ICA  Vol.6 No.3 , August 2015
Sigma-Point Filters in Robotic Applications
Abstract: Sigma-Point Kalman Filters (SPKFs) are popular estimation techniques for high nonlinear system applications. The benefits of using SPKFs include (but not limited to) the following: the easiness of linearizing the nonlinear matrices statistically without the need to use the Jacobian matrices, the ability to handle more uncertainties than the Extended Kalman Filter (EKF), the ability to handle different types of noise, having less computational time than the Particle Filter (PF) and most of the adaptive techniques which makes it suitable for online applications, and having acceptable performance compared to other nonlinear estimation techniques. Therefore, SPKFs are a strong candidate for nonlinear industrial applications, i.e. robotic arm. Controlling a robotic arm is hard and challenging due to the system nature, which includes sinusoidal functions, and the dependency on the sensors’ number, quality, accuracy and functionality. SPKFs provide with a mechanism that reduces the latter issue in terms of numbers of required sensors and their sensitivity. Moreover, they could handle the nonlinearity for a certain degree. This could be used to improve the controller quality while reducing the cost. In this paper, some SPKF algorithms are applied to 4-DOF robotic arm that consists of one prismatic joint and three revolute joints (PRRR). Those include the Unscented Kalman Filter (UKF), the Cubature Kalman Filter (CKF), and the Central Differences Kalman Filter (CDKF). This study gives a study of those filters and their responses, stability, robustness, computational time, complexity and convergences in order to obtain the suitable filter for an experimental setup.
Cite this paper: Al-Shabi, M. (2015) Sigma-Point Filters in Robotic Applications. Intelligent Control and Automation, 6, 168-183. doi: 10.4236/ica.2015.63017.

[1]   Hatamleh, K., Al-Shabi, M., Khasawneh, Q.A. and Al-Asal, M.A. (2014) Application of SMC and NLFC into a PRRR Robotic. ASME 2014 International Mechanical Engineering Congress and Exposition, Montreal, 14-20 November 2014, Paper No. IMECE2014-39136.

[2]   Al-Shabi, M. and Hatamleh, K. (2014) The Unscented Smooth Variable Structure Filter Application into a Robotic Arm. ASME 2014 International Mechanical Engineering Congress and Exposition, Montreal, 14-20 November 2014, Paper No. IMECE2014-40118.

[3]   Al-Shabi, M., Hatamleh, K. and Asad, A. (2013) UAV Dynamics Model Parameters Estimation Techniques: A Comparison Study. 2013 IEEE Jordan Conference on Applied Electrical Engineering and Computing Technologies, Amman, 3-5 December 2013.

[4]   Ali, Z., Deriche, M. and Landolsi, M. (2009) Sigma Point Kalman Filters For Multipath Xhannel Estimation In CDMA Networks. Proceedings of the 2009 6th International Symposium on Wireless Communication Systems, Tuscany, 7-10 September 2009, 423-427.

[5]   Ambadan J. and Tang, Y.M. (2009) Sigma-Point Kalman Filter Data Assimilation Methods for Strongly Non-Linear Systems. Journal of the Atmospheric Sciences, 66, 261-285.

[6]   Anderson B. and Moore, J. (1979) Optimal Filtering. Prentice-Hall.

[7]   Bar-Shalom, T. Li X.and Kirubarajan, T. (2001) Estimation with Applications to Tracking and Navigation—Theory, Algorithm and Software. John Wiley & Sons, Inc.

[8]   Grewal M. and Andrews, A. (2001) Kalman Filtering—Theory and Practice Using MATLAB. John Wiley & Sons, Inc.

[9]   Barker, A., Brown, D. and Martin, W. (1995) Bayesian estimation and the Kalman Filter. Computers & Mathematics with Applications, 30, 55-77.

[10]   Welch, G. and Bishop, G. (2006) An Introduction to the Kalman Filter. Department of Computer Science, University of North Carolina, Chapel Hill, TR 95-041.

