arXiv subjects

Claire J. Tomlin

Publications and source records attributed to Claire J. Tomlin.

At least 19 recordsLinked to original sources

arXiv2026

From Global to Local: Hierarchical Probabilistic Verification for Reachability Learning

Hamilton-Jacobi (HJ) reachability provides formal safety guarantees for nonlinear systems. However, it becomes computationally intractable in high-dimensional settings, motivating learning-based approximations that may introduce unsafe errors or overly optimistic safe sets. In this work, we propose a hierarchical probabilistic verification framework for reachability learning that bridges offline global certification and online local refinement. We first construct a coarse, conservative under-approximation of the reach-avoid safe set using scenario optimization, providing an efficient global probabilistic certificate. We then introduce an online local refinement module that is structurally anchored to the global certificate and recovers regions excluded by the global verification. This refinement reduces conservatism while focusing computation on critical regions of the state space. Integrated with a condition-based switching mechanism between a learned reachability policy and a model-based controller, the proposed framework improves success rates in goal-reaching tasks with safety constraints, as demonstrated in drone-racing simulation experiments.

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arXiv2026

Reachability-Guided Sequential Quadratic Programming-Guarded Model Predictive Path Integral for Safe Nonlinear Predictive Control

Safe robot control often requires combining long-horizon performance optimization with hard state and input constraints, but existing approaches tend to address this tradeoff partially. Sampling-based model predictive control (MPC) methods such as model predictive path integral (MPPI) are effective in handling nonlinear and nonconvex environments, yet their finite-sample rollouts and unconstrained weighted-average update can return an unsafe control. Deterministic nonlinear MPC can explicitly incorporate constraints, but its real-time safety and performance depends strongly on warm starts and local convergence. Hamilton--Jacobi reachability (HJR) provides rigorous safety certificates, but offline value-function computation remains practical only for reduced-order models. We propose ReSQ-MPPI, a reachability-informed generation--refinement architecture that combines these complementary strengths. An HJR value function computed offline for a reduced-order model guides online MPPI sampling toward safe, promising trajectory candidates. The MPPI solution is then refined by a small number of sequential quadratic programming (SQP) iterations in a full-order MPC problem. A key observation is that the standard MPPI inference step is an unconstrained weighted least-squares problem; ReSQ-MPPI replaces it with a constrained MPC refinement that recovers the MPPI update when it is feasible and minimally modifies it otherwise. Simulations in cluttered navigation and autonomous racing environments demonstrate that ReSQ-MPPI improves safety and performance over standalone MPPI, MPC, and reachability-filtered sampling-based control baselines.

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arXiv2026

A Forward Reachability Perspective on Control Barrier Functions and Discount Factors in Reachability Analysis

Control invariant sets are crucial for various methods that aim to design safe control policies for systems whose state constraints must be satisfied over an indefinite time horizon. In this article, we explore the connections among reachability, control invariance, and Control Barrier Functions (CBFs). Unlike prior formulations based on backward reachability concepts, we establish a strong link between these three concepts by examining the inevitable Forward Reachable Tube (FRT), which is the set of states such that every trajectory reaching the FRT must have passed through a given initial set of states. First, our findings show that the inevitable FRT is a robust control invariant set if it has a continuously differentiable boundary. If the boundary is not differentiable, the FRT may lose invariance. We also show that any robust control invariant set including the initial set is a superset of the FRT if the boundary of the invariant set is differentiable. Next, we formulate a differential game between the control and disturbance, where the inevitable FRT is characterized by the zero-superlevel set of the value function. By incorporating a discount factor in the cost function of the game, the barrier constraint of the CBF naturally arises in the Hamilton-Jacobi (HJ) equation and determines the optimal policy. The resulting FRT value function serves as a CBF-like function, and conversely, any valid CBF is also a forward reachability value function. We further prove that any $C^1$ supersolution of the HJ equation for the FRT value functions is a valid CBF and characterizes a robust control invariant set that outer-approximates the FRT. Building on this property, finally, we devise a novel method that learns neural control barrier functions, which learn an control invariant superset of the FRT of a given initial set.

