotes conventional and downwash-aware control allocation, respectively.
0 5 10 15 20 25
Time (s)
0
0.5
1
1.5
2
2.5
Orientation (rad)
rollref
roll
pitchref
pitch
yawref
yaw
(a) Four (C): Attitude.
0 5 10 15 20 25
Time (s)
-0.2
-0.1
0
0.1
0.2
Position(m)
x y z
(b) Four (C): Position.
0 5 10 15 20 25
Time (s)
0
0.2
0.4
0.6
0.8
1
Thrust Forces(N)
0.8
1
1.2
1.4
Thrust Efficiency
T1
T3
T2
T4
(c) Four (C): Thrust.
0 5 10 15 20 25
Time (s)
-2
-1
0
1
2
Tilting Angles (rad)
1
3
2
4
(d) Four (C): Tilting Angles.
0 5 10 15 20 25
Time (s)
-2
-1
0
1
2
Twisting Angles (rad)
1
3
2
4
(e) Four (C): Twisting Angles.
0 5 10 15 20 25
Time (s)
0
0.5
1
1.5
2
2.5
Orientation (rad)
rollref
roll
pitchref
pitch
yawref
yaw
(f) Four (D): Attitude.
0 5 10 15 20 25
Time (s)
-0.2
-0.1
0
0.1
0.2
Position(m)
x y z
(g) Four (D): Position.
0 5 10 15 20 25
Time (s)
0
0.2
0.4
0.6
0.8
1
Thrust Forces(N)
0.8
1
1.2
1.4
Thrust Efficiency
T1
T3
T2
T4
(h) Four (D): Thrust.
0 5 10 15 20 25
Time (s)
-2
-1
0
1
2
Tilting Angles (rad)
1
3
2
4
(i) Four (D): Tilting Angles.
0 5 10 15 20 25
Time (s)
-2
-1
0
1
2
Twisting Angles (rad)
1
3
2
4
(j) Four (D): Twisting Angles.
(k) Four (D): Video frames.
Fig. 8: Experiment: Comparison of conventional and downwash-aware control allocation on the over-actuated UAV platform. C
and D denotes conventional and downwash-aware control allocation, respectively.
VII. CON CL U SI ON
We presented the downwash-aware control allocation
framework of over-actuated UAVs, which makes synergy
of downwash effect avoidance and thrust efficiency main-
tenance. The downwash avoidance constraint and thrust
efficiency index were derived and incorporated into the
10484
nullspace-based allocation framework. In simulation, the
proposed downwash-aware and original nullspace-based al-
location frameworks were studied and compared on two
different over-actuated platforms. These frameworks were
further implemented on our customized UAV platforms in
experiment for demonstration. Both simulation and experi-
ment verified that our proposed framework fully explores the
allocation space and finds the desired allocation solution that
could both avoid downwash effect and maintain high thrust
efficiency, significantly improving the control performance.
ACK NO WL EDG EME NT
The authors thank Dr. Tengyu Liu, Nan Jiang, Zihang
Zhao, Hao Liang, Zeyu Zhang, Zhen Chen, Yifei Dong at
BIGAI for discussions and help on hardware design, motion
capture system, and figures; Dr. Pengkang Yu at UCLA for
his help on control framework of Crazyflie. In particular,
Yao Su wants to thank the love, patience, and care from his
girlfriend Mengmeng, and wishes the best of her surgery.
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10485
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Thesis
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Multirotor copters with full six DoF maneuvering are often overactuated with redundancy. By having redundant actuation, the multirotors can have better fault-tolerance, energy-efficiency, and payload carrying capability. And they can have an infinite number of solutions when doing control allocation due to their redundancy. However, how to take full advantage of input redundancy to improve control performance of overactuated UAV under some constraints is still an open problem. And insufficient usage of input redundancy may dramatically influence the control performance of overactuated UAV in some cases. For example, (i) on tilt-rotor quadcopter platform, the maximum range of attitude is limited caused by input saturation. (ii) the twist and tilt rotor quadcopter platform cannot track a vertical rotation trajectory due to kinematic-singularity. The contributions of this dissertation are addressing the important issues in the copter's redundant actuator control: Firstly, an add-on controller is designed which utilizes the auxiliary inputs at low-level control to do dynamic compensation, thus it can attenuate unknown disturbance and track the reference trajectory with faster response and better performance. Secondly, a novel nullspace based allocation framework is proposed in high-level control, which can take input constraints and explore the nullspace of the allocation matrix to find exact solutions. It is also very robust about controller sampling frequency and measurement noise. Lastly, a fault-tolerant control (FTC) is designed which takes advantage of input redundancy in both high-level and low-level control, and can maintain the stability of whole platform under propeller failure. All these works are validated in both simulation and real-world experiment.
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