ATTITUDE DETERMINATION AND CONTROL OF MICROSATELLITE USING TYPE-1 AND TYPE-2 FUZZY LOGIC

Transcription

1 BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de Universitatea Tehnică Gheorghe Asachi din Iaşi Tomul LIX (LXIII), Fasc. 1, 2013 SecŃia AUTOMATICĂ şi CALCULATOARE ATTITUDE DETERMINATION AND CONTROL OF MICROSATELLITE USING TYPE-1 AND TYPE-2 FUZZY LOGIC BY ALI ASAEE, SAEED BALOCHIAN * and SAEED HESHMATI Islamic Azad University, Gonabad, Iran, Department of Electrical Engineering, Gonabad Branch Received: February 25, 2013 Accepted for publication: March 28, 2013 Abstract. Attitude controlling has been investigated from various points of view and different controllers have been designed. In addition to respecting microsatellite limitations, control algorithm needs to have high reliability in noise. This is because of the nonlinearity, complexity and uncertainty satellite model. It is essential to use smart controllers in microsatellite control systems which don t have high accuracy attitude sensors and whose control operator does not have high capability facing turbulent torque. Since control input is a torque, attitude control system operators are torque generators. For optimal maneuver, the controlling torque applied to the nonlinear satellite system must have proper size and quality. Reaction wheels, thrusters, magnetic coils and control moment gyroscopes (CMG) are some of the most important satellite operators. In this study we have used fuzzy controllers of type 1 and 2 in three axis control with high accuracy. Finally, simulation results are given to show the effectiveness of the proposed method. Key words: attitude determination, fuzzy type 1 and 2 control, rotation, Euler angle Mathematics Subject Classification: 93C42, 94D05, 37M05. * Corresponding author; saeed.balochian@gmail.com

2 52 Ali Asaee, Saeed Balochian and Saeed Heshmati 1. Introduction Satellite attitude represents its orientation in space with respect to different coordinate systems. The system for attitude determination and control is one of the most important subsystems of a satellite, since the accuracy of its mission depends on this subsystem capability. In other words, this subsystem represents the satellite visual sense and feeling in space. This is very important in small satellites. The algorithm and control system needs to be as simple and fast as possible and with Low-volume hardware and software. These are all because of small size limitation of these satellites. Thus micro satellite designing engineers are trying to use the techniques and parts overcoming the limitation and optimizing system output rate. Generally the reasons of attitude controlling subsystem presence are as follows (Sidi, 2007; Overby, 2004): a) In communication satellites, antennas are required to be focused to a certain point on earth with high accuracy. b) In earth observing satellites, cameras are required to be focused to a certain point in earth so give an acceptable video coverage of the region. c) In orbital maneuvers desired attitude must be provided. d) For maximum use of solar energy solar cells must be pointed to the sun. A satellite attitude survey includes attitude controlling and the way of it and also prediction of its next move. The general block diagram of attitude determination and control is shown in Fig. 1. Fig. 1 Block diagram of attitude determination and control. A satellite attitude determination can be done toward the sun, stars, the earth and magnetic field. This is done by using different sensors and operators. Sensors send the satellite position errors to the satellite central processor system, that uses the controlling algorithm to produce controlling rule and sends signals to actuators. The actuators produce required torque to control satellite attitude (Larson & Wertz, 1999; Overby, 2004). Attitude control means placing the satellite in a specific predetermined direction. This consists of attitude stabilization and control maneuver. In attitude stabilization the goal is keeping the satellite in its position and in

