Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map

Robotica

Volumn 26, Issue 4, 2008, Pages 419-434

  Park, Wooram ^a   Liu, Yan ^a   Zhou, Yu ^b   Moses, Matthew ^a   Chirikjian, Gregory S ^a  

^a Johns Hopkins University   (United States)

^b Stony Brook University   (United States)

Author keywords

Exponential map; Lie groups; Nonholonomic motion planning; State estimation

Indexed keywords

ARTIFICIAL NOISES; CAMBRIDGE UNIVERSITY; COMPUTATIONAL MODELS; CONFIGURATION SPACES; CURRENT STATE; DENSITY ESTIMATION; EUCLIDEAN MOTIONS; EXPONENTIAL MAPS; EXTERNAL NOISE; FOKKER PLANCK EQUATION (FPE); FOURIER; HIGH PROBABILITY; INTERNAL NOISE; LIE ALGEBRAS; LIE GROUPS; NEEDLE STEERING; NON HOLONOMIC SYSTEMS; NUMERICAL APPROXIMATIONS; OPERATIONAL PROPERTIES; PHYSICAL SYSTEMS; PLAN PATHS; PROBABILISTIC PATH PLANNING; PROBABILITY DENSITIES; REPRESENTATION MATRICES; STOCHASTIC MANNER; THREE-DIMENSIONAL (3D) SPACE;

ALGEBRA; AUTOMATION; CURRENT DENSITY; ESTIMATION; FOKKER PLANCK EQUATION; FOURIER TRANSFORMS; KINEMATICS; MATRIX ALGEBRA; MODELS; MOTION PLANNING; PROBABILITY; RECONNAISSANCE AIRCRAFT; STOCHASTIC PROGRAMMING; THEOREM PROVING; THREE DIMENSIONAL; TIMING JITTER; TRAJECTORIES; WAVE FUNCTIONS;

EQUATIONS OF MOTION;

EID: 47949133348     PISSN: 02635747     EISSN: 14698668     Source Type: Journal    
DOI: 10.1017/S0263574708004475     Document Type: Article

References (84)

