Enhanced LQR Control for Unmanned Helicopter in Hover

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Abstract-Real time adaptability is of central importance for linearization and optimization", but the implementation on the control of Unmanned Helicopter flying ...
Enhanced LQR Control for Unmanned Helicopter in Hover Zhe Jiang, Jianda Han ,Yuechao Wang and Qi Song Abstract-Real time adaptability is of central importance for the control of Unmanned Helicopter flying under different circumstances. In this paper, an active model is employed to handle the time varying uncertainties involved in the helicopter dynamics during flight. In the scheme, a normal LQR control designed from a simplified model at hovering is enhanced by means of Unscented-Kalman-Filter (UKF) based estimation, which tries to online capture the error between the simplified model and the full dynamics. This is intended to achieve adaptive performance without the need of adjusting the controller modes or parameters along with the changing dynamics of helicopter. Simulations with respect to a model helicopter are conducted to verify both the UKF-based estimation and the enhanced LQR control. Results are also

demonstrated with the normal LQR control with the active model enhancement.

I. INTRODUCTION T he Dynamics of an unmanned helicopter is strongly nonlinear, inherently unstable, highly coupled and forms a multiple input multiple output (MIMO) non-minimum phase system with time varying parameters. At the same time, the dynamics is also influenced by the turbulence from tail rotor and lateral wind. A helicopter also has multiple flight modes, such as hovering, forward, backward, sideslip, up and downward flights as well as other aggressive maneuvers combining the basic patterns. A helicopter has 6 degree-offreedom (DOF), but it is usually controlled by 4 actuator inputs, which means that the control of each DOF is coupled but independent. All of the above mentioned issues complicate the helicopter dynamics and make it impossible to build high fidelity model for a helicopter [1]. Helicopter flight control system design has been dominated by classical control techniques. But, recent years have seen a growing interest in applications of nonlinear control theory when a nonlinear model is deployed for the controller design. This is mainly due to the increasing attempts on making a helicopter to be unmanned and autonomous. Feedback linearization [2] and State Dependent Riccati Equation (SDRE) method [4] tried to handle the nonlinear dynamics of a helicopter by the way of "on-line Manuscript received December 9, 2005. This work was partially supported by the National High Technology Research and Development

linearization and optimization", but the implementation on real system was constrained by the non-minimum phase characteristics, the computational complexity as well as the lack of robustness [3]. On the other hand, robust control, such

is

as H-nfinity control [5], another algorthm proposed for helicopter, but it Is difficult to make an optimal balance between robustness and conservation, which usually leads to an over- conservative controller to guarantee robustness. Besides the control algorithms explicitly based on off-line model, online modeling, which tries to build a high fidelity model on board and in real time, becomes an important '

direction for the high precision control of vehicles with nonlinear and time-varying dynamics. In [3], the algorithm known as active model estimation plays an important role in autonomous control architecture because it provides necessary information required by robust control, trajectory generation, mission planning, and control reconfiguration for fault accommodation, to the autonomous system that adapts over time. The information which an active model can provide includes [6]: a) state estimation, which provides necessary information for full state feedback control; b) predictive model, which builds online model for adaptive control and fault detection and identification; and c) uncertainty bound, which forms the basis of robust control. Neural Networks (NN) and NN-based self learning have been proposed as one of the most important approaches for active modeling of unmanned vehicle [7-9]. However, the problems involved in NN, such as training data selection, online convergence, robustness, reliability and real-time implementation, limit its extensive application in real

systems.

Most recently, researches are focusing on the sequential estimation and its applications on active modeling and model-reference control [10]. The classical state estimator for nonlinear system is the extended Kalman Filter (EKF). Although widely used, EKF's have some deficiencies, mainly due to its linearizing the nonlinear dynamics. The UKF, on the other hand, has the same computational complexity with the EKF, but directly use the nonlinear models instead of linearizing it. UKF does not need to calculate Jacobians or Hessians and can achieve the second-order accuracy (the accuracy of EKF is first-order). Therefore, UKF is well suited

Program under the grant 2003AA421020. online aplcto. 7 T;,,----41, 1,'D-1-+;-, fo onin of Shenyang Institute Z. Jiang was with the Robotics Laboratory, time-varying dynamics. Automation, Chinese Academy of Sciences, Nanta street 1 14#, Shenyang,

Tfor

110016, China. He is now with the Graduate School, Chinese Academy of Sciences, Beijing 100864, China (phone: ±86-024-23970721; fax: ±86-024-2397002 1; e-mail: zha (~ic). J.D. Han, Y.C. Wang and Q. Song are with the Robotics Laboratory, Shenyang Institute of Automation, Chinese Academy of Sciences, China.(e-mail: idhanKX7sin, Xcwa ~siac, son im~sacn)

1438

application

with fast nonlinearytm innnniea systemsihfs

In this paper, a normal LQR control designed from a

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ens

f

Unscented-Kalman-Filter (UKF) based estimation, which tries to online capture the error between the simplified model

and the real dynamics. This is intended to achieve adaptive performance without need of adjusting the controller modes or parameters along with the changing dynamics of helicopter. Simulation results are presented with respect to the dynamics of a model helicopter to verify both the UKF-based estimation and the enhanced LQR control.

