Spacecraft Attitude Determination and Control System for an Earth Observation Satellite

Transcription

1 Spacecraft Attitude Determination and Control System for an Earth Observation Satellite Mario Rodriguez, Rafael Vazquez Escuela Técnica Superior de Ingeniería, University of Seville, Seville, Spain, Spacecraft optical payloads usually demand high levels of accuracy in order to track ground targets in relative motion, requiring pointing errors to remain in the arc-second range. The attitude control system has to achieve this level of performance while at the same time dealing with perturbation torques and imperfect attitude measurements. This paper addresses the complete analysis and design of a three-axis quaternion-based ADCS algorithm, from mission planning to real-time simulation using reaction wheels, gyroscopes, and direction measurements. A set of ground targets is considered; whenever any of the targets is at sight, the optical axis of the satellite shall maneuver to lock on the target and track it as the Earth rotates and the satellite moves on a certain sun-synchronous orbit. If no target is accessible, the satellite maneuvers to an attitude profile which maximizes solar array illumination. The proposed output-feedback control algorithm constructs a solution for the motion planning and then solves the tracking problem by combining the efficiency of the Finite Horizon Linear Quadratic Regulator (LQR) and the accuracy of the Multiplicative Extended Kalman Filter (MEKF). Realistic simulations including uncertainties and perturbations show the effectiveness of the algorithm. MATLAB -STK software integration provides a virtual visualization allowing intuitive evaluation and analysis of the problem. Nomenclature J a, J t Reaction wheels axial and transverse moment of inertia q Attitude Quaternion ω Body angular velocity ω w Reaction wheels angular velocities h Reaction wheels angular momentum n Orbital Velocity Control Torque u i I. Introduction Earth Observation satellites constitute a sizeable part of the present and projected satellite population. By observing the Earth from (typically low) orbit they are able to provide data for applications ranging from environmental monitoring to cartography. Current instrument capabilities are able to provide high levels of resolution provided that rather stringent pointing requirements are satisfied. Typically many targets need to be observed, resulting in a tighter schedule yielding shorter access windows and needing faster slewing maneuvers, raising the performance requirements and control power needs. Thus, there is a need for accurate Attitude Determination and Control Systems (ADCSs) that are able to deliver precise and fast tracking capabilities for targets on the ground. Many algorithms for ADCS exist in the literature. 1, 2 In this work, we explore a tracking algorithm composed of three parts. First, we develop a path planner that constructs attitude maneuvers to switch from one target to another. Subsequently, we use a finite-horizon LQR algorithm to make the maneuver asymptotically stable, at least for small deviations and perturbations. The particular ADCS architecture selected in this work for attitude control is a three axis momentum exchange control system, based on three reaction wheels placed on the principal axes of the spacecraft. Finally, we formulate an attitude estimator using integration of noisy inertial and vector measurements from a set of gyroscopes and a star tracker; our formulation is based on the Multiplicative Extended Kalman Filter. A quaternion approach is used in all three parts to obtain global attitude representation. We also provide some simulations where we explore how to tune the LQR parameters and investigate the robustness of the proposed design with respect to reaction wheel saturations. Additionally, the use of Matlab and STK integration has been applied to obtain visualizations of the different functions of the ADCS system in several scenarios. Aerospace Engineer, Aircraft and Spacecraft, mariorolu@gmail.com Associate Professor, Dpto. Ing. Aeroespacial, rvazquez1@us.es 1 of 11