1. D. H. Titterton and J.L.Weston, Strapdown Inertial Navigation Technology (Peter Peregrinus Ltd., London, UK, 1997).
2. A. Lawrence, Modern Inertial Technology (Springer-Verlag, New York, USA, 1993).
3. C. Jekeli, Inertial Navigation Systems with Geodetic Applications (Walter de Giny ter, Berlin; New York, 2001).
4. A. Makadia and K. Daniilidis, "Rotation estimation from spherical images," IEEE Trans. Pattern Anal. Mach. Intell. 28, 1170-1175 (2006).
5. E. Bayro-Corrochano and Y. Zang, "The motor extended Kalman filter a geometric approach for rigid motion estimation," J. Math. Imaging and Vis. 13, 205-228 (2000).
6. G. S. Chirikjian and A. B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis (CRC Press, Boca Raton, FL, 2001).
7. I. M. Gelfand, R. A. Minios and Z. Ya. Shapiro, Representations of the Rotation and Lorentz Groups and their Applications (Macmillan, New York, 1963).
8. D. Gurarie, Symmetry and Laplacians: Introduction to Harmonic Analysis, Group Representations and Applications (Elsevier Science Publisher, The Netherlands, 1992).
9. G. W. Mackey, The Theory of Unitary Group Representations (The University of Chicago Press, Chicago, 1976).
10. D. K. Maslen, Fast Transforms and Sampling for Compact Groups (Ph.D. Dissertation, Department of Mathematics, Harvard University, May 1993).
11. D. K. Maslen and D. N. Rockmore, "Generalized FFTs - a survey of some recent results," DIMACS Series Discrete Math. Theor. Comput. Sci. 28, 183-237 (1997).
12. W. Miller, Lie Theory and Special Functions (Academic Press, New York, 1968).
13. W. Miller, Jr., "Some applications of the representation theory of the Euclidean group in three-space," Commun. Pure App. Math. 17, 527-540 (1964).
14. M. A. Naimark, Linear Representations of the Lorentz Group (Macmillan, New York, 1964).
15. F. Peter and H. Weyl, "Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe," Math. Ann. 97, 735-755 (1927).
16. Sugiura, M., Unitary Representations and Harmonic Analysis, 2nd ed. (North-Holland, Amsterdam, 1990).
17. M. E. Taylor, Noncommutative Harmonic Analysis (Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1986).
18. D. A. Varshalovich, A. N. Moskalev and V. K. Khersonskii, Quantum Theory of Angular Momentum (World Scientific, Singapore, 1988).
19. N. J. Vilenkin and A. U. Klimyk, Representation of Lie Groups and Special Functions, vols. 1-3 (Kluwer Academic Publishers, Dordrecht, Holland 1991).
20. E. Wigner, "On Unitary Representations of the Inhomogeneous Lorentz Group," Ann. Math. 40(1), 149-204 (1939).
21. Ď. P. Zelobenko, Compact Lie Groups and their Representations (Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1973).
22. P. T. Kim, "Deconvolution density estimation on SO(N)," Ann. Stat. 26(3), 1083-1102 (1998).
23. B. W. Silverman, "Kernel density estimation using the fast Fourier transform," Appl. Stat. 31, 93-99 (1982).
24. H. Hendriks, "Nonparametric estimation of a probability density on a Riemannian manifold using Fourier expansions," Ann. Stat. 18(2), 832-849 (1990).
25. B. W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman and Hall, London, 1985).
26. L. I. Aramanovitch, "Quaternion nonlinear filter for estimation of rotating body attitude," Math. Methods Appl. Sci. 18(15), 1239-1255 (1995).
27. I. Y. Bar-Itzhack and Y. Oshman, "Attitude determination from vector observations - quaternion estimation," IEEE Trans. Aerosp. Electron. Syst. 21(1), 128-136 (1985).
28. I. Y. Bar-Itzhack and M. Idan, "Recursive attitude determination from vector observations-Euler angle estimation," J. Guid. Control Dyn. 10(2), 152-157 (1987).
29. W. K. Burns, ed., Optical Fiber Rotation Sensing (Academic Press, Boston, 1994).
30. D. G. Carta and D. H. Lackowski, "Estimation of orthogonal transformations in strapdown inertial systems," IEEE Trans. Autom. Control AC-17, 97-100 (1972).
31. S. Chaudhuri and S.S. Karandikar, "Recursive methods for the estimation of rotation quaternions," IEEE Trans. Aerosp. Electron. Syst. 32(2), 845-854 (1996).
