  Modern nonlinear control theory seeks to develop feedback controllers that
endow systems with properties such as safety and stability. The guarantees
ensured by these controllers often rely on accurate estimates of the system
state for determining control actions. In practice, measurement model
uncertainty can lead to error in state estimates that degrades these
guarantees. In this paper, we seek to unify techniques from control theory and
machine learning to synthesize controllers that achieve safety in the presence
of measurement model uncertainty. We define the notion of a Measurement-Robust
Control Barrier Function (MR-CBF) as a tool for determining safe control inputs
when facing measurement model uncertainty. Furthermore, MR-CBFs are used to
inform sampling methodologies for learning-based perception systems and
quantify tolerable error in the resulting learned models. We demonstrate the
efficacy of MR-CBFs in achieving safety with measurement model uncertainty on a
simulated Segway system.
