In recent years, lower limb exoskeleton has attracted extensive attention in academic and engineering research. In the rehabilitation motion training scenario, it is of concern that the exoskeleton should have the ability to control its own leg movements in order to assist a patient with natural and anthropomorphic gaits. A patient with paralysis in leg, is incapable of controlling and coordinating with the exoskeleton well enough to produce a desirable gait. Thus, a critical problem in this scenario is that the exoskeleton needs to track a desired gait in order for the patient to adopt and produce different walking patterns. To this end, this paper proposes an adaptive tracking controller for the exoskeleton. Although many adaptive control methods exist, a traditional control design utilizes time-triggered method, namely, the system data is periodically sampled and controller parameters are updated at specific time instances. To implement the update mechanism, in this paper, an aperiodically adaptive controller based on policy iteration is developed by integrating event triggering mechanism and reinforcement learning. Further, in order to achieve online learning and adaptation, an actor-critic neural network is employed, and a novel event triggered tuning law is designed to learn the controller signals. We conduct simulations and perform experiments with a lower limb rehabilitation exoskeleton robot. Experimental results demonstrate that the proposed event triggered control strategy can reduce control updates when compared with many traditional time triggered control methods, for a guaranteed control performance.