3 Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, attitude control maneuver is changing from one state to another. In microsatellite attitude control system (includes operators and controlling algorithm) must be chosen in a way that consumes less power so desired optimal satellite life for its mission is guaranteed. So far, various control methods have been used to control the satellite attitude. The first attitude control system was designed in Mercury and Vestok satellites returning to Earth s atmosphere. Most of the controlling systems were based on the thruster before the discovery of the photoelectric cells using solar energy to provide electrical power. James Wertz is one of the ones that analyses determination and control attitude of a satellite in his book (Wertz, 1978). One of the most important parameters that specify the dynamics of the attitude is the system inertia matrix. In a real case the satellite mass properties may be inaccurate or change over cargo movements. Thus nonlinear attitude control needs to be adaptable in unknown system parameters and resistant to external disturbances. Attitude detection of a rigid spacecraft with an imprecise inertial matrix in external disturbances has been investigated by adaptive control in reference (Luo et al., 2005). It reveals that in no disturbed situation, for all initial values of the Euler angles, the inverse optimal adaptive attitude controller converges well to the desired position. The approach of applying MIMO feedback for designing the attitude control system of a satellite based on a nonlinear model with inconclusive parameters in the presence of disturbances is presented in reference (Nudehi et al., 2008). A bang-bang fuzzy controller for a satellite with reaction wheel operator is designed in reference (Nagi et al., 2012). However the environment model is not mentioned. An adaptive fuzzy system in combination with attitude controlling H 2 H has is used by Cheng and Shu for a nonlinear system with uncertain and imprecise inertia matrix in external disturbance in (Cheng & Shu, 2009). In our current paper we first investigate the nonlinear modeling of a satellite attitude and its dynamic. Then three fuzzy controllers for Euler angles are designed. Finally the designed controller is evaluated through MATLAB simulation. 2. Modeling a Microsatellite The aim of the attitude determination system is to calculate the attitude of the body coordinate with respect to a specified coordinate system. Thus firstly orbital motion coordinate systems will be mentioned. Then the rotation between these systems will be discussed. Finally situational dynamics of microsatellite will be investigated. Different coordinate systems are illustrated in Fig. 2.

4 54 Ali Asaee, Saeed Balochian and Saeed Heshmati Fig. 2 Different coordinate systems Rotation Since Newton s laws only work in inertial coordinate systems, motion equations cannot be written in a body fix coordinate system. Hence determining the relationship between the components of a vector is needed. There are several ways to show the rotation between two coordinates system such as (Kaplan, 2006). Direction Cosine Matrix Euler angle Euler angles/axis Quaternions Gibbs vectors 2.2. Quaternions The main reason of using quaternions instead of Euler angle is avoiding singularity. Quaternions, usually denoted by q, extend the complex numbers by having a real part η and three imaginary parts indicated by vector ε (Wertz, 1978). If the coordinate system rotates with angle θ around vector λ, it can be written: η ε1 ε θ θ 1 η= cos, ε = ε 2 sin,. 2 = λ q= (1) 2 ε 2 ε3 ε 3

5 Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, Quaternions satisfy the relationship q q = 1 that means: T η + ε + ε + ε = (2) 2.3. Attitude Equations Suppose that a rigid body is moving in an inertial coordinate system; this motion can be described by the center of mass translational motion and the rotational motion of object around the center of mass Attitude Dynamics Assume the A vector in I and B systems. The following equation is known as the Coriolis law (Sidi, 2007): d d A = A + ω A. (3) I B dt dt This equation shows that the A vector observed rate of changes from the fixed system I is equal to its rate of changes in the mobile system B (it moves with the speed v toward I system) plus the poduct ω A. The angular momentum of the whole body is: m i ( ω r) h= r m i i i 2 2 = i ωx ( yi + zi ) mi ωy yi ximi ωz xi zimi mi mi m i j ωy ( xi + zi ) mi ωx yi ximi ωz yi zimi mi mi m i k ωz ( xi + yi ) mi ωx zi ximi ωy yi zimi. mi mi m i In this equation ωx, ωy, ω z are the particle angular velocities around i, 2 2 Ixx = yi + zi mi, Ixy = yi ximi and j, k. The momentum definitions are ( ) I = x z m. There is a same definition for other axes. The simplified form of xz i i i m i eq. (4) is as follows: m i m i (4)

