## 1 Introduction
### Literature Review  
Heuristic strategies aim at adjusting the movements of vehicles to cross an intersection without collisions; they generally adopt the rule of first come, first served (FCFS). Typical heuristic strategies that have been proposed in the literature include resource reservation [6; 7], fuzzy logic [8; 9], and virtual platooning [10; 11]. For instance, [PERSON] et al. [6] proposed a centralized resource reservation method to coordinate vehicles at a signal-free intersection with a single straight lane in each direction. This work was further extended in [7], which considered turning and improved the performance of the central controller. The central controller frequently interacted with vehicles and scheduled the time slots and space tiles, which suffered heavy communication burdens and the risk of deadlocks. Note that the existing heuristic strategies only regulate vehicles to pass through intersections without conflicts, but are incapable of further optimizing the trajectories of vehicles. In addition to collision avoidance, the trajectory planning strategies drive vehicles to a better traffic performance. For example, [PERSON] et al. [12] presented a trajectory optimization algorithm to minimize the length of potential conflicting overlapping trajectories. Considering vehicle jitter, acceleration, and expected velocity, [PERSON] et al. [13] formulated a multi-objective trajectory optimization problem. By using a Monte Carlo tree search, [PERSON] et al. [14] planned the trajectories of vehicles through the global-optimal passing orders. Planning-based strategies have been shown to perform better than heuristic strategies under high traffic demand [15]. However, the trajectory planning strategies have a weakness in real-time practical implementation since the computation of the formulated problems suffers the curse of dimensionality. Several recent studies on the cooperation of vehicles at signal-free intersections introduced the concept of multi-agent, i.e., consensus, control [16; 17]. The coordinated vehicles are regulated to approach a desired state of motion to avoid collisions at intersections. For instance, [PERSON] et al. [18] proposed a coordinated optimization trajectory algorithm by formulating a mixed-integer non-linear program (MILP), which aims at forcing vehicles to reach consensus. This approach pushed the distributed vehicle-level solutions toward global optimality, but the formulated MILP problems suffered a heavy computational burden. [PERSON] et al. [11] presented a virtual consensus-based platoon control approach by transforming the two-dimensional vehicle cluster at the intersection into a one-dimensional virtual platoon in a virtual lane. However, this approach was not evaluated in the receding horizon framework. For those problems of vehicle collaboration at signal-free intersections, receding horizon control (RHC), also known as model predictive control (MPC) [19], has been utilized to dynamically capture predictions on vehicle states and future trajectories to optimize the coordination systems in real time. In the vehicle-intersection coordination framework proposed in [20], a conflict risk indicator was integrated into the MPC controller. [PERSON] et al. [21] formulated the corridor-level vehicle coordination problem in the RHC framework, where vehicle collision was avoided by using a consensus algorithm to force vehicles to track the reference speed. Note that the existing research on the cooperation of vehicles at signal-free intersections is only confined to the fully connected AV environment. Since the mixed driving environment containing AVs and MVs will stay with us for an extended period, it is es sential to establish control mechanisms for cooperating with both types of vehicles. In the current study, traffic signal lights still have to be used to regulate the mixed autonomy traffic flow at intersections [22; 23]. To jointly optimize the signal timing and vehicle trajectories, a complex MILP is constructed [24].