基于平坦性的多挂车车辆轨迹高效计算方法
Efficient flatness-based computation of trajectories for vehicles with many trailers
AI总结：
针对带多挂车车辆的前馈转向输入计算难题，提出结合四要素的算法，其复杂度为O(r³)且效率远超SymPy和贝尔多项式方法，可实现20个挂车的泊车轨迹计算。
AI中文摘要：
我们通过微分平坦性解决带n个挂车车辆的前馈转向输入的实际计算问题。标准方法（平坦输出的迭代符号求导）随n的扩展极具挑战性：即使使用计算机代数软件，表达式规模也会超指数增长，当n约为5时就难以处理。我们提出一种结合四个要素的算法：角度中间变量、子图中递归的因式分解、重新缩放的高阶乘积/商法则，以及基于截断形式幂级数而非Faa di Bruno公式的组合法则。我们将所提方法与（i）SymPy中的直接符号求导、（ii）通过贝尔多项式计算的Faa di Bruno公式进行基准测试。两种参考方法均遇到明显的计算瓶颈：SymPy在导数阶数r=14时达到20秒超时，而贝尔多项式递归在r>24时超过60秒超时。所提方法在导数阶数下的算术复杂度为O(r³)，在r=40时仍保持在1毫秒以内，且在参考方法成功的范围内达到机器精度的数值一致性。作为端到端示例，计算并动画展示了20个挂车的泊车操作。
英文摘要：
We address the practical computation of feedforward steering inputs for a car with n trailers via differential flatness. The standard route (iterated symbolic differentiation of the flat output) scales prohibitively with n: even with computer-algebra software the expression sizes grow super-exponentially and become intractable beyond n approx. 5. We propose an algorithm that combines four ingredients (angular intermediate variables, factorisation of the recursion in submaps, rescaled higher-order product/quotient rules, and a composition rule based on truncated formal power series instead of Faa di Bruno's formula). We benchmark the proposed method against (i) direct symbolic differentiation in SymPy and (ii) Faa di Bruno's formula evaluated via Bell polynomials. Both reference methods hit a clear computational wall: SymPy reaches a 20 s timeout at derivative order r = 14, while the Bell-polynomial recursion exceeds a 60 s timeout beyond r = 24. The proposed approach has O(r3) arithmetic complexity in the derivative order and remains under one millisecond up to r = 40, with numerical agreement to machine precision in the regime where the reference methods succeed. As an end-to-end illustration, a 20-trailer parking manoeuvre is computed and animated.