6 56 Ali Asaee, Saeed Balochian and Saeed Heshmati h= i ω I ω I ω I + j ω I ω I ω I x xx y xy z xz y yy x yx z yz + k ω I ω I ω I z zz x zx y zy. (5) The matrix forms of eq. (5) is as follows Ixx Ixy Ixz ωx h= I yx I yy I yz ω y = Iω. (6) I zx Izy I zz ω z In symmetric mode we have I = I, I = I, I = I. Matrix I is xy yx xz zx yz zy known as inertia matrix. With proper selection of the axes, the inertia matrix can be transformed to a diagonal form. The center of these axes is the center of mass. They are known as main inertia axes and are reached by a fix rotation from primary fixed body system. The relationship between angular momentum and applied torque is: τ = hɺ = hɺ + ω h (7) I This equation is known as Euler s moment equation (Sidi, 2007; Wertz, 1978). By assuming X B, YB, Z B as inertia axis, finally we will have: B ( ) τ = I ɺ ω + ω ω I I, (8) x x x y z z y τ = I ɺ ω + ω ω ( I I ), (9) y y y x z x z τ = I ɺ ω + ω ω ( I I ). (10) z z z x y y x These are nonlinear equations and do not have analytical solutions (Sidi, 2007). In this equations applied torques are control and disturbance torques Attitude Kinetic Kinetic equations describe the satellite orientation and, as mentioned, there are various ways of describing it. One of them is using quaternions (Sidi, 2007; Wertz, 1978). where d 1 q= Ω ɺ q, (11) dt 2

7 Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, ωz ω y ωx ωz 0 ωx ωy Ωɺ = ωy ωx 0 ω. (12) z ωx ω y ωz 0 The relationship between quaternions and Euler s angles is: φ= asin( 2 q q q q ), (13) ( q1q3 q2q4 ) θ = atan , (14) q 1 q2 q3 + q 4 2( q1q 2 q3q4 ) ψ = atan (15) q1 + q2 + q3 + q 4 where: atan 2 is a tangent function with bounds between 180 and 180 instead of 90 and Disturbance Torques Applying on the Satellite Here are disturbance sources that must be considered (Kaplan, 2006): gravity gradient, magnetic disturbance, aerodynamic drag, solar radiation pressure Gravitational Moment Applying on the Satellite The Earth's gravity gradient torque is one of the most important external torques applying on satellites in orbits that are near the Earth. An asymmetrical body towards the Earth's gravity field tends to set the axis with the minimum inertia in field direction. The amount of this torque can be reached from the following equation (Kaplan, 2006). 2 3 ωo ( I z I y ) a23a 33 τ 3 ω ( ) 3 ω ( ) B 2 2 B B g = o Iz I x a13a33 = o RO 3 IRO3 2 3 ωo ( Ix I y ) a13a23 (16) 2.5. Reaction Wheel Reaction wheels are preferred in attitude control systems with high control accuracy and medium rate maneuver due to constant smooth control

8 58 Ali Asaee, Saeed Balochian and Saeed Heshmati with minimum disturbance torque noise. The amount of torque reached by reaction wheel is 0.01 to 1 N/m. Within a spacecraft if a symmetric rotatory object accelerates around its rotational axis, angular torque will be produced. It may have Initial fixed momentum of h r. As this is considered for internal spacecraft, its increase does not change total momentum of system instead exchange the momentum (negative on) with the spacecraft (h = h r +h B ). This is the angular momentum conservation principle. In order to control the attitude in space three reaction wheels are needed. The required angular momentum can be reached from accelerating electrical motors rotors which their rotational axis are in same vector of X B, Y B, Z B body axis (Ge & Cheng, 2006). As control operator in this micro satellite is a reaction wheel. In satellite model simulation there are used the Norwegian satellite parameters (Overby, 2004). 3. Fuzzy Control The word fuzzy means ambiguous, dumb, inaccurate, confusing, confused, tangled or invalid. When fuzzy systems are used in controllers, they are depicted as fuzzy controllers. The fuzzy controllers are nonlinear controllers with a specific structure. They provide successful application of fuzzy theory in practical cases. Zadeh introduced type-2 fuzzy sets as a generalization of ordinary fuzzy sets. They will be called as type-1 fuzzy systems. In a type-2 fuzzy set, the degree of membership for each point is a normal fuzzy number which can vary in range of [0, 1]. These sets are appropriate for membership functions with uncertainty. In the other word, the approach of a type-1 fuzzy set is modeling the uncertainties by a number between 0 and 1 that is a crisp number. In fact the degree of belonging of each element to a type-1 fuzzy set is an exact crisp number while the degree of belonging of each element to a type-2 fuzzy set can be a verbal (linguistic) amount (Mendel, 2007). In this study, a fuzzy control is used in a closed-loop system. It is shown in Fig. 3. Fig. 3 Block diagram of a fuzzy controller in a closed-loop system.

