Special Section on Space Telescopes

Space-based telescopes for actionable refinement of ephemeris pathfinder mission

[+] Author Affiliations
Lance M. Simms, Willem De Vries, Vincent Riot, Scot S. Olivier, Alex Pertica, Brian J. Bauman, Don Phillion, Sergei Nikolaev

Lawrence Livermore National Laboratory, Physical and Life Sciences, 7000 East Avenue, MS L210, Livermore, California 94550

Opt. Eng. 51(1), 011004 (Jan 19, 2012). doi:10.1117/1.OE.51.1.011004
History: Received July 20, 2011; Revised October 2, 2011; Accepted October 4, 2011
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Open Access Open Access

Abstract.  The Space-based Telescopes for Actionable Refinement of Ephemeris (STARE) program will collect the information needed to help satellite operators avoid collisions in space by using a network of nanosatellites to determine more accurate trajectories for selected space objects orbiting the Earth. In the first phase of the STARE program, two pathfinder cube-satellites (CubeSats) equipped with an optical imaging payload are being developed and deployed to demonstrate the main elements of the STARE concept. We first give an overview of the STARE program. The details of the optical imaging payload for the STARE pathfinder CubeSats are then described, followed by a description of the track detection algorithm that will be used on the images it acquires. Finally, simulation results that highlight the effectiveness of the mission are presented.

Figures in this Article

Space is becoming an increasingly crowded place. The Space Surveillance Network (SSN), operated by the United States Air Force (USAF), currently tracks over 20,000 manmade objects larger than 10cm in orbit around the Earth, and the NASA Debris Office estimates that as many as 300,000 objects >1cm are present in low Earth orbit (LEO) alone.1 Each year, several satellites typically disintegrate spontaneously into many pieces, further fueling the space debris population. As the Cosmos-Iridium collision of February 2009 proved, the consequences of this overcrowding can be both disastrous and expensive. Worse still, some experts predict that these incidents could cascade and render entire orbital regimes unsafe for satellites.2

A principal reason that the USAF tracks space objects is to warn satellite operators when they may need to maneuver their satellites to prevent a collision with another space object. However, the level of positional accuracy maintained by the SSN for the complete set of tracked space objects is insufficient to predict collisions with an adequate degree of certainty, and multiple false alarms occur daily as a result. Because of the high false alarm rate—approximately one per month for the average, active satellite, or approximately 10,000 false alarms per expected collision—satellite operators typically choose not to maneuver their satellites based on these warnings. This is where the STARE program comes in.

The objective of the STARE program is to collect the information necessary to provide satellite operators with actionable collision warnings. What is needed is improved accuracy in the knowledge of orbital trajectories for those space objects that are predicted to pass close to an active satellite. For instance, an accuracy of 100 m in these orbital trajectories at the time of closest approach would reduce the collision false alarm rate to approximately one per decade for the average, active satellite, or 100 false alarms per expected collision. This lower collision false alarm rate, and the corresponding higher likelihood of collision for each warning, would enable satellite operators to respond effectively by repositioning satellites (approximately once during the lifetime of the typical satellite) to avoid potential collisions.

The question then becomes how to reach the necessary level of accuracy in the knowledge of these orbital trajectories. A detailed architecture study performed at Lawrence Livermore National Laboratory using a sophisticated high-performance computer modeling and simulation environment for Space Situational Awareness (SSA) has recently concluded that a network of 12 small, inexpensive, space-based, optical imaging sensors can be used to achieve the accuracy needed to provide actionable satellite collision warnings to all maneuverable satellites in low Earth orbit (LEO).3,4

To demonstrate the essential elements of the STARE concept, the initial phase of this program involves the development and deployment of two pathfinder nanosatellites. These satellites will use a standard cube-satellite (or CubeSat) format and will be launched into a low earth orbit (most likely a 480×780km orbit with an inclination of 60 deg) where they will be used to image space objects at optical wavelengths. The images will be processed, along with Global Positioning Service (GPS) data, to refine the trajectory of the targets. The latter process is highlighted in circles 1–3 of Fig. 1. If successful, the mission will pave the way for a small constellation of similar satellites capable of refining ephemerides for all of the satellites and debris pieces in LEO involved in close approaches. The nanosats will provide actionable information, allowing satellite owners to move their assets, as in circles 4–5.

Graphic Jump LocationF1 :

High-level diagram illustrating the STARE concept. In the pathfinder mission, steps 1–3 will be executed for verification that the orbits of the chosen targets are indeed refined. In the full constellation, steps 4–5 will allow satellite owners to move their assets if a collision is deemed probable.

It is beyond the scope of this paper to address all the details (target selection, ground communication, etc.) of the STARE pathfinder mission. Nevertheless, an attempt will be made to motivate the mission in Sec. 2 and provide a sufficient overview of it in Sec. 3. The rest of the paper is devoted to the optical payload, the acquisition and processing of track data, and the anticipated results for the mission. In Sec. 4, the optics and imager implemented in the payload are described, followed by the algorithm used to detect stars and satellite tracks in the target images in Sec. 5. Section 6 discusses the orbital refinement expected for a number of targets.

There are several reasons why a space-based observing platform is preferable for orbital refinement. For instance, loss of observing time due to cloud coverage is not an issue and neither is image degradation from atmospheric turbulence. And mobile telescopes inherently have better sky coverage than their fixed, ground-based partners. However, the most important reason is observing efficiency.

Because LEO satellites maintain such a close proximity to earth, a typical fixed observatory is only able to observe them for <2h after sunset and before dawn. And that is just for satellites that pass overhead. In contrast, an orbiting satellite in a 90-min orbit is able to see a very large fraction of the full LEO population many times a day.

