The paper considers the problem of controlling Connected and Automated Vehicles (CAVs) traveling through a three-entry roundabout so as to jointly minimize both the travel time and the energy consumption while providing speed-dependent safety guarantees, as well as satisfying velocity and acceleration constraints. We first design a systematic approach to dynamically determine the safety constraints and derive the unconstrained optimal control solution. A joint optimal control and barrier function (OCBF) method is then applied to efficiently obtain a controller that optimally track the unconstrained optimal solution while guaranteeing all the constraints. Simulation experiments are performed to compare the optimal controller to a baseline of human-driven vehicles showing effectiveness under symmetric and asymmetric roundabout configurations, balanced and imbalanced traffic rates and different sequencing rules for CAVs.
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