arXiv subjects

Claire J. Tomlin

Publications and source records attributed to Claire J. Tomlin.

At least 37 records · Page 2Linked to original sources

arXiv2024

Recursively Feasible Probabilistic Safe Online Learning with Control Barrier Functions

Learning-based control has recently shown great efficacy in performing complex tasks for various applications. However, to deploy it in real systems, it is of vital importance to guarantee the system will stay safe. Control Barrier Functions (CBFs) offer mathematical tools for designing safety-preserving controllers for systems with known dynamics. In this article, we first introduce a model-uncertainty-aware reformulation of CBF-based safety-critical controllers using Gaussian Process (GP) regression to close the gap between an approximate mathematical model and the real system, which results in a second-order cone program (SOCP)-based control design. We then present the pointwise feasibility conditions of the resulting safety controller, highlighting the level of richness that the available system information must meet to ensure safety. We use these conditions to devise an event-triggered online data collection strategy that ensures the recursive feasibility of the learned safety controller. Our method works by constantly reasoning about whether the current information is sufficient to ensure safety or if new measurements under active safe exploration are required to reduce the uncertainty. As a result, our proposed framework can guarantee the forward invariance of the safe set defined by the CBF with high probability, even if it contains a priori unexplored regions. We validate the proposed framework in two numerical simulation experiments.

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arXiv2024

Safe Connectivity Maintenance of Underactuated Multi-Agent Networks in Dynamic Oceanic Environments

Autonomous multi-agent systems are increasingly being deployed in environments where winds and ocean currents have a significant influence. Recent work has developed control policies for single agents that leverage flows to achieve their objectives in dynamic environments. However, in multi-agent systems, these flows can cause agents to collide or drift apart and lose direct inter-agent communications, especially when agents have low propulsion capabilities. To address these challenges, we propose a hierarchical multi-agent control approach that allows arbitrary single-agent performance policies that are unaware of other agents to be used in multi-agent systems while ensuring safe operation. We first develop a safety controller using potential functions, solely dedicated to avoiding collisions and maintaining inter-agent communication. Next, we design a low-interference safe interaction (LISIC) policy that trades off the performance policy and the safety control to ensure safe and performant operation. Specifically, when the agents are at an appropriate distance, LISIC prioritizes the performance policy while smoothly increasing the safety controller when necessary. We prove that under mild assumptions on the flows experienced by the agents, our approach can guarantee safety. Additionally, we demonstrate the effectiveness of our method in realistic settings through an extensive empirical analysis with simulations of fleets of underactuated autonomous surface vehicles operating in dynamic ocean currents where these assumptions do not always hold.

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arXiv2024

Safety Filters for Black-Box Dynamical Systems by Learning Discriminating Hyperplanes

Learning-based approaches are emerging as an effective approach for safety filters for black-box dynamical systems. Existing methods have relied on certificate functions like Control Barrier Functions (CBFs) and Hamilton-Jacobi (HJ) reachability value functions. The primary motivation for our work is the recognition that ultimately, enforcing the safety constraint as a control input constraint at each state is what matters. By focusing on this constraint, we can eliminate dependence on any specific certificate function-based design. To achieve this, we define a discriminating hyperplane that shapes the half-space constraint on control input at each state, serving as a sufficient condition for safety. This concept not only generalizes over traditional safety methods but also simplifies safety filter design by eliminating dependence on specific certificate functions. We present two strategies to learn the discriminating hyperplane: (a) a supervised learning approach, using pre-verified control invariant sets for labeling, and (b) a reinforcement learning (RL) approach, which does not require such labels. The main advantage of our method, unlike conventional safe RL approaches, is the separation of performance and safety. This offers a reusable safety filter for learning new tasks, avoiding the need to retrain from scratch. As such, we believe that the new notion of the discriminating hyperplane offers a more generalizable direction towards designing safety filters, encompassing and extending existing certificate-function-based or safe RL methodologies.

