Adaptive Robust Nonlinear Control of UAV Helicopters in Three-Dimensional Space Using RBF Neural Network

Adaptive Robust Nonlinear Control of UAV Helicopters in Three-Dimensional Space Using RBF Neural Network

Azin Yazdekhasti and Khoshnam Shojaei Department of Electrical Engineering, Najafabad Branch, Islamic Azad University,

Esfahan, Iran

{Azin.yazdekhasti & khoshnam.shojaee}@gmail.com

Abstract - This paper aims to propose an efficient control algorithm for the unmanned aerial vehicle (UAV) helicopter with underactuated dynamics. An intelligent control system is proposed by using radial basis function neural network (RBFNN). Moreover, based on sliding-mode approach, the adaptive tuning laws of RBFNN can be derived. Then, the developed RBFNN control system is applied to an UAV helicopter for achieving desired trajectory tracking. From the simulation results, the control scheme has been shown to achieve favorable control performance for the UAV helicopter with underactuated dynamics even it is subjected to parameter uncertainties and external disturbance.

Index Terms - Unmanned aerial vehicle helicopter, Radial basis function neural network, Sliding mode control, Adaptive robust.

I.INTRODUCTION

Regarding their versatility, reliability, maneuverability, low cost, small size, and so on and so forth, unmanned helicopters are invaluable for many civil and military applications. The unmanned helicopters are widely used in many application fields because of their small size and superior flight characteristics, such as vertical take-off, landing, and hovering where human intervention may be restricted. They can be used in search and rescue after big natural disasters, patrol and surveillance, filming movies, suppression of smuggling, inspection of power lines, large bridges and so on. For unmanned helicopter control, it is essential to produce moments and forces on the helicopter such that the desired regulated state is achieved and so that the helicopter can track a desired trajectory [1].

In order to develop the controllers for such unmanned helicopters, an approximate linearization-based control scheme that transforms the system into linear form has been utilized in [1]. A model for the helicopter independent of an accompanying control scheme has been introduced in [2]. In [3-10], results on the dynamic modeling and control of helicopters are presented. In these papers, several controller design methods, such as PID, adaptive nonlinear control, neural network control, model predictive control, and fuzzy control are discussed.

Most UAV autopilots use classical proportional-integral- derivative (PID) controllers and ad-hoc methods to tune the controller gains during the flight. This methodology is not the best one because it has high risks and there are a lot of limitations in the UAVs performances and robustness. In order

to design good controllers and to improve the system reliability and robustness, simulations through flight tests have been performed. An output feedback control scheme with a neural network (NN)-based controller using feedback linearization has been implemented in [4]. An inner and outer loop control using pseudo-control hedging has been employed in [4]. A backstepping-based controller for the helicopter has been introduced in [5]. The control schemes for Lyapunov- based control of helicopter UAVs have been generated in [6]

and [7]. In [8], the stabilization of the UAV is made by a controller which has been designed by using the H_ technique and the  synthesis technique.

Since equations of helicopter UAV are nonlinear, there are a lot of techniques for the control of flying objects:

dynamic inversion, nonlinear predictive control or techniques which use neural networks [11]. By using the first method, the system nonlinearities are cancelled and a closed loop linear system is obtained. The disadvantage of this method is that we have to be completely aware of all the nonlinearities and their derivatives. This is a serious problem because the aerodynamic forces and moments, which describe the system, can not be modelled at a high precision degree [2].

Backstepping makes use of a recursive procedure that breaks down the control problem for the full system into a sequence of designs for lower order systems [12, 13]. The advantages of the backstepping method are: 1) backstepping relaxes time-scale separation requirements by including transients in the virtual controls explicitly in the control formulation; 2) the designer may use Lyapunov functions as the design progresses and he may discriminate between which nonlinearities cancel, and which plant dynamics to exploit. The control of UAVs is studied, using the backstepping method, in many research papers. For example, Azinheira and Moutinho present in their paper [5] a backstepping-based controller with input saturations, applicable for the hover flight of an unmanned aerial vehicle. In order to cope with limitations due to reduced actuation, saturations are introduced in the control design, and the stability of the modified control solution is verified. Hemanshu et al. obtained a novel backstepping based velocity control method for unmanned helicopters [14].

As can be seen, many experts and scholars have begun to research and develop the small unmanned helicopter, but the dynamics of the helicopter UAV are nonlinear, strong coupled with each other, and underactuated which makes the control

design very challenging. To solve this problem, the main contribution of this paper puts forward a idea that applies new adaptive sliding mode control (SMC) based on RBFNN which has a fast response, high control precision, good robustness, strongly adaptability and does not need accurate mathematics model to the small unmanned helicopter control system.

