A ROBUST ATTITUDE MEASURING SYSTEM FOR AGILE SATELLITES. J. Treurnicht W.H. Steyn. University of Stellenbosch, Stellenbosch, South Africa

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1 A ROBUST ATTITUDE MEASURING SYSTEM FOR AGILE SATELLITES J. Treurnicht W.H. Steyn University of Stellenbosch, Stellenbosch, South Africa Abstract: A novel attitue measuring sensor is presente to be use on agile satellites without any angular fiel of view restrictions. The sensor is base on low-cost, meium resolution attitue vector measurements, obtaine from geomagnetic etectors (magnetometers) an coarse sun sensors (solar cells). The vector measurements chosen, have an unrestricte fiel of view an a reasonably high response banwith as require uring agile satellite manoeuvres. Lowcost soli state angular rate sensors with moerate to high rift bias are use to propagate the attitue between vector measurements. All sensor outputs are valiate by a pre-filtering process to eliminate spurious or faulty sensors, before feeing the ata into an extene Kalman filter (EKF). The EKF is use to estimate the attitue quaternion an angular rate sensor (gyroscope) bias vectors. Practical results from a rotating platform is presente to verify the theoretical an simulation performance. This new sensor has irect application in satellite attitue estimation owing to it s robust an goo performance. Keywors: Micro Satellites; Attitue Estimation; Extene Kalman Filter (EKF). 1. INTRODUCTION The orientation of micro satellites may be robustly etermine using two non-coincient vector measurements. A combination of a geomagnetic etector (3-axis magnetometer) an a coarse sun sensor array (solar cells) are a particularly suite pair for a polar or sun synchronous orbit. The combination allows low-cost continuous unrestricte satellite attitue measurements uring the important ay part of the orbit with reasonable accuracy of better than 1. The ynamic performance of an agile satellite may be enhance using gyroscopes. Unfortunately, these gyroscopes are expensive an power hungry, which limite their use to ate on smaller satellites. The availability of MEMS gyroscopes at very low cost (about USD3) makes this option now feasible. The performance an robustness can be extene by using multiples of these gyroscopes per axis. A novel robust attitue Extene Kalman Filter (EKF) is constructe using an innovation mixing mechanism. This scheme allows the flexible use of multiple sensor vectors without a substantial increase in the complexity of the gain calculation of the EKF.. SENSORS The main attitue sensors consist of: Coarse Sun Sensor (CSS), consisting of 6 solar cells, one per satellite facet see Figure 1. Vector measurements are generate by calculating the current ifference from opposite sies. A unit vector is generate from the 3-axis measurement an a confience flag c s

2 is generate base on the measurement size. The current from the solar cell is moulate by the cosine of the incience angle from normal. Deviations from this ieal response occur when the incience angle on the panel is small. Correction for the earth Albeo effect, using a earth moel, is escribe in [6]. Let the measurement from the gyro u be relate to the actual rate ω, bias rate b an measurement noise η 1 u = ω + b + η 1 Our best estimate of the rate is then ˆω = u ˆb The bias rate equation is escribe by a ranom walk process ḃ = η Fig. 1. Coarse Sun Sensor Configuration A 3-axis magnetometer (Honeywell HMS3 or similar). A unit vector is generate from the 3-axis measurement an a confience flag c m is generate base on the status (ON, vali communications etc.) an measurement size. (Multiple) low-cost MEMS gyros (Analog Devices ADXRS15 or similar), one or more per axis. Depening on the complexity of the temperature an offset compensation, the gyro rift performance can approach 7 /hr. The formation of the measurement vectors are illustrate in Figure. Compose the 6-element state vector as consisting of the vector part of the error quaternion δq 1... δq 3 an the gyro rift-rate bias vector δb x... δb z δq 1 (t) δq (t) x(t) = δq 3 (t) δb x (t) δb y (t) δb z (t) The true attitue quaternion is given by (note the rotate sequence) q = ˆq δq where enotes a quaternion multiplication see [], section IV. The state equation for the system is given by x(t) = f(x(t), t) + g(x(t), t)w(t) t with w(t) = system noise with covariance matrix Q(t). The state error vector an covariance matrix are efine by x(t) = x(t) ˆx(t) P (t) = E[ x(t) x T (t)] Fig.. Forming the Sensor Measurement Vectors The sun vector can be upate at the esire rate, but the magnetometer measurements will typically be interleave between magneto-torquer pulses (time-slice), so a practical magnetometer measurement upate perio can be 1 sec. 3. EKF FORMULATION Analysis follows that of the Multiplying EKF (MEKF) from [], [3] an [5]. so the state error vector satisfies the ifferential equation with x(t) = F (t) x(t) + G(t)w(t) t F (t) = x G(t) = g(ˆx(t), t) f(x(t), t) ˆx(t) The preicte covariance matrix satisfies the ifferential equation t P (t) = F (t)p (t) + P (t)f T (t) + G(t)Q(t)G T (t) For the sixth-orer boy reference moel, we get

