Content uploaded by Yao Su

Author content

All content in this area was uploaded by Yao Su on Dec 28, 2022

Content may be subject to copyright.

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

9νﬀ“„W

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

u“„řN

i“1

B

iR

R

R Tiˆz

ˆz

ˆz

řN

i“1pd

d

diˆB

iR

R

R Tiˆz

ˆz

ˆzq“„J

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

k“1

Vpzi,j,k, ri,j,k qti,j ,@j“1,¨ ¨ ¨ ,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

ﬁ

ﬃ

ﬂ»

—

–

∆ti,1

∆ti,2

∆ti,3

∆ti,4

ﬁ

ﬃ

ﬂ,(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 b“a{?2, where ais

the distance of each propeller to the quadcopter center. cτ

is a constant deﬁned as cτ“Kτ{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

u“„W

BR

R

RpřN

i“1

B

iR

R

R∆Tiˆz

ˆz

ˆzq

řN

i“1pd

d

diˆB

iR

R

R∆Tiˆz

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´1qˆ1by 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 i“1¨ ¨ ¨ Ndo

for j“1¨ ¨ ¨ Ndo

if i‰jthen

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

i“1

B

iR

R

R Tiˆz

ˆz

ˆz}

řN

i“1Ti

“}J

J

Jξpα

α

α, β

β

βqT

T

T}

řN

i“1Ti

“}u

u

up1 : 3,1q}

řN

i“1Ti

, 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 (ηf“1). 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

i“1Ti); 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, ﬁve, 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

9νﬀ“„1

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

ud“„mW

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

νﬀ“„u

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

N‰TPR3Nˆ1,(17)

where

F

F

Fipαi, βi, Tiq “ Ti»

–

sin βi

´sin αicos βi

cos αicos βiﬁ

ﬂ.(18)

With F

F

F, we can transform the nonlinear allocation problem

to a linear one,

u

u

ud“„J

J

Jξpα

α

α, β

β

βq

J

J

Jνpα

α

α, β

β

βqT

T

T“W 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´6qˆ1is 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

X“X

X

X0

∆X

X

X“W

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

X“X

X

X0

∆X

X

XěO

O

Omin.(22)

The physical constraints of the platform are designed as

X

X

Xmin ´X

X

Xoď∆X

X

XďX

X

Xmax ´X

X

Xo,(23)

∆X

X

Xmin ď∆X

X

Xď∆X

X

Xmax.(24)

The objective function is designed as

∆X

X

XTQ

Q

Q1∆X

X

X`s

s

sTQ

Q

Q2s

s

s`Z

Z

ZTQ

Q

Q3Z

Z

Z`P

P

PT∆X

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

PT““0

0

01ˆ2Nγ1

1

11ˆN‰PR1ˆ3N.(26)

Then we have

P

P

PTpX

X

Xo`∆X

X

Xq “ γ

N

ÿ

i“1

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

X“X

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

i“bF2

ix `F2

iy `F2

iz,(31)

αd

i“atan2p´Fiy, Fiz q,(32)

βi“asinpFix

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

i“kP αeα`kI α żeαdt `kDα

9eα,

:

βd

i“kP β eβ`kIβ żeβdt `kD β

9

eβ,

(34)

where kr¨sαand kr¨sβare constant PID gains, and

eα“αd

i´αe

i,

eβ“βd

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

i“BJx

i

:αd

icos βi,

My

i“BJy

i

:

βd

i,

Mz

i“BJx

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

r´?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.

REF ER E NC ES

[1] T. Anzai, M. Zhao, X. Chen, F. Shi, K. Kawasaki, K. Okada,

and M. Inaba, “Multilinked multirotor with internal communication

system for multiple objects transportation based on form optimiza-

tion method,” in International Conference on Intelligent Robots and

Systems (IROS), 2017.

[2] B. Li, L. Ma, D. Huang, and Y. Sun, “A ﬂexibly assembled

and maneuverable reconﬁgurable modular multi-rotor aerial vehicle,”

IEEE/ASME Transactions on Mechatronics (TMECH), 2021.