2 We consider the following reconnaissance mission in this work. As a primary goal, the satellite shall track the city of Seville, every day at dawn. This city is the first of a target schedule, which also includes cities like Cape Town, Yakutat and Sydney. When none of the scheduled targets are visible, the satellite maneuvers to track a quasi-inertially-fixed attitude maximizing solar array output. Therefore, this attitude reference profile can be split into three attitude segments: target tracking, sun tracking and slewing maneuvers between those (see figure 1). The slewing maneuver is accomplished through a minimal energy rotation. The structure of this paper is as follows. We start with Section 2, where the attitude dynamic and kinematic models are described. Section 3 follows with the precise description of the tracking problem. Section 4 briefly describes the LQR controller. Section 5 introduces the key ideas behind the MEKF observer. Section 6 follows with the combined formulation of the LQR-MEKF algorithm, and finally, section 7 shows some simulation results. Appendix describes the interaction between Matlab and STK software used in this work. II. The model Since this mission yields a wide variety of attitude ranges and rates, a nonsingular attitude representation system must be adopted. Euler angles provide an intuitive attitude representation, but they imply trigonometric functions usage and not every attitude is well defined. Direction Cosine Matrix (DCM) is an useful tool for reference frame translations, but it is not an efficient figure when numerical integration is involved. The Euler angle-axis representation describes a rotation through the rotation axis e, and the angle rotated about it, θ. It is an intuitive attitude representation but not well defined when the rotation angle is zero. Amongst the wide variety of attitude representations available, the attitude quaternion is chosen in this work. It is an computationally efficient and non-singular attitude representation, along with the drawback of being a not very intuitive way of understanding the attitude of the satellite its mathematical form resembles a four-dimension complex number. The attitude quaternion encodes the Euler axis-angle representation as q = [ e sin θ/2 cos θ/2 ] T. Please note that it obeys the unit length constraint. The main advantage of this representation lies in how kinematic equations are written, as they only involve products q = 1 [ ] ω 2 q, (1) 0 where represents the quaternion product. 3 On the other hand, system dynamics are represented with the Euler equations d Γ dt = M = d Γ A dt + ω Γ, (2) B where A denotes the J2000 inertial reference frame, B is the body axes frame, and M is the perturbation torque. Given the altitude of the sun-synchronous orbit at hand, the most important applicable perturbation is Gravity Gradient (GG). In terms of the attitude quaternion, its components are 4 M 1 = 6n 2 (I 2 I 3 ) (q 1 q 4 + q 2 q 3 ) ( 1 2q1 2 2q2 2 ) M 2 = 6n 2 (I 3 I 1 ) (q 1 q 3 q 2 q 4 ) ( 1 2q1 2 2q2 2 ) (3) M 3 = 12n 2 (I 1 I 2 ) (q 1 q 3 q 2 q 4 ) (q 1 q 4 + q 2 q 3 ). The system angular momentum Γ in Euler equations (2) can be decomposed in the angular momentums of spacecraft and reaction wheels, as in Γ i = I i ω i + h i. Taking into account the motion equations of the reaction wheels 5 u i = J a ω i + ḣi and after some algebra, the model is represented with spacecraft dynamics equations (4), attitude kinematics equations (5) and reaction wheels motion dynamics (6). (I 1 J a ) ω 1 = (I 2 I 3 )ω 2 ω 3 h 3 ω 2 + h 2 ω 3 u 1 + M 1 (I 2 J a ) ω 2 = (I 3 I 1 )ω 3 ω 1 h 1 ω 3 + h 3 ω 1 u 2 + M 2 (4) (I 3 J a ) ω 3 = (I 1 I 2 )ω 1 ω 2 h 2 ω 1 + h 1 ω 2 u 3 + M 3 q 4 q 3 q 2 q 1 ω 1 q = 1 q 3 q 4 q 1 q 2 ω 2 2 q 2 q 1 q 4 q 3 ω 3 (5) q 1 q 2 q 3 q 4 0 (I 1 J a )ḣ1 = J a [(I 2 I 3 )ω 2 ω 3 h 3 ω 2 + h 2 ω 3 ] + I 1 u 1 (I 2 J a )ḣ2 = J a [(I 3 I 1 )ω 3 ω 1 h 1 ω 3 + h 3 ω 1 ] + I 2 u 2 (6) (I 3 J a )ḣ3 = J a [(I 1 I 2 )ω 1 ω 2 h 2 ω 1 + h 1 ω 2 ] + I 3 u 3 2 of 11

3 The resulting equations are a highly non-linear system x = f( x)+g( u), where x is the state vector and u is the control torque vector delivered by the actuators. The state x is a 10x1 vector which includes the angular velocity [ ] T vector, the attitude quaternion and the angular momentums of the reaction wheels, x = ω q h. III. The tracking problem In this work, a linear controller (LQR) is calculated and applied to the former non-linear system. To accomplish this, we must first linearize around a reference state which describes the ideal motion of the satellite to carry on the reconnaissance mission. The reference trajectory x ref is built according to the following key ideas: Solar array illumination: Most spacecraft solar arrays can rotate autonomously about their axis. In order to maximize illumination, the attitude quaternion shall be that whose Y body axis (the solar arrays axis) is perpendicular to the Sun vector, q ref (t) / Y body r (7) Ground target tracking: The optical axis of the