BIAS MOMENTUM SATELLITE MAGNETIC ATTITUDE CONTROL BASED ON GENETIC ALGORITHMS. Jianting Lv, Guangfu Ma, Dai Gao

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1 IAS MOMENUM SAELLIE MAGNEIC AIUDE CONROL ASED ON GENEIC ALGORIHMS Jianting Lv, Guangfu Ma, Dai Gao Department of Control Science and Engineering, Harbin Institute of echnolog, Harbin 5 Abstract: o solve the magnetic attitude control problem of a bias momentum satellite, a controller design method based on genetic algorithms (GAs is addressed. he pitch loop of the satellite is controlled b a bias momentum wheel, while the roll/aw loops are stabilized b two magnetic torquers which are installed along their respective aes. Momentum unloading is achieved through the two magnetic torquers. Sum of squares of three aes attitude errors and angular velocit errors via appropriate weighting serves as objective function that used in GA optimization, and the minimization of objective function is our performance inde. Simulation results demonstrate that high pointing and stabilit accurac can be obtained using the proposed optimized control scheme. Copright 6 UACC ewords: ias Momentum satellite; attitude control; magnetic control; genetic algorithms. INRODUCION Attitude control sstems pla an important role in the operation of satellite. Since the magnetic attitude control sstems are relativel lightweight, require low power and are inepensive, the use of magnetic torquers for attitude control has become an active research topic (arr, ; Lovera, 5; Wisniewski,. For bias momentum satellite, union control schemes using bias momentum wheels and magnetic torque rods have been developed through the ears (Cavallo, et al., 993; Jan, 5; Fichter, 996. Such sstem configuration has both the advantage of a minimum number of low cost actuators and the fact that the adopted actuators have no nominall fuel consumption in performing attitude and angular momentum control. Such magnetic attitude control is an attractive alternative for satellites subject to cost and weight constraints. Satellite attitude control sstems design must take into account a number of performance issues, such as stabilit accurac, pointing accurac. Man project designers appl the traditional PD control scheme of which the control parameters are often obtained from method of trial and error, thus the drawbacks of this kind of design obviousl cost much time and efforts, and hardl to guarantee the optimal control parameters. Genetic Algorithms (GAs are search algorithms based on the mechanics of natural selection and natural genetics (Goldberg, 989. GAs can be adopted in solving these problems b encoding the parameters of the controllers into a chromosome, and defining a fitness measure as a function over the performance demands, thus formulating the design problem as the minimization of an objective function with respect to the controller parameters. his paper deal with the controller design for a three-ais stabilized bias momentum satellite, that is equipped with one momentum wheel and two magnetic torquers. Here a controller design method based on GAs is addressed. he GA optimization process is introduced to turn the control parameters for attitude control to achieve good results. he paper is organized as follows: in Section we briefl state the dnamics equation of the satellite; the satellite attitude controller design approach is presented in Section 3; the proposed decimal GA

