Stellar Positioning System (Part II): Improving Accuracy During Implementation

Transcription

1 Stellar Positioning System (Part II): Improving Accuracy During Implementation Drew P. Woodbury, Julie J. Parish, Allen S. Parish, Michael Swanzy, Ron Denton, Daniele Mortari, and John L. Junkins Texas A&M University, College Station, Texas, July 6, 2009 Abstract This paper discusses the implementation of a Stellar Positioning System as well as techniques for error mitigation in experimentation and data post-processing. The hardware used during the development and testing of the Stellar Positioning System is described. Star-centroiding, star-identification, attitude estimation, and the local gravity vector were used by the Stellar Positioning System to determine latitude and longitude. Image processing, attitude filtering, and focal length estimation are presented as techniques to further improve the capabilities of the system. The resulting prototype was tested in three different locations and the results demonstrate accuracy of the Stellar Positioning System to be within 50 meters for short time intervals. Ph.D. Student, Aerospace Engineering, TAMU-3141, Student Member AIAA. Ph.D. Student, Aerospace Engineering, TAMU-3141, Student Member AIAA. M.S. Student, Computer Engineering, TAMU M.S. Student, Aerospace Engineering, TAMU Ph.D. Student, Computer Science, TAMU Associate Professor, Aerospace Engineering, TAMU-3141, AIAA Associate Fellow. Distinguished Professor, Aerospace Engineering, TAMU-3141, AIAA Fellow. 1

2 1 Introduction For centuries, the stars have been used as a means of position determination for navigation purposes. With today s precise clocks and high quality imaging capabilities, it is possible to accurately determine position using similar methods to those used by early navigators. Taking advantage of these capabilities and highly precise star catalogs, the Stellar Positioning System (SPS) was developed as a modern application of ancient celestial navigation techniques. While the theory has been developed to determine local latitude and longitude position from inter-star angles [1], there are additional obstacles to overcome when implementing these concepts in hardware. Since position determination requires a large number of measurements, error creeps into the algorithm from several sources such as the physical environment and the hardware itself. Fortunately, many of these errors can be mitigated using a range of techniques. This paper will discuss in detail the hardware and software components used to implement and validate the positioning system theory developed and presented in a companion paper [3]. A series of techniques will then be described that increase the accuracy of the overall system by reducing noise produced by the hardware and software. Experimental results are presented based on tests of the completed system at multiple locations in the United States. Results indicate that the SPS technology is particularly suitable to estimate the geographical position in non-global Positioning System (GPS) environments, such as the Moon and other planets. It can, however, be considered a valid backup system on Earth when and where the GPS signal is accidentally or intentionally made unavailable. 2 Hardware The current version of the SPS consists of a high quality astronomical camera, two inclination (i.e., tilt) sensors, a notebook computer, and all necessary cables. The various aspects of the 2

3 SPS hardware will be discussed in this section. 2.1 Astronomical Camera The camera used to validate the SPS theory is a Santa Barbara Instrument Group Model ST- 8XME camera. The camera s CCD has dimensions of 1,530 1,020 pixels (1.56 Megapixels) and each pixel is 9 microns square. The CCD is thermoelectrically cooled with temperature regulation to within ±0.1 Celsius and has a high quantum efficiency ( 85%). The integration time is adjustable from 110 milliseconds to 3, 600 seconds with 10 millisecond resolution. Integration times used during the tests are between 200 and 500 milliseconds. The camera system includes a USB 1.1 interface for control and image extraction. The download of a full frame image requires approximately 3.7 seconds. Pictures of the camera are shown in Figs. 1 and 2. Figure 1: Top View of Camera (with Inclinometers Attached) 2.2 Lenses The camera is equipped to accept C-mount lenses. Several high quality lenses were tested during the initial phase of the project. Manual-focus lenses were chosen to reduce the possible 3