spacecraft Z body shall point in the satellite-target direction, q ref (t) / Z body r sat-tar (8) Slew manoeuver: Rotation is performed by application of a certain rotation quaternion q rot (t) to the last quaternion of the former attitude profile q 1. This rotation quaternion is itself a rotation composition of q 1 and q 2, the first attitude quaternion of the next attitude profile, and an euler angle rate law. This law depends primarily on the available slewing time. q tr (t) = q rot (t) q 1 (9) Figure 1: Targets along its groundtrack. Black groundtrack portions indicate access to nearby target station Once the reference attitude quaternion is concatenated, the reference angular velocities ω ref and reaction wheels angular momentum h ref are derived. The reference state vector satisfies the non-linear system equations, xref = f( x ref ) + g( u ref ). If state errors are introduced in the formulation as δx = x x ref, δu = u uref, (10) a linear variable time (LVT) system can be derived by δx = f( x) x δx + g( u) ref u δu, (11) ref }{{}}{{} A(t) B(t) where A(t) and B(t) are the system matrices, and vary with time along with the reference trajectory. In equation (10), the error vector δx can be obtained by subtraction in its δω and δh components. However, the attitude quaternion error is obtained by a quaternion product. The state error is, then, δω = ω ω ref δx = x x ref δq = q q ref δh = h h ref 3 of 11

4 where the superscript denotes the quaternion conjugate. It is convenient to represent the attitude error in terms of the Gibbs vector a [ ] 1 a δq( a) =, (12) 4 + a 2 2 and will play an important rule in the MEKF estimator. IV. The Finite Horizon Linear Quadratic Regulator (LQR) Since the system from previous section δx = A δx + B δu is a linear time-variant system, the finite horizon LQR variant is employed. Table 1 summarizes the formulation. Variable time control feedback in equation (15) is fed with the optimal feedback gain K(t). This gain matrix is calculated by the Riccati equation (13). Differential Riccati equation: Ṡ = AT S + SA SBR 1 B T S + Q (13) t T, S(T ) given. Optimal feedback gain: Time-varying feedback: K(t) = R 1 B(t) T S(t) (14) δu = K(t) δx (15) Table 1: LQR summary Interestingly enough, the differential Riccati equation (13) must be solved backwards in time, from a given final condition S(T) in the finite horizon. This fact yields important implications in the algorithm architecture, as will be discussed later. This differential matrix equation is solved for the system matrices A(t) and B(t), and the weighting matrices Q(t), for the state error level desired; R(t), for the control input necessary; and S(t), for the final state error. In modern control design, weighting matrices are selected by the engineer. As an initial guess for this matrices, the Bryson s rule 6 is adopted. According to this rule, the diagonal elements of this matrices are inversely proportional to the maximum acceptable value of the square state error, control input and final error, respectively 1 Q ii = max [x i (t)] 2 i = 1, 2,..., 9 1 S ii = max [x i (T )] 2 i = 1, 2,..., 9 (16) 1 R jj = max [u j (t)] 2 j = 1, 2, 3. As will be shown later through simulation, the LQR has important guaranteed robustness properties. V. The Multiplicative Extended Kalman Filter (MEKF) So far, the formulation described is based on perfect knowledge of the state. Unfortunately, the ADCS system can only process information from a set of sensors, which are not perfectly accurate, nor provide measurements with infinite availability. The estimate of state error is then based upon a state estimation, δx(t) ˆ = ˆ x x ref. This sensor-fusion estimation is achieved in this work by the MEKF observer. The model considers inertial measurements of the satellite angular velocity ω from gyroscopes, which are considered to have an unlimited bandwidth, and vector measurements from a star tracker. The latter provides accurate attitude measurements in discrete time, every given refresh period. The discrete-continuous filtering is performed in three phases: propagation, update and reset. Propagation phase integrates inertial measurements provided by the gyroscopes from the last vector observation, through the kinematic equations (5). The Gaussian processes theory provides the matrix differential equation which allows the covariance matrix P propagation, P = F P + P F T + GQG T. (17) 4 of 11