TT.

e \

NM

C.G

HELICOPTER DYNAMICS

A. Full Dynamics The helicopter model can be described as a 6-DOF rigid body with external forces and moments originating from the main and tail rotors, empennage and fuselage drag. The forces and moments acting on the helicopter are shown along with the main helicopter variables in Fig. 1. The rigid body equations of motion for a helicopter with respect to the body coordinate system are given by the Newton-Euler equations shown below [4]. TM

T

Fig. 2. Helicoptermodel: topview In (1) and (2), R3 (V, ,0) is the transformation matrix from body coordinate to the inertial frame. The free body diagram of helicopter is as shown in Fig. 2. The force terms in the x, y, and z direction are denoted by X, Y, and Z respectively. The moment terms in roll, pitch and yaw direction are denoted by R, M, and N, respectively. The subscripts M and T denote main rotor and tail rotor The inertia term along each axis is presented by respectively. F,f and Iz respectively, and the cross inertia terms are [V_;__ Ix1 E IX 1Y, neglected. hM, hT, YM, IM and /T are the distances between the helicopter C.G. and the acting points relative forces. =The forces and moments generated by the main rotor are controlled by TM, TT , a,, and b,,, in which a,, and b,, are the longitudinal and lateral tilt of the tip path plane of the 'fs main rotor with respect to the shaft, respectively. The forces C.G. / and moments can be expressed as mg ~ ~~~~~~~~~~~~~~~d .

y,v,O,q

A-~ X, u, 0,

/,~

t z,w1,v,r

Fig. 1. Forces and moments acting on helicopter

|

v

= vr - wq + gsinXb+ X/Im =

wp-ur+gcos0isinO+Y/m

w = uq - vp +

q

= =

si

= =

tV =

RM = -(

YM =TMsinblS

MM

X,_YT,TT

w)

-

m, =-~~~~~abc+4zrVc

NfNM

QT

YMIM YTIT

TT and the moments QM. QT can

-3m2R0O - 2m5 )(m2RO

I C =XM R = RM +YMhM +ZMYM+YThT M =MM +MT -XMhM +ZM/M

=-QM cos als cos bls

n

where

Z =ZM

as -QMsinbls a5

R

horizontal stabilizer and vertical fin, the external forces and moments acting on a helicopter are:

YM

I

be calculated by using Eq.3 and 4 where the subscript M or T has been omitted for simplicity. For more details about (3) and (4), please see references [1 1, 12, 13]. 3 R3 T R3-° m3O+ 36 (R2-Ro) 2 33 F 2 ( 3 / 2 ) (3) 8AQ2 l15 20 2 m2R 5)M2R

-

(| *> ±)T sO )(u v By omitting the fuselage drag and the influences of

Y

da

MT= -QT

In above, the forces TM,

pr(Iz Ix )/II + M / IX pq Ix -Iy)I'z +N/Iz .qsin~4tanO+rcos~btanO + rcosotanO p +p qsinotanO qcos0-rsino qsinosecO + rcososecO

b1b - QMsinalS

ds

-TM cos als cos bls NM

ZM

gcososO+zZIIm

=R3(Vf

XM = -TMsinals

(2)~ m3 = PQabc + 4rTVc 2 m= 2 V cm2

m5Q=m1-

1439

+

5)3/2 ) )y

m2 = 8zQ2abc M4 = m20

2

__ =-__

c

Q=n1n7 3 -

21 (R3 -R )+ Cd/ (R4 -R ) FT= R (p Q2abc)O + 2/I~~~~~~~/I5 2 4 ~~~~~~~6 + 8m m4R+m )3/2 3(R3 R3) 2n{(15m2R2212m4mR + 1015M3 (4) 5 5 4 (p Q2 a2b 2C2)O

- (15m3R4 1 5m4

4m5R0 +

+ 2-2i5)(n2+o

4R0

5)

Substituting (7), (8) and (9) into (1) and (2), we can obtain the simplified model denoted as: x (10) fsimp (X, U)

)3/2

=

- (3m2R0 - 2m5 )(m2R0 + m5 )3/2 )}

Where n

PQ2bC~~~~~~~~~ Q2 bc Vn6

P p

2

iml

V

n3

a

8rQ 2

Q

n,

aVm

4AQ2

J

-

4

aO

=

n3n4 - n5m1 +

n4 =ml + m5 5

where the definitions of x and u are the same as those of (5)