The latter point is made more concrete in Fig. 2, which compares the observing efficiency of hypothetical ground-based and spaced-based networks of orbital refinement telescopes. The spaced-based networks, shown in empty circles and triangles, consist of a series of nanosats arranged in three separate planes (either six or eight per plane, respectively). Each nanosat is equipped with an 8-cm telescope and a detector with moderate read noise. The ground-based network consists of 48 telescopes spread out more or less evenly across the globe in longitude, with the latitude varying between 45 and +45deg. The telescopes were allowed to be placed at a location regardless of whether there was solid ground there, and no obstruction due to cloud cover was considered, making it a highly idealized scenario. Each of them has a diameter of 0.5 m and is assumed to have a state-of-the-art detector having only a few electrons read noise. Distinction is also made between the period around the full moon (squares), where observing time is more limited than the period around the new moon (filled triangles).

Graphic Jump LocationF2 :

A plot showing the superior observing efficiency of a space-based constellation of telescopes over a ground-based one. Each point shows the percentage of 760 total observations (152 LEO targets; 5 observations per target) that each telescope network has made after the time shown on the x-axis. The empty circles and triangles show the efficiency for a series of space-based telescopes arranged in 3 planes (6 and 8 per plane, respectively). The squares and filled triangles show the results for a network of 48 ground-based 0.5 m telescopes equipped with low read-noise detectors spread evenly around the globe in longitude and between 45 and +45 in latitude. The filled triangles show a scenario during New Moon and squares shows the same scenario during Full Moon. Targets are assumed to be 1-m diameter spheres in the case of a satellite and 10 cm in the case of debris.

The plot in Fig. 2 shows the completeness fraction for a set of 152 targets that need to be observed. A total of five observations per target are required, making the total amount of observations 760. The target positions are based on real catalog entries and an observation date of August 16, 2010. One can see that the three plane-8 satellites per plane arrangement reaches about 80% efficiency in 8 h. While the idealized ground-based network has similar performance during these first 10 hours, it lags behind noticeably during the remainder of time. This is because the observation locations are fixed. They cannot reach the remaining targets and thus level out before reaching 90%. The spaced-based system, on the other hand, is able to reach these targets and make virtually all the required observations during the 24-h period.

The spaced-based network, with its smaller telescopes and mediocre detectors is thus able to outperform an idealized set of large telescopes with low-noise detectors on the ground. This is the primary motivation for the STARE constellation.

STARE Pathfinder Mission Goals

The primary objective of the STARE pathfinder mission is to demonstrate the feasibility of using nanosatellites to refine the orbital parameters of selected space objects to a level of accuracy that is useful for providing actionable collision warnings to satellite operators. As discussed in Sec. 1, it can be shown that 100 m is a useful level of positional accuracy for the orbital trajectories of space objects that are predicted to pass close to an active satellite. This value has therefore been adopted as the goal for the refined orbital positional accuracy of space objects observed by the STARE pathfinder CubeSats.

Table 1 presents some other technical goals for the STARE pathfinder mission. Operational goals for the pathfinder mission, which are expected to launch in summer of 2012, include achieving a mission lifetime of at least one month and completing observations of at least 30 space objects.

Table Grahic Jump Location
Table 1A list of the goals for the STARE pathfinder mission. The second row refers to the minimum limit at which accuracy can be considered valid.
Choice of Orbital Regime

One of the primary constraints on the STARE mission is the size of the satellites themselves. The 3U CubeSat limits the diameter of the primary optic to <10cm, which, along with the characteristics of the sensor, limits the maximum distance and relative velocity of the targets. With the pathfinder Cypress IBIS-5B CMOS sensor, the observations are limited to distances of 100km and relative velocities of 3km/s. For the full STARE constellation, a better sensor choice will allow distances on the order of 1000 km and relative velocities as high as 7km/s. Note that scheduling for the full STARE constellation will take into account everything that factors into the calculation of target signal-to-noise: distance, size, relative velocity, solar illumination, and observation geometry. As such, the CubeSat is not strictly limited to a fixed distance or relative velocity.

On the basis of signal-to-noise calculations with these considerations in place, an orbital platform for the pathfinder satellites has been simulated to maximize the number of observation opportunities,4 with the following criteria constituting a valid observation (i.e., one capable of reducing the size of the uncertainty ellipsoid of the target to the desired level):

  1. A maximum separation of <100km
  2. A relative tangential velocity of <3km/s
  3. A solar separation angle of >30deg (i.e., a solar exclusion angle of 30 deg)
  4. An Earth exclusion angle of 85 deg
  5. A lunar exclusion angle of 1 deg

These criteria, along with considerations of downlink opportunities, solar panel orientation and attitude control, drag-limited orbital lifetime, and GPS signal coverage, limit the number of useful orbital regimes for the STARE pathfinder satellites.

Examining the close conjunctions from the simulations occurring over a one-week period with the cataloged objects in LEO shows that a 700-km polar orbit with an inclination of 90deg is optimal for the STARE constellation. In particular, a sun-synchronous orbit of 98 deg simplifies satellite attitude control with respect to solar panel power generation. The orbits of the STARE pathfinder satellites will differ slightly from this optimum. Although not yet precisely known, they are scheduled to have a perigee near 480 km, an apogee near 780 km, and an inclination of about 60 deg.

The STARE Satellite

As shown in Fig. 3, the 3U STARE CubeSats consist of two main components: a Colony II Bus supplied by Boeing and an optical payload developed at LLNL. The Colony II Bus is essentially the brain of the satellite. It handles communication with the ground, controls the solar panels, distributes power to the various components, and provides attitude control with an on-board star tracker/reaction wheel system. It also contains a nonvolatile flash file system consisting of two SD cards, one being the RAID mirror of the other, that will be used to store images and telemetry data. Proper functioning of the bus, particularly in the attitude determination and control system (ADCS), is critical to success of the mission. Note that although the jitter and smear levels in the ADCS of the Colony II will not be precisely known until the satellites are launched, they are estimated at 72 and 188arcsec, respectively. These values are high, but simulations show that accurate star positions and track end points can still be obtained for a large number of targets and fields (see Sec. 5).

