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Volume 27 Issue 11-12 - Publication Date: 1 November 2008
 
Nonparametric Second-order Theory of Error Propagation on Motion Groups
 
Y. Wang Department of Mechanical Engineering, The College of New Jersey, Ewing, NJ 08628, USA and G. S. Chirikjian Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
 
Error propagation on the Euclidean motion group arises in a number of areas such as in dead reckoning errors in mobile robot navigation and joint errors that accumulate from the base to the distal end of kinematic chains such as manipulators and biological macromolecules. We address error propagation in rigid-body poses in a coordinate-free way. In this paper we show how errors propagated by convolution on the Euclidean motion group, SE(3), can be approximated to second order using the theory of Lie algebras and Lie groups. We then show how errors that are small (but not so small that linearization is valid) can be propagated by a recursive formula derived here. This formula takes into account errors to second order, whereas prior efforts only considered the first-order case. Our formulation is non-parametric in the sense that it will work for probability density functions of any form (not only Gaussians). Numerical tests demonstrate the accuracy of this second-order theory in the context of a manipulator arm and a flexible needle with bevel tip.
 
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