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Fractional-Order Complementary Filters for Small Unmanned Aerial System Navigation

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Abstract

Orientation estimation is very important for development of unmanned aerial systems (UASs), and is performed by combining data from several sources and sensors. Kalman filters are widely used for this task, however they typically assume linearity and Gaussian noise statistics. While these assumptions work well for high-quality, high-cost sensors, it does not work as well for low-cost, low-quality sensors. For low-cost sensors, complementary filters can be used since no assumptions are made with regards to linearity and noise statistics. In this article, the history and basics of complementary filters are included with examples, the concepts of filtering based on fractional-order calculus are applied to the complementary filter, and the efficacy of non-integer-order filtering on systems with non-Gaussian noise is explored with good success.

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References

  1. Paparazzi: The Free Autopilot (an opensource project). http://paparazzi.enac.fr/. Accessed 4 Oct 2013

  2. Anderson, W., Fritze, E.: Instrument approach system steering computer. Proc. Inst. Radio Eng. 41(2), 219–228 (1953). doi:10.1109/JRPROC.1953.274209

    Google Scholar 

  3. Bachmann, E.R., Duman, I., Usta, U.Y., McGhee, R.B., Yun, X.P., Zyda, M.J.: Orientation tracking for humans and robots using inertial sensors. In: Proc. of the IEEE International Symposium on Computational Intelligence in Robotics and Automation, pp. 187–194 (1999). doi:10.1109/CIRA.1999.810047

  4. Bachmann, E.R., McGhee, R.B., Yun, X., Zyda, M.J.: Inertial and magnetic posture tracking for inserting humans into networked virtual environments. In: Proc. of the ACM Symposium on Virtual Reality Software and Technology VRST, p. 9. ACM Press (2001). doi:10.1145/505008.505011

  5. Bachmann, E.R., McKinney, D., McGhee, R.B., Zyda, M.J.: Design and implementation of MARG sensors for 3-DOF orientation measurement of rigid bodies. In: Proc. of the IEEE International Conference on Robotics and Automation, pp. 1171–1178 (2003). doi:10.1109/ROBOT.2003.1241751

  6. Baerveldt, A.J., Klang, R.: A low-cost and low-weight attitude estimation system for an autonomous helicopter. In: Proc. of the IEEE International Conference on Intelligent Engineering Systems, pp. 391–395 (1997). doi:10.1109/INES.1997.632450

  7. Baldwin, G., Mahony, R., Trumpf, J., Hamel, T., Cheviron, T.: Complementary filter design on the special euclidean group SE(3). In: Proc. of the European Control Conference, vol. 1, pp. 3763–3770 (2007)

  8. Barton, J.D.: Fundamentals of small unmanned aircraft flight. Johns Hopkins APL Tech. Dig. 31(2), 132–149 (2012)

    Google Scholar 

  9. Bohannan, G.: Analog realization of a fractional control element—revisited. In: Proc. of the 41st IEEE Int. Conf. on Decision and Control, Tutorial Workshop, vol. 1, pp. 27–30 (2002)

  10. Brown, R.G., Hwang, P.Y.C.: Introduction to Random Signals and Applied Kalman Filtering, 2nd edn. No. 4 in Wiley, TK5102.5.B696. John Wiley & Sons (1997). doi:10.1521/ijgp.2010.60.4.455

  11. Bryson, M., Sukkarieh, S.: Vehicle model aided inertial navigation for a UAV using low-cost sensors. In: Proc. of the Australasian Conference on Robotics and Automation (2006)

  12. Butler, H., de Hoon, C.: Fractional-order filters for active damping in a lithographic tool. Control Eng. Pract. 21(4), 413–419 (2013). doi:10.1016/j.conengprac.2012.12.011

    Article  Google Scholar 

  13. Chao, H., Coopmans, C., Di, L., Chen, Y.: A comparative evaluation of low-cost IMUs for unmanned autonomous systems. In: Proc. of the 2010 IEEE Conference on Multisensor Fusion and Integration, pp. 211–216. IEEE (2010). doi:10.1109/MFI.2010.5604460

  14. Chen, Y., Moore, K.L.: Discretization schemes for fractional order differentiators and integrators. IEEE Trans. Circuits Syst. I: Fund. Theory Appl. 49(3), 363–367 (2002)

    Article  MathSciNet  Google Scholar 

  15. Chen, Y., Vinagre, B.M., Podlubny, I.: Continued fraction expansion approaches to discretizing fractional order derivatives—an expository review. Nonlinear Dyn. 38(1–4), 155–170 (2004). doi:10.1007/s11071-004-3752-x

    Article  MATH  MathSciNet  Google Scholar 

  16. Coopmans, C., Di, L., Jensen, A., Dennis, A., Chen, Y.: Improved architecture designs for a low cost personal remote sensing platform: flight control and safety. In: Proc. of the ASME IDETC/CIE 2011, 1st Small Unmanned Aerial Vehicle Technologies and Applications (SUAVTA), 2011 ASME/IEEE International Conference on Mechatronic and Embedded Systems and Applications (MESA11) (2011)