Graphic Jump LocationF3 :

The Colony II Bus, on the left, is the backbone of the STARE Satellite. The bus communicates with the Optical Payload, shown on the right, via an RS-422 connection.

Connected to the Bus with an RS-422 link is the optical payload. The payload contains the optical elements, the visible CMOS imager and its carrier board, an OEMV-1G GPS receiver and antenna, and several additional interface components. A dedicated Marvell PXA 270 microprocessor in the payload handles communication with the Colony II Bus and orchestrates the image acquisition and processing along with retrieval of concurrent GPS data. Further discussion of the optical payload can be found in Sec. 4.

Observing Strategy

The STARE pathfinder mission will only have one dedicated ground station for communication with the satellites. This station, located at the Naval Postgraduate School, will allow for 2min of data transfer per day at 9600 baud. The downlink is thus limited to 1Mb of data per day, which is close to the size of the 1280×1024 images generated by the payload.

Fortunately, the vast majority of the 1,310,720 pixels in an image will contain only detector noise and sky background; thus, they are of no use. Note that it should be emphasized that the telescope will not be tracking the targets. Rather, it will stare at the fixed stars and wait for the target to pass through the field of view. The stars will then appear as point sources in the image, and the track will be an extended object. With a 4-in. aperture and 1-s exposure, the limiting magnitude will be 12. The information that is needed is the following:

  1. Precise position and time of satellite at time of observation. This information is contained in the GPS logs that are recorded simultaneously with the image capture. Each GPS log is approximately 200 to 300 bytes.
  2. Stellar Positions (in detector coordinates). The positions of the stars will give very accurate pointing of the satellite once matched up to cataloged stellar positions. The location and flux of the 100 brightest stars in the image will be recorded.
  3. Track End-point Positions (in detector coordinates). Along with the timing and angular information from items 1 and 2, the track end points reveal exactly where the target was at the start and end of the observation (in the transverse plane).

Although there will be capability to download a full, raw image from the payload to the ground (a typical image averages 600–700 kB in size once compressed) for diagnostic and calibration purposes, the three pieces of information above are what will be routinely received on the ground. The GPS data will be logged from the on-board receiver, and the star and track data will be extracted from the images by using the algorithm described in Sec. 5, which will run in the XScale PXA 270 payload microprocessor.

Of course, this all relies on the assumption that the images contain a track and a suitable number of stars to yield an astrometric solution. To ensure this is the case, the satellite will be commanded to point toward a given target (when it is passing through a field with an ample number of bright stars) and begin acquiring images at the calculated time of conjunction. In a typical observing sequence, ten consecutive 1-s exposures will be taken along with their corresponding timestamps. The ten-image allotment should guarantee that one or two images contain the track even with the 1000 m uncertainty of its orbital elements. Note that it is preferable to have the entire streak and both track end points recorded in one image. However, refinement is still possible if two segments of the streak are captured in two separate images (i.e., one image showing a segment of the satellite entering the detector and the other image showing a segment leaving the detector).

Although the optical payload contains many components, the heart of it is comprised of a reflective Cassegrain telescope and a CMOS imager at its focus. Each will be discussed in turn.

The Telescope

A wide field of view is obviously beneficial for STARE since it (i) increases the chance that the entire streak will be captured in one exposure and (ii) increases the maximum velocity the target can have relative to a STARE satellite. To obtain a wide field with minimal aberrations in the small 10×10×10cm space offered in the CubeSat payload is a challenge, though.

The telescope (shown in Fig. 4) is a Cassegrain design, which can be optionally modified with corrective lenses near the focal plane. The lenses will help to reduce aberrations at the field edges. The telescope delivers an approximately f/2.5 beam, and with the 8.6×6.9mm imager, this equates to a field of view of about 2.08×1.67deg. Other details of the telescope are provided in the caption of Fig. 4, and the expected performance will be discussed in Sec. 4.2.

Graphic Jump LocationF4 :

The Cassegrain optical system employed in the STARE telescope. The telescope consists of two reflective conics. Optionally, a set of corrector lenses may be placed near the detector to reduce aberrations at the field edges. With a 225-mm focal length and 85-mm aperture, the system will yield a resolution of 29μrad/pixel across the Field of View, which corresponds to 6.1 arcsec/pixel. At a range of 100 km, this is 2.9m/pixel. The entire field size is 2.08×1.67deg. A baffling system, not shown in the figure, will reduce stray light.

Another challenge of the optical system is that there is no focusing mechanism (at least for the pathfinder mission). Thermal expansion and contraction in the space environment are thus of great concern. The telescope is designed to have a depth of focus of 10  μm, and an Invar support structure will be used to provide stiffness under changing temperatures. Preliminary thermal calculations show that focus will be maintained over the 20 to +60°C range expected in orbit.

The Imager

After collecting the star and track light, the telescope will focus it onto a Cypress IBIS5-B-1300 CMOS imager.5 This sensor, which has a 1280×1024 format with 6.7 μm pixels, is mainly intended for video rate imaging. It was chosen because Boeing is able to provide it in a fully integrated system (which includes the PXA 270 microprocessor) that will facilitate communications with the Colony II Bus and save a great deal of development time and expenses, allowing the pathfinder satellites to be completed in time for launch. In the full constellation of refinement satellites, the Cypress IBIS5-B-1300 will be replaced with a low-noise, high-performance imager.

The IBIS5-B-1300 has been tested extensively in the laboratory to verify that it will meet the mission requirements. The characteristics measured for the sensor at Lawrence Livermore National Laboratory, along with the ones specified by Cypress, are shown in Table  2. These values can be used to predict the signal to noise delivered by the optical system for a range of scenarios that might be encountered.4 As a specific case, a spherical piece of debris with a radius of r=0.3m and a reflectivity of 50% (albedo of 0.5) is considered. The object produces Lambertian scattering of the incident V-band portion of sunlight. A range of relative velocities are considered, these determining the dwell time per pixel and the number of photoelectrons received by a given pixel along the track. And for the detector, a quantum-efficiency (QE)/fill factor product of 0.22 photoelectrons/photon is assumed.