3 [ω(t) ] 1 F (t) = I G(t) = I I 3 3 We can integrate F (t) numerically to get the iscrete state space matrix Φ(k) Φ(k) = I F (k)δt +... For a small sample perio (δt 1), we can approximate 1 ω z δt ω y δt.5δt ω z δt 1 ω x δt.5δt Φ(k) ω y δt ω x δt 1.5δt Innovation The innovation is etermine from the crossprouct also use by [4] an [5]. Let the innovation i m (k) be given by the crossprouct of the measurement v m (k) an the reference vector ˆv b (k) in boy coorinates i m (k) = v m (k) ˆv b (k) = v m (k) A(ˆq)ˆv r (k) 3.3 State Upate The error quaternion is calculate from the innovation vector i m an quaternion gain submatrix K q δ q (k) = K q (k) i m (k) an then the error quaternion is normalise δq 4 (k) = [ ] δ q (k) 1 δ q (k), δq(k) = δq 4 (k) The bias vector is calculate from the innovation vector i m an bias gain submatrix K b δb(k) = K b (k) i m (k) The correcte [ state is now forme from ˆq + ] ˆx + (k) (k) = ˆb+ with (k) ˆq + (k) = δq(k) ˆq(k) an ˆb+ (k) = ˆb(k) + δb(k) [ ˆq ] The preicte state ˆx (k+1) = (k+1) ˆb is forme using the quaternion integration from [1] [ ˆq (k + 1) = I cos ˆωδt ] ˆωδt + sin ˆq Ωˆω + (k), with ˆω(k) = ˆω x + ˆω y + ˆω z, an the bias states are simply propagate as ˆb (k + 1) = ˆb + (k) After some manipulation, the measurement sensitivity H(k) becomes H(k) = [ H a (ˆv b ) 3 3 ] with ˆv bz + ˆv by ˆv bxˆv by ˆv bxˆv bz H a (ˆv b ) = ˆv byˆv bx ˆv bx + ˆv bz ˆv byˆv bz ˆv bzˆv bx ˆv bzˆv by ˆv by + ˆv bx 3.4 Block iagram A block iagram combining the above mentione equations is shown in Figure Gain an Covariance Calculation The covariance upate P + (k) an gain K(k) are calculate by K(k) =P (k)h T (k) [ H(k)P (k)h T (k) + R(k) ] 1 P + (k) =(I K(k)H(k))P (k) with Q(k) = T G(t)Q(t)GT (t) t. The covariance preictor P (k + 1) for the EKF is given by P (k + 1) = Φ(k)P + (k)φ T (k) + Q Fig. 3. MEKF Block Diagram

