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Connected and automated vehicles (CAVs) are gaining momentum, especially in the potential to improve road safety and reducing energy consumption and emissions. Lane-change maneuver is one of the most important conventional parts of automated driving. We address the problem of optimally CAVs to accomplish an automated lane-change and eliminate potential collision during the lane-change process on a curved road. Drivers’ safety, comfort, convenience, and fuel economy are also engaged in trajectory planning. We assume that the centripetal motion displacement and the rotational angular displacement meet the requirement of odd-order polynomial constrains. Then, the polynomial coefficient of the trajectory can be reduced and the mathematical model of virtual trajectory for lane-change can be designed based on the models of centripetal displacement and angular displacement by applying the above constrains and boundary conditions.