Abstract—In the group formation in leader-follower strategy due

Abstract—In this
article, the problem of controlling a team of Quadrotors that cooperatively
grasps and transports a common payload is considered. In this regard, a novel decentralized
nonlinear robust optimal controller is designed based on Second Order Sliding
Mode (SOSMC) approach which is capable of facing external disturbances. The
method combines control vectors, calculated for each Quadrotor using
Moore-Penrose theory, with individual control vectors aiming at cooperative
grasping of an object. A second order sliding mode controller is used to obtain
the individual control vectors while using a nonlinear observer based on
Extended Kalman-Bucy Filter (EKBF) to estimate the unmeasured states. This approach
leads to robustness against model uncertainties along with high flexibility in
designing the control parameters to have an optimal solution for the nonlinear
dynamic of the system. Design of the controller is based on Lyapunov techniques
which can provide the stability of the end-effecter during tracking of the
desired trajectory. Finally, simulation results are given to illustrate the
effectiveness of the proposed method.

Keywords—Quadrotor, Second Order Sliding Mode Control, Extended
Kalman-Bucy Filter, Cooperative Decentralized Control, External Disturbances

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1.        
Introduction

The cooperative control of multiple
vehicle systems despite its wide range of practical applications requires
tackling important theoretical and practical challenges which have attracted
many researchers in recent years. Some research works have focused on
centralized and leader-follower schemes to create interaction mechanisms
between the object and cooperative Quadrotors 1-6. Nevertheless, these methods
suffer from several disadvantages including high computational complexity of
centralized methods in large-scale systems and possibility of disappearing the
group formation in leader-follower strategy due to not receiving the position
of leader by the followers.

        In
this regard, there is a plethora of research in cooperative multi-robot
controller design based on decentralized control methods which can solve a
significant number of problems in cooperative control strategies and benefit
also from the advantages such as decreased number of sensors and fast
performance. In 7, the problem of cooperation by a team of ground robots is
addressed under quasi-static assumptions based on decentralize controllers considering
a unique solution to robot and object motion. Transporting a payload by aerial
manipulation using cables based on decentralized control is studied in 8,9.
In these papers, the focus is on finding robot configurations and ensuring
static equilibrium of the payload at a desired position while taking into
account the constraints on the tension. In addition, cooperative aerial towing
has been studied in 10. 11 employs bilinear matrix inequalities to present
optimal solutions for decentralized nonlinear multi-agent systems. Despite the
numerous advantages of decentralized methods there are still some critical
issues related to the performance of the control system in presence of
disturbances and uncertainties.

        Sliding mode control
(SMC) is a well-known control strategy robust against disturbances and dynamic
uncertainties 12-16. In this method, the chattering phenomenon can pose
problems; therefore, it should be avoided or at least reduced. A high order
sliding mode concept can effectively reduce the chattering while keeping the
invariant characteristics in the sliding mode. This control strategy shifts the
switching function into higher order, whereas in standard SMC the switching
function is in the first order. This high order (second order) SMC has been
applied in a quadcopter 17-18. In 19, a robust second order sliding mode
controller was proposed for the attitude stabilization of a four-rotor
helicopter. This controller was able to overcome the chattering phenomena in
classical sliding mode control while preserving the invariance property of
sliding mode. In addition, using an equivalent approach, the performance of
control system in 20 was improved by using a second-order sliding mode
control (2-SMC) for second-order uncertain plants. An adaptive second order
sliding mode controller (SOSMC) with a nonlinear sliding surface was introduced
in 21 and a linear switching surface was proposed for similar underactuated
systems in 22. Wind as a disturbance source has also been considered in the
flight process of the quadrotor to demonstrate the robustness of the control
algorithm 23,24.

