Influence Analysis of Star Sensors Sampling Frequency on Attitude Determination Accuracy

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1 Sensors & ransducers Vol. Special Issue June pp. -8 Sensors & ransducers by IFSA Influence Analysis of Star Sensors Sampling Frequency on Attitude Determination Accuracy Yuanyuan Jiao Xiaogang Pan Jiongqi Wang Yong Li Haiyin Zhou College NineNational University of Defense echnology Changsha China School of Surveying & Spatial Information Systems he University of New South Wales Sydney Australia Department of Mathematics and Systems Science National University of Defense echnology Changsha China s: Received: 5 April /Accepted: June /Published: 8 June Abstract: he star sensor has been widely used as an important and accurate attitude measurement sensor in classical satellite attitude determination systems. his paper analyses the influence of star sensor s sampling frequency on attitude determination accuracy within an Extended Kalman filter (EKF). Simulations are used to validate the theoretical analysis and determine the parameters of the influence function. he results show that in most scenarios the influence of the star sensor s sampling frequency on the attitude determination accuracy can be expressed by the characteristic parameter in the influence function which is not affected by other accuracy influence factors. Copyright IFSA. Keywords: Star sensor Attitude determination Sampling frequency Influence analysis.. Introduction High accuracy attitude determination plays an important role in earth-orientation and satellite control []. As input of the attitude determination algorithms the accuracy of the attitude measurement is regarded as the most important influence factor in determining the satellite s attitude. Since star sensor has the highest measurement accuracy [ ] it is the most widely used sensor in satellite attitude determination systems. Over the past decade the problem of attitude determination using star sensors has been reported in the research literatures such as determining the attitude by a set of observation vectors of star sensors [-6] and different types of filtering methods using the observation vectors of star sensors together with the measurement of angular velocity [7-9] deep analysis of the star sensor s measurement model as well as the characteristics of measurement errors [ Article number P_SI_86 -]. In addition to the aspects above it has been found that the sampling frequency of the star sensor is an important factor that has significant influence on the accuracy of attitude solution. Farrenopf gave an analytical solution for the steady-state covariance of attitude angle s estimation error for a single-axis Kalman filter []. Marley and Reynolds then modified Farrenopf s results to include the effect of gyro output noise [] and derived the relationship between the star sensor s sampling interval and the standard deviation of the angular estimation. Based on such wor this paper derives the analytical representation of the influence of the star sensor s sampling frequency on the widely-used EKF-based attitude determination accuracy we defined. he quaternion-based EKF in this paper combines the measurements from three star sensors and gyros. Since star sensor s measurement accuracy is high the following attitude determination method is

2 Sensors & ransducers Vol. Special Issue June pp. -8 commonly used. hat is the angular velocity information is used to predict the attitude when the star sensor s measurement data is absent. he attitude is updated when the star sensor s measurement is available. Obviously the star sensor s sampling frequency determines the time span of prediction and updating in the process of attitude determination. herefore a high sampling frequency can help to increase the attitude determination accuracy. Nevertheless since the time of exposure and the time of internal circuit and algorithmic processing are restricted a heavier cost has to be paid for the increase of the star sensor s sampling frequency during design and manufacturing [5]. herefore analyzing the influence of star sensor s sampling frequency on the attitude determination accuracy can support the requirement analysis for increasing the star sensor s sampling frequency. he paper is organized as follows. Following the introduction the measurement model of attitude sensors and the equations of the EKF-based attitude determination system are presented. he form of the influence function of the star sensor s sampling frequency is then analyzed in Section from a theoretical point of view. In Section simulations are used to validate the theoretical analysis and determine the parameters of the influence function. Furthermore the parameters in the influence function are estimated and validated in the simulation tests under different conditions. Finally the paper is concluded in Section 5.. Equations of Attitude Determination System.. Measurement Model of Gyros and State Equation of EKF he gyros measurement model is first introduced. It is written as [9]. g bi b g b b () where g is the measured angular velocity. bi is the coordinate of the inertial angular velocity of the satellite in the body frame. b denotes the gyro