发表机构
University of Tübingen（图宾根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出一种有界悬链线求解器，通过将问题简化为单一超越方程并证明收敛性，实现平均6.9微秒的实时求解，比先前方法快40倍，并已在ArduPilot飞行控制器上验证。
AI 中文摘要
在真实条件下，对于非静止的系绳多旋翼无人机，模拟系绳气动阻力对无人机施加的力变得至关重要，而在线使用场景对最大求解时间设置了硬性限制。在先前的工作中，一种准解析悬链线系绳模型使用通用求根器达到了0.51毫秒的平均求解时间，但没有最坏情况保证或已证明的收敛性。在本工作中，我们通过将悬链线边值问题简化为一个关于良态未知数的单一超越方程，重新构造了内部求解器。我们推导了一个闭式括号，并证明了单调性、凸性以及根的存在性和唯一性，这些共同保证了求解器的收敛性。我们进一步提出了一种双区域初始猜测，其近似真实根的误差在3.4%以内，并将平均迭代次数比教科书式初始化减少了68.0%，降至2.36次。基于混合求根方法rtsafe（牛顿-拉夫森法配合二分法回退，给出有界迭代次数），我们实现了一个专门变体，利用问题结构省略不必要的检查同时保持正确性，这带来了高达1.3倍的加速。使用所提出的求解器，完整系绳模型实现了近乎恒定的求解时间，平均为6.9微秒，最坏情况为7.7微秒，比先前方法的优化重实现快40倍，同时与其相对偏差为8.7e-9。由于重新构造未改变底层物理模型，先前工作的实验验证可直接沿用。我们进一步通过直接在无人机飞行控制器上的ArduPilot中运行的Lua实现，展示了其适用于嵌入式、资源受限平台的特性，平均求解时间为0.74毫秒，远在调度预算之内。
英文摘要
For non-stationary tethered multirotor UAVs in real-world conditions, simulating the forces imposed on the drone by the aerodynamic drag of the tether becomes crucial, with online use cases placing a hard bound on the maximum solve time. In previous work, a quasi-analytical catenary tether model reached a mean solve time of 0.51 ms using a general-purpose root finder, but without any worst-case guarantees or proven convergence. In this work, we reformulate the inner solver by reducing the catenary boundary-value problem to a single transcendental equation in one well-conditioned unknown. We derive a closed-form bracket and prove monotonicity and convexity as well as existence and uniqueness of the root, which together guarantee convergence of the solver. We further propose a two-regime initial guess which approximates the true root within 3.4% and reduces the mean iteration count by 68.0% to 2.36 compared to the textbook initialization. Building on the hybrid root-finding method rtsafe (Newton-Raphson with bisection fallback giving bounded iteration counts), we implement a specialized variant that exploits the problem structure to omit unnecessary checks while retaining correctness, which gives up to 1.3 times speedup. With the proposed solver the full tether model achieves a nearly constant solve time of 6.9 us on average and 7.7 us at worst, a 40 times speedup over an optimized re-implementation of the previous method, while agreeing with it to a relative deviation of 8.7e-9. Because the reformulation leaves the underlying physical model untouched, the experimental validation of the previous work carries over unchanged. We further demonstrate its suitability for embedded, resource-constrained platforms with a Lua implementation running directly in ArduPilot on a drone's flight controller, where it stays well inside the scheduling budget with a mean solve time of 0.74 ms.
