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Control and Experiments of a Novel Tiltable-Rotor Aerial Platform

Comprising Quadcopters and Passive Hinges

Lecheng Ruana,Chen-Huan Pib,Yao Sua,Pengkang Yua,Stone Chengband Tsu-Chin Tsaoa

aMechanical and Aerospace Engineering Department, University of California, Los Angeles, Los Angeles

bMechanical Engineering Department, National Yang Ming Chiao Tung University, HsinChu

ARTICLE INFO

Keywords:

Multirotor

Overactuation

Six DoF control

Optimal control allocation

Flight experiment

ABSTRACT

This paper presents the control and experiment of a novel multirotor aerial platform, which is capable

of full actuation for six Degree of Freedom (DoF) motions. The platform is actuated by a number

of tilting-thrust modules, each consisting of a regular quadcopter and a mechanically passive hinge.

The platform in this paper has four such actuator modules, making an over-actuated system that

requires input allocations in the feedback control. In addition to the common least-square method

that minimizes the sum of squares of the thrusts, we propose a control allocation that minimizes

the maximum thrust in a closed-form analytical solution for eﬃcient real time computation. This

allocation can achieve larger inclination angles than that by the least-square method, when thrust

forces are insuﬃcient to overcome the gravity for all poses. Simulation and real world experiments are

presented to demonstrate the control of the aerial platform for six DoF motion tracking and disturbance

rejection.

1. Introduction

Multirotors, among various unmanned aerial vehicles

(UAVs) that have caught interests in research and appli-

cations recently, have advantages in mechanical simplic-

ity, high agility, vertical take-oﬀ and landing(VTOL), and

hovering capability. Traditional multirotor platforms usually

align multiple propellers in the same direction to eﬀectively

compensate for the gravity during the ﬂight. This conﬁg-

uration is underactuated, but is proven diﬀerential-ﬂat [1]

for the independent control of a four DoF subspace [2,3],

which is adequate for position-oriented applications such

as surveillance, rescue, delivery, inspection and mobile net-

work construction, etc. [4,5,6,7,8,9,10,11,12].

The regular multirotors cannot control position and at-

titude independently due to the intrinsic underactuation.

This limits their applications where six DoF control is re-

quired, such as the exploration in complicated terrains [13]

or multi-pose interaction with the environment [14]. These

requirements have motivated the designs of fully actuated

multirotor platforms, which are categorized into two major

classes [15]. The ﬁrst class [16,17,18,19] deploys at

least six propellers at various ﬁxed orientations for six DoF

actuation, and inherits the mechanical simplicity of regu-

lar multirotors. However, the thrust capability in diﬀerent

DoFs has large disparities and can not be changed once the

propeller directions are ﬁxed. Therefore, the conﬁgurations

must be designed to meet the requirements of speciﬁc appli-

cations [16,17].

The second class actively controls the attitude of each

propeller [20,21,22,23,24], referred to as the tiltable-rotor

aerial platforms in the rest of this paper. This concept was

ORCID(s):

1The ﬁrst two authors contributed equally to this work. CP. Pi and S.

Cheng were visiting UCLA during this work.

2The supplementary video of real-world experiments is available at

https://www.youtube.com/watch?v=ERbJaznHi-c.

ﬁrst proposed and realized by [20], where the propeller an-

gles of a quadrotor platform are actively controlled by servo

motors. The enhanced ﬂexibility and conﬁgurability of these

platforms are accompanied by signiﬁcantly increased me-

chanical complexity and weights. Therefore, only a limited

number of designs have been realized and experimentally

validated [20,21,24]. The tiltable-rotor platforms introduce

three issues that are not existent in regular quadcopters.

The ﬁrst issue is that the aerodynamic drag torques of the

propellers, when at diﬀerent cant angles, cannot cancel one

another like those of the quadcopters. The second issue is

that the tilting-thrust actuators exert reaction torques to the

central frame when tilting. The third issue is that the gyro-

scopic torques occur for each propeller rotor when changing

directions. In the control system design, these three issues

have been commonly treated as disturbances or unmodeled

dynamics.

Most fully actuated multirotor platforms have input re-

dundancies and require allocation schemes to map the de-

sired torque and force to the physical inputs, usually the

thrust force and orientation of each propeller. To this end,

the least-square approach is commonly used for its simplicity

and robustness [21,22]. This approach is based on a linear

force decomposition, where the least-square method is ap-

plied to some intermediate variables, followed by a nonlinear

mapping from the optimal solution to the physical input

variables. As such, the optimal solution for the intermediate

variables may not carry a physical meaning. In view of

this, [20] formulate the allocation optimization based on the

physical input variables to account for physical constraints,

but the gradient-based numerical iterations in every control

update require signiﬁcant real-time computation.

Regular quadcopters can provide variable total thrust

force while simultaneously regulating their attitude [1], thus

can be used to replace the rotor and tilting motor pair.

Cables [25,26] or universal joints [27] have been used to

L.Ruan et al.:Preprint submitted to Elsevier Page 1 of 14

Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges

connect multiple quadcopters to the tool frame for cooper-

ative manipulations. However, the limited range of the ten-

sioned cables or spherical joints used to connect quadcopters

and the tool frame signiﬁcantly constrains the achievable

attitudes. Our group has proposed modular vectored thrust

units made of quadcopters and passive mechanisms without

angle range limits or constraints for two realizations, one

DoF hinge [28,29] or two DoF gimbal [30,31,32]. These

modular vectored thrust units overcome the constraints of

the limited orientations between the quadcopters and the

main frame in [25,26,27], and enable creating new mul-

tirotor aerial platforms for unique capabilities and perfor-

mance. Although having advantageous thrust eﬃciency at

large attitude angles, the two DoF gimbal [30] exhibit higher

complexity, weight and structural compliance than the one

DoF hinge, and will encounter kinematic singularity when

its two axes are co-linear.

In this paper, we present a novel low-complexity and

simple-to-build fully actuated tiltable-rotor aerial platform [28,

29]. Our presented platform has unlimited joint angle ranges

and greatly reduces the mechanical complexity, compared to

existing tiltable-rotor conﬁgurations where the tilting of each

rotor is actuated by a servo motor. An important feature of

our tiltable-rotor platform is the elimination of the propeller

drags, gyroscopic momentums and tilting reaction torques,

which had been treated as disturbances or unmodeled dy-

namics in other tiltable-rotor platforms, because of the

paired propellers rotating in opposite directions and the zero-

torque transmission in the passive hinges. We have presented

the preliminary work on the control experiment using a

heuristic allocation method in [29]. In this paper, we will

present closed-form optimal solutions for min-max input

allocations for higher thrust eﬃciency, and comprehensive

simulation and experiments to demonstrate the platform

and controller’s capabilities. Our contributions, signiﬁcantly

extended from [29], are highlighted in the followings:

(1) We will analyze the dynamics of our aerial platform

and provide a comprehensive comparison with other

related conﬁgurations to show the advantages of elim-

inating the aerodynamic drag forces, gyroscopic mo-

mentums and tilting reaction torques that are regarded

as disturbances in tiltable rotor conﬁgurations driven

by propeller-rotor pairs, and furthermore, the unique

high-bandwidth auxiliary torque inputs to perform

control compensations and actuator failure recovery,

which are unavailable to other tiltable rotor realiza-

tions or the two DoF gimbal conﬁguration.

(2) We will formulate and analytically solve the min-

max optimal allocation to suppress the maximum

required thrust and thus achieve a larger operational

space under input saturation compared with existing

allocations. This method will also facilitate real-time

implementation without numerical iterations for solv-

ing optimization. We will also provide a quantitative

stability criterion for the presented hierarchical con-

trol architecture.

(3) We will provide simulation and experimental valida-

tions to demonstrate the platform’s capability of full

actuation, the eﬀectiveness of the min-max optimal

allocation to achieve a larger operational space subject

to the maximum thrust limits, and the stability and dis-

turbance rejection capability of the presented control

architecture.

The rest of this paper is organized in the following

structure. Section 2illustrates the mechanical design and

dynamics of the presented platform, and conduct a compre-

hensive comparison with existing tiltable-rotor multirotor

aerial platforms. Section 3presents the hierarchical control

architecture and the stability analysis of the platform, while

elaborating the formulation of the min-max optimal allo-

cation and its analytical solution. Section 4demonstrates

the simulation and experiment veriﬁcations. The paper is

discussed and concluded in Section 5and Section 6respec-

tively.

