  Useful robot control algorithms should not only achieve performance
objectives but also adhere to hard safety constraints. Control Barrier
Functions (CBFs) have been developed to provably ensure system safety through
forward invariance. However, they often unnecessarily sacrifice performance for
safety since they are purely reactive. Receding horizon control (RHC), on the
other hand, consider planned trajectories to account for the future evolution
of a system. This work provides a new perspective on safety-critical control by
introducing Forward Invariance in Trajectory Spaces (FITS). We lift the problem
of safe RHC into the trajectory space and describe the evolution of planned
trajectories as a controlled dynamical system. Safety constraints defined over
states can be converted into sets in the trajectory space which we render
forward invariant via a CBF framework. We derive an efficient quadratic program
(QP) to synthesize trajectories that provably satisfy safety constraints. Our
experiments support that FITS improves the adherence to safety specifications
without sacrificing performance over alternative CBF and NMPC methods.