4 Figure 2: Side View of Camera (with Inclinometers Attached) uncertainty associated with an automatic zoom capability. Nikkor lenses were selected (see Table 1) as the best quality option given budget constraints. As shown in the table, a range of focal lengths were selected. This range allows for camera calibration testing with respect to, as well as independent of, focal length. Also, lenses were chosen with the largest aperture available to minimize integration time. Brand Focal Length Minimum f# Lens Type Nikkor 24 f2.0 Wide Angle Nikkor 35 f1.4 Wide Angle Nikkor 50 f1.2 Standard Angle Nikkor 85 f1.4 Telephoto Table 1: Photographic Properties of SPS Lenses 2.3 Inclinometers The gravity direction measurement was done using a pair of Wyler Zerotronic ±1 inclination sensors mounted orthogonal to one another as well as to the optical axis of the camera (see Figs. 1 and 2). The sensors transfer the measurements to the computer via a transceiver/converter, shown in Fig. 3. These sensors were chosen for their high precision and range of accuracy. The sensors measure the gravity direction and convert it into a range of angles (±1 ) based on an internal calibration. The error in the sensors is minimized 4

5 (0.5%) by aligning the optical axis as close to zenith as possible. Error can be further reduced by obtaining multiple measurements from the inclination sensors and filtering them from noise. The current system can measure the inclination angle in approximately 300 milliseconds. The overall orientation of the system requires two measurements, one from each inclinometer, resulting in a set of measurements every 600 milliseconds. Figure 3: Transciever/Converter for Inclinometers. 2.4 Time The time is sampled from the Windows XP system clock on the computer. The clock is updated using Network Time Protocol (NTP) from the National Institute of Standards and Technology (NIST) Internet Time Service (ITS). This service provides accuracy to within 20 milliseconds. The Windows XP system clock provides a 15 millisecond granularity. In the current version of the system, no additional interpolation is used to resolve the time to a step size smaller than 15 milliseconds. 5

6 3 Software The theory developed to determine position is summarized in a flow chart as shown in Fig. 4 and is presented in detail in the companion paper [3]. The required measurements are collected by driver programs and then processed using MATLAB code. The images are first searched for stars using centroiding and star-identification algorithms. The vector positions of the stars in the body and inertial frames are the result of this part of the algorithm. The time measurement is then used to construct the Inertial-to-Greenwich Direction Cosine Matrix (DCM), R G/I, which is used to evaluate the star position vectors in the Greenwich frame. Next, the inclination measurements are used to estimate the direction of the local gravity vector. The body and Greenwich star location vectors, together with the local gravity direction, are used to solve for the local latitude and longitude through the inner product solution for position determination [3]. Finally, a GPS receiver is used to collect a true measurement of the latitude and longitude for comparison and calibration of the SPS. 3.1 Centroiding The locations of a set of stars, B r, are needed to calculate the vector positions of the stars in the body (i.e., camera) frame. 1 The primary responsibility of the centroiding algorithm is to best estimate the photocenter for each potential star in the image array. Figure 5 shows a sample image array while Fig. 6 shows a typical plot of a star s energy distribution on a section of the imaging array. Although the majority of bright spots in star field images are in fact stars, the possibility of imaging planets and satellites or of having faulty bright pixels ( hot pixels) in the array exists. For these reasons, the bright spots are not assumed to be stars until they have been positively identified as such. The general procedure of the centroiding algorithm is given as follows. 1 A superscript preceding a star position vector indicates the frame in which the vector is described (B is body, I is inertial, and G is Greenwich) 6

7 Figure 4: Basic SPS Local Position Estimation Algorithm. 1. Locate the brightest pixel in the image array: (x k, y k ). 2. Using a topological search, identify all pixels surrounding (x k, y k ) with intensity values greater than the gray-level threshold. 3. Determine the centroid location (ˆx k, ŷ k ) using Eq. (1). 4. Reset the intensity value for each pixel used in the (ˆx k, ŷ k ) centroid calculation to 0. NOTE: This step ensures that the k th star will not affect other centroid calculations. 5. Repeat steps 1-4 for all remaining bright spots in the image. ˆx k = x k + n n x ij I ij i=1 j=1 n n I ij ŷ k = y k + n n y ij I ij i=1 j=1 n n i=1 j=1 i=1 j=1 I ij (1) Here, n is the number of pixels that surround the (x k, y k ) pixel that have an intensity greater than the gray-level threshold, (ˆx k, ŷ k ) is the star centroid location, I ij is the pixel 7