2. Platform

2.1. Structure and Dynamics

The mechanical structure of the presented platform is

demonstrated in Figure 1. The central frame is composed

of four carbon ﬁber tubes perpendicularly installed on the

central connector. A regular quadcopter is passively hinged

to each carbon ﬁber tube by a connector with an embedded

nylon bearing. The relative motion of the quadcopter with

respect to the central frame is restrained merely to the rota-

tion along the tube, denoted as 𝛼𝑖for quadcopter 𝑖.𝑙refers to

the platform arm length, 𝑎the quadcopter arm length, and 𝑟

the propeller radius of each quadcopter. It has been analyzed

in [29] that each quadcopter in the presented platform has

more than the full functionalities of a propeller with active

tiltable cant angle, as in the previous works [20,21,22].

The platform dynamics is equivalent to a rigid body

with exerted forces and torques of varying magnitudes and

directions [29]. As shown in Figure 1, the world inertial

frame, body frame and quadcopter frames are deﬁned as

𝑊∶ {𝑂;𝑥

𝑥

𝑥, 𝑦

𝑦

𝑦, 𝑧

𝑧

𝑧},

𝐵∶ {𝑂𝐵;𝑥

𝑥

𝑥𝐵, 𝑦

𝑦

𝑦𝐵, 𝑧

𝑧

𝑧𝐵},

𝑄𝑖∶ {𝑂𝑄𝑖;𝑥

𝑥

𝑥𝑄𝑖, 𝑦

𝑦

𝑦𝑄𝑖, 𝑧

𝑧

𝑧𝑄𝑖},

(1)

respectively, where 𝑖= 0,1,2,3and 𝑗= 0,1,2,3refers to

the orders of the quadcopters and propellers.

Primary Inputs: It has been stated in [29] that the four

propeller-rotor pairs in quadcopter 𝑖collectively generate

four independent inputs in its quadcopter frame 𝑄𝑖as

𝑈

𝑈

𝑈𝑖=𝑇𝑖𝑀𝑥

𝑖𝑀𝑦

𝑖𝑀𝑧

𝑖𝑇=𝑇

𝑇

𝑇𝑄Ω

Ω

Ω𝑖,(2)

where 𝑇𝑖refers to the collective total thrust force, 𝑀𝑥

𝑖,𝑀𝑦

𝑖,

𝑀𝑧

𝑖the collective total torques along the 𝑥𝑄𝑖,𝑦𝑄𝑖and 𝑧𝑄𝑖

directions. Ω

Ω

Ω𝑖is the propeller speed vector where each ele-

ment can be controlled independently, and 𝑇

𝑇

𝑇𝑄is a constant

matrix dependent merely on the quadcopter and propeller

L.Ruan et al.:Preprint submitted to Elsevier Page 2 of 14

Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges

(a) The presented tiltable-rotor aerial platform.

(b) The tilting-thrust actuator comprising of regular

quadcopter and passive hinge.

Figure 1: Conﬁguration of the presented tiltable-rotor aerial platform. Regular quadcopters are mounted on mechanically passive

hinges as tilting-thrust actuators. The world frame 𝑊is deﬁned under the North-East-Down (NED) convention. The body frame

𝐵origin is located at the geometric center of the central connector. 𝑥𝐵and 𝑦𝐵are aligned with the carbon ﬁber tubes. The

quadcopter frames 𝑄𝑖is attached with each quadcopter. 𝑦𝑄𝑖is aligned with the hinge axis, and 𝑧𝑄𝑖crosses the quadcopter

geometric center.

properties. The propeller speed is usually, if not always,

considered to be of zero-order dynamics as the inertia of

the propeller-rotor pairs is neglectable compared with the

quadcopter or frame inertia. Therefore, the four inputs in 𝑈

𝑈

𝑈

are regarded as the primary inputs of the platform dynamics.

Translational Dynamics: The translational motion of

the platform is actuated by the four total thrust forces 𝑇𝑖

on each quadcopter. Denoting 𝑚and 𝐺

𝐺

𝐺as the platform

mass and the gravitational acceleration, and deﬁning the

platform center-of-mass position as 𝜉

𝜉

𝜉=𝑥𝑦𝑧𝑇, the

translational motion can be calculated by Newton’s Second

Law as

𝜉

𝜉

𝜉=1

𝑚

𝑊𝑅

𝑅

𝑅𝐵(

3

𝑖=0

𝐵𝑅

𝑅

𝑅𝑄𝑖𝑇

𝑇

𝑇𝑖) + 𝐺

𝐺

𝐺, (3)

where each total thrust force 𝑇𝑖on quadcopter 𝑖is trans-

formed to the platform body frame 𝐵by 𝐵𝑅

𝑅

𝑅𝑄𝑖, summed

up and transformed collectively to the world frame 𝑊by

𝑊𝑅

𝑅

𝑅𝐵. Notice that the transformation matrix 𝐵𝑅

𝑅

𝑅𝑄𝑖from each

quadcopter to the platform body frame is a function of the

corresponding tilting angle 𝛼𝑖, and matrix 𝑊𝑅

𝑅

𝑅𝐵is a function

of the platform attitude, usually denoted by a set of Euler

angles 𝜂

𝜂

𝜂=𝜙 𝜃 𝜓 𝑇, as stated in [29].

Rotational Dynamics: Denote the total external torque

exerted on the platform as 𝜏

𝜏

𝜏in the platform body frame 𝐵,

and deﬁne the angular velocity vector in the platform body

frame 𝐵as 𝜈

𝜈

𝜈=𝑝 𝑞 𝑟𝑇, the rotational dynamics can

be simply written as

𝜈

𝜈

𝜈=𝐼

𝐼

𝐼−1(−𝜈

𝜈

𝜈× (𝐼𝜈

𝐼𝜈

𝐼𝜈) + 𝜏

𝜏

𝜏).(4)

The total controlled external torque is dependent on the

primary inputs in 𝑈

𝑈

𝑈𝑖and mainly consists of two parts. The

ﬁrst part is generated collectively by the total thrust forces

𝑇𝑖of the four quadcopters as

𝜏

𝜏

𝜏𝑇=

−𝑐𝛼0𝑙0𝑐𝛼2𝑙0

0𝑐𝛼1𝑙0 −𝑐𝛼3𝑙

𝑠𝛼0𝑙 𝑠𝛼1𝑙 𝑠𝛼2𝑙 𝑠𝛼3𝑙

𝑇0

𝑇1

𝑇2

𝑇3

,(5)

where sin ⋅and cos ⋅are denoted as 𝑠⋅and 𝑐⋅for simplicity.

The second part is related to the total torques of each quad-

copter in the primary input set, 𝑀𝑥

𝑖,𝑀𝑦

𝑖and 𝑀𝑧

𝑖. It can be

observed in Figure 1(b) that 𝑀𝑦

𝑖is along the direction of the

passive hinge and cannot be exerted on the platform. The

relative motions along the directions of 𝑀𝑥

𝑖and 𝑀𝑧

𝑖with

respect to the platform are constrained, so these torques can

be directly transferred to the central frame, whose total eﬀect

can be calculated as

𝜏

𝜏

𝜏𝑀=

3

𝑖=0

𝐵𝑅

𝑅

𝑅𝑄𝑖

𝑀𝑥

𝑖

0

𝑀𝑧

𝑖

.(6)

Therefore, the total external torque exerted on the presented

platform can be represented by the total controlled external

torque

𝜏

𝜏

𝜏=𝜏

𝜏

𝜏𝑇+𝜏

𝜏

𝜏𝑀.(7)

Tilting Dynamics: It can be observed that transition

matrices 𝐵𝑅

𝑅

𝑅𝑄𝑖are involved in both translational and rota-

tional dynamics bu changing the directions of the primary

inputs 𝑈

𝑈

𝑈𝑖.𝐵𝑅

𝑅

𝑅𝑄𝑖are merely dependent on the tilting angles

of quadcopters 𝛼𝑖, which are controlled by the primary inputs

𝑀𝑦

𝑖as

𝛼𝑖=𝑀𝑦

𝑖−𝑠(𝜋

2𝑖)𝑝 −𝑐(𝜋

2𝑖)𝑞. (8)

L.Ruan et al.:Preprint submitted to Elsevier Page 3 of 14

Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges

Table 1

Comparisons of Tiltable-Rotor Platforms (The best performance under each index is bolded.)