5 Whenever a new vector measurement is available, the update phase estimates the attitude and angular velocity [ estimation errors, and accordingly corrects both the Kalman state vector x MEFK = a ] T b and the covariance matrix P. Finally, the reset phase updates the attitude quaternion estimate, and resets to zero the Kalman state vector. For the algorithm at hand, the sensors output is artificially generated. For the gyroscopes, the Farrenkopf s model is employed, 7 { ωgyr (t) = ω(t) + η 1 (t) + b (18) b(t) = η 2 (t), where η 1 (t) and η 2 (t) are white gaussian noises, tipically featured in most commercial gyroscopes. The MEKF will filter the noisy output signal of the gyroscope by estimation of the gyroscope error and measurement error. Since the attitude error estimate is written in terms of products (hence the name of the filter), it is parameterized in terms of the Gibbs vector in equation (12). This filter will obtain a statistically optimal estimate by minimization of the estimation error covariance matrix P, [ ] P (t) = E (x(t) ˆx(t)) (x(t) ˆx(t)) T (19) which shall be positive-semidefinite and symmetric. More information on the MEKF can be found in F. Landis Markley s work. 8 VI. Tracking with Imperfect Information Algorithm It consist in the application of the LQR controller together with the MEKF estimator. Figure (2) sketches the algorithm developed in this work. Once set the finite horizon time limits, the reference track during this time lapse yields an optimal gain matrix schedule through offline calculation of the Riccatti equation, backwards in time. This previously calculated and stored optimal gain K(t) feeds the online control loop, multiplied by the estimate of the state error. The resulting control input and the gravity gradient perturbations feed the satellite non-linear motion equations. The real state vector is artificially stained with sensor noises, so that realistic sensor outputs are filtered in the MEKF estimator, thus closing the online loop. Please note that the attitude error will be an addition of estimation error and control error. Figure 2: Tracking with Imperfect Information algorithm diagram Similarly to the MEKF algorithm, the online architecture of the problem is divided in three phases of propagation, measurement and reset. Propagation of the angular velocity is not necessary in this algorithm, since the real value of ω is obtained with eqs. (4). The gyro output is then calculated by Farrenkopf s model. Now, within the propagation phase, the real attitude quaternion q is obtained through integration of the non-linear model equations (4), (5) and (6), so that it can feed the GG equations (3) with real attitude information. The attitude quaternion estimate ˆq and the covariance matrix P are propagated until the next star tracker measurement is available. A Runge Kutta method is used in the propagation phase. The Dormand and Prince routine integrates the 50 resulting differential equations in each propagation phase. This fact, along with the complexity of random noise functions applied, makes the algorithm computationally expensive. Nonetheless, real implementation of the algorithm 5 of 11

6 would not need artificial generation of noise and propagation of the real attitude quaternion, then it is assumed to be computationally affordable. VII. Simulation Results In order to assess the ADCS performance, the attitude quaternion, control torques and pointing error will be evaluated. The pointing error, also known as Absolute Pointing Error (APE) according to ESA 9, is the Euler angle of the rotation quaternion between the reference attitude and the simulated attitude, i.e., APE(t) = 2 arccos δq 4 (t), (20) where δq 4 (t) is the scalar component of δq(t) = q ref (t) q sim (t). In addition to the APE, the ADCS performance can be appraised according to the energy index, which provides a measurement of the energy needed by the actuators. According to Junkins, 5 the energy index is and is measured in Joules. E = T 0 u(t) ω w (t) dx = N i=1 j=1 3 u ij ωij w dx [J], (21) The energy index can also be understood as the integral of the instantaneous power delivered to the reaction wheels P. The power P will be introduced for evaluation purposes as well. A. Tracking with Perfect Information Some simulation results are summarized here to analyze the LQR controller performance. In order to isolate the control problem, in this section we assume perfect knowledge of the state vector. As mentioned in section IV., the Finite Horizon LQR regulator can be tuned for higher accuracy or affordability, depending on the selected weighting matrices Q(t), R(t) and S(t). The ADCS performance will be analyzed for several tunings. For the sake of simplicity, we consider