32. H. Comisel, M. Forster, E. Georgescu, M. Ciobanu, V. Truhlik and J. Vojta, "Attitude estimation of a near-earth satellite using magnetometer data," Acta Astronautica 40(11), 781-788 (1997).
33. J. L. Crassidis, E. G. Lightsey and F. L. Markley, "Efficient and optimal attitude determination using recursive global positioning system signal operations," J. Guid. Control Dyn. 22(2), 193-201 (1999).
34. J. L. Crassidis and F. L. Markley, "Predictive filtering for attitude estimation without rate sensors," J. Guid. Control Dyn. 20(3), 522-527 (1997).
35. E. J. Lefferts, F. L. Markley and M. D. Shuster, "Kalman filtering for spacecraft attitude estimation," J. Guid Control Dyn. 5(5), 417-429 (1982).
36. T. E. Duncan, "AnEstimation problemin compact Lie groups," Syst. Control Lett. 10, 257-263 (1998).
37. J. T. -H. Lo and L. R. Eshleman, "Exponential Fourier densities on SO(3) and optimal estimation and detection for rotational processes," SIAM J. Appl. Math. 36(1), 73-82 (1979).
38. J. T.-H. Lo, "Optimal estimation for the satellite attitude using star tracker measurements," Automatica 22(4), 477-482 (1986).
39. Y. Oshman and F. L. Markley, "Minimal-parameter attitude matrix estimation from vector observations," J. Guid. Control Dyn. 21(4), 595-602 (1998).
40. Y. Oshman and F. L. Markley, "Sequential attitude and attitude-rate estimation, using integrated-rate parameters," J. Guid. Control Dyn. 22(3), 385-394 (1999).
41. F. C. Park, J. Kim and C. Kee, "Geometric descent algorithms for attitude determination using the global positioning system," J. Guid. Control Dyn. 23(1), 26-33 (2000).
42. P. G. Savage, "Strapdown inertial navigation integration algorithm design part 1: Attitude algorithms," J. Guid. Control Dyn. 21(1), 19-28 (1998).
43. P. G. Savage, "Strapdown inertial navigation integration algorithm design part 2: Velocity and position algorithms," J. Guid. Control Dynamics 21(2), 208-221 (1998).
44. M. D. Shuster and S. D. Oh, "Three-axis attitude determination form vector observations," J. Guid. Control Dyn. 4(1), 70-77 (1981).
45. R. B. Smith, ed., Fiber Optic Gyroscopes, SPIE Milestone Series, vol. MS 8 (SPIE Optical Engineering Press, Washington, DC, 1989).
46. A. S. Willsky, "Some Estimation Problems on Lie Groups" Geometric Methods in System Theory (D. Q. Mayne and R. W. Brockett, eds.) (Reidel Publishing Company, Dordrecht-Holland, 1973) pp. 305-313.
47. A. S. Willsky, Dynamical Systems Defined on Groups: Structural Properties and Estimation (PhD Dissertation, Department of Aeronautics and Astronautics, MIT, 1973).
48. H. E. Moses, "Irreducible representations of the rotation group in terms of the axis and angle of rotation," Ann. Phys. 42, 343-346 (1967).
49. H. E. Moses, "Irreducible representations of the rotation group in terms of the axis and angle of rotation," Ann. Phys. 37, 224-226 (1966).
50. H. E. Moses, "Irreducible representations of the rotation group in terms of Euler's theorem," Il Nuovo Cimento 40(4), 1120-1138 (1965).
51. D. Applebaum and H. Kunita, "Levy flows on manifolds and Levy processes on Lie groups," J. Math. Kyoto. Univ. 33/34, 1103-1123 (1993).
52. R. W. Brockett, "Notes on Stochastic Processes on Manifolds," In Systems and Control in the Twenty-First Century (C. I. Byrnes et al. eds.) (Birkhäuser, Boston, 1997) pp. 75-101.
53. A. D. Fokker, "Die Mittlere Energie rotierender elektrischer Dipole in Strahlungs Feld," Ann. Physik. 43, 810-820 (1914).
54. C. W. Gardiner, Handbook of Stochastic Methods, 2nd ed. (Springer-Verlag, Berlin, 1985).
55. K. Itô, "Brownian Motions in a Lie Group," Proceedings of the Japan Academy, vol. 26, 4-10 (1950).
56. K. Itô, "Stochastic differential equations in a differentiable manifold," Nagoya Math. J. 1, 35-47 (1950).
57. K. Itô, "Stochastic differential equations in a differentiable manifold (2)," Sci. Univ. Kyoto Math., Series A 28(1), 81-85 (1953).
58. H. Kunita, Stochastic Flows and Stochastic Differential Equations (Cambridge University Press, Cambridge, UK, 1997).
59. J. McConnell, Rotational Brownian Motion and Dielectric Theory (Academic Press, New York, 1980).
