4 research outputs found
Fast Reachable Set Approximations via State Decoupling Disturbances
With the recent surge of interest in using robotics and automation for civil
purposes, providing safety and performance guarantees has become extremely
important. In the past, differential games have been successfully used for the
analysis of safety-critical systems. In particular, the Hamilton-Jacobi (HJ)
formulation of differential games provides a flexible way to compute the
reachable set, which can characterize the set of states which lead to either
desirable or undesirable configurations, depending on the application. While HJ
reachability is applicable to many small practical systems, the curse of
dimensionality prevents the direct application of HJ reachability to many
larger systems. To address computation complexity issues, various efficient
computation methods in the literature have been developed for approximating or
exactly computing the solution to HJ partial differential equations, but only
when the system dynamics are of specific forms. In this paper, we propose a
flexible method to trade off optimality with computation complexity in HJ
reachability analysis. To achieve this, we propose to simplify system dynamics
by treating state variables as disturbances. We prove that the resulting
approximation is conservative in the desired direction, and demonstrate our
method using a four-dimensional plane model.Comment: in Proceedings of the IEE Conference on Decision and Control, 201
Sampling-Based Approximation Algorithms for Reachability Analysis with Provable Guarantees
The successful deployment of many autonomous systems in part hinges on providing rigorous guarantees on their performance and safety through a formal verification method, such as reachability analysis. In this work, we present a simple-to-implement, sampling-based algorithm for reachability
analysis that is provably optimal up to any desired approximation accuracy. Our method achieves computational efficiency by judiciously sampling a finite subset of the state space and generating an approximate reachable set by conducting reachability analysis on this finite set of states. We prove that the reachable set generated by our algorithm approximates the ground-truth
reachable set for any user-specified approximation accuracy. As a corollary to our main method, we introduce an asymptoticallyoptimal, anytime algorithm for reachability analysis. We present simulation results that reaffirm the theoretical properties of our algorithm and demonstrate its effectiveness in real-world inspired scenariosNational Science Foundation (U.S.
Backward Reachability Analysis of Perturbed Continuous-Time Linear Systems Using Set Propagation
Backward reachability analysis computes the set of states that reach a target
set under the competing influence of control input and disturbances. Depending
on their interplay, the backward reachable set either represents all states
that can be steered into the target set or all states that cannot avoid
entering it -- the corresponding solutions can be used for controller synthesis
and safety verification, respectively. A popular technique for backward
reachable set computation solves Hamilton-Jacobi-Isaacs equations, which scales
exponentially with the state dimension due to gridding the state space. In this
work, we instead use set propagation techniques to design backward reachability
algorithms for linear time-invariant systems. Crucially, the proposed
algorithms scale only polynomially with the state dimension. Our numerical
examples demonstrate the tightness of the obtained backward reachable sets and
show an overwhelming improvement of our proposed algorithms over
state-of-the-art methods regarding scalability, as systems with well over a
hundred states can now be analyzed.Comment: 16 page