4 4. SENSOR MIXING Consier the case in Figure 4 where n sensor measurements (unit vectors) are referre to the boy, an n innovation sequences i... i n 1 are create. Fig. 4. Innovation base sensor mixing scheme A number of strategies might be consiere for combining the iniviual innovation sequences into one combine innovation sequence. A weighte scheme was selecte i(k) = w i (k)i i (k), w i (k) = 1 i=1 i=1 The number of vectors in this application are insufficient for implementing a complete fault tolerant scheme. However, it still allows isturbance rejection through the choices of weights. The sensor confience inicators which checks the total CSS sensor current an magnetic fiel against a esign winow, can be use to temporary isable or reuce the effect of a faulty measurement vector. In the normal case the weights can be base on the known sensor noise covariances R i. If the two vector measurements are inepenent, then the weights may then calculate from w i = R i j=1 1 Rj, R 1 = j=1 R 1 j This strategy is simple to implement,an it can accommoate known changes in sensor noise. For the two-sensor case, we have w 1 = R 1 [ R1 + R ] 1, w = R [ R1 + R ] 1, R =R 1 R [R 1 + R ] 1 The performance may be further enhance by moulating the sensor noise covariances at known positions. For example, the geomagnetic measurements close to the poles an the sun sensor measurements at low incience angles on the solar cells will lea to increase errors for affecte vector measurement. 5. PRACTICAL RESULTS The EKF escribe was implemente on a emonstrator consisting of rotating base limite in rotation aroun the Z-axis only. The harware on this base consiste of yaw measurement ADXL15 MEMS gyros, 4 solar panels an a HMR3 magnetometer. The measurements were integrate using a Cygnal USB microcontroller at a sample perio of.1 sec. The attitue EKF was run on a PC in real-time using a C- program. The reference IGRF an sun vectors are upate using the measurement position latituelongitue coorinates an local time. No Albeo compensation was performe for these experiments. Compensation for the MEMS gyros were limite to first-orer compensation for temperature an offset. 5.1 Measure Performance The gyro bias estimation performance of the attitue control system, for an arbitary platform attitue rate profile is shown in Figure 5 for the combine sensor innovation strategy. Yaw angle in eg Bias in mra/s Estimate Bias Rate Measure Rate Fig. 5. Typical bias rate estimation Note that the bias estimate inclues effects such as rate calibration errors etc. (the rate scale factor error changes the sign of the bias error in Figure 5). The typical yaw angle estimation of the attitue control system, with the base rotating at a constant spee about -3 mra/sec an with the combine sensor innovation strategy, is shown in Figure CONCLUSION The configuration use in this paper uses the MEKF concepts efine in [] an [3] with the

5 [6] S. Theil, P. Appel an A Schleicher (3) Low cost, Goo Accuracy - Attitue etermination using Magnetometer an simple Sun sensor, SSC3-XI-7, 17th Annual AIAA/USU Conference on Small Satellites. Angle error in eg Fig. 6. Typical yaw angle estimation error of the estimate yaw angle relative to the integral of the line-of-sight rate, using both vector measurements an with the platform moving at a constant rate. cross-prouct innovation in [4] an [5]. The two sensor vector measurements from the coarse sun sensors an magnetometer similar to [6] is use. A low-cost gyro is ae to exten the tracking performance for an agile satellite, even when the magnetometer ata rate is low. Although these gyros are not suitable for the preiction of attitue for longer than a few minutes, their low power an high reliability augments the other sensors to provie a smooth an robust attitue estimate. A novel metho of combining the innovation from the sensors is presente that allows an arbitrary number of sensor measurements to be use simultaneously without a substantial increase in processing. This enables practical on-line EKF gain calculation instea of less accurate fixe-gain banke filters. REFERENCES [1] J.R. Wertz (1978), Spacecraft Attitue Determination an Control, Kluwer. [] E.J. Lefferts, F.L. Markley an M.D. Shuster (198) Kalman Filtering for Spacecraft Attitue Estimation, J Guiance, No. 5, pages [3] F.L. Markley (3) Attitue Error Representations for Kalman Filtering, J Guiance, Control an Dynamics, Vol 6, No., pages [4] M.L. Psiaki, F. Martel an P.M. Pal (199) Three-Axis Attitue Determination via Kalman Filtering of Magnetometer Data, J Guiance, Vol 13, No. 3, pages [5] W.H. Steyn (1994), Full Satellite State Determination from Vector Observations, WH Steyn, 13th IFAC Symposium, Automatic Control in Aerospace, pages