Advancement: A geometric approach for learning reach sets

Shadi Haddad
Applied Mathematics Ph.D. Student
Applied Mathematics Ph.D.
Virtual Event
Abhishek Halder

Join us on Zoom: Passcode: 328589

Description: Reachability analysis is a method to guarantee the performance of safety-critical applications such as automated driving and robotics against uncertainties. The main object of study is the reach sets, defined as the set of states that the controlled dynamics may reach in a future time, given the set-valued description of uncertainties. We develop the theory and algorithms for learning the reach sets of full state feedback linearizable systems - an important class of nonlinear control systems common in vehicular applications such as automobiles and drones. These reach sets, in very general settings, are compact but nonconvex. The new idea we propose is to perform an exact computation of these reach sets in the associated Brunovsky normal coordinates, and then transform the sets back to the original coordinates via known diffeomorphisms. Our algorithms exploit learning-theoretic ideas to provide probabilistic guarantees on the computed sets.

As a by-product of our analysis, we uncover the exact geometry of the integrator reach sets with compact set-valued inputs. The exact results include the closed-form parametric and implicit formula for the boundaries, volumes, and widths of the integrator reach sets. These results on integrators should be of independent interest to serve as benchmarks for quantifying the conservatism in reach set computation algorithms.