Formal Verification and Control with Conformal Prediction

In this survey, we design formal verification and control algorithms for autonomous systems with practical safety guarantees using conformal prediction (CP), a statistical tool for uncertainty quantification. We focus on learning-enabled autonomous systems (LEASs) in which the complexity of learning-enabled components (LECs) is a major bottleneck that hampers the use of existing model-based verification and design techniques. Instead, we advocate for the use of CP, and we will demonstrate its use in formal verification, systems and control theory, and robotics. We argue that CP is specifically useful due to its simplicity (easy to understand, use, and modify), generality (requires no assumptions on learned models and data distributions, i.e., is distribution-free), and efficiency (real-time capable and accurate). We pursue the following goals with this survey. First, we provide an accessible introduction to CP for non-experts who are interested in using CP to solve problems in autonomy. Second, we show how to use CP for the verification of LECs, e.g., for verifying input-output properties of neural networks. Third and fourth, we review recent articles that use CP for safe control design as well as offline and online verification of LEASs. We summarize their ideas in a unifying framework that can deal with the complexity of LEASs in a computationally efficient manner. In our exposition, we consider simple system specifications, e.g., robot navigation tasks, as well as complex specifications formulated in temporal logic formalisms. Throughout our survey, we compare to other statistical techniques (e.g., scenario optimization, PAC-Bayes theory, etc.) and how these techniques have been used in verification and control. Lastly, we point the reader to open problems and future research directions.

Statistical Reachability Analysis of Stochastic Cyber-Physical Systems under Distribution Shift

Reachability analysis is a popular method to give safety guarantees for stochastic cyber-physical systems (SCPSs) that takes in a symbolic description of the system dynamics and uses set-propagation methods to compute an overapproximation of the set of reachable states over a bounded time horizon. In this paper, we investigate the problem of performing reachability analysis for an SCPS that does not have a symbolic description of the dynamics, but instead is described using a digital twin model that can be simulated to generate system trajectories. An important challenge is that the simulator implicitly models a probability distribution over the set of trajectories of the SCPS; however, it is typical to have a sim2real gap, i.e., the actual distribution of the trajectories in a deployment setting may be shifted from the distribution assumed by the simulator. We thus propose a statistical reachability analysis technique that, given a user-provided threshold $1-\epsilon$, provides a set that guarantees that any reachable state during deployment lies in this set with probability not smaller than this threshold. Our method is based on three main steps: (1) learning a deterministic surrogate model from sampled trajectories, (2) conducting reachability analysis over the surrogate model, and (3) employing {\em robust conformal inference} using an additional set of sampled trajectories to quantify the surrogate model's distribution shift with respect to the deployed SCPS. To counter conservatism in reachable sets, we propose a novel method to train surrogate models that minimizes a quantile loss term (instead of the usual mean squared loss), and a new method that provides tighter guarantees using conformal inference using a normalized surrogate error. We demonstrate the effectiveness of our technique on various case studies.

Recursively Feasible Shrinking-Horizon MPC in Dynamic Environments with Conformal Prediction Guarantees

In this paper, we focus on the problem of shrinking-horizon Model Predictive Control (MPC) in uncertain dynamic environments. We consider controlling a deterministic autonomous system that interacts with uncontrollable stochastic agents during its mission. Employing tools from conformal prediction, existing works derive high-confidence prediction regions for the unknown agent trajectories, and integrate these regions in the design of suitable safety constraints for MPC. Despite guaranteeing probabilistic safety of the closed-loop trajectories, these constraints do not ensure feasibility of the respective MPC schemes for the entire duration of the mission. We propose a shrinking-horizon MPC that guarantees recursive feasibility via a gradual relaxation of the safety constraints as new prediction regions become available online. This relaxation enforces the safety constraints to hold over the least restrictive prediction region from the set of all available prediction regions. In a comparative case study with the state of the art, we empirically show that our approach results in tighter prediction regions and verify recursive feasibility of our MPC scheme.

Reactive Temporal Logic-based Planning and Control for Interactive Robotic Tasks

Robots interacting with humans must be safe, reactive and adapt online to unforeseen environmental and task changes. Achieving these requirements concurrently is a challenge as interactive planners lack formal safety guarantees, while safe motion planners lack flexibility to adapt. To tackle this, we propose a modular control architecture that generates both safe and reactive motion plans for human-robot interaction by integrating temporal logic-based discrete task level plans with continuous Dynamical System (DS)-based motion plans. We formulate a reactive temporal logic formula that enables users to define task specifications through structured language, and propose a planning algorithm at the task level that generates a sequence of desired robot behaviors while being adaptive to environmental changes. At the motion level, we incorporate control Lyapunov functions and control barrier functions to compute stable and safe continuous motion plans for two types of robot behaviors: (i) complex, possibly periodic motions given by autonomous DS and (ii) time-critical tasks specified by Signal Temporal Logic~(STL). Our methodology is demonstrated on the Franka robot arm performing wiping tasks on a whiteboard and a mannequin that is compliant to human interactions and adaptive to environmental changes.

