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Hamilton Jacobi (HJ) Reachability is a formal verification tool widely usedin robotic safety analysis. Given a target set as unsafe states, a dynamicalsystem is guaranteed not to enter the target under the worst-case disturbanceif it avoids the Backward Reachable Tube (BRT). However, computing BRTs suffersfrom exponential computational time and space complexity with respect to thestate dimension. Previously, system decomposition and projection techniqueshave been investigated, but the trade off between applicability to a widerclass of dynamics and degree of conservatism has been challenging. In thispaper, we propose a State Dependency Graph to represent the system dynamics,and decompose the full system where only dependent states are included in eachsubsystem, and "missing" states are treated as bounded disturbance. Thus for alarge variety of dynamics in robotics, BRTs can be quickly approximated inlower-dimensional chained subsystems with the guaranteed-safety propertypreserved. We demonstrate our method with numerical experiments on the 4DQuadruple Integrator, and the 6D Bicycle, an important car model that wasformerly intractable.

Guaranteed-Safe Approximate Reachability via State Dependency-Based Decomposition