## 1 Introduction  
With the rapid increase in population and transportation demand, intersections have become major bottlenecks in urban road networks as well as one of the most dangerous types of urban traffic infrastructures. According to an estimate from the Texas A&M Transportation Institute (TTI), due to traffic congestion, the population in the United States will consume 10 billion more hours commuting and purchase an extra 3.6 billion gallons of fuel at a total congestion cost of $237 billion by 2025 [1]. Besides, it has been reported that 21.5% of fatalities and 40% of accidents in the United States occur at intersections [2]. In order to keep the balance between the growth in travel demand and the supply of transportation capacity, policy-makers and researchers have adopted various approaches to optimize road infrastructures and enhance road capacity in the past decades, including left-turn waiting areas [3], variable approach lanes [4], dynamic lane assignments [5], and integrated waiting areas [6]. Nevertheless, congestion has gotten worse over the last several years. The reason why these approaches failed to significantly alleviate congestion is that they only optimized the structure of road macroscopically but ignored the impact of microscopic driving behaviors and traffic orders. Actually, drivers' unnecessary acceleration and deceleration operations before reaching an intersection will cause speed fluctuations, which affect traffic efficiency, ride comfort, and energy consumption. Furthermore, the frequent start-stop cycle caused by traffic lights at signalized intersections will make drivers tired, agitated, or even angry and lead to a more aggressive and dangerous driving behavior called road rage, which may further aggravate congestion [7]. Recent technological advances in sensing, computing, and wireless communication have laid the foundation for the development of an approach based on Connected and Automated Vehicles (CAVs), which was named Cooperative, Connected, and Automated Mobility (CCAM) in Europe [8]. CAVs equipped with advanced onboard sensors, controllers, actuators, communicators, and other devices can perceive local and global environmental information and make intelligent decisions [9]. With the application of Vehicle-to-Everything (V2X) technologies, sensing and computing resources can be shared, and then vehicle information, including speed, acceleration, origin, and destination, can be obtained before the vehicle enters an intersection [10]. Furthermore, the behaviors and trajectories of vehicles can be optimized to coordinate with other vehicles for global traffic improvements [11]. It is expected that signal controllers will no longer be needed for the management and control at intersections in a future fully CAV-based environment [12; 13; 14]. With the development of the Cooperative Vehicle Infrastructure System (CVIS), CAVs can realize autonomous traffic organization and management to ensure driving safety, improve traffic efficiency, save energy, and reduce emissions [15; 16]. Therefore, a new traffic organization method called signal-free intersections comes into being. Challenges in such a new organization method include three aspects:  
1. First, traffic flows coming from all directions will enter the intersection at the same time, and their trajectories may form some conflict points in the central region. Therefore, the conflict resolution approach needs to guarantee that all vehicles can pass conflict points alternately rather than simultaneously. Furthermore, the number of vehicles entering the intersection at the same time will increase along with the traffic flow, which motivates us to consider resolution approaches suitable for different occasions, including two-vehicle collision and multi-vehicle collision. 2. Second, the spatio-temporal trajectory of CAVs needs to be optimized in the whole process, which starts at the approach road, goes through the intersection, and ends at the departure road.