Stable-BC: Controlling Covariate Shift with Stable Behavior Cloning

Behavior cloning is a common imitation learning paradigm. Under behavior cloning the robot collects expert demonstrations, and then trains a policy to match the actions taken by the expert. This works well when the robot learner visits states where the expert has already demonstrated the correct action; but inevitably the robot will also encounter new states outside of its training dataset. If the robot learner takes the wrong action at these new states it could move farther from the training data, which in turn leads to increasingly incorrect actions and compounding errors. Existing works try to address this fundamental challenge by augmenting or enhancing the training data. By contrast, in our paper we develop the control theoretic properties of behavior cloned policies. Specifically, we consider the error dynamics between the system's current state and the states in the expert dataset. From the error dynamics we derive model-based and model-free conditions for stability: under these conditions the robot shapes its policy so that its current behavior converges towards example behaviors in the expert dataset. In practice, this results in Stable-BC, an easy to implement extension of standard behavior cloning that is provably robust to covariate shift. We demonstrate the effectiveness of our algorithm in simulations with interactive, nonlinear, and visual environments. We also conduct experiments where a robot arm uses Stable-BC to play air hockey. See our website here: https://collab.me.vt.edu/Stable-BC/

Parameterized Fast and Safe Tracking (FaSTrack) using Deepreach

Fast and Safe Tracking (FaSTrack) is a modular framework that provides safety guarantees while planning and executing trajectories in real time via value functions of Hamilton-Jacobi (HJ) reachability. These value functions are computed through dynamic programming, which is notorious for being computationally inefficient. Moreover, the resulting trajectory does not adapt online to the environment, such as sudden disturbances or obstacles. DeepReach is a scalable deep learning method to HJ reachability that allows parameterization of states, which opens up possibilities for online adaptation to various controls and disturbances. In this paper, we propose Parametric FaSTrack, which uses DeepReach to approximate a value function that parameterizes the control bounds of the planning model. The new framework can smoothly trade off between the navigation speed and the tracking error (therefore maneuverability) while guaranteeing obstacle avoidance in a priori unknown environments. We demonstrate our method through two examples and a benchmark comparison with existing methods, showing the safety, efficiency, and faster solution times of the framework.

SAFE-GIL: SAFEty Guided Imitation Learning

Behavior Cloning is a popular approach to Imitation Learning, in which a robot observes an expert supervisor and learns a control policy. However, behavior cloning suffers from the "compounding error" problem - the policy errors compound as it deviates from the expert demonstrations and might lead to catastrophic system failures, limiting its use in safety-critical applications. On-policy data aggregation methods are able to address this issue at the cost of rolling out and repeated training of the imitation policy, which can be tedious and computationally prohibitive. We propose SAFE-GIL, an off-policy behavior cloning method that guides the expert via adversarial disturbance during data collection. The algorithm abstracts the imitation error as an adversarial disturbance in the system dynamics, injects it during data collection to expose the expert to safety critical states, and collects corrective actions. Our method biases training to more closely replicate expert behavior in safety-critical states and allows more variance in less critical states. We compare our method with several behavior cloning techniques and DAgger on autonomous navigation and autonomous taxiing tasks and show higher task success and safety, especially in low data regimes where the likelihood of error is higher, at a slight drop in the performance.

Imposing Exact Safety Specifications in Neural Reachable Tubes

Hamilton-Jacobi (HJ) reachability analysis is a verification tool that provides safety and performance guarantees for autonomous systems. It is widely adopted because of its ability to handle nonlinear dynamical systems with bounded adversarial disturbances and constraints on states and inputs. However, it involves solving a PDE to compute a safety value function, whose computational and memory complexity scales exponentially with the state dimension, making its direct usage in large-scale systems intractable. Recently, a learning-based approach called DeepReach, has been proposed to approximate high-dimensional reachable tubes using neural networks. While DeepReach has been shown to be effective, the accuracy of the learned solution decreases with the increase in system complexity. One of the reasons for this degradation is the inexact imposition of safety constraints during the learning process, which corresponds to the PDE's boundary conditions. Specifically, DeepReach imposes boundary conditions as soft constraints in the loss function, which leaves room for error during the value function learning. Moreover, one needs to carefully adjust the relative contributions from the imposition of boundary conditions and the imposition of the PDE in the loss function. This, in turn, induces errors in the overall learned solution. In this work, we propose a variant of DeepReach that exactly imposes safety constraints during the learning process by restructuring the overall value function as a weighted sum of the boundary condition and neural network output. This eliminates the need for a boundary loss during training, thus bypassing the need for loss adjustment. We demonstrate the efficacy of the proposed approach in significantly improving the accuracy of learned solutions for challenging high-dimensional reachability tasks, such as rocket-landing and multivehicle collision-avoidance problems.

