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This paper investigates the issue of robust relative orbit synchronization control for spacecraft cluster subjects to space perturbation. First, the relative orbit dynamics of the leader-follower spacecraft cluster is established. Combined with P-type learning algorithm and sliding mode control (SMC) strategy, a novel distributed learning SMC (LSMC) strategy with the consensus algorithm is explored such that all the follower spacecrafts converge to the target relative position and their velocities tend to the leader spacecraft. The distributed P-type learning algorithm is used to update and compensate space perturbation for all the follower spacecrafts. The proposed LSMC approach has the potential to achieve higher control accuracy than the traditional SMC one while it has the lower computation burden than the distributed adaptive SMC one. The stability analysis of the proposed distributed LSMC strategy is provided in detail. Finally, a numerical example is simulated to illustrate the effectiveness and superiority of the proposed distributed LSMC-based relative orbit synchronization control for spacecraft cluster. [ABSTRACT FROM AUTHOR]