Accuracy performance of star trackers - a tutorial

Accuracy Performance of Star Trackers–A Tutorial CARL CHRISTIAN LIEBE Jet Propulsion Laboratory California Institute of Technology An autonomous star tracker is an avionics instrument used to provide the absolute 3-axis attitude of a spacecraft utilizing star observations. It consists of an electronic camera and associated processing electronics. The processor has the capability to perform star identification utilizing an internal star catalog stored in firmware and to calculate the attitude quaternion autonomously. Relevant parameters and characteristics of an autonomous star tracker are discussed in detail. Manuscript received March 5, 2001; revised January 7, 2002; released for publication January 28, 2002. IEEE Log No. T-AES/38/2/11444. Refereeing of this contribution was handled by J. T. Burnett. The research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology and was sponsored by the National Aeronautics and Space Administration. References herein to any specific commercial product, process or service by trademark, manufacturer, or otherwise, does not constitute or imply its endorsement by the United States Government or the Jet Propulsion Laboratory, California Institute of Technology. Author’s address: Jet Propulsion Laboratory, California Institute of Technology, MS 198-138, 4800 Oak Grove Dr., Pasadena, CA 91109-8099, E-mail: (carl.c.liebe@jpl.nasa.gov). 0018-9251/02/$17.00 c° 2002 IEEE I. INTRODUCTION Star trackers have recently undergone a generation shift. First generation star trackers were characterized by acquiring a few bright stars and outputting the focal plane coordinates of these stars to the spacecraft computer. The coordinates were not related to inertial space and therefore no attitude information was provided directly. The star identification and the attitude determination had to be done externally to the star tracker. In the last decade, the availability of space qualified powerful microcomputers has allowed the star observations to be compared against a stored firmware star catalog. Thus the rotation from an inertial based coordinate system to a star tracker based coordinate system can be calculated directly [1]. Autonomous star trackers are rapidly becoming the preferred attitude determination instruments onboard most spacecraft. This is primarily due to simple integration of the self-contained autonomous unit and low cost compared with first generation star trackers or other instruments for attitude determination. The market volume over the next 10 years could be several hundred units. This has attracted many vendors to the market. An open literature/web search shows more than 15 star tracker vendors offering star trackers [2—16] at rapidly decreasing costs. Basically, a star tracker is an electronic camera1 connected to a microcomputer. Using a sensed image of the sky, stars can be located and identified. The orientation of the spacecraft can be determined based on these observations. An autonomous star tracker automatically performs pattern recognition of the star patterns in the field of view (FOV) and calculates the attitude with respect to the celestial sphere. Fig. 1 shows a sketch of a modern star tracker [23]. A typical (year 2001) autonomous star tracker has a mass of 1—7 kg and consumes 5—15 W of power. The accuracy is in the arcsecond range and update rates are typically 0.5—10 Hz [24]. It is anticipated that the ongoing technology development will decrease the power consumption and mass of autonomous star trackers significantly. However, since the number of photons from the stars is finite, no major improvements in the update rate or the accuracy is anticipated in the future. The operation of a star tracker is described in this work. The performance of the star tracker depends on the sensitivity to starlight, FOV, the accuracy of the star centroiding, the star detection threshold, the number of stars in the FOV, the internal star catalog, 1Typically a Charge Coupled Device (CCD) focal plane array is used. However, Charge Injected Devices (CID) [17, 18] and Active Pixel Sensors (APS) [19, 20] have also been utilized. APS has the advantage of low power consumption and logic integrated on the focal plane itself (e.g. centroiding and A/D conversion [21, 22]). CIDs are inherently very radiation tolerant. IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 38, NO. 2 APRIL 2002 587 Fig. 1. Sketch of an autonomous star tracker. and the calibration. These topics are discussed in detail. The accuracy of the star tracker is typically divided into different components such as line of sight stability, relative accuracy, calibration errors, noise equivalent angle (NEA), S-curve errors and algorithmic errors. These categories and measurements of the NEA and the relative accuracy utilizing a telescope drive are discussed. II. STAR TRACKER OPERATION An autonomous star tracker typically