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Unmanned aerial vehicles (UAVs) are anticipated to be widely deployed in future wireless communications, due to their advantages of high mobility and easy deployment. However, the broadcast nature of air-to-ground line-of-sight wireless channels brings a new challenge to the information security of UAV-ground communication. This paper tackles such a challenge in the physical layer by exploiting the mobility of UAV via its trajectory design. We consider a UAV-ground communication system with multiple potential eavesdroppers on the ground, where the information on the locations of the eavesdroppers is imperfect. We formulate an optimization problem, which maximizes the average worst case secrecy rate of the system by jointly designing the robust trajectory and transmit power of the UAV over a given flight duration. The nonconvexity of the optimization problem and the imperfect location information of the eavesdroppers make the problem difficult to be solved optimally. We propose an iterative suboptimal algorithm to solve this problem efficiently by applying the block coordinate descent method, $\mathcal {S}$ -procedure, and successive convex optimization method. Simulation results show that the proposed algorithm can improve the average worst case secrecy rate significantly, as compared to two other benchmark algorithms without robust design. [ABSTRACT FROM AUTHOR]