Koopman operator theory offers a rigorous treatment of dynamics, emerging as a robust alternative for learning-based control in robotics. By representing nonlinear dynamics as a linear, higher-dimensional operator, it provides a fresh lens for modeling complex systems. Its ability to support incremental updates and low computational cost makes it particularly appealing for real-time applications and online learning. This review delves deeply into the foundations, systematically bridging theoretical principles to practical robotic applications. We explain mathematical underpinnings, approximation approaches for inputs, data collection strategies, and lifting function design. We explore how Koopman models unify tasks like model-based control, state estimation, and motion planning. The review surveys cutting-edge research across domains ranging from aerial and legged platforms to manipulators, soft robots, and multi-agent networks. We also present advanced theoretical topics and reflect on open challenges and future research directions. To support adoption, we provide a hands-on tutorial with code athttps://github.com/sunnyshi0310/KoopmanRobo/tree/main.

Establishing stability certificates for closed-loop systems under reinforcement learning (RL) policies is essential to move beyond empirical performance and offer guarantees of system behavior. Classical Lyapunov methods require a strict stepwise decrease in the Lyapunov function but such certificates are difficult to construct for learned policies. The RL value function is a natural candidate but it is not well understood how it can be adapted for this purpose. To gain intuition, we first study the linear quadratic regulator (LQR) problem and make two key observations. First, a Lyapunov function can be obtained from the value function of an LQR policy by augmenting it with a residual term related to the system dynamics and stage cost. Second, the classical Lyapunov decrease requirement can be relaxed to a generalized Lyapunov condition requiring only decrease on average over multiple time steps. Using this intuition, we consider the nonlinear setting and formulate an approach to learn generalized Lyapunov functions by augmenting RL value functions with neural network residual terms. Our approach successfully certifies the stability of RL policies trained on Gymnasium and DeepMind Control benchmarks. We also extend our method to jointly train neural controllers and stability certificates using a multi-step Lyapunov loss, resulting in larger certified inner approximations of the region of attraction compared to the classical Lyapunov approach. Overall, our formulation enables stability certification for a broad class of systems with learned policies by making certificates easier to construct, thereby bridging classical control theory and modern learning-based methods.

This paper considers the problem of coordinating vehicular traffic at an intersection and on the branches leading to it in order to minimize a combination of total travel time and energy consumption. We propose a provably safe hierarchical-distributed solution to balance computational complexity and optimality of the system operation. In our design, a central intersection manager communicates with vehicles heading toward the intersection, groups them into clusters (termed bubbles) as they appear, and determines an optimal schedule of passage through the intersection for each bubble. The vehicles in each bubble receive their schedule and implement local distributed control to ensure system-wide inter-vehicular safety while respecting speed and acceleration limits, conforming to the assigned schedule, and seeking to optimize their individual trajectories. Our analysis rigorously establishes that the different aspects of the hierarchical design operate in concert and that the safety specifications are satisfied. We illustrate its execution through a suite of simulations and compare its performance against optimized signal-based coordination over a wide range of system parameters.

A recent body of exciting work seeks to shed light on the behavior of accelerated methods in optimization via high-resolution differential equations. These differential equations are continuous counterparts of the discrete-time optimization algorithms, and their convergence properties can be characterized using the powerful tools provided by classical Lyapunov stability analysis. An outstanding question of pivotal importance is how to discretize these continuous flows while maintaining their convergence rates. This paper provides a novel approach through the idea of opportunistic state-triggered control. We take advantage of the Lyapunov functions employed to characterize the rate of convergence of high-resolution differential equations to design variable-stepsize forward-Euler discretizations that preserve the Lyapunov decay of the original dynamics. The philosophy of our approach is not limited to forward-Euler discretizations and may be combined with other integration schemes.

This paper considers games of incomplete information, where one player (the deceiver) has an informational advantage over the other (the mark) and intends to employ it for belief manipulation. We use the formalism of hypergames to represent the asymmetric information available to players. This framework allows us to formalize various notions of belief manipulation that revolve around the idea of the deceiver being able to make the mark believe that a particular action has lost its advantageous character. In the case when the deceiver does not mind revealing information to the mark as the game evolves, we provide a necessary condition and a sufficient condition for deceivability. In the case when the deceiver acts in a stealthy way, i.e., restricts its actions to those that do not contradict the belief of the mark, we fully characterize when deception is possible and design to find a sequence of deceiving actions. Our correctness guarantees for this strategy are based on a precise characterization of the acyclic structure of subjective hypergames. An example illustrates our results.

In this paper, we study the feature-based map merging problem in robot networks. While in operation, each robot observes the environment and builds and maintains a local map. Simultaneously, each robot communicates and computes the global map of the environment. Communication between robots is range-limited. We propose a dynamic strategy, based on consensus algorithms, that is fully distributed and does not rely on any particular communication topology. Under mild connectivity conditions on the communication graph, our merging algorithm, asymptotically, converges to the global map. We present a formal analysis of its convergence rate and provide accurate characterizations of the errors as a function of the timestep. The proposed approach has been experimentally validated using real visual information.

