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Backstepping Control with Sliding Mode. Estimation for a Hexacopter. Carlos A. Arellano-Muro, Luis F. Luque-Vega, B. Castillo-Toledo, Alexander G. Loukianov.
2013 10th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) Mexico City, Mexico. September 30-October 4, 2013

Backstepping Control with Sliding Mode Estimation for a Hexacopter Carlos A. Arellano-Muro, Luis F. Luque-Vega, B. Castillo-Toledo, Alexander G. Loukianov Automatic Control Department CINVESTAV-IPN Campus Guadalajara, Av. Del Bosque 1145, Col. El Bajío, 45019. Zapopan, Jalisco, México Email: {carellano,lluque,toledo,louk}@gdl.cinvestav.mx

but the wind parameters resulting from the aerodynamic forces and moments, which are time dependent functions, are considering in this paper, which is the main contribution of this work. These wind parameters are estimated via a sliding mode observer based on the super twisting algorithm in order to ensure convergence in finite time, this will bring robustness to the complete system against external disturbances and moreover, against parameter variations. This paper is organized as follows: in section II, the nonlinear dynamic model of the hexarotor is described considering external disturbances. Based on this nonlinear model a backstepping control is designed considering Lyapunov stability conditions. In section IV an estimator for wind parameters is designed using the super twisting algorithm. Simulation results are shown in section V. Finally, section VI shows the conclusions of this work.

Abstract— This paper presents the dynamic nonlinear model and the design of a controller for a hexarotor aerial robot. The control algorithm is based on the backstepping technique and it is applied to the trajectory tracking problem. The wind parameters resulting from the aerodynamic forces and moments acting on the hexacopter are estimated via a sliding mode observer with the super twisting algorithm in order to ensure robustness against aerial external disturbances and parameter variations. The performance and effectiveness of the proposed controller are tested in a simulation study taking into account external disturbances which are function of time. Keywords—hexacopter; nonlinear control; backstepping; sliding mode; parameter estimation.

I.

INTRODUCTION

Nowadays, Unmanned Aerial Vehicles (UAV’s) are expected to be an important research field for the automatic control community due to the fact that they have a large number of civil as well as military applications [1]. Multirotors have captured the attention since they offers some important advantages over other aircrafts. One of the aerial vehicles with a strong potential is the hexa-rotor. Hexa-rotors have numerous advantages over quadrotors since they can offer more power and speed due to the properly position of the rotors. Finally, hexa-rotors have additional redundancy over quadrotors [2]. In order to meet the requirements for achieving autonomous flight several control methods have been applied to the hexa-rotor such as input- output feedback linearization [3], backstepping [4], among others. It is important to mention that control design carried out for a quadrotor can be applied to the hexa-rotor or any other multirotor since they are modeled as a rotating rigid body with six degrees of freedom. The Hexa-rotor dynamics are affected by both matched and unmatched perturbations. To overcome the mentioned problem a sliding mode disturbance observer has been designed in Moktari et al. [5] where it is clear that this controller is not robust in face of wind disturbances that affect the forces dynamic model of the displacement (x,y). In this work, the backstepping technique is applied to the trajectory tracking problem following the structure of [11]

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II.

DYNAMIC MODELING OF A HEXAROTOR

The hexarotor configuration has six rotors which generate the propeller forces !!   ! = 1,2,3,4,5,6 . Each rotor consists of a brushless DC motor and a fixed-pitch propeller. This rotorcraft is constituted by three rotors which rotate clockwise (1,3,5), and three rotating counterclockwise (2,4,6). In order to increase the altitude of the aircraft it is necessary to increase the rotor speeds altogether with the same quantity. Forward motion is accomplished by increasing the speed of the rotors (3,4,5) while simultaneously reducing the same value for forward rotors (1,2,6). For leftward motion the speed of rotors (5,6) is increased while the speed of rotors (2,3) is reduced. Backward and rightward motion can be accomplished similarly. Finally, yaw motion can be performed by speeding up or slowing down the clockwise rotors depending on the desired angle direction. Fig. 1 depicts the coordinate axes in order to describe the kinematic model of the hexacopter. Vector !!! corresponds to the x axis of the earth fixed reference frame, ! ! , while !!! is the x axis expressed in the body reference frame ! ! . The space orientation of the hexarotor from ! ! to ! ! is given by the following

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2013 10th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) Mexico City, Mexico. September 30-October 4, 2013 where !!"#! and !!"#! are the forces and torques produced by propeller system, respectively. !!"#$ and !!"#$ are the aerodynamic forces and moments acting on the UAV, respectively, and !!"#$ = !" Θ ! ! is the gravity effect forces with ! = !"# 0,0, ! !/! ! . These variables are defined as !!"#! = 0 !!"#! = !

