Reachability of Koopman Linearized Systems Using Random Fourier Feature Observables and Polynomial Zonotope Refinement

Bak, Stanley; Bogomolov, Sergiy; Hencey, Brandon; Kochdumper, Niklas; Lew, Ethan; Potomkin, Kostiantyn

doi:10.1007/978-3-031-13185-1_24

Cited by 6 publications

(3 citation statements)

References 42 publications

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“…In the experiments below, the main lifting functions used are squared-exponential Random Fourier Features (SERFFs) [31], which take in any number of vector-space inputs and generate a user-specified number of sinusoidal functions. SERFFs are common lifting functions for Koopman methods used for their experimentally determined high performance [1], [32], and for their connection to the squared-exponential kernel [1], [33].…”

Section: Experiments and Resultsmentioning

confidence: 99%

Koopman Linearization for Data-Driven Batch State Estimation of Control-Affine Systems

Guo

Vassili

Forbes

et al. 2022

IEEE Robot. Autom. Lett.

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We present a framework for model-free batch localization and SLAM. We use lifting functions to map a controlaffine system into a high-dimensional space, where both the process model and the measurement model are rendered bilinear. During training, we solve a least-squares problem using groundtruth data to compute the high-dimensional model matrices associated with the lifted system purely from data. At inference time, we solve for the unknown robot trajectory and landmarks through an optimization problem, where constraints are introduced to keep the solution on the manifold of the lifting functions. The problem is efficiently solved using a sequential quadratic program (SQP), where the complexity of an SQP iteration scales linearly with the number of timesteps. Our algorithms, called Reduced Constrained Koopman Linearization Localization (RCKL-Loc) and Reduced Constrained Koopman Linearization SLAM (RCKL-SLAM), are validated experimentally in simulation and on two datasets: one with an indoor mobile robot equipped with a laser rangefinder that measures range to cylindrical landmarks, and one on a golf cart equipped with RFID range sensors. We compare RCKL-Loc and RCKL-SLAM with classic model-based nonlinear batch estimation. While RCKL-Loc and RCKL-SLAM have similar performance compared to their model-based counterparts, they outperform the model-based approaches when the prior model is imperfect, showing the potential benefit of the proposed data-driven technique.

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Section: Experiments and Resultsmentioning

confidence: 99%

Koopman Linearization for Data-Driven Batch State Estimation of Control-Affine Systems

Guo

Vassili

Forbes

et al. 2022

IEEE Robot. Autom. Lett.

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“…The Koopman framework has been successfully applied to many applications, including control [4,5], state estimation [6] and recently, formal verification [7]. This article is a journal extension of our paper presented at CAV 2022, which advanced the state-of-the-art in formal verification using reachability analysis on Koopman operator linearized systems [8]. First, extend the paper to use advancements in explicit map kernel approximations beyond random Fourier features [9], namely Gaussian quadrature features [10,11].…”

Section: Introductionmentioning

confidence: 92%

Reachability of Koopman Linearized Systems Using Explicit Kernel Approximation and Polynomial Zonotope Refinement

Bak

Bogomolov

Hencey

et al. 2023

Preprint

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Koopman operator linearization approximates nonlinear systems of differential equations with higher-dimensional linear systems. For formal verification using reachability analysis, this is an attractive conversion, as highly scalable methods exist to compute reachable sets for linear systems. However, two main challenges are present with this approach, both of which are addressed in this work. First, the approximation must be sufficiently accurate for the result to be meaningful, which is controlled by the choice of observable functions during Koopman operator linearization. We formulate a more systematic approach for observables generation for black box systems using a two explicit kernel approximation methods: random Fourier features and Gaussian quadrature features. We also utilize a data-dependent reweighting to reduce the number of features by roughly 2-3 times, consequently speeding up reachability analysis while maintaining comparable accuracy. Second, although the dynamics of the higher-dimensional system is linear, simple convex initial sets in the original space can become complex non-convex initial sets in the linear system due to the nonlinear observable functions. We overcome this using a combination of Taylor model arithmetic and polynomial zonotope refinement. Compared with prior work, the result is more efficient, more systematic, and more accurate.

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“…1 and ν is the number of intersecting pairs. To efficiently check if a reachable set represented by a polynomial zonotope intersects an obstacle represented by a polytope, the polynomial zonotope refinement algorithm [36] can be used. This algorithm recursively splits the polynomial zonotope along the longest generator until the intersection with the polytope can either be proven or disproven using zonotope enclosures of the split polynomial zonotopes.…”

Section: Safety Shieldmentioning

confidence: 99%

Provably Safe Reinforcement Learning via Action Projection Using Reachability Analysis and Polynomial Zonotopes

Kochdumper

Krasowski

Wang

et al. 2023

IEEE Open J. Control. Syst.

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While reinforcement learning produces very promising results for many applications, its main disadvantage is the lack of safety guarantees, which prevents its use in safety-critical systems. In this work, we address this issue by a safety shield for nonlinear continuous systems that solve reach-avoid tasks. Our safety shield prevents applying potentially unsafe actions from a reinforcement learning agent by projecting the proposed action to the closest safe action. This approach is called action projection and is implemented via mixed-integer optimization. The safety constraints for action projection are obtained by applying parameterized reachability analysis using polynomial zonotopes, which enables to accurately capture the nonlinear effects of the actions on the system. In contrast to other state-of-the-art approaches for action projection, our safety shield can efficiently handle input constraints and dynamic obstacles, eases incorporation of the spatial robot dimensions into the safety constraints, guarantees robust safety despite process noise and measurement errors, and is well suited for high-dimensional systems, as we demonstrate on several challenging benchmark systems.

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Reachability of Koopman Linearized Systems Using Random Fourier Feature Observables and Polynomial Zonotope Refinement

Cited by 6 publications

References 42 publications

Koopman Linearization for Data-Driven Batch State Estimation of Control-Affine Systems

Koopman Linearization for Data-Driven Batch State Estimation of Control-Affine Systems

Reachability of Koopman Linearized Systems Using Explicit Kernel Approximation and Polynomial Zonotope Refinement

Provably Safe Reinforcement Learning via Action Projection Using Reachability Analysis and Polynomial Zonotopes

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