drift. g and b are independent Gaussian white-noise processes with E( ( t )) E( ( t) ( t)) I ( tt) ( g b) is the Dirac-delta function. he EKF state equation can be derived from the quaternion based inematic equation and it is where ( t t) constructed by the first three independent components of the error quaternion q [67] which is defined to represent the rotation from the estimated attitude ˆq to the true attitude q. For convenience the first three independent components are denoted as q. With the state vector consisting of q and the estimation error of gyro drift b the state equation for X 6 [ q b ] is given by [96]: where F () X( ) F( ) X( ) W( ) [ ˆ ].5I ˆ ˆ bi () [ ˆ ] ˆ ˆ ˆ ˆ.5 g W (). b he corresponding discrete form of Equation () is X ( ) ( ItF( )) X( ) ( ) Φ( ) X( ) ( ) where ( ) f g () () b g f ( s) ( s) ( s) ds (5) g b() s ds (6) Q E( ( ) ( ) ) t ( t ) I t I t I t I b g b b b.. Measurement Model of Star Sensors and Measurement Equation of EKF (7) Assumed that three star sensors are mounted on the three axes of the body frame of a satellite the measurement equation of the star sensors optical axes is given as [ 6 7]: where Z( ) h( q l) V ( ) (8)

3 Sensors & ransducers Vol. Special Issue June pp. -8 l ( ) iz h Abi q l bz hql ( ) liz h Abi ( q) lbz iz bi ( ) l l h A q bz In the above equations A bi ( q) is the transfer matrix from the body frame to the inertial frame and q is the corresponding attitude quaternion. lbzj l izj ( j ) are the coordinates of the star sensors optical axes in the body frame and inertial frame respectively. V ( ) is the measurement random noise sequence its covariance matrix is R. Based on the star sensors measurement model the measurement equation for X 6 [ q b ] is written as [6 7] Y H X( ) V (9) where ˆ ˆ ˆ () Abi( q) bz () Abi( q) bz () Abi( q) bz Y Z l Z l Z l Z ( j) is the j th element of Z( ) H ˆ bi( q)[ l ] A bz H ( ˆ H Abi q)[ lbz] H ( ˆ bi q)[ lbz] A. Analysis of the Influence of Star Sensor s Sampling Frequency.. According to the formula of the EKF-based attitude determination algorithm [6] together with Equation. (9) and Equation. ()-(7) one can see that the influence of the star sensor s sampling frequency on the EKF-based attitude determination method is mainly represented by the state transformation matrix Φ and covariance matrix of process noiseq. Before analyzing the influence of star sensor s sampling frequency a criterion is defined to measure the accuracy of attitude determination. P / is the covariance matrix of estimate error. Since only the accuracy of the satellite attitude is of concern the trace of the first three dimensions of the covariance matrix P / can be used as the measure of attitude estimation accuracy. For convenience the sub-matrix constructed by the first three dimensions of matrix P is denoted as P. he trace of P is denoted as tr( P) which is the criterion of measuring the accuracy of attitude determination. According to the formulation of the EKF P( / ) P( / ) K( ) H( ) P( / ) Φ P Φ Q -( P / ) H () R H() Φ( / )( P / ) Φ ( / ) Q ) ( / ) ( / ) ( / ) ( Denote Ψ Φ / ( / ) Φ ( / ) Θ H ( ) H( ) R r I. he accuracy metric tr( P( / )) written as () can be tr( P( / )) tr( Ψ/ P( / )) tr( Q ) tr( P( / ) ΘR Ψ/ P( / )) tr( P( / ) ΘR Q ) () Combining with Equation () one can obtain Ψ / t ˆ t ˆ t t ˆ ˆ t ˆ ˆ t ˆ ˆ ˆ ˆ ˆ ˆ t t.5 t t t ˆˆ ˆˆ ˆ ˆ t t t.5t.5 () Considering that the rotating angular velocity of the satellite is usually small Ψ / ( ) can be ii approximated by a diagonal matrix. Denote the element in the ith row and the jth column of attitude matrix A as aij i j. Since the directions of the star sensors optical axes are aligned with the axes of the body frame one can obtain the observation matrix by combining them with the form of the observation matrix hus Θ hen a a H a a a a a a H a a a a a a H a a. a a H H ()

4 Sensors & ransducers Vol. Special Issue June pp. -8 tr( P( / )) p ( ) p ( ) Q ii ii ii ii i i i 8 ( p( ) ii p( ) iiii) 8 p( ) iiqii r i r i () Considering that the Kalman filter will converge to the steady-state one can assume that the covariance after processing all these measurements P ( / ) is identical with the covariance P ( / ) []. Furthermore using the average value of the first three diagonal components of matrix P as the accuracy index y Equation () can be rewritten as 8 8 y y ii Qii y ii y Qii i i r i r i (5) Substituting Equation (-7) and Equation () one obtains 8 6 ( (( ˆ ˆ ˆ ).75) t ) y ( bt gt r r r (( ˆ ˆ ˆ ).75) t ) y( bt gt) (6) he solution of this quadratic equation represents the relationship between the accuracy index y and the star sensor s sampling interval t. It is a polynomial with respect to t. Nevertheless its concrete form is too complex to use. Since the order of t in this solution is between - and we use the p equation y pt p p as the form of approximated solution i.e. the influence function model where p is. It can be derived from Equation (6) as t approaches to. Since the star sensor s sampling frequency is x / t the influence function model of the star sensor s sampling frequency and the square root of y (used as the final accuracy index) is as follows: p y px / p. Notice that when the star sensors mounting direction is changed if they are still orthogonal