  This work explores the nature of augmented importance sampling in
safety-constrained model predictive control problems. When operating in a
constrained environment, sampling based model predictive control and motion
planning typically utilizes penalty functions or expensive optimization based
control barrier algorithms to maintain feasibility of forward sampling. In
contrast the presented algorithm utilizes discrete embedded barrier states in
augmented importance sampling to apply feedback with respect to a nominal state
when sampling. We will demonstrate that this approach of safety of discrete
embedded barrier states in augmented importance sampling is more sample
efficient by metric of collision free trajectories, is computationally feasible
to perform per sample, and results in better safety performance on a cluttered
navigation task with extreme un-modeled disturbances. In addition, we will
utilize the theoretical properties of augmented importance sampling and safety
control to derive a new bound on the free energy of the system.