60. H. P. McKean, Jr., "Brownian motions on the 3-dimensional rotation group," Mem. Coll. Sci. Univ. Kyoto, Series A 33(1), 2538 (1960).
61. M. Planck, "Uber einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie," Sitz. ber. Berlin A Akad. Wiss. 1917, 324-341 (1917).
62. H. Risken, The Fokker-Planck Equation, Methods of Solution and Applications, 2nd ed. (Springer-Verlag, Berlin, 1989).
63. P. H. Roberts and H. D. Ursell, "Random, walk on a sphere and on a Riemannian manifold," Phil. Trans. R Soc. London A252, 317-356 (1960).
64. N. G. van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, Amsterdam, 1981).
65. K. Yosida, "Integration of Fokker-Planck's equation in a compact Riemannian space," Arkiv für Matematik 1(9), 71-75 (1949).
66. D. J. Higham, "An algorithmic introduction to numerical simulation of stochastic differential equations," SIAM Rev. 43(3), 525-546 (2001).
67. R. M. Murray, Z. Li and S. S. Sastry, A Mathematical Introduction to Robotic Manipulation (CRC Press, Ann Arbor MI, 1994).
68. P. Smith, T. Drummond and K. Roussopoulos, "Computing MAP Trajectories by Representing, Propagating and Combining PDFs over Groups," Proceedings of the 9th IEEE International Conference on Computer Vision, vol. 2, Nice, France (2003) pp. 1275-1282.
69. S. Su and C. S. G. Lee, "Manipulation and propagation of uncertainty and verification of applicability of actions in assembly tasks," IEEE Trans. Syst. Man Cybern. 22(6), 1376-1389 (1992).
70. Y. Zhou, G. S. Chirikjian, "Probabilistic Models of Dead-Reckoning Error in Nonholonomic Mobile Robots," Proceedings of ICRA'03, Taipei, Taiwan (Sep. 14-19, 2003) pp. 1594-1599.
71. Y. Wang and G. S. Chirikjian, "Error propagation on the Euclidean group with applications to manipulator kinematics," IEEE Trans. Robot. 22(4), 591-602 (2006).
72. Y. Wang and G. S. Chirikjian, "Worksapce generation of hyperredundant manipulators as a diffusion process on SE(N)," IEEE Trans. Robot. Autom. 20(3), 339-408 (2004).
73. Y. Wang and G. S. Chirikjian, "A New Potential Field Method for Robot Path Planning," Proceedings of ICRA'00, San Francisco, CA (Apr. 2000) pp. 977-982.
74. H. Choset, W. Burgard, S. Hutchinson, G. Kantor, L. E. Kavraki, K. Lynch and S. Thrun, Principles of Robot Motion: Theory, Algorithms, and Implementation (MIT Press, Cambridge, MA, USA, 2005).
75. G. S. Chirikjian and I. Ebert-Uphoff, "Numerical convolution on the Euclidean group with applications to workspace generation," IEEE Trans. Robot. Autom. 14(1), 123-136 (1998).
76. I. Ebert-Uphoff and G. S. Chirikjian, "Inverse Kinematics of Discretely Actuated Hyper-Redundant Manipulators Using Workspace Densities," Proceedings of ICRA'96, Minneaoplis, MN, pp. 139-145. Minneapolis, MN, USA.
77. R. Mason and J. W. Burdick, "Trajectory Planning Using Reachable-State Density Functions," 2002 IEEE International Conference on Robotics and Automation, Washington, DC (May 2002) pp. 273-280.
78. A. B. Kyatkin and G. S. Chirikjian, "Computation of robot configuration and workspaces via the Fourier transform on the discrete motion group," Int. J. Robot. Res. 18(6), 601-615 (1999).
79. Z. Li and J. F. Canny, eds., Nonholonomic Motion Planning (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993).
80. J. -P. Laumond, ed., Robot Motion Planning and Control (Springer, New York, USA, 1998).
81. G. S. Chirikjian and J. W. Burdick, "A modal approach to hyper-redundant manipulator kinematics," IEEE Trans. Robot. Autom. 10, 343-54 (1994).
82. R. J .Webster III, J. S. Kim, N. J. Cowan, G. S. Chirikjian and A. M. Okamura, "Nonholonomic modeling of needle steering," Int. J. Robot. Res. 25, 509-525 (2006).
83. W. Park, J. S. Kim, Y. Zhou, N. J. Cowan, A. M. Okamura and G. S. Chirikjian, "Diffusion-Based Motion Planning for a Nonholonomic Flexible Needle Model," IEEE International Conference on Robotics and Automation (2005) pp. 4611-4616. Barcelona, Spain.
84. R. Alterovitz, T. Simeon and K. Goldberg, "The stochastic motion roadmap: A sampling framework for planning with Markov motion uncertainty," Proc. Robotics: Science and Systems III, June 2007, Atlanta, CA.