In-Orbit Magnetometer Calibration for a Spinning Satellite

Transcription

1 In-Orit Magnetometer Caliration for a Spinning Satellite Halil Ersin Söen and Shin-ichiro Saai Japan Aerospace Exploration Agenc (JAXA), Sagamihara, 5-50 Japan Magnetometers are commonl used attitude sensors for Earth oriting spinning spacecraft. Specificall for spinning small satellites the are ideal sensors as the are cheap, small and lightweight. If the magnetometers are error free, attitude of the satellite can e estimated with a sufficient accurac. Howeer an in-orit caliration algorithm is required for using magnetometers onoard the small satellites. The susstems are closel located on these satellites and magnetometer errors ar due to the electromagnetic interference. In this stud, two simple algorithms for real time estimation of magnetometer errors for spinning spacecraft are proposed. First two quasi-measurements for the ias estimation are uilt enefiting from the spin dnamics. A pseudo-linear Kalman filter (KF) with statedependent measurement model and a linear KF with linearized measurement model are designed. The filters are tested for a hpothetical nanosatellite ia numerical simulations. The results are discussed in detail and suggestions for implementation of the algorithms are gien. I. Introduction PIN-stailization is the simplest method of stailizing the attitude of a spacecraft. In the earl era of space Sexploration it was widel emploed. Toda, as the numer of micro and nanosatellites increases, interest in spinstailization reies, mainl ecause of its simplicit that maes the method suitale for the small satellites,. The spin-stailization is used for the nanosatellites either for the whole mission duration or temporaril to conduct an experiment, especiall to de-orit the spacecraft. Magnetometers are widel used onoard the small satellites as the are lightweight, inexpensie and reliale. Howeer a real-time magnetometer caliration algorithm is required for correcting their measurements. Using a realtime algorithm is specificall adantageous as the susstems are closel located on small satellites and the magnetometer errors ma ar due to the electromagnetic interference 3. Magnetometer caliration for spinning spacecraft is usuall not treated distinctl from the common caliration algorithms which are used also for three-axis stailized spacecraft. For example for the ST-5 and THEMIS missions of NASA 4, the Two-Step algorithm 5 is used. The Two-Step method is a fast and roust algorithm for estimating the magnetometer errors ut it is onl applicale as a atch method, not in real-time. A real-time attitude independent magnetometer caliration approach is deried in Ref.6 and also used for a spinning spacecraft in Ref.7. This algorithm wors well onl with an Unscented Kalman Filter (UKF), which comes with a computational cost, and has conergence issues for spinning spacecraft 7. In Ref.8 magnetometers for a spinning projectile are calirated in real-time with a sliding-window filter formulated in the EKF framewor. Despite eing used for a spinning projectile, the filter algorithm does not mae use of an information that can e extracted from the spin motion. Besides, the measurements within an N-sized window should e ept in the filter memor. This increases the computational load. A caliration method for magnetometers on a spinning spacecraft is discussed in Ref.9. It is one of the few studies that are intended solel for spinning spacecraft magnetometers and maing adantage of the spin dnamics. As a drawac this method requires on-ground processing with an excessie amount of data collected from different periods. In fact, it is proposed for correcting the readings from scientific magnetometers rather than calirating the attitude sensors. Aerospace Project Research Associate, Institute of Space and Astronautical Science (ISAS), Yoshinodai 3-- Chuo-u, Sagamihara 5-50 Japan Associate Professor, Institute of Space and Astronautical Science (ISAS), Japan Aerospace Exploration Agenc (JAXA), Yoshinodai 3-- Chuo-u, Sagamihara 5-50 Japan.

2 In this paper we propose two simple Kalman filter (KF) algorithms for real-time attitude-independent magnetometer ias estimation for spinning spacecraft. First we show that it is possile to enefit from the spin motion of the spacecraft to uild two quasi-measurements for the ias estimation. This consideral improes the conergence characteristics of the filter. The second contriution is using a pseudo-linear KF algorithm to estimate the magnetometer ias without an need for attitude nowledge. The formulation for a linear KF that uses centered ariales is also gien. The performance of the KF algorithms is inestigated using the simulated data for a nanosatellite. II. Magnetometer Measurements for a Spinning Spacecraft The measurement equation for three-axis magnetometer (TAM) at th time step is, B A H, () B, where, A is the attitude matrix for transformation from the inertial frame to the spacecraft od frame, is the ias ector and B is the Gaussian zero-mean measurement noise with coariance B as B N 0, B. Consider a spin-stailized spacecraft rotating aout od Z axis with an angular rate. The od Z axis is called as spin axis and the plane normal to this axis is referred as spin plane. The assumptions are: - There is no dnamic error (e.g. coning) for the spacecraft or these errors hae een alread estimated and corrected. - Each sensor of the TAM is aligned with the od axes. - Bias is the onl magnetometer error and there is no scaling, nonorthagonalit or sensor misalignment for the measurements. The measurements for the spin plane magnetometers hae spin-modulated dnamics. In an ideal case, where there is no ias, the measurements in the spin plane oscillate aout a zero-mean (Fig.) 9. Besides, the phase difference etween the signals measured two sensors is exactl 90. Howeer when the ias exists, the mean of oscillation for the spin plane measurements will shift depending on the magnitude of the ias in these measurement terms (Fig ). Regarding this fact, aeraging the measurements for the spin plane magnetometers within a window gies, ˆ B B, (a) x x, xi, N i N ˆ B B. (),, i N i N Here N is the window size. ˆx and ˆ are nois estimates for the corresponding ias terms. If the window size is large, the noise in the ias estimates, ˆx, ˆ can e reduced and the estimation accurac can e improed. But apparentl this is not practical for real-time applications. Instead we can eep the window size small and use the aeraged alues as measurements for a KF algorithm: x xi, x Bx N i N, i B N i N B, (3a) B. (3) Bx B, (4a) Bx N i N B N i N are Gaussian white noises as. (4) Here Bx and B Bx, N 0, Bx, / N. Thus far we hae two measurements for estimating the ias terms for the magnetometers in the spin plane; we need a third measurement to estimate the ias for the magnetometer aligned in the spin direction. As clearl seen in Fig. and Fig., there is no spin modulation for this magnetometer measurements and aeraging procedure is not