Providing Safety Assurances for Systems with Unknown Dynamics

As autonomous systems become more complex and integral in our society, the need to accurately model and safely control these systems has increased significantly. In the past decade, there has been tremendous success in using deep learning techniques to model and control systems that are difficult to model using first principles. However, providing safety assurances for such systems remains difficult, partially due to the uncertainty in the learned model. In this work, we aim to provide safety assurances for systems whose dynamics are not readily derived from first principles and, hence, are more advantageous to be learned using deep learning techniques. Given the system of interest and safety constraints, we learn an ensemble model of the system dynamics from data. Leveraging ensemble uncertainty as a measure of uncertainty in the learned dynamics model, we compute a maximal robust control invariant set, starting from which the system is guaranteed to satisfy the safety constraints under the condition that realized model uncertainties are contained in the predefined set of admissible model uncertainty. We demonstrate the effectiveness of our method using a simulated case study with an inverted pendulum and a hardware experiment with a TurtleBot. The experiments show that our method robustifies the control actions of the system against model uncertainty and generates safe behaviors without being overly restrictive. The codes and accompanying videos can be found on the project website.

On Safety and Liveness Filtering Using Hamilton-Jacobi Reachability Analysis

Hamilton-Jacobi (HJ) reachability-based filtering provides a powerful framework to co-optimize performance and safety (or liveness) for autonomous systems. Under this filtering scheme, a nominal controller is minimally modified to ensure system safety or liveness. However, the resulting controllers can exhibit abrupt switching and bang-bang behavior, which is not suitable for applications of autonomous systems in the real world. This work presents a novel, unifying framework to design safety and liveness filters through reachability analysis. We explicitly characterize the maximal set of control inputs that ensures safety (or liveness) at a given state. Different safety filters can then be constructed using different subsets of this maximal set along with a projection operator to modify the nominal controller. We use the proposed framework to design three safety filters, each balancing performance, computation time, and smoothness differently. The proposed filters can easily handle dynamics uncertainties, disturbances, and bounded control inputs. We highlight their relative strengths and limitations by applying these filters to autonomous navigation and rocket landing scenarios and on a physical robot testbed. We also discuss practical aspects associated with implementing these filters on real-world autonomous systems. Our research advances the understanding and potential application of reachability-based controllers on real-world autonomous systems.

Verification of Neural Reachable Tubes via Scenario Optimization and Conformal Prediction

Learning-based approaches for controlling safety-critical systems are rapidly growing in popularity; thus, it is important to assure their performance and safety. Hamilton-Jacobi (HJ) reachability analysis is a popular formal verification tool for providing such guarantees, since it can handle general nonlinear system dynamics, bounded adversarial system disturbances, and state and input constraints. However, its computational and memory complexity scales exponentially with the state dimension, making it intractable for large-scale systems. To overcome this challenge, neural approaches, such as DeepReach, have been used to synthesize reachable tubes and safety controllers for high-dimensional systems. However, verifying these neural reachable tubes remains challenging. In this work, we propose two verification methods, based on robust scenario optimization and conformal prediction, to provide probabilistic safety guarantees for neural reachable tubes. Our methods allow a direct trade-off between resilience to outlier errors in the neural tube, which are inevitable in a learning-based approach, and the strength of the probabilistic safety guarantee. Furthermore, we show that split conformal prediction, a widely used method in the machine learning community for uncertainty quantification, reduces to a scenario-based approach, making the two methods equivalent not only for verification of neural reachable tubes but also more generally. To our knowledge, our proof is the first in the literature to show a strong relationship between conformal prediction and scenario optimization. Finally, we propose an outlier-adjusted verification approach that uses the error distribution in neural reachable tubes to recover greater safe volumes. We demonstrate the efficacy of the proposed approaches for the high-dimensional problems of multi-vehicle collision avoidance and rocket landing with no-go zones.