        In this paper, a novel
decentralized robust optimal controller is designed for a group of UAVs in a
formation which enables manipulation of a common payload in three dimensions.
Our cooperative control algorithm is comprised of two parts. In the first part,
fundamental control vectors, which guarantee the optimal performance of the
coupled system, are determined using Moore-Penrose theory. In the second part,
individual control vectors are designed for each Quadrotor which can guarantee
the robustness of the coupled system against uncertainties and disturbances.
Moreover, a combination of second order sliding mode controller (SOSMC) with an optimal cooperative algorithm is employed to control all of the Quadrotors
such that a soft touching occurs between the object being carried and the
cooperative Quadrotors. A nonlinear filter equivalent to well-known Extended Kalman-Bucy filter (EKBF) in a continuous-time stochastic
system has also been developed in our work aiming to
improve the estimation accuracy and robustness of the system.

        The proposed controller successfully rejects the external
disturbances (e.g. wind effect) for cooperative Quadrotors. The advantages of
using this approach are ranging from simple implementation and flexible
parameter learning to design criteria customization. The simulation
results show the effectiveness and superior performance of the proposed control
strategy for the cooperative Quadrotors grasping and transporting a common
payload in various maneuvers.

The subsequent parts
of this paper are as follows: at first developing a model for a
single Quadrotor and modeling of a team of UAV’s rigidly attached to a payload
is discussed in Sec.2. Sec.3 proposes cooperative Control Algorithm laws defined with respect to the
payload that stabilizes the payload along three dimensional trajectories based
on a SOSMC incorporated with optimal Moore-Penrose theory. We
review and analyze experimental study with teams of Quadrotors cooperatively
grasping, stabilizing, and transporting payloads in Sec.4. It includes
point-to-point path planning and trajectory tracking. Finally, Sec.5 concludes the paper with some future improvement
suggestions.

2.                 
Mathematical Modeling and Coordinate Reference Frame

Fig.1 shows the configuration of our quadrotor system that can fly in
all directions and has no limit on maneuver. Two main reference frames are
considered in order to analyze the dynamic: model frame,

, and the body-fixed reference frame,

. In this nomenclature, ?

and

 are the position and orientation vectors in reference

.
The three Euler angels are named roll angle

,
pitch angle

 and yaw angle

.

 is the thrust forced
produced by four rotors. The full Quadrotor
dynamic model with the

  motions as
outcome of a pitch, roll and yaw rotation is 25:

(1)

The model (1)
can be rewritten in a state-space form,

, by
introducing

which is state vector
of the system:

(2)

Finally, the
control inputs are calculated as:

(3)

In this equation

 and

 represent the
thrust and drag coefficient, respectively. The input

 is related to
the total thrust; while the inputs

 are related to
the rotations of the Quadrotor and

 are external
disturbances,

 denotes the
inertia of the z-axis and

  is the overall
residual propeller angular speed. Finally,
the equation of motion for the
coupled system is obtained as follows 25:

(4)

where

 is the net body force and

,

,

 are the body moments in the

 direction. Also,

 is the total force from each Quadrotor and  

,

,

 are moments about each of the Quadrotor’s body
frame axes.

3. The Cooperative Control
Algorithm

In this section, the
perfect state cooperative guidance controller of chasers and target UAV presented
in 26 will be developed to achieve a decentralized control strategy with superior results regarding state estimation. The proposed control algorithm includes three parts. In the first part, the
control fundamental vectors for n Quadrotors using the Moore-Penrose and EKBF methods are
determined. The
second part is composed of determining individual control vectors by Second
Order Sliding Mode Controller (SOSMC). Finally, in last part, the cooperative
control algorithm can be extracted by combining the individual control vectors
and the control fundamental vectors.

3.1. Control Fundamental Vectors of
Cooperative Control Strategy

After determining the set of all force
vectors of the coupled system in (4), four equations each one with 4n unknown variables can
be described as:

(5)

where

 includes 4 control input vectors for each

Quadrotor:

(6)

and

 is a constant vector containing the states of translational and
rotational subsystems, and is obtained from EKBF in Theorem 1.