Robust STL Control Synthesis under Maximal Disturbance Sets

This work addresses maximally robust control synthesis under unknown disturbances. We consider a general nonlinear system, subject to a Signal Temporal Logic (STL) specification, and wish to jointly synthesize the maximal possible disturbance bounds and the corresponding controllers that ensure the STL specification is satisfied under these bounds. Many works have considered STL satisfaction under given bounded disturbances. Yet, to the authors' best knowledge, this is the first work that aims to maximize the permissible disturbance set and find the corresponding controllers that ensure satisfying the STL specification with maximum disturbance robustness. We extend the notion of disturbance-robust semantics for STL, which is a property of a specification, dynamical system, and controller, and provide an algorithm to get the maximal disturbance robust controllers satisfying an STL specification using Hamilton-Jacobi reachability. We show its soundness and provide a simulation example with an Autonomous Underwater Vehicle (AUV).

Risk-Aware Robotics: Tail Risk Measures in Planning, Control, and Verification

The need for a systematic approach to risk assessment has increased in recent years due to the ubiquity of autonomous systems that alter our day-to-day experiences and their need for safety, e.g., for self-driving vehicles, mobile service robots, and bipedal robots. These systems are expected to function safely in unpredictable environments and interact seamlessly with humans, whose behavior is notably challenging to forecast. We present a survey of risk-aware methodologies for autonomous systems. We adopt a contemporary risk-aware approach to mitigate rare and detrimental outcomes by advocating the use of tail risk measures, a concept borrowed from financial literature. This survey will introduce these measures and explain their relevance in the context of robotic systems for planning, control, and verification applications.

Conformalized Adaptive Forecasting of Heterogeneous Trajectories

This paper presents a new conformal method for generating simultaneous forecasting bands guaranteed to cover the entire path of a new random trajectory with sufficiently high probability. Prompted by the need for dependable uncertainty estimates in motion planning applications where the behavior of diverse objects may be more or less unpredictable, we blend different techniques from online conformal prediction of single and multiple time series, as well as ideas for addressing heteroscedasticity in regression. This solution is both principled, providing precise finite-sample guarantees, and effective, often leading to more informative predictions than prior methods.

Conformal Predictive Programming for Chance Constrained Optimization

Motivated by the advances in conformal prediction (CP), we propose conformal predictive programming (CPP), an approach to solve chance constrained optimization (CCO) problems, i.e., optimization problems with nonlinear constraint functions affected by arbitrary random parameters. CPP utilizes samples from these random parameters along with the quantile lemma -- which is central to CP -- to transform the CCO problem into a deterministic optimization problem. We then present two tractable reformulations of CPP by: (1) writing the quantile as a linear program along with its KKT conditions (CPP-KKT), and (2) using mixed integer programming (CPP-MIP). CPP comes with marginal probabilistic feasibility guarantees for the CCO problem that are conceptually different from existing approaches, e.g., the sample approximation and the scenario approach. While we explore algorithmic similarities with the sample approximation approach, we emphasize that the strength of CPP is that it can easily be extended to incorporate different variants of CP. To illustrate this, we present robust conformal predictive programming to deal with distribution shifts in the uncertain parameters of the CCO problem.

Multi-Modal Conformal Prediction Regions by Optimizing Convex Shape Templates

Conformal prediction is a statistical tool for producing prediction regions for machine learning models that are valid with high probability. A key component of conformal prediction algorithms is a non-conformity score function that quantifies how different a model's prediction is from the unknown ground truth value. Essentially, these functions determine the shape and the size of the conformal prediction regions. However, little work has gone into finding non-conformity score functions that produce prediction regions that are multi-modal and practical, i.e., that can efficiently be used in engineering applications. We propose a method that optimizes parameterized shape template functions over calibration data, which results in non-conformity score functions that produce prediction regions with minimum volume. Our approach results in prediction regions that are multi-modal, so they can properly capture residuals of distributions that have multiple modes, and practical, so each region is convex and can be easily incorporated into downstream tasks, such as a motion planner using conformal prediction regions. Our method applies to general supervised learning tasks, while we illustrate its use in time-series prediction. We provide a toolbox and present illustrative case studies of F16 fighter jets and autonomous vehicles, showing an up to $68\%$ reduction in prediction region area.

Robust Conformal Prediction for STL Runtime Verification under Distribution Shift

Cyber-physical systems (CPS) designed in simulators behave differently in the real-world. Once they are deployed in the real-world, we would hence like to predict system failures during runtime. We propose robust predictive runtime verification (RPRV) algorithms under signal temporal logic (STL) tasks for general stochastic CPS. The RPRV problem faces several challenges: (1) there may not be sufficient data of the behavior of the deployed CPS, (2) predictive models are based on a distribution over system trajectories encountered during the design phase, i.e., there may be a distribution shift during deployment. To address these challenges, we assume to know an upper bound on the statistical distance (in terms of an f-divergence) between the distributions at deployment and design time, and we utilize techniques based on robust conformal prediction. Motivated by our results in [1], we construct an accurate and an interpretable RPRV algorithm. We use a trajectory prediction model to estimate the system behavior at runtime and robust conformal prediction to obtain probabilistic guarantees by accounting for distribution shifts. We precisely quantify the relationship between calibration data, desired confidence, and permissible distribution shift. To the best of our knowledge, these are the first statistically valid algorithms under distribution shift in this setting. We empirically validate our algorithms on a Franka manipulator within the NVIDIA Isaac sim environment.