This paper presents LQG-Obstacles, a new concept that combines linear-quadratic feedback control of mobile robots with guaranteed avoidance of collisions with obstacles. Our approach generalizes the concept of Velocity Obstacles  to any robotic system with a linear Gaussian dynamics model. We integrate a Kalman filter for state estimation and an LQR feedback controller into a closed-loop dynamics model of which a higher-level control objective is the "control input". We then define the LQG-Obstacle as the set of control objectives that result in a collision with high probability. Selecting a control objective outside the LQG-Obstacle then produces collision-free motion. We demonstrate the potential of LQG-Obstacles by safely and smoothly navigating a simulated quadrotor helicopter with complex non-linear dynamics and motion and sensing uncertainty through three-dimensional environments with obstacles and narrow passages.