A Scalable Method for Optimal Path Planning on Manifolds via a Hopf-Lax Type Formula

TL;DR

The paper tackles minimal-time path planning for points constrained to a manifold that is the graph of a smooth function $M:\mathbb{R}^n\to\mathbb{R}$. It recasts the problem via dynamic programming, yielding a Hamilton-Jacobi-Bellman equation with Hamiltonian $H(x,p,t)= v(x,t)\sqrt{p^T A(x) p}-1$ where $A(x)= I - \frac{\nabla M(x) \nabla M(x)^T}{1+|\nabla M(x)|^2}$, and introduces a time-reversed, Hopf-Lax-type representation to avoid grid-based PDE solves. A primal-dual hybrid gradient (PDHG) algorithm, aided by a Cholesky factorization $A(x)=L(x)L(x)^T$ and a variable change $w=L(x)^T p$, solves the resulting saddle-point problem efficiently and in a grid-free fashion. The method demonstrates near real-time performance up to $n=30$ dimensions, with parallelizable computations across trajectories, offering a scalable and interpretable approach to high-dimensional PDE-based path planning on manifolds.

Abstract

We consider the problem of optimal path planning on a manifold which is the image of a smooth function. Optimal path-planning is of crucial importance for motion planning, image processing, and statistical data analysis. In this work, we consider a particle lying on the graph of a smooth function that seeks to navigate from some initial point to another point on the manifold in minimal time. We model the problem using optimal control theory, the dynamic programming principle, and a Hamilton-Jacobi-Bellman equation. We then design a novel primal dual hybrid gradient inspired algorithm that resolves the solution efficiently based on a generalized Hopf-Lax type formula. We present examples which demonstrate the effectiveness and efficiency of the algorithm. Finally, we demonstrate that, because the algorithm does not rely on grid-based numerical methods for partial differential equations, it scales well for high-dimensional problems.

Figures (5)

• Figure 1: A particle traveling on a manifold, and its projection down to $\mathbb R^{n}$ (represented by the plane). Here $\boldsymbol x(t)$ is the red dot and $(\boldsymbol x(t),z(t))$ is the black dot. The red arrow is the control variable $\boldsymbol a(t)$, which is a direction vector in $\mathbb R^n$. This choice of $\boldsymbol a(t)$ induces a motion (black arrow) along the manifold.
• Figure 2: The approximate solution $u$ provided by Algorithm \ref{['Algorithm1']} in the case that $M \equiv 0, v \equiv 1$, and $x_f = (0,0,\ldots, 0)$. The state space is $\mathbb R^{10}$ and the $x$ values are 10 randomly generated members of $[-1,1]^{10}.$ In this case, the exact solution is $u(x,t) = \left\lvert x \right \rvert.$ Note that our approximation error is never larger than $3.8 \times 10^{-2}$, even though $\Delta t = 0.1$.
• Figure 3: Optimal paths on the graph of $M(x,y) = a\sin(\pi x)\cos(\pi y)$ for constant $a$. Top panel: $a=1$, optimal paths are plotted from 20 randomly chosen points in $[-1,1]\times [-1,1]$ to the point $(1,1)$. Bottom left panel: $a =1$, plotted is the optimal path from $(-1,-1)$ to $(1,1)$. Bottom right panel: $a = 3$, plotted is the optimal path from $(-1,-1)$ to $(1,1)$. Notice that when $a$ is larger, the path is more apt to stay in the flat regions, avoiding the hills and valleys, since travel distance along the manifold is $(1+\left\lvert \nabla M(x,y) \right \rvert^2)^{1/2}$ per unit distance in the $xy$-plane.
• Figure 4: Optimal paths on the manifold $M(x,y) = 2\text{exp}(-(x^2+y^2))$. The final point is $x_f = (1,1)$ in each image, and there are 20 initial points are randomly selected in $[-0.85,-0.65]\times [-0.85,-0.65]$ (the random selection for each image is done separately). In the left panel, the velocity is $v \equiv 1$, so that, as one may expect, roughly half of the paths travel either direction around the large hill in the middle. In the right panel, the velocity is $v(x,y) = 1+(x-1)^2$, meaning much faster travel is allowed in the left hand side of the image. Because of this, each particle travels up in $y$ before traveling $right$ in $x$.
• Figure 5: Left panel: an optimal path from $x = (-0.9,-1,-1,\ldots,-1)$ to $x_f = (1,1,1,\ldots,1)$ on the graph of $M(x) = \text{exp}(-\left\lvert x \right \rvert^2)$ in 25 dimensions. Plotted are each coordinate of the solution. The gray dotted line is the value $u(x,t) \approx 11.30$ resolved by the algorithm, which is an approximation of the optimal travel time. We see that all coordinates do indeed reach $1$ at roughly that time and then stay there. Right panel: the average CPU time required to resolve the optimal path as a function of the dimension. Averages are taken over ten trials and the bars represent the standard deviation in the results. These results seem to indicate that the dimensional scaling for the algorithm is at very least subexponential, and looks approximately linear for the dimensions 10-30.

Theorems & Definitions (6)

• Theorem 2.1
• proof
• Definition 3.1
• Theorem 3.1
• Remark 3.1
• proof