The internet engineering task force working group has proposed IPv6 Routing Protocol for Low-power and Lossy (RPL) networks for the energy-constrained networks on the Internet of things. This protocol has limitations and does not pay particular attention to mobile nodes. We introduce a new method to improve the RPL protocol with the aim to provide continuous connectivity for mobile nodes. Using additional fields to RPl control packets, a parent is responsible for mobility detection and informs a mobile node for maintaining the energy of it and reduces network overheads. A mobile node chooses a new parent based on metrics namely expected transmission count, received signal strength indicator and residual energy, before disconnection from previous parent. Memory has also been defined that maintains a list of previous parents and when selecting a new parent, a mobile node gets help from this list to select stable parents. They used the Cooja simulator to evaluate the performance of the proposed protocol and the results compared with the RPL and mRPL from the aspect of packet delivery ratio, network overhead, end-to-end delay, energy consumption, and end-to-end throughput under different scenarios. The simulation results showed the effectiveness of the proposed method.