标题： 基于PGD的从二维轨道中心线优化三维雪车轨道中心线以用于仿真应用 显示英文标题

标题： PGD-based optimization of 3D bobsleigh track centerlines from 2D centerlines for simulation applications

摘要： 雪车赛道的中心线定义了其几何形状，对于仿真建模至关重要。 为了降低雪车训练成本，利用雪车赛道的中心线构建一个能紧密复制真实比赛环境的虚拟环境是一个有前景的解决方案。 然而，公开可用的中心线数据通常有限，仅基于二维（2D）中心线构建训练系统是不精确的。 为了解决这一实际问题，本文提出了一种基于二维（2D）中心线数据生成三维（3D）赛道中心线的方法。 结合国际赛道设计规范，该方法制定了一种优化问题，考虑了总赛道长度、高度差、坡度约束和几何连续性。 使用投影梯度下降（PGD）算法来求解该优化问题。 生成的三维中心线与真实赛道数据进行比较，结果表明该方法可以从原始或缩放的二维数据中再现现实的中心线趋势。 对于选定的赛道段，真实二维数据的总长度、高度差和平均坡度的相对误差分别在1.7%、3.2%和4.1%以内，而缩放数据的相对误差分别为1.1%、3.5%和4.3%。 所有坡度值均在允许范围内。 此外，通过调整分段或修改成本函数中高度差的权重，可以生成适用于不同比赛的各种中心线风格。 在不同的分段和权重因子下，最大误差分别达到4.4%、4.8%和9.8%，以及4.4%、4.8%和10.0%。 所提出的方法为支持雪车赛道中心线设计提供了一个灵活且高效的工具。 显示英文摘要

摘要： The centerline of a bobsleigh track defines its geometry and is essential for simulation modeling. To reduce bBobsleigh training costs, leveraging the centerline of the bobsleigh track to construct a virtual environment that closely replicates real competitive settings presents a promising solution. However, publicly available centerline data are typically limited and it is imprecise to construct a training system solely based on 2-dimensional (2D) centerline. To address this practical issue, this paper proposes a method for generating a 3-dimensional (3D) track centerline based on 2D centerline data. Incorporating international track design regulations, the method formulates an optimization problem that considers total track length, height difference, slope constraints, and geometric continuity. A Projected Gradient Descent (PGD) algorithm is used to solve the optimization problem. The generated 3D centerlines are compared with real track data, and the results show that the method can reproduce realistic centerline trends from original or scaled 2D data. For the selected track segment, the relative errors in total length, height difference, and average slope are within 1.7%, 3.2% and 4.1%, respectively, for real 2D data and within 1.1%, 3.5% and 4.3% respectively for scaled data. All slope values remain within the allowable limits. Moreover, by adjusting the segmentation or modifying the weight of height difference in the cost function, various centerline styles applicable to different competitions can be generated. Under different segmentation and weight factors, the maximum errors reach up to 4.4%, 4.8%, and 9.8%, and 4.4%, 4.8%, and 10.0%, respectively. The proposed method provides a flexible and efficient tool for supporting bobsleigh track centerline design.