! ! !!! !! ,     ! !!!! −1 ! !! ! ! ! −!! − ! + ! + !! + ! ! ! !

0

!

! !



!! !

,

!! + !! − !! − !!

!!"#$ = !"# !! , !! , !! , !!"#$ = !!!! !!  Ω×!! −1

!!!

!!

!!"#$ = !"# !! , !! , !! ,

Fig. 1. Hexarotor orientation using Euler angles.

where ! is the distance from the center of mass to the rotor shaft, ! is the drag factor, !! is the rotor inertia, !! is the ! velocity of the rotor !, !! = !!! ! ! are the aerodynamic ! functions, !! are the aerodynamic coefficients where ! = !, !, !, !, !, ! , ! is the air density, and ! = Ω − Ω!"# which is the hexarotor velocity with respect to the air [7].

transformation velocity and rotation velocity matrices, respectively C ψ C θ   ! Θ = SψCθ −Sθ

CψSθSφ − SψCφ SψSθSφ + CψCφ CθSφ

CψSθCφ + SψSφ SψSθCφ − CψSφ CθCφ

and −Sθ ! Θ = CθSφ CθCφ

0 Cφ −Sφ

To simplify the calculations it can be assumed that Θ ≈ Ω, consideration that can be done if the perturbations from hover flight are small [8]. Therefore, using (1) and (2) we yield to the equations describing the dynamics of the hexarotor expressed in the earth reference frame are described as

1 0 0

Where Θ = !"# ψ, θ, φ represents the Euler angles expressed in ! ! . The mathematical model of the hexarotor can be derived according to Newton’s laws of motion in the same manner as for ordinary aircraft [1], the model ideally includes the gyroscopic effects resulting from both the rigid body rotation in space, and the six propeller's rotation [6]. Thus the dynamics equations of motion are described by:

1 ! + !!"                   ! ! 1 ! = C!S!S! − S!C! ! + !!"                   ! ! 1 ! = −! + C ! C ! ! + !!                                                   ! !                     3 !! −!! !! ! ! = !! − !! + !! + !!                       !! !! !! !! −!! !! ! − !! + !! + !!                     ! = !! !! !! !! !! −!! 1 + !! + !!                                                       ! = !! !! !! ! = C!S!C! + S!S!

!!"# = !!! + Ω×!!! !!"# = !Ω + Ω×!Ω                                                       1 Where ! and ! = !"#$ !! , !! , !! are the mass and the inertia matrix of the hexarotor, Ω = !"# !, !, ! and !! = !"# !, !, ! are the angular and translational velocity of the airframe expressed in ! ! , respectively. The external forces and moments Σ!!"# and Σ!!"# are described by

with !! = !!" , !!" , !!  

!!"# = !!"#! − !!"#$ − !!"#$

       ! ! = !! , !! , !!

=

1 ! Θ !!"#$ ,                           !

= ! !! !!"#$                                                                    (4)

! = −!! + !! − !! + !! − !! + !!

!!"# = !!"#! − !!"#$                                                                     2

The abbreviations S(·) and C(·) denote sin(·) and cos (·), respectively.

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!

!

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2013 10th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) Mexico City, Mexico. September 30-October 4, 2013 !! = !! + !! + !! + !! + !! + !!

which is connected to a sliding mode observer as it is shown in Fig. 2.

3 ! + !! − !! − !!                   2 !                             5 1 !! = 2!! + !! − !! − 2!! − !! + !! 2 !! = −!! + !! − !! + !! − !! + !!                         !! =

Assuming that thrust and drag are proportional to the square of the rotor speed, then the force generated by the ith rotor is given by !! = !!!!   ! = 1,2,3,4,5,6 , where ! is the thrust factor. Moreover, !! and ! ! are the resulting aerodynamic forces and moments acting on the UAV, respectively. III.