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arXiv2023

Optimality Guarantees for Particle Belief Approximation of POMDPs

Partially observable Markov decision processes (POMDPs) provide a flexible representation for real-world decision and control problems. However, POMDPs are notoriously difficult to solve, especially when the state and observation spaces are continuous or hybrid, which is often the case for physical systems. While recent online sampling-based POMDP algorithms that plan with observation likelihood weighting have shown practical effectiveness, a general theory characterizing the approximation error of the particle filtering techniques that these algorithms use has not previously been proposed. Our main contribution is bounding the error between any POMDP and its corresponding finite sample particle belief MDP (PB-MDP) approximation. This fundamental bridge between PB-MDPs and POMDPs allows us to adapt any sampling-based MDP algorithm to a POMDP by solving the corresponding particle belief MDP, thereby extending the convergence guarantees of the MDP algorithm to the POMDP. Practically, this is implemented by using the particle filter belief transition model as the generative model for the MDP solver. While this requires access to the observation density model from the POMDP, it only increases the transition sampling complexity of the MDP solver by a factor of $\mathcal{O}(C)$, where $C$ is the number of particles. Thus, when combined with sparse sampling MDP algorithms, this approach can yield algorithms for POMDPs that have no direct theoretical dependence on the size of the state and observation spaces. In addition to our theoretical contribution, we perform five numerical experiments on benchmark POMDPs to demonstrate that a simple MDP algorithm adapted using PB-MDP approximation, Sparse-PFT, achieves performance competitive with other leading continuous observation POMDP solvers.

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arXiv2023

Out of Distribution Detection via Domain-Informed Gaussian Process State Space Models

In order for robots to safely navigate in unseen scenarios using learning-based methods, it is important to accurately detect out-of-training-distribution (OoD) situations online. Recently, Gaussian process state-space models (GPSSMs) have proven useful to discriminate unexpected observations by comparing them against probabilistic predictions. However, the capability for the model to correctly distinguish between in- and out-of-training distribution observations hinges on the accuracy of these predictions, primarily affected by the class of functions the GPSSM kernel can represent. In this paper, we propose (i) a novel approach to embed existing domain knowledge in the kernel and (ii) an OoD online runtime monitor, based on receding-horizon predictions. Domain knowledge is provided in the form of a dataset, collected either in simulation or by using a nominal model. Numerical results show that the informed kernel yields better regression quality with smaller datasets, as compared to standard kernel choices. We demonstrate the effectiveness of the OoD monitor on a real quadruped navigating an indoor setting, which reliably classifies previously unseen terrains.

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arXiv2023

Stranding Risk for Underactuated Vessels in Complex Ocean Currents: Analysis and Controllers

Low-propulsion vessels can take advantage of powerful ocean currents to navigate towards a destination. Recent results demonstrated that vessels can reach their destination with high probability despite forecast errors. However, these results do not consider the critical aspect of safety of such vessels: because of their low propulsion which is much smaller than the magnitude of currents, they might end up in currents that inevitably push them into unsafe areas such as shallow areas, garbage patches, and shipping lanes. In this work, we first investigate the risk of stranding for free-floating vessels in the Northeast Pacific. We find that at least 5.04% would strand within 90 days. Next, we encode the unsafe sets as hard constraints into Hamilton-Jacobi Multi-Time Reachability (HJ-MTR) to synthesize a feedback policy that is equivalent to re-planning at each time step at low computational cost. While applying this policy closed-loop guarantees safe operation when the currents are known, in realistic situations only imperfect forecasts are available. We demonstrate the safety of our approach in such realistic situations empirically with large-scale simulations of a vessel navigating in high-risk regions in the Northeast Pacific. We find that applying our policy closed-loop with daily re-planning on new forecasts can ensure safety with high probability even under forecast errors that exceed the maximal propulsion. Our method significantly improves safety over the baselines and still achieves a timely arrival of the vessel at the destination.

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arXiv2023

Sparse tree search optimality guarantees in POMDPs with continuous observation spaces

Partially observable Markov decision processes (POMDPs) with continuous state and observation spaces have powerful flexibility for representing real-world decision and control problems but are notoriously difficult to solve. Recent online sampling-based algorithms that use observation likelihood weighting have shown unprecedented effectiveness in domains with continuous observation spaces. However there has been no formal theoretical justification for this technique. This work offers such a justification, proving that a simplified algorithm, partially observable weighted sparse sampling (POWSS), will estimate Q-values accurately with high probability and can be made to perform arbitrarily near the optimal solution by increasing computational power.

