Some Useful Results for Closed-Form Propagation of Error in Vehicle Odometry

Abstract

Odometry can be modelled as a nonlinear dynamical system. The linearized error propagation equations for both deterministic and random errors in the odometry process have time varying coefficients and therefore may not be easy to solve. However, the odometry process exibits a property here called "commutable dynamics" which makes the transition matrix easy to compute. As a result, an essentially closed form solution to both deterministic and random linearized error propagation is available. Examination of the general solution indicates that error expressions depend on a few simple path functionals which are analogous to the moments of mechanics and equal to the first two coefficients of the power and Fourier series of the path followed. The resulting intuitive understanding of error dynamics is a valuable tool for many problems of mobile robotics. Required sensor performance can be computed from tolerable error, trajectories can be designed to minimize error for operation or to maximize it for calibration and evaluation purposes. Optimal estimation algorithms can be implemented in nearly closed form for small footprint embedded applications, etc.