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

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)