This paper addresses the problem of safely navigating a mobile robot with limited sensing capability and limited information about stationary obstacles. We consider two sensing limitations: blind spots between sensors and limited sensing range. We identify a set of constraints on the sensors' readings whose satisfaction at time t guarantees collision-freedom during the time interval [t, t+Δt]. Here Δt is a parameter whose value is bounded by a function of the maximum velocity of the robot and the range of the sensors. The constraints are obtained under assumptions about minimum internal angle and minimum edge length of polyhedral obstacles. We apply these constraints in the switching logic of the Simplex architecture to obtain a controller that ensures collision-freedom. Experiments we have conducted are consistent with these claims. To the best of our knowledge, our study is the first to provide runtime assurance that an autonomous mobile robot with limited sensing can navigate without collisions with only limited information about obstacles.