Conﬁgurations C1: Propeller-rotor + Motor C2: Quad + 2 DoF Gimbal C3: Quad + 1 DoF Hinge

References [20], [21] [30] This paper, [29]

Mechanical Complexity High in overall structure High in gimbal mechanism Low

Propeller Drag Disturbance Yes No No

Fast Auxiliary Inputs No Neglectable Yes

Actuator Failure Recovery No No Yes

Tilting Reaction Disturbance Yes No No

Propeller Gyroscopic Eﬀect High Low No

Thrust Eﬃciency Medium High Medium

2.2. Comparisons of Tiltable-rotor Platforms

To date, there are two major conﬁgurations of tiltable-

rotor multirotor aerial platforms. One is realized by installing

an additional actuator to actively tilt each propeller-rotor pair

(denoted as C1) [20,21], and the other is to mount regular

quadcopters onto the platform with passive hinges to unify

tilting and thrusting actuations. The latter conﬁguration has

two DoF gimbal realization (denoted as C2), as presented

in [30], and one DoF hinge realization (denoted as C3), as

presented in this paper and [29]. This section compares these

three types of platforms in various aspects, as presented in

Table 1. Detailed explanations are as follows.

Mechanical Complexity: C1 requires a special design

to install the tilting actuators on the platform and house the

propeller-rotor pairs on the tilting actuators. C2 only needs to

design the passive gimbal, but the complexity of the gimbal

mechanism is high. C3 only requires the design of the hinge

connector with no inner mechanism, and is easily scalable

for frames and quadcopters of various sizes.

Propeller Drag Disturbance: Similar to (6), each

propeller-rotor pair of C1 or each quadcopter of C2 also

generates a torque that is directly transferred to the platform,

as

𝜏

𝜏

𝜏𝑀=

3

𝑖=0

𝐵𝑅

𝑅

𝑅𝑄𝑖

0

0

𝑀𝑧

𝑖

.(9)

The 𝑀𝑧

𝑖in C2 and C3 is a primary input that can be

controlled independently based on (2), thus may be set to

zero at all times upon requirements. However, the 𝑀𝑧

𝑖in C1

is determined by the corresponding thrust force on the same

propeller-rotor pair as

𝑀𝑧

𝑖= (𝜅𝜏∕𝜅𝑇)𝑇𝑖,(10)

where 𝜅𝜏and 𝜅𝑇are propeller related constants. This is

called the propeller drag torque, which can cancel each other

or be used to control the yaw angle for regular quadcopters

(as the 𝐵𝑅

𝑅

𝑅𝑄𝑖for regular quadcopter is the identity matrix),

but is usually regarded as disturbances in the tiltable-rotor

platforms [20,21] as the propeller axes are not co-linear.

Fast Auxiliary Inputs & Actuator Failure Recovery:

It can be observed by comparing (6) and (9) that 𝑀𝑥

𝑖

can directly adjust the total external torque exerted on the

platform 𝜏

𝜏

𝜏for C3. This is called the fast auxiliary input

as it has much faster dynamics (zero’s order for any given

𝛼𝑖, feedthrough) compared with the 𝜏

𝜏

𝜏𝑇in (5) (second order

dynamics as 𝛼need to be controlled by (8) to generate

desired 𝜏

𝜏

𝜏𝑇). This property has been found of great signif-

icance in our following works for the tracking of highly-

dynamic reference signals [33] and ﬂight recovery under

certain actuator failures [34,35]. On the other hand, C1 does

not have auxiliary inputs according to (9) and (10). C2 only

have 𝑀𝑧

𝑖, but the magnitude (therefore the control eﬀort) is

much smaller than 𝑀𝑥

𝑖, and is usually not suﬃcient to handle

the tracking or failure recovery tasks in [33,34].

Tilting Reaction Disturbance: The tilting actuator of

C1 is installed on the platform. Therefore, when a tilting

torque is exerted on the propeller-rotor pair from the tilting

actuator, a reaction torque 𝜏𝑅

𝜏𝑅

𝜏𝑅shall be exerted reversely on

the platform as

𝜏

𝜏

𝜏=𝜏

𝜏

𝜏𝑇+𝜏

𝜏

𝜏𝑀+𝜏

𝜏

𝜏𝑅.(11)

The reaction torque 𝜏𝑅

𝜏𝑅

𝜏𝑅is dynamic and exists whenever

any 𝛼𝑖≠0, thus can only be regarded as disturbance. Unlike

C1, the actuator tilting of C2 or C3 is controlled by 𝑀𝑦

𝑖

(and 𝑀𝑥

𝑖for C2), which is generated by the interaction of

the propellers and the air, thus does not have tilting reaction

disturbance 𝜏𝑅

𝜏𝑅

𝜏𝑅on the platform.

Propeller Gyroscopic Eﬀect: The propeller gyroscopic

eﬀect comes from the the torque 𝜏

𝜏

𝜏𝐺generated from the

change of total angular momentum 𝐿

𝐿

𝐿𝐺as

𝜏

𝜏

𝜏𝐺=𝑑𝐿

𝐿

𝐿𝐺

𝑑𝑡 .(12)

In C1, the total angular momentum of each actuator is

𝐿

𝐿

𝐿𝐺=𝐼𝑃𝜔

𝜔

𝜔, (13)

where 𝐼𝑃is the propeller inertia and 𝜔

𝜔

𝜔the angular velocity.

In C2 and C3, the total angular momentum of each actuator

(quadcopter) collects each propeller’s angular momentum as

𝐿

𝐿

𝐿𝐺=𝐼𝑝

3

𝑗=0

𝜔

𝜔

𝜔𝑗.(14)

In C3, for the primary inputs 𝑇𝑖and 𝑀𝑦

𝑖, the angular ve-

locities of 𝜔

𝜔

𝜔0and 𝜔

𝜔

𝜔1are always of identical magnitude and

L.Ruan et al.:Preprint submitted to Elsevier Page 4 of 14

opposite direction, e.g. 𝜔

𝜔

𝜔0+𝜔

𝜔

𝜔1≡0.𝜔

𝜔

𝜔2and 𝜔

𝜔

𝜔3have the

same relation. Therefore, 𝐿

𝐿

𝐿𝐺≡0and thus

𝜏

𝜏

𝜏𝐺≡0(15)

for C3, e.g. the propeller gyroscopic eﬀect is eliminated. The

total angular momentum of C2 generally does not have this

property as the magnitude of each angular velocity is diﬀer-

ent when the quadcopter tilts along both axes. However, the

total angular momentum of C2 shall be much smaller than

C1 as the two oppositely rotating propeller pairs cancel a

large portion of the angular momentum, so the gyroscopic

eﬀect shall be also attenuated.

Thrust Eﬃciency: The thrust eﬃciency is deﬁned as

the percentage of thrust used to compensate for platform

gravity. For C1 and C3, the eﬃciency decreases for larger

platform attitudes, but is generally higher compared with

regular quadcopters when allocated properly. However, it

has been proved and experimentally validated in [30] that

C2 can maintain almost constant full eﬃciency for arbitrary

attitudes.

3. Control

3.1. Dynamic Model Reformulation

The nonlinear platform dynamic model is reformulated

in this section to accommodate linear control techniques.

In this paper, only primary inputs 𝑇𝑖and 𝑀𝑦

𝑖are used in

control. 𝑀𝑥

𝑖and 𝑀𝑧

𝑖are assigned zero at all time. Assuming

that the gyroscopic eﬀect of the central frame is neglectable,

the translational and rotational dynamics are written in the

matrix form

𝜉

𝜉

𝜉

𝜈

𝜈

𝜈=1

𝑚

𝑊𝑅

𝑅

𝑅𝐵0

0

0

0

0

0𝐼

𝐼

𝐼−1

𝑄𝐽

𝐽

𝐽𝜉

𝐽

𝐽

𝐽𝜈𝑇

𝑇

𝑇+𝐺

𝐺

𝐺

0

0

0,(16)

where

𝐽

𝐽

𝐽𝜉=

−𝑠𝛼00𝑠𝛼20

0𝑠𝛼10 −𝑠𝛼3

𝑐𝛼0𝑐𝛼1𝑐𝛼2𝑐 𝛼3

,

𝐽

𝐽

𝐽𝜂=

−𝑐𝛼0𝑙0𝑐𝛼2𝑙0

0𝑐𝛼1𝑙0 −𝑐𝛼3𝑙

𝑠𝛼0𝑙 𝑠𝛼1𝑙 𝑠𝛼2𝑙 𝑠𝛼3𝑙

,

𝑇

𝑇

𝑇=𝑇0𝑇1𝑇2𝑇3𝑇.

