  Motion planning is a fundamental problem and focuses on finding control
inputs that enable a robot to reach a goal region while safely avoiding
obstacles. However, in many situations, the state of the system may not be
known but only estimated using, for instance, a Kalman filter. This results in
a novel motion planning problem where safety must be ensured in the presence of
state estimation uncertainty. Previous approaches to this problem are either
conservative or integrate state estimates optimistically which leads to
non-robust solutions. Optimistic solutions require frequent replanning to not
endanger the safety of the system. We propose a new formulation to this problem
with the aim to be robust to state estimation errors while not being overly
conservative. In particular, we formulate a stochastic optimal control problem
that contains robustified risk-aware safety constraints by incorporating
robustness margins to account for state estimation errors. We propose a novel
sampling-based approach that builds trees exploring the reachable space of
Gaussian distributions that capture uncertainty both in state estimation and in
future measurements. We provide robustness guarantees and show, both in theory
and simulations, that the induced robustness margins constitute a trade-off
between conservatism and robustness for planning under estimation uncertainty
that allows to control the frequency of replanning.