[11]   Maybeck, P. (1979) Stochastic Models, Estimation, and Contro. Mathematics in Science and Engineering, Volume 141, Part 1, Academic Press, Waltham, iii-xix, 1-423.

[12]   Kalman, R. (1960) A New Approach to Linear Filtering and Prediction Problems. ASME Journal of Basic Engineering, 82, 35-45.

[13]   Ormsby, C., Raquet, J. and Maybeck, P. (2006) A Generalized Residual Multiple Model Adaptive Estimator of Parameters and States. Mathematical and Computer Modelling, 43, 1092-1113.

[14]   Negenborn, R. (2003) Robot Localization and Kalman Filters—On Finding Your Position in a Noisy World. MS Thesis, Utrecht University, Utrecht.

[15]   Simon, D. (2006) Optimal State Estimation: Kalman, H [Infinity] and Nonlinear Approaches. Wiley-Interscience.

[16]   Lary, D. and Mussa, H. (2004) Using an Extended Kalman Filter Learning Algorithm for Feed-Forward Neural Networks to Describe Tracer Correlations. Atmospheric Chemistry and Physics Discussion, 4, 3653-3667.

[17]   Leu, G. and Baratti, R. (2000) An Extended Kalman Filtering Approach with a Criterion to Set Its Tuning Parameters—Application to a Catalytic Reactor. Computers & Chemical Engineering, 23, 1839-1849.

[18]   Shojaie, K., Ahmadi, K. and Shahri, A. (2007) Effects of Iteration in Kalman Filter Family for Improvement of Estimation Accuracy in Simultaneous Localization and Mapping. IEEE/ASME International Conference on Advanced Intelligent Mechatronics, Zurich, 4-7 September 2007, 1-6.

[19]   Zhang, Y., Zhou, D. and Duan, G. (2006) An Adaptive Iterated Kalman Filter. IMACS Multiconference on Computational Engineering in Systems Applications, Beijing, 4-6 October 2006, 1727-1730.

[20]   Hyland, J. (2002) An Iterated-Extended Kalman Filter Algorithm for Tracking Surface and Sub-Surface Targets. OCEANS’02 MTS/IEEE, 3, 1283-1290.

[21]   Dungate, D., Theobald, R. and Nurse, F. (1999) Higher-Order Kalman Filter to Support Fast Target Tracking in a Multi-Function Radar System. IEE ColloquiumTarget Tracking: Algorithms and Applications, London, 11-12 November 1999, 14/1-14/3.

[22]   Bayard, D. and Kang, B. (2003) A High-Order Kalman Filter for Focal Plane Calibration of NASA’s Space Infrared Telescope Facility (SIRTF). AIAA Guidance, Navigation and Control Conference and Exhibit, Austin, 11-14 August 2003.

[23]   Athans, M., Wishner, R. and Bertolini, A. (1968) Suboptimal State Estimation for Continuous-Time Nonlinear Systems from Discrete Noisy Measurements. IEEE Transactions on Automatic Control, 13, 504-514.

[24]   Al-Shabi, M. (2012) The General Toeplitz/Observability Smooth Variable Structure Filter: Fault Detection and Parameter Estimation. LAP Lambert Academic Publishing, Saarbrücken.

[25]   Nguyen, H. and Walker, E. (1996) A First Course in Fuzzy Logic. CRC Press, Boca Raton.

[26]   Yager, R. and Zadeh, L. (1992) An Introduction to Fuzzy Logic Applications in Intelligent Systems. Kluwer Academic, location.

[27]   Carrasco, R., Cipriano, A. and Carelli, R. (2005) Nonlinear State Estimation in Mobile Robots Using a Fuzzy Observer. Processing of the 16th IFAC World Congress, Vol. 16, Part 1, Czech Republic.

[28]   Simon, D. (2003) Kalman Filtering for Fuzzy Discrete Time Dynamic Systems. Applied Soft Computing Journal, 3, 191-207.

