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A Level Set Approach to Online Sensing and Trajectory Optimization with Time Delays

Matthew R. Kirchner

TL;DR

The paper tackles online trajectory optimization for systems experiencing time delays from sensing and computation, addressing the lack of full environmental information by integrating sensing with real-time planning. It introduces a generalized Hopf formula-based level-set method to solve Hamilton–Jacobi PDEs without grid-based discretization, and extends this framework to time-delayed, time-varying Hamiltonians. A Newton-based procedure computes minimal hitting times to a goal set, enabling efficient re-planning as new information arrives. The authors apply the method to an online sensing scenario where a robot estimates a wireless channel with an unknown transmitter location using a non-parametric Gaussian-process channel model, demonstrating improved channel estimates and trajectories as delays are compensated.

Abstract

Presented is a method to compute certain classes of Hamilton-Jacobi equations that result from optimal control and trajectory generation problems with time delays. Many robotic control and trajectory problems have limited information of the operating environment a priori and must continually perform online trajectory optimization in real time after collecting measurements. The sensing and optimization can induce a significant time delay, and must be accounted for when computing the trajectory. This paper utilizes the generalized Hopf formula, which avoids the exponential dimensional scaling typical of other numerical methods for computing solutions to the Hamilton-Jacobi equation. We present as an example a robot that incrementally predicts a communication channel from measurements as it travels. As part of this example, we introduce a seemingly new generalization of a non-parametric formulation of robotic communication channel estimation. New communication measurements are used to improve the channel estimate and online trajectory optimization with time-delay compensation is performed.

A Level Set Approach to Online Sensing and Trajectory Optimization with Time Delays

TL;DR

The paper tackles online trajectory optimization for systems experiencing time delays from sensing and computation, addressing the lack of full environmental information by integrating sensing with real-time planning. It introduces a generalized Hopf formula-based level-set method to solve Hamilton–Jacobi PDEs without grid-based discretization, and extends this framework to time-delayed, time-varying Hamiltonians. A Newton-based procedure computes minimal hitting times to a goal set, enabling efficient re-planning as new information arrives. The authors apply the method to an online sensing scenario where a robot estimates a wireless channel with an unknown transmitter location using a non-parametric Gaussian-process channel model, demonstrating improved channel estimates and trajectories as delays are compensated.

Abstract

Presented is a method to compute certain classes of Hamilton-Jacobi equations that result from optimal control and trajectory generation problems with time delays. Many robotic control and trajectory problems have limited information of the operating environment a priori and must continually perform online trajectory optimization in real time after collecting measurements. The sensing and optimization can induce a significant time delay, and must be accounted for when computing the trajectory. This paper utilizes the generalized Hopf formula, which avoids the exponential dimensional scaling typical of other numerical methods for computing solutions to the Hamilton-Jacobi equation. We present as an example a robot that incrementally predicts a communication channel from measurements as it travels. As part of this example, we introduce a seemingly new generalization of a non-parametric formulation of robotic communication channel estimation. New communication measurements are used to improve the channel estimate and online trajectory optimization with time-delay compensation is performed.

Paper Structure

This paper contains 9 sections, 50 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Solutions to the Hamilton– Jacobi Equation with the Hopf Formula
  3. Viscosity Solutions with the Hopf Formula
  4. Time-Optimal Control to a Goal Set
  5. Online Trajectory Optimization With Delay
  6. An Example of Online Sensing and Trajectory Optimization
  7. Channel Estimation with Unknown Emitter Location
  8. Results
  9. Planar Motion Example

Figures (2)

  • Figure 1: The channel signal estimates from measurements as a vehicle traverses a known path. The vehicle path is shown in red and the ground-truth transmitter location shown as a black diamond. Best viewed in color.
  • Figure 2: Three optimization cycles of trajectories and their corresponding channel estimates and optimal trajectories. Best viewed in color.