9 Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, Since the three axes of the body have separate dynamics, the design for each axis can be done independently. Thus three separate controllers are used to control the attitude. We expect that appropriate control torques to maneuver and stabilize the satellite by reaction wheel mechanism for each controller. The desired attitude is achieved through the torque that each controller can apply around a satellite axis. The simplest torque control law is based on Euler s angles errors. For the Roll, Pitch and Yaw angle, the obtained values are compared with the reference values providing the Euler s angles errors. Then each error signal with its derivative is given to the corresponding controller and it produces appropriate control signal. We are going to design the controller using both types of fuzzy sets. The type-1 will be designed first, then the type Type-1 Fuzzy Controller The type-1 fuzzy control system includes a fuzzifier, a rule base, an inference engine and a defuzzifier. The fuzzification means defining fuzzy sets for input and output variables. The basic knowledge of each variable definition scope is needed to define these sets. We assume that if Euler s angles go out of the range [-1, 1], the system will be out of control. The maximum of error changes is assumed in range of [-0.3, 0.3]. Here, for each input variable (error and its derivative) five type-1 fuzzy sets with triangular and trapezoidal membership functions are defined, abbreviated as NB for big negatives, NS for small negatives, Z for zero, PB for big positives and PS for small positives. Therefore 25 fuzzy rules exist in the rule base. In Figs. 4 and 5 membership functions for the two inputs are shown. Fig. 4 The first input membership functions. Fig. 5 The second input (error derivative) membership functions.

10 60 Ali Asaee, Saeed Balochian and Saeed Heshmati The output of each type-1 fuzzy controllers is a torque that must be applied to its axis. The value of this torque is considered in the range of [-0.4, 0.4] because of practical limitations. Seven membership functions are considered for that. Memberships of output variable are shown in Fig. 6. (NB for big negative, NM for medium negative, NS for small negative, Z for zero, PB for big positive, PM for medium positive, PS for small positives). Fig. 6 The output membership functions. In the next step some fuzzy rules are inserted to the rule base. Fuzzy rule base is the heart of a fuzzy system; i.e. the other elements of fuzzy system are effectively and efficiently used to implement them (Wang, 1997). Used rules are presented in Table 1. Table 1 Rule Base We used Mamdani minimum Inference Engine, Singleton fuzzifier and Center Gravity defuzzifier in the controller of the system. The control surface is shown in Fig. 7.

11 Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, Fig. 7 Control level. After designing the controller, its performance is checked and tested in MATLAB Type-2 Fuzzy Controller The design of a type-2 fuzzy controller is similar to the design of the type-1 controller that was mentioned above. Here we investigate the Roll angle controller. (The process of designing the controllers for the Yaw and Pitch angles is same). The fuzzifing step is defining fuzzy sets for input and output variables. The basic knowledge of each variable definition scope is needed to define these sets. We assume that if Euler s angles go out of the range [-1, 1], the system will be out of control. The maximum of error changes is assumed in range of [-0.3, 0.3]. Here, for each input variable (error and its derivative) five type-1 fuzzy sets with triangular and trapezoidal membership functions are defined. Uncertainty in membership functions must be considered in type-2 fuzzy sets. Hence in membership function figures there are narrow picks instead of lines. The width of pick shows the uncertainty (Figs. 8 and 9). There will be 25 fuzzy rules in rule base by defining membership functions for inputs. Fig. 8 The first input membership functions with uncertainty.

