Quantifier elimination for robot positioning with landmarks of uncertain position

We consider a mobile robot equipped with a camera that moves in a 2D environment. We aim to compute a domain for the pose (position and orientation) r = (x, y, θ) in which we are sure the robot is situated [1, 3]. The robot knows landmarks positions Xw(xw, yw) in the world frame, and measures their projection u in the image (represented in normalized coordinates, assuming known camera calibration parameters). The perspective projection of a world point in the camera frame is given as follows: { u = f(r, Xw) = 1 zc ((xw − x) cos(θ)− (yw − y) sin(θ)) zc = (xw − x) sin(θ) + (yw − y) cos(θ) > 0 (1)