Refining Control Barrier Functions through Hamilton-Jacobi Reachability

Abstract

Learning-based control algorithms have led to major advances in robotics at the cost of decreased safety guarantees. Recently, neural networks have also been used to characterize safety through the use of barrier functions for complex nonlinear systems. Learned barrier functions approximately encode and enforce a desired safety constraint through a value function, but do not provide any formal guarantees. In this paper, we propose a local dynamic programming (DP) based approach to "patch" an almost-safe learned barrier at potentially unsafe points in the state space. This algorithm, HJ-PATCH, obtains a novel barrier that provides formal safety guarantees, yet retains the global structure of the learned barrier. Our local DP based reachability algorithm, HJ-PATCH, updates the barrier function "minimally" at points that both (a) neighbor the barrier safety boundary and (b) do not satisfy the safety condition. We view this as a key step to bridging the gap between learning-based barrier functions and Hamilton-Jacobi reachability analysis, providing a framework for further integration of these approaches. We demonstrate that for welltrained barriers we reduce the computational load by 2 orders of magnitude with respect to standard DP-based reachability, and demonstrate scalability to a 6-dimensional system, which is at the limit of standard DP-based reachability.

Constructing Control Lyapunov-Value Functions Using Hamilton-Jacobi Reachability Analysis

“…The main drawback to this method is the heavy computation requirement (i.e., "curse of dimensionality") due to dynamic programming. Future work to address this issue includes decomposition of the system [17], using hand-tuned CLF candidates to warm-start the computation [28], adaptive grids [19], and data-driven approaches [18]. Other directions of interest include incorporating disturbances, exploring systems with multiple isolated equilibrium points, finding connections with black-box models, and tuning γ 's online to achieve different stabilizability properties.…”

Differentiable Safe Controller Design Through Control Barrier Functions

“…1). By leveraging techniques from [38], we update CBF h q recursively using Hamilton-Jacobi reachability. When the process terminates, we obtain a valid CBF h q,q ′ on C q \BackUnsafe q,q ′ .…”

“…Lemma 1. (Adapting Theorem 1 from [38]) The refined CBF h q,q ′ is valid upon convergence, i.e., there exists an extended class K ∞ function α q,q ′ (•) such that: sup u∈Uq ∂h q,q ′ (x) ∂x [f q (x)+g q (x)u] ≥ −α q,q ′ (h q,q ′ (x)), (7) for all x ∈ C q,q ′ .…”

“…Our initial h q can be considered as a good warmstarting for the CBF refinement, which can accelerate the convergence of the recursive update. However, note that this dynamic programming (DP)-based CBF refinement is generally limited to low-dimensional systems [38]. This is typically the case since spatially discretized DP recursion results in exponential computational complexity w.r.t.…”

“…Remark 2. From [38], [39], we know that the converged CBF recovers a valid control barrier-value function (CBVF), and that CBVF recovers the largest controlled invariant set. Hence, our refined CBF-based method is not conservative under the given multiple local CBFs.…”