(17)

The rotation inertia matrix is denoted as 𝐼

𝐼

𝐼𝑄. In the

dynamic model (16), 𝑇𝑖and 𝛼𝑖are controlled by the platform

primary inputs 𝑇𝑖and 𝑀𝑦

𝑖, and thus considered as inputs,

but nonlinearly coupled. However, when deﬁning the tilted

thrust force vector

𝐹

𝐹

𝐹=𝐹𝑠0𝐹𝑐0…𝐹𝑠3𝐹𝑐3𝑇,(18)

where

𝐹𝑠𝑖 =𝑠𝛼𝑖𝑇𝑖, 𝐹𝑐𝑖 =𝑐𝛼𝑖𝑇𝑖,(19)

the dynamic equation (16) becomes linear with respect to 𝐹

𝐹

𝐹,

and can be rewritten as

𝜉

𝜉

𝜉

𝜈

𝜈

𝜈=1

𝑚

𝑊𝑅

𝑅

𝑅𝐵0

0

0

0

0

0𝐼

𝐼

𝐼−1

𝑄𝑊

𝑊

𝑊 𝐹

𝐹

𝐹+𝐺

𝐺

𝐺

0

0

0,(20)

where 𝑊

𝑊

𝑊is a full-rank constant matrix

𝑊

𝑊

𝑊=

−1 0 0 0 1 0 0 0

0 0 1 0 0 0 −1 0

0 1 0 1 0 1 0 1

0 −𝑙000𝑙0 0

0 0 0 𝑙0 0 0 −𝑙

𝑙0𝑙0𝑙0𝑙0

,(21)

and can be obtained by calculating the vector 𝐽

𝐽

𝐽𝜉

𝐽

𝐽

𝐽𝜈𝑇

𝑇

𝑇and

rewriting each element as a linear combination of the ele-

ments in 𝐹

𝐹

𝐹.

The new dynamic model (20) adopts tilted thrust forces

𝐹

𝐹

𝐹as inputs. The inputs of the original model (16)𝑇𝑖and 𝛼𝑖

can be retrieved by

𝑇𝑖=𝐹2

𝑠𝑖 +𝐹2

𝑐𝑖 ,(22)

𝛼𝑖=atan2(𝐹𝑠𝑖, 𝐹𝑐 𝑖), 𝛼𝑖∈ [−𝜋 , 𝜋).(23)

The dynamic model (20) has the standard form of an

over-actuated robot, thus can be further simpliﬁed by the

feedback linearization technique by deﬁning the position and

attitude virtual input vectors 𝑢

𝑢

𝑢𝜉and 𝑢

𝑢

𝑢𝜈, and assigning

𝐹

𝐹

𝐹=𝑊

𝑊

𝑊†𝑚(𝑊𝑅

𝑅

𝑅𝐵)𝑇0

0

0

0

0

0𝐼

𝐼

𝐼𝑄(𝑢

𝑢

𝑢𝜉

𝑢

𝑢

𝑢𝜈−𝐺

𝐺

𝐺

0

0

0),(24)

where 𝑊

𝑊

𝑊†is the Moore-Penrose pseudo inverse of 𝑊

𝑊

𝑊such

that

𝑊

𝑊

𝑊 𝑊

𝑊

𝑊†=𝐼

𝐼

𝐼6.(25)

Notice that 𝐹

𝐹

𝐹is in the range space of 𝑊

𝑊

𝑊†according to (24),

and thus a least-square allocation [21,22].

Substituting (24) into (20) yields the ﬁnal system dy-

namic model

𝜉

𝜉

𝜉

𝜈

𝜈

𝜈=𝑢

𝑢

𝑢𝜉

𝑢

𝑢

𝑢𝜈.(26)

3.2. Hierarchical Control

As stated in the previous section, the platform dynamics

has three representations, of which the progressive relation-

ships can be clariﬁed by the evolution of system inputs.

Recall that the primary inputs of the platform are deﬁned

in (2), which are assumed to be controlled in a feedthrough

manner.

Representation 1 is shown in Equation (16), where

inputs 𝑇𝑖are primary, and inputs 𝛼𝑖are controlled by primary

inputs 𝑀𝑦

𝑖through the second-order tilting dynamics (8).

Representation 2 is stated in (20), which uses the tilted

thrust forces 𝐹

𝐹

𝐹in (18) as new inputs to circumvent the

L.Ruan et al.:Preprint submitted to Elsevier Page 5 of 14

Figure 2: (a) The presented hierarchical control architecture. The high-level position and attitude controllers generate virtual

inputs for the integrator dynamics, then transformed to the desired thrust forces and tilting angles of the four quadcopters by

the feedback linearization and the nonlinear allocation. The low-level controllers track the desired thrust force and tilting angle

onboard each quadcopter. (b) The nominal loop is an equivalent representation of Figure 2(a). The two dashed paths are virtual

and cancel each other. The system uncertainty and approximations are represented by the uncertainty Δ

Δ

Δ.

nonlinear coupling between 𝑇𝑖and 𝛼𝑖in Representation 1.

Representation 3 is demonstrated in (26), which absorbs

the inertia, coordinate rotation and gravity into the virtual

inputs 𝑢

𝑢

𝑢𝜉and 𝑢

𝑢

𝑢𝜈in (24), so that the platform dynamics is

reduced to integrator dynamics.

A hierarchical control architecture is designed to cover

the transitions among the three representations, as shown in

Figure 2(a). The controller is composed of high level and low

level parts.

In the high level, LQRi controllers are adopted for posi-

tion and attitude tracking. Deﬁne the augmented error state

vectors as

𝐸

𝐸

𝐸𝜉=

∫𝑡

0𝑒

𝑒

𝑒𝜉𝑑𝑡

𝑒

𝑒

𝑒𝜉

𝑒

𝑒

𝑒𝜉

, 𝐸

𝐸

𝐸𝜈=

∫𝑡

0𝑒

𝑒

𝑒𝜂𝑑𝑡

𝑒

𝑒

𝑒𝜂

𝑒

𝑒

𝑒𝜈

,(27)

where the errors are calculated by

𝑒

𝑒

𝑒𝜉=𝜉

𝜉

𝜉𝑑−𝜉

𝜉

𝜉,

𝑒

𝑒

𝑒𝜂=1

2[(𝑊𝑅

𝑅

𝑅𝐵)𝑇(𝑊𝑅

𝑅

𝑅𝑑

𝐵)−(𝑊𝑅

𝑅

𝑅𝑑

𝐵)𝑇(𝑊𝑅

𝑅

𝑅𝐵)]∨,(28)

according to [36], and the operator [⋅]∨refers to the vee

mapping from Lie algebra 𝔰𝔬(3) to ℝ3. The position and

attitude virtual inputs are then designed by

𝑢

𝑢

𝑢𝜉=

𝜉

𝜉

𝜉𝑑+𝐾

𝐾

𝐾𝜉𝐸

𝐸

𝐸𝜉,

𝑢

𝑢

𝑢𝜈=

𝜈

𝜈

𝜈𝑑+𝐾

𝐾

𝐾𝜈𝐸

𝐸

𝐸𝜈,(29)

where the state feedback gain matrices𝐾

𝐾

𝐾𝜉, 𝐾

𝐾

𝐾𝜈are calculated

by solving the Riccati equations for error dynamics on the

augmented error state vectors 𝐸

𝐸

𝐸𝜉and 𝐸

𝐸

𝐸𝜈. The virtual inputs

are then transformed into 𝐹

𝐹

𝐹by the feedback linearization

(26), and further into the desired 𝑇𝑖and 𝛼𝑖by the nonlinear

transformations (22) and (23).