3 applicale to get ˆz estimate (or nois measurement). Instead, we use the fact that the magnitude of the magnetic field must e same in the inertial and spacecraft od frames in an ideal case. The procedure starts with eliminating the attitude dependence in Eq.. We transpose the terms and compute the square 5, ( ) B, B, H B B B. (5) Fig. Magnetometer measurements in ideal case. Fig. Magnetometer measurements when there is ias in the measurements. Dashed line is for zero leel and dotted line is for the mean of spin planar measurements. Here, Next we define the measurement as the difference of the magnetic field magnitudes in two frames, B H B. (6) B, B, ( B ), (7) N, for especiall a satellite in is the measurement noise, which ma e assumed as Gaussian noise as low-earth orit since B 5 B,. Regarding R here B, tr( R), (8a) T 4( B ) R ( B ) tr R. (8) As a result we hae three measurements to estimate all three magnetometer ias terms. Two of these equations are deried ased on the information that is inherent to the spin motion of the spacecraft. The third equation is 3

4 otained using the magnitude difference etween the reference and measured magnetic field ectors. Using these measurements a KF can e designed to estimate the ias terms in real-time. How we treat the nonlinearit in the third measurement equation determines the tpe of the filter we use for ias estimation. III. Kalman Filtering for Magnetometer Bias Estimation The sstem equation for the constant magnetometer ias ector is, 0. (9) The Eq.(6) is nonlinear in terms of the ias ector. So a linear KF algorithm cannot e used unless this equation is further processed to sae from nonlinearit. In this stud we examine two KF algorithms that differ depending on the method for treating the nonlinearit. A. Pseudo-Linear Kalman Filter for Magnetometer Bias Estimation The first option is to leae Eq.(6) as it is and uild a pseudo-linear measurement model for the KF as, x 0 0 x Bx 0 0. (0) B Bx x B Bz z z This measurement model is used together with the process model (Eq.9) in a pseudo-linear KF to estimate the magnetometer ias ector. The algorithm is an ordinar linear KF that operates on a linear model ut the measurement matrix gien in Eq.(0) is state-dependent. Note that the measurement ariance in Eq.(8) is a function of the estimated ias terms and it can e calculated using the estimates from the preious step, ˆ. B. Linear Kalman Filter for Magnetometer Bias Estimation Another possiilit for the KF measurement model is to use centering method 5,6 to linearize the Eq.(6). In this case the method leads to B () as the third measurement equation instead of Eq.(6). Then the measurement equation for the KF ecomes, x 0 0 x Bx 0 0 B. () B x, B, B z, z The centered ariales are defined as, where,, in B B B, in in (3) in. (4) In fact, a linear KF running with Eq.() as a part of the measurement model is similar in essence with appling a sequential algorithm for ias estimation using the centering approach 6. The gien algorithm differs onl as it applies centering within a moing window and adds Eq.(3) inherent to spinning spacecraft to the measurement model. Both the pseudo-linear KF and the linear KF can e initialized using a atch of data. Spin period of the spacecraft is one of the factors determining the moing window and data size. 4