Fig. 2. Control-observer scheme proposed in this work.

A. Backstepping Control for Rotation Subsystem

BACKSTEPPING CONTROL DESIGN

Backstepping is a recursive Lyapunov-based scheme proposed in the beginning of 1990s. The idea is to design a controller recursively by considering some of the state variables as “virtual controls” and designing for them intermediate control laws [9]. The control design will be carried considering first that external disturbances resulting from the aerodynamic forces and moments are known. Then, a sliding mode observer will be designed to estimate these external disturbances. First, let us define the roll angle error as !! = !!! − !!                                                                        (7)

The dynamic nonlinear model (3) can be used to describe the system in its state space representation form [9] ! = ! !, !, !                                                                           6 with the state vector ! = !! , !! , … , !!" ! . !! = ! !! = !

!! = ! !! = !

!! = ! !! = !

!! = ! !! = !

 !! = ! !!" = !

 !!! = ! !!" = !  

!! !! !! !! + !! !! ! + !! + !! !! !! !! !! !! + !! !! ! + !! + !! !!   !! !! !! !! + !! + !! !! !! 1 !(!, !, !) = !! − ! + C !! C !! !         ! ! !!" 1 !!" + C !! S !! C !! + S !! S !! ! ! ! !!" 1 !!" + C !! S !! S !! − S !! C !! ! ! ! With !(!, !) = !! !, ! , !! !, ! , … , !!" (!, !) !! − !! !! !! = , !! = − , !! !! !! − !! ! !! = , !! = , !! !!

!

And derive the dynamics of the new coordinate!! = !!! − !! . Our objective here is to design a virtual control !!∗ which makes lim!→! !! → 0. Consider a control Lyapunov function !! = whose derivative is: !! = !! !! = !! !!! − !!                                              (9) we introduce the virtual control !!∗ from (8), !!∗ = !!! + !! !!                                                                (10) Changing the variable, and replace !!∗ by (10) !! = !!∗ − !! = !!! + !! !! − !! ,  

and

!! = !!! + !! !! − !!                                                  (11)

!! − !! !! ,  !! = , !! = !! !! ! 1 !! = , !! = !! !!

the term !! !! is added to stabilize !! . Deriving (11) we yield !! = !!! + !! !! − !! !! !! − !! !! ! − !! !! − !!    (12) Now, in order to make lim!→! !! → 0we choose the control Lyapunov function

The main objective of the controller is to ensure the asymptotic convergence of the variable state vector !! , !! , !! , !!! ! to the reference trajectories !!! , !!! , !!! , !!!! ! . The control-observer scheme proposed in this work is logically divided in a position and a rotation controller

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1 ! ! > 0                                                                              (8) 2 !

1 !! = !! + !!! > 0                                                      (13) 2 Taking its derivate and replace from (11)

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2013 10th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) Mexico City, Mexico. September 30-October 4, 2013 ∗ Then, !!" = !!" − !!" = !!! + !! !! − !!" derivative is

!! = !! + !! !!  

!! = !! + !! !!! + !! !! − !! !! !! − !! !! ω − b! U! − A !                                                                                                                                                                                                (14)

!!" = !!! + !! !! −

where !! = −!! !!! + !! !! is the augmented Lyapunov function derivate, then

and

its

!! ! − A!"                                      (23) ! !

!

! Moreover, !!" = !! + !!" > 0   and !

!! = −!! !!! + !! !! + !!! + !! !! − !! !! !! − !! !! ω − b! U! − A !                                                                                     15

!!" = !! + !!" !!! + !! !! −

!! ! − A!"              (24) ! !

Using (21) and (53)

Thus, using (15), the control !! is: !! = !!!! !!! + !! !! − !! !! !! − !! !! ω + z! + k ! z! − A !                                                                                                                  (16)

!! ! − A!"         ! !                                                                                                                                                                                        (25)

The same procedure is followed to design !! and !! in order to control the pitch and yaw angle, respectively.

where !! = !!" /! + C !! S !! C !! + S !! S !! . It is easy to see from (25) that the fictitious control !!∗ can be considered as the orientation responsible for the motion in ! position, then ! ! + !! !! + !! + !!" !!" − A!"        (26) !!∗ = !! !!