In the low level, although 𝑇𝑖are primary inputs, 𝛼𝑖

need to be controlled through the second-order tilting dy-

namics (8). Double-loop PID controllers are used on each

quadcopter for the high-bandwidth tracking of tilting angle

trajectories, as elaborated in [28].

The robust stability of the presented hierarchical con-

trol architecture has not been quantitatively addressed, even

though veriﬁed on various tiltable-rotor aerial platforms

through simulations and experiments [21,22,30]. This is

mainly due to the high nonlinearity within the dynamics.

In practice, the low-level dynamics are usually designed to

be much faster than the high-level dynamics, so the overall

robustness is dominated by the high-level control [37]. This

section aims to develop a quantitative criterion for robust sta-

bility, under certain approximations of the system dynamics.

An alternative representation of the hierarchical control

architecture is demonstrated in Figure 2(b). It can be ob-

served that the nominal loop is equivalent to the structure

in 2(a). Here 𝑃

𝑃

𝑃is the integrator dynamics (26), 𝑃

𝑃

𝑃𝑎the low-

level tracking dynamics for 𝑇𝑖and 𝛼𝑖,𝐶

𝐶

𝐶the position and

attitude controllers.

and

†refers to the forward and

inverse feedback linearization, and

and

−1 the forward

and inverse nonlinear transformations between 𝐹

𝐹

𝐹and 𝑇𝑖&

𝛼𝑖. The two dashed paths are virtual and cancel each other,

thus do not aﬀect the overall dynamics.

L.Ruan et al.:Preprint submitted to Elsevier Page 6 of 14

The feedback linearization block

can be explicitly

calculated as

=𝑊

𝑊

𝑊†𝑄

𝑄

𝑄, (30)

where

𝑄

𝑄

𝑄=1

𝑚

𝑊𝑅

𝑅

𝑅𝐵0

0

0

0

0

0𝐼

𝐼

𝐼−1

𝑄.(31)

The nonlinear transformation

can be approximated with

a variational approach. When the system is disturbed by a

small signal,

𝛿𝐹𝑠𝑖

𝐹𝑐𝑖 =𝑠𝛼𝑖𝑐𝛼𝑖𝑇𝑖

𝑐𝛼𝑖−𝑠𝛼𝑖𝑇𝑖𝛿𝑇𝑖

𝛼𝑖≜

𝑖𝛿𝑇𝑖

𝛼𝑖,(32)

where 𝛿refers to the variation operator. Therefore, the non-

linear transformation

can be approximated by the linear

transformation

=𝑑𝑖𝑎𝑔(

0,

1,

2,

3).(33)

The nominal loop gain 𝐿

𝐿

𝐿is then calculated by

𝐿

𝐿

𝐿= (

𝐺

𝐺

𝐺

†)(

−1𝑃

𝑃

𝑃𝑎

−𝐼

𝐼

𝐼8)

=𝑊

𝑊

𝑊†𝑄

𝑄

𝑄𝐺

𝐺

𝐺𝑄

𝑄

𝑄−1𝑊

𝑊

𝑊

−1(𝑃

𝑃

𝑃𝑎−𝐼

𝐼

𝐼8)

,(34)

where𝐺

𝐺

𝐺denotes the diagonal closed-loop high-level dynam-

ics

𝐺

𝐺

𝐺= (𝐼

𝐼

𝐼6+𝐶𝑃

𝐶𝑃

𝐶 𝑃 )−1𝐶𝑃

𝐶𝑃

𝐶𝑃 =𝐺𝑝𝐼

𝐼

𝐼30

0

0

0

0

0𝐺𝑎𝐼

𝐼

𝐼3.(35)

Here each DoF of position and attitude is represented by the

identical scalar transfer functions 𝐺𝑝and 𝐺𝑎respectively.

Therefore, considering slow varying platform attitude, we

obtain

𝑄

𝑄

𝑄𝐺

𝐺

𝐺𝑄

𝑄

𝑄−1 =𝐺𝑝(1

𝑚

𝑊𝑅

𝑅

𝑅𝐵)𝐼

𝐼

𝐼3(𝑚𝑊𝑅

𝑅

𝑅𝑇

𝐵) 0

0

0

0

0

0𝐺𝑎𝐼

𝐼

𝐼−1

𝑄𝐼

𝐼

𝐼𝑄=𝐺

𝐺

𝐺.

(36)

The low-level dynamics 𝑃

𝑃

𝑃𝑎is diagonal, where each element

is either the feedthrough thrust tracking dynamics, or the

tilting angle tracking dynamics, denoted as 𝑃𝑎. Consider the

worst-case scenario where all channels are represented by

the relatively slow dynamics 𝑃𝑎, then

−1(𝑃

𝑃

𝑃𝑎−𝐼

𝐼

𝐼8)

= (𝑃𝑎− 1)

−1𝐼

𝐼

𝐼8

= (𝑃𝑎− 1)𝐼

𝐼

𝐼8.(37)

The system approximations and uncertainties are repre-

sented by the uncertainty block Δ

Δ

Δin Figure 2(b). Deﬁning

the uncertainty upper bound

Δ

Δ

Δ≤Δ𝑈⋅𝐼

𝐼

𝐼8,(38)

the robust stability criterion can be obtained by the small

gain theorem

𝑇

𝑇

𝑇𝐶Δ

Δ

Δ∞≤𝑊

𝑊

𝑊†

𝐺𝑝Δ𝑈

1+𝐺𝑝(1−𝑃𝑎)

⋅𝐼

𝐼

𝐼30

0

0

0

0

0𝐺𝑎Δ𝑈

1+𝐺𝑎(1−𝑃𝑎)

⋅𝐼

𝐼

𝐼3

𝑊

𝑊

𝑊∞

< 𝐼

𝐼

𝐼8,

(39)

where 𝑇

𝑇

𝑇𝐶refers to the complementary sensitivity function

from 𝑆1to 𝑆2in Figure 2(b). It can be observed that

the proposed criterion reduced to the qualitative stability

criterion [37] when 𝑃𝑎is much faster than 𝐺𝑝and 𝐺𝑎, which

echos with the previous works [21,30]. Robust stability

of the control architecture also indicates its capability of

disturbance rejection.

3.3. Optimal Allocator to Suppress Maximum

Required Thrust

Using eight DoF input 𝐹

𝐹

𝐹to represent the six DoF virtual

inputs 𝑢

𝑢

𝑢𝜖and 𝑢

𝑢

𝑢𝜈has multiple solutions, including the least-

square allocation (24) widely used in previous works [21,30,

31]. Actually, it can be observed from (20) that adding any

components from the nullspace of 𝑊

𝑊

𝑊

(𝑊

𝑊

𝑊)={𝑥

𝑥

𝑥∈ℝ8∶𝑥

𝑥

𝑥=𝜎1𝑣

𝑣

𝑣1+𝜎2𝑣

𝑣

𝑣2,∀𝜎1, 𝜎2∈ℝ},(40)

where

𝑣

𝑣

𝑣1=−1 1 −1 1 0

0

01×4𝑇,

𝑣

𝑣

𝑣2=0

0

01×4 −1 1 −1 1𝑇(41)

to 𝐹

𝐹

𝐹will generate a new allocation without inﬂuencing the

resulting virtual inputs 𝑢

𝑢

𝑢𝜖and 𝑢

𝑢

𝑢𝜈.

However, these diﬀerent allocations do inﬂuence the

calculated thrust forces 𝑇𝑖, according to (22). In a practical

platform, the maximum thrust force provided by each quad-

copter is physically constrained, and controller failure could

occur when the required thrust forces exceed this limitation.

This saturation eﬀect also limits the operational space of the

platform as the thrust eﬃciency decreases with larger atti-

tude angle [22]. Therefore, this section provides the design

process of an optimal allocator to minimize the maximum

required thrust and thus achieve a larger operational space

of tilting angle platforms (including the previous platforms

[20,21]) under thrust limitation, compared to the least-

square allocation (24).