5 C. Simulations The algorithms are tested for a hpothetical spinning nanosatellite whose orit is assumed to e a low Earth orit 5 with a small eccentricit of e 6.4 0, an inclination of i 74 and approximate altitude of 6m. The satellite spins aout od Z axis with a spin rate of 7.5 rpm. For the magnetometer measurements, the sensor noise is characterized zero mean Gaussian white noise with a standard deiation of B 300nT. We assume that the magnetometers are free of errors other than the ias. The ias error is assumed to e constant and true alues are gien in Tale. Simulations are run for 8000s, which is aout 3 orits for the satellite and the sampling time (oth for filter propagation and magnetometer measurements) is t s. The geomagnetic field is simulated using the International Geomagnetic Reference Field model 0. 8 The process noise coariance matrix for all ias terms is Q 0 nt. The filter initialized with ias 7 estimates set to 0nT. The initial coariance matrix is diagonal and corresponding alues are P,0 0 (nt). The window size N is selected as 8 samples which is exactl spin period. Tale summarizes the results for 00 runs executed with each KF algorithm. The results for a pseudo-linear KF that runs using onl the scalar measurement (Eq.6) are also gien. This helps understanding the adantages of including Eq.(3) in the measurement model. The mean the filter estimates at the final time are gien together with their 3 ounds. The pseudo-linear KF performs est and including the Eq.(3) in the measurement model improes the estimation accurac. Although the mean alues are close for the single measurement pseudo-linear KF and full three measurements pseudo-linear KF, 3 ounds, which are smaller for the latter one, shows that including quasimeasurements increases the estimation accurac. Tale Results for magnetometer ias estimation. Bias Term Truth Single Measurement Pseudo-Linear KF Linear KF x 5000 nt nt z 4000 nt Comparing the pseudo-linear KF and the linear KF, the pseudo-linear KF conerges to the true alues faster. Fig.3 and 4 show estimation results for a representatie run. Using the nonlinear measurement (Eq.6) instead of its linear form (Eq.) specificall quicens the conergence of ˆz estimation, which is for the spin direction magnetometer. This is due to more accurate measurement modeling with the nonlinear equation. Moreoer, the measurement noise in Eq.() is not uncorrelated although we ignore the correlation in practice 5. In contrast the measurement ariance for nonlinear equation (Eq.8) can e calculated accuratel using the estimates from the preious step, ˆ. Fig. 3 Pseudo-Linear KF estimates. 5

6 Fig. 4 Linear KF estimates. Varing the window size, N, for mean calculation and centering requires retuning for the filters. Yet the pseudolinear KF is more roust to such changes. The linear KF requires slightl more computation due to centering operation ut this is negligile. Compared to more complex algorithms for real-time ias estimation such as the UKF oth algorithm require er few computations. IV. Conclusion Two simple real-time attitude-independent magnetometer ias estimation algorithms were deeloped for spinning spacecraft. In addition to the well-nown attitude-independent scalar measurement for magnetometer caliration, two auxiliar measurements were deried maing use of the spin dnamics. Simulations showed that using these axillar measurements increases the ias estimation accurac as well as improing algorithm s conergence speed. The pseudo-linear Kalman filter, which uses nonlinear form of the scalar measurement equation, is the most roust algorithm in terms of oth oerall accurac and conergence properties. The other enefit of the algorithms is haing er few computational requirements for real-time implementation. Acnowledgments This research is supported JSPS KAKENHI Grant Numer 6K833. References Nielsen, T., Weston, C., Fish, C. and Bingham, B., Dice: Challenges of Spinning Cuesats, 37 th Annual AAS Guidance & Control Conference, AAS 4-068, Brecenridge, CO, USA, 04. Slainsis, A, Ehrpais, H., Kuuste H, Sünter, I., Viru, J., Kütt, J., Kulu, E., and Noorma, M., Flight Results of ESTCue- Attitude Determination Sstem, Journal of Aerospace Engineering., Vol. 9, No., 40504, Springmann, J. C., and Cutler, J. W., Attitude-Independent Magnetometer Caliration with Time-Varing Bias, Journal of Guidance, Control, and Dnamics, Vol. 35, No.4, 0, pp Natanson, G., Sedla, J. E., Hashmall, J. A, and Harman, R. Mission Experience Using New Attitude Tools for Spinning Spacecraft, AIAA/AAS Astrodnamics Specialist Conference, AIAA , Hawaii, 008, pp Alonso, R., and Shuster, M. D., TWOSTEP: A fast roust algorithm for attitude-independent magnetometer-ias determination, The Journal of the Astronautical Sciences, Vol. 50, No.4, 00, pp Crassidis, J. L., Lai, K.-L., and Harman, R. R., Real-Time Attitude-Independent Three-Axis Magnetometer Caliration, Journal of Guidance, Control, and Dnamics, Vol. 8, No., 005, pp Dos Santos, D.A., and Waldmann, J., Attitude and Angular Rate Estimation from Vector Measurements of Magnetometer and Sun Sensor for a Low-Cost Satellite, 0 th International Congress of Mechanical Engineering, Gramado, RS, Brazil, Grandallet, B., Zemouche, A., Boutae, M., and Change, S., Real-Time Attitude-Independent Three-Axis Magnetometer Caliration for Spinning Projectiles: A Sliding Window Approach, IEEE Transactions on Control Sstems Technolog, Vol., No., 04, pp Farrell, W. M., Thompson, R. F., Lepping, R. P., and Brnes, J. B., A method of calirating magnetometers on a spinning spacecraft, IEEE Transactions on Magnetics, Vol. 3, No., 995, pp Théault E., et al., International Geomagnetic Reference Field: the th Generation, Earth, Planets and Space, Vol. 67, No.79, 05. 6