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arXiv2026

Active Calibration of Reachable Sets Using Approximate Pick-to-Learn

Reachability computations that rely on approximate or learned models require calibration in order to uphold confidence about their guarantees. Calibration generally involves sampling scenarios inside the reachable set. However, producing reasonable probabilistic guarantees may require many samples, which can be costly. To remedy this, we propose that calibration of reachable sets be performed using active learning strategies. In order to produce a probabilistic guarantee on the active learning, we adapt the Pick-to-Learn algorithm, which produces generalization bounds for standard supervised learning, to the active learning setting. Our method, Approximate Pick-to-Learn, treats the process of choosing data samples as maximizing an approximate error function. Conformal prediction is used to ensure that the approximate error is close to the true model error. We demonstrate our technique for a simulated drone racing example in which learning is used to provide an initial guess of the reachable tube. Our method requires fewer samples to calibrate the model and provides more accurate sets than the baselines while also producing a novel generalization bound.

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arXiv2026

Robust Direct Data-Driven Hamiltonian for Safe Set Computation under Measurement Noise and Disturbances

Safe set computation is a fundamental challenge in safety-critical control systems, especially in direct data-driven settings where safety analysis is performed directly from noise-affected measurements, without explicit modeling. A recently proposed method, Data-Driven Hamiltonian (DDH), enables reachability analysis directly from measurements, without relying on prior knowledge of the underlying system dynamics. This paper extends the DDH framework to a robust setting that accounts for measurement noise, exogenous disturbances, and sampling-induced state-velocity estimation error. A Robust Data-Driven Hamiltonian (R-DDH) is derived from noisy measurements and shown to yield a certified lower bound on the exact Hamiltonian. This results in a provable under-approximation of the value function and an inner approximation of the associated safe set. The gap between the data-driven and exact Hamiltonians is quantified, and it is shown to converge to zero with more data in a noise-free setting with additive disturbances. The effectiveness of the approach is shown through two case studies: a constrained double integrator and an aircraft taxiing system with a nonlinear closed-loop controller operating under perceptual uncertainty.

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arXiv2026

Characterization and Analysis of Emergency Landing Flight Envelopes with Graded Safety Specifications

Emergency landing flight envelope analysis traditionally adopts a binary notion of safety, whereby a trajectory is safe only if state constraints are satisfied pointwise in time. In practice, ensuring a successful landing requires recognizing that aircraft operation spans a continuum in the state space from the nominal to the critical regime. Between these regimes lies a degraded regime of states outside nominal operation that may be visited only for limited durations. Safety is therefore inherently graded, in the sense that limited exposure to degraded states may be tolerated, and must be assessed using a trajectory-dependent criterion rather than a purely pointwise-in-time one. This paper develops a Hamilton-Jacobi reachability framework for analyzing emergency landing flight envelopes under this graded notion of safety. Safety is encoded through a soft constraint defined by a designer-specified continuous violation cost function that assigns zero cost in the nominal regime and larger cost to more safety-critical off-nominal states. We introduce a general class of state- and time-dependent violation cost functions and establish monotonicity and continuity properties that characterize how the flight envelope varies with the cost of off-nominal operation. These results provide a principled sensitivity analysis linking safety conservativeness to operational capability. Building on this analysis, we propose a synthesis algorithm for parameterized violation cost functions in this class. The algorithm provably converges to the least conservative parameter under which a prescribed off-nominal safety requirement is satisfied. Numerical results for a fixed-wing emergency landing scenario under propulsion failure demonstrate the sensitivity properties and validate the algorithm.

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arXiv2026

Inverse Safety Filtering: Inferring Constraints from Safety Filters for Decentralized Coordination

Safe multi-agent coordination in uncertain environments can benefit from learning constraints from other agents. Implicitly communicating safety constraints through actions is a promising approach, allowing agents to coordinate and maintain safety without expensive communication channels. This paper introduces an online method to infer constraints from observing the safety-filtered actions of other agents. We approach the problem by using safety filters to ensure forward safety and exploit their structure to work backwards and infer constraints. We provide sufficient conditions under which we can infer these constraints and prove that our inference method converges. This constraint inference procedure is coupled with a decentralized planning method that ensures safety when the constraint activation distance is sufficiently large. We then empirically validate our method with Monte Carlo simulations and hardware experiments with quadruped robots.