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arXiv2023

Online and Offline Learning of Player Objectives from Partial Observations in Dynamic Games

Robots deployed to the real world must be able to interact with other agents in their environment. Dynamic game theory provides a powerful mathematical framework for modeling scenarios in which agents have individual objectives and interactions evolve over time. However, a key limitation of such techniques is that they require a-priori knowledge of all players' objectives. In this work, we address this issue by proposing a novel method for learning players' objectives in continuous dynamic games from noise-corrupted, partial state observations. Our approach learns objectives by coupling the estimation of unknown cost parameters of each player with inference of unobserved states and inputs through Nash equilibrium constraints. By coupling past state estimates with future state predictions, our approach is amenable to simultaneous online learning and prediction in receding horizon fashion. We demonstrate our method in several simulated traffic scenarios in which we recover players' preferences for, e.g., desired travel speed and collision-avoidance behavior. Results show that our method reliably estimates game-theoretic models from noise-corrupted data that closely matches ground-truth objectives, consistently outperforming state-of-the-art approaches.

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arXiv2022

Compositional Learning-based Planning for Vision POMDPs

The Partially Observable Markov Decision Process (POMDP) is a powerful framework for capturing decision-making problems that involve state and transition uncertainty. However, most current POMDP planners cannot effectively handle high-dimensional image observations prevalent in real world applications, and often require lengthy online training that requires interaction with the environment. In this work, we propose Visual Tree Search (VTS), a compositional learning and planning procedure that combines generative models learned offline with online model-based POMDP planning. The deep generative observation models evaluate the likelihood of and predict future image observations in a Monte Carlo tree search planner. We show that VTS is robust to different types of image noises that were not present during training and can adapt to different reward structures without the need to re-train. This new approach significantly and stably outperforms several baseline state-of-the-art vision POMDP algorithms while using a fraction of the training time.

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arXiv2022

On the Computational Consequences of Cost Function Design in Nonlinear Optimal Control

Optimal control is an essential tool for stabilizing complex nonlinear systems. However, despite the extensive impacts of methods such as receding horizon control, dynamic programming and reinforcement learning, the design of cost functions for a particular system often remains a heuristic-driven process of trial and error. In this paper we seek to gain insights into how the choice of cost function interacts with the underlying structure of the control system and impacts the amount of computation required to obtain a stabilizing controller. We treat the cost design problem as a two-step process where the designer specifies outputs for the system that are to be penalized and then modulates the relative weighting of the inputs and the outputs in the cost. To characterize the computational burden associated to obtaining a stabilizing controller with a particular cost, we bound the prediction horizon required by receding horizon methods and the number of iterations required by dynamic programming methods to meet this requirement. Our theoretical results highlight a qualitative separation between what is possible, from a design perspective, when the chosen outputs induce either minimum-phase or non-minimum-phase behavior. Simulation studies indicate that this separation also holds for modern reinforcement learning methods.

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arXiv2022

Risk-sensitive safety analysis using Conditional Value-at-Risk

This paper develops a safety analysis method for stochastic systems that is sensitive to the possibility and severity of rare harmful outcomes. We define risk-sensitive safe sets as sub-level sets of the solution to a non-standard optimal control problem, where a random maximum cost is assessed via Conditional Value-at-Risk (CVaR). The objective function represents the maximum extent of constraint violation of the state trajectory, averaged over a given percentage of worst cases. This problem is well-motivated but difficult to solve tractably because the temporal decomposition for CVaR is history-dependent. Our primary theoretical contribution is to derive computationally tractable under-approximations to risk-sensitive safe sets. Our method provides a novel, theoretically guaranteed, parameter-dependent upper bound to the CVaR of a maximum cost without the need to augment the state space. For a fixed parameter value, the solution to only one Markov decision process problem is required to obtain the under-approximations for any family of risk-sensitivity levels. In addition, we propose a second definition for risk-sensitive safe sets and provide a tractable method for their estimation without using a parameter-dependent upper bound. The second definition is expressed in terms of a new coherent risk functional, which is inspired by CVaR. We demonstrate our primary theoretical contribution via numerical examples.

