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/

翻译：在满足约束条件的前提下，为高维机器人系统高效生成最优轨迹是一项具有挑战性的任务。本文提出PHLAME方法，该方法应用伪谱配点法与空间向量代数，高效求解用于轨迹优化的仿射几何热流偏微分方程。与哈密顿-雅可比-贝尔曼偏微分方程等传统方法在整个状态空间上求解函数不同，AGHF偏微分方程的解定义在二维域上，从而避免了状态空间维度扩展带来的计算困难，因此其求解规模更具效率。求解AGHF通常采用直线法，该方法离散化AGHF偏微分方程的一个变量，将偏微分方程转化为常微分方程组，并采用标准时间积分方法求解。尽管功能强大，但该方法需要精细离散化才能生成精确解，且需计算AGHF偏微分方程，这对高维系统而言计算代价高昂。PHLAME通过采用伪谱方法克服了这一缺陷，该方法减少了获得高精度解所需的目标函数计算次数，从而能高效扩展至高维机器人系统。为进一步提升计算速度，本文给出了AGHF及其雅可比矩阵的解析表达式，二者均可通过刚体动力学算法高效计算。PHLAME在多种动力学系统（含/不含障碍物场景）中进行了测试，并与多种先进技术进行了对比。实验表明，PHLAME可在约5秒内为44维状态空间系统生成轨迹，速度显著优于当前最先进技术。项目页面详见：https://roahmlab.github.io/PHLAME/