Detecting and Mitigating System-Level Anomalies of Vision-Based Controllers

Autonomous systems, such as self-driving cars and drones, have made significant strides in recent years by leveraging visual inputs and machine learning for decision-making and control. Despite their impressive performance, these vision-based controllers can make erroneous predictions when faced with novel or out-of-distribution inputs. Such errors can cascade to catastrophic system failures and compromise system safety. In this work, we introduce a run-time anomaly monitor to detect and mitigate such closed-loop, system-level failures. Specifically, we leverage a reachability-based framework to stress-test the vision-based controller offline and mine its system-level failures. This data is then used to train a classifier that is leveraged online to flag inputs that might cause system breakdowns. The anomaly detector highlights issues that transcend individual modules and pertain to the safety of the overall system. We also design a fallback controller that robustly handles these detected anomalies to preserve system safety. We validate the proposed approach on an autonomous aircraft taxiing system that uses a vision-based controller for taxiing. Our results show the efficacy of the proposed approach in identifying and handling system-level anomalies, outperforming methods such as prediction error-based detection, and ensembling, thereby enhancing the overall safety and robustness of autonomous systems.

Hamilton-Jacobi Reachability Analysis for Hybrid Systems with Controlled and Forced Transitions

Hybrid dynamical systems with non-linear dynamics are one of the most general modeling tools for representing robotic systems, especially contact-rich systems. However, providing guarantees regarding the safety or performance of such hybrid systems can still prove to be a challenging problem because it requires simultaneous reasoning about continuous state evolution and discrete mode switching. In this work, we address this problem by extending classical Hamilton-Jacobi (HJ) reachability analysis, a formal verification method for continuous non-linear dynamics in the presence of bounded inputs and disturbances, to hybrid dynamical systems. Our framework can compute reachable sets for hybrid systems consisting of multiple discrete modes, each with its own set of non-linear continuous dynamics, discrete transitions that can be directly commanded or forced by a discrete control input, while still accounting for control bounds and adversarial disturbances in the state evolution. Along with the reachable set, the proposed framework also provides an optimal continuous and discrete controller to ensure system safety. We demonstrate our framework in simulation on an aircraft collision avoidance problem, as well as on a real-world testbed to solve the optimal mode planning problem for a quadruped with multiple gaits.

Discovering Closed-Loop Failures of Vision-Based Controllers via Reachability Analysis

Machine learning driven image-based controllers allow robotic systems to take intelligent actions based on the visual feedback from their environment. Understanding when these controllers might lead to system safety violations is important for their integration in safety-critical applications and engineering corrective safety measures for the system. Existing methods leverage simulation-based testing (or falsification) to find the failures of vision-based controllers, i.e., the visual inputs that lead to closed-loop safety violations. However, these techniques do not scale well to the scenarios involving high-dimensional and complex visual inputs, such as RGB images. In this work, we cast the problem of finding closed-loop vision failures as a Hamilton-Jacobi (HJ) reachability problem. Our approach blends simulation-based analysis with HJ reachability methods to compute an approximation of the backward reachable tube (BRT) of the system, i.e., the set of unsafe states for the system under vision-based controllers. Utilizing the BRT, we can tractably and systematically find the system states and corresponding visual inputs that lead to closed-loop failures. These visual inputs can be subsequently analyzed to find the input characteristics that might have caused the failure. Besides its scalability to high-dimensional visual inputs, an explicit computation of BRT allows the proposed approach to capture non-trivial system failures that are difficult to expose via random simulations. We demonstrate our framework on two case studies involving an RGB image-based neural network controller for (a) autonomous indoor navigation, and (b) autonomous aircraft taxiing.