Theorem 1: A Nonlinear Observer Based on EKBF 27, 28

Consider a nonlinear stochastic system with uncorrelated Gaussian process
and measurement noise as follows:

(7)

where

 is state vector
and

 is the
measurements vector.

 denotes the state
noise and the vector

 represents the
measurement noise. Then states of the system can be estimated by an optimal
(minimum variance) recursive algorithm designed by determining the filtering
and prediction equations. In the filtering step, the mean

 at a given instant
of time

 is calculated by
state estimation:

(8)

in which the initial conditions are:

(9)

and estimation of error covariance matrix is determined as:

(10)

and Kalman filter gain is computed as:

(11)

where

 represents the
covariance matrix of the sensor noise. In prediction step, we have:

(12)

(13)

where,

 is the Jacobean
matrix of

 evaluated at

.

Now, consider the following cost function to minimize the control
inputs 26:

(14)

in which

 are the weight factors. By using the mathematical theory of Moore-Penrose inverse, an optimal vector to reach the
desired values can be written as:

(24)

where:

 ,

(15)

and the
columns of the matrix

 are the control fundamental vectors

. Thus,
the optimal control input vector can be rewritten as follows:

(16)

Finally,
after calculating the four fundamental control vectors, the cooperative
control vectors for n collaboration Quadrotors is obtained as follows:

(17)

in which

 are the control fundamental vectors including
the total force, roll, pitch and yow moments generated for the coupled system of n Quadrotors.

3.2. Individual Control Vectors of
Cooperative Control Strategy

Now, to complete the cooperative control algorithm, individual control vectors are
determined by using nonlinear second order sliding mode controller (SOSMC) strategy in order to generate a
robust solution.

3.2.1 Second Order Sliding Mode Control of Quadrotors

        In this paper,
determining individual
control vectors of Cooperative Control Strategy based on second order sliding
mode techniques is desired to preform asymptotic
position and attitude tracking of the quadrotors. Using tracking errors of
state variable(s), a switching sliding surface is considered for the whole
subsystem (rotational or transitional subsystems). The sliding manifolds based on second order sliding mode control
are chosen as
follows 20, 21:

(18)

where the coefficients

. The time derivatives of the sliding manifolds of rotational subsystem are
obtained;

(19)

By making

and

the corresponding control laws are designed;

(20)

where the coefficients of the exponential approach laws

 Now for transitional subsystem, the
objective is to guarantee the state variables

 converge to the desired values

, respectively. The time derivatives of the sliding manifolds of rotational
subsystem are obtained;

(21)

By making

and

the corresponding control laws are designed;

(22)

where the coefficients of the exponential approach laws

. In addition, the terms

 are:

(23)

To synthesize a stabilizing control law by second
order sliding mode, the sufficient condition for the
stability of the system

 must be verified. Time derivative of

 satisfying

where ? is the positive value

 . So, a Lyapunov function can be written as

, that one can describe as 29:

(24)

The chosen
law for the attractive surface is the time derivative of

(25)

Therefore, according to (25) the time derivative of final Lyapunov function

is as:

(26)

Thus, under the control laws, all
the system state trajectories can reach, and thereafter, stay on the
corresponding sliding surfaces, respectively. Finally using (26) the input control
vectors extracted from second sliding mode approach can be expressed as
follows:

(27)

Remark 1. Note that
signum function which appears in second order sliding mode controllers and filters
can cause chattering phenomena, or high-frequency oscillations of control
variables. This problem can be avoided by replacing discontinuous signum function
with appropriate continuous approximation, like;

.

Finally
individual control vectors are generated by SOSMC method which has been
explained in the (28)
and they are represented by parameters

 that are equivalent of

 . Thus the individual control vectors are
derived as follows:

(29)

In
these equations

 is the state vector. Therefore:

(30)

and:

(31)

3.3. Cooperative Control Vectors

Now, from (17) the cooperative control vectors for each
independently designed Quadrotor can be described as follows:

(32)