This paper addresses online modification of reference trajectories for multiple robots with collision avoidance. First, a smooth mapping dependent on the relative distances between the robots is proposed to coordinate collision-free motion from given reference trajectories. The trajectories modified by the mapping make the robots avoid collisions with each other, and are smooth enough to be tracked by the robots. Moreover, when the reference trajectories are separated from each other, the mapping is identical and the modified trajectories remain unchanged from the original ones. Next, a value function is considered consisting of the distances between the original and modified reference trajectories. From the viewpoint of minimizing the value function, a way to choose design parameters for the mapping is specified. Finally, a simulation for three mobile robots demonstrates the effectiveness of the proposed method.