The optimal allocator is obtained by designing the

nullspace components of 𝐹

𝐹

𝐹to minimize the maximum

required thrust

(𝜎𝑜𝑝𝑡

1, 𝜎𝑜𝑝𝑡

2) = arg min

(𝜎1,𝜎2)(max

𝑖𝑇𝑖),(42)

given that each element of 𝐹

𝐹

𝐹under translational dynamics

can be written as a function of 𝜎1and 𝜎2as

𝐹𝑠0= − 𝜁𝑥

2−𝜎1, 𝐹𝑐0= − 𝜁𝑧

4−𝜎2,

𝐹𝑠1=𝜁𝑦

2+𝜎1, 𝐹𝑐1= − 𝜁𝑧

4+𝜎2,

𝐹𝑠2=𝜁𝑥

2−𝜎1, 𝐹𝑐2= − 𝜁𝑧

4−𝜎2,

𝐹𝑠3= − 𝜁𝑦

2+𝜎1, 𝐹𝑐3= − 𝜁𝑧

4+𝜎2,

(43)

where 𝜁𝑥,𝜁𝑦and 𝜁𝑧refers to the three DoF elements

of virtual input 𝑢

𝑢

𝑢𝜁. As the platform is symmetric in the

L.Ruan et al.:Preprint submitted to Elsevier Page 7 of 14

𝑥𝐵and 𝑦𝐵directions, we constrain the scenario within the

𝑥𝑏𝑂𝐵𝑧𝐵plane for simplicity. The total thrust forces of the

four quadcopters can be explicitly calculated by (22) as

𝑇0=(𝜁𝑥

2+𝜎1)2+ ( 𝜁𝑧

4+𝜎2)2,

𝑇1=𝑇3=𝜎2

1+ ( 𝜁𝑧

4−𝜎2)2,

𝑇2=(− 𝜁𝑥

2+𝜎1)2+ ( 𝜁𝑧

4+𝜎2)2,

(44)

which are convex functions of 𝜎1and 𝜎2. Therefore, (42) is

an unconstrained convex optimization, thus having a unique

global optimal solution. The rest of this section will show

that this solution has a closed form and can be calculated

with the geometric interpretation.

Deﬁne

ℎ(𝜎1, 𝜎2) = max

𝑖𝑇𝑖,(45)

then it is a piece-wise function with respect to the nullspace

(𝑊

𝑊

𝑊)as

ℎ(𝜎1, 𝜎2) =

𝑇0,(𝜎1, 𝜎2) ∈ 0

𝑇1,(𝜎1, 𝜎2) ∈ 1

𝑇2,(𝜎1, 𝜎2) ∈ 2

,(46)

where 0refers to the region that 𝑇0≥𝑇1and 𝑇0≥𝑇2, or

0= {(𝜎1, 𝜎2) ∶ 𝜎1≥0, 𝜎2≥−𝜁𝑥

𝜁𝑧

𝜎1−𝜁2

𝑥

4𝜁𝑧

}; (47)

2refers to the region that 𝑇2≥𝑇0and 𝑇2≥𝑇1, or

2= {(𝜎1, 𝜎2) ∶ 𝜎1≤0, 𝜎2≥𝜁𝑥

𝜁𝑧

𝜎1−𝜁2

𝑥

4𝜁𝑧

}; (48)

and 1can be calculated by the set subtraction operation

1=ℝ2−0−2.(49)

The min-max optimization (42) then becomes the com-

parison of the minimal values in these three diﬀerent regions,

as

(𝜎𝑜𝑝𝑡

1, 𝜎𝑜𝑝𝑡

2) = arg min

(𝜎1,𝜎2){ℎ0, ℎ1, ℎ2},(50)

where

ℎ0= min 𝑇0,(𝜎1, 𝜎2) ∈ 0

ℎ1= min 𝑇1,(𝜎1, 𝜎2) ∈ 1

ℎ2= min 𝑇2,(𝜎1, 𝜎2) ∈ 2

.(51)

Each minimal value can be geometrically interpreted as

a minimal distance, as shown in Figure 3, where the points

𝐶0,𝐶1,𝐶2are deﬁned as

𝐶0= (− 𝜁𝑥

2,−𝜁𝑧

4),

𝐶1= (0,𝜁𝑧

4),

𝐶2= ( 𝜁𝑥

2,−𝜁𝑧

4),

(52)

(a) When 𝑢

𝑢

𝑢𝜁satisﬁes (56), ℎ0,ℎ1

and ℎ2obtain the same minimal

value at the same point in the

nullspace.

(b) When 𝑢

𝑢

𝑢𝜁satisﬁes (58), ℎ0

and ℎ2obtain the same minimal

value at the same point in the

nullspace.

Figure 3: Geometric interpretation of the presented min-max

optimization.

in the (𝜎1, 𝜎2)space, and 0,1and 2are marked in

cyan, green and pink respectively. According to (44), ℎ0

is equivalent to the minimal distance between 𝐶0and an

arbitrary point 𝑃∈0, as

ℎ0= min

𝑃∈0

𝐶0𝑃=𝐶0𝑃0.(53)

Similarly,

ℎ1= min

𝑃∈1

𝐶1𝑃=𝐶1𝑃1,(54)

ℎ2= min

𝑃∈2

𝐶2𝑃=𝐶2𝑃2.(55)

Here 𝑃0,𝑃1and 𝑃2stands for the optimal point in the

nullspace (𝜎1, 𝜎2)where the minimal values ℎ0,ℎ1and ℎ2

are obtained. It can be observed from Figure 3that the

optimal point varies with a diﬀerent selection of 𝑢

𝑢

𝑢𝜁. When

−𝜁𝑧

4≤−𝜁2

𝑥

4𝜁𝑧

,(56)

ℎ0,ℎ1and ℎ2obtain the same minimal value at the same

point

(𝜎𝑜𝑝𝑡

1, 𝜎𝑜𝑝𝑡

2) = 𝑃0=𝑃1=𝑃2= (0,−𝜁2

𝑥

4𝜁𝑧

),(57)

as shown in Figure 3(a).

On the other hand, when

−𝜁𝑧

4>−𝜁2

𝑥

4𝜁𝑧

,(58)

ℎ0and ℎ2obtain the same minimal value at the same point

(𝜎𝑜𝑝𝑡

1, 𝜎𝑜𝑝𝑡

2) = 𝑃0=𝑃2= (0,−𝜁𝑧

4),(59)

as shown in Figure 3(b).

Therefore, the optimal solution for the min-max opti-

mization (42) is

(𝜎𝑜𝑝𝑡

1, 𝜎𝑜𝑝𝑡

2) = (0,−𝜁2

𝑥∕(4𝜁𝑧)) when (56) holds

(0,−𝜁𝑧∕4) when (58) holds .(60)

L.Ruan et al.:Preprint submitted to Elsevier Page 8 of 14

Figure 4: Experiment setup. The motion capture system

measures the position and attitude of the platform central

frame, and send the data to the target PC through Ethernet,

where the position and attitude controllers in addition to the

input allocator run. The inner-loop reference signals are sent to

each quadcopter through 2.4G radio. The inner-loop tracking

controllers run onboard each quadcopter.

The corresponding optimal allocation can be expressed

as

𝑇𝑑

𝑖=𝐹2

𝑠𝑖 + (𝐹𝑐𝑖 + (−1)𝑖+1 𝜎𝑜𝑝𝑡

2)2,(61)

for the desired thrust forces and

𝛼𝑑

𝑖=atan2(𝐹𝑠𝑖, 𝐹𝑐 𝑖 + (−1)𝑖+1𝜎𝑜𝑝𝑡

2),(62)

for the desired tilting angles, where 𝐹𝑠𝑖 and 𝐹𝑐𝑖 refers to the

least-square allocation results.

4. Simulations and Experiments

To demonstrate the eﬀectiveness of the presented con-

ﬁguration and the control/allocation scheme, simulations

and experiments are conducted. This section concludes the

results of three tests: (1) The independent tracking of six DoF

trajectories to verify the full actuation functionality of the

presented conﬁguration; (2) The maximum reachable incli-

nation angle under the proposed allocation in comparison

with the regular least-square allocation to verify our control

advantages on suppressing the maximum required thrust and

thus enlarging the operational space under input saturation;

(3) The recovery under external impulse disturbance to

verify the stability of the control architecture.

4.1. Setups

Simulations and experiments are conducted on the proto-

type built in UCLA MacLab [29], where Crazyﬂie 2.1 [38],

of which the key parameters are tested in [39], are selected

as the quadcopter module. The prototype critical parameters

are listed in Table 2. For experiments, the central controller

runs on the Ubuntu 16.04 operating system on the target PC.