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arXiv2026

A Spectral Perspective on Stochastic Control Barrier Functions

Stochastic control barrier functions (SCBFs) provide a safety-critical control framework for systems subject to stochastic disturbances by bounding the probability of remaining within a safe set. However, synthesizing a valid SCBF that explicitly reflects the true safety probability of the system, which is the most natural measure of safety, remains a challenge. This paper addresses this issue by adopting a spectral perspective, utilizing the linear operator that governs the evolution of the closed-loop system's safety probability. We find that the dominant eigenpair of this Koopman-like operator encodes fundamental safety information of the stochastic system. The dominant eigenfunction is a natural and valid SCBF, with values that explicitly quantify the relative long-term safety of the state, while the dominant eigenvalue indicates the global rate at which the safety probability decays. A practical synthesis algorithm is proposed, termed power-policy iteration, which jointly computes the dominant eigenpair and an optimized backup policy. The method is validated using simulation experiments on safety-critical dynamics models.

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arXiv2026

EigenSafe: A Spectral Framework for Learning-Based Probabilistic Safety Assessment

We present EigenSafe, an operator-theoretic framework for safety assessment of learning-enabled stochastic systems. In many robotic applications, the dynamics are inherently stochastic due to factors such as sensing noise and environmental disturbances, and it is challenging for conventional methods such as Hamilton-Jacobi reachability and control barrier functions to provide a well-calibrated safety critic that is tied to the actual safety probability. We derive a linear operator that governs the dynamic programming principle for safety probability, and find that its dominant eigenpair provides critical safety information for both individual state-action pairs and the overall closed-loop system. The proposed framework learns this dominant eigenpair, which can be used to either inform or constrain policy updates. We demonstrate that the learned eigenpair effectively facilitates safe reinforcement learning. Further, we validate its applicability in enhancing the safety of learned policies from imitation learning through robot manipulation experiments using a UR3 robotic arm in a food preparation task.

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arXiv2026

System-Level Analysis of Module Uncertainty Quantification in the Autonomy Pipeline

Modern autonomous systems with machine learning components often use uncertainty quantification to help produce assurances about system operation. However, there is a lack of consensus in the community on what uncertainty is and how to perform uncertainty quantification. In this work, we propose that uncertainty measures should be understood within the context of overall system design and operation. To this end, we present two novel analysis techniques. First, we produce a probabilistic specification on a module's uncertainty measure given a system specification. Second, we propose a method to measure a system's input-output robustness in order to compare system designs and quantify the impact of making a system uncertainty-aware. In addition to this theoretical work, we present the application of these analyses on two real-world autonomous systems: an autonomous driving system and an aircraft runway incursion detection system. We show that our analyses can determine desired relationships between module uncertainty and error, provide visualizations of how well an uncertainty measure is being used by a system, produce principled comparisons between different uncertainty measures and decision-making algorithm designs, and provide insights into system vulnerabilities and tradeoffs.

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arXiv2025

A Hamilton-Jacobi Reachability Framework with Soft Constraints for Safety-Critical Systems

Traditional reachability methods provide formal guarantees of safety under bounded disturbances. However, they strictly enforce state constraints as inviolable, which can result in overly conservative or infeasible solutions in complex operational scenarios. Many constraints encountered in practice, such as bounds on battery state of charge in electric vehicles, recommended speed envelopes, and comfort constraints in passenger-carrying vehicles, are inherently soft. Soft constraints allow temporary violations within predefined safety margins to accommodate uncertainty and competing operational demands, albeit at a cost such as increased wear or higher operational expenses. This paper introduces a novel soft-constrained reachability framework that extends Hamilton-Jacobi reachability analysis for the formal verification of safety-critical systems subject to both hard and soft constraints. Specifically, the framework characterizes a subset of the state space, referred to as the soft-constrained reach-avoid set, from which the system is guaranteed to reach a desired set safely, under worst-case disturbances, while ensuring that cumulative soft-constraint violations remain within a user-specified budget. The framework comprises two principal components: (i) an augmented-state model with an auxiliary budget state that tracks soft-constraint violations, and (ii) a regularization-based approximation of the discontinuous Hamilton-Jacobi value function associated with the reach-avoid differential game studied herein. The effectiveness of the proposed framework is demonstrated through numerical examples involving the landing of a simple point-mass model and a fixed-wing aircraft executing an emergency descent, both under wind disturbances. The simulation results validate the framework's ability to simultaneously manage both hard and soft constraints in safety-critical settings

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arXiv2025

When are safety filters safe? On minimum phase conditions of control barrier functions