according to the form of observation matrix the analysis result is the same as described above. But if they are not orthogonal then the diagonal element is unchanged but the off-diagonal elements of Θ are the cosine values of the angles between every pair of optical axes. Commonly the angle is still nearly 9 degree hence compared to the diagonal elements the off-diagonal elements of Θ are very small. hen Θ can still be assumed to be a diagonal matrix. his will not affect the above result significantly.. Simulated Experiments Four sets of experiments with different conditions are used to validate the theoretical analysis. It is nown that the estimated errors of the quaternion and attitude angles have a linear relationship when the estimated error is small. Meanwhile the estimated accuracy of the attitude angles is commonly employed to measure the performance. herefore instead of the accuracy index represented by error quaternion the average of the attitude angles estimated error is used as the metric in the simulations. his replacement will not change the form of the influence function... Experiment () he standard deviation of the gyros measurement noise and constant drift noise is.5deg/ h and.deg/ h.he gyro g b constant drift is assumed to be b [ ] deg/h. () he measurement accuracy of the star sensor is assumed to be ( ). he mounting direction of the star sensors axes align with the body frame s three axes. Under the above assumed conditions the experiment is conducted by choosing experimental points in which the star sensor s sampling frequency is varied from. Hz to Hz. According to the experimental results the influence of the star sensor s sampling frequency on EKF-based attitude determination accuracy is shown in Fig.. attitude determination accuracy (arc sec) sampling frequency of star sensor (Hz) Fig.. EKF based attitude determination accuracy with different star sensor s sampling frequency. (Star sensor s measurement accuracy is arc sec.). As can be seen from Fig. the curve has a negative exponential form. his characterizes the influence of the star sensor s sampling frequency on attitude determination accuracy which is also

5 Sensors & ransducers Vol. Special Issue June pp. -8 consistent with the results of theoretical analysis i.e. this undetermined influence function can be modeled as the negative exponential function p y p x p (7) he tas is now to estimate the parameters i.e. and p. From Equation (7) the model can be transformed into the form ln y ln p pln x. hen the parameter p can be estimated as p =-.5 in this experiment by using the principle of least squares fitting. herefore the inverse.5 proportional function model y p x can be / p used as star sensor s sampling frequency influence function. Similarly the parameter p can be estimated as p =.8 using the experimental results. hus the influence function between the star sensor s sampling frequency x and the EKF-based attitude determination accuracy y under the experimental condition has been established i.e..5 y.8 / x. In order to verify the validity of this influence function the curve of the influence function derived above is plotted in Fig.. In addition the curve based on the experimental results (shown in Fig. ) is also plotted in Fig. for easy comparison. Correspondingly the fitting errors for all experimental points are listed in the first row of able. able. he influence function s fitting errors with different measurement accuracy (arc sec). Exp Star sensor arc sec Star sensor 8 arc sec Star sensor arc sec From Fig. it is evident that the curves of the influence function and experimental results almost overlap. Moreover data in the first row of able also shows that the fitting error at every experimental point limits. his means that the derived influence function model can effectively reflect the relationship between the star sensor s sampling frequency and the EKF-based attitude determination accuracy under this experimental condition. Attitude Determination Accuracy (arcsec) Experimental Results Influence Function Sampling Frequency of Star-Sensor (HZ) Fig.. Comparison between the influence function and experiment results. (Star sensor s measurement accuracy is arc sec.)... Experiment his experiment mainly investigates the influence function in relation to different star sensor s measurement accuracy. With the same experimental conditions as in Experiment experiments are conducted with the measurement accuracies of 8 ( ) and ( ). he same method as in Experiment is used to determine the parameters of the influence function. When the measurement accuracy is 8 ( ) the.5 influence function is y.897 / x. When the measurement accuracy is ( ) it becomes.5 y 7.7 / x. Similarly for different measurement errors the fitting errors between influence functions and the experimental results at different experimental points are listed in rows and of able. As seen from the able when the measurement accuracy is 8 ( ) the fitting errors at every experimental point are mostly smaller than. ( ). When the measurement error is ( ) the fitting errors are all bounded by.5 ( ). his reflects the fact that our derived influence functions are consistent with the experimental results even with different measurement errors. It also verifies the fact that the form of the influence function we analyzed can 5