The remaining parts of this paper are organized as follows. First, in Section II, the model of the UAV helicopter is described to be used in our further derivations. Then, in Section III, the controller design and Analysis has been developed for this UAV helicopter in detail. The simulation results are presented in Section IV, and finally, the paper is concluded in Section V.

II.HELICOPTER DYNAMIC MODEL

Consider the helicopter which is shown in Fig. 1 with six degrees of freedom defined in the inertial coordinate frame OE, where its position coordinates are given by [x y z]^T and its orientation described as roll, pitch, and yaw, respectively, is given by Euler angles  [  ]^T. The kinematics of the helicopter are given by

 

R v

  _{. (1)}

 

T  

  _{. (2)}

Where the translational rotation and rotational transformation matrices from the body fixed frame to the inertial coordinate frame are defined as [2]

c c c c

c c c

c c

s c s s s s s c

R s c s s s s s s c

s s c

           

           

    

    

 

 

   

 

  

 

(3)

c c

1 1

0 ^{s c}

s t c t

T c s

   

 

 

 

 

 

 

  

 

 

 

(4)

where s_, c_, and t_denote the ^sin

 

^{ , cos}

 

^{ , and tan}

 

^

functions, respectively. The equations of motion are expressed in the body-fixed frame Ob associated with the helicopter’s center of mass. The dynamics of the helicopter is given by the Newton–Euler equation in the body fixed coordinate system and can be written as follow [10]:

0 0 0

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0

0 0 0

0

x y

x z

d

y x x x x

y y y

y T

z

z z z M z

v mI v

j v mg

j j U

j Q

j

j Q

  



 



 



       

       

         

         

        

         

          

 

         

        

 













(5)

Where m is a positive scalar denoting the mass of the helicopter, Jdiag j



x^,jy^,jz



³ is the positive definite inertia matrix, vvx vy vz^T^{3 1} represents the translational velocity vector, x y z^T^{3 1} represents the body angular velocity vector, g is the gravitational acceleration, QM

and QT are the unknown nonlinear aerodynamic effects taken into consideration for modeling of the helicopter originally found in [2], d represents unknown bounded disturbances such that _d _M for all time where _M is a known positive constant, and finally U^{6 1} is the control input vector given by

1

11 2

22 3

33

0 0 0 0

0 0 0 0

1 0 0 0 ˆ

0 0 0 ˆ

0 0 0 ˆ

0 0 0

u

U p

p p







 

 

 

 

 

 

 

  

 

 

 

    

 

 

 

where u providing the thrust in the z-direction, ^ˆ₁, ^ˆ₂, and ^ˆ₃ providing the rotational torques in x-, y-, and z- directions, respectively and p_iiare positive definite constants. It should be noted that () denotes the vector cross product. In this section, the dynamic model of the helicopter with six degrees of freedom, six outputs and four inputs has been considered. By defining a new augmented variable asu , the methodology for the controller design will now be addressed in the following section.

Fig. 1: Helicopter Dynamics

III.CONTROL DESIGN AND ANALYSIS

In this section, we will present an adaptive robust controller which stabilizes System (5). This controller has been derived from a class of Lyapunov based controllers that guarantee finite time stabilization. The overall control objective for the unmanned helicopter is to track a desired trajectory_d[x_d y_d z_d]^T. In figure 2, the entire adaptive sliding mode control scheme is illustrated. In this scheme to overcome parameter uncertainties and external disturbance, the radial basis function neural network is employed. The strategy is to define a sliding surface and design a RBF neural network with learning rules and a robust adaptive term to force the tracking error and sliding surface to be uniformly ultimately bounded (UUB) in spite of the presence of the system uncertainties and external disturbances. In this scheme, the weights of the RBF neural network can be adaptively adjusted for the compensation of uncertain dynamics and external disturbances.

Fig. 2: Control Scheme

In order to demonstrate stability, define the tracking error as e_ _dand sliding surface as

1 2 3

p p p p

Se^  e^  e 



e dt

where _i ’s are the constant matrices . The input signal guaranteed the closed loop system performances may be obtained by:

1 2 3

ˆ ˆ ˆ

Eq Cor RBFNN

u u u









 

 

    

 

 

 

.

Where u_Eq is the equivalent control law for sliding phase motion, u_Cor is the corrective control for the reaching phase

motion, and u_RBFNN is adaptively adjusted term for the compensation of uncertain dynamics. The control objective is to guarantee that the state trajectory can converge to the sliding surface. So, u_Eqshould be chosen as common methods to satisfy S0 in nominal system without considering the uncertain dynamics and external disturbances and also the ideal expression of corrective term will be:



1



( )



2



( )



3



( )

Cor x x y y z z

u   k k T S  k k T S  k k T S Where the controller parameters k1, k2, k3, kx, ky, and kz are real positive. The aforementioned corrective term ensures the stability of the system which requires a reasonable range of uncertainties during the sliding mode control. In general, the corrective control in SMC is characterized by the charting phenomena provide by the discontinuous term, to eliminate the chattering we will replace it by a continuous function T(.) giving by:

4 4

( ) 1 tanh(2 )

1

z z

T z e z

e





  



In order to design a control law without explicit knowledge about the uncertain dynamics and external disturbances, an adaptive RBFNN is introduced to approximate uncertainty. And here u_RBFNN is the adaptive neural network system used to approximate u_ideal compensated the effect of these uncertain dynamics and external disturbances. It should be noted that the ideal RBFNN weights in matrices W_RBFNN^ideal , that are needed to best approximate a given nonlinear function which is difficult to determine. In fact, they may not even be unique. However, all one needs to know for controls purposes is that, for a specified value of some ideal approximating RBFNN weights exist.

Fig. 3: The structure of RBFNN

The structure of RBFNN is shown in Figure3. RBFNN is a forward network with three layers: input layer, implicit layer, and output layer. The mapping from input to output is nonlinear, while it is linear from implicit layer to output.

RBFNN can approximate nonlinear function locally, so its

learning rate is fast and the local minimization problem can be avoided.

Assume there are inputs and implicits nodes in the

RBF network. Let



1 2



T

X  x x  xn be the input vector;

then denote the radial basis vector



1 2



T

H h h  hm , where ℎ is Gaussian function defined as

2

2 2

( )

i i

X c b

h Xi e

  

 

 

 

 ,

where



1 2



T

i i i im

c  c c  c is the pre defined central vector of the ith node and > 0 is the basis width parameter of the th node. Define the weight vector of the RBF neural network as

RBFNN

W ; then in this study, the output of the RBFNN can be expressed as follows:

     

^T



^{T T}



RBFNN RBFNN

u t  W t H   

Consider the Lyapunov function defined by :

  

¹

 ^

^{2 2 2}

^

1 1 1

( ) ( ) ( ) ( )

2 2 ^N 2

T T

x y z

V S t S t  tr w t F ^ w t  K K K . (6) Where weight estimates errors w t( ) _{define as}

ˆ

 

^ideal

RBFNN RBFNN

W t W . By assuming following parameter (0)

(0) (0) (0)

x xo

y yo

z zo

F B

k k

k k

k k

w W







 

 

^ideal

 

RBFNN RBFNN

u t u t 

   

1

ideal

ideal RBFNN

u t u t 



to make tracking error e_

 

t , controller parameters ( ), ( ), ( )

x y z

k t k t k t , and the NN weight estimates errorsw t( ) _are UUB and also S and ˆ_RBFNN( )

F

W t with practical bounds given respectively by the right hand sides of

 



 

  ^{ }  

^{ }



2 1

2 2 2

1 2 3

4 2

B

x f x y f y z f z

W S

k k k k k k k k k k k k

 

  

 

 

 

 

 

 

 



  

, (7)

 

²

ˆ 2 1

2 4

B B RBFNN

F

W

W W    , (8)

it is easily conclude that the following adaptation rule should be considered.

 

ˆ ^{T T T} ˆ

RBFNN N w fN N RBFNN

W^ F H

 

 

 

S G k F S W ^{. (9)}

x f x

 

x x

k  k k T S S _{. (10)}

 

y f y y y

k  k k T S S . (11)

 

z f z z z

k  k k T S S _{. (12)}

It should be emphasized that by considering the aforementioned adaptation rule, differentiating (6) with respect to time yields

    ^{ }

   

2 2

1 2

2 3

2

1

( ) ( )

( )

ˆ ˆ 2

x f x x x y f y y y

z f z z z

RBFNN B RBFNN

fN F F

V k k k k T S S k k k k T S S

k k k k T S S

S k W W W  

    



  

 

    

    

 

  

 

     

     

     

 

   

   

  



IV.SIMULATION STUDY

According to the proposed control method, the simulation is performed using MATALB/Simulink. We choose the following parameters: g9.8m s² , m9.6kg , Q_M 0.5,

T 0.1

Q  , j_x 0.4kg m. ², j_y 0.56kg m. ² ,k₁ 10, k₂ 10,

3 10

k  , k_xo 50 , k_yo50 , k_zo50 , j_z 0.29kg m. ² ,

11 1.1

p  , p₂₂1.1, p₃₃1.1,  ₁ 1,  ₂ 0.33,  ₃ 0.04,

f 10

k  , k_fN 0.1 , FN eye

 

²⁴ , and G_wones(3, 4) . Figure 4 demonstrates the helicopter's ability to follow a trajectory in three dimensions. The desired trajectory is defined as



^0.4

 

^0.4



[5 1 sin 5 1 cos 50]

25 25

t t T

d e  t e  t

   ^ ^  ^ ^

   

in meters. The figure shows that the helicopter can follow the desired circular trajectory after a transient response.

