arXiv subjects

Timothy L. Molloy

Publications and source records attributed to Timothy L. Molloy.

At least 19 recordsLinked to original sources

arXiv2026

Semi-Explicit Solutions to the Prying-Pedestrian Surveillance-Evasion Differential Game and Extensions to Two Pursuers

In [1], the authors recently proposed and solved a surveillance-evasion differential game in which an agile pursuer (the prying pedestrian) seeks to remain within a given surveillance range of a less agile evader for as long as possible while the evader seeks to escape as quickly as possible. In this paper, we provide initial results that extend this game from the 1 versus 1 (1v1) setting to a 2 versus 1 (2v1) setting with two pursuers and one evader. By deriving and exploiting semi-explicit or geometric reinterpretations of the existing 1v1 results, we derive partial solutions to the 2v1 game for the case of static pursuers and for the case of an evader that is at least twice as fast as the pursuers. While the 2v1 results of this paper build on the 1v1 results of [1], a different solution approach is developed to avoid a coordinate transformation that reduces the 1v1 game to two dimensions but which is ineffective at simplifying the 2v1 game. Beyond enabling progress on the 2v1 game, our new approach yields new geometric interpretations of the optimal pursuer and evader strategies in the 1v1 game, and opens further possible extensions.

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arXiv2026

Equivariant Filter for High Performance Image Tracking using an Event Camera

Image tracking is the problem of estimating the transformation that relates a moving image of a scene to an original reference image. The problem is important in control of autonomous vehicles or robots, where the image encodes information about the motion of the camera or environment, as well as in pure computer vision applications. In this paper, we present an equivariant filter design for high performance tracking of planar image transformations using an event camera. The design exploits the Asynchronous Event Blob (AEB) tracker (Wang et al., 2024) to extract feature-position measurements from the raw event stream, and an equivariant filter to compute an affine image translation and rotation using the special Euclidean group symmetry. The equivariant filter incorporates an equivalent-measurement update step that de-correlates the (highly temporally correlated) feature-position measurements provided by the AEB tracker. We evaluate the design experimentally using two datasets involving general and fast rotational motion. We benchmark results against direct optimisation (estimating the relative transformation from the raw blob tracks), and a covariance intersection approach for overcoming data correlation. Our design provides smooth image tracking for features moving up to 7000 pixels per second on the image plane.

cs.CV↗

arXiv2026

From open-loop representations to closed-loop feedback implementations in differential games: A numerical case study

Solutions to pursuit-evasion and surveillance-evasion differential games are typically computed and expressed using open-loop representations, with the synthesis of feedback strategies significantly less common. We propose a numerical scheme for obtaining feedback strategies for the recently introduced prying-pedestrian surveillance-evasion differential game. The scheme involves computing feedback strategies as input-output maps approximated via neural networks trained using data obtained from open-loop representations of solutions. Simulations show the effectiveness of neural networks trained with an appropriate learning-loss function. Since optimal feedback strategies are discontinuous, as a second contribution, the potential loss/gain of individual players is subsequently studied for players using sample-and-hold feedback compared to continuous-time feedback.

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arXiv2025

An Extended Kalman Filter for Systems with Infinite-Dimensional Measurements

This article examines state estimation in discrete-time nonlinear stochastic systems with finite-dimensional states and infinite-dimensional measurements, motivated by real-world applications such as vision-based localization and tracking. We develop an extended Kalman filter (EKF) for real-time state estimation, with the measurement noise modeled as an infinite-dimensional random field. When applied to vision-based state estimation, the measurement Jacobians required to implement the EKF are shown to correspond to image gradients. This result provides a novel system-theoretic justification for the use of image gradients as features for vision-based state estimation, contrasting with their (often heuristic) introduction in many computer-vision pipelines. We demonstrate the practical utility of the EKF on a public real-world dataset involving the localization of an aerial drone using video from a downward-facing monocular camera. The EKF is shown to outperform VINS-MONO, an established visual-inertial odometry algorithm, in some cases achieving mean squared error reductions of up to an order of magnitude.

