Khalil Ghorbal

On Provably Safe Obstacle Avoidance for Autonomous Robotic Ground Vehicles

Nowadays, robots interact more frequently with a dynamic environment outside limited manufacturing sites and in close proximity with humans. Thus, safety of motion and obstacle avoidance are vital safety features of such robots. We formally study two safety properties of avoiding both stationary and moving obstacles: (i) passive safety, which ensures that no collisions can happen while the robot moves, and (ii) the stronger passive friendly safety in which the robot further maintains sufficient maneuvering distance for obstacles to avoid collision as well. We use hybrid system models and theorem proving techniques that describe and formally verify the robot’s discrete control decisions along with its continuous, physical motion. Moreover, we formally prove that safety can still be guaranteed despite location and actuator uncertainty.

A Formally Verified Hybrid System for the Next-Generation Airborne Collision Avoidance System

The Next-Generation Airborne Collision Avoidance System ACASi¾?X is intended to be installed on all large aircraft to give advice to pilots and prevent mid-air collisions with other aircraft. It is currently being developed by the Federal Aviation Administration FAA. In this paper we determine the geometric configurations under which the advice given by ACAS X is safe under a precise set of assumptions and formally verify these configurations using hybrid systems theorem proving techniques. We conduct an initial examination of the current version of the real ACAS X system and discuss some cases where our safety theorem conflicts with the actual advisory given by that version, demonstrating how formal, hybrid approaches are helping ensure the safety of ACAS X. Our approach is general and could also be used to identify unsafe advice issued by other collision avoidance systems or confirm their safety.

Characterizing Algebraic Invariants by Differential Radical Invariants

We prove that any invariant algebraic set of a given polynomial vector field can be algebraically represented by one polynomial and a finite set of its successive Lie derivatives. This so-called differential radical characterization relies on a sound abstraction of the reachable set of solutions by the smallest variety that contains it. The characterization leads to a differential radical invariant proof rule that is sound and complete, which implies that invariance of algebraic equations over real-closed fields is decidable. Furthermore, the problem of generating invariant varieties is shown to be as hard as minimizing the rank of a symbolic matrix, and is therefore NP-hard. We investigate symbolic linear algebra tools based on Gaussian elimination to efficiently automate the generation. The approach can, e.g., generate nontrivial algebraic invariant equations capturing the airplane behavior during take-off or landing in longitudinal motion.

A Method for Invariant Generation for Polynomial Continuous Systems

This paper presents a method for generating semi-algebraic invariants for systems governed by non-linear polynomial ordinary differential equations under semi-algebraic evolution constraints. Based on the notion of discrete abstraction, our method eliminates unsoundness and unnecessary coarseness found in existing approaches for computing abstractions for non-linear continuous systems and is able to construct invariants with intricate boolean structure, in contrast to invariants typically generated using template-based methods. In order to tackle the state explosion problem associated with discrete abstraction, we present invariant generation algorithms that exploit sound proof rules for safety verification, such as differential cut

Formal verification of obstacle avoidance and navigation of ground robots

{\text {DC}}

A Hierarchy of Proof Rules for Checking Differential Invariance of Algebraic Sets

, and a new proof rule that we call differential divide-and-conquer

Invariance of Conjunctions of Polynomial Equalities for Algebraic Differential Equations

{\text {DDC}}