8 Figure 5: Example of a Night Sky Image Array Starlight PSF Plot Figure 6: Star Energy Distribution 8

9 intensity value of the (i, j) pixel, and (x ij, y ij ) are the x and y coordinates of the (i, j) pixel. These centroid locations, together with the focal length, f, and CCD offsets, (x 0, y 0 ), of the camera, provide the components of the star locations in the body frame, B r k, via the collinearity equation. B r k = 1 (ˆxk x 0 ) 2 + (ŷ k y 0 ) 2 + f 2 (ˆx k x 0 ) (ŷ k y 0 ) (2) f 3.2 Pyramid: Star Identification and Attitude Determination The star identification algorithm represents one of the most vital components of the local position determination algorithm. Without star identification, the attitude of the camera body frame with respect to the inertial frame cannot be determined by the SPS. Similarly, false star identifications will generate erroneous local position output. The star identification algorithm compares the inter-star angles of stars in a given star field image to those in a star catalog. The most advanced implementation of this type of star identification is the Pyramid algorithm [4]. The Pyramid algorithm uses a minimum of four (if available) of the B r k vectors from the centroiding algorithm to find a set of (measured) inter-star angles. This set of angles is compared to existing values in a star catalog. The angles of the pyramid geometry between the four stars is generally unique, so finding a match to this geometry in the star catalog provides the names of the stars in the image as well as their inertial frame reference vectors, I r k, via their established right ascension and declination angles. Given the position vectors of the image stars in the inertial and body frames, I r k and B r k respectively, the attitude between the inertial and body frames can be calculated using an attitude determination algorithm. In other words, the optimal solution for R B/I can be found such that B r k = R I B/I r k. Among the several existing algorithms (see Ref. [5] for a survey), the Second EStimator of the Optimal Quaternion (ESOQ-2) [6] was selected for 9

10 its computational speed and because a flight-tested implementation of it was available. This algorithm finds the optimal quaternion that satisfies both B r k = R B/I I r k and the Wahba optimality criterion [7]. The resulting quaternion can then be used to construct R B/I [8]. As for the star catalog, the Tycho-2 catalog [9] was selected. This catalog contains star directions with accuracy of ±0.6 micro-arcseconds (at J2000) and provides the proper motion with error of ±0.25 micro-arcseconds per year. 2 The star catalog error is much smaller than the error associated with the instrument body vectors, which are based on the image centroiding results and the optical system parameter calibrations. Consequently, the reference vector errors are assumed to be negligible for the purpose of this analysis. 3.3 Position Determination In order to implement the algorithm outlined in Fig. 4, the current time must be sampled and used to construct the inertial to Greenwich DCM, R G/I. This coordinate transformation is used to write the star position vectors in the Greenwich frame: G r k = R I G/I r k. Measurements from the inclinometers are then used to estimate the local gravity vector direction using a optimal cones intersection technique [10]. These are the three pieces of information needed to solve the position determination problem. 4 Position Determination Enhancement Techniques While the procedure described in the previous sections provide adequate results, additional techniques can be used to further improve the estimated position. Image processing, attitude filtering, and focal length estimation techniques, were used to increase the resulting accuracy. The methods described in this section are designed to minimize measurement noise and to improve position estimate accuracy. 2 This small error results from the multitude of accurate sensor measurements taken from Earth-bound and orbiting observatories over a period of years. 10

11 4.1 Image Processing Image processing techniques were implemented to minimize errors produced by the centroiding algorithm. Since centroiding accuracy affects both attitude estimation and star identification, centroid error minimization is critical. These techniques convert the original images taken by the astronomical camera (see Fig. 7) to data arrays easier to process by the onboard algorithms Dark Frame Subtraction Sixteen separate images were taken with the camera while the shutter remained closed (i.e., dark frames). The sixteen values for each pixel were then averaged to produce a master dark frame. This image contained the noise produced by the CCD and camera electronic system due to internal dark current as well as thermal effects. This process was repeated for each integration time and temperature setting used during testing. After a night image was taken by the camera, the corresponding master dark frame was subtracted from that image, pixel by pixel. If any values dropped below the allowable threshold, they were set to the minimum threshold value (typically zero). The resulting image contained only values produced by photons hitting the CCD from external sources and was consistent with the previous image format. This subtraction provided a means for star centroids to be more easily distinguished from background noise by the centroiding algorithm (see Fig. 8). Reference [11] discusses alternative image processing techniques to the dark frame subtraction method presented above including flat-fielding and taking bias images. While these techniques could be implemented to improve the accuracy of certain imaging systems, it was found that applying the basic dark frame subtraction procedure provided adequate results. As such, this was the method used during testing. 11