  17. Coopmans, C., Han, Y.: AggieAir: An integrated and effective small multi-UAV command, control and data collection architecture. In: Proc. of the ASME IDETC/CIE 2009, 1st Small Unmanned Aerial Vehicle Technologies and Applications (SUAVTA), 2009 ASME/IEEE International Conference on Mechatronic and Embedded Systems and Applications (MESA09). San Diego, CA, USA (2009)

  18. Coopmans, C., Stark, B., Coffin, C.M.: A payload verification and management framework for small uav-based personal remote sensing systems. In: Proc. of the 2012 Int. Symposium on Resilient Control Systems (ISRCS2012), pp. 184–189. IEEE (2012). doi:10.1109/ISRCS.2012.6309316

  19. Crassidis, J.L., Markley, L.F., Cheng, Y.: Survey of nonlinear attitude estimation methods. J. Guid. Control. Dyn. 30(1), 12–28 (2007)

    Article  Google Scholar 

  20. Gordon, N., Percival, J., Robinson, M.: The Kalman–Levy filter and heavy-tailed models for tracking maneuvering targets. In: Proc. of the International Conference on Information Fusion, pp. 1024–1031 (2003)

  21. Gorenflo, R., Mainardi, F.: Fractional calculus and stable probability distributions. Arch. Mech. 50(3), 1–10 (1998)

    MathSciNet  Google Scholar 

  22. Hamel, T., Mahony, R.: Attitude estimation on SO[3] based on direct inertial measurements. In: Proc. of the IEEE International Conference on Robotics and Automation, pp. 2170–2175 (2006). doi:10.1109/ROBOT.2006.1642025

  23. Higgins, W.: A comparison of complementary and Kalman filtering. IEEE Trans. Aerosp. Electron. Syst. AES 11(3), 321–325 (1975)

    Article  Google Scholar 

  24. Jensen, A., Neilson, B., McKee, M., Chen, Y.: Thermal remote sensing with an autonomous unmanned aerial remote sensing platform for surface stream temperatures. In: Proc. of the International Geoscience and Remote Sensing Symp. (IGARSS), pp. 5049–5052 (2012)

  25. Jensen, A.M., Chen, Y., McKee, M., Hardy, T., Barfuss, S.L.: AggieAir—a low-cost autonomous multispectral remote sensing platform: new developments and applications. In: Geoscience and Remote Sensing Symposium, 2009 IEEE International, IGARSS 2009, vol. 4, pp. IV–995–IV–998 (2009). doi:10.1109/IGARSS.2009.5417547

  26. Jun, M., Roumeliotis, S.I., Sukhatme, G.S.: State estimation of an autonomous helicopter using Kalman filtering. In: Proc. of the IEEE/RSJ International Conference on Intelligent Robots and Systems, vol. 3, pp. 1346–1353 (1999). doi:10.1109/IROS.1999.811667

  27. Kingston, D.B., Beard, A.W.: Real-time attitude and position estimation for small UAVs using low-cost sensors. In: Proc. of the AIAA Unmanned Unlimited Technical Conference, Workshop and Exhibit, pp. 2004–6488 (2004)

  28. Krishna, B.T.: Studies on fractional order differentiators and integrators: a survey. Signal Process. 91, 386–426 (2011)

    Article  MATH  Google Scholar 

  29. Krobka, N.I.: Differential methods of identifying gyro noise structure. Gyrosc. Navig. 2(3), 126–137 (2011)

    Article  Google Scholar 

  30. Mahony, R., Hamel, T., Pflimlin, J.M.: Complementary filter design on the special orthogonal group SO(3). In: Proc. of the IEEE Conference on Decision and Control and European Control Conference, pp. 1477–1484 (2005)

  31. Mahony, R., Hamel, T., Pflimlin, J.M.: Nonlinear complementary filters on the special orthogonal group. IEEE Trans. Autom. Control 53(5), 1203–1218 (2008). doi:10.1109/TAC.2008.923738

    Article  MathSciNet  Google Scholar 

  32. Mahony, R., Hamel, T., Trumpf, J., Lageman, C.: Nonlinear attitude observers on SO(3) for complementary and compatible measurements: a theoretical study. In: Proc. of the IEEE Conference on Decision and Control held jointly with the Chinese Control Conference, pp. 6407–6412 (2009). doi:10.1109/CDC.2009.5399821

  33. Monje, C.A., Chen, Y., Vinagre, B., Xue, D., Feliu, V.: Fractional Order Systems and Control—Fundamentals and Applications (Advanced Industrial Control Series). Springer-Verlag (2010)

