Study on Motion Trajectory Problem Based on Archimedean Spiral and Collision Detection

This study investigates the motion characteristics of the traditional cultural heritage “Bandenglong” ( Bench Dragon) , a performance involving intricate spiral trajectories and coordinated multi-node movements, which demands high precision in modeling and optimization. To enhance safety, aesthetic performance, and operational efficiency, we propose a comprehensive framework integrating Archimedean spiral trajectory modeling, collision detection algorithms, and multi-stage dynamic path optimization. By employing numerical methods such as the central difference scheme, Newton-Raphson iteration, and binary search, alongside simulation techniques, we innovatively transform. collision constraints into optimization problems and develop a selective collision detection approach. A multi-phase adaptive optimization algorithm is introduced to dynamically adjust parameters, ensuring global optimality while addressing subproblems like spiral transition smoothing and collision avoidance. The established mathematical model accurately describes the spiral motion of the dragon’s head, body, and tail, resolving multi-scale modeling challenges in complex dynamic systems. Beyond its application to the Bandenglong, this framework offers novel insights and methodological support for diverse fields, including celestial orbital design, DNA double-helix structural analysis, and robotic path planning, thereby bridging cultural heritage preservation with modern technological advancements.