!!" = −!! !!! + !! !!" + !!" !!! + !! !! −

!! = !!!! !!! + !! !! − !! !! !! − !! !! ! + !! + !! !! − !!                                                                                                                  (17) !! = !!!! !!! + !! !! − !! !! !! + !! + !! !! − !!            (18) with

Then, it is possible to find the desired pitch angle !!! from (6) in order to track the reference of ! position, this angle is given by

!! = !!! − !!                             !! = !!! + !! !! − !! !! = !!! − !!                             !! = !!! + !! !! − !!

!!! = sin!!

B. Backstepping Control for Traslational Subsystem

!!∗ − sin !! sin !!                              (27) cos !! cos !!

3) Latitudinal Position (y-position)

1) Altitude control The altitude is obtained using the same procedure stated in III.A. Therefore we obtain ! !! = ! + !! !! + ! + !! + !! !! − !! cos !! cos !! !!                                                                                                                                                                                      (19)

The control for y position is obtained using the same procedure stated in III.B.2. Since the fictitious control !!∗ can be considered as the orientation responsible for the motion in ! position, we obtain ! ! + !!! !!! + !!! + !!" !!"              (28) !!∗ = !! !!!

with

with !! = !!! − !!                          

!! = !!" /! + C !! S !! S !! − S !! C !!

!! = !!! + !! !! − !!

!!! = !!!! − !!! !!" = !!!! + !!! !!! − !!" Finally, replacing !! in (28) by !!! from (27), it is possible to find the desired roll angle !!! as

2) Longitudinal Control (x-position) Let us define the x position error as !! = !!! − !!                                                                  (20)

!!! = sin!! !!∗ sin !! − !!∗ cos !!                    (29)     This closes the controller design, in the following section the sliding mode observer for wind parameters and parameter variations is explained in detail.

!

Considering a Lyapunov function !! = !!! > 0 !

!! = !! !! = !! !!! − !!"                                      (21) ∗ From (21) the virtual control input !!"

IV.

∗ !!" = !!! + !! !!                                                            (22)

IEEE Catalog Number CFP13827-ART ISBN 978-1-4799-1461-6 978-1-4799-1461-6/13/$31.00 ©2013 IEEE

WIND PARAMETERS ESTIMATION

To achieve robustness of the closed-loop system with respect to external disturbances a robust first order exact

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2013 10th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) Mexico City, Mexico. September 30-October 4, 2013 differentiator [12] is implemented. Since the aerodynamic forces !! and ! ! in (4) are used to derive the control law they are estimated, among this term, any model uncertainties will be also contained in the term estimated [10]. Let us define

shows the transitory of the position of the hexarotor. Finally, the position in the earth reference frame is displayed in Fig. 6.

!! = !! !, ! − !!                                                        (30) where !! = −!! !! !/! !"#$ !! + !! ,              ! = 2,4,6,8,10,12                   !! = −!! !"#$ !!                                                                                                                          (31) moreover, !! = !! !, ! − !! and the estimation error is defined as !! = !! − !! . Therefore, !! = !! − !! = !! − !! The error dynamics !! converges to zero in finite time with sufficiently large and positive gain constants !! > 0 and !! > 0, then !! !" = !!                                                                        (31) Fig. 3. Wind parameter and uncertainty model estimation.

Therefore, the estimation of !! and ! ! can be obtained from (31) as !! !" = !!                                                                      (32) Moreover, in the event of model uncertainties and parameter variations in the aerial system, the estimation variable !! will comprises these terms; giving the controller robustness against this situations. V.

SIMULATION RESULTS

To probe the estimator, include perturbations as:                          !! = 1 + 0.5 sin(0.2!)                                  !! = 0.2 + 0.1 sin(0.2!)                                      !! = 0.4 + 0.2 sin 0.2!           !! = 2                                                !!" = 0.5 cos 0.3!                                                !!" = 4.5! !! !!!.!

! /!!  

Fig. 4. Transitory of Euler angles.

