Using an Unscented Kalman Filter (UKF) as the nonlinear estimator within a Global Positioning System/Inertial Navigation System (GPS/INS) sensor fusion algorithm for attitude estimation, various methods of calculating the matrix square root were discussed and compared. Specifically, the diagonalization method, Schur method, Cholesky method, and five different iterative methods were compared. Additionally, a different method of handling the matrix square root requirement, the square-root UKF (SR-UKF), was evaluated. The different matrix square root calculations were compared based on computational requirements and the sensor fusion attitude estimation performance, which was evaluated using flight data from an Unmanned Aerial Vehicle (UAV). The roll and pitch angle estimates were compared with independently measured values from a high quality mechanical vertical gyroscope. This manuscript represents the first comprehensive analysis of the matrix square root calculations in the context of UKF. From this analysis, it was determined that the best overall matrix square root calculation for UKF applications in terms of performance and execution time is the Cholesky method. 1. Introduction The improvement of microprocessors and sensors has increased civilian use of Unmanned Aerial Vehicles (UAVs) for various applications, many of which requiring an accurate estimate of the aircraft attitude [1, 2]. For example, attitude estimation is an important requirement for UAV remote sensing applications such as 3D mapping with direct georeferencing [3] or constructing large mosaics [4]. Due to cost and weight restrictions [5, 6], high-quality military grade inertial navigation systems may not be practical [6, 7]. Therefore, attitude estimation algorithms that rely only on low-cost sensors have become essential for civilian applications. A popular approach to the attitude estimation problem involves fusing together information from a low-cost Inertial Navigation System (INS) with information from a Global Positioning System (GPS) receiver [8, 9]. Various formulations and analyses of GPS/INS sensor fusion exist in the literature [10–12], including a detailed comparison by the authors with respect to attitude estimation performance and computational cost [13]. The Unscented Kalman Filter (UKF) [14] is a well-known nonlinear estimator for use within GPS/INS sensor fusion [10, 11, 15]. Within this effort, the UKF is used to obtain an accurate estimate of the aircraft attitude, in particular the roll and pitch angles. For this type of problem, an alternative nonlinear
References
[1]
L. Changchun, S. Li, W. Hai-bo, and L. Tianjie, “The research on unmanned aerial vehicle remote sensing and its applications,” in Proceedings of the IEEE International Conference on Advanced Computer Control (ICACC '10), pp. 644–647, Shenyang, China, March 2010.
[2]
A. M. Jensen, M. Baumann, and Y. Chen, “Low-cost multispectral aerial imaging using autonomous runway-free small flying wing vehicles,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS '08), pp. 506–509, Boston, Mass, USA, July 2008.
[3]
M. Nagai, T. Chen, R. Shibasaki, H. Kumagai, and A. Ahmed, “UAV-borne 3-D mapping system by multisensor integration,” IEEE Transactions on Geoscience and Remote Sensing, vol. 47, no. 3, Article ID 4783021, pp. 701–708, 2009.
[4]
T. Suzuki, Y. Amano, and T. Hashizume, “Vision based localization of a small UAV for generating a large mosaic image,” in Proceedings of the SICE Annual Conference (SICE '10), pp. 2960–2964, Taipei, Taiwan, October 2010.
[5]
S. Dascalu, “Remote sensing using autonomous UAVs suitable for less developed countries,” The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Enschede, Netherlands, vol. 34, no. 30, 2006.
[6]
G. Zhou and D. Zang, “Civil UAV system for earth observation,” in Proceedings of the IEEE International Geoscience and Remote Sensing Symposium (IGARSS '07), Barcelona, Spain, 2007.
[7]
J. Everaerts, “The use of unmanned aerial vehicles (UAVs) for remote sensing and mapping,” The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Beijing, vol. 36, no. B1, pp. 1187–1192, 2008.
[8]
M. S. Grewal, L. R. Weill, and A. P. Andrew, Global Positioning, Inertial Navigation & Integration, John Wiley & Sons, New York, NY, USA, 2nd edition, 2007.
