Design of an Adaptive-Neural Network Attitude Controller of a Satellite using Reaction Wheels

Transcription

1 Journal of Applied and Computational echanics, Vol., No., (5), DOI:.55/jacm Design of an Adaptive-Neural Network Attitude Controller of a Satellite using Reaction heels Abbas Ajorkar, Alirea Falab, Farhad Fani Saberi 3 and ansour Kabganian 4 Department of echanical Engineering, Amirkabir Universit of Technolog Hafe Street, Tehran, , Iran, ajorkar@aut.ac.ir, Department of echanical Engineering, Amirkabir Universit of Technolog Hafe Street, Tehran, , Iran, afalab@aut.ac.ir, 3 Space Science and Technolog Institute, Amirkabir Universit of Technolog Hafe Street, Tehran, , Iran, f.saber@aut.ac.ir, 4 Department of echanical Engineering, Amirkabir Universit of Technolog Hafe Street, Tehran, , Iran, kabgan@aut.ac.ir, Received June 4; revised August 4; accepted for publication August 4. Corresponding author: Alirea Falab, afalab@aut.ac.ir Abstract In this paper, an adaptive attitude control algorithm is developed based on neural network for a satellite using four reaction wheels in a tetrahedron configuration. Then, an attitude control based on feedback lineariation control is designed and uncertainties in the moment of inertia matri and disturbances torque have been considered. In order to eliminate the effect of these uncertainties, a multilaer neural network with back-propagation law is designed. In this structure, the parameters of the moment of inertia matri and eternal disturbances are estimated and used in feedback lineariation control law. Finall, the performance of the designed attitude controller is investigated b several simulations. Kewords: Attitude Control, Adaptive-neural network control, Satellite, Reaction wheel.. Introduction A spacecraft requires several subsstems to achieve its mission successfull. Attitude determination and control sstem (ADCS) is one of the most crucial subsstems of a spacecraft. The major tasks of attitude control sstem are to provide the capabilit of high maneuverabilit, high pointing stabilit and attitude accurac for the satellite. This subsstem has different parts including, sensors, actuators, control algorithm and electronic control board. The control algorithm is an important part of ADCS that provides commands for actuators. Difficult problems in the design of control algorithm for such a comple dnamic sstem are due to the inherent nonlinearities of the model because of large angle maneuvers, uncertainties and unknown environmental disturbances. Among the research that has been done recentl, there are a number of techniques that can deal with the control problems of such a comple dnamic sstem from classical PID control to adaptive control e.g. optimal control [,, 3], sliding mode control [4], adaptive control [5, 6, 7, 8], robust control [9, ] such as variable structure control (VSC) which are designed based on Euler angle errors or quaternion error vector. Generall, thrusters [] are used for quick and large angle maneuvers and reaction wheels are used for slow and high precision maneuvers. Three reaction wheels, with each one's rotational ais parallel to one of the satellite's bod aes, make up the simplest control sstem []. As the three bod aes' dnamics are separated, the design can be carried out independentl for each ais. However, if one of the assemblies becomes damaged then the satellite's attitude can no longer be adequatel controlled. For this reason, a fourth RA is installed in order to increase the reliabilit of the entire sstem. The additional wheel is installed with its ais off the three principal aes, enabling torque control about an of those aes. Thus, the incapacit of an of the RAs aligned with the satellite's principal aes can be compensated b the