6 Sensors & ransducers Vol. Special Issue June pp. -8 effectively represent the influence of star sensor s sampling frequency on the EKF-based attitude determination accuracy. Fig. illustrates that the curves of the influence functions relating the star sensor s sampling frequency and attitude determination accuracy with different measurement accuracies i.e. arc sec ( ) 8 arc sec ( ) and arc sec ( ). Attitude Determination Accuracy (arcsec) arcsec 8 arcsec arcsec Sampling Frequency of Star-Sensor (HZ) Fig.. EKF based attitude determination accuracy with different star sensor s sampling frequency and measurement accuracies. Based on the results in Fig. the forms of the influence functions are the same as Equation (7). More interestingly after repeating every experiment several times it is found that the degree of the variable x in the influence function is usually -.5 which is consistent with the theoretical analysis. According to the properties of the exponential function family this parameter determines the function s shape. In other words it is the intrinsic characteristic of the influence relating the star sensor s sampling frequency and the attitude determination accuracy. It doesn t change with respect to star sensor s measurement accuracy. hus it can be considered the characteristic parameter. he other parameter is changed for different simulation conditions. It may depend on other factors such as the measurement accuracy. It is referred to as the scale parameter. As seen from Fig. the scale parameter increases with decrease of measurement accuracy. From Experiment the conclusion can be drawn that the characteristic parameter of the influence function is -.5 and it determines the influence trend of the star sensor s sampling frequency on the EKF-based attitude determination accuracy. he scale parameter however is closely related to other factors such as the measurement accuracy... Experiment his experiment is to validate whether the influence law changes with different values of other factors such as gyros measurement accuracy and star sensor s mounting direction. First the influence of gyros measurement accuracy is investigated. All inds of gyros noise are changed to. / h and other conditions remained the same as in Experiment. hen the mounting direction of the star sensors is changed as: lbz l bz cos sin sin cos ( ) cos cos sin With changing the gyros measurement accuracy and the mounting direction of star sensor Fig. depicts curves of influence function between star sensor s sampling frequency and attitude determination accuracy. attitude determination accuracy (arc sec) sampling frequency of star sensor (Hz) changed star sensors' mounting direction arc sec in experiment changed gyros' measurement accuracy Fig.. he influence relations between star sensor s sampling frequency and attitude determination accuracy with gyro s measurement accuracy and star sensors mounting direction changed. Fig. shows that when the gyro s measurement accuracy and the star sensor s mounting direction are changed the form of the influence function relating star sensor s sampling frequency and attitude p determination accuracy is still y px. A similar method as in Experiment is used to estimate the parameters in the influence functions they are.5.5 y.885 / x and y.786 / x respectively. Furthermore the fitting errors between the influence function and the experimental data at each experimental point are calculated and the results are listed in able. As seen from able when the gyro s measurement accuracy and the mounting direction of the star sensor are changed the maximum fitting error between the influence functions of the star sensor s sampling frequency and the experimental results is about. ( ). 6

7 Sensors & ransducers Vol. Special Issue June pp. -8 able. he influence function s fitting errors with different simulation conditions (arc sec). Exp Changed gyros accuracy Changed star sensors mounting direction ogether with the previous two experiments the results indicate that simulated experiments with different conditions show the validation of influence function model and its estimated parameters. Concretely the results show that: () the influence function model constructed can effectively reflect the influence of the star sensor s sampling frequency on attitude determination accuracy for different situations. () he influence of the star sensor s sampling frequency on the attitude determination accuracy is mostly represented by the characteristic parameter p =-.5. In most scenarios this parameter does not change with other influence factors... Experiment he first three groups of experiments are focus on theoretical analysis validation. hus the simulated attitude data is created by attitude inematic equation with an initial quaternion parameter and a set of angular velocity information. In this group of experiment attitude data of on orbit satellite is derived from SK software. hen this on orbit data is used to verify the previous conclusions deeply. he concrete parameters in SK are listed as follows: for a GEO satellite the semi major axis is m eccentricity is degree inclination is