Table Grahic Jump Location
Table 2Characterization of the Cypress IBIS5-B-1300 imager.

The outcomes of the various scenarios, shown in Table  3, indicate that the STARE mission goals can indeed be achieved. The track recognition algorithm presented in Sec. 5 can tolerate a signal-to-noise-ratio (SNR) as low as 2.5 and still allow for orbital refinement. Again, it is important to note that a high-quality imager will greatly increase the signal-to-noise for each case. This will vastly open up the list of potential targets.

Table Grahic Jump Location
Table 3Signal-to-noise ratio (SNR) for the r=0.3m target objects described in the text. The average value of the read noise (58e) and dark current shot noise (35e) were used to calculate the total noise of about 68e (sky background noise is negligible). The calculation assumes the telescope is not drifting or rotating, an issue that will be touched upon in Sec. 5.
Satellite Recognition and End-point Determination

The issue of autonomously detecting satellite and airplane tracks in images is by no means a new one. For decades, these tracks have been nothing more than a nuisance for astronomers—foreground artifacts that must be removed in the preprocessing of data—and several methods for getting rid of them have been discussed in the literature. For instance, the recognition-by-adaptive-subdivision (RAST)6 algorithm removes satellite streaks using a geometric approach that assumes the tracks are straight lines and Storkey et al. use the random-sampling-and-consensus (RANSAC)7 algorithm to allow for removal of curved tracks and scratches as well.

Neither of these methods are concerned with accurately determining where the track starts and ends in the image, however. Levesque presents an algorithm for accurate end-point detection from ground-based images, but this again relies on the track being straight.8 Because the attitude of the STARE satellites will not be precisely controlled, the telescope may be rotating about the pointing axis, which could potentially produce tracks with a large and unknown curvature. A novel algorithm that can deliver subpixel end-point determination for tracks with arbitrary curvature is therefore required. It should be emphasized that the algorithm is not concerned with detection of faint streaks, but rather high-fidelity end-point determination for streaks with ample SNR.

To avoid confusion while describing the algorithm in this section, the term satellite will be reserved for the STARE CubeSat. The debris or satellite being imaged will be referred to as the target.

Target tracks in STARE images

During a STARE observation, the satellite will stare at a fixed-star background and allow the target to streak across the field of view. Changes in the orientation of the satellite during the observation are unwanted because they could potentially reduce the dwell time per pixel of the stars and the target (the exception being the case where the motion of the satellite causes inadvertent rate tracking of the target). But rotation of the satellite about the two axes perpendicular to the telescope pointing is of less concern because it simply adds to the transverse velocity component of the target and causes the stars to streak in a uniform manner across the detector. It will not produce curvature in the streak left by the target. Note that a simplification has been made by approximating the path of the target as a straight line during the exposure, which it will not be due to the motion of the satellite during the one second exposure and the curved orbit of the target.

Rotation about the pointing axis, on the other hand, could potentially induce significant curvature. If the satellite has a rotational velocity of θ˙ about the pointing axis, which will be taken as z, and the target has velocity components (vx, vy, vz) and coordinates of Display Formula

x=xo+vxt,y=yo+vyt,z=zo+vzt,(1)
with respect to the satellite center of mass, then the location of the target in the detector coordinate system is given by Display Formula
x=(xo+vxt)cos(θ˙t)+(yo+vyt)sin(θ˙t),y=(xo+vxt)sin(θ˙t)+(yo+vyt)cos(θ˙t),(2)
where the primes represent the mapping of object space to pixel space and rotation of the satellite about the x-and y-axis has been folded into the components vx and vy. Note that Mapping the detector coordinate system to equatorial coordinates will be achieved by the astrometric solution to the star field being observed. An equatorial angles-only determination of the target orbit can then be achieved.

One can gain an appreciation for the form of Eq. (2) by considering that for the case of xo=yo=0, it is the parametric representation of a spiral. Angular velocities of the telescope above 0.1deg/s are not anticipated; thus a spiral pattern should never be observed in STARE images. But θ˙=0.1deg/s is large enough to make a Hough transform ineffective for basic detection and create an error as large as two pixels for a track that extends all the way across the image if a global linear fit is used.

Fortunately, fitting the entire track is not necessary. As long as θx˙, θy˙, and θz˙ are known reasonably well (this information will be available from calibration data taken before the observation), the track end points (xo,yo), (xf,yf) are sufficient to refine the orbit of the target. The primary intent of the STARE algorithm is to find these coordinates.

STARE End-Point Determination Algorithm

The following subsections follow the numbering in Fig. 5, which gives an overview of the STARE algorithm.

Graphic Jump LocationF5 :

Flow diagram for the various steps used in the STARE end-point detection algorithm.

Image correction

Before the images are searched for stars and tracks, they must first be cleaned. Because the STARE algorithm identifies stars and tracks as a contiguous set of pixels above a noise threshold, T, preprocessing of the data is crucial to its success. The basic steps of the image correction, shown in box 1 of Fig. 5, are as follows:

  1. Sky image subtraction. The 10 raw images acquired during an observation sequence will be slightly offset from each other so that a given pixel sees sky background most of the time. A median filter is used on these 10 images to produce a sky image. Subtracting this sky image from a raw image very accurately removes both dark-current and sky background.
  2. Bad pixel masking and interpolation. Bad pixels are problematic for thresholding. These pixels can easily be mapped during routine calibration of the detector and stored as a mask in nonvolatile memory. They are first zeroed in each of the background subtracted images and then corrected with a nearest-neighbor filter that uses the local values and gradients across the pixel to fill in a reasonable value.
  3. Low-pass filter. The corrected image is optionally smoothed using a Gaussian kernel with a full width at half maximum (FWHM) on the order of 1–2 pixels. The smoothing helps ensure that tracks are contiguous. If the bad pixel density becomes excessive, the kernel can be extended at the expense of increasing the error in end-point estimation.

