On the Difficulty of Intersection Checking with Polynomial Zonotopes

by   Yushen Huang, et al.

Polynomial zonotopes, a non-convex set representation, have a wide range of applications from real-time motion planning and control in robotics, to reachability analysis of nonlinear systems and safety shielding in reinforcement learning. Despite this widespread use, a frequently overlooked difficulty associated with polynomial zonotopes is intersection checking. Determining whether the reachable set, represented as a polynomial zonotope, intersects an unsafe set is not straightforward. In fact, we show that this fundamental operation is NP-hard, even for a simple class of polynomial zonotopes. The standard method for intersection checking with polynomial zonotopes is a two-part algorithm that overapproximates a polynomial zonotope with a regular zonotope and then, if the overapproximation error is deemed too large, splits the set and recursively tries again. Beyond the possible need for a large number of splits, we identify two sources of concern related to this algorithm: (1) overapproximating a polynomial zonotope with a zonotope has unbounded error, and (2) after splitting a polynomial zonotope, the overapproximation error can actually increase. Taken together, this implies there may be a possibility that the algorithm does not always terminate.We perform a rigorous analysis of the method and detail necessary conditions for the union of overapproximations to provably converge to the original polynomial zonotope.


page 1

page 2

page 3

page 4


A Fully Polynomial Time Approximation Scheme For A NP-Hard Problem

We present a novel feasibility criteria for the intersection of convex s...

Is a Finite Intersection of Balls Covered by a Finite Union of Balls in Euclidean Spaces ?

Considering a finite intersection of balls and a finite union of other b...

A tractable class of binary VCSPs via M-convex intersection

A binary VCSP is a general framework for the minimization problem of a f...

Regular Model Checking Upside-Down: An Invariant-Based Approach

Regular model checking is a well-established technique for the verificat...

Polynomial Logical Zonotopes: A Set Representation for Reachability Analysis of Logical Systems

In this paper, we introduce a set representation called polynomial logic...

Efficient Backward Reachability Using the Minkowski Difference of Constrained Zonotopes

Backward reachability analysis is essential to synthesizing controllers ...

Please sign up or login with your details

Forgot password? Click here to reset