12 Figure 7: Raw Image Prior to Image Process- Figure 8: Sample Image after Dark Frame ing Subtraction Gaussian Filter Often after the subtraction of the master dark frame, a star centroid will be significantly pixelated (see Fig. 9). Since the centroiding algorithm assumes that each star centroid has a Gaussian (i.e., normal) distribution, this pixelation can create unwanted effects in the centroid calculation. A Gaussian filter was applied over the entire image following the master dark frame subtraction to smooth out the remaining centroids. The two parameters used by the Gaussian filter were tuned based on simulated numerical tests. The result was solid and well-distributed centroids for the centroiding algorithm (see Fig. 10) Figure 9: Centroid before Gaussian Filter Figure 10: Centroid after Gaussian Filter 12

13 4.2 Attitude Filtering Separate developments at the University of Texas at Austin [12] and at Texas A&M University [13] have shown that the classic Multiplicative Kalman Filter (MKF) framework to estimate the attitude [14] can be modified to guarantee a unit-norm updated quaternion. This approach eliminates the errors caused by the brute force normalization performed during the quaternion update phase of the MKF. The resulting Constrained-MKF, or Quaternion Constrained Kalman Filter (QCKF), provides improved results over the MKF as shown in Fig. 11 and was used in place of the MKF in the SPS software to remove measurement noise and to provide a more precise attitude estimate. Figure 11: Error Comparison Between QCKF and MKF The angular velocity estimation used for the SPS varied slightly from the traditional [14], because there are no gyros present in the system. In a typical MKF or QCKF, a gyro is modeled as ω = ω + β + η v (3) β = η u (4) where ω is the measured angular velocity, ω is the true angular velocity, β is the gyro drift, β is the gyro drift rate and η v N(0, σvi ) (5) 13

14 η u N(0, σ 2 ui 3 3 ) (6) where σ v and σ u are the standard deviation of the measurement noise for angular velocity and rate of gyro bias, respectively. In the case of no gyros, however, many of these terms are removed. As there is no sensor to measure angular velocity directly, ω is set to zero. Since Earth s angular velocity is known with great accuracy and is presumed constant, σ v is also set to zero. Thus, to satisfy Eq. (3), β must be set equal to ω. This leaves σ u as a tuning parameter. To summarize ω = 0, β = ω, and σ v = Focal Length Estimation In order to accurately determine position vectors for stars in the body frame, B r k, the camera s focal length and CCD offsets must be known with high precision. For a typical camera optical system, the focal length will be on the order of tens of millimeters while the CCD offsets will be on the order of pixels or tens of microns. Thus, the focal length is the dominant measurement and the CCD offsets can be presumed to be equal to zero [15]. Based on this assumption, the Gaussian Least Squares Differential Correction (GLSDC) method was implemented to estimate the focal length of the optical system. While GLSDC requires a good initial estimate to converge properly, it is assumed that this value is either supplied by the lens manufacturer or found during a laboratory calibration. Based on the assumption that the CCD offsets are zero, the inter-star angle of the measured stars can be written as cos ϑ ij = b T i b j = x i x j + y i y j + f 2 (x 2 i + y2 i + f 2 )(x 2 j + y2 j + f 2 ) (7) where b i is defined by Eq. (2). 3 In the ideal case (x 0 = y 0 = 0), the observed angles should be equal to the cataloged angles cos ϑ ij = r T i r j which are, in turn, almost free of error when 3 In these equations b i = B r i and r i = I r i. 14

15 using an accurate star catalog such as the Tycho-2. From the definitions above a function, F ij, can be defined as F ij = cos ϑ ij = r T i r j = N ij D i D j (8) where N ij = x i x j + y i y j + f 2 D i = x 2 i + y2 i + f 2 D j = x 2 j + y2 j + f 2 (9) By linearizing about the estimated value, ˆf, the inter-star angle can be redefined as cos ϑ ij F ij ( ˆf) + df ij f (10) df where f = ˆf + f. The error, ε ij, can be defined as ε ij = F ij ( ˆf) cos ϑ ij (11) Equations (10) and (11) can be used in conjunction with the GLSDC model described in Crassidis and Junkins [14] to develop an algorithm to estimate the focal length. The recursive update equation for this algorithm is then f k+1 = f k (H T k W H k ) 1 H T k W ε ijk (12) where H k = df ij df and W is a weighting matrix. 5 Results As the SPS team is based at Texas A&M University the majority of the data collection occurred in and around College Station, TX. Two additional sites outside of the local area 15