12 62 Ali Asaee, Saeed Balochian and Saeed Heshmati Fig. 9 The second input membership functions with uncertainty. The output of each fuzzy controller is a torque that must be applied to its axis. The value of this torque is considered in the range of [-0.5, 0.5] because of practical limitations. Seven triangular membership functions with uncertainty are considered for that (Fig. 10). The control level is shown in Fig. 11. Fig. 10 The output (torque) membership functions with uncertainty. a b Fig. 11 The control level from different angles.

13 Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, In the next step some fuzzy rules will be defined. The value of control signal will be calculated by them. The value of the error and its derivative must be considered too. The rules are the ones used in type-1 fuzzy controller. After designing the controller, its performance is checked and tested in MATLAB. 4. Simulation The controller performance can be investigated from 2 points of view: attitude maneuver and stability. In the following they will be discussed. For better visualization and understanding of fuzzy controller type-1 response, it will be compared to an ordinary PID controller. A) Attitude Stabilization: This means returning to the equilibrium point Θ= ( φ, θ, ψ ) = (0,0,0). It is called zero input state. In this state the Euler s angles are equal to the initial value Θ 0= ( φ0, θ0, ψ 0). We suppose a controller applies a proper torque to lead the value of these angles to zero. In Figs. 12 and 13 the response of attitude control system by fuzzy controller type 1 and PID are shown respectively. Fig. 12 The attitude control system output at attitude stability state using fuzzy control type 1. Fig. 13 The attitude control system output at attitude stability state by PID control.

14 64 Ali Asaee, Saeed Balochian and Saeed Heshmati Figs. 14 and 15 show the torque produced by fuzzy controller type 1 and PID controller in attitude stability state. Fig. 14 The torque produced by fuzzy controller type 1 at attitude stability state. Fig. 15 The torque produced by PID controller at attitude stability state. Now the other state will be discussed. B) Maneuver Attitude: It is the ability of setting the satellite in the predefined desire attitude. It is considered that the value of Euler s angles is zero value. We are going to rotate the satellite in a direction so that the angles reach to 10 degree (In this case the reference value is 10). In Figs. 16 and 17 the response of attitude control system using fuzzy controller and PID controller in attitude maneuver state are shown, respectively.

15 Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, Fig. 16 The attitude control system output using type-1 fuzzy control at maneuver state. Fig. 17 The attitude control system output using PID control at maneuver state. Figs. 18 and 19 show the torque produced by type-1 fuzzy controller and PID controller in attitude maneuver state.

16 66 Ali Asaee, Saeed Balochian and Saeed Heshmati Fig. 18 The torque produced by the type-1 fuzzy controller in maneuver state. Fig. 19 The torque produced by PID controller in maneuver state. According to Euler s torque equations each external disturbance apply to the body adds an Angular Momentum to the satellite system. Therefore the satellite attitude changes with increase in Angular Momentum without active attitude control. C) Adding noise to system. In the next step performance of fuzzy controller type one in disturbances is evaluated by applying noise to the system is at its maneuver state. The noise is applied as a torque. Figs. 20 and 21 show the output and produced torque of the attitude control system by fuzzy control type 1 at maneuver state in presence of the noise.

17 Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, Fig. 20 The output of the attitude control system by fuzzy control type 1 at maneuver state in presence of the noise. Fig. 21 The produced torques by type-1fuzzy control at Maneuver state in the presence of noise. Now the response of attitude control system by fuzzy control type 1 at maneuver state will be discussed. In Fig. 22 the torque produced by three controls (Roll, Pitch and Yaw) are shown. Fig. 22 The attitude control system output by fuzzy control type 2 at maneuver state.