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arXiv2022

Stability and Robustness of a Hybrid Control Law for the Half-bridge Inverter

Hybrid systems combine both discrete and continuous state dynamics. Power electronic inverters are inherently hybrid systems: they are controlled via discrete-valued switching inputs which determine the evolution of the continuous-valued current and voltage state dynamics. Hybrid systems analysis could prove increasingly useful as large numbers of renewable energy sources are incorporated to the grid with inverters as their interface. In this work, we explore a hybrid systems approach for the stability analysis of power and power electronic systems. We provide an analytical proof showing that the use of a hybrid model for the half-bridge inverter allows the derivation of a control law that drives the system states to desired sinusoidal voltage and current references. We derive an analytical expression for a global Lyapunov function for the dynamical system in terms of the system parameters, which proves uniform, global, and asymptotic stability of the origin in error coordinates. Moreover, we demonstrate robustness to parameter changes through this Lyapunov function. We validate these results via simulation. Finally, we show empirically the incorporation of droop control with this hybrid systems approach. In the low-inertia grid community, the juxtaposition of droop control with the hybrid switching control can be considered a grid-forming control strategy using a switched inverter model.

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arXiv2022

Koopman-Based Neural Lyapunov Functions for General Attractors

Koopman spectral theory has grown in the past decade as a powerful tool for dynamical systems analysis and control. In this paper, we show how recent data-driven techniques for estimating Koopman-Invariant subspaces with neural networks can be leveraged to extract Lyapunov certificates for the underlying system. In our work, we specifically focus on systems with a limit-cycle, beyond just an isolated equilibrium point, and use Koopman eigenfunctions to efficiently parameterize candidate Lyapunov functions to construct forward-invariant sets under some (unknown) attractor dynamics. Additionally, when the dynamics are polynomial and when neural networks are replaced by polynomials as a choice of function approximators in our approach, one can further leverage Sum-of-Squares programs and/or nonlinear programs to yield provably correct Lyapunov certificates. In such a polynomial case, our Koopman-based approach for constructing Lyapunov functions uses significantly fewer decision variables compared to directly formulating and solving a Sum-of-Squares optimization problem.

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arXiv2022

Computation of Regions of Attraction for Hybrid Limit Cycles Using Reachability: An Application to Walking Robots

Contact-rich robotic systems, such as legged robots and manipulators, are often represented as hybrid systems. However, the stability analysis and region-of-attraction computation for these systems are often challenging because of the discontinuous state changes upon contact (also referred to as state resets). In this work, we cast the computation of region-ofattraction as a Hamilton-Jacobi (HJ) reachability problem. This enables us to leverage HJ reachability tools that are compatible with general nonlinear system dynamics, and can formally deal with state and input constraints as well as bounded disturbances. Our main contribution is the generalization of HJ reachability framework to account for the discontinuous state changes originating from state resets, which has remained a challenge until now. We apply our approach for computing region-of-attractions for several underactuated walking robots and demonstrate that the proposed approach can (a) recover a bigger region-of-attraction than state-of-the-art approaches, (b) handle state resets, nonlinear dynamics, external disturbances, and input constraints, and (c) also provides a stabilizing controller for the system that can leverage the state resets for enhancing system stability.

