In this paper, the distributed displacement-based formation and leaderless maneuver guidance control problems of multi-space-robot systems are investigated by introducing event-triggered control update mechanisms. A distributed formation and leaderless maneuver guidance control framework is first constructed, which includes two parallel controllers, namely, an improved linear quadratic regulator and a distributed consensus-based formation controller. By applying this control framework, the desired formation configuration of multi-space-robot systems can be achieved and the center of leaderless formation can converge to the target point globally. Second, a pull-based event triggering mechanism is introduced to the distributed formation controller, which makes the control update independent of the events of neighboring robots, avoids unnecessary control updates, and saves the extremely limited energy of space robots. Furthermore, the potential Zeno behaviors have been excluded by proving a positive lower bound for the inter-event times. Finally, numerical simulation verifies the effectiveness of the theoretical results.