板凳龙的路径优化设计与速度控制

Bench Dragon’s Path Optimisation Design and Speed Control

摘要: 本研究专注于传统民俗活动中板凳龙表演的路径优化及速度控制问题，旨在构建基于位置递推与速度传导机制的物理模型，以增进表演的协调性与观赏性。研究过程中，运用欧几里得距离公式及速度传递关系，借助Python构建迭代模型，用以计算龙头在60秒内的位移轨迹以及龙尾的响应速度。针对螺旋盘入过程中潜在的碰撞场景，将螺旋线等效为圆弧轨迹，对螺距为55厘米时关键位置处的曲率半径、龙头位置及到达时间展开分析。基于碰撞判据，构建龙头进入掉头区域时的最小盘入螺距模型。引入三次B样条曲线进行路径优化，通过选取合理切点位置缩短S形调头路径长度，确保路径的连续性与安全性，并验证节点间距的无碰撞条件。在限定全队速度不超过2 m/s的约束下，结合变分法与二分法确定龙头的最大安全速度，构建函数$v_{max}\left(v_0\right)$ 开展速度传导分析，从而得出满足全队安全要求的最优龙头速度上限。

Abstract: This study focuses on the path optimization and speed control of bench dragon dance performances in a traditional folk activity, aiming to construct a physical model based on position recursion and speed conduction mechanisms to enhance performance coordination and aesthetic appeal. Using the Euclidean distance formula and speed transfer relationships, an iterative model was built with Python to calculate the displacement trajectory of the dragon head over 60 seconds and the response speed of the dragon tail. For potential collision scenarios during spiral coiling, the spiral path was approximated as a circular arc trajectory to analyze the curvature radius, dragon head position, and arrival time at key points when the pitch is 55 centimeters. A minimum coiling pitch model was developed for the dragon head entering the turning area based on collision criteria. Cubic B-spline curves were introduced for path optimization, shortening the S-shaped turning path through reasonable tangent point selection to ensure path continuity and safety while verifying collision-free conditions for node spacing. Under the constraint that the entire team’s speed does not exceed 2 m/s, the calculus of variations and bisection method were combined to determine the maximum safe speed of the dragon head. A function $v_{max}\left(v_0\right)$ was constructed for speed conduction analysis, ultimately deriving the optimal upper limit for the dragon head’s speed that ensures the safety of the entire team.