Object detection

After the image is corrected, it is searched for contiguous sets of pixels that have a value above T. This step is shown in box 2 of Fig 5. With both real and simulated images, typically T=3.5×RN, where RN is the read noise of the detector, produces good results. The read noise will dominate both the sky noise and dark-current shot noise with the STARE 1-s integration times.

Once a contiguous set of pixels has been identified, it is characterized as a star, track, or unknown object (such as a delta or Compton scattered worm) based on its ellipticity (e) and the number of pixels (N) it contains. Using a cut of e>0.8 and N>20 should effectively identify all real tracks. A perfectly straight track should have e=1; the margin e=0.81.0 allows for curvature and the possibility of overlapping stars or cosmic rays. The chance of a muon hit producing a track of >20 pixels long is extremely low.

Confusion of cosmic rays and stars is more troublesome. Because the optical system produces a subpixel point-spread function (PSF), most stars will actually appear as 1–4 pixel points rather than the nice Gaussian profiles encountered in astronomy applications. On the basis of previous space-based measurements, though, a significant amount of 1–4 pixel cosmic ray events are not expected in the STARE 1-s exposures.9,10 At geomagnetic latitudes of <50deg, 0.706 events per exposure are expected, and at >50deg this number may go up to 12. With these rates, an astrometric solution from the list of star centroids is possible even with the contamination.

Iterative local fitting at track end points (transverse degree of freedom)

The next step, step 3, is to find the end points for each of the tracks identified in step 2. As previously mentioned, applying a global linear fit to the track to find its end points may result in large errors. But a local linear fit to the track at each end point can still help in constraining their possible locations. The question that then arises is how many pixels to use in the fit. If too many are used, the curvature of the track will force the slope toward the global average. If too few are used, then the estimate is vulnerable to detector noise, bad pixels, etc.

One might consider using the second derivative as a criterion, Display Formula

d2ydx2=2θ˙[vxsin(θ˙t)+vycos(θ˙t))θ˙2((xo+vxt)cos(θ˙t)+(yo+vyt)sin(θ˙t)]2θ˙[vxcos(θ˙t)vysin(θ˙t))θ˙2((xo+vxt)sin(θ˙t)+(yo+vyt)cos(θ˙t)],(3)
(note that any change in the angular velocity has been ignored, θ¨=0). But this expression requires accurate knowledge of xo, yo, vx, and vy, which will not be known.

A solution to the problem is to use an iterative weighted least-squares fit to each track end-point until the root-mean-square (rms) deviation of distance from the included track pixels to the line is below a certain threshold, σDmax. Starting with all Npix=N pixels identified in the track, a line is fit using the expression: Display Formula

m=i=0Npixx2i=0NpixIyi=0NpixIxi=0NpixIxyNpixi=0NpixIx2(i=0NpixIx)2,b=Npixi=0NpixIxyi=0NpixIxi=0NpixIyNpixi=0NpixIx2(i=0NpixIx)2,(4)
where I is the pixel intensity and the indices on x, y, and I have been left out for notational convenience. Then, the distance of the track points to the line is calculated using Display Formula
D=I(mxy+b)Imaxm2+12,(5)
where Imax is the maximum pixel intensity for the Npix pixels used in the fit. If the rms of this value, σD, is below the threshold σDmax then the fit is considered valid. If not, n pixels are removed from the end of the track opposite to the one being fit and the above procedure is repeated. Thus, at the j’th iteration, the track end will be fit with Npix=Nn*j pixels. A minimum number of pixels to be used in the fit Npix=Nmin is also incorporated; the value will be based on the calibration data obtained during the mission.

The threshold σDmax and whether the intensity weighting in Eq. (5) is used will depend on the actual PSF of the STARE optical system. Figure 6 shows results for a simulated track where θ˙=1.0°deg/s and σDmax=0.50 was used without weighted fitting. The eventual error in end-point estimation was <0.1 pixels in both x and y.

Graphic Jump LocationF6 :

Example of the local fitting at each end-point. (a) shows the track fit in red when all pixels were used, (b) when the left 200 pixels were used, and (c) when the right when the right 170 pixels were used.

Matched filter at track end points (longitudinal degree of freedom)

Once the track has been fit at each end point, the path the target took along the detector near that point is well approximated. What is left is to determine precisely where the target was along this path at the start (or end) of the exposure (step 4). Simply recording the first or last pixel with a value above T will obviously result in errors. Accurately determining the location of the target requires taking into account the PSF of the optical system and the kernel used in the low pass filter of step 1.

To do this, a region of interest (ROI) around the roughly estimated end point that spans R×R pixels is first considered. An example ROI with R=7 is shown in Fig. 7(a). The goal is to reproduce this ROI with a simulated one obtained by convolving a line segment with a filter that matches the PSF and kernel described previously. The form of the line segment is already known from the fit obtained in step 3, and its length will indicate exactly where the end point is located.

Graphic Jump LocationF7 :

Illustration of the matched filter process. (a) shows an ROI taken from a corrected raw image. (b) shows a simulated ROI, where a line segment of length L1 has been convolved with a match filter to attempt to reproduce the real track in (a). In (c) the length has been extended to L2 as part of the iterative process. And in (d), the entire simulated ROI has been spanned to produce a residual at all R·r grid points. The first and last 10 points appear flat because the edges of the ROI are ignored due to convolution edge effects. The real track length Lreal is evident at the minimum of the residual curve.

After dividing each simulated pixel into r subpixels, a line segment of length L=1/r is created at the edge of the simulated ROI from which the track emerges. The segment is convolved with the filter to produce a track in the simulated ROI, as shown in Fig. 7(b). The simulated ROI is then subtracted from the real one, and the residual is squared. The length of the line segment is increased by 1/r and the process is repeated so that after R·r iterations, there will be a set of R·r residuals. The minimum of these, as shown in Fig. 7(d), indicates where the end-point is located.