operates in two modes: 1) initial attitude acquisition and 2) tracking mode. The difference between the modes is whether approximate attitude knowledge is available. In initial attitude acquisition, the task is to perform pattern recognition of the star pattern in the FOV. Many algorithms have been developed to perform this function [25—31]. Typically, the identification can be accomplished in a few seconds. Normal operating mode (tracking mode) assumes that the present attitude is close to the last attitude update (» less than 1 s ago). The task is much easier since the star tracker only has to track previously identified stars at known positions. A typical star tracker image is shown in Fig. 2.2 It is observed that the image is slightly defocused. This is to overcome the sampling theorem, which otherwise limits the accuracy to 0.38 pixel.3 Different algorithms for calculating star centroids exist [23, 32, 33]. The calculated star centroids for this image is shown in Fig. 3. Centroiding is discussed further in Section IV. Attitude and rate information from previous exposures is used in the tracking mode to predict star positions in the next exposure. Therefore, it is not necessary to digitize the entire image. Some star trackers utilize special hardware that only digitizes specific small windows in the image at predicted star positions. An example is shown in Fig. 4. This technique reduces the processor load significantly. The positions of the star centroids on the focal plane can be transformed into unit vectors in a star 2The image has been acquired by a JPL developmental APS based star tracker [19]. 3 R 0:5 ¡0:5 R 0:5 ¡0:5 p x2 + y2dxdy = 0:38: Fig. 2. Star tracker image. Fig. 3. The brightest centroids. Fig. 4. The star image from Fig. 2. This Image is only digitized around predicted star positions. 588 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 38, NO. 2 APRIL 2002 tracker based coordinate system utilizing a pinhole model. A star tracker pinhole model is sketched in Fig. 5. A straight line is extended through the equivalent pinhole and the star image on the focal plane. The distance from the equivalent pinhole to the focal plane is the focal length of the lens system. The boresight of the star tracker can be defined as a line passing the geometric center of the focal plane and the equivalent pinhole of the lens system. In order for the star tracker attitude to be useful, the star tracker internal coordinate system must be known relative to an external reference. The vendor will supply a rotation from the internal coordinate system to an external alignment cube4 or similar alignment surfaces. This rotation is determined using real sky observations and simultaneous measurements of the alignment cube or a test setup with artificial star stimulation and simultaneous readings on the alignment cube. In a simple pinhole model, the equations for transforming the centroid coordinates on the focal plane into unit vectors in a star tracker based coordinate system are 0B@ ij k 1CA= 0BBBBBBBBBBBBB@ cos(atan2(x¡ x0,y¡ y0)) ¢ cos 0@¼ 2 ¡ atan 0@sµx¡ x0 F ¶2 + µ y¡ y0 F ¶21A1A sin(atan2(x¡ x0,y¡ y0)) ¢ cos 0@¼ 2 ¡ atan 0@sµx¡ x0 F ¶2 + µ y¡ y0 F ¶21A1A sin 0@¼ 2 ¡ atan 0@sµx¡ x0 F ¶2 + µ y¡ y0 F ¶21A1A 1CCCCCCCCCCCCCA (1) where (x,y) is the focal plane coordinate, (x0,y0) is the intersection of the focal plane and the optical axis, F is the focal length of the optical system, and atan2 is four quadrant inverse tangent. The unit vectors of the stars in an inertial based coordinate system (e.g. J2000) are also known from the firmware star catalog. The average rotation (quaternion or direction cosine matrix) from the inertial based coordinate system to the star tracker based coordinate system can therefore be calculated. Typically the Quest algorithm is used for this [34]. III. STAR LIGHT SENSITIVITY, DETECTION THRESHOLD, AVERAGE NUMBER OF STARS IN THE FOV AND SKY COVERAGE The average number of stars in the FOV and the brightness of the stars are very important to the accuracy and the percentage of the sky where the star 4An alignment cube is a small, high precision cube with reflecting surfaces mounted on the star tracker. Using an autocollimator it is possible to establish a coordinate system referring to this cube and the star tracker. Fig. 5. Sketch of a pinhole star tracker. The dashed line is the boresight of the star tracker. The left sketch shows the star tracker. The center sketch shows x-ray image of the star tracker. In this sketch, focal plane is visible. In the right sketch, the lens system is replaced with equivalent pinhole. tracker operates. Therefore, this section calculates star light sensitivity and shows how to determine the star detection threshold. Also, equations to calculate the average number of stars in the FOV are shown. Finally, simulations of the percentage of the sky where the star tracker will operate are performed. Many bright stars