## 4 Speed Guidance Strategy Considering the Uncertainty of Leading Vehicles
### Optimal Speed Adjustment Method under Uncertain Disturbances  
During the speed guidance process, uncertainties introduced by HDVs necessitate explicit consideration of queuing at intersections and disturbances caused by the lead HDV. Relying solely on predictions of their crossing decisions at intersections is insufficient to derive the optimal guided speeds for the corresponding CAVs. Hence, this study employs optimal control theory and a rolling optimization model to refine the process, yielding target speeds tailored to these exceptional circumstances.  
2. Velocity Rolling Optimization under Leading Vehicle Disturbances Ahead  
When a CAV approaches an intersection and there is an HDV ahead of it (as depicted in Figure 7), uncertainties inherent in the behavior of the HDV prevent the CAV from accurately predicting the HDV trajectory or the time it will take to clear the intersection. Consequently, the CAV movement is constrained by the actions of the leading vehicle. In this scenario, the CAV must overcome the dynamic disturbances caused by the preceding HDV and adopt a path constraint that differs from those applied under queuing conditions. Specifically, the path constraint in this context is dynamically time-varying, implying:  
\[x_{i}(t)\leq x_{i-1}(t)+x_{d} \tag{30}\]  
where: \(x_{i}(t)\) is the real-time location of the current CAV and \(x_{i-1}(t)\) is the real-time position of the front vehicle of the current CAV \(i-1\) (the vehicle \(i-1\) is a driving vehicle).  
Since some of the dynamic path constraints \(x_{i-1}(t)+x_{d}\) cannot be learned in advance, to determine the optimal speed trajectory of the CAV in this driving scenario, the control strategy was optimized through the speed rolling time domain in this study, and the principle is shown in Figure 9.  
Figure 8: Example of the queuing path constraints.  
Figure 9: Schematic of the rolling time domain optimization.  
In this study, the idea of rolling time domain optimization for dynamic planning of the target vehicle speed trajectory, as shown in Figure 9, is used. Each moment \(t_{i}^{n}\) is determined for a long time period \(\left[t_{i}^{n},t_{j}^{n}\right]\) as a time window, each short time period \(t_{s}^{n}\) is determined as a sub-time window, the optimal control strategy at this time \(t_{s}^{n}\) is determined and executed, and the calculation process is repeated (\(t_{i}^{n+1}=t_{i}^{n}+t_{s}^{n}\)) in the next moment, executed in the corresponding range until the CAV passes through the parking line.  
In order to ensure the safety of the driving scenario, assuming that there is a CAV \(i\) and artificial driving on an HDV in the moment \(t_{i}^{n}\) and CAV \(i\), and that they have the same speed and position (\(v_{i}^{n},x_{i}^{n}\)), by the second chapter, the ACC is improved using a car-following model to determine its speed track \(A_{i}^{n}(t)\). If the trajectory \(A_{i}^{n}(t)\) above the trajectory \(R_{i}^{n}(t)\) is set \(R_{i}^{n}(t)\) to the CAV speed trajectory limit, on the contrary, the trajectory \(R_{i}^{n}(t)\) above the trajectory \(A_{i}^{n}(t)\) will be set to the intelligently made vehicle speed trajectory limit, as shown in Figure 10. For CAV, the actual execution speed track is \(E_{A,\text{R}}^{n}(t)\):  
\[E_{A,\text{R}}^{n}(t)=\text{min}[A_{i}^{n}(t),R_{i}^{n}(t)] \tag{31}\]  
The min represents that the CAV's actual execution speed track has the minimum value between the two.  
The velocity rolling horizon optimization strategy is continually executed through iterative processes until the target CAV crosses the stop line or, during execution, it is detected that the preceding HDV has joined the queue in front of the stop line; in this next instance, the velocity control strategy is determined using the optimization control method tailored for queuing constraints.