Results for Simulated and Real Images

The results from testing the STARE algorithm on real images obtained by ground-based telescopes are encouraging. For these images, a median sky frame and bad pixel map could not be obtained, but subtraction of the mode sufficed for image correction. In Fig. 8, tracks found in three separate Oceanit images are shown after being analyzed by the algorithm. The ends of the green line segment indicate where the extracted end points are located. Although there are no official coordinates for these reported in the Oceanit data, inspection by eye shows that they line up well with the locations expected from the 1.9-pixel FWHM PSF.

Graphic Jump LocationF8 :

End-point determination for satellite track detected in three separate Oceanit images. While precise end-point coordinates are not available for comparison as they are in the simulated images, the reported end points match up well with what we expect based on the PSF of the system.

Extensive testing on simulated tracks and star fields has also been performed. These tests are useful because the measured end point can be compared to the true end point to determine the accuracy of the algorithm as a function of track length, orientation, brightness, etc. To comprehensively measure the error in the estimated end points, a 10 h run was performed in which 400 images were generated and analyzed. Real star fields were sampled and then tracks with random orientation and length were generated in a number of different brightness intervals. As a proxy for brightness, the quantity of photons per micron, which is the x-axis of Fig. 9, was chosen. The reason for this is that a track of a given brightness will produce varying signal-to-noise ratios, depending on how it is oriented relative to the detector. For instance, if a track is centered over the boundary between a row of pixels, then it will produce roughly half the SNR as it would when centered directly over one of the two rows.

Graphic Jump LocationF9 :

Plot showing the total end-point error from a run of 400 tracks of random lengths, orientation, and brightness. The y-axis shows the total end-point error and the x-axis shows photons per micron, both of which are described in the text. At 250 photons per micron, the SNR ranges from 2 to 4. At 600 photons per micron, the SNR ranges roughly from 6 to 12. These values depend on the orientation of the track relative to pixel boundaries.

On the y-axis of Fig. 9 is the total error in the end point estimate, Err=xerr2+yerr2, where xerr and yerr are simply the difference between the real and measured coordinates. The plot shows that at a level of 600 photons per micron, the error approaches a near constant value of Err=0.14. This is expected from the choice of r=10 for the simulated grid, which should produce an error of roughly 0.1 pixels for each coordinate (the step in length at each iteration is L=0.1 pixels). The value of 600 photons per micron corresponds to a SNR in the range of 6–12, depending on the track orientation. One can see that at a value of 250 photons per micron, which is roughly a SNR of 2–4, the error is slightly larger; but it is still subpixel and will serve well for the purpose of orbital refinement.

Of course, these are highly idealized numbers. A number of other errors—global positioning system measurement errors, timing errors, attitude control uncertainty, etc.—come into play in the game of orbital refinement. The simulations have neglected these. Also, the simulations ignore the low fill factor of the CMOS detector that will be used for the STARE pathfinder mission. Because the pixels are not sensitive over their entire area, information is lost every time the target spot passes over the pixel boundaries, and this alone can produce 0.3–0.7 pixel errors. Note that while the STARE pathfinder satellites will suffer from this problem, the future STARE constellation CubeSats will carry high-quality sensors that will not. As long as these systematics remain reasonably well behaved, though, the subpixel results provided by the STARE algorithm will allow for orbital refinement.

Once a set of star coordinates and track end points are available, the process of refining the orbits of the target can be carried out. How exactly this is done is beyond the scope of this paper, but the basic process involves using the extracted right ascension (RA) and Declination (DEC) of the target as input to a sequential differential least-squares orbit-refinement algorithm. The algorithm yields an updated state vector consisting of six orbital elements and a covariance matrix for these orbital elements.

To measure the improvement gained with the new state vectors, the operation of the STARE CubeSat has been simulated in an end-to-end fashion, beginning with image generation based on the current specifications for the Cypress IBIS5-B-1300 imager. The simulations assume a 3.6 arcsec angle measurement error in the end points of the streak for each target. This angle measurement error encompasses the following contributions: fitting error to the streak in the image, absolute timing errors, absolute camera position errors (better than 10 m with GPS), pixel to astronomical coordinate transformation errors, and static camera distortion correction residuals.

The initial covariance for all objects has been kept constant over all modeled observations. In the Keplerian coordinate system, all off-axis diagonal elements are 0, the angle errors on the diagonal are 109, and the semi-major axis uncertainty is set to 106m2. As for the targets, they have been selected over a range of distances and relative velocities. The orbital parameters for the STARE CubeSat vary in altitude for the Sun-synchronous (i=98deg) orbits. Also included are 45-deg inclinations for comparison (the 60-deg inclinations for the pathfinder mission should fall somewhere in between these extremes). All the targets are selected from a current North American Aerospace Defense Command (NORAD) catalog. The smallest size objects in this catalog are on the order of 10cm2.

Single Observations

For the case where only a single observation of each target has been made, the results are listed in Table  4. The size of the uncertainty ellipsoid is listed at the time of observation, and has not been propagated to the time of conjunction. This means that it has been reduced as much as possible given the observation and that it subsequently will increase in size again (through propagation uncertainties). Thus, one should not infer from the values in the eight columns that they meet (or fail) the 100-m accuracy goal of the STARE mission.

Table Grahic Jump Location
Table 4Results for a single observation of each target. The first two columns show the altitude and inclination of the STARE CubeSat making the observation. The target name is shown in the third column, and the fourth and fifth columns show the distance to and relative velocity of the target, respectively. The seventh column indicates the size of the uncertainty ellipsoid before the STARE orbital refinement, and the eight column shows this same number afterwards.

On the basis of these results, it appears that the level of orbital refinement is not strongly correlated with distance and relative velocity, although a longer streak does help slightly. The orbital regime of the imaging satellite relative to the target does not matter, aside from the fact that low relative velocity observations are very rare if the satellite and target orbits differ significantly.