In this paper we present an algorithm for merging visual maps in a robot network. Along the operation, each robot observes the environment and builds and maintains its local map. Simultaneously, the robots communicate and build a global map of the environment. The communication between the robots is limited, and, at every time instant, each robot can only exchange data with its neighboring robots. We provide a distributed solution to the problem which does not rely on any particular communication topology and is robust to changes in the topology. Each robot computes and tracks the global map based on local interactions with its neighbors. Our contribution is the extension of distributed sensor fusion ideas to the problem of dynamic map merging. Under mild connectivity conditions on the communication graph, this algorithm asymptotically converges to the global map. The real experiments have been carried out with visual information, which is of special interest in robotics.

This paper describes a strategy to select optimal motions of multi robot systems equipped with cameras in such a way that they can successively improve the observation of the environment. We present a solution designed for omnidirectional cameras, although the results can be extended to conventional cameras. The key idea is the selection of a finite set of candidate next positions for every robot within their local landmark-based stochastic maps. In this way, the cost function measuring the perception improvement when a robot moves to a new position can be easily evaluated on the finite set of candidate positions. Then, the robots in the team can coordinate based on these small pieces of information. The proposed strategy is designed to be integrated with a map merging algorithm where robots fuse their maps to get a more precise knowledge of the environment. The interest of the proposed strategy for uncertainty reduction is that it is suitable for visual sensing, allows an efficient information exchange, presents a low computational cost and makes the robot coordination easier.

This paper presents a coordination algorithm for mobile autonomous robots. Relying on distributed sensing, the robots achieve rendezvous, i.e., they move to a common location. Each robot is a point mass moving in a simply connected, nonconvex, unknown environment according to an omnidirectional kinematic model. It is equipped with line-of-sight limited-range sensors, i.e., it can measure the relative position of any object (robots or environment boundary) if and only if the object is within a given distance and there are no obstacles in between. The perimeter minimizing algorithm is designed using the notions of robust visibility, connectivity-preserving constraint sets, and proximity graphs. The algorithm provably achieves rendezvous if the interagent sensing graph is connected at any time during the evolution of the group. Simulations illustrate the theoretical results and the performance of the proposed algorithm in asynchronous setups and with measurement errors, control errors, and nonzero robot size. Simulations to illustrate the importance of visibility constraints and comparisons with the optimal centralized algorithm are also included.

In this paper, we explore the impact of reference node, or "landmark", placement on the accuracy of the coordinate systems built using topology-based localization techniques. Such techniques employ landmarks to which each node computes its hop-count distance. A node's coordinates is given by the hop-count distance to all landmarks. To our knowledge, our paper is the first to study the impact of landmark placement on the accuracy of the resulting coordinate system. We show that placing landmarks on the periphery of the topology yields more accurate coordinate systems when compared to placing landmarks in the interior of the topology. Nevertheless, our simulation results also show that, in general, if enough landmarks are used, random landmark placement yields comparative performance to placing landmarks on the boundary randomly or equally spaced. This is an important result since boundary placement (especially at equal distances) may turn out to be infeasible and/or prohibitively expensive (in terms of power consumption as well as processing and communication overhead). This is also the first study to consider not only uniform, synthetic topologies, but also, non-uniform topologies resembling more concrete deployments.

This paper investigates the behavior of a group of autonomous robots evolving in a polygonal environment according to a "move away from the closest neighbor" heuristic. We demonstrate that this distributed coordination algorithm optimizes an aggregate cost function that measures how uniformly distributed are the robots in their environment. Our technical approach based on non-smooth analysis and computational geometry unveils a sphere-packing problem. The algorithm is implemented in a testbed of indoor mobile robots equipped with sonar. We develop novel approaches for improving single point sonar scan performance. These algorithms are then shown to have improved reliability, resolution and speed in distributed environments as compared to other scanning methods.

This paper presents control and coordination algorithms for groups of vehicles. The focus is on autonomous vehicle networks performing distributed sensing tasks, where each vehicle plays the role of a mobile tunable sensor. The paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies. The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.

This paper presents motion planning algorithms for underactuated systems evolving on rigid rotation and displacement groups. Motion planning is transcribed into (low-dimensional) combinatorial selection and inverse-kinematics problems. We present a catalog of solutions for all underactuated systems on SE(2), SO(3) and SE(2) /spl times/ /spl Ropf/ classified according to their controllability properties.

This paper describes decentralized control laws for the coordination of multiple vehicles performing spatially distributed tasks. The control laws are based on a gradient descent scheme applied to a class of decentralized utility functions that encode optimal coverage and sensing policies. These utility functions are studied in geographical optimization problems and they arise naturally in vector quantization and in sensor allocation tasks. The approach exploits the computational geometry of spatial structures such as Voronoi diagrams.