This is a long research that I started in 1999. Specifically, at that time, my goal was to derive an analytic expression of the covariance matrix that characterizes the odometry error of a wheeled robot in terms of the traveled trajectory and starting by an odometry error model. This problem was approached at the end of 90's by the robotics community and only solved by introducing approximations on the robot motion. In 2001 I found, for the first time, the exact analytic expression. The covariance matrix can be expressed as a double integral on the trajectory (the trajectory as computed by the odometry, hence available from the measurements). In particular, the trajectory is expressed as the scalar function that provides the orientation in terms of the curve length. The main results can be found here. Later, starting from 2013, I realized that this computation is much more general and has a fundamental physics interest. Specifically, what I computed in that work was the analytic expression of the statistics up to the second order of an overdamped 2D Brownian motion under the unicycle constraint. I was captured by the following challenge: extend my former computation to include the computation of the statistics up to any order. From a mathematics point of view, this is really a hard challenge. It corresponds to solve a Fokker-Plank equation where the detailed balance is not satisfied. Note that I did not introduce any hypothesis on the trajectory (e.g., the trivial case of circular motion, for which analytic solutions can be found by using standard approaches). I proceeded by a direct integration of the Langevin equation, by using several tricks borrowed from the Feynman path integral to quantum mechanics. Each moment of the statistics is expressed as a multiple integral of the deterministic motion (i.e., the known motion that would result in absence of noise). For the special case when the ratio between the linear and angular speed is constant, the multiple integrals can be easily solved and expressed as the real or the imaginary part of suitable analytic functions. All the analytic results and derivations can be found here and here.