Multiple Observations

The conjunction analysis code used for the single-observation simulations can be used to select observing opportunities for the same target, but separated by an integer amount of orbits. If focus is placed on the 700 km sun-synchronous orbit, many targets that are in a polar orbit will pass closely overhead several times per day. In Table 5, the refinement results for a randomly selected set of targets from a 100-day modeled campaign are listed. The second observation (typically, 90 min after the initial one) significantly improves the accuracy of the orbit. Adding in more observations continues this trend and also keeps the uncertainty ellipsoid small, as it will start expanding only after the last observation. Table 5 lists the instantaneous reduced uncertainty ellipsoid, with the offset since the first observation (in minutes) directly underneath.

Table Grahic Jump Location
Table 5Results for multiple observations for a selected set of targets. The first row in each entry lists the size of the uncertainty ellipsoid. The second row of each entry lists the time of the observation (in minutes) relative to the first observation.

As shown in Fig. 10, the effect of adding more observations over a 24-h period is beneficial. Initially, large accuracy improvements are seen but, beyond 10 observations, very little is gained. The fitting error was set to a lower value (1 arcsec) for this plot to illustrate the effect of increasing the number of observations on the overall orbital accuracy. Also, the GPS positional accuracy of 10 m was ignored. In reality, this GPS error must be added to the values in the plot in quadrature. Both this and a more realistic fitting error (3 to 4 arcsec for the pathfinder mission) will level the plot off earlier than the 10 observations in Fig. 10 and also at a higher plateau.

Graphic Jump LocationF10 :

The size of the uncertainty ellipsoid for two satellites versus the number of observations made over a 24-h period. Note that this plot assumes a 1 arcsec fitting error for the track end points to illustrate the effect of adding progressively more observations. With a higher error (3.6 arcsec), the leveling off after 10 observations occurs much earlier.

Even with the slightly higher fitting error of the STARE pathfinder mission, it is clear that repeated observations (in the range of 5–10) over a 24-h period will yield sufficiently accurate orbits. This increased accuracy will greatly reduce the number of false collisions, providing satellite operators with actionable information. For the full STARE constellation, improvements in detector capability and the ADCS of the CubeSat will lower the fitting error further, allowing for even more precise positional information.

The STARE pathfinder mission will pave the way for a constellation of satellites that will provide refined orbital information for satellites and debris in orbit around the Earth. These “space traffic cams” will drastically lower the number of false collision warnings, allowing satellite operators to take action when their assets are in danger. The motivation for the future STARE constellation has been covered, as well as the goals of the STARE pathfinder mission and how they will be met. An overview of the optics and detector that will be used in the CubeSat-based prototypes has been provided, along with a description of the algorithm that will extract the target track end points and star positions from the images they acquire. Lastly, the results of end-to-end simulations were presented and they show that the STARE mission is capable of meeting its goals and making space a safer place.

This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

“Committee on SpaceDebris,” Orbital Debris: A Technical Assessment. ,  National Academy Press ,  Washington DC  (1995).
Kessler  D. J., Cour-Palais  B. G., “Collision frequency of artificial satellites: the creation of a debris belt,” J.Geophys. Res.. 83, , 2637 –2646 (June 1978).CrossRef
Phillion  D. et al., “Large-scale simulation of a process for catalouging small orbital debris,” in  Proc. of Advanced Maui Optical and Space Surveillance Technologies Conf.  (September, 14–17, 2010).
Vries  W. De, “Collision avoidance through orbital refinement using imaging from a constellation of nanosatellites,” (in print).
Cypress Semiconductor Corp., IBIS5-B-1300 CYII5FM1300AB 1.3 MP CMOS Image Sensor Manual, 198 Champion Court, San Jose, CA ( August 2007). Document Number: 38-05710 Rev. *C.
Ali  H., Lampert  C., Breuel  T., “Satellite tracks removal in astronomical images,” Lect. Notes Comp. Sci.. 4225, , 892 –901 (2006)  0302-9743 CrossRef.
Storkey  A. J. et al., “Cleaning sky survey data bases using hough transform and renewal string approaches,” Mon. Notices Royal Astron. Soc.. 347, , 36 –51 (january 2004). 0035-8711 CrossRef
Levesque  M., “Automatic reacquisition of satellite positions by detecting their expected streaks in astronomical images,”  Advanced Maui Optical and Space Surveillance Technologies Conf.  (2009).
Shaw  D., Shaw  P., “Cosmic ray rejection in STIS CCD images,” Tech. Rep. ( June 1998).
AMS Collaboration, “Protons in near earth orbit,” Phys. Lett. B. 472, , 215 –226 (January 2000). 0370-2693 CrossRef

Biographies and photographs of the authors not available.

© 2012 Society of Photo-Optical Instrumentation Engineers

Citation

Lance M. Simms ; Willem De Vries ; Vincent Riot ; Scot S. Olivier ; Alex Pertica, et al.
"Space-based telescopes for actionable refinement of ephemeris pathfinder mission", Opt. Eng. 51(1), 011004 (Jan 19, 2012). ; http://dx.doi.org/10.1117/1.OE.51.1.011004


Figures

Graphic Jump LocationF1 :

High-level diagram illustrating the STARE concept. In the pathfinder mission, steps 1–3 will be executed for verification that the orbits of the chosen targets are indeed refined. In the full constellation, steps 4–5 will allow satellite owners to move their assets if a collision is deemed probable.

Graphic Jump LocationF2 :

A plot showing the superior observing efficiency of a space-based constellation of telescopes over a ground-based one. Each point shows the percentage of 760 total observations (152 LEO targets; 5 observations per target) that each telescope network has made after the time shown on the x-axis. The empty circles and triangles show the efficiency for a series of space-based telescopes arranged in 3 planes (6 and 8 per plane, respectively). The squares and filled triangles show the results for a network of 48 ground-based 0.5 m telescopes equipped with low read-noise detectors spread evenly around the globe in longitude and between 45 and +45 in latitude. The filled triangles show a scenario during New Moon and squares shows the same scenario during Full Moon. Targets are assumed to be 1-m diameter spheres in the case of a satellite and 10 cm in the case of debris.