18 68 Ali Asaee, Saeed Balochian and Saeed Heshmati Here the performance of type-2 fuzzy controller will be investigated in presence of the noise. Figs. 23 and 24 show the output of the attitude control system and its effect. Fig. 23 The attitude control system output by type-2 fuzzy control at maneuver state in presence of the noise. Fig. 24 The produced torque of attitude control system by type-2 fuzzy control at maneuver state in presence of the noise. 5. Conclusions It can be seen that the transient response when using the type-1 fuzzy controller has less fluctuations in comparison with the PID. The settling time is smaller too. Both controllers can lead the steady state error to zero. According to achieved diagrams, the type-1 fuzzy control at maneuver state performs well while the PID performance is inefficient and the system output fluctuates completely.

19 Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 1, The results reveal that the fuzzy controller can control the system in the presence of noise. There is not a significant difference between transient and steady state characteristics in presence or absence of the noise. It has been shown in the figures that fuzzy control as an active controller can control the satellite in disturbance. They are caused by aerodynamic drag, solar radiation, solar wind torques, propulsion disturbing torque and etc. The mentioned controllers in designing fuzzy controllers are independent. The range of torque changes according to their maximum and minimum, problem assumptions and practical limits is acceptable and achievable. The performance of Roll, Pitch and Yaw controllers is acceptable in type 1 and 2. The response in type-2 has less settling time and a reaches to the desired value sooner in comparison with type-2. Although Roll controller output has some fluctuations, Pitch and Yaw controllers response improves. Therefore it can be concluded that type-2 fuzzy controller performance is better than in the case of type-1. REFERENCES Cheng C.H., Shu S.L., Application of Fuzzy Controllers for Spacecraft Attitude Control. IEEE Transactions on Aerospace and Electronic Systems, 45, 2, , Ge S., Cheng H., A Comparative Design of Satellite Attitude Control System with Reaction Wheel. First NASA/ESA Conference on Adaptive Hardware and Systems AHS 2006, , Kaplan C., Leo Satellites: Attitude Determination and Control Components; Some Linear Attitude Control Technique. M. Sc. Thesis, Middle East Technical University, Turkey, Larson W.J., Wertz J.R., Space Mission Analysis and Design. Third Edition, Microcosm Press, Hawthorne. CA, Luo W., Chu Y.C., Ling K.V., Inverse Optimal Adaptive Control for Attitude Tracking of Spacecraft. IEEE Transactions on Automatic Control, 50, 11, , Mendel J.M., Advances in Type-2 Fuzzy Sets and System. Information Sciences, 177, 1, , Nagi F., Zulkarnain A.T., Nagi J., Tuning Fuzzy Bang Bang Relay Controller for Satellite Attitude Control System. Aerospace Science and Technology, 26, 1, 76 86, Nudehi S.S., Farooq U., Alasty A., Issa J., Satellite Attitude Control Using Three Reaction Wheels. American Control Conference ACC 08, , Overby E.J., Attitude Control for the Norwegian Student Satellite ncube. M. Sc. Thesis, Norwegian University of Science and Technology, Sidi M.J., Spacecraft Dynamics and Control: A Practical Engineering Approach. Cambridge University Press, Wang L.X., A Course in Fuzzy Systems and Control. Prentice-Hall, Wertz J.R., Spacecraft Attitude Determination and Control. Springer, 1978.

20 70 Ali Asaee, Saeed Balochian and Saeed Heshmati DETERMINAREA ATITUDINII ŞI CONTROLUL UNUI MICROSATELIT UTILIZÂND LOGICA FUZZY DE TIP 1 ŞI 2 (Rezumat) De-a lungul anilor, determinarea atitudinii satelińilor a fost investigată prin multe metode, iar pentru controlul acesteia au fost proiectate diverse regulatoare. În ceea ce priveşte controlul microsatelińilor, datorită dimensiunilor acestora apar restricńii suplimentare, de exemplu fiind necesar ca algoritmii de control să fie insenzitivi la zgomote. Acest lucru se datorează faptului că, de regulă, microsatelińii nu sunt dotańi cu senzori de atitudine cu precizie înaltă. În studiul de fańă pentru controlul pe trei axe sunt utilizate cu mare acurateńe regulatoare fuzzy de tipul 1 şi 2. Rezultatele obńinute prin simulare dovedesc eficacitatea metodei propuse.