The control commands are sent to each quadcopter via the

Crazy Radio PA antennas through 2.4 G radio. Quadcopter

controllers run on the onboard STM32. An Optitrack motion

Table 2

Critical Parameters of the Prototype for Simulations and

Experiments.

Parameter Value

𝑚0.16 kg

𝑙0.14 m

𝑎4.60 cm

𝐼𝑥𝑥 1.46 × 10−3 kg⋅m2

𝐼𝑦𝑦 1.46 × 10−3 kg⋅m2

𝐼𝑧𝑧 2.77 × 10−3 kg⋅m2

𝑇𝑀0.55N

Table 3

Six DoF RMS Tracking Errors of Test 1 in Simulation (S) and

Experiment (E).

𝑥(mm) 𝑦(mm) 𝑧(mm) roll(rad) pitch(rad) yaw(rad)

S 2.60 2.63 1.62 0.008 0.007 0.006

E 9.00 6.90 6.13 0.017 0.026 0.030

capture system is used for position and attitude measure-

ments in the indoor environment, and communicates with

the target PC through Ethernet. The PC-quadcopter com-

munication rate is set to 100 𝐻𝑧. The outer-loop controller

runs at 100 𝐻 𝑧. Quadcopter controllers run at 500 𝐻 𝑧 to

ensure fast inner-loop response. The system is demonstrated

in Figure 4.

The simulation model includes the gyroscopic eﬀect of

the frame, motor inner dynamics and saturation of motor

speed, which are neglected in the model for controller de-

sign. Oﬀsets of quadcopters’ center of mass with respect to

the hinge axis are calculated by quadcopters’ free responses

along passive hinges. A communication delay of 20 𝑚𝑠 is

added in the simulation model [28]. Representative sensor

noises are also included [30].

4.2. Test 1: Independent Tracking of Six DoF

Trajectories

The major desired functionality of the presented plat-

form conﬁguration is full actuation, e.g. the capability to

track six DoF trajectories independently. Test 1 designs

six reference trajectories for the six DoF respectively, each

of which contains multiple line segments connected with

non-diﬀerentiable connecting points to explore the tracking

performance in both low and high frequencies. Overlay of

snapshots in test 1 are demonstrated in Figure 6.

The tracking performances in simulation and experiment

are shown in Figure 5. Consistency of the results indicates

the validation of the presented dynamic model and the con-

troller. The Rooted-Mean-Square (RMS) tracking errors are

summarized in Table 3. The performance is comparable with

the state-of-the-art works [20] and [21]. Furthermore, com-

pared with the controller previously designed on the same

prototype [29], the RMS error under the same trajectory is

signiﬁcantly decreased, especially in the 𝑥(95% in S, 83%

in E) and the 𝑦(94% in S, 83% in E) directions.

L.Ruan et al.:Preprint submitted to Elsevier Page 9 of 14

(a) S: Position. (b) E: Position.

(c) S: Attitude. (d) E: Attitude.

(e) S: Thrust Forces. (f) E: Thrust Forces.

(g) S: Tilting Angles. (h) E: Tilting Angles.

Figure 5: Test 1 results: tracking six DoF independent trajec-

tories in simulation (S) and experiment (E).

Figure 6: Overlay of snapshots in the experiment of test

1, where the prototype in each DoF tracks an independent

trajectory. 1

○-4

○indicate the corresponding regions in Figure5.

4.3. Test 2: Maximum Inclination Angle

All strict allocation methods from virtual inputs 𝑢

𝑢

𝑢𝜉and

𝑢

𝑢

𝑢𝜈by exploring the nullspace (40) are supposed to be equiv-

alent under the ideal case where thrust forces do not have

limitations. However, that is not the case in practice. As the

thrust eﬃciency of tilting cant angle multirotor platforms

decreases with increasing platform attitude [22], suppressing

the maximum required thrust force will result in a larger

operational space under the same thrust limit.

This test compares the operational spaces under the

proposed min-max allocation and the regular least-square

allocation by testing the maximum inclination angle in the

pitch direction. During the test, the prototype initiates by

hovering at a ﬁxed point with zero attitude. The prototype

pitch angle then increases by tracking a ramp trajectory until

ﬂight failure occurs, when either position or attitude errors

exceed the tolerance boundaries [28]. The largest pitch angle

before failure is deﬁned as the maximum inclination angle

under one certain allocation method.

Test 2 is conducted in both simulation and experiment,

and the results are shown in Figure 7. It can be observed

that (1) the maximum inclination angle of the proposed al-

location method shows an increase compared with the least-

square allocation in both simulation (0.11 rad or 17%) and

experiment (0.14 rad or 33%); (2) the proposed allocation,

compared with the least-square allocation, does suppress

the maximum required thrust force for the same desired

pitch angle; (3) the simulation results match with the exper-

iment results. These observations verify that the proposed

allocation method suppresses the maximum required thrust

force and results in a larger operational space under thrust

limitations in contrast with the least-square allocation.

4.4. Test 3: Recovery under Impulse Disturbance

The stability of the presented hierarchical control archi-

tecture can be demonstrated by the response of an impulse

disturbance, as shown in Figure 8. The disturbance is created

by artiﬁcially injecting additive signals to the original inputs

in the experiment, as shown in Figure 8(a). The thrust forces

𝑇1and 𝑇3are biased by 0.02 N and -0.05 N from 1s to

1.3s respectively, which is equivalent to injecting an impulse

disturbance force and impulse disturbance torque simultane-

ously to the platform. It can be observed that both position

and attitude in all six DoF are deviated and recovered to

the initial states, indicating the stability of the controller

and its robustness under external disturbances. The impulse

response also shows that the low-level dynamics is much

faster than the high-level position and attitude controllers.

5. Discussion

This paper presented a novel conﬁguration of fully actu-

ated multirotor aerial platform, which replaces the propeller-

rotor pair and tilting motor in the existing conﬁgurations [20,

21,22] with regular quadcopter mounted on a passive hinge.

The presented conﬁguration largely reduces the diﬃcul-

ties of design and prototyping of tiltable-rotor aerial plat-

forms, and also eliminates the disturbances from propeller

drag, gyroscopic eﬀect and tilting reaction, which are all

inevitable for the aforementioned tiltable-rotor platforms.

One unique functionality of the presented conﬁguration is

the fast auxiliary input, which has been proved to be eﬀective

on improving the tracking performance of high-bandwidth

references [33] and recovering from actuator failure [34].

L.Ruan et al.:Preprint submitted to Elsevier Page 10 of 14

0 5 10 15 20 25

Time (s)

0

0.2

0.4

0.6

0.8

Orientation (rad)

Maximum angle

0.64rad

roll pitch yaw

(a) S(LS): Attitude.

0 5 10 15 20 25

Time (s)

0

0.2

0.4

0.6

0.8

Orientation (rad)

Maximum angle

0.75rad

roll pitch yaw

(b) S(MO): Attitude.

0 5 10 15 20 25

Time (s)

0

0.2

0.4

0.6

0.8

Orientation (rad)

Maximum angle

0.42rad

roll pitch yaw

(c) E(LS): Attitude.

0 5 10 15 20 25

Time (s)

0

0.2

0.4

0.6

0.8

Orientation (rad)

Maximum angle

0.56rad

roll pitch yaw

(d) E(MO): Attitude.

(e) S(LS): RMS Error. (f) S(MO): RMS Error. (g) E(LS) RMS Error. (h) E(MO): RMS Error.

0 5 10 15 20 25

Time (s)

0.2

0.3

0.4

0.5

0.6

Thrust (N)

T0T1T2T3

(i) S(LS): Thrust Forces.

0 5 10 15 20 25

Time (s)

0.2

0.3

0.4

0.5

0.6

Thrust (N)

T0T1T2T3

(j) S(MO): Thrust Forces.

0 5 10 15 20 25

Time (s)

0.2

0.3

0.4

0.5

0.6

Thrust Forces(N)

T0T1T2T3

(k) E(LS): Thrust Forces.

0 5 10 15 20 25

Time (s)

0.2

0.3

0.4

0.5

0.6

Thrust Forces(N)

T0T1T2T3

(l) E(MO): Thrust Forces.

0 5 10 15 20 25

Time (s)

-2

-1

0

1

2

3

Tilting Angles (rad)

0

d

0

1

d

1

2

d

2

3

d

3

(m) S(LS): Tilting Angles.