                         

And the parameters of hexarotor used are: m = 1.830  kg , d = 0.30  m , !! = !! = 21.6×10!! , !! = 43.2×10!! , !! = 3.357×10!!  !"  ! ! , ! = 2.98×10!! ,! = 1.14×10!! , and ! = 9.81  !/! ! . The initial conditions are zero except for the yaw angle where !! = 0.5. The controller parameters are: !!,!,!,!,!,! = 5 , !! = 0.9 , !!,!,!! = 3,      !!" = 10, !!" = 23, and for wind parameter estimation !! = 8,      !! = 0.4, ! = 2,4,6 ,      !! = 8,      !! = 0.3,      (! = 8,10,12). The estimation results of the angles and translation are in Fig 3, and from 13.5 seconds to introduce increased proportional mass and inertia to 1kg more. The waypoints to following in the coordinates (!, !, !) are: (0,0,2) in 5 seconds, (2,0,2) in same time, (2,2,2), (0,2,2) and (0,0,2) whit the same interval of time. In the Fig 4 shows the transitory of the angles roll pith and yaw. Fig. 5

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Fig. 5. Transitory of translation

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2013 10th International Conference on Electrical Engineering, Computing Science and Automatic Control (CCE) Mexico City, Mexico. September 30-October 4, 2013 [6]

Samir Bouabdallah, Pierpaolo Murrieri, and Roland Siegwart, “Design and control of an indoor micro quadrotor”, Proceeding of the IEEE International Conference on Robotics and Automation (ICRA), vol. 5, p.4393-4398, May 2004. [7] Gessow, A. and Myers G. C., Aerodynamics of the Helicopter, fifth printing, Frederick Unger Publishing Co., New York, 1978. [8] S. Bouabdallah, R. Siegwart, “Full control of a quadrotor”, IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), San Diego, CA, USA, 2007. [9] Wang Li, and Wang Qing-Lin p, “The feedback linearization based on backstepping technique”, IEEE International Conference on Intelligent Computing and Intelligent Systems (ICIS ), Shangai, 2009. [10] L. Luque-Vega, B. Castillo-Toledo, and Alexander G. Loukianov, Robust block second order sliding mode control for a quadrotor, Advances in Guidance and Control of Aerospace Vehicles using Sliding Mode Control and Observation Techniques, Journal of the Franklin Institute vol. 349, issue 2, March 2012, p. 719-739, 2010. [11] Bouabdallah, Samir, and Roland Siegwart. "Backstepping and sliding-mode techniques applied to an indoor micro quadrotor." Proceedings of the IEEE International Conference on Robotics and Automation, (ICRA), April 2005. [12] Arie Levant. "Robust exact differentiation via sliding mode technique", Automatica vol. 34, issue 3, March 1998, p. 379384.

Fig. 6. Position of the quadrotor in the earth reference frame.

VI. CONCLUSIONS In this paper, a controller-observer scheme based on the backstepping technique has been proposed for the hexacopter. The wind parameters resulting from the aerodynamic forces and moments have been estimated via a sliding mode observer based on the super twisting algorithm. This control law has led to satisfactory results in terms of trajectory tracking due to the robustification of the backstepping technique since the controller is able to withstand external disturbances and parameter variations. Simulations show the good performance of the proposed controller. ACKNOWLEDGMENTS This project was supported by the National Council on Science and Technology (CONACYT), Mexico REFERENCES [1]

[2]

[3] [4]

[5]

Armando S. Sanca, P. J. Alsina, and Jés de Jesus F. Cerqueira, “Dynamic modeling with nonlinear inputs and backstepping control for a hexarotor micro-aerial vehicle”, Latin American Robotics Symposium and Intelligent Robotics Meeting, São Paulo, Brasil, 2010. Richard Voyles, and Guangying Jiang, “Hexarotor uav platform enabling dextrous interaction with structures - preliminary work”, IEEE International Symposium on Safety, Security, and Rescue Robotics (SSRR), Texas, USA, 2012 Radek Baránek, and František Šolc, “Modelling and control of a hexa-copter”, 13th International Carpathian Control Conference (ICCC), High Tatras, Slovakia, 2012. B. Bijnens, Q.P. Chu, G.M. Voorsluijs, and J.A. Mulder, “Adaptive feedback linearization flight control for a helicopter uav”, AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, California, USA, August 2005. A. Mokhtari, and A. Benallegue, “Dynamic feedback controller of Euler angles and wind parameters estimattion for a quadrotor unmanned aerial vehicle” American Control Conference (ACC), Denver, June 2003.

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