Our robot was triggered by the motion of the preceding robot in front of the ultrasonic sensor mounted on the right side of ours. The preceding robot had a very low profile, so we mounted the sensor low to the ground.
Our robot then triggered the subsequent robot by driving into a touch sensor mounted on its rear. As a result, we constructed a front bumper of sorts for our robot to push the touch sensor with. Due to the instability caused by the narrow wheel base of the subsequent robot, we deliberately placed our robot just within range of it. This meant that it bumped the button without shoving the robot forward at all.