[3] M. J. Gerber and T.-C. Tsao, “Twisting and tilting rotors for high-

efﬁciency, thrust-vectored quadrotors,” Journal of Mechanisms and

Robotics, vol. 10, no. 6, p. 061013, 2018.

[4] A. F. S¸enkul and E. Altu˘

g, “System design of a novel tilt-roll rotor

quadrotor uav,” Journal of Intelligent & Robotic Systems, vol. 84,

no. 1, pp. 575–599, 2016.

[5] H.-N. Nguyen, S. Park, J. Park, and D. Lee, “A novel robotic platform

for aerial manipulation using quadrotors as rotating thrust generators,”

Transactions on Robotics (T-RO), vol. 34, no. 2, pp. 353–369, 2018.

[6] C. Pi, L. Ruan, P. Yu, Y. Su, S. Cheng, and T. Tsao, “A simple six

degree-of-freedom aerial vehicle built on quadcopters,” in Proceedings

of IEEE Conference on Control Technology Applications (CCTA),

2021.

[7] P. Yu, Y. Su, M. J. Gerber, L. Ruan, and T.-C. Tsao, “An over-actuated

multi-rotor aerial vehicle with unconstrained attitude angles and high

thrust efﬁciencies,” IEEE Robotics and Automation Letters (RA-L),

vol. 6, no. 4, pp. 6828–6835, 2021.

[8] N. Michael, D. Mellinger, Q. Lindsey, and V. Kumar, “The grasp

multiple micro-uav testbed,” IEEE Robotics and Automation Magazine

(RA-M), vol. 17, no. 3, pp. 56–65, 2010.

[9] Y. Su, L. Ruan, P. Yu, C.-H. Pi, M. J. Gerber, and T.-C. Tsao, “A fast

and efﬁcient attitude control algorithm of a tilt-rotor aerial platform

using inputs redundancies,” IEEE Robotics and Automation Letters

(RA-L), vol. 7, no. 2, pp. 1214–1221, 2021.

[10] H. Lee, M. Jeong, C. Kim, H. Lim, C. Park, S. Hwang, and H. Myung,

“Low-level pose control of tilting multirotor for wall perching tasks

using reinforcement learning,” in International Conference on Intelli-

gent Robots and Systems (IROS), 2021.

[11] W. Zhang, M. Brunner, L. Ott, M. Kamel, R. Siegwart, and J. Nieto,

“Learning dynamics for improving control of overactuated ﬂying

systems,” IEEE Robotics and Automation Letters (RA-L), vol. 5, no. 4,

pp. 5283–5290, 2020.

[12] R. Yang, L. Zheng, J. Pan, and H. Cheng, “Learning-based predictive

path following control for nonlinear systems under uncertain distur-

bances,” IEEE Robotics and Automation Letters (RA-L), vol. 6, no. 2,

pp. 2854–2861, 2021.

[13] P. Yu, An Over-Actuated Multi-Rotor Aerial Platform and Iterative

Learning Control Applications. PhD thesis, University of California,

Los Angeles, 2022.

[14] Y. Su, P. Yu, M. Gerber, L. Ruan, and T.-C. Tsao, “Nullspace-based

control allocation of overactuated uav platforms,” IEEE Robotics and

Automation Letters (RA-L), vol. 6, no. 4, pp. 8094–8101, 2021.

[15] W. Khan, M. Nahon, and R. Caverly, “Propeller slipstream model for

small unmanned aerial vehicles,” in AIAA modeling and simulation

technologies (MST) conference, 2013.

[16] Y. Zheng, S. Yang, X. Liu, J. Wang, T. Norton, J. Chen, and Y. Tan,

“The computational ﬂuid dynamic modeling of downwash ﬂow ﬁeld

for a six-rotor uav,” Frontiers of Agricultural Science and Engineering,

vol. 5, no. 2, pp. 159–167, 2018.

[17] K. P. Jain, T. Fortmuller, J. Byun, S. A. M¨

akiharju, and M. W.

Mueller, “Modeling of aerodynamic disturbances for proximity ﬂight

of multirotors,” in International Conference on Unmanned Aircraft

Systems (ICUAS), 2019.

[18] R. Miyazaki, R. Jiang, H. Paul, K. Ono, and K. Shimonomura, “Air-

borne docking for multi-rotor aerial manipulations,” in International

Conference on Intelligent Robots and Systems (IROS), 2018.