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arXiv2025

Two-Player Dynamic Potential LQ Games with Sequentially Revealed Costs

We investigate a novel finite-horizon linear-quadratic (LQ) feedback dynamic potential game with a priori unknown cost matrices played between two players. The cost matrices are revealed to the players sequentially, with the potential for future values to be previewed over a short time window. We propose an algorithm that enables the players to predict and track a feedback Nash equilibrium trajectory, and we measure the quality of their resulting decisions by introducing the concept of \emph{price of uncertainty}. We show that under the proposed algorithm, the price of uncertainty is bounded by horizon-invariant constants. The constants are the sum of three terms; the first and second terms decay exponentially as the preview window grows, and another depends on the magnitude of the differences between the cost matrices for each player. Through simulations, we illustrate that the resulting price of uncertainty initially decays at an exponential rate as the preview window lengthens, then remains constant for large time horizons.

math.OC↗

arXiv2025

ZORMS-LfD: Learning from Demonstrations with Zeroth-Order Random Matrix Search

We propose Zeroth-Order Random Matrix Search for Learning from Demonstrations (ZORMS-LfD). ZORMS-LfD enables the costs, constraints, and dynamics of constrained optimal control problems, in both continuous and discrete time, to be learned from expert demonstrations without requiring smoothness of the learning-loss landscape. In contrast, existing state-of-the-art first-order methods require the existence and computation of gradients of the costs, constraints, dynamics, and learning loss with respect to states, controls and/or parameters. Most existing methods are also tailored to discrete time, with constrained problems in continuous time receiving only cursory attention. We demonstrate that ZORMS-LfD matches or surpasses the performance of state-of-the-art methods in terms of both learning loss and compute time across a variety of benchmark problems. On unconstrained continuous-time benchmark problems, ZORMS-LfD achieves similar loss performance to state-of-the-art first-order methods with an over $80$\% reduction in compute time. On constrained continuous-time benchmark problems where there is no specialized state-of-the-art method, ZORMS-LfD is shown to outperform the commonly used gradient-free Nelder-Mead optimization method.

cs.LG↗

arXiv2025

ISC-POMDPs: Partially Observed Markov Decision Processes with Initial-State Dependent Costs

We introduce a class of partially observed Markov decision processes (POMDPs) with costs that can depend on both the value and (future) uncertainty associated with the initial state. These Initial-State Cost POMDPs (ISC-POMDPs) enable the specification of objectives relative to a priori unknown initial states, which is useful in applications such as robot navigation, controlled sensing, and active perception, that can involve controlling systems to revisit, remain near, or actively infer their initial states. By developing a recursive Bayesian fixed-point smoother to estimate the initial state that resembles the standard recursive Bayesian filter, we show that ISC-POMDPs can be treated as POMDPs with (potentially) belief-dependent costs. We demonstrate the utility of ISC-POMDPs, including their ability to select controls that resolve (future) uncertainty about (past) initial states, in simulation.

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arXiv2024

Prying Pedestrian Surveillance-Evasion: Minimum-Time Evasion from an Agile Pursuer

A new surveillance-evasion differential game is posed and solved in which an agile pursuer (the prying pedestrian) seeks to remain within a given surveillance range of a less agile evader that aims to escape. In contrast to previous surveillance-evasion games, the pursuer is agile in the sense of being able to instantaneously change the direction of its velocity vector, whilst the evader is constrained to have a finite maximum turn rate. Both the game of kind concerned with conditions under which the evader can escape, and the game of degree concerned with the evader seeking to minimize the escape time whilst the pursuer seeks to maximize it, are considered. The game-of-degree solution is surprisingly complex compared to solutions to analogous pursuit-evasion games with an agile pursuer since it exhibits dependence on the ratio of the pursuer's speed to the evader's speed. It is, however, surprisingly simple compared to solutions to classic surveillance-evasion games with a turn-limited pursuer.