  34. Newman, M.E.J.: Power laws, Pareto distributions and Zipfs law. Contemp. Phys. 46(5), 323–351 (2005)

    Article  Google Scholar 

  35. Nolan, J.P.: Stable Distributions—Models for Heavy Tailed Data. Birkhauser, Boston (2013)

    Google Scholar 

  36. Oldham, K.B., Spanier, J.: The Fractional Calculus, vol. 17. Dover (1974)

  37. Oliveira, P., Kaminer, I., Pascoal, A.: Navigation System Design Using Time-varying Complementary Filters. IEEE Transactions on Aerospace and Electronic Systems 36(4), 1099–1114 (2000)

    Article  Google Scholar 

  38. Oustaloup, A., Levron, F., Mathieu, B., Nanot, F.M.: Frequency-band complex noninteger differenciator: characterization and systhesis. IEEE Trans. Control Syst. I: Fundam. Theory Appl. 47(1), 25–39 (2000)

    Google Scholar 

  39. Pflimlin, J.M., Hamel, T., Souères, P.: Nonlinear Attitude and Gyroscope’s Bias Estimation for a VTOL UAV. Int. J. Syst. Sci. 38(3), 197–210 (2007). doi:10.1080/00207720601110798

    Article  MATH  Google Scholar 

  40. Plummer, A.R.: Optimal complementary filters and their application in motion measurement. Proc. Inst. Mech. Eng. Part I: J. Syst. Control Eng. 220(6), 489–507 (2010)

    Google Scholar 

  41. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  42. Podlubny, I., Petráš, I., O’Leary, P., Dorčák, L., Vinagre, B.M.: Analogue Realizations of Fractional Order Controllers, vol. 29 (2002)

  43. Rehbinder, H., Hu, X.: Drift-free attitude estimation for accelerated rigid bodies. Automatica 40(4), 653–659 (2004). doi:10.1016/j.automatica.2003.11.002

    Article  MATH  MathSciNet  Google Scholar 

  44. Roberts, J.M., Corke, P.I., Buskey, G.: Low-cost flight control system for a small autonomous helicopter. In: Proc. of the IEEE International Conference on Robotics and Automation, vol. 1, pp. 546–551. Australian Robotics Automation Association (2002). doi:10.1109/ROBOT.2003.1241651

  45. Salcudean, S.: A Globally convergent angular velocity observer for rigid body motion. IEEE Trans. Autom. Control 36(12), 1493–1497 (1991). doi:10.1109/9.106169

    Article  MATH  MathSciNet  Google Scholar 

  46. Shaw, F.R., Srinivasan, K.: Bandwidth enhancement of position measurements using measured acceleration. Mech. Syst. Signal Process. 4(1), 23–38 (1990). doi:10.1016/0888-3270(90)90038-M

    Article  Google Scholar 

  47. Sheng, H., Chao, H., Coopmans, C., Chen, Y., Mckee, M.: Low-cost UAV-based thermal infrared remote sensing: platform, calibration and applications. In: Proc. of the 2010 IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, pp. 38–43. Qingdao, ShanDong, China (2010). doi:10.1109/MESA.2010.5552031

  48. Sheng, H., Chen, Y., Qiu, T.: Fractional Processes and Fractional-Order Signal Processing: Techniques and Applications. Springer (2012)

  49. Shiau, J.K., Huang, C.X., Chang, M.Y.: Noise Characteristics of MEMS Gyro’s Null Drift and Temperature Compensation. Appl. Sci. Eng. 15(3), 239–246 (2012)

    Google Scholar 

  50. Valério, D.: Ninteger (2008). http://www.mathworks.com/matlabcentral/fileexchange/8312-ninteger. Accessed 4 Oct 2013

  51. Veillette, M.: STBL: Alpha Stable Distributions for MATLAB (2012)

  52. Wirkler, W.H.: Aircraft Course Stabilizing Means (1951)

  53. Zimmermann, M., Sulzer, W.: High bandwidth orientation measurement and control based on complementary filtering. In: Proc. of the SYROCO IFAC Symposium on Robot Control. Vienna, Austria (1991)

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Authors and Affiliations

  1. Center for Self-Organizing and Intelligent Systems (CSOIS), Utah State University, Logan, UT, 84322, USA

    Calvin Coopmans

  2. Utah Water Research Laboratory, Utah State University, Logan, UT, 84322, USA

    Austin M. Jensen

  3. Mechatronics, Embedded Systems and Automation (MESA) Lab, University of California, Merced, Merced, CA, 95343, USA

    YangQuan Chen

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  1. Calvin Coopmans
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  2. Austin M. Jensen
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  3. YangQuan Chen
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Correspondence to Calvin Coopmans.

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Coopmans, C., Jensen, A.M. & Chen, Y. Fractional-Order Complementary Filters for Small Unmanned Aerial System Navigation. J Intell Robot Syst 73, 429–453 (2014). https://doi.org/10.1007/s10846-013-9915-6

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  • DOI: https://doi.org/10.1007/s10846-013-9915-6

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