degree argument of perigee is degree longitude of ascending node is 6 degree true anomaly is degree. he propagator used is J perturbation. Reference attitude is used nadir alignment with ECI velocity. he initial quaternion is he initial angular velocity is.876 deg/sec deg/sec deg/sec. he attitude data is derived with.5 Hz Hz Hz Hz 8 Hz and Hz. he attitude estimation results are plotted by real line in Fig. 5. he model y p x p p =.5 is still used as the influence function then p =.799 can be obtained as in previous experiments. For comparison the results computed by the influence function directly are also drawn in Fig. 5. he corresponding fitting error is listed in able. hey are basically within % of attitude estimated error. attitude determination accuracy (arc sec) Experimental Results Influence Function 6 8 sampling frequency of star sensor (Hz) Fig. 5. Comparison between the influence function and experiment results (SK). hese results show that the form of influence function which describes the relationship between star sensors s sampling frequency and attitude determination accuracy is reasonable. And the characteristic parameter in the influence function p =-.5 is valid even with the on orbit simulated data. able. he influence function s fitting errors with SK (arc sec). Exp. 5 6 SK Conclusions his paper analyzed the influence of the star sensor s sampling frequency on the EKF-based attitude determination accuracy. heoretical analysis was used to construct the form of the influence function and the simulation experiments were employed to determine the parameters of this 7

8 Sensors & ransducers Vol. Special Issue June pp. -8 function. he influence functions relating the star sensor s sampling frequency and attitude determination accuracy under different conditions were determined. he influence function was verified by the precise fitting between the influence function and the experimental data in simulation experiments under different conditions. he results show that the influence on attitude determination accuracy of the star sensor s sampling frequency is mostly represented by the characteristic parameter in the influence function i.e. p =-.5. he scale parameter p in the influence function changes with other influence factors while the characteristic parameter p is a constant. Acnowledgements his wor is supported by NSFC China (No and 6). he authors would lie to than reviewers and Professor Chris Rizos in University of New South Wales for their constructive suggestions. References []. M. E. Pittelau Kalman Filtering for Spacecraft System Alignment Calibration Journal of Guidance Control and Dynamics Vol. No. 6 pp []. S. u Satellite Attitude Dynamic and Control Chinese Astronautics Press Beijing 5. []. L. Blarre J. Ouanine L. Oddos-Marcel et al. High accuracy Sodern Star racers: Recent improvements proposed on SED6 and HYDRA star racers in Proceedings of the AIAA Guidance Navigation and Control Conference Keystone CO United States 66 6 pp. -8. []. M. D. Shuster S. D. Oh hree-axis Attitude Determination from Vector Observations Journal of Guidance Control and Dynamics Vol. No. 98 pp [5]. I. Y. Bar-Itzhac and Y. Oshman Attitude Determination from Vector Observations: Quaternion Estimation IEEE ransactions on Aerospace and Electronic Systems Vol. AES- No. 985 pp [6]. Y. Cheng J. L. Crassidis and F. L. Marley Attitude Estimation for Large Field-of-View Sensors AAS Paper [7]. M. L. Psiai Attitude-Determination Filtering via Extended Quternion Estimation Journal of Guidance Control and Dynamics Vol. No. pp. 6-. [8]. I. Kim J. Kim and Y. Kim Angular Rate Estimator Using Disturbance Accommodation echnique AIAA Paper 89. [9]. J. L. Crassidis F. L. Marley and Y. Cheng Survey of Nonlinear Attitude Estimation Methods Journal of Guidance Control and Dynamics Vol. No. 7 pp. -8. []. M. D. Shuster Kalman Filtering of Spacecraft Attitude and the QUES Model Journal of the Astronautical Sciences Vol. 8 No. 99 pp []. Y. Liu Y. Chen Star-sensor measurement model and its application to the spacecraft attitude determination system Journal of Astronautics Vol. No. pp []. Y. Jiao H. Zhou X. Li J. Wang X. Pan Cone Measurement Error Model of Star-sensor s Optic Axis and Its Application Journal of Astronautics Vol. No. 9 pp. 8-. []. R. L. Farrenopf Analytic Steady-State Accuracy Solutions for wo Common Spacecraft Attitude Estimators Journal of Guidance and Control Vol. No 978 pp []. F. L. Marley and R. G. Reynolds Analytic Steady-State Accuracy of a Spacecraft Attitude Estimator Journal of Guidance Control and Dynamics Vol. No. 6 pp [5]. G. Ju and J. L. Junins Overview of Star racer echnology and its rends in Research and Development in Proceedings of the Advances in the Astronautical Sciences he John L. Junins Astrodynamics Symposium Vol. 5 AAS--85 pp [6]. L. Li Research on the Satellite Autonomous Navigation and Attitude Determination echnology Postdoctoral Dissertation of Space Science and Application Research Center Chinese Academy of Science. [7]. E. J. Lefferts F. L. Marley and M. D. Shuster Kalman Filtering for Spacecraft Attitude Estimation Journal of Guidance Control and Dynamics Vol. 5 No pp Copyright International Frequency Sensor Association (IFSA). All rights reserved. ( 8