Optimality and robustness in path-planning under initial uncertainty

Abstract

Classical deterministic optimal control problems assume full information about the controlled process. The theory of control for general partially-observable processes is powerful, but the methods are computationally expensive and typically address the problems with stochastic dynamics and continuous (directly unobserved) stochastic perturbations. In this paper we focus on path planning problems which are in between -- deterministic, but with an initial uncertainty on either the target or the running cost on parts of the domain. That uncertainty is later removed at some time $T$, and the goal is to choose the optimal trajectory until then. We address this challenge for three different models of information acquisition: with fixed $T$, discretely distributed and exponentially distributed random $T$. We develop models and numerical methods suitable for multiple notions of optimality: based on the average-case performance, the worst-case performance, the average constrained by the worst, the average performance with probabilistic constraints on the bad outcomes, risk-sensitivity, and distributional-robustness. We illustrate our approach using examples of pursuing random targets identified at a (possibly random) later time $T$.

Figures (14)

• Figure 1: Time-optimal path planning. (A): a contour plot of the speed $f = 1.4+0.6\cos(2\pi x)\sin(2\pi y)$ on a unit square domain with one rectangular obstacle (shown in white) at $(0.45,0.55)\times(0.15,0.85).$ The starting point $\bm{\bm{x}_0^{}} = (0.3,0.2)$ is shown by an orange dot. The approach finds optimal trajectories to all target locations, but we highlight four potential targets (shown by orange stars and numbered clockwise, starting from the top): $\bm{\hat{x}}_1 = (0.5, 0.95), \bm{\hat{x}}_2 = (0.9,0.5), \bm{\hat{x}}_3 = (0.5, 0.05),$ and $\bm{\hat{x}}_4 = (0.1, 0.5).$ (B): a contour plot of $u(\bm{x};\bm{\bm{x}_0^{}})$ and the optimal trajectories (shown by dashed red lines) to these four targets.
• Figure 2: Fixed certainty time. (A): $T=0.08.$ (B): $T=0.4.$ The speed $f$, domain geometry, starting position $\bm{\bm{x}_0^{}}$ (orange dot) and four targets $\bm{\hat{x}}_1, \ldots, \bm{\hat{x}}_4$ (orange stars) are the same as in Figure \ref{['fig:determ']}. The corresponding target probabilities (clockwise, starting at the top) are $\bm{\hat{p}} = (0.2, 0.3, 0.2, 0.3).$ Boundaries of $\Omega^{}_T$ (in yellow) are superimposed on a contour plot of $q.$ The time optimal path from $\bm{\bm{x}_0^{}}$ to the optimal waypoint $\bm{s} = \mathop{\mathrm{arg\,min}}\limits_{\Omega^{}_T} q(\bm{x})$ is shown in red, with dotted red lines showing time-optimal trajectories from $\bm{s}$ to each $\bm{\hat{x}}_i$. In the left subfigure, the $\bm{s}$-to-$\bm{\hat{x}}_4$ trajectory overlaps the already traversed $\bm{\bm{x}_0^{}}$-to-$\bm{s}$ trajectory.
• Figure 3: Random $T$ with possible values $T_1 = 0.08$ and $T_2 = 0.4$. The basic setup is the same as in Figure \ref{['fig:FixedT']}, but the optimal behavior is heavily dependent on $T$'s probability distribution. (A): $p_1 = 0.9, \, p_2 = 0.1.$ (B): $p_1 = 0.55, \, p_2 = 0.45.$ (C): $p_1 = 0.1, \, p_2 = 0.9.$
• Figure 4: Exponentially distributed $T$. Each subfigure shows the level sets of $u^{\lambda},$ starting position $\bm{\bm{x}_0^{}}$ (in orange), optimal waiting position $\bm{s}$ (in cyan), motionless local minima of $u^\lambda$ (in magenta) and the optimal trajectory (in red). (A): with $\lambda = 2.5$ an early target identification is not very likely, and the optimal trajectory leads to the global minimum of $q$. (B): with $\lambda = 20$ still heading to the global minimum although the nearby local minimum is already motionless. (C): with $\lambda = 30$ reaching the global minimum before the target identification is less unlikely, and it is optimal to head toward the local minimum nearby.
• Figure 5: Worst-case optimal planning. (A): the same ("Fixed $T=0.4$") setting as in Figure \ref{['fig:FixedT']}B. The cyan diamond represents a $\bm{\bar{s}}$ and the red curve is the time-optimal trajectory, with level sets of $\bar{q}$ in the background. (B): the same ("Exponentially distributed $T$ with $\mathbb{E}[T] = 0.4$") setting as in Figure \ref{['fig:exponential']}A except that $q$ is replaced by $\bar{q}$. Level sets of $u^\lambda$ are shown in the background.

Theorems & Definitions (5)

• remark 1
• remark 2
• remark 3
• remark 4
• remark 5