Topology of UAV ad hoc networks is more dynamic due to fast movement of nodes compared with other traditional ones. The performance of traditional routing algorithms is not able to satisfy efficient communication for multi-UAVs carrying out missions, and this paper proposes an on-demand routing algorithm based on link quality prediction. The link quality is chosen as a standard to choose route and calculated by sensing the stability of a link and the congestion level of a link. Meanwhile, relevant parameters are predicted by Grey-WNN prediction model, and then they are used to estimate the stability of a link and the congestion level of a link. After that, the link quality can be evaluated in advance. The process of routing discovery and routing maintenance is optimized according to the predicted link quality, thus avoiding the negative influence on swarm network caused by the high-dynamic characteristics of UAVs. The simulation results show that the algorithm improves the network performance obviously, decreases the number of route fracture, reduces the route overhead and average delay increases the packet delivery ratio when nodes are moving at a high speed.