  In this paper, we consider the problem of tracking a reference trajectory for
a simplified car model based on unicycle kinematics, whose position only is
measured, and where the control input and the measurements are corrupted by
independent Gaussian noises. To tackle this problem we devise a novel
observer-controller: the invariant Linear Quadratic Gaussian controller (ILQG).
It is based on the Linear Quadratic Gaussian controller, but the equations are
slightly modified to account for, and to exploit, the symmetries of the
problem. The gain tuning exhibits a reduced dependency on the estimated
trajectory, and is thus less sensitive to misestimates. Beyond the fact the
invariant approach is sensible (there is no reason why the controller
performance should depend on whether the reference trajectory is heading west
or south), we show through simulations that the ILQG outperforms the
conventional LQG controller in case of large noises or large initial
uncertainties. We show that those robustness properties may also prove useful
for motion planning applications.
