  In this paper, we address the real-time risk-bounded safety verification
problem of continuous-time state trajectories of autonomous systems in the
presence of uncertain time-varying nonlinear safety constraints. Risk is
defined as the probability of not satisfying the uncertain safety constraints.
Existing approaches to address the safety verification problems under
uncertainties either are limited to particular classes of uncertainties and
safety constraints, e.g., Gaussian uncertainties and linear constraints, or
rely on sampling based methods. In this paper, we provide a fast convex
algorithm to efficiently evaluate the probabilistic nonlinear safety
constraints in the presence of arbitrary probability distributions and long
planning horizons in real-time, without the need for uncertainty samples and
time discretization. The provided approach verifies the safety of the given
state trajectory and its neighborhood (tube) to account for the execution
uncertainties and risk. In the provided approach, we first use the moments of
the probability distributions of the uncertainties to transform the
probabilistic safety constraints into a set of deterministic safety
constraints. We then use convex methods based on sum-of-squares polynomials to
verify the obtained deterministic safety constraints over the entire planning
time horizon without time discretization. To illustrate the performance of the
proposed method, we apply the provided method to the safety verification
problem of self-driving vehicles and autonomous aerial vehicles.
