  The increasing complexity of modern robotic systems and the environments they
operate in necessitates the formal consideration of safety in the presence of
imperfect measurements. In this paper we propose a rigorous framework for
safety-critical control of systems with erroneous state estimates. We develop
this framework by leveraging Control Barrier Functions (CBFs) and unifying the
method of Backup Sets for synthesizing control invariant sets with robustness
requirements -- the end result is the synthesis of Measurement-Robust Control
Barrier Functions (MR-CBFs). This provides theoretical guarantees on safe
behavior in the presence of imperfect measurements and improved robustness over
standard CBF approaches. We demonstrate the efficacy of this framework both in
simulation and experimentally on a Segway platform using an onboard
stereo-vision camera for state estimation.