2 optimization procedure to tune and optimize the performance of the controller in the former section is presented in Section 4; Numerical simulations are carried on and demonstrate the effectiveness of the controller and GA approach in Section 5. We conclude the paper in Section 6.. DYNAMICS EQUAIONS OF HE SAELLIE he attitude dnamics for a satellite with internal angular momentum can be epressed b the Euler s equations Iω ω Iω h h ( ( c d Where 3 ω is the satellite bod angular velocit vector, I is the inertia matri, c is the control torques vector, d is the eternal disturbance torques vector, and h is the wheel angular momentum. For a pitch momentum bias sstem with a momentum wheel aligned along the satellite pitch ais, we have h h ( And we note that h hn Δ h (3 Where h n ( h n < is the pitch momentum bias, Δh is the small variations in wheel momentum for pitch control, and h n Δh. Under the definition on 3-- rotation, the attitude kinematics are represented b ω ϕcosθ ψcosϕsinθ ω ω θ ψ sinϕ ω z ψcosθcosϕ ϕsinθ (4 cosθ sinψ sinθsinϕcosψ ω cosϕcosψ sinθ sinψ cosθsinϕcosψ Where ω, ω, and ω z are the angular velocit components of the bod frame with respect to the inertial frame, decomposed in the bod frame aes; ω is the orbit angular velocit;ϕ, θ, and ψ are the roll, pitch, and aw angles, respectivel. Suppose the attitude of the pitch loop has been well controlled, that is h fluctuates within a relativel small scope ( Δ h is ver small. Hence to the rollaw coupling loops, h can be treated as a constant, let h hn, and when the satellite var in a small angle, using equations (, (3, and (4, the eq.( can be written as I ϕ ω hϕ hψ I θ h d I ψ ω hψ h ϕ n n c d z n n cz dz (5 As seen from (5, since the linear pitch dnamics is decoupled from the roll/aw dnamics, the pitch controller and roll/aw controller are design separatel. 3. ANALYSIS AND DESIGN OF AIUDE CONROL 3. Pitch control ased on the simplified model (5, the pitch loop equation controlled onl b momentum biased wheel is: I θ h (6 d d includes the eternal disturbance torque of the pitch ais. h θ θ, (6 can be written as Let p d I θ pθ dθ d. his is a classical secondorder control sstem. We can get control parameters through the genetic algorithm optimization. U c p( τ ps m PD Controller Wheel L m R Is Fig.. lock diagram of the pitch loop 3. Roll/aw control s h Δh h s c Unload unload Attitude control of roll/aw loop utilize magnetic actuation, in which the mechanical torque required for attitude control is generated b the magnetic interaction between the geomagnetic field and magnetic torquers. Here Magnetic torquers can be used for attitude and also for wheel momentum unloading. he control torque generated b magnetic torquers can be epressed as c mc mu M S( m (7 Where mc is the control torque generated from the roll/pitch aes magnetic torquers, and mu is the unloading torque of the bias momentum wheel generated from the roll/aw aes magnetic torquers; M [ M ] Mz is magnetic control moment, [ ] z is the vector formed with the components of the Earth s magnetic field in the bod frame of reference, and S( is given b d

3 z S ( z (8 M is the control magnetic moment vector which is split into two parts Mc Mu M Mc M u (9 M cz M uz Where Mc is control moment on roll/aw ais; M u is unloading moments for pitch bias angular momentum. For a momentum bias satellite using two-ais magnetic torquers, we have c z Mc c z h ( cz M cz For a desired torque c r to be applied on the satellite, we must generate the magnetic momentum vector M. he matri in ( does have an inverse, so M c r h z r ( M cz rz he control torques in roll/aw loop can be calculated b the formula ( ϕ ϕ ( ψ ψ r pz dz rz p d ( Substituting ( into (, we have the following control law Mc ( pψ dψ Mcz ( pzϕ dzϕ (3 Magnetic torquers generate magnetic moments whose interactions with the earth s magnetic field produce the torques necessar to remove the ecess momentum. he control equation for momentum unloading is as following, see (Sidi, 997 k Mu ( Δh (4 Where Δ h h h Δh nom is the ecess momentum to be removed, h is the wheel s momentum vector, h nom is the nominal wheel momentum vector, and k is the unloading control gain. 4. CONROLLER DESIGN VIA GENEIC ALGORIHM Genetic algorithms are search and optimization techniques based on the mechanics of natural selection and natural genetics. GAs manipulate a population of potential solutions which are called chromosomes or individuals. Chromosomes are encoded representations of all the parameters of the solution. We evaluate chromosomes according to their lever of fitness.o evolve chromosomes that encode better solutions, the GA emplos the genetic operators of selection, crossover, and mutation for manipulating the chromosomes in a population. he selection mechanism for parent chromosomes takes the fitness of the parents into account, ensuring that the better solutions have a higher chance to procreate. Newl generated chromosomes in time replace the eisting ones. hrough this process the population will converge to a best solution. he real-coded GA is chosen for this stud because the processed parameters are multidimensional ones and var in large scopes. he real-coded GA can simplif the encoding/decoding procedure and provide high precision results. he proposed method is described in the following form. Step : Give a reasonable range of control parameters. Select coding method. Set up GA parameters: crossover probabilit, mutation probabilit, population sizes, and maimum number of generations. Step : Initialize the population of GA. Step 3:Calculate the objective function for each chromosome. he best controller parameters for this eample obtained b the minimization of the objective function f (. Here we define the objective function f ( as 3 t 3 t f f ( αe ( αe f( i t t i t t hat is sum of squares of three aes attitude errors and angular velocit errors via appropriate weighting serves as objective function which is used in GA optimization. α and α are set mainl according to order of three aes attitude errors and angular velocit errors. he objective function is used to provide a measure of how individuals have performed in the problem domain. Another function, the fitness function, is normall used to transform the objective function value into a measure of relative fitness (.A.De Jong, 975. Fitness values are derived from objective function values through a scaling or ranking function.