[19] J. L. Brinkman, B. Davis, and C. E. Johnson, “Post-movement

stabilization time for the downwash region of a 6-rotor uav for remote

gas monitoring,” Heliyon, vol. 6, no. 9, p. e04994, 2020.

[20] J. A. Preiss, W. H¨

onig, N. Ayanian, and G. S. Sukhatme, “Downwash-

aware trajectory planning for large quadrotor teams,” in International

Conference on Intelligent Robots and Systems (IROS), 2017.

[21] G. Shi, W. H¨

onig, Y. Yue, and S.-J. Chung, “Neural-swarm: Decentral-

ized close-proximity multirotor control using learned interactions,” in

International Conference on Robotics and Automation (ICRA), 2020.

[22] G. Shi, W. H¨

onig, X. Shi, Y. Yue, and S.-J. Chung, “Neural-

swarm2: Planning and control of heterogeneous multirotor swarms

using learned interactions,” Transactions on Robotics (T-RO), 2021.

[23] M. Ryll, H. H. B¨

ulthoff, and P. R. Giordano, “A novel overactuated

quadrotor unmanned aerial vehicle: Modeling, control, and experimen-

tal validation,” IEEE Transactions on Control Systems Technology,

vol. 23, no. 2, pp. 540–556, 2014.

[24] M. Kamel, S. Verling, O. Elkhatib, C. Sprecher, P. Wulkop, Z. Taylor,

R. Siegwart, and I. Gilitschenski, “The voliro omniorientational hexa-

copter: An agile and maneuverable tiltable-rotor aerial vehicle,” IEEE

Robotics and Automation Magazine (RA-M), vol. 25, no. 4, pp. 34–44,

2018.

[25] M. Zhao, K. Okada, and M. Inaba, “Enhanced modeling and control

for multilinked aerial robot with two dof force vectoring apparatus,”

IEEE Robotics and Automation Letters (RA-L), vol. 6, no. 1, pp. 135–

142, 2020.

[26] M. Santos, L. Hon´

orio, A. Moreira, M. Silva, and V. Vidal, “Fast real-

time control allocation applied to over-actuated quadrotor tilt-rotor,”

Journal of Intelligent & Robotic Systems, vol. 102, no. 3, pp. 1–20,

2021.

[27] T. A. Johansen, T. I. Fossen, and S. P. Berge, “Constrained nonlinear

control allocation with singularity avoidance using sequential quadratic

programming,” IEEE Transactions on Control Systems Technology,

vol. 12, no. 1, pp. 211–216, 2004.

[28] M. A. da Silva Ferreira, M. F. T. Begazo, G. C. Lopes, A. F.

de Oliveira, E. L. Colombini, and A. da Silva Sim˜

oes, “Drone

reconﬁgurable architecture (dra): A multipurpose modular architecture

for unmanned aerial vehicles (uavs),” Journal of Intelligent & Robotic

Systems, vol. 99, pp. 517–534, 2020.

[29] L. Ruan, Independent position and attitude control on multirotor aerial

platforms. PhD thesis, University of California, Los Angeles, 2020.

[30] M. Ryll, D. Bicego, and A. Franchi, “Modeling and control of fast-hex:

A fully-actuated by synchronized-tilting hexarotor,” in International

Conference on Intelligent Robots and Systems (IROS), 2016.

[31] Y. Su, Compensation and Control Allocation with Input Saturation

Limits and Rotor Faults for Multi-Rotor Copters with Redundant

Actuations. PhD thesis, University of California, Los Angeles, 2021.

[32] L. Ruan, C. Pi, Y. Su, P. Yu, S. Cheng, and T. Tsao, “Control

and experiments of a novel tiltable-rotor aerial platform comprising

quadcopters and passive hinges,” Mechatronics (submitted), 2022.

[33] Y. Su, P. Yu, M. Gerber, L. Ruan, and T. Tsao, “Fault-tolerant control

of overactuated multirotor uav platform under propeller failure,”

IEEE/ASME Transactions on Mechatronics (TMECH) (submitted),

2022.

10485