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This paper focuses on the cooperative driving challenges of connected and automated vehicles (CAVs) at single-lane roundabouts. First, a geometric path planning method is proposed for CAVs navigating a single-lane roundabout. Based on this method, a vehicle roundabout model is established. Four potential traffic scenarios for CAVs are established, and the optimal arrival times at conflict points are analyzed. By correlating the optimal arrival times at conflict points with the optimal entry times into the roundabout, the multi-vehicle coordination problem in complex intersections is simplified to a speed control issue during entry. Utilizing the principles of optimal control and Pontryagin minimization, two speed optimization strategies are proposed. Finally, MATLAB is employed for simulation analysis. The results indicate that the control strategy proposed in this paper enables the system to clearly identify potential conflicts between vehicles and implement an optimal control strategy, ensuring that vehicles can navigate the roundabout efficiently in terms of time and fuel without collisions. Additionally, the minimum time interval is established at 0.2 seconds to completely prevent vehicle collisions. In this study, the fusion problem involving two vehicles at a single conflict point is further expanded to encompass multiple vehicles at multiple conflict points. Thus, the efficient scheduling of multiple vehicles in single-lane roundabouts is realized.