Generating optimal trajectories for high-dimensional robotic systems in a time-efficient manner while adhering to constraints is a challenging task. To address this challenge, 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 works by discretizing one variable of the AGHF PDE, effectively converting the PDE into a system of ordinary differential equations (ODEs) that can be solved using standard time-integration methods. Though powerful, this method requires a fine discretization to generate accurate solutions and still requires evaluating the AGHF PDE which can be 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. The proposed method 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 $\sim3$ seconds, much faster than current state-of-the-art techniques.

翻译：为高维机器人系统生成满足约束条件的最优轨迹，同时保证时间效率，是一项具有挑战性的任务。为解决这一挑战，本文提出了PHLAME方法，该方法应用伪谱配置法与空间向量代数，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偏微分方程方法在整个状态空间上求解函数不同，AGHF偏微分方程的求解规模效率更高，因为其解定义在二维域上，从而避免了状态空间规模带来的计算困难。求解AGHF通常采用直线法，该方法通过对AGHF偏微分方程的一个变量进行离散化，将偏微分方程有效地转化为一个常微分方程组，从而可以使用标准的时间积分方法求解。尽管功能强大，但该方法需要精细的离散化才能生成精确解，并且仍然需要计算AGHF偏微分方程，这对于高维系统而言计算成本高昂。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函数求值次数，从而能够高效扩展到高维机器人系统。为了进一步提高计算速度，本文给出了AGHF及其雅可比矩阵的解析表达式，两者均可使用刚体动力学算法高效计算。所提出的PHLAME方法在多种动力学系统上进行了测试，包括有障碍物和无障碍物的场景，并与多种先进技术进行了比较。PHLAME可在约3秒内为44维状态空间系统生成轨迹，速度远超当前最先进的技术。