16 were chosen to show the validity of the concepts and methods presented above (Venice, FL and Lake Anna, VA). Initial testing of the system was performed using all four lenses described in Table 1. After comparing each of the lenses individually, a single lens was selected for further testing at each of the additional sites. 5.1 Lens Testing The lenses were tested by allowing each lens to take 100 images over the course of a single evening. The settings used for each of the lenses is shown in Table 2. Focal Length (mm) Aperture Focus Setting Integration Time (ms) Table 2: Lens Testing Parameters A summary of the maximum and average position errors along with their respective standard deviations are in Table 3. The position determination results based on the methods described above are displayed in Fig. 12. Based on these results, the current maximum and average errors are around 50 m and 20 m, respectively. Since all four lenses produced similar results, it can be deduced that the current error in the system is caused by something other than the optical system. The scatterplots, however, indicate that systematic errors still exist in the SPS because of the observed correlation between the latitude and longitude estimates. Focal Length (mm) Maximum Error (km) Average Error (km) Standard Deviation (km) Table 3: Summary of Lens Testing Position Error Results 16

17 Latitude (deg) Longitude (deg) Occurences (a) 24mm Error (km) Latitude (deg) Longitude (deg) Occurences (b) 35mm Error (km) Latitude (deg) Occurences Longitude (deg) (c) 50mm Error (km) Latitude (deg) Longitude (deg) Occurences (d) 85mm Error (km) Figure 12: Plots of Position Error in College Station, TX for All Lenses Tested 17

18 5.2 Venice, FL Results The 50 mm lens was used in Florida. An aperture of 2.0, a focus setting of 1.6, and an integration time of 250 ms were used. As previous tests were over short periods of time, the tests in both Florida and Virginia were designed to observe the effects of longer measurement periods. An average error of km with a standard deviation of km and a maximum error of km was determined from 500 images taken over a period of two hours. The specific position errors are shown in Fig. 13. Latitude (deg) Longitude (deg) Occurences Error (km) Figure 13: Position Error in Measurements Taken in Venice, FL These results show an increase in error when compared to the measurements taken in College Station. It can be seen in the data point plot, however, that there is a large clustering of data points with a number of outliers. This seems to suggest that the larger errors are caused by other factors than the normal system operation. When the position determination errors are compared to fluctuations in the inclinometers, the answer to this problem becomes clear (Figure 14). The spikes in the error values correspond to fluctuations in the inclinometer angle measurements. These fluctuations were caused by floor deflections due to temperature variations of the wooden dock floor. Based on this comparison it is noticed that small disturbances to the inclinometers created large angular measurement variations in the system ( arcseconds). Starting around image 250 both inclinometers begin to change their values slowly with time. This is the result of an internal drift in the inclinometers themselves. 18

19 X Inclinometer (") Y Inclinometer (") Error (km) Test # Test # Test # Figure 14: Comparison of Inclinometer Fluctuations to Position Error for Venice,FL 5.3 Lake Anna, VA Results The 85 mm lens was tested in Virginia. An aperture of 2.0, a focus setting of 5.2, and an integration time of 250 ms were used. An average error of 3.93 km with a standard deviation of 1.81 km and a maximum error of 8.11 km was determined from 1,800 images taken over a period of 7 hours. The results are shown in Figs. 15 and 16. Latitude (deg) Longitude (deg) Occurences Error (km) Figure 15: Position Error in Measurements from Lake Anna, VA While this error is significantly larger than any of other tests, it is also the longest test run to date. Here large spikes in the position error were again caused by fluctuations in the inclinometer angle readings. Drift in the inclinometers is also observed again, but it was determined that this slow drift was too small to cause the large drift in the longitude 19

20 X Inclinometer (") Y Inclinometer (") Error (km) Test # Test # Test # Figure 16: Comparison of Inclinometer Fluctuations to Position Error for Lake Anna, VA measurements. Note that when a random 100 image subset is analyzed, results similar to those contained in Table 3 are achieved. Thus, the larger error must be caused by a drift in the overall system. Possible causes of the longitudinal drift include the noise in the angular velocity estimates and full system calibration errors. Small fluctuations in the attitude due to noise and other external causes affect the angular velocity estimate, β, in the QCKF. Although small, these fluctuations compound over time. The angular velocity for celestial bodies is known with high accuracy. As such, special filtering techniques can be devised to reduce these fluctuations since the quaternion traces a great circle in 4-D space. Moreover, the Aerospace Engineering Department at Texas A&M University is not equipped with a state-of-the-art laboratory to fully calibrate the SPS. Improved calibration equipment along with a dedicated timing system would further reduce sources of error in the system and enhance position estimates. 20