these weighting matrices constant in time and diagonal. The control parameters will be then their diagonal elements Q ii, R ii and S(T ) ii The selected simulation scenario is the following. From an initial attitude which yields a large APE off the initial reference attitude, the ADCS shall stabilize the Sun pointing attitude profile. Once the first target is at sight, it locks and tracks the target by maneuvering from the initial attitude reference profile. Figure 3: APE pointing error for R ii = 10 k, k = 1, 3, 5, 7, during initial stabilization maneuver (a) and Sun pointing attitude profile between two slewing maneuvers (b) Once the target is off sight, it maneuvers back to the sun pointing attitude profile. Sweeping the values of Q ii and R ii, the APE angular error as a function of time is presented in figures 3 and 4.a. As expected, the smaller R ii and/or higher Q ii is, the smaller the average APE results. Initial stabilization maneuver takes less time to yield acceptable values of APE, and slewing maneuvers are more accurate, allowing longer periods of ground target observation and Sun tracking. Please note that the second Sun pointing attitude profile does not necessarily lock the same attitude quaternion as the first Sun pointing attitude profile, since it is only necessary to suffice the condition in equation (7) 6 of 11

7 Figure 4: APE angular error (a) and angular velocity of one reaction wheel (b) during the initial stabilization phase, for several Q ii Figure 4.b shows that increasing values of Q ii lead to higher reaction wheels angular velocity, thus increasing the performance requirements of the ADCS system. Sweeping for several diagonal parameters of the final state error weighting matrix S(T ), figure 5 shows that it only affects the final instants of the simulation, while rendering no effect on the rest of the mission profile. Figure 5: APE angular error for increasing values of S(T ) qq, during the initial stabilization phase (a), a slewing maneuver (b) and final state surroundings (c). In a) and b), all the APE functions overlap. This fact could lead to neglect the S(T ) effect on the overall performance of the ADCS. Nonetheless, as mentioned in section IV., the architecture of the problem makes it necessary to store the previously calculated optimal gain K(t) in a memory buffer. Then, every certain amount of time, must be recalculated and then fed to the system. Then, small S(T ) ii parameters will generate bad initial conditions for the next control stage, thus reducing the ADCS performance periodically. However, this architecture can be seized if we consider different tuning parameters depending on the attitude profile. We would require higher precision in ground target tracking, holding cheap Sun tracking and slewing maneuvers. Figure 6 shows a real time simulation where high accuracy is tuned only for target tracking, while the rest of the simulation is tuned for control economy (blue). Figure 6: Realtime simulation for economic, accurate and mixed ADCS tunings This figure also shows a full-time affordably tuned ADCS (red), and a full-time accurate ADCS (green). Simulation results show that the mixed controller saves up to a 26% control energy, holding the same accuracy as the precise ADCS during the earth observation mission profile. 7 of 11

8 B. Limited Control Torque As previously mentioned, a tight target schedule can lead to fast slewing maneuvers, raising the control power needs. Unfortunately, reaction wheels motors cannot deliver unlimited torque, as this parameter compromises the reaction wheel dimensions, weight and power consumption. Thus, a very demanding attitude reference profile along with a too accurately tuned LQR can lead to control torques too high to be delivered by the actuators. In this section, we simulate this situation. The control torque is artificially truncated under the maximum necessary level. Figure 7: Simulation under limited torque conditions, with overshooting up to t 7000s Figure 7 shows the attitude quaternion and the APE during a slewing maneuver to a ground tracking attitude profile with limited control output. The slewing maneuver commanded by the reference profile (from 6700 to approximately 6800 s) is too demanding for the reaction wheel motors. Then, wide overshooting takes place, with a maximum APE about 40 degrees. In spite of the violation of the small errors hypothesis based upon which the tracking linearization is formulated, the LQR robustness allows the attitude quaternion to track the reference after 300 seconds of overshooting. Once