in the sky have surface temperatures close to that of the Sun. Therefore, sensitivity to stars of spectral class G2 are calculated in this section. The calculation can easily be done for stellar surface temperatures other than 5800K. The Sun has an apparent magnitude MV =¡26:7 and the solar flux is 1:3 KW/m2. Therefore the Sun is 2:526:7 = 4:2 ¢ 1010 times brighter than a magnitude 0 star. The incident energy from an MV = 0 star on an area of 1 mm2 has the same relative spectral characteristic as a black body radiator and the total power is (1:3 KW/m2£ 10¡6 m2)=4:2 ¢1010 = 2:96 ¢ 10¡14 W. The radiation from a black body at a given wavelength and temperature is given by [35] I(¸,T) = 2 ¢¼ ¢ h ¢ c2 ¸5 ¢ (eh¢c=¸¢kB ¢T¡1) (2) where h= 6:626 ¢ 10¡34 J ¢ s, c is the speed of light= 2:997 ¢108 m/s, and kB is Boltzmans constant= 1:38 ¢10¡23 J/K. ¸ is the wavelength in m and T is the temperature (in Kelvin). LIEBE: ACCURACY PERFORMANCE OF STAR TRACKERS–A TUTORIAL 589 Fig. 6. The power influx from a 5800K, MV = 0 star on an area of 1 mm2. Fig. 7. The power influx from a star on the focal plane. Fig. 8. The photon influx from a star on the focal plane. The absolute spectral characteristics of the influx from an MV = 0 star on an area on 1 mm 2 is found by multiplying a constant to (2) so the total power is 2:96 ¢ 10¡14 W/mm2. The spectral influx is shown in Fig. 6. Typically, star tracker optics are coated to minimize reflections, chromatic distortion, and restrict the wavelengths that reach the focal plane because red light diffuse inside the focal plane [36]. In this calculation, it is assumed that the optical system transmits wavelengths in the 400—800 nm band. The resulting power influx on the focal plane is shown in Fig. 7. The energy of a photon is [35] E = hc ¸ (3) where E is the energy in Joules, ¸ is the wavelength in m, h= 6:626 ¢ 10¡34 J ¢ s, and c is the speed of light = 2:997 ¢ 108 m/s. It is possible to express the power influx in terms of photons/s by dividing the power influx by the photon energy. The result is shown in Fig. 8. The fraction of photons that are converted into photoelectrons on the focal plane is called the absolute quantum efficiency (QE) [36]. In Fig. 9 is shown the QE for a typical focal plane array (Kodak KAF-401) [37]. The QE is multiplied with the photon influx. The result is shown in Fig. 10. Fig. 9. The QE for a typical silicon focal plane array. Fig. 10. The detected photons from a star. The result of summing over the wavelengths in Fig. 10 is 19100 photoelectrons. This means that a star tracker that has an exposure time of 1 s and a lens aperture area on 1 mm2 will generate 19100 photoelectrons for a magnitude 0 star of spectral class G2. The number is similar to experimental data [19], [38]. As an example, a star tracker has a 3 cm lens aperture and a 200 ms exposure time. The star tracker images an MV = 4 star. This star tracker generates 19100 photoelectrons s ¢mm2 ¢ 1 2:54¡0 ¢0:2 s exposure ¢¼ ¢ 15 mm2 = 69235photoelectrons exposure : (4) The star detection threshold is also important and is therefore discussed. The principal contributors to the background signal noise are typically: read noise, inhomogeneity of the dark current, and the dark current noise itself. It is possible to estimate the background noise as the standard deviation of all pixel values in a dark frame. A focal plane consists of up to 106 pixels and it is therefore reasonable to set the detection threshold for a star signal to be the average background pixel value +5 times the standard deviation to avoid false positives. A star is detected if the brightest pixel in the star is above the threshold.5 The brightest pixel of a star depends on the point spread function (PSF) and the position of the star. If the star image has a Gaussian PSF radius of 0.5 pixels 5It is assumed that stars are detected by sifting the image for pixels above the threshold value. It is possible to improve the detection threshold by sifting the image with a 2 by 2 pixel window or to subtract an “average” background image. However, these approaches require additional processor and memory resources. 590 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 38, NO. 2 APRIL 2002 and it is centered on a pixel, 29.4%6 of the signal will be contained in the brightest pixel. However, if the radius of the Gaussian PSF is 1 pixel and the star is centered on the boundary between 4 pixels then the brightest pixel will only contain 12.7%7 of the signal. The detection threshold for stars is therefore: detection limit = Apixel + 5 ¢¾pixel ¢ 1R 1 0 R 1 0 1 2 ¢¼ ¢¾PSF e¡x2+y2=2¢¾PSFdxdy (5) where Apixel is the mean value of the pixels, ¾pixel is the standard deviation of the pixel values in a dark frame, and ¾PSF is the PSF radius in pixels (assuming a Gaussian PSF). The average number of stars in the FOV is also essential to the accuracy and sky coverage of a star tracker. It is assumed that the FOV is