Modeling And Analysis of Motion Path and Collision Detection for The Bench Dragon Using Spiral Equations and Differential Algorithms

This study establishes a model for the motion path and collision detection of the bench dragon by analyzing spiral and motion trajectories, describing the system's motion characteristics through mathematical models and differential equations, and verifying the model's reliability. Given the current lack of systematic research in this area, this paper pioneers a relevant model for detailed analysis. Initially, a differential equation model for θ and t was developed based on the spiral equation and polar coordinate arc length formula, with the improved Runge-Kutta algorithm used for iterative solving to determine connection point positions and finite difference methods applied for velocity calculations. The Separating Axis Theorem was then introduced for collision detection, and a function relating pitch to turning space was initialized, with traversal of possible pitches and step lengths revealing a minimum collision pitch of 0.452m. Finally, geometric methods derived the positions and velocities of connection points on the shortest path, demonstrating that the turning path length is fixed and unoptimizable. The model successfully generated motion paths, collision times, and minimum pitch under varying conditions, with results visualized in images and tables, effectively capturing the bench dragon's motion characteristics while ensuring reliability through numerical validation and error analysis. Future work may focus on algorithmic optimization to enhance computational efficiency.