## 2 Materials and Methods
### Decision-Making Model
#### 2.3.2 The Principle of Least Action  
The car can be thought of as an energy conservation mechanism during the ideal driving process, which is to say, disregarding road friction. The attraction potential energy G exists to provide an attractive force that prompts the vehicle to travel to the target point. As this potential energy G was released, the vehicle's kinetic energy gradually increased, where the release and absorption of energy were in accordance with the law of energy conservation. The concept of least action has been extensively used in the study of electromagnetic fields, gravitational fields, and classical dielectric mechanics. It is a fundamental variational principle for particles and continuous systems. The principle is first applied by [15] in the field of trajectory generation for autonomous driving. In [15], by imagining all possible trajectories that the system can imagine, computing the actions (functions of the trajectory) for each trajectory, finding the true dynamic trajectory of the system between the initial and final configurations in a specified time, and choosing an action that makes the actions partially stationary (traditionally called \"least\"), then the true trajectories are those with the fewest actions. If we use \(S\) to represent the action, the principle can be expressed as \(\delta S=0\).  
The primary objective of automated driving systems is to maximize gains and minimize losses, similar to the primary objective of human driving. Whereas gains stand in for effectiveness and safety, and losses for improper driving and violations. The minimum action can be described as follows if we use the kinetic energy of the vehicle along with the force and potential energy produced by interaction with other traffic participants as the safety criterion and the trip time as the efficiency requirement. The primary objective of automated driving systems is to maximize gains and minimize losses, similar to the primary objective of human driving. Whereas gains stand in for effectiveness and safety, and losses for improper driving and violations. The minimum action can be described as follows if we use the kinetic energy of the vehicle along with the force and potential energy produced by interaction with other traffic participants as the safety criterion and the trip time as the efficiency requirement.  
\[S=\int_{t_{0}}^{T}Ldt=\int_{t_{0}}^{T}(K-V)dt, \tag{21}\]  
where \(K\) and \(V\) stand for the vehicle's kinetic and potential energy, respectively, and L is the Lagrangian quantity. Ref. [12] defines an accident as an abnormal transfer of a vehicle's potential energy, which is followed by the vehicle's elastic and plastic deformation. As a result, the vehicle's kinetic energy can be thought of as its own risk, and the process of accelerating the vehicle from a stop to a particular speed can be thought of as the accumulation of the vehicle's risk. The relationship between the vehicle and the goal pointas well as the relationship between the vehicle and the surrounding traffic environment are both included in the potential energy \(V\). As an illustration, Figure 4 illustrates how, in the adjacent lane, the potential energy associated with the target point always decreases while the kinetic energy increases. In contrast, the potential energy associated with the vehicle and the surrounding traffic environment increases and then decreases as the risk field state changes as a result of oncoming traffic coming from the opposite direction, while the kinetic energy appropriately releases the energy before absorbing it. Additionally, the times \(t_{0}\) and \(T\), which in this calculation stand in for the start and finish times, respectively, signify the efficiency of the passage as well as the accumulation of risk. Therefore, the safer and more effective the driving procedure, the less the value of \(S\).