[9]
E. Kaplan and C. Heagarty, Understanding GPS Principles and Applications, Arttech House, Norwood, Mass, USA, 2nd edition, 2006.
[10]
J. L. Crassidis, “Sigma-point kalman filtering for integrated GPS and inertial navigation,” in Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, San Francisco, Calif, USA, 2005.
[11]
R. D. Van Merwe, E. A. Wan, and S. I. Julier, “Sigma-point kalman filters for nonlinear estimation and sensor-fusion—applications to integrated navigation,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, pp. 1735–1764, Providence, RI, USA, August 2004.
[12]
T. Fiorenzani, et al., “Comparative Study of Unscented Kalman Filter and Extended Kalman Filter for Position/Attitude Estimation in Unmanned Aerial Vehicles,” IASI-CNR, R. 08-08, 2008.
[13]
J. Gross, Y. Gu, M. Rhudy, S. Gururajan, and M. Napolitano, “Flight test evaluation of GPS/INS sensor fusion algorithms for attitude estimation,” IEEE Transactions on Aerospace Electronic Systems. In press.
[14]
S. J. Julier and J. K. Uhlmann, “New extension of the Kalman filter to nonlinear systems,” in Signal Processing, Sensor Fusion, and Target Recognition VI, vol. 3068 of SPIE Proceedings Series, pp. 182–193, July 1997.
[15]
J. Gross, et al., “A comparison of extended kalman filter, sigma-point kalman filter, and particle filter in GPS/INS sensor fusion,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, Toronto, Canada, 2010.
[16]
R. E. Kalman and R. S. Bucy, “New results in linear filtering and prediction theory,” Journal of Basic Engineering, vol. 83, pp. 95–108, 1961.
[17]
N. El-Sheimy, E. H. Shin, and X. Niu, “Kalman filter face-off: extended vs. Unscented kalman filters for integrated GPS and MEMS inertial,” in Proceedings of the International Symposium on Global Navigation Satellite Systems (GNSS '06), pp. 48–54, 2006.
[18]
M. Rhudy, Y. Gu, J. Gross, and M. Napolitano, “Sensitivity analysis of EKF and UKF in GPS/INS sensor fusion,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, Portland, Ore, USA, 2011.
[19]
R. van der Merwe and E. A. Wan, “The square-root unscented Kalman filter for state and parameter-estimation,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, pp. 3461–3464, Salt Lake City, Utah, USA, May 2001.
[20]
N. J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 2008.
[21]
J. Wendel, J. Metzger, R. Moenikes, A. Maier, and G. F. Trommer, “A Performance comparison of tightly coupled GPS/INS navigation systems based on extended and sigma point kalman filters,” Journal of the Institute of Navigation, vol. 53, no. 1, pp. 21–31, 2006.
[22]
N. B. Stastny, R. A. Bettinger, and F. R. Chavez, “Comparison of the extended and unscented Kalman filters for angles based relative navigation,” in Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Honolulu, Hawaii, USA, August 2008.
[23]
Y. Wu, D. Hu, M. Wu, and X. Hu, “Unscented Kalman filtering for additive noise case: augmented vs. non-augmented,” in Proceedings of the American Control Conference (ACC '05), pp. 4051–4055, Portland, Ore, USA, June 2005.
[24]
F. Sun, G. Li, and J. Wang, “Unscented kalman filter using augmented state in the presence of additive noise,” in Proceedings of the IITA International Conference on Control, Automation and Systems Engineering (CASE '09), pp. 379–382, Zhangjiajie, China, July 2009.
[25]
M. C. VanDyke, J. L. Schwartz, and C. D. Hall, “Unscented kalman filtering for spacecraft attitude state and parameter estimation,” in Proceedings of the AIAA Space Flight Mechanics Meeting, Maui, Hawaii, USA, 2004.
[26]
S. A. Banani and M. A. Masnadi-Shirazi, “A new version of unscented kalman filter,” World Academy of Science, Engineering, and Technology, vol. 20, pp. 192–197, 2007.
