A new concept of avoidance map is introduced which can effectively execute collision avoidance among UAVs. It is based upon representation of the effect of control effort by pairs of UAVs on achievement of avoidance, thus partitioning the control effort space into avoidance and collision regions. To do this, it is demonstrated that collision avoidance can be carried out by applying constant acceleration control inputs to UAVs. Also, the duration for which a UAV needs to execute avoidance maneuver can be determined by monitoring the location of the origin coordinates of the control effort space on the partitioned region. The idea is developed in an intuitive way and is extended to multiple UAVs. Several examples are given to demonstrate the simplicity and effectiveness of the concept.