2 68 Abbas Ajorkar et. al., Vol., No., 5 torque capabilities of the fourth wheel. In this paper, four reaction wheels in a tetrahedron configuration as satellite actuator have been considered in such a wa that all four wheels have the abilit to create torque control around all the aes. In recent ears, the use of intelligent sstems such as neural networks, fu sstems, fu- neural network sstems, etc. to estimate variables and parameters of the sstem is increased. A lot of work has been performed to estimate sstem s states b the neural network. However, little attention has been given to the estimation of the unknown parameters of the sstem using neural network. In [3], a Hopfield neural network is designed to estimate the unknown parameters of the nonlinear sstem of an aircraft. The method emploed was based on the lineariation of the sstem and was implemented for the sstems with simple dnamics. In [4], a Hopfield neural network is used to estimate the unknown parameters of a robotic arm but estimation error percentage is significant. In this paper, an adaptive-neural network attitude control algorithm is developed using four reaction wheels in a tetrahedron configuration with uncertainties in the moment of inertia matri and disturbance torques. In this algorithm, a simple idea to estimate unknown parameters of a satellite is used based on multilaer neural network. The neural network is the inverse of sstem dnamics, so that the states of the sstem are used as input to the neural network and the control torque is estimated b the network. In this structure, the unknown parameters are used instead of neural network s weights and b using error back propagation update law; unknown parameters converge to their true values. In addition, feedback lineariation control is used to attitude control of the satellite.. Dnamics and Kinematics of satellite For mathematical modeling, we suppose spacecraft as a rigid bod. The general equation to describe the attitude motion of a rigid bod in space (Euler equations) in the presence of momentum echange devices (e.g., reaction wheels) is described as follows [5]: dh dt i dh dt B h. h B h. h w h here is angular velocit of spacecraft with respect to bod reference frame. The angular momentum of the entire sstem ( h ) will be divided b the angular momentum of the spacecraft ( h B ) and the angular momentum of the reaction wheel ( h w ). The terms dh / dt i and dh / dt B represent the time derivative of angular momentum with respect to the inertia reference frame and the bod reference frame, respectivel. In equation () h I, where I represents the moment of inertia matri with respect to bod reference frame. B replacing h I and assuming that XB, YB, ZB are the principal aes of inertia, the final equations are obtained as follow: I ( I I ) I ( ) I ( I I ( I I ) I I ) I ( ( The angular velocit of spacecraft in orbit can be epressed as follows: I B OR B I B B B OR w I here B OR B is angular velocit of satellite with respect to the inertia reference frame, B B is angular I velocit of spacecraft with respect to the orbital reference frame and B OR is angular velocit of orbital reference frame respect to the inertia reference frame that all the vectors are described in the bod reference frame. hen we choose the order of aes transformation as (with aes order of rotation 3 ), where,, are the Euler angles, the kinematics equations of the sstems can be epressed as follows: sin( ) (4) sin( ) tan( ) cos( ) tan( ) cos( ) cos( ) sin( ) cos( ) sin( ) cos( ) tan( )sin( ) cos( ) cos( ) ) ) () () (3) 3. Reaction wheels arrangement To provide high roll and pitch control torque as well as high speed slew capabilit, we consider a rigid spacecraft model with a set of four reaction wheels which are arranged in a pramid configuration about the spacecraft aw ais as shown in Fig.. Journal of Applied and Computational echanics, Vol., No., (5), 67-73

3 Design of an Adaptive-Neural Network Attitude Controller of a Satellite using Reaction heels 69 Z, Yaw R R4 R Y. Pitch 9-β α X, Roll The wheel control torque vector can define as follow: h h h Fig.. Arrangement of reaction wheels h Ch a where matri (C) is the orientation matri of the four-wheel pramid configuration and (5) h a is torques generated b the reaction wheels. The orientation matri represented b a 3 4 matri (C) which is considered as follows: cos sin sin sin cos sin sin sin (6) C sin sin cos sin sin sin cos sin cos cos cos cos where C i s (i =,, 3, 4) are the wheel aial vectors and and are considered as 45 and The energ cost b the four-wheel pramid configuration as is shown in Fig.. is the smallest [6] Adaptive-neural network control design For the design of adaptive-neural network control, first the feedback lineariation controller is designed and then a neural network is used to estimate the sstem s uncertaint. 4. Feedback lineariation control design For feedback lineariation design, we choose the states equal to P which denote the integration of the angular velocities. e choose the control moment as follows: (7) ( I I ) T ~ d ~ Tc I( P d K P K P ( I I ) I Td ( I I ) Td where, K I and K I are the coefficient matrices for the desired dnamic model of controlled sstem and is a positive constant. B placing the control law in the dnamics of the sstem, the sstem dnamic error is ero (eq. (8)). That means the eponential stabilit is guaranteed. ~ ~ ~ (8) P P P 4. Neural network design To estimate the unknown parameters of the sstem, the multilaer neural network with back propagation update law is used. Simple structure and a small number of parameters to be adjusted are the main advantages of the designed neural network. As it is shown in Fig., the states of the sstem are inputs of the network and multilaer neural network control estimates control torques of equation (9) as equation (). Journal of Applied and Computational echanics, Vol., No., (5), 67-73

