In the case of disasters such as earthquakes or Nuclear/Biological/Chemical(NBC) terrorist attacks, mobile robots, called “rescue robots,” that can work in dangerous environments instead of rescue crews in rescue missions, can be of great help. However, realizing such robot systems requires many types of technologies. In particular, path planning is an important technology that provides a mobile robot with autonomous navigation to a target destination with collision avoidance. To avoid evacuees, the robot should consider the motion of people in the near future. In this research, we propose a collision avoidance method that estimates the motions and personal spaces of the evacuees. The method consists of three steps: “estimation,” “conversion,” and “planning.” In the estimation step, the future positions of evacuees are estimated by considering their planned motions and personal spaces. Then, in the conversion step, a time axis is added to construct a 3D time-space coordinate system. Finally, in the planning step, a distance-time transform is applied to plan a safe 3D path from the robot's current position to the desired goal. The proposed method has been implemented on our rescue robot simulator, and some simulation experiments were conducted to verify its usefulness.