Fig. 4: 3-D perspective of position during the circular maneuver

Fig. 5: Helicopter position vs. time for the circular maneuver.

Fig. 6: Tracking Errors vs. time for the circular maneuver.

Fig. 7: Evolution of the control signal with respect to Figure 3.

Figure 5 shows the actual and desired trajectories with respect to time during hovering. As expected, they track the target values despite the bounded uncertainties and disturbance. Figure 6 shows that the tracking errors converge near zero. The evolution of the control signal is shown in

figure 7 under the initial condition^

 

^{0 [2 2 55]T}. Figure 8 shows the controller parameters versus the time.

Fig. 8: Evolution of the controller parameters vs. time.

Fig. 9: Helicopter position vs. time for the circular maneuver.

Fig. 10: Evolution of the control signal with respect to Figure 9.

3

In comparison with classical sliding mode control, to show the efficacy of the proposed method, figure 9 shows the actual and desired trajectories with respect to time during hovering under the same initial conditions when the classical sliding mode control is used to control. In this case, the evolution of the control signal is shown in figure 10. Based on these figures, it should be conclude that the proposed method achieves favorable control performance by reducing the control signal chattering for the UAV helicopter with underactuated dynamics in presence of the system uncertainties and external disturbances.

V.CONCLUSIONS

In this paper, to deal with the parameter uncertainties of the unmanned aerial vehicle helicopter with underactuated dynamics, an adaptive robust controller was successfully designed based on sliding mode, radial basis function neural network, and Lyapunov stability theory. Simulation results show that the proposed control method achieves favorable control performance by adjusting the adjustable gain of controller parameter. Besides, the adaptive algorithm is simple, easy to achieve and has good adaptability and robustness against the parameter uncertainties and external disturbances. Furthermore in comparison with classical sliding mode control, a smooth version of the adaptive sliding mode controller is used to reduce the control chattering.

REFERENCES

[1] T.J. Koo and S. Sastry, “Output tracking control design of a helicopter model based on approximate linearization”, in Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, FL, 1998, pp. 3635- 3640.

[2] B. F. Mettler, M. B. Tischlerand, and T. Kanade, System identification modelling of a small-scale rotorcraft for flight control design, International Journal of Robotics Research, Vol. 20, 2000, pp. 795-807.

[3] H. Chao, Y. Cao, and Y. Chen, “Autopilots for small fixed-wing unmanned air vehicle: A survey”, IEEE International Conference on Mechatronics and Automation, 2007; pp. 3144-3149.

[4] N. Hovakimyan, N. Kim, A. J. Calise, J.V.R. Prasad, “Adaptive output feedback for high-bandwidth control of an unmanned helicopter”, in Proceedings of AIAA Guidance, Navigation, and Control Conference, Montreal, Canada, 2001, pp. 1-11

[5] J. R. Azinheira and A. Moutinho, “Hover Control of an UAV With Backstepping Design Including Input Saturations”, IEEE Transactions on Control Systems Technology, vol. 16 (3), 2008, pp. 517-526.

[6] E. Frazzoli, M. A. Dahleh and E. Feron, ”Trajectory tracking control design for autonomous helicopters using a backstepping algorithm”, in Proceedings of American Control Conference, Chicago, IL, 2000, pp. 4102-4107.

[7] B. Ahmed, H. R. Pota and M. Garratt, “Flight control of a rotary wing UAV using backstepping”, International Journal of Robust and Nonlinear Control, vol. 20, 2010, pp. 639-658.

[8] Y. C. Paw, Synthesis and Validation of Flight Control for UAV, Dissertation submitted to the Faculty of the Graduate School of the University of Minnesota, 2009.

[9] Y. C. Paw and G. J. Balas, “Uncertainty modeling, analysis and robust flight control design for a small UAV system”, AIAA Guidance, Navigation, and Control Conference, Honolulu, HI, August 2008, No.

2008-7434.

[10] M. Lungu and R. Lungu, “Adaptive backstepping flight control for a mini UAV”, International Journal of Adaptive Control and Signal Processing, vol. 27(8), 2013, pp. 635-650.

[11] F. L. Lewis, M. D. Darren, and T. A. Chaouki, Robot Manipulator Control Theory and Practice, CRC Press, 2003.

[12] H. K. Khalil, Nonlinear Systems, Prentice-Hall, Inc., New Jersey, 2nd ed., 1996.

[13] P. V. Kokotovic, “The Joy of Feedback: Nonlinear and Adaptive”, IEEE Control Systems, June 1992, pp. 7-17.

[14] R. Hemanshu, A. Bilal, and M. Garratt, “Velocity Control of a UAV using Backstepping Control”, Proceedings of the 45^th IEEE Conference on Decision & Control, San Diego, CA, USA, 2006; pp. 5894-5899.