the control input needs are smaller than the reaction wheels motors output torque, the controller tracks the reference as if it did not overshoot in the first place. This fact highlights the need to model an attitude reference for the slewing maneuvers, since steep changes in the reference could lead to this kind of scenario. C. MEKF In this section we briefly assess the MEKF algorithm implemented. In order to evaluate the estimation error, the Attitude Measurement Error (AME) is similarly defined as its control counterpart, AME(t) = 2 arccos δq 4 (t), (22) where δq 4 (t) is the scalar component of δq(t) = q ref (t) q est (t). We analyze the AME versus several values of the gyroscope error variances η 1 and η 2, ranging values found in real spacecraft grade inertial units. Figure 8 shows the mean AME along the simulation period as a function of (η 1, η 2 ). AME MEKF rapidly increases with η 1, while it remains almost constant with η 2. But most importantly is shown a huge increase in AME MEKF for small values of η 1 and large values of η 2. It is found that this region of the (η 1, η 2 ) plane yields bigger roundoff errors within the Runge Kutta integration method, thus making the covariance matrix P lose its assumed symmetry. This fact dramatically decreases the LAPACK number of the inverse matrix involved in the Kalman gain calculation, leading to the MEKF instability. In this work, it is proposed to force the covariance matrix symmetry in equation (17), by P = F P + P F T + GQG T = F P + P T F T + Q = F P + (F P ) T + Q. (23) in which the PLATO exoplanetary system explorer inertial unit lies within, see figure 8.b 8 of 11

9 Figure 8: AME MEKF (η 1, η 2 ), instability (a); projection over (η 1, η 2 ) plane (b) This symmetrization solution removes the MEKF instability with very little additional computational cost. D. Tracking with Imperfect Information Simulation results are shown here to demonstrate the complete ADCS algorithm detailed in section VI. In order to understand this simulation, several angular errors must be defined: Absolute measurement error AME Euler angle of the rotation quaternion between the attitude estimate and the actual attitude. It is represented as the light blue line. Fictitious angular error Euler angle of the rotation quaternion between the attitude estimate and the reference attitude quaternion. It determines how much control input needs to be delivered to the actuators. It is represented as the red line. Angular update error Euler angle of the rotation quaternion between the attitude estimate before and after the star tarcker measurement update. It is represented as the green line. Absolute pointing error APE Now, the APE measures the angular error between the actual attitude quaternion and the reference attitud quaternion. Thus, it is the ultimate performance figure for the ADCS. Represented as the blue line, it is can be considered as the qualitative addition of the fictitious angular error and the absolute measurement error. Figure 9: Relevant angular errors in the tracking with imperfect information problem, during a sun tracking attitude profile Figure 9 shows a Sun pointing attitude profile. The AME increases with time up to the next star tracker 9 of 11

10 measurement update, delivered every 5 seconds, when it suddenly decreases. The APE evolves in a similar way, according to the accuracy-tuned LQR. Each star tracker update yields a sudden increase in the fictitious error, due to the steep change of the attitude estimate. Since the reference attitude is (almost) constant for this attitude profile, the fictitious error increase can be understood as sudden increments of the absolute value of the control input. As a result, slight delays of the actual attitude with respect to the attitude estimate take place. Figure 10 shows the same simulation, during a ground tracking attitude profile (a), compared to the perfect information algorithm in the same situation (b). It clearly depicts the high frequency noise induced by the estimation technique. This perturbation is known as Relative Pointing Error (RPE) 9, also known in the field as jitter. This disturbance may have an important impact on the spacecraft optic systems, and its mitigation is left as future work. In spite of the RPE perturbation, this simulation features the noise characteristics of the HRG gyroscopes of the Figure 10: Relevant angular errors in the tracking with imperfect information problem, during a ground target tracking attitude profile SSIRU inertial unit, onboard Herschel Space Telescope, and the Cryosat