0 5 10 15 20 25

Time (s)

-2

-1

0

1

2

3

Tilting Angles (rad)

0

d

0

1

d

1

2

d

2

3

d

3

(n) S(MO): Tilting Angles.

0 5 10 15 20 25

Time (s)

-2

-1

0

1

2

3

Tilting Angles (rad)

0

d

0

1

d

1

2

d

2

3

d

3

(o) E(LS): Tilting Angles.

0 5 10 15 20 25

Time (s)

-2

-1

0

1

2

3

Tilting Angles (rad)

0

d

0

1

d

1

2

d

2

3

d

3

(p) E(MO): Tilting Angles.

Figure 7: Test 2 results: reaching the maximum inclination angle under the least-square (LS) allocation and the proposed min-max

optimal (MO) allocation, in simulation (S) and experiment (E).

012345

Time (s)

-0.1

0

0.1

Disturbance (N)

T0

dist T1

dist T2

dist T3

dist

(a) Disturbances.

012345

Time (s)

-0.2

0

0.2

Position(m)

xref

x

yref

y

zref

z

(b) Position.

012345

Time (s)

-0.2

0

0.2

0.4

0.6

Orientation (rad)

rollref

roll

pitchref

pitch

yawref

yaw

(c) Attitude.

012345

Time (s)

0.3

0.4

0.5

0.6

Thrust Forces(N)

T0T1T2T3

(d) Thrust forces.

012345

Time (s)

-1

-0.5

0

0.5

1

Tilting Angles (rad)

0

d

0

1

d

1

2

d

2

3

d

3

(e) Tilting angles.

Figure 8: Test 3 results: Recovery under an impulse disturbance

in both position and attitude at hovering in experiment.

However, one common limitation of all one DoF tiltable-

rotor aerial platforms, including [20,21] and the presented

one, is that the thrust eﬃciency decreases when the atti-

tude becomes larger. Therefore, when the thrust force is

limited, the operational space will be constrained. This can

be improved by either algorithms or mechanisms. As for

the algorithm, this paper proposed an min-max optimal

allocation method that ﬁts all one DoF tiltable-rotor aerial

platforms. The new allocation method can improve the op-

erational space under thrust saturation compared with the

least-square method [21]. As for the mechanism, [27] ap-

plied universal joints to connect multiple quadcopters, but

the operational space is still constrained by the limited range

of the joint. [30] proposed a two DoF passive gimbal joint for

universal rotation and achieves higher thrust eﬃciency, but

increased the mechanical complexity and structural compli-

ance, and sacriﬁced the auxiliary inputs for tracking perfor-

mance improvement and actuator failure recovery.

The hierarchical control architecture applied in this

paper has been widely used for tiltable-rotor aerial plat-

forms [21,22,30,40]. However, the stability and robustness

L.Ruan et al.:Preprint submitted to Elsevier Page 11 of 14

remain challenging issues. Apart from the robustness from

the high-level LQRi controllers when the low-level tracking

bandwidth is suﬃcient, a quantitative criterion for robust

stability is proposed in this paper to take into account the

overall eﬀect of high-level and low-level dynamics, thus

can be used to design robust controllers to handle model

uncertainty and disturbance rejection. However, it should

be noticed that the disturbance rejection capability can be

better addressed by explicitly introducing disturbance in the

controller design, as presented in [41], which can be one

important direction for the future works.

The tracking performance of the proposed controller

reaches the level of the state-of-the-arts [20,21]. However,

compared with regular quadcopters, the performance still

has room for improvement. This can be one promising future

direction of this work, to elevate the accuracy either by

more advanced control algorithms [42] or estimation of

disturbances and uncertainties [43].

6. Conclusion

The presented tilting-thrust actuator module is simple

in the mechanical design and the quad-conﬁguration aerial

platform in this paper has the full actuation capability for six

DoF motions. The dynamics of aerial platforms made up of

these modules, when properly conﬁgured can be modeled as

a single rigid body for the control design. The hierarchical

control is eﬀective with the communication latency included

in the outer loop control design and the fast inner loop actu-

ator dynamics. A quantitative stability criterion is presented

for controller design. Our actuator modules with quadcopters

mounted on passive hinges have the advantages that the

passive hinges do not transmit to the platform – the hinge

rotation torque, the rotor’s aerodynamic drag torque, and

the rotor’s gyroscopic torque when changing orientations.

As such the accurate multi-body dynamics of the platform

facilitates the model-based control without these terms con-

sidered as disturbances or unmodeled dynamics.

The optimal control allocation for the over-actuated

aerial platform, in the least-square or min-max formulation,

has been experimentally validated to be eﬀective until hitting

the thrust saturation constraints. Both closed form solutions

are eﬃcient in computation without resorting to numerical

iterations. Subject to the thrust saturation limits against

the total weight, the maximum inclination angle achieved

by the min-max allocation is larger than that achieved by

the least-square allocation. Any arbitrary platform attitude

angle would have been achievable had the thrust force

been suﬃciently large to overcome gravity, thanks to the

unlimited range of the passive hinge rotations.

The general agreements in the simulation and experi-

mental results suggest that the modeling is adequate for the

model-based control design with the unmodeled factors in

check.

Acknowledgement

The authors would like to appreciate Dr. Wenzhong Yan

and Dr. Ankur Mehta for the access and assistance in the

motion capture system.

Declaration of Competing Interests

The authors declare that they have no known competing

ﬁnancial interests or personal relationships that could have

appeared to inﬂuence the work reported in this paper.

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L.Ruan et al.:Preprint submitted to Elsevier Page 13 of 14

Lecheng Ruan received the B.S. honor degree

from the School of Mechatronic Engineering,

Harbin Institute of Technology in 2015, and the

Ph.D. degree from the Department of Mechanical

and Aerospace Engineering, University of Cali-

fornia, Los Angeles in 2020. He is now aﬃliated

with Beijing Institute for General Artiﬁcial Intelli-

gence and Peking University. His research interests

include control and optimization, mechatronics,

robotics, perception and signal processing.

Chen-Huan Pi received the B.S. degree in me-

chanical engineering and the Ph.D. degree in

control science and engineering at the Institute

of Mechanical Engineering from National Chiao

Tung University, HsinChu in 2015 and 2021 re-

spectively. He worked as a visiting researcher in

Aerospace Engineering Department, University of

California, Los Angeles in 2019-2020. His re-

search interests include intelligent control of multi-

rotor unmanned aerial vehicles.

Yao Su received the B.S. degree from the School

of Mechatronic Engineering, Harbin Institute of

Technology in 2016, and the M.S. and Ph.D. de-

grees from the Department of Mechanical and

Aerospace Engineering, University of California,

Los Angeles in 2017 and 2021. He is now a re-

search scientist at Beijing Institute for General Ar-

tiﬁcial Intelligence. His research interests include

robotics, control, planning, and optimization.

Pengkang Yu received the B.Eng. degree in Me-

chanical Engineering from Hong Kong University

of Science and Technology, Hong Kong, in 2016.

He received the M.S. degree and the Ph.D. degree

in Mechanical Engineering from the University

of California, Los Angeles in 2017 and 2022 re-

spectively. His research interests include control,

optimization, planning, robotics and mechatronics.

Stone Cheng received the B.Sc and M.Sc. degrees

in Control Engineering from the National Chiao

Tung University, HsinChu, in 1981 and 1983, re-

spectively, and the Ph.D. degree in electrical engi-

neering from Manchester University, UK in 1994.

He is currently a Professor with the Department

of Mechanical Engineering, National Yang Ming

Chiao Tung University. His current research inter-

ests include motion control, reinforcement learn-

ing, and the wide-band-gap semiconductor power

device.

Tsu-Chin Tsao received the B.S. degree in engi-

neering from National Taiwan University, Taipei,

in 1981, and the M.S. and Ph.D. degrees in me-

chanical engineering from the University of Cal-

ifornia, Berkeley, in 1984 and 1988, respectively.

He is currently a Professor with the Mechanical and

Aerospace Engineering Department, University of

California, Los Angeles. His research interests in-

clude precision motion control, mechatronics, and

robotics. Professor Tsao is a Fellow of ASME and

a Senior Member of IEEE.

L.Ruan et al.:Preprint submitted to Elsevier Page 14 of 14