In emerging control applications involving multiple and complex tasks, safety filters are gaining prominence as a modular approach to enforcing safety constraints. Among various methods, control barrier functions (CBFs) are widely used for designing safety filters due to their simplicity, imposing a single linear constraint on the control input at each state. In this work, we focus on the internal dynamics of systems governed by CBF-constrained control laws. Our key observation is that, although CBFs guarantee safety by enforcing state constraints, they can inadvertently be "unsafe" by causing the internal state to diverge. We investigate the conditions under which the full system state, including the internal state, can remain bounded under a CBF-based safety filter. Drawing inspiration from the input-output linearization literature, where boundedness is ensured by minimum phase conditions, we propose a new set of CBF minimum phase conditions tailored to the structure imposed by the CBF constraint. A critical distinction from the original minimum phase conditions is that the internal dynamics in our setting is driven by a nonnegative virtual control input, which reflects the enforcement of the safety constraint. We include a range of numerical examples, including single-input, multi-input, linear, and nonlinear systems, validating both our analysis and the necessity of the proposed CBF minimum phase conditions.

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arXiv2025

Maximizing Seaweed Growth on Autonomous Farms: A Dynamic Programming Approach for Underactuated Systems Navigating on Uncertain Ocean Currents

Seaweed biomass presents a substantial opportunity for climate mitigation, yet to realize its potential, farming must be expanded to the vast open oceans. However, in the open ocean neither anchored farming nor floating farms with powerful engines are economically viable. Thus, a potential solution are farms that operate by going with the flow, utilizing minimal propulsion to strategically leverage beneficial ocean currents. In this work, we focus on low-power autonomous seaweed farms and design controllers that maximize seaweed growth by taking advantage of ocean currents. We first introduce a Dynamic Programming (DP) formulation to solve for the growth-optimal value function when the true currents are known. However, in reality only short-term imperfect forecasts with increasing uncertainty are available. Hence, we present three additional extensions. Firstly, we use frequent replanning to mitigate forecast errors. Second, to optimize for long-term growth, we extend the value function beyond the forecast horizon by estimating the expected future growth based on seasonal average currents. Lastly, we introduce a discounted finite-time DP formulation to account for the increasing uncertainty in future ocean current estimates. We empirically evaluate our approach with 30-day simulations of farms in realistic ocean conditions. Our method achieves 95.8\% of the best possible growth using only 5-day forecasts.This demonstrates that low-power propulsion is a promising method to operate autonomous seaweed farms in real-world conditions.

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arXiv2025

Resolving Conflicting Constraints in Multi-Agent Reinforcement Learning with Layered Safety

Preventing collisions in multi-robot navigation is crucial for deployment. This requirement hinders the use of learning-based approaches, such as multi-agent reinforcement learning (MARL), on their own due to their lack of safety guarantees. Traditional control methods, such as reachability and control barrier functions, can provide rigorous safety guarantees when interactions are limited only to a small number of robots. However, conflicts between the constraints faced by different agents pose a challenge to safe multi-agent coordination. To overcome this challenge, we propose a method that integrates multiple layers of safety by combining MARL with safety filters. First, MARL is used to learn strategies that minimize multiple agent interactions, where multiple indicates more than two. Particularly, we focus on interactions likely to result in conflicting constraints within the engagement distance. Next, for agents that enter the engagement distance, we prioritize pairs requiring the most urgent corrective actions. Finally, a dedicated safety filter provides tactical corrective actions to resolve these conflicts. Crucially, the design decisions for all layers of this framework are grounded in reachability analysis and a control barrier-value function-based filtering mechanism. We validate our Layered Safe MARL framework in 1) hardware experiments using Crazyflie drones and 2) high-density advanced aerial mobility (AAM) operation scenarios, where agents navigate to designated waypoints while avoiding collisions. The results show that our method significantly reduces conflict while maintaining safety without sacrificing much efficiency (i.e., shorter travel time and distance) compared to baselines that do not incorporate layered safety. The project website is available at https://dinamo-mit.github.io/Layered-Safe-MARL/