TERMA HE5AS star tracker. Simulation results show an achieved average pointing error of 0.7 arc-seconds. VIII. Conclusion Advanced techniques for attitude control and estimation theories are implemented. The proposed algorithm provides an heuristic yet precise platform of analysis and dimensioning of a three axis estimation and control system. Fully customizable from orbital parameters to ground-station targets scheduling, provides accurate information on sensors and actuator performances required for a certain mission. For a given assembly of satellite and sensors/actuators, it yields quantitative and qualitative information on how it could perform on a certain observation mission, and give hints on the optical system performance required. The robustness of the algorithm is demonstrated under some adverse situations. As future work, implementation of a jitter reduction system can be considered. Also, the overall performance of the control system could be greatly improved with the implementation of energy/momentum wheels. 10 The wheel cluster configuration accomplishes both attitude actuation and kinetic energy storage, thus rendering the traditional chemical batteries unnecessary. Appendix: MATLAB -STK software integration As it has been pointed out above, the attitude quaternion is an efficient mean for attitude representation, but it makes difficult to understand the geometry underlying the formulation. Geometric understanding of the problem is utterly important by means of validation, mission planning and software debugging. This drawback is avoided with three dimensional motion representation in the MATLAB -STK simulation platform developed. The operation of the developed algorithm is sketched in figure CreaEscenario.m Sets the TCP/IP link between MATLAB and STK. Generates the STK scenario simulation parameters, and creates several auxiliary objects necessary for the simulation. 2. Orbita.m Calculates the desired orbit. The resulting keplerian elements are fed to the STK propagator J2Perturbation. This propagator takes into account the westward nodal regression typical of sun-synchronous orbits. It returns the calculated keplerian elements and the epoch. 3. Referencia.m With the keplerian elements calculated by Orbita.m, the target schedule and the slewing maneuver time, this script, calculates the reference state vector as explained above. It also animates the reference satellite representation in STK. 10 of 11

11 Figure 11: Flux diagram of the MATLAB-STK attitude simulation platform 4. Riccati.m With the reference state vector calculated by Referencia.m and the spacecraft mass parameters, it solves the differential Riccati equation (13), and returns the optimal gain matrix K(t). This script does not interact with STK. 5. E C Actitud.m This script is fed with the optimal gain matrix K(t) calculated by Riccati.m, the characteristic error variances of the sensors (gyros and star tracker), and the refresh rate of the star tracker. This routine solves the final tracking problem with imperfect information, by implementation of propagation/integration, update and reset phases. Along with several Matlab representations, this code sends both real (simulated) and estimated attitude quaternions for their representation in STK, q sim (t) and q est (t). References 1 Sidi, M. J., Spacecraft Dynamics and Control: A Practical Engineering Approach, Vol. 7 of Cambridge Aerospace Series, Cambridge University Press, New York, NY., A. H. de Ruiter, Christopher Damaren, J. R. F., Spacecraft Dynamics and Control: An Introduction, Wiley, Lefferts, E. J., Markley, F. L., and Shuster, M. D., Kalman Filtering for Spacecraft Attitude Estimation, Journal of Guidance, Control, and Dynamics, Vol. 5, No. 5, 1982, pp Wie, B., Space Vehicle Dynamics and Control,, Inc., Junkins, J. L., J. D. Turner, J., and Ho, Y.-C., Optimal Spacecraft Rotational Maneuvers, Elsevier, New York, Bryson, A. E., Control of Spacecraft and Aircraft, Princeton University Press, Princeton, NJ., Farrenkopf, R. L., Analytic Steady-State Accuracy Solutions for Two Common Spacecraft Attitude Estimators, Journal of Guidance and Control, Vol. 1, No. 4, 1978, pp Markley, F. L., Attitude Error Representations for Kalman Filtering, Journal of Guidance, Control, and Dynamics, Vol. 26, No. 2, 2003, pp ESA, Herschel/Planck System Requirements Specifications, SCI-PT-RS-05991, Vol. 3, No. 3, Tsiotras, P., Shen, H., and Hall, C., Satellite Attitude Control and Power Tracking with Energy/Momentum Wheels, Journal of Guidance, Control, and Dynamics, Vol. 24, No. 1, 2001, pp of 11