4 7 Abbas Ajorkar et. al., Vol., No., 5 I ( I I ) I ( ) d I ( I I ) I ( ) d I ( I I ) I ( ) d c c c ( ( ( 3 3 ) I ) I 3 ) I ( ( ( ) ) 4 5 ) 6 (9) () c X Neural Network Parameter Estimation c e Fig.. Neural network diagram In this network, the unknown parameters are considered as network s weights, which are updated over the time and are converged to their true values. eight updates are be done using the error between output of the network (estimated control torques) and the control torque generated b feedback lineariation controller. The back propagation update law is considered as follows: J J e net () ef X ex e net here, J is the cost function of neural network which is defined based on the difference between the estimated torque and real torque control (equation ()). In equation (), f represents derivative of activation function; f is equal to one, since the activation function is assumed to be linear. T () J e e Simulation and results In this section, to demonstrate the performance of adaptive-neural network controller and effect of uncertaint and eternal disturbances, the results of simulations are presented. In all simulations, desired Euler angles are considered as:,, (3) d d d oreover, the satellite dnamic model, reaction wheels, controller parameters and initial conditions are summaried in Table. Saturation torque of reaction wheels are also considered equal to U m.7n. m. Table. Simulation parameters Subsstem Parameter Quantit (wheel s moment of inertia).3 Kg.m Reaction heels Satellite Dnamics Adaptive Controller I U m (saturation torque).7n. m I I I 5 7 Kg. m deg rad / sec K K Journal of Applied and Computational echanics, Vol., No., (5), 67-73

5 Design of an Adaptive-Neural Network Attitude Controller of a Satellite using Reaction heels 7 Fig. 3 to 5 show the result of simulation for roll, pitch and aw aes respectivel for adaptive-neural network attitude control of satellite. Fig. 6 shows that the tracking error for all aes is converging to ero. Fig. 7, Fig. 8. and Fig. 9 show the output torque of reaction wheels and estimation of moment of inertia and disturbances torque respectivel. As it is shown in Fig., since the eternal disturbances for all aes are considered as d.sin( t ), the neural network estimates the unknown disturbances correctl. Fig. shows the parameter estimation b a classical adaptive controller [7]. According to this figure, the controller estimates parameters in 5 seconds; hence b using the proposed adaptive-neural network, the estimation has been decreased to 6 seconds. 4 Reference Pitch 8 Roll(deg) 6 4 Reference Roll Pitch(deg) Fig. 3. Performance of adaptive- neural network for roll ais Fig. 4. Performance of adaptive- neural network for pitch ais 5 Yaw Pitch Roll Yaw(deg) 5 Error(deg) 5 5 Reference Yaw T orque(n.m) Fig. 5. Performance of adaptive- neural network for Yaw ais Reaction heel Reaction heel Reaction heel 3 Reaction heel 4 Inertia umentum (Kg.m^) Fig. 6. Tracking error of desired Euler angles for adaptiveneural network controller 95 I_ I_ I_ Fig. 7. Output torque of reaction wheels Fig. 8. Estimation of moment of inertia parameters b adaptive-neural network controller Journal of Applied and Computational echanics, Vol., No., (5), 67-73

6 7 Abbas Ajorkar et. al., Vol., No., 5 Disturbance Torque(N.m) - - Eternal Disturbance Estimated Disturbance Estimated Disturbance Estimated Disturbance Disturbance Torque(N.m) Eternal Disturbance Estimated Disturbance Estimated Disturbance Estimated Disturbance Fig. 9. Estimation of eternal disturbances b adaptiveneural network controller Fig.. Estimation of eternal disturbances b adaptiveneural network controller I 8 I 5. Conclusion I I I I Fig.. Estimation of moment of inertia parameters b classical adaptive controller [7] In this paper, an attitude control algorithm in the presence of four reaction wheels in a tetrahedron configuration as the satellite actuator was designed. Then adaptive-neural network attitude control using feedback lineariation controller and multilaer neural network with back propagation update rule as a parameter estimator was designed in the presence of uncertainties in the moment of inertia matri and disturbances torque. Results of simulation of designed attitude controller show that using the adaptive-neural network controller not onl can decrease the attitude tracking error and increase accurac of sstem s performance, but also estimates the unknown parameters of the sstem correctl. References [] Staiar, A. J., Investigation of Flow Phenomena in a transonic Fan Rotor Using Laser Anemometr, ASE Journal of Engineering for Gas Turbines and Power, Vol. 7, No., pp , 985. [] ers, R. H. and ontgomer, D. C., Response Surface ethodolog: Process and product optimiation using designed eperiments, John ile & Sons, New York, 995. [3] Guinta, A. A., Aircraft ultidisciplinar Design Optimiation Using Design of Eperimental Theor and Response Surface odeling ethods, Ph. D. Thesis, Department of Aerospace Engineering, Virginia Poltechnic Institute and State Universit, Blacksburg, VA, 997. [4] Jameson, A., Schmidt,., and Turkel, E., Numerical Solutions of the Euler Equation b Finite Volume ethods Using Runge-Kutta Time Stepping Schemes, AIAA 8-59, 98. [5] Denton, J. D., Xu, L., The Effects of Lean and Sweep on Transonic Fan Performance, ASE Turbo Epo, Journal of Applied and Computational echanics, Vol., No., (5), 67-73

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