In this paper, a new model predictive control method for position regulation and vibration suppression in electrical drives with elastic transmission, is presented. The control methodology enables the drive's safety and physical limitations to be directly incorporated into control synthesis so that potential future constraint violations are anticipated and prevented. The proposed predictive position controller is realized in an explicit form allowing easy implementation on control platforms with relatively low power and memory demands. The simulation results show that the controller is very effective in regulating shaft position for a wide-range of operating points and load disturbances while respecting the drive's torque and speed constraints thus avoiding possibly dangerous and/or harmful situations during operation. Preliminary experimental results confirm the simulation study.