## 2 Local Path Generation Based on Spatial Perception
### Dynamic Control Point Determination
#### 2.1.2 Determination of Control Points for Local Obstacle Avoidance Path  
For the safe and comfortable execution of obstacle avoidance, it is imperative to institute a constraint on the spatial distance between the vehicle and the obstacles, known as the spatiotemporal safety constraint. Concurrently, ensuring the path's smoothness and stability mandates restrictions on its curvature alterations. This restriction is termed the curvature constraint for obstacle avoidance. Considering both the vehicle's spatial position and the spatial data of obstacles, dynamic control points are established by jointly considering spatiotemporal safety and curvature constraints. Figure 1 illustrates a straightforward lane-changing scenario for obstacle avoidance. Throughout this process, to prevent collisions, the intelligent driving vehicle must maintain the requisite safety distance. This ensures that the longitudinal distance between its current spatial position and that of nearby obstacles remains sufficiently expansive. Several assumptions are made for this analysis: (1) No obstacles exist on the neighboring lanes. (2) Obstacles ahead of the intelligent driving vehicle maintain a constant, lower speed. The longitudinal safety distance, often determined using TTC (time-to-collision) [31], serves as a measure for calculating the collision safety distance and assessing the safety level. This distance is influenced by the relative longitudinal velocity between the intelligent driving vehicle and the obstacle ahead [32; 33; 34]. Thus, the longitudinal safety distance between the vehicle's real-time spatial position and the forward obstacle's position can be denoted as \(d_{0}\), and its value can be derived from Equation (1). \[d_{0}=\left\{ \begin{aligned} &(v_{0}-v_{ob})\alpha_{0}+l_{0}\\ & l_{0}\end{aligned} \right. \tag{1}\]  
where \(l_{0}\) denotes the appropriate following distance between vehicles, \(v_{0}\) signifies the current speed of the intelligent driving sightseeing vehicle, and \(v_{ob}\) represents the speed of the obstacle ahead. The term \(\alpha_{0}\) indicates the minimum safe time interval to avoid a collision. To ensure trajectory curvature continuity while mitigating computational demands, we harness the features of B-spline curves. Specifically, these curves facilitate the translation of constraints on the trajectory curvature imposed by actuators into limitations regarding segment lengths and angles. Alongside the constraint of longitudinal safety distance for obstacle avoidance, these determinants collaboratively dictate the positioning of the B-spline curve control points. For a B-spline curve characterized by its linear segments, its maximum curvature, denoted as \(k_{max}\), is steered by two pivotal parameters [35]: the segment length \(d_{1}\) and the segment angle \(\beta\). Within the context of the aforementioned obstacle avoidance lane-change scenario, \(d_{1}\) stands for the obstacle avoidance curvature constraint distance. The relationship between \(d_{1}\) and \(\beta\) is shown in the control line segment model in Figure 2. As long as any two control line segments satisfy Equation (2), the curvature of the B-spline curve is guaranteed. \[d_{1}\geq\frac{1}{6}\frac{\sin\beta}{\kappa_{max}}\bigg{(}\frac{1-\cos\beta}{ 8}\bigg{)}^{-1.5} \tag{2}\]  
Figure 1: Obstacle safety distance. The characteristics of Nth-degree B-spline curves vary with their degree. As the curve's degree escalates, its derivatives' order augments, leading to an increase in the zeros of the curve. This proliferation of zeros can result in a curve exhibiting multiple extrema, translating into more pronounced peaks and troughs. However, a curve with a diminished degree can more closely approximate its control points [36]. Given their second-order derivative continuity, cubic B-spline curves are optimal for trajectory planning. Specifically, quasi-uniform cubic B-spline curves, which typically necessitate at least four control points, are preferred to craft a seamless, smooth curve.