This paper deals with the security robot motion planning in order to stealthily approach the backside of the invader based on an active prediction planning execution (APPE) strategy. The stealth navigation is needed in the security system because the invader will try to run away from the robot when it detects the robot. We propose an algorithm for making the robot to approach the invader stealthily within a desired range. We predict the long-term motion of the invader and plan the security robot motion by using detection map. It represents the region that the robot can move stealthily on a certain path. The robot motion can be separately planned as a geometric path and a speed profile using detection map. The path is planned on the predetermined roadmap. The speed profile is planned so that the robot is not detected by the invader. The simulation results demonstrate that our algorithm is efficient for shortening the distance between the robot and the invader when the invader first detects the robot. Our algorithm is compared with the case that does not consider the stealth condition of the robot and the grid-based method.