用快速算法解决高维机器人轨迹优化难题，5秒完成44维系统规划。
Bring the Heat: Rapid Trajectory Optimization with Pseudospectral Techniques and the Affine Geometric Heat Flow Equation
- 结合伪谱法与几何热流方程，将复杂轨迹问题简化为二维求解。
- 44维系统仅需约5秒生成最优轨迹，速度远超现有方法。
- 适合需要实时路径规划的复杂机器人系统研发者使用。
在满足约束条件下高效生成高维机器人系统的最优轨迹是一项挑战性任务。本文提出PHLAME，利用伪谱配点法与空间向量代数，高效求解仿射几何热流（AGHF）偏微分方程以实现轨迹优化。与传统基于哈密顿-雅可比-贝尔曼（HJB）方程的方法不同，AGHF的解定义于二维域，避免了状态空间维度爆炸问题。通常采用行方法（MOL）将AGHF离散化为常微分方程组，但需精细网格且计算成本高。PHLAME采用伪谱法，大幅减少函数求值次数，在保证精度的同时显著提升效率。此外，本文推导出AGHF及其雅可比矩阵的解析表达式，借助刚体动力学算法实现快速计算。在含障碍物与无障碍物的多种动力系统上测试表明，PHLAME可在约5秒内完成44维状态空间的轨迹生成，优于当前主流方法。
原文摘要 · Abstract (English)
Generating optimal trajectories for high-dimensional robotic systems in a time-efficient manner while adhering to constraints is a challenging task. This paper introduces PHLAME, which applies pseudospectral collocation and spatial vector algebra to efficiently solve the Affine Geometric Heat Flow (AGHF) Partial Differential Equation (PDE) for trajectory optimization. Unlike traditional PDE approaches like the Hamilton-Jacobi-Bellman (HJB) PDE, which solve for a function over the entire state space, computing a solution to the AGHF PDE scales more efficiently because its solution is defined over a two-dimensional domain, thereby avoiding the intractability of state-space scaling. To solve the AGHF one usually applies the Method of Lines (MOL), which discretizes one variable of the AGHF PDE, and converts the PDE into a system of ordinary differential equations (ODEs) that are solved using standard time-integration methods. Though powerful, this method requires a fine discretization to generate accurate solutions and requires evaluating the AGHF PDE which is computationally expensive for high-dimensional systems. PHLAME overcomes this deficiency by using a pseudospectral method, which reduces the number of function evaluations required to yield a high accuracy solution thereby allowing it to scale efficiently to high-dimensional robotic systems. To further increase computational speed, this paper presents analytical expressions for the AGHF and its Jacobian, both of which can be computed efficiently using rigid body dynamics algorithms. PHLAME is tested across various dynamical systems, with and without obstacles and compared to a number of state-of-the-art techniques. PHLAME generates trajectories for a 44-dimensional state-space system in $\sim5$ seconds, much faster than current state-of-the-art techniques. A project page is available at https://roahmlab.github.io/PHLAME/
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