21 6 Conclusions This paper demonstrates the feasibility of the Stellar Positioning System to estimate geographical position by inter-star angles, local gravity measurements, and accurate time knowledge. The algorithm and enhancement methods presented provide the necessary techniques for determining position to within tens of meters over short time intervals. Results also indicate, however, that there are longer time varying effects which must be taken into account. Furthermore, the Stellar Positioning System is a prototype and suffers from calibration errors due to the use of off-the-shelf components. These errors can be mitigated by fabricating a custom prototype of the system described. Nevertheless, this proof-of-concept provides a foundation for further development of a position determination system that could be used for navigation on Earth or other celestial bodies. Acknowledgements The authors would like to thank Mark Whorton of NASA-MSFC for providing the contract and grant necessary to perform this research. References [1] Swanzy, M. J., Analysis and Demonstration: A Proof-of-Concept Compass Star Tracker, Master s thesis, Texas A&M University, [2] Parish, J., Parish, A., Swanzy, M., Woodbury, D., Mortari, D., and Junkins, J. L., Stellar Positioning System (Part I): Applying Ancient Theory to a Modern World, AIAA Guidance, Navigation, and Control Conference, August 2008, Honolulu, Hawaii. 21

22 [3] Parish, J., Parish, A., Swanzy, M., Woodbury, D., Mortari, D., and Junkins, J. L., The Stellar Positioning System (Part I): An Autonomous Position Determination Solution, submitted to Navigation. [4] Mortari, D., Samaan, M. A., Bruccoleri, C., and Junkins, J. L., The Pyramid Star Identification Technique, Navigation, Vol. 51, No. 3, Fall 2004, pp [5] Markley, F. L. and Mortari, D., Quaternion Attitude Estimation using Vector Observations, Journal of the Astronautical Sciences, Vol. 48, No. 2/3, April September 2000, pp , Special Issue: The Richard H. Battin Astrodynamics Symposium. [6] Mortari, D., Second Estimator of the Optimal Quaternion, Journal of Guidance, Control, and Dynamics, Vol. 23, No. 5, September October 2000, pp [7] Wahba, G., A Least-Squares Estimate of Satellite Attitude, SIAM Review, Vol. 7, No. 3, 1965, pp [8] Schaub, H. and Junkins, J. L., Analytical Mechanics of Space Systems, AIAA Education Series, Reston, VA, [9] Høg, E., Fabricius, C., Makarov, V., Urban, S., Corbin, T., Wycoff G. Bastian, U., Schwekendiek, P., and Wicenec, A., The Tycho-2 Catalog of the 2.5 Million Brightest Stars, Astronomy and Astrophysics, Vol. 355, 2000, pp. L27 L30. [10] Mortari, D. and Singla, P., Optimal Cones Intersection Technique, ACTA Astronautica, Vol. 59, No. 6, September 2006, pp [11] Berry, R. and Burnell, J., The Handbook of Astronomical Image Processing, Willmann- Bell, 2nd ed., [12] Zanetti, R. and Bishop, R., Quaternion Estimation and Norm Constrained Kalman Filtering, American Institute of Aeronatuics and Astronautics, AIAA Paper , August

23 [13] Majji, M. and Mortari, D., Quaternion Constrained Kalman Filter, Paper AAS of the 2008 AAS Space Flight Mechanics Meeting Conference, January [14] Crassidis, J. and Junkins, J., Optimal Estimation of Dynamic Systems, Chapman & Hall/CRC Press, Boca Raton, FL, [15] Samaan, M. A., Mortari, D., and Junkins, J. L., Non Dimensional Star Identification for Un Calibrated Star Cameras, Journal of the Astronautical Sciences, Vol. 54, No. 1, January March 2006, pp [16] Woodbury, D., Parish, J., Parish, A., Swanzy, M., Denton, R., Mortari, D., and Junkins, J. L., Stellar Positioning System (Part II): Overcoming Error During Implementation, AIAA Guidance, Navigation, and Control Conference, August 2008, Honolulu, Hawaii. 23