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Particle filtering on the Euclidean group: framework and applications

Published online by Cambridge University Press:  01 November 2007

Junghyun Kwon ,
Minseok Choi ,
F. C. Park  and
Changmook Chun
Junghyun Kwon
Affiliation:
School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-742, Korea
Minseok Choi
Affiliation:
School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-742, Korea
F. C. Park*
Affiliation:
School of Mechanical and Aerospace Engineering, Seoul National University, Seoul 151-742, Korea
Changmook Chun
Affiliation:
Intelligent Robotics Research Center, Korea Institute of Science and Technology, Seoul 136-791, Korea
*
*Corresponding author. E-mail: fcp@snu.ac.kr
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Summary

We address general filtering problems on the Euclidean group SE(3). We first generalize, to stochastic nonlinear systems evolving on SE(3), the particle filter of Liu and West for simultaneous estimation of the state and covariance. The filter is constructed in a coordinate-invariant way, and explicitly takes into account the geometry of SE(3) and P(n), the space of symmetric positive definite matrices. Some basic results for bilinear systems on SE(3) with linear and quadratic measurements are also derived. Three examples—GPS attitude estimation, needle tip location, and vision-based robot end-effector pose estimation—are presented to illustrate the framework.

Keywords

Particle filteringEuclidean groupLie group
Type
Article
Information
Robotica , Volume 25 , Issue 6 , November 2007 , pp. 725 - 737
Copyright
Copyright © Cambridge University Press 2007

