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

## TL;DR

This paper introduces a level set method using the generalized Hopf formula for real-time trajectory optimization in robotics, effectively handling time delays in sensing and control, demonstrated through communication channel estimation.

## Contribution

It presents a novel application of the generalized Hopf formula for online Hamilton-Jacobi equation solutions with time delays, including a new non-parametric approach for robotic communication channel estimation.

## Key findings

- Efficient computation of Hamilton-Jacobi equations with time delays.
- Successful online trajectory optimization with delay compensation.
- Improved robotic communication channel estimation from incremental measurements.

## 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.