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

TL;DR

This work tackles constructing a full 3D bobsleigh track centerline from a 2D centerline under international design rules to enable realistic, low-cost simulation environments for training. It casts track generation as a constrained optimization that minimizes errors in total length $L$, height difference $H$, and curvature alignment $(K_{pk}-K_{sk})$, while matching the average slope $\bar{g}$ and enforcing slope bounds $g_{k_{min}}\le g_k \le g_{k_{max}}$, solved by Projected Gradient Descent (PGD). Validation on real tracks and proportionally scaled data shows generated centerlines closely track true geometry, with relative errors in $L$, $H$, and $\bar{g}$ typically under a few percent and slopes remaining within design limits. The method provides a flexible tool to generate diverse centerline styles for different competitions and can broaden the applicability of simulation-based training, with future work exploring stylistic variants and alternative optimization approaches.

Abstract

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.

Figures (7)

• Figure 1: Transition from a 2D centerline to a 3D centerline.
• Figure 2: Numerical procedure.
• Figure 3: The comparison of the track height variation and slope along the distance for the generated and actual track centerlines.
• Figure 4: The comparison of trajectory differences between the generated and actual track centerlines.
• Figure 5: The comparison between the generated and actual $x$ and $y$ coordinates of the track centerlines.