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arXiv2021

Safety and Liveness Guarantees through Reach-Avoid Reinforcement Learning

Reach-avoid optimal control problems, in which the system must reach certain goal conditions while staying clear of unacceptable failure modes, are central to safety and liveness assurance for autonomous robotic systems, but their exact solutions are intractable for complex dynamics and environments. Recent successes in reinforcement learning methods to approximately solve optimal control problems with performance objectives make their application to certification problems attractive; however, the Lagrange-type objective used in reinforcement learning is not suitable to encode temporal logic requirements. Recent work has shown promise in extending the reinforcement learning machinery to safety-type problems, whose objective is not a sum, but a minimum (or maximum) over time. In this work, we generalize the reinforcement learning formulation to handle all optimal control problems in the reach-avoid category. We derive a time-discounted reach-avoid Bellman backup with contraction mapping properties and prove that the resulting reach-avoid Q-learning algorithm converges under analogous conditions to the traditional Lagrange-type problem, yielding an arbitrarily tight conservative approximation to the reach-avoid set. We further demonstrate the use of this formulation with deep reinforcement learning methods, retaining zero-violation guarantees by treating the approximate solutions as untrusted oracles in a model-predictive supervisory control framework. We evaluate our proposed framework on a range of nonlinear systems, validating the results against analytic and numerical solutions, and through Monte Carlo simulation in previously intractable problems. Our results open the door to a range of learning-based methods for safe-and-live autonomous behavior, with applications across robotics and automation. See https://github.com/SafeRoboticsLab/safety_rl for code and supplementary material.

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arXiv2021

Robust Control Barrier-Value Functions for Safety-Critical Control

This paper works towards unifying two popular approaches in the safety control community: Hamilton-Jacobi (HJ) reachability and Control Barrier Functions (CBFs). HJ Reachability has methods for direct construction of value functions that provide safety guarantees and safe controllers, however the online implementation can be overly conservative and/or rely on chattering bang-bang control. The CBF community has methods for safe-guarding controllers in the form of point-wise optimization using quadratic programs (CBF-QP), where the CBF-based safety certificate is used as a constraint. However, finding a valid CBF for a general dynamical system is challenging. This paper unifies these two methods by introducing a new reachability formulation inspired by the structure of CBFs to construct a Control Barrier-Value Function (CBVF). We verify that CBVF is a viscosity solution to a novel Hamilton-Jacobi-Isaacs Variational Inequality and preserves the same safety guarantee as the original reachability formulation. Finally, inspired by the CBF-QP, we propose a QP-based online control synthesis for systems affine in control and disturbance, whose solution is always the CBVF's optimal control signal robust to bounded disturbance. We demonstrate the benefit of using the CBVFs for double-integrator and Dubins car systems by comparing it to previous methods.

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arXiv2021

Pointwise Feasibility of Gaussian Process-based Safety-Critical Control under Model Uncertainty

Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) are popular tools for enforcing safety and stability of a controlled system, respectively. They are commonly utilized to build constraints that can be incorporated in a min-norm quadratic program (CBF-CLF-QP) which solves for a safety-critical control input. However, since these constraints rely on a model of the system, when this model is inaccurate the guarantees of safety and stability can be easily lost. In this paper, we present a Gaussian Process (GP)-based approach to tackle the problem of model uncertainty in safety-critical controllers that use CBFs and CLFs. The considered model uncertainty is affected by both state and control input. We derive probabilistic bounds on the effects that such model uncertainty has on the dynamics of the CBF and CLF. We then use these bounds to build safety and stability chance constraints that can be incorporated in a min-norm convex optimization-based controller, called GP-CBF-CLF-SOCP. As the main theoretical result of the paper, we present necessary and sufficient conditions for pointwise feasibility of the proposed optimization problem. We believe that these conditions could serve as a starting point towards understanding what are the minimal requirements on the distribution of data collected from the real system in order to guarantee safety. Finally, we validate the proposed framework with numerical simulations of an adaptive cruise controller for an automotive system.

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arXiv2021

Analyzing Human Models that Adapt Online

Predictive human models often need to adapt their parameters online from human data. This raises previously ignored safety-related questions for robots relying on these models such as what the model could learn online and how quickly could it learn it. For instance, when will the robot have a confident estimate in a nearby human's goal? Or, what parameter initializations guarantee that the robot can learn the human's preferences in a finite number of observations? To answer such analysis questions, our key idea is to model the robot's learning algorithm as a dynamical system where the state is the current model parameter estimate and the control is the human data the robot observes. This enables us to leverage tools from reachability analysis and optimal control to compute the set of hypotheses the robot could learn in finite time, as well as the worst and best-case time it takes to learn them. We demonstrate the utility of our analysis tool in four human-robot domains, including autonomous driving and indoor navigation.

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