Blending HJ reachability and Control Barrier functions

This project works towards developing new theory and tools that connect two popular approaches for safety analysis of nonlinear systems: Hamilton-Jacobi (HJ) reachability and control barrier functions (CBFs).

Both of these methods ultimately produce a safety certificate function across the state space that encodes two key pieces of information for safety analysis and control: the output of the function provides the current safety level at a particular state, and the gradient of the function informs the required control action that will maintain safety. Despite this similarity, these two techniques (HJ reachability and CBFs) have been largely developed independently.

HJ reachability is the more general approach, and directly solves for the maximal safe region within which a system may operate. However, this technique struggles with scalability. A CBF generally has a more conservative safe region than HJ reachability, but provides a useful quadratic program online controller that smoothly trades off between the desired performance control and the necessary safe control. For systems of specific forms, the CBF can be constructed analytically and can therefore be very scalable, but finding a valid CBF is in general challenging for nonlinear dynamic systems. There is active research on learning CBF approximations for general system dynamics, and augmenting approximate CBFs with hand-tuned or learned “backup” controllers.

We are interested in drawing connections between these two methods so that we can combine the theory and tools developed in each community. Specifically, we are working on the following thrusts:

1) Direct construction of CBFs.

We have demonstrated how a CBF-like function with the largest possible safe operating region can be constructed directly using HJ reachability tools. We are actively working to identify the conditions under which one can directly compute CBFs for general nonlinear dynamical systems. By identifying the theoretical connection between these two fields, we can now combine the tools from both!

2) Fast approximations with rigorous convergence to safety guarantees.

There are many recent research efforts on efficiently approximating CBFs using data-driven approaches. However, these CBFs approximations may not be valid. In the videos below, the same CBF approximation is too conservative in some cases (not reaching the goal), and too optimistic in others (intersecting the floor)

In our recent work, we showed that by using fast CBF approximations as initializations for HJ reachability, scalable safety guarantees can be computed for general nonlinear dynamical systems. See below what happens when we initialize the HJ reachability computation with the CBF approximation:

3) Adapting safety online.

There is active research on adapting control policies for CBFs based on information learned online. Similarly, recent work from our lab has shown how to update a reachability-based safety analysis online. We propose to apply safe learning to CBFs using these tools from both methods to update safety analyses efficiently with convergence guarantees. Moreover, we will explore how to adapt the safe set under large changes in system representations (e.g. changes in the state space).

Relevant Papers:

S Tonkens, SL Herbert. Refining Control Barrier Functions through Hamilton-Jacobi Reachability. IEEE Conference on Intelligent Robots and Systems, 2022.

JJ Choi, D Lee, K Sreenath, CJ Tomlin, SL Herbert. Robust Control Barrier-Value Functions for Safety-Critical Control. IEEE Conference on Decision and Control, 2021.