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arXiv2025

Data-Driven Hamiltonian for Direct Construction of Safe Set from Trajectory Data

In continuous-time optimal control, evaluating the Hamiltonian requires solving a constrained optimization problem using the system's dynamics model. Hamilton-Jacobi reachability analysis for safety verification has demonstrated practical utility only when efficient evaluation of the Hamiltonian over a large state-time grid is possible. In this study, we introduce the concept of a data-driven Hamiltonian (DDH), which circumvents the need for an explicit dynamics model by relying only on mild prior knowledge (e.g., Lipschitz constants), thus enabling the construction of reachable sets directly from trajectory data. Recognizing that the Hamiltonian is the optimal inner product between a given costate and realizable state velocities, the DDH estimates the Hamiltonian using the worst-case realization of the velocity field based on the observed state trajectory data. This formulation ensures a conservative approximation of the true Hamiltonian for uncertain dynamics. The reachable set computed based on the DDH is also ensured to be a conservative approximation of the true reachable set. Next, we propose a data-efficient safe experiment framework for gradual expansion of safe sets using the DDH. This is achieved by iteratively conducting experiments within the computed data-driven safe set and updating the set using newly collected trajectory data. To demonstrate the capabilities of our approach, we showcase its effectiveness in safe flight envelope expansion for a tiltrotor vehicle transitioning from near-hover to forward flight.

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arXiv2024

Constraint-Guided Online Data Selection for Scalable Data-Driven Safety Filters in Uncertain Robotic Systems

As the use of autonomous robots expands in tasks that are complex and challenging to model, the demand for robust data-driven control methods that can certify safety and stability in uncertain conditions is increasing. However, the practical implementation of these methods often faces scalability issues due to the growing amount of data points with system complexity, and a significant reliance on high-quality training data. In response to these challenges, this study presents a scalable data-driven controller that efficiently identifies and infers from the most informative data points for implementing data-driven safety filters. Our approach is grounded in the integration of a model-based certificate function-based method and Gaussian Process (GP) regression, reinforced by a novel online data selection algorithm that reduces time complexity from quadratic to linear relative to dataset size. Empirical evidence, gathered from successful real-world cart-pole swing-up experiments and simulated locomotion of a five-link bipedal robot, demonstrates the efficacy of our approach. Our findings reveal that our efficient online data selection algorithm, which strategically selects key data points, enhances the practicality and efficiency of data-driven certifying filters in complex robotic systems, significantly mitigating scalability concerns inherent in nonparametric learning-based control methods.

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arXiv2024

Infinite-Horizon Reach-Avoid Zero-Sum Games via Deep Reinforcement Learning

In this paper, we consider the infinite-horizon reach-avoid zero-sum game problem, where the goal is to find a set in the state space, referred to as the reach-avoid set, such that the system starting at a state therein could be controlled to reach a given target set without violating constraints under the worst-case disturbance. We address this problem by designing a new value function with a contracting Bellman backup, where the super-zero level set, i.e., the set of states where the value function is evaluated to be non-negative, recovers the reach-avoid set. Building upon this, we prove that the proposed method can be adapted to compute the viability kernel, or the set of states which could be controlled to satisfy given constraints, and the backward reachable set, or the set of states that could be driven towards a given target set. Finally, we propose to alleviate the curse of dimensionality issue in high-dimensional problems by extending Conservative Q-Learning, a deep reinforcement learning technique, to learn a value function such that the super-zero level set of the learned value function serves as a (conservative) approximation to the reach-avoid set. Our theoretical and empirical results suggest that the proposed method could learn reliably the reach-avoid set and the optimal control policy even with neural network approximation.

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arXiv2024

Gait Switching and Enhanced Stabilization of Walking Robots with Deep Learning-based Reachability: A Case Study on Two-link Walker

Learning-based approaches have recently shown notable success in legged locomotion. However, these approaches often lack accountability, necessitating empirical tests to determine their effectiveness. In this work, we are interested in designing a learning-based locomotion controller whose stability can be examined and guaranteed. This can be achieved by verifying regions of attraction (RoAs) of legged robots to their stable walking gaits. This is a non-trivial problem for legged robots due to their hybrid dynamics. Although previous work has shown the utility of Hamilton-Jacobi (HJ) reachability to solve this problem, its practicality was limited by its poor scalability. The core contribution of our work is the employment of a deep learning-based HJ reachability solution to the hybrid legged robot dynamics, which overcomes the previous work's limitation. With the learned reachability solution, first, we can estimate a library of RoAs for various gaits. Second, we can design a one-step predictive controller that effectively stabilizes to an individual gait within the verified RoA. Finally, we can devise a strategy that switches gaits, in response to external perturbations, whose feasibility is guided by the RoA analysis. We demonstrate our method in a two-link walker simulation, whose mathematical model is well established. Our method achieves improved stability than previous model-based methods, while ensuring transparency that was not present in the existing learning-based approaches.

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