Technical Brief

Stochastic Optimal Motion Planning for the Attitude Kinematics of a Rigid Body With Non-Gaussian Uncertainties

[+] Author and Article Information
Taeyoung Lee

Assistant Professor
Department of Mechanical and Aerospace Engineering,
George Washington University,
Washington, DC 20052
e-mail: tylee@gwu.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received January 31, 2014; final manuscript received June 6, 2014; published online October 21, 2014. Assoc. Editor: Jongeun Choi.

J. Dyn. Sys., Meas., Control 137(3), 034502 (Oct 21, 2014) (7 pages) Paper No: DS-14-1054; doi: 10.1115/1.4027950 History: Received January 31, 2014; Revised June 06, 2014

This paper investigates global uncertainty propagation and stochastic motion planning for the attitude kinematics of a rigid body. The Fokker–Planck equation on the special orthogonal group is numerically solved via noncommutative harmonic analysis to propagate a probability density function along flows of the attitude kinematics. Based on this, a stochastic optimal control problem is formulated to rotate a rigid body while avoiding obstacles within uncertain environments in an optimal fashion. The proposed intrinsic, geometric formulation does not require the common assumption that uncertainties are Gaussian or localized. It can be also applied to complex rotational maneuvers of a rigid body without singularities in a unified way. The desirable properties are illustrated by numerical examples.

Copyright © 2015 by ASME
Your Session has timed out. Please sign back in to continue.


Donev, A., Torquato, S., and Stillinger, F., 2005, “Neighbor List Collision-Driven Molecular Dynamics Simulation for Nonspherical Hard Particles. I. Algorithmic Details,” J. Comput. Phys., 202(2), pp. 737–764. [CrossRef]
LaValle, S., 2006, Planning Algorithms, Cambridge University Press, Cambridge, UK.
LaValle, S., and Sharma, R., 1995, “A Framework for Motion Planning in Stochastic Environments: Modeling and Analysis,” Proceedings of the IEEE Conference on Robotics and Automation, pp. 3057–3062. [CrossRef]
LaValle, S., and Sharma, R., 1995, “A Framework for Motion Planning in Stochastic Environments: Applications and Computational Issues,” Proceedings of the IEEE Conference on Robotics and Automation, pp. 3063–3068. [CrossRef]
Alterovitz, R., Simeon, T., and Goldberg, K., 2007, “The Stochastic Motion Roadmap: A Sampling Framework for Planning With Markov Motion Uncertainty,” Robotics: Science and Systems, MIT Press, pp. 246–253.
Kalakrishnan, M., Chitta, S., Theodorou, E., Paster, P., and Schaal, S., 2011, “STOMP: Stochastic Trajectory Optimization for Motion Planning,” Proceedings of the IEEE Conference on Robotics and Automation, pp. 4569–4574.
Bullo, F., and Lewis, A., 2005, “Modeling, Analysis, and Design for Simple Mechanical Control Systems,” Geometric Control of Mechanical Systems (Texts in Applied Mathematics), Vol. 49, Springer-Verlag, New York.
Bhat, S., and Bernstein, D., 2000, “A Topological Obstruction to Continuous Global Stabilization of Rotational Motion and the Unwinding Phenomenon,” Syst. Control Lett., 39(1), pp. 66–73. [CrossRef]
Ge, Q., and Ravani, B., 1994, “Computer Aided Geometric Design of Motion Interpolants,” ASME J. Mech. Des., 116(3), pp. 756–762. [CrossRef]
Srinivasan, L., and Ge, Q., 1998, “Fine Tuning of Rational b-Spline Motions,” ASME J. Mech. Des., 120(1), pp. 46–51. [CrossRef]
Belta, C., and Kumar, V., 2005, “Geometric Methods for Multi-Robot Optimal Motion Planning,” Handbook of Geometric Computing, Springer, New York, pp. 679–706.
Park, W., Liu, Y., Zhou, Y., Moses, M., and Chirikjian, G., 2008, “Kinematic State Estimation and Motion Planning for Stochastic Nonholonomic System Using the Exponential Map,” Robotica, 26(4), pp. 419–434. [CrossRef] [PubMed]
Sugiura, M., 1990, Unitary Representations and Harmonic Analysis, Kodansha, Tokyo, Japan.
Kirillov, A., Soucek, V., and Neretin, Y., 1994, Representation Theory and Noncommutative Harmonic Analysis I: Fundamental Concepts, Springer, New York.
Diaconis, P., 1988, Group Representations in Probability and Statistics, Institute of Mathematical Statistics, Beachwood, OH.
Emery, M., 1989, Stochastic Calculus in Manifolds, Springer, New York.
Lo, J. T. H., and Eshleman, L. R., 1979, “Exponential Fourier Densities on SO(3) and Optimal Estimation and Detection for Rotational Processes,” SIAM J. Appl. Math., 36(1), pp. 73–82. [CrossRef]
Hendriks, H., 1990, “Nonparametric Estimation of a Probability Density on a Riemannian Manifold Using Fourier Expansions,” Ann. Stat., 18(2), pp. 832–849. [CrossRef]
Marsden, J., and Ratiu, T., 1999, Introduction to Mechanics and Symmetry (Texts in Applied Mathematics), 2nd ed., Vol. 17, Springer-Verlag, New York.
Chirikjian, G., 2012, Stochastic Models, Information Theory, and Lie Groups, Vol. 2, Birkhäuser, New York.
Chirikjian, G., and Kyatkin, A., 2001, Engineering Applications of Noncommutative Harmonic Analysis, CRC Press, Boca Raton, FL.
Peter, F., and Weyl, H., 1927, “Die Vollständigkeit der Primitiven Darstellungen Einer Geschlossenen Kontinuierlichen Gruppe,” Math. Ann., 97(1), pp. 735–755. [CrossRef]
Fisher, R., 1953, “Dispersion on a Sphere,” Proc. R. Soc. Lond. A., 217, pp. 295–305. [CrossRef]
Lee, T., Leok, M., and McClamroch, N. H., 2008, “Global Symplectic Uncertainty Propagation on SO(3),” Proceedings of the IEEE Conference on Decision and Control, pp. 61–66. [CrossRef]
Censi, A., Calisi, D., Luca, A., and Oriolo, G., 2008, “A Bayesian Framework for Optimal Motion Planning With Uncertainty,” Proceedings of the IEEE Conference on Robotics and Automation, pp. 1798–1805. [CrossRef]
Lawrence, T., Zhou, J., and Tits, A., 1994, “User's Guide for CFSQP Version 2.0: A C Code for Solving (Large Scale) Constrained Nonlinear (Minimax) Optimization Problems, Generating Iterates Satisfying All Inequality Constraints,” Institute for Systems Research, University of Maryland, College Park, MD, Technical Report No. TR-94-16.
Moses, H., 1966, “Irreducible Representations of the Rotation Group in Terms of the Axis and Angle of Rotation,” Ann. Phys., 37(2), pp. 224–226. [CrossRef]


Grahic Jump Location
Fig. 1

Attitude uncertainty propagation (diffusion)

Grahic Jump Location
Fig. 2

Attitude uncertainty propagation (diffusion and advection)

Grahic Jump Location
Fig. 3

Stochastic motion planning of a rigid body: (ε = 0.3)

Grahic Jump Location
Fig. 4

Stochastic motion planning of a rigid body: (ε = 0.1)




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In