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

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  • Beijing Institute for General Artificial Intelligence

Abstract and Figures

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.
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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
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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
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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 𝑠and 𝑐for 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
𝑖𝑠(𝜋
2𝑖)𝑝 𝑐(𝜋
2𝑖)𝑞. (8)
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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= (𝜅𝜏𝜅𝑇)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 𝛼𝑖0, 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
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Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges
opposite direction, e.g. 𝜔
𝜔
𝜔0+𝜔
𝜔
𝜔10.𝜔
𝜔
𝜔2and 𝜔
𝜔
𝜔3have the
same relation. Therefore, 𝐿
𝐿
𝐿𝐺0and thus
𝜏
𝜏
𝜏𝐺0(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=𝐹𝑠0𝐹𝑐0𝐹𝑠3𝐹𝑐3𝑇,(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
𝑊𝑚(𝑊𝑅
𝑅
𝑅𝐵)𝑇0
0
0
0
0
0𝐼
𝐼
𝐼𝑄(𝑢
𝑢
𝑢𝜉
𝑢
𝑢
𝑢𝜈𝐺
𝐺
𝐺
0
0
0),(24)
where 𝑊
𝑊
𝑊is 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
𝑊according 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
Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges
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
𝜉𝑑𝜉
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 3. 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
Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges
The feedback linearization block
can be explicitly
calculated as
=𝑊
𝑊
𝑊𝑄
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
𝑐𝛼𝑖𝑠𝛼𝑖𝑇𝑖𝛿𝑇𝑖
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
𝑊𝑄
𝑄
𝑄𝐺
𝐺
𝐺𝑄
𝑄
𝑄−1𝑊
𝑊
𝑊
−1(𝑃
𝑃
𝑃𝑎𝐼
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
𝑃𝑎𝐼
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
ΔΔ𝑈𝐼
𝐼
𝐼8,(38)
the robust stability criterion can be obtained by the small
gain theorem
𝑇
𝑇
𝑇𝐶Δ
Δ
Δ𝑊
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
𝑥8𝑥
𝑥
𝑥=𝜎1𝑣
𝑣
𝑣1+𝜎2𝑣
𝑣
𝑣2,𝜎1, 𝜎2},(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
2𝜎1, 𝐹𝑐0= 𝜁𝑧
4𝜎2,
𝐹𝑠1=𝜁𝑦
2+𝜎1, 𝐹𝑐1= 𝜁𝑧
4+𝜎2,
𝐹𝑠2=𝜁𝑥
2𝜎1, 𝐹𝑐2= 𝜁𝑧
4𝜎2,
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
Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges
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
4𝜎2)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
(𝜎1, 𝜎2) = max
𝑖𝑇𝑖,(45)
then it is a piece-wise function with respect to the nullspace
(𝑊
𝑊
𝑊)as
(𝜎1, 𝜎2) =
𝑇0,(𝜎1, 𝜎2) 0
𝑇1,(𝜎1, 𝜎2) 1
𝑇2,(𝜎1, 𝜎2) 2
,(46)
where 0refers to the region that 𝑇0𝑇1and 𝑇0𝑇2, or
0= {(𝜎1, 𝜎2) 𝜎10, 𝜎2𝜁𝑥
𝜁𝑧
𝜎1𝜁2
𝑥
4𝜁𝑧
}; (47)
2refers to the region that 𝑇2𝑇0and 𝑇2𝑇1, or
2= {(𝜎1, 𝜎2) 𝜎10, 𝜎2𝜁𝑥
𝜁𝑧
𝜎1𝜁2
𝑥
4𝜁𝑧
}; (48)
and 1can be calculated by the set subtraction operation
1=202.(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){0, 1, 2},(50)
where
0= min 𝑇0,(𝜎1, 𝜎2) 0
1= min 𝑇1,(𝜎1, 𝜎2) 1
2= 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), 0,1
and 2obtain the same minimal
value at the same point in the
nullspace.
(b) When 𝑢
𝑢
𝑢𝜁satisfies (58), 0
and 2obtain 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), 0
is equivalent to the minimal distance between 𝐶0and an
arbitrary point 𝑃0, as
0= min
𝑃0
𝐶0𝑃=𝐶0𝑃0.(53)
Similarly,
1= min
𝑃1
𝐶1𝑃=𝐶1𝑃1,(54)
2= min
𝑃2
𝐶2𝑃=𝐶2𝑃2.(55)
Here 𝑃0,𝑃1and 𝑃2stands for the optimal point in the
nullspace (𝜎1, 𝜎2)where the minimal values 0,1and 2
are obtained. It can be observed from Figure 3that the
optimal point varies with a different selection of 𝑢
𝑢
𝑢𝜁. When
𝜁𝑧
4𝜁2
𝑥
4𝜁𝑧
,(56)
0,1and 2obtain 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)
0and 2obtain 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
Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges
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 kgm2
𝐼𝑦𝑦 1.46 × 10−3 kgm2
𝐼𝑧𝑧 2.77 × 10−3 kgm2
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
Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges
(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
Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges
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
Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges
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
Control and Experiments of a Novel Tiltable-Rotor Aerial Platform Comprising Quadcopters and Passive Hinges
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
ResearchGate has not been able to resolve any citations for this publication.
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