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We present a predictive runtime monitoring technique for estimating future vehicle positions and the probability of collisions with obstacles. Vehicle dynamics model how the position and velocity change over time as a function of external inputs. They are commonly described by discrete-time stochastic models. Whereas positions and velocities can be measured, the inputs (steering and throttle) are not directly measurable in these models. In our paper, we apply Bayesian inference techniques for real-time estimation, given prior distribution over the unknowns and noisy state measurements. Next, we pre-compute the set-valued reachability analysis to approximate future positions of a vehicle. The pre-computed reachability sets are combined with the posterior probabilities computed through Bayesian estimation to provided a predictive verification framework that can be used to detect impending collisions with obstacles. Our approach is evaluated using the coordinated-turn vehicle model for a UAV using on-board measurement data obtained from a flight test of a Talon UAV. We also compare the results with sampling-based approaches. We find that precomputed reachability analysis can provide accurate warnings up to 6 seconds in advance and the accuracy of the warnings improve as the time horizon is narrowed from 6 to 2 seconds. The approach also outperforms sampling in terms of on-board computation cost and accuracy measures.

We propose a predictive runtime monitoring approach for linear systems with stochastic disturbances. The goal of the monitor is to decide if there exists a possible sequence of control inputs over a given time horizon to ensure that a safety property is maintained with a sufficiently high probability. We derive an efficient algorithm for performing the predictive monitoring in real time, specifically for linear time invariant (LTI) systems driven by stochastic disturbances. The algorithm implicitly defines a control envelope set such that if the current control input to the system lies in this set, there exists a future strategy over a time horizon consisting of the next N steps to guarantee the safety property of interest. As a result, the proposed monitor is oblivious of the actual controller, and therefore, applicable even in the presence of complex control systems including highly adaptive controllers. Furthermore, we apply our proposed approach to monitor whether a UAV will respect a “geofence” defined by a geographical region over which the vehicle may operate. To achieve this, we construct a data-driven linear model of the UAVs dynamics, while carefully modeling the uncertainties due to wind, GPS errors and modeling errors as time-varying disturbances. Using realistic data obtained from flight tests, we demonstrate the advantages and drawbacks of the predictive monitoring approach.