Ensuring robot safety can be challenging; user-defined constraints can miss edge cases, policies can become unsafe even when trained from safe data, and safety can be subjective. Thus, we learn about robot safety by showing policy trajectories to a human who flags unsafe behavior. From this binary feedback, we use the statistical method of conformal prediction to identify a region of states, potentially in learned latent space, guaranteed to contain a user-specified fraction of future policy errors. Our method is sample-efficient, as it builds on nearest neighbor classification and avoids withholding data as is common with conformal prediction. By alerting if the robot reaches the suspected unsafe region, we obtain a warning system that mimics the human's safety preferences with guaranteed miss rate. From video labeling, our system can detect when a quadcopter visuomotor policy will fail to steer through a designated gate. We present an approach for policy improvement by avoiding the suspected unsafe region. With it we improve a model predictive controller's safety, as shown in experimental testing with 30 quadcopter flights across 6 navigation tasks. Code and videos are provided.

Neural networks are increasingly used in robotics as policies, state transition models, state estimation models, or all of the above. With these components being learned from data, it is important to be able to analyze what behaviors were learned and how this affects closed-loop performance. In this article, we take steps toward this goal by developing methods for computing control invariant sets and regions of attraction (ROAs) of dynamical systems represented as neural networks. We focus our attention on feedforward neural networks with the rectified linear unit (ReLU) activation, which are known to implement continuous piecewise-affine (PWA) functions. We describe the reachable polyhedral marching (RPM) algorithm for enumerating the affine pieces of a neural network through an incremental connected walk. We then use this algorithm to compute exact forward and backward reachable sets, from which we provide methods for computing control invariant sets and ROAs. Our approach is unique in that we find these sets incrementally, without Lyapunov-based tools. In our examples, we demonstrate the ability of our approach to find nonconvex control invariant sets and ROAs on tasks with learned van der Pol oscillator and pendulum models. Further, we provide an accelerated algorithm for computing ROAs that leverages the incremental and connected enumeration of affine regions that RPM provides. We show this acceleration to lead to a $15\times $ speedup in our examples. Finally, we apply our methods to find a set of states that are stabilized by an image-based controller for an aircraft runway control problem.

In this article, we provide finite-sample performance guarantees for control policies executed on stochastic robotic systems. Given an open- or closed-loop policy and a finite set of trajectory rollouts under the policy, we bound the expected value, value at risk, and conditional value at risk of the trajectory cost, and the probability of failure in a sparse cost setting. The bounds hold, with user-specified probability, for any policy synthesis technique and can be seen as a postdesign safety certification. Generating the bounds only requires sampling simulation rollouts, without assumptions on the distribution or complexity of the underlying stochastic system. We adapt these bounds to also give a constraint satisfaction test to verify the safety of the robot system. We provide a thorough analysis of the bound sensitivity to sim-to-real distribution shifts and provide results for constructing robust bounds that can tolerate some specified amount of distribution shift. Furthermore, we extend our method to apply when selecting the best policy from a set of candidates, requiring a multihypothesis correction. We show the statistical validity of our bounds in the Ant, Half-cheetah, and Swimmer MuJoCo environments and demonstrate our constraint satisfaction test with the Ant. Finally, using the 20-degree-of-freedom MuJoCo Shadow Hand, we show the necessity of the multihypothesis correction.

We present a method for computing exact reachable sets for deep neural networks with rectified linear unit (ReLU) activation. Our method is well-suited for use in rigorous safety analysis of robotic perception and control systems with deep neural network components. Our algorithm can compute both forward and backward reachable sets for a ReLU network iterated over multiple time steps, as would be found in a perception-action loop in a robotic system. Our algorithm is unique in that it builds the reachable sets by incrementally enumerating polyhedral cells in the input space, rather than iterating layer-by-layer through the network as in other methods. If an unsafe cell is found, our algorithm can return this result without completing the full reachability computation, thus giving an anytime property that accelerates safety verification. In addition, our method requires less memory during execution compared to existing methods where memory can be a limiting factor. We demonstrate our algorithm on safety verification of the ACAS Xu aircraft advisory system. We find unsafe actions many times faster than the fastest existing method and certify no unsafe actions exist in about twice the time of the existing method. We also compute forward and backward reachable sets for a learned model of pendulum dynamics over a 50 time step horizon in 87s on a laptop computer. Algorithm source code: https://github.com/StanfordMSL/Neural-Network-Reach.