Abstract
In this paper the motor algebra for linearizing the 3D Euclidean motion of lines is used as the oretical basis for the development of a novel extended Kalman filter called the motor extended Kalman filter (MEKF). Due to its nature the MEKF can be used as online approach as opposed to batch SVD methods. The MEKF does not encounter singularities when computing the Kalman gain and it can estimate simultaneously the translation and rotation transformations. Many algorithms in the literature compute the translation and rotation transformations separately. The experimental part demonstrates that the motor extended Kalman filter is an useful approach for estimation of dynamic motion problems. We compare the MEKF with an analytical method using simulated data. We present also an application using real images of a visual guided robot manipulator; the aim of this experiment is to demonstrate how we can use the online MEKF algorithm. After the system has been calibrated, the MEKF estimates accurately the relative position of the end-effector and a 3D reference line. We believe that future vision systems being reliably calibrated will certainly make great use of the MEKF algorithm.
Similar content being viewed by others
References
K.S. Arun, T.S. Huang, and S.D. Blostein, “Least-Squares fitting of two 3Dpoint sets,” IEEE Trans. Pattern Anal. Machine Intell., Vol. 9, No. 5, pp. 698-700, 1987.
A.J. Azarbayejani, H. Bradley, and A. Pentland, “Recursive estimation of structure and motion using relative orientation constraints,” In IEEE Conference on Computer Vision and Pattern Recognition, Los Alamitos, CA, June 1993, pp. 294-299.
I.Y. Bar-Itzhack and Y. Oshman, “Attitude determination from vector observations: Quaternion estimation,” IEEE Trans. on Aerospace and Electronic Systems, Vol. 21, No. 1, pp. 128-135, 1985.
E. Bayro-Corrochano, “The geometry and algebra of kinematics,” In Geometric computing with clifford algebra, G. Sommer (Ed.), Springer-Verlag, 2000.
E. Bayro-Corrochano, S. Buchholz, and G. Sommer, “Self-organizing clifford neural network,” in Proceedings of the International Conference of Neural Networks, ICNN'96, June 3-6, 1996, Washington D.C., USA, Vol. 1, pp. 120-125.
E. Bayro-Corrochano, K. Daniilidis, and G. Sommer, “Motor algebra for 3D kinematics: The case of the hand-eye calibration,” Journal of Mathematical Imaging and Vision, October 2000, Vol. 13(??), pp. 79-99.
J. Laseby and E. Bayro-Corrochano, “Analysis and computation of projective invariants from multiple views in the geometric algebra framework,” IJPRAI, Vol. 13, No. 8, 1999, pp. 1105-1121.
W.K. Clifford, “Preliminary sketch of bi-quaternions,” in Proc. London Math. Soc., Vol. 4, pp. 381-395, 1873.
W.K. Clifford, “Applications of Grassmann's extensive algebra,” Am. J. Math. Vol. 1, pp. 350-358, 1878.
K. Daniilidis and E. Bayro-Corrochano, “The dual quaternion approach to hand-eye calibration,” IEEE Proceedings of ICPR'96 Viena, Austria, Vol. I, Aug. 1996, pp. 318-322.
H. Grassmann, “Der Ort der Hamilton'schen Quaternionen in der Ausdehnungslehre,” Math. Ann., Vol. 12, p. 375, 1877.
D. Hestenes, “Space-time algebra,” Gordon and Breach, 1966.
D. Hestenes, New Foundations for Classical Mechanics, D. Reidel: Dordrecht, 1986.
D. Hestenes and G. Sobczyk, Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics, D. Reidel: Dordrecht, 1984.
R. Horaud and F. Dornaika, “Hand-eye calibration,” Intern. Journal of Robotics Research, Vol. 14, pp. 195-210, 1995.
E.J. Lefferts, F.L. Markley, and M.D. Shuster, “Kalman filtering for spacecraft attitude estimation,” AIAA Journal on Guidance, Control, and Dynamics, Vol. 5, pp. 417-429, 1982.
R.E. Kalman, “A new approach to linear filtering and prediction problems,” Trans ASME Journal of Basic Engineering, Vol. 82, pp. 35-45, 1960.
S. Kunze, “Ein Hand-Auge-System zur visuell basierten Lokalisierung und Identifikation von Objekten,” Ms. Thesis, Christian-Albrechts-Universität zu Kiel, Institut für Informatik und Praktische Mathematik, 1999.
P. Maybeck, Stochastic Models, Estimation and Control, Vol. 1, Academic Press: New York, 1979.
R. Sabata and J.K. Aggarwal, “Estimation of motion from a pair of range images: A review,” CVGIP: Image Understanding, Vol. 54, pp. 309-324, 1991.
H.W. Sorenson, “Kalman filtering techniques,” in Advances in Control Systems Theory and Applications, C.T. Leondes (Ed.), Vol. 3, Academic Press: New York, pp. 218-292, 1966.
Z. Zhang and O. Faugeras, 3D Dynamic Scene Analysis. Springer-Verlag: Berlin 1992.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bayro-Corrochano, E., Zhang, Y. The Motor Extended Kalman Filter: A Geometric Approach for Rigid Motion Estimation. Journal of Mathematical Imaging and Vision 13, 205–228 (2000). https://doi.org/10.1023/A:1011293515286
Issue Date:
DOI: https://doi.org/10.1023/A:1011293515286