4 Step 4: Calculate the fitness value for each individual, if the search goal is achieved, or an allowable generation is attained, stop and return; else go to step 5. Step 5: Selection and reproduction. he individuals with better fitness will be selected into net generation. Appl crossover and mutation. Step 6: Repeat step 4. Flow chart of the proposed method is shown in Fig.. Start Parameter coding Initial Population Generation Objective Function Calculation Fitness Calculation Ending Condition? N Selection, Crossover, & Mutation New generation Y Return Controller & Actuator Fig.. Flow chart of the proposed method 5. SIMULAION EXAMPLES Satellite od Simulink Model We use an eample to demonstrate the validit of the method. he numerical values of satellite parameters are assumed as flowing: ϕ( θ( ψ( ϕ( θ( ψ (. / s For this eample the following GA parameters are defined in able. able GA parameters GA Parameter Value or Method Generation number 3 Individual number 4 Parameter number 6 Coding method Real-valued coding Selection method Stochastic universal sampling Crossover and/or recombination Discrete recombination Probabilit mutation.67 Fitness assignment Rank-based fitness assignment Generation gap.9 Fig. 3. shows the progress of the objective function during the 3 generations. In the nd generation, the best individual is found b GA. k 4 is chosen in this eample. Using the controller parameters turned b GA, the results of the simulation, attitude angles, attitude velocities, momentum wheel torque, wheel momentum, and magnetic moments are shown in the figure 4, 5, 6, 7, and 8, respectivel. As can be seen form the figures, high control accurac is achieved and the maimum magnetic momentum can provide the control torques of the roll/aw loops. able lists the controller parameters. he significant digit is -4. able Controller parameters p d p d pz dz Orbit attitude is 85 km, and inclination is he moments of inertia are 6 8 Ι ( kg m ias angular momentum h 6( N m s. Ma magnetic momentum is ( A m. Disturbance torques are d.5cos( ω t.5sin( ωot 3 o 4 sin( ωot 5 ( N m n Fig. 3. Progress of the objective function Whereω.8 rad / s. he initial attitude angle and initial angular velocit of satellite are

5 Fig. 4. Histor of attitude angles Fig. 8. Histor of magnetic moment 6. CONCLUSION In this paper, magnetic attitude control problem of a three-ais stabilized bias momentum satellite has been considered, and an optimal controller design method based on genetic algorithm is proposed. he creative combination of control methods and GAs can result in a powerful tool that is able to address real engineering control problems. REFERENCES Fig. 5. Histor of attitude angular velocities Fig. 6. Histor of bias momentum wheel s torque Fig. 7. Histor of bias momentum wheel s angular momentum arr S. Leonard (. NPSA Magnetic Attitude Control Sstem. In: 6th Annual AIAA/USU Conference on Small Satellites,, Utah, -7. Cavallo, A., G. De Maria, F. Ferrara and P.Nistri. (993. A Sliding Manifold Approach to Satellite Attitude Control. In: th World Congress IFAC,993, Sidne, Goldberg, D.E. (989. Genetic Algorithms in Search, Optimization and Machine Learning. Addison- Wesle. Jan, Y.W., Chiou, J.C. (5. Attitude Control Sstem for ROCSA-3 Microsatellite. In: A Conceptual Design[J], Acta Astronautica. Vol. 56, A.De Jong. (975. Analsis of the ehaviour of a Class of Genetic Adaptive Sstems, PhD hesis. Dept. of Computer and Communication Sciences, Universit of Michigan, Ann Arbor. M.J. Sidi. (997. Spacecraft Dnamics and Control. Cambridge Universit Press. Marco Lovera. (5. Global Magnetic Attitude Control of Inertiall Pointing Spacecraft. Journal of Guidance, Control, and Dnamics. Vol. 8, No.5, September-October Walter Fichter, Michael Surauer. (996. Magentic Attitude and Angular Momentum Control Design For Momentum ias Satellites in Low Earth Orbits. 3th World Congress IFAC,996, San Francisco, USA, Wisniewski, R. (. Linear ime-varing Approach to Satellite Attitude Control Using Onl Electromagnetic Actuation. Journal of Guidance, Control, and Dnamics. Vol. 3(4, 64-64