circular and is A deg wide. The fraction of the sky that is covered by the FOV is then: 1¡ cos µ A 2 ¶ 2 : (6) The number of stars brighter than a given magnitude can be estimated by plotting the number of stars brighter than a given magnitude versus magnitude. This was done for the PPM star catalog in an Excel spreadsheet. The relationship was found to be N = 6:57 ¢ e1:08¢M (7) where N is the total number of stars on the sky and M is the visual magnitude (MV). The average number of stars in the FOV is then: NFOV = 6:57 ¢ e1:08¢M ¢ 1¡ cos µ A 2 ¶ 2 : (8) As an example, a star tracker has a detection threshold of MV = 5. The star tracker FOV is 15 ±. On average, the star tracker will detect 6.2 stars in the FOV. Sky coverage is defined as the percentage of the sky where the star tracker will operate. Ideally a star tracker should be able to operate at all attitudes, but two stars must, as a minimum be present in the FOV to calculate the attitude solution. The stars are not distributed homogeneously on the sky–there are many stars in the galactic plane and relatively fewer at the galactic poles. Therefore, there are areas on the sky, where there are not enough stars for some star trackers to operate. Fig. 11 show a simulation of the percentage of sky where there are more than one star in the FOV as a function of the FOV size for different detection 6 R 0:5 ¡0:5 R 0:5 ¡0:5(1=2 ¢¼ ¢ 0:5)e1x 2+y2=2¡0:5dxdy: 7 R 1 0 R 1 0 (1=2 ¢¼ ¢ 1)e1x2+y2=2¡1dxdy: Fig. 11. The theoretical sky coverage as a function of FOV and detection threshold. Fig. 12. ROI and the border of ROI of detected star. thresholds. As an example, a star tracker with a FOV on 10± and a detection threshold on MV = 4 will only have more than 1 star in the FOV in 87% of the time. The theoretical sky coverage is therefore 87%. It should be noted that this is the theoretical limit. Normally, a star tracker requires more than 2 stars in the FOV to operate, depending on the implementation. The initial attitude acquisition will typically require much more stars. Most star trackers are designed to have sky coverage close to 100%. IV. STAR CENTROIDING Star trackers utilize subpixel centroiding to increase the accuracy. In a focused image a star appears as a point source, so all the photoelectrons from a star are generated in a single pixel. However, if the optics is slightly defocused a star will occupy several pixels. This facilitates calculating the center of the star with subpixel accuracy. This section discusses an algorithm for calculating centroids and making centroiding accuracy simulations. Initially, the image is sifted for pixels that are above a given threshold. Once a pixel is detected, a region of interest (ROI) window is aligned with the detected pixel in the center. The average pixel value on the border is calculated and subtracted from all pixels in the ROI as shown in Fig. 12. The centroid (xcm,ycm) and DN (brightness) are calculated from the background-subtracted pixels in the ROI. DN = ROIend¡1X x=ROIstart+1 ROIend¡1X y=ROIstart+1 image(x,y) (9) LIEBE: ACCURACY PERFORMANCE OF STAR TRACKERS–A TUTORIAL 591 Fig. 13. The simulation of the centroiding accuracy as function of the number of photoelectrons. Fig. 14. Simulation of the centroiding accuracy as a function of the size of the PSF. xcm = ROIend¡1X x=ROIstart+1 ROIend¡1X y=ROIstart+1 x ¢ image(x,y) DN (10) ycm = ROIend¡1X x=ROIstart+1 ROIend¡1X y=ROIstart+1 y ¢ image(x,y) DN : (11) The centroiding accuracy is proportional to the square root of the number of photoelectrons generated by the star. In Fig. 13 is shown the centroiding accuracy as a function of the number of detected photoelectrons for various sizes of the PSF radius. In the simulation, the only noise source is the photon noise. A Gaussian PSF, 100% QE, and a 9£ 9 pixel ROI have been assumed. In Fig. 14 the centroiding accuracy is shown as a function of the size of the PSF radius for stars with different number of photoelectrons. In the simulation only photon noise is included, a QE on 100%, a Gaussian PSF, and an ROI on 9£ 9 pixels are assumed. It is observed that the centroiding accuracy does not have a strong dependence on the size of the PSF radius as long as it has a radius of more than 0.5 pixels. The reason for this is that the accuracy is limited by the sampling theorem when the PSF radius is less than 0.5 pixels. In real implementations it is also important to have a small ROI because of pixel inhomogeneity and noise, that is not included in this simulation. However, it must be ensured that the star signal does not fall outside the ROI. In Fig. 15 is shown a simulation of the centroiding accuracy as a function of the noise level. As an Fig. 15. Simulation of the centroiding accuracy. Fig. 16. Typical error tree for a star tracker. example, a star is known to have a read/dark current noise on 100 photoelectrons. The brightness of the star is 100000 photoelectrons. The average centr