The philosophy, architecture, and capabilities of the COntinuous Reachability Analyzer (CORA) are presented. CORA is a toolbox that integrates various vector and matrix set representations and operations on them as well as reachability algorithms of various dynamic system classes. The software is designed such that set representations can be exchanged without having to modify the code for reachability analysis. CORA has a modular design, making it possible to use the capabilities of the various set representations for other purposes besides reachability analysis. The toolbox is designed using the object oriented paradigm, such that users can safely use methods without concerning themselves with detailed information hidden inside the object. Since the toolbox is written in MATLAB, the installation and use is platform independent.

An approach for formally verifying the safety of automated vehicles is proposed. Due to the uniqueness of each traffic situation, we verify safety online, i.e., during the operation of the vehicle. The verification is performed by predicting the set of all possible occupancies of the automated vehicle and other traffic participants on the road. In order to capture all possible future scenarios, we apply reachability analysis to consider all possible behaviors of mathematical models considering uncertain inputs (e.g. sensor noise, disturbances) and partially unknown initial states. Safety is guaranteed with respect to the modeled uncertainties and behaviors if the occupancy of the automated vehicle does not intersect that of other traffic participants for all times. The applicability of the approach is demonstrated by test drives with an automated vehicle of the Robotics Institute at Carnegie Mellon University.

Given an initial set of a nonlinear system with uncertain parameters and inputs, the set of states that can possibly be reached is computed. The approach is based on local linearizations of the nonlinear system, while linearization errors are considered by Lagrange remainders. These errors are added as uncertain inputs, such that the reachable set of the locally linearized system encloses the one of the original system. The linearization error is controlled by splitting of reachable sets. Reachable sets are represented by zonotopes, allowing an efficient computation in relatively high-dimensional space.

Abstract-Safety of planned paths of autonomous cars with respect to the movement of other traffic participants is considered. Thereto, the stochastic occupancy of the road by other vehicles is predicted. The prediction considers uncertainties originating from the measurements and the possible behaviors of other traffic participants. In addition, the interaction of traffic participants as well as the limitation of driving maneuvers due to the road geometry is considered. The result of the presented approach is the probability of a crash for a specific trajectory of the autonomous car. The presented approach is efficient as most intensive computations are performed offline, resulting in a lean online algorithm for real-time application.

A new technique for computing the reachable set of hybrid systems with nonlinear continuous dynamics is presented. Previous work showed that abstracting the nonlinear continuous dynamics to linear differential inclusions results in a scalable approach for reachability analysis. However, when the abstraction becomes inaccurate, linearization techniques require splitting of reachable sets, resulting in an exponential growth of required linearizations. In this work, the nonlinearity of the dynamics is more accurately abstracted to polynomial difference inclusions. As a consequence, it is no longer guaranteed that reachable sets of consecutive time steps are mapped to convex sets as typically used in previous works. Thus, a non-convex set representation is developed in order to better capture the nonlinear dynamics, requiring no or much less splitting. The new approach has polynomial complexity with respect to the number of continuous state variables when splitting can be avoided and is thus promising when a linearization technique requires splitting for the same problem. The benefits are presented by numerical examples.

Abstract-Numerical experiments for motion planning of road vehicles require numerous components: vehicle dynamics, a road network, static obstacles, dynamic obstacles and their movement over time, goal regions, a cost function, etc. Providing a description of the numerical experiment precise enough to reproduce it might require several pages of information. Thus, only key aspects are typically described in scientific publications, making it impossible to reproduce results-yet, reproducibility is an important asset of good science. Composable benchmarks for motion planning on roads (CommonRoad) are proposed so that numerical experiments are fully defined by a unique ID; all information required to reconstruct the experiment can be found on the CommonRoad website. Each benchmark is composed by a vehicle model, a cost function, and a scenario (including goals and constraints). The scenarios are partly recorded from real traffic and partly hand-crafted to create dangerous situations. We hope that CommonRoad saves researchers time since one does not have to search for realistic parameters of vehicle dynamics or realistic traffic situations, yet provides the freedom to compose a benchmark that fits one's needs.

a b s t r a c tThe computation of reachable sets for hybrid systems with linear continuous dynamics is addressed. Zonotopes are used for the representation of reachable sets, resulting in an algorithm with low computational complexity with respect to the dimension of the considered system. However, zonotopes have drawbacks when being intersected with transition guards which determine the discrete behavior of the hybrid system. For this reason, in the proposed approach, reachable sets are represented by polytopes within guard sets as an intermediate step in order to enclose them by zonotopes afterwards. Different methods for the conservative conversion from zonotopes to polytopes and vice versa are proposed and numerically evaluated.

This paper presents a numerical procedure for the reachability analysis of systems with nonlinear, semi-explicit, index-1 differential-algebraic equations. The procedure computes reachable sets for uncertain initial states and inputs in an overapproximative way, i.e. it is guaranteed that all possible trajectories of the system are enclosed. Thus, the result can be used for formal verification of system properties that can be specified in the state space as unsafe or goal regions. Due to the representation of reachable sets by zonotopes and the use of highly scalable operations on them, the presented approach scales favorably with the number of state variables. This makes it possible to solve problems of industry-relevant size, as demonstrated by a transient stability analysis of the IEEE 14-bus benchmark problem for power systems.