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References

1.Arulampalam, M. S., Maskell, S., Gordon, N. and Clapp, T., “A tutorial on particle filter for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50 (2), 174–188, (2002).CrossRefGoogle Scholar
2.Banovac, F., Tang, J., Xu, S., Linch, D., Chung, H. Y., Levy, E. B., Chang, T., McCullough, M. F., Yaniv, Z., Wood, B. J. and Cleary, K., “Precision targeting of liver lesions using a novel electromagnetic navigation device in physiologic phantom and swine,” Med. Phy. 32 (8), 269–2705, (2005).CrossRefGoogle ScholarPubMed
3.Blake, A., Isard, M. and MacCormick, J., “Statistical Models of Visual Shape and Motion,” In: Sequential Monte Carlo Methods in Practice (Doucet, A., Freitas, N. and Gordon, N., eds.) (Springer-Verlag, New York, 2001) pp. 339–357.CrossRefGoogle Scholar
4.Brockett, R. W., “The Geometry of the Conditional Density Functions,” In: Analysis and Optimization of Stochastic Systems (Jacobs, O. L. R. et al. , eds.) (Academic, New York, 1980) pp. 399–409.Google Scholar
5.Chirikjian, G. S. and Kyatkin, A. B., Engineering Applications of Noncommutative Harmonic Analysis (CRC, Boca Raton, 2000.CrossRefGoogle Scholar
6.Chiuso, A. and Soatto, S., “Monte Carlo Filtering on Lie Groups,” Proceedings of the 39th IEEE Conference on Decision and Control 1, (2000) pp. 304–309.Google Scholar
7.Crouch, P. and Grossman, R., “Numerical integration of ordinary differential equations on manifolds,” J. Nonlinear Sci. 3, 1–33, (1993).CrossRefGoogle Scholar
8.Doucet, A., Freitas, N. and Gordon, N., eds., Sequential Monte Carlo Methods in Practice (Springer-Verlag, New York, 2001.CrossRefGoogle Scholar
9.Duncan, T. E., Probability Densities for Diffusion Processes With Applications to Nonlinear Filtering Theory Ph.D. dissertation (Stanford, CA: Stanford University, 1967).CrossRefGoogle Scholar
10.Fletcher, P. and Joshi, S., “Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors,” Proceedings of the ECCV'04 Workshop Computer Vision Approaches to Medical Image Analysis (2004), pp. 87–98.Google Scholar
11.Hartley, R. and Zisserman, A., Multiple View Geometry in Computer Vision (Cambridge University Press: Cambridge, U.K. 2000).Google Scholar
12.Higuchi, T., “Self organizing time series model,” In: Sequential Monte Carlo Methods in Practice (Doucet, A., Freitas, N. and Gordon, N., eds.) pp. 429–444, Springer-Verlag, New York 2001.CrossRefGoogle Scholar
13.Iserles, A. and Munthe-Kaas, H., “Lie Group Methods,” InActa Numerica (2000), Cambridge University Press, Cambridge 2000.Google Scholar
14.Kitagawa, G., “A self-organizing state-space model,” J. Amer. Statist. Assoc., 93, 1203–1215, (1998).Google Scholar
15.Kloeden, P. E. and Platen, E., Numerical Solution of Stochastic Differential Equations, Berlin: Springer, 1999.Google Scholar
16.Lenglet, C., Rousson, M., Deriche, R. and Faugeras, O., “Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing,” J. Math. Imaging Vis., 25 (3), 423–444 (2006).CrossRefGoogle Scholar
17.Liu, J. and West, M., “Combined parameter and state estimation in simulation-based filtering,” In: Sequential Monte Carlo Methods in Practice (Doucet, A., Freitas, N. and Gordon, N., eds.) 97–223, Springer-Verlag, New York (2001).Google Scholar
18.Moakher, M., “Means and averaging in the group of rotations,” SIAM J. Matrix Analysis Appl., 24 (1), 1–16 (2002).CrossRefGoogle Scholar
19.Mortensen, R. E., Optimal control of continuous time stochastic systems, Ph.D. dissertation Berkeley, CA: University of California (1966).CrossRefGoogle Scholar
20.Munthe-Kaas, H., “Higher order Runge-Kutta methods on manifolds,” Appl. Numer. Math., 29, 115–127, (1999).CrossRefGoogle Scholar
21.Okazawa, S., Ebrahimi, R., Chuang, J., Salcudean, S. E. and Rohling, R., “Hand-held steerable needle device,” IEEE/ASME Trans. Mechatron, 10 (3), 85–296 (2005).CrossRefGoogle Scholar
22.Park, F. C., “Distance metrics on the rigid-body motions with applications to mechanism design,” ASME J. Mech. Design, 17 (1), 48–54 (1995).CrossRefGoogle Scholar
23.Park, F. C., Kim, J. and Kee, C., “Geometric descent algorithms for GPS-based attitude determination,” AIAA J. Guidance, Control, Dynam, 23 (1), 26–33 (2000).CrossRefGoogle Scholar
24.Park, W., Kim, J. S., Zhau, Y., Cowan, N. J., Okamura, A. M. and Chirikjian, G. S., “Diffusion-based motion planning for a nonholonomic flexible needle model,” Proceedings of the IEEE International Conference on Robotics and Automation, pp. 4611–4616 (2005).Google Scholar
25.Pitt, M. and Shephard, N., “Filtering via simulation: Auxiliary particle filters,” J. Amer. Statistical Assoc., 94 (446), 590–599 (1999).CrossRefGoogle Scholar
26.Sastry, S., Nonlinear Systems. New York: Springer, 1999.CrossRefGoogle Scholar
27.Smith, S. T., “Covariance, subspace and intrinsic Cramer-Rao bounds,” IEEE Trans. Signal Process, 53 (5), 1610–1630 (2005).CrossRefGoogle Scholar
28.Solomon, S. B., Magee, C., Acker, D. E. and Venbrux, A. C., “TIPS placement in swine, guided by eletromagnetic real-time needle tip locaization displayed on previously acquired 3-D CT,” Cardiovasc Intervent Radiol, 22, 411–414, (1999).CrossRefGoogle Scholar
29.Spekowius, G. and Wendler, T., eds., Advances in Healthcare Technology: Shaping the Future of Medical Care. New York: Springer-Verlag, 2006.CrossRefGoogle Scholar
30.Srivastava, A., “Bayesian Filtering for Tracking Pose and Location of rigid targets,” Proceedings of SPIE Aerosense, Orlando, FL, pp. 160–171. (April, 2000).CrossRefGoogle Scholar
31.Srivastava, A. and Ericlassen, K, “Monte Carlo extrinsic estimators of manifold-valued parameters,” IEEE Trans. Signal Process, 50 (2), 299–308 (2002).CrossRefGoogle Scholar
32.Srivastava, A., Grenander, U., Jensen, G. R. and Miller, M.I., “Jump-diffusion markov processes on orthogonal groups for object pose estimation,” J. Statistical Planning and Inference, 103 (1/2), 15–37 (2002).CrossRefGoogle Scholar
33.Storvik, G., “Particle filters in state space models with the presence of unknown static parameters,” IEEE Trans. Signal Process, 50 (2), 281–289 (2002).CrossRefGoogle Scholar
34.Thrun, S., Burgard, W. and Fox, D., Probabilistic Robotics. Cambridge, MA: MIT Press, 2005.Google Scholar
35.Vallot, L. C., “Filtering for Silinear Systems,” M.S. Thesis, MIT, Cambridge, MA, 1981.Google Scholar
36.Wang, Y. and Chirikjian, G. S., “Workspace generation of hyper-redundant manipulators as a diffusion process on SE(N),” IEEE Trans. Robot. Autom., 20 (3), 399–408 (2004).CrossRefGoogle Scholar
37.Wang, Y., Zhou, Y., Maslen, D. K. and Chirikjian, G. S., “Solving the phase-noise Fokker-Planck equation using the motion group Fourier transform,” IEEE Trans. Commun., 54 (5), 868–877 (2006).CrossRefGoogle Scholar
38.Wang, Y. and Chirikjian, G. S., “Error propagation on the Euclidean group with application to manipulator kinematics,” IEEE Trans. Robot., 22 (4), 591–602 (2006).CrossRefGoogle Scholar
39.Webster, R. J., Kim, J.-S., Cowan, N. J., Chirikjian, G. S. and Okamura, A., “Nonholonomic modeling of needle steering,” Int. J. Robot. Res., 25 (5–6), 509–525 (2006).CrossRefGoogle Scholar
40.Yau, S. S.-T. and Lai, Y. T., “Explicit solution of DMZ equation in nonlinear filtering via solution of ODEs,” IEEE Trans. Autom. Control, 48 (3), 505–508 (2003).CrossRefGoogle Scholar
41.Zakai, M., “On the optimal filtering of diffusion processes,” Z. Warsch, Verw. Geb., 11, 230–243, (1969).CrossRefGoogle Scholar
42.Zhou, Y. and Chirikjian, G. S., “Probabilistic Models of Dead-Reckoning Error in Nonholonomic Mobile Robots,” Proceedings of the IEEE International Conference Robotics and Automation, pp. 1594–1599 (2003).Google Scholar
43.Zhou, Y. and Chirikjian, G. S., “Nonholonomic Motion Planning Using Diffusion of Workspace Density Functions,” Proceedings of the International Symposium on Advances in Robot Dynamics and Control (Washington, D.C., Nov. 16–21, 2003).CrossRefGoogle Scholar