[29]   Matia, F., Jimenez, A., Rodriguez-Losada, D. and Al-Hadithi, B.M. (2004) A Novel Fuzzy Kalman Filter for Mobile Robots Localization. IPMU 2004, 10th International Conference on Information Processing and Management of Uncertainty in Knowledge Based Systems, Volume II, Perugia.

[30]   Chen, Z. (2003) Bayesian Filtering: From Kalman Filters to Particles and Beyond. McMaster University.

[31]   Van Der Merwe, R. and Wan, E. (2004) Sigma-Point Kalman Filters for Integrated Navigation. Proceedings of the Annual Meeting, Institute of Navigation, 641-654.

[32]   Wang, L., Wang, L., Liao, C. and Liu, J. (2009) Sigma-Point Kalman Filter Application on Estimating Battery SOC. 5th IEEE Vehicle Power and Propulsion Conference, Dearborn, 7-10 September 2009, 1592-1595.

[33]   Sadhu, S., Mondal, S., Srinivasan, M. and Ghoshal, T. (2006) Sigma Point Kalman Filter for Bearing Only Tracking. Signal Processing, 86, 3769-3777.

[34]   Schenkendorf, R., Kremling, A. and Mangold, M. (2009) Optimal Experimental Design with the Sigma Point Method. IET Systems Biology, 3, 10-23.

[35]   Tang, X., Zhao, X. and Zhang, X. (2008) The Square-Root Spherical Simplex Unscented Kalman Filter for State and Parameter Estimation. 9th International Conference on Signal Processing, Beijing, 26-29 October 2008, 260-263.

[36]   Kim, J. and Shin, D. (2005) Joint Estimation of Time Delay and Channel Amplitude by Simplex Unscented Filter without Assisted Pilot in CDMA Systems. The 7th International Conference on Advanced Communication Technology, 1, 233-238.

[37]   Julier, S. (2003) The Spherical Simplex Unscented Transformation. Proceedings of the American Control Conference, 3, 2430-2434.

[38]   Gadsden, S., Al-Shabi, M., Arasaratnam, I. and Habibi, S. (2010) Estimation of an Electrohydrostatic Actuator Using a Combined Cubature Kalman and Smooth Variable Structure Filter. International Mechanical Engineering Congress and Exposition (IMECE), American Society of Mechanical Engineers, Vancouver, British Columbia.

[39]   Gadsden, A., Al-Shabi, M., Arasaratnam, I. and Habibi, S. (2014) Combined Cubature Kalman and Smooth Variable Structure Filtering: A Robust Nonlinear Estimation Strategy. Signal Processing, 96, 290-299.

[40]   Nrgaard, M., Poulsen, N. and Ravn, O. (2000) New Developments in State Estimation for Nonlinear Systems. Automatica, 36, 1627-1638.

[41]   Zhang, U., Gao, F. and Tian, L. (2008) INS/GPS Integrated Navigation for Wheeled Agricultural Robot Based on Sigma-Point Kalman Filter. 2008 Asia Simulation Conference-7th International Conference on System Simulation and Scientific Computing, Beijing, 10-12 October 2008, 1425-1431.

[42]   Zhu, J.H., Zheng, N.N., Yuan, Z.J. and Zhang, Q. (2009) A SLAM Algorithm Based on the Central Difference Kalman Filter. 2009 IEEE Intelligent Vehicles Symposium, Xi’an, 3-5 June 2009, 123-128.

[43]   Sadati, N. and Ghaffarkhah, A. (2007) POLYFILTER: A New State Estimation Filter for Nonlinear Systems. International Conference on Control, Automation and Systems, Seoul, 17-20 October 2007, 2643-2647.

[44]   Henrici, P. (1964) Elements of Numerical Analysis. John Wiley and Sons, New York.

[45]   Van Der Merwe, R. (2004) Sigma Point Kalman Filters for Probabilistic Inference in Dynamic State-Space Models. PhD Thesis, OGI School of Science & Engineering, Oregon Health & Science University, USA.