[27]
B. Meini, “The matrix square root from a new functional perspective: theoretical results and computational issues,” SIAM Journal on Matrix Analysis and Applications, vol. 26, no. 2, pp. 362–376, 2004.
[28]
J. Gross, Y. Gu, and M. R. Napolitano, “A systematic approach for extended kalman filter tuning and low-cost inertial sensor calibration within a GPS/INS application,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference, Toronto, Canada, 2010.
[29]
J. Jarrell, Y. Gu, B. Seanor, and M. Napolitano, “Aircraft attitude, position, and velocity determination using sensor fusion,” in Proceedings of the AIAA Guidance, Navigation and Control Conference and Exhibit, Honolulu, Hawaii, USA, August 2008.
[30]
D. Simon, Optimal State Estimation, Wiley, New York, NY, USA, 2006.
[31]
D. A. McQuarrie, Mathematical Methods for Scientists and Engineers, chapter 10, University Science Books, Sausalito, Calif, USA, 1st edition, 2003.
[32]
M. M. Konstantinov, Perturbation Theory for Matrix Equations, Elsevier Science B.V., Amsterdam, The Netherlands, 2003.
[33]
N. J. Higham, “Newton’s method for the matrix square root,” Mathematics of Computation, vol. 46, no. 174, pp. 537–549, 1986.
[34]
S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, “Approximating the logarithm of a matrix to specified accuracy,” SIAM Journal on Matrix Analysis and Applications, vol. 22, no. 4, pp. 1112–1125, 2001.
[35]
L. N. Trefethen and D. Bau, Numerical Linear Algebra, SIAM, Philadelphia, Pa, USA, 1st edition, 1997.
[36]
N. J. Higham, “Analysis of the cholesky decomposition of a semi-definite matrix,” in Reliable Numerical Computation, pp. 161–185, Oxford University Press, Oxford, UK, 1990.
[37]
N. J. Higham, Accuracy and Stability of Numerical Algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa, USA, 2nd edition, 2002.
[38]
D. W. Tufts and C. D. Melissinos, “Simple, effective computation of principal eigenvectors and their eigenvalues and application to high-resolution estimation of frequencies,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 34, no. 5, pp. 1046–1053, 1986.
[39]
J. Roskam, Airplane Flight Dynamics and Automatic Flight Controls, chapter 1, DARcorporation, Lawrence, Kan, USA, 2003.
[40]
F. L. Lewis and V. L. Syrmos, Optimal Control, chapter 6, Wiley & Sons, New York, NY, USA, 2nd edition, 1995.
[41]
Y. Gu, B. Seanor, G. Campa, M. R. Napolitano, S. Gururajan, and L. Rowe, “Autonomous formation flight: Hardware development,” in Proceedings of the 14th Mediterranean Conference on Control and Automation (MED '06), Ancona, Italy, June 2006.
[42]
B. Seanor, Y. Gu, M. R. Napolitano, G. Campa, S. Gururajan, and L. Rowe, “3-Aircraft formation flight experiment,” in Proceedings of the 14th Mediterranean Conference on Control and Automation (MED '06), Ancona, Italy, June 2006.
[43]
Y. Gu, B. Seanor, G. Campa et al., “Design and flight testing evaluation of formation control laws,” IEEE Transactions on Control Systems Technology, vol. 14, no. 6, pp. 1105–1112, 2006.
[44]
M. G. Perhinschi, J. Burken, and G. Campa, “Comparison of different neural augmentations for the fault tolerant control laws of the WVU YF-22 model aircraft,” in Proceedings of the 14th Mediterranean Conference on Control and Automation (MED '06), Ancona, Italy, June 2006.
[45]
J. Gross, Y. Gu, B. Seanor, S. Gururajan, and M. R. Napolitano, “Advanced Research Integrated Avionic (ARIA) system for fault-tolerant flight research,” in Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, Chicago, Ill, USA, August 2009.
[46]
E. Kreyszig, Advanced Engineering Mathematics, chapter 20, Wiley & Sons, New York, NY, USA, 9th edition, 2006.