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arXiv2024

Optimal Linear Filtering for Discrete-Time Systems with Infinite-Dimensional Measurements

Systems equipped with modern sensing modalities such as vision and lidar gain access to increasingly high-dimensional measurements with which to enact estimation and control schemes. In this article, we examine the continuum limit of high-dimensional measurements and analyze state estimation in linear time-invariant systems with infinite-dimensional measurements but finite-dimensional states, both corrupted by additive noise. We propose a linear filter and derive the corresponding optimal gain functional in the sense of the minimum mean square error, analogous to the classic Kalman filter. By modeling the measurement noise as a wide-sense stationary random field, we are able to derive the optimal linear filter explicitly, in contrast to previous derivations of Kalman filters in distributed-parameter settings. Interestingly, we find that we need only impose conditions that are finite-dimensional in nature to ensure that the filter is asymptotically stable. The proposed filter is verified via simulation of a linearized system with a pinhole camera sensor.

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arXiv2024

Extended Kalman Filtering for Recursive Online Discrete-Time Inverse Optimal Control

We formulate the discrete-time inverse optimal control problem of inferring unknown parameters in the objective function of an optimal control problem from measurements of optimal states and controls as a nonlinear filtering problem. This formulation enables us to propose a novel extended Kalman filter (EKF) for solving inverse optimal control problems in a computationally efficient recursive online manner that requires only a single pass through the measurement data. Importantly, we show that the Jacobians required to implement our EKF can be computed efficiently by exploiting recent Pontryagin differentiable programming results, and that our consideration of an EKF enables the development of first-of-their-kind theoretical error guarantees for online inverse optimal control with noisy incomplete measurements. Our proposed EKF is shown to be significantly faster than an alternative unscented Kalman filter-based approach.

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arXiv2023

Exactly Optimal Bayesian Quickest Change Detection for Hidden Markov Models

This paper considers the quickest detection problem for hidden Markov models (HMMs) in a Bayesian setting. We construct an augmented HMM representation of the problem that allows the application of a dynamic programming approach to prove that Shiryaev's rule is an (exact) optimal solution. This augmented representation highlights the problem's fundamental information structure and suggests possible relaxations to more exotic change event priors not appearing in the literature. Finally, this augmented representation also allows us to present an efficient computational method for implementing the optimal solution.

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arXiv2023

Regret Analysis of Online LQR Control via Trajectory Prediction and Tracking: Extended Version

In this paper, we propose and analyze a new method for online linear quadratic regulator (LQR) control with a priori unknown time-varying cost matrices. The cost matrices are revealed sequentially with the potential for future values to be previewed over a short window. Our novel method involves using the available cost matrices to predict the optimal trajectory, and a tracking controller to drive the system towards it. We adopted the notion of dynamic regret to measure the performance of this proposed online LQR control method, with our main result being that the (dynamic) regret of our method is upper bounded by a constant. Moreover, the regret upper bound decays exponentially with the preview window length, and is extendable to systems with disturbances. We show in simulations that our proposed method offers improved performance compared to other previously proposed online LQR methods.

math.OC↗

arXiv2023

Smoother Entropy for Active State Trajectory Estimation and Obfuscation in POMDPs

We study the problem of controlling a partially observed Markov decision process (POMDP) to either aid or hinder the estimation of its state trajectory. We encode the estimation objectives via the smoother entropy, which is the conditional entropy of the state trajectory given measurements and controls. Consideration of the smoother entropy contrasts with previous approaches that instead resort to marginal (or instantaneous) state entropies due to tractability concerns. By establishing novel expressions for the smoother entropy in terms of the POMDP belief state, we show that both the problems of minimising and maximising the smoother entropy in POMDPs can surprisingly be reformulated as belief-state Markov decision processes with concave cost and value functions. The significance of these reformulations is that they render the smoother entropy a tractable optimisation objective, with structural properties amenable to the use of standard POMDP solution techniques for both active estimation and obfuscation. Simulations illustrate that optimisation of the smoother entropy leads to superior trajectory estimation and obfuscation compared to alternative approaches.

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arXiv2023

Entropy-Regularized Partially Observed Markov Decision Processes

We investigate partially observed Markov decision processes (POMDPs) with cost functions regularized by entropy terms describing state, observation, and control uncertainty. Standard POMDP techniques are shown to offer bounded-error solutions to these entropy-regularized POMDPs, with exact solutions possible when the regularization involves the joint entropy of the state, observation, and control trajectories. Our joint-entropy result is particularly surprising since it constitutes a novel, tractable formulation of active state estimation.

