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

Published online by Cambridge University Press:  01 July 2008

Wooram Park ,
Yan Liu ,
Yu Zhou ,
Matthew Moses  and
Gregory S. Chirikjian
Wooram Park
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA. E-mail: wpark7@jhu.edu
Yan Liu
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA. E-mail: wpark7@jhu.edu
Yu Zhou
Affiliation:
Department of Mechanical Engineering, Stony Brook University, Stony Brook, New York 11794-2300
Matthew Moses
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA. E-mail: wpark7@jhu.edu
Gregory S. Chirikjian*
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA. E-mail: wpark7@jhu.edu
*
*Corresponding author. E-mail: greg@jhu.edu
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Summary

A nonholonomic system subjected to external noise from the environment, or internal noise in its own actuators, will evolve in a stochastic manner described by an ensemble of trajectories. This ensemble of trajectories is equivalent to the solution of a Fokker–Planck equation that typically evolves on a Lie group. If the most likely state of such a system is to be estimated, and plans for subsequent motions from the current state are to be made so as to move the system to a desired state with high probability, then modeling how the probability density of the system evolves is critical. Methods for solving Fokker-Planck equations that evolve on Lie groups then become important. Such equations can be solved using the operational properties of group Fourier transforms in which irreducible unitary representation (IUR) matrices play a critical role. Therefore, we develop a simple approach for the numerical approximation of all the IUR matrices for two of the groups of most interest in robotics: the rotation group in three-dimensional space, SO(3), and the Euclidean motion group of the plane, SE(2). This approach uses the exponential mapping from the Lie algebras of these groups, and takes advantage of the sparse nature of the Lie algebra representation matrices. Other techniques for density estimation on groups are also explored. The computed densities are applied in the context of probabilistic path planning for kinematic cart in the plane and flexible needle steering in three-dimensional space. In these examples the injection of artificial noise into the computational models (rather than noise in the actual physical systems) serves as a tool to search the configuration spaces and plan paths. Finally, we illustrate how density estimation problems arise in the characterization of physical noise in orientational sensors such as gyroscopes.

Keywords

State estimationLie groupsnonholonomic motion planningexponential map
Type
Article
Information
Robotica , Volume 26 , Special Issue 4: Geometry in Robotics: Sensing , July 2008 , pp. 419 - 434
Copyright
Copyright © Cambridge University Press 2008

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