Graphic Jump LocationF3 :

The Colony II Bus, on the left, is the backbone of the STARE Satellite. The bus communicates with the Optical Payload, shown on the right, via an RS-422 connection.

Graphic Jump LocationF4 :

The Cassegrain optical system employed in the STARE telescope. The telescope consists of two reflective conics. Optionally, a set of corrector lenses may be placed near the detector to reduce aberrations at the field edges. With a 225-mm focal length and 85-mm aperture, the system will yield a resolution of 29μrad/pixel across the Field of View, which corresponds to 6.1 arcsec/pixel. At a range of 100 km, this is 2.9m/pixel. The entire field size is 2.08×1.67deg. A baffling system, not shown in the figure, will reduce stray light.

Graphic Jump LocationF5 :

Flow diagram for the various steps used in the STARE end-point detection algorithm.

Graphic Jump LocationF6 :

Example of the local fitting at each end-point. (a) shows the track fit in red when all pixels were used, (b) when the left 200 pixels were used, and (c) when the right when the right 170 pixels were used.

Graphic Jump LocationF7 :

Illustration of the matched filter process. (a) shows an ROI taken from a corrected raw image. (b) shows a simulated ROI, where a line segment of length L1 has been convolved with a match filter to attempt to reproduce the real track in (a). In (c) the length has been extended to L2 as part of the iterative process. And in (d), the entire simulated ROI has been spanned to produce a residual at all R·r grid points. The first and last 10 points appear flat because the edges of the ROI are ignored due to convolution edge effects. The real track length Lreal is evident at the minimum of the residual curve.

Graphic Jump LocationF8 :

End-point determination for satellite track detected in three separate Oceanit images. While precise end-point coordinates are not available for comparison as they are in the simulated images, the reported end points match up well with what we expect based on the PSF of the system.

Graphic Jump LocationF9 :

Plot showing the total end-point error from a run of 400 tracks of random lengths, orientation, and brightness. The y-axis shows the total end-point error and the x-axis shows photons per micron, both of which are described in the text. At 250 photons per micron, the SNR ranges from 2 to 4. At 600 photons per micron, the SNR ranges roughly from 6 to 12. These values depend on the orientation of the track relative to pixel boundaries.

Graphic Jump LocationF10 :

The size of the uncertainty ellipsoid for two satellites versus the number of observations made over a 24-h period. Note that this plot assumes a 1 arcsec fitting error for the track end points to illustrate the effect of adding progressively more observations. With a higher error (3.6 arcsec), the leveling off after 10 observations occurs much earlier.

Tables

Table Grahic Jump Location
Table 1A list of the goals for the STARE pathfinder mission. The second row refers to the minimum limit at which accuracy can be considered valid.
Table Grahic Jump Location
Table 2Characterization of the Cypress IBIS5-B-1300 imager.
Table Grahic Jump Location
Table 3Signal-to-noise ratio (SNR) for the r=0.3m target objects described in the text. The average value of the read noise (58e) and dark current shot noise (35e) were used to calculate the total noise of about 68e (sky background noise is negligible). The calculation assumes the telescope is not drifting or rotating, an issue that will be touched upon in Sec. 5.
Table Grahic Jump Location
Table 4Results for a single observation of each target. The first two columns show the altitude and inclination of the STARE CubeSat making the observation. The target name is shown in the third column, and the fourth and fifth columns show the distance to and relative velocity of the target, respectively. The seventh column indicates the size of the uncertainty ellipsoid before the STARE orbital refinement, and the eight column shows this same number afterwards.
Table Grahic Jump Location
Table 5Results for multiple observations for a selected set of targets. The first row in each entry lists the size of the uncertainty ellipsoid. The second row of each entry lists the time of the observation (in minutes) relative to the first observation.

References

“Committee on SpaceDebris,” Orbital Debris: A Technical Assessment. ,  National Academy Press ,  Washington DC  (1995).
Kessler  D. J., Cour-Palais  B. G., “Collision frequency of artificial satellites: the creation of a debris belt,” J.Geophys. Res.. 83, , 2637 –2646 (June 1978).CrossRef
Phillion  D. et al., “Large-scale simulation of a process for catalouging small orbital debris,” in  Proc. of Advanced Maui Optical and Space Surveillance Technologies Conf.  (September, 14–17, 2010).
Vries  W. De, “Collision avoidance through orbital refinement using imaging from a constellation of nanosatellites,” (in print).
Cypress Semiconductor Corp., IBIS5-B-1300 CYII5FM1300AB 1.3 MP CMOS Image Sensor Manual, 198 Champion Court, San Jose, CA ( August 2007). Document Number: 38-05710 Rev. *C.
Ali  H., Lampert  C., Breuel  T., “Satellite tracks removal in astronomical images,” Lect. Notes Comp. Sci.. 4225, , 892 –901 (2006)  0302-9743 CrossRef.
Storkey  A. J. et al., “Cleaning sky survey data bases using hough transform and renewal string approaches,” Mon. Notices Royal Astron. Soc.. 347, , 36 –51 (january 2004). 0035-8711 CrossRef
Levesque  M., “Automatic reacquisition of satellite positions by detecting their expected streaks in astronomical images,”  Advanced Maui Optical and Space Surveillance Technologies Conf.  (2009).
Shaw  D., Shaw  P., “Cosmic ray rejection in STIS CCD images,” Tech. Rep. ( June 1998).
AMS Collaboration, “Protons in near earth orbit,” Phys. Lett. B. 472, , 215 –226 (January 2000). 0370-2693 CrossRef

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