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arXiv2022

Minimum-Time Escape from a Circular Region for a Dubins Car

We investigate the problem of finding paths that enable a robot modeled as a Dubins car (i.e., a constant-speed finite-turn-rate unicycle) to escape from a circular region of space in minimum time. This minimum-time escape problem arises in marine, aerial, and ground robotics in situations where a safety region has been violated and must be exited before a potential negative consequence occurs (e.g., a collision). Using the tools of nonlinear optimal control theory, we show that a surprisingly simple closed-form feedback control law solves this minimum-time escape problem, and that the minimum-time paths have an elegant geometric interpretation.

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arXiv2021

Active Trajectory Estimation for Partially Observed Markov Decision Processes via Conditional Entropy

In this paper, we consider the problem of controlling a partially observed Markov decision process (POMDP) in order to actively estimate its state trajectory over a fixed horizon with minimal uncertainty. We pose a novel active smoothing problem in which the objective is to directly minimise the smoother entropy, that is, the conditional entropy of the (joint) state trajectory distribution of concern in fixed-interval Bayesian smoothing. Our formulation contrasts with prior active approaches that minimise the sum of conditional entropies of the (marginal) state estimates provided by Bayesian filters. By establishing a novel form of the smoother entropy in terms of the POMDP belief (or information) state, we show that our active smoothing problem can be reformulated as a (fully observed) Markov decision process with a value function that is concave in the belief state. The concavity of the value function is of particular importance since it enables the approximate solution of our active smoothing problem using piecewise-linear function approximations in conjunction with standard POMDP solvers. We illustrate the approximate solution of our active smoothing problem in simulation and compare its performance to alternative approaches based on minimising marginal state estimate uncertainties.

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arXiv2021

Smoothing-Averse Control: Covertness and Privacy from Smoothers

In this paper we investigate the problem of controlling a partially observed stochastic dynamical system such that its state is difficult to infer using a (fixed-interval) Bayesian smoother. This problem arises naturally in applications in which it is desirable to keep the entire state trajectory of a system concealed. We pose our smoothing-averse control problem as the problem of maximising the (joint) entropy of smoother state estimates (i.e., the joint conditional entropy of the state trajectory given the history of measurements and controls). We show that the entropy of Bayesian smoother estimates for general nonlinear state-space models can be expressed as the sum of entropies of marginal state estimates given by Bayesian filters. This novel additive form allows us to reformulate the smoothing-averse control problem as a fully observed stochastic optimal control problem in terms of the usual concept of the information (or belief) state, and solve the resulting problem via dynamic programming. We illustrate the applicability of smoothing-averse control to privacy in cloud-based control and covert robotic navigation.

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arXiv2021

Intelligent Reference Curation for Visual Place Recognition via Bayesian Selective Fusion

A key challenge in visual place recognition (VPR) is recognizing places despite drastic visual appearance changes due to factors such as time of day, season, weather or lighting conditions. Numerous approaches based on deep-learnt image descriptors, sequence matching, domain translation, and probabilistic localization have had success in addressing this challenge, but most rely on the availability of carefully curated representative reference images of the possible places. In this paper, we propose a novel approach, dubbed Bayesian Selective Fusion, for actively selecting and fusing informative reference images to determine the best place match for a given query image. The selective element of our approach avoids the counterproductive fusion of every reference image and enables the dynamic selection of informative reference images in environments with changing visual conditions (such as indoors with flickering lights, outdoors during sunshowers or over the day-night cycle). The probabilistic element of our approach provides a means of fusing multiple reference images that accounts for their varying uncertainty via a novel training-free likelihood function for VPR. On difficult query images from two benchmark datasets, we demonstrate that our approach matches and exceeds the performance of several alternative fusion approaches along with state-of-the-art techniques that are provided with prior (unfair) knowledge of the best reference images. Our approach is well suited for long-term robot autonomy where dynamic visual environments are commonplace since it is training-free, descriptor-agnostic, and complements existing techniques such as sequence matching.

cs.CV↗