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.

Robotic search and rescue, exploration, and inspection require trajectory planning across a variety of domains. A popular approach to trajectory planning for these types of missions is ergodic search, which biases a trajectory to spend time in parts of the exploration domain that are believed to contain more information. Most prior work on ergodic search has been limited to searching simple surfaces, like a 2D Euclidean plane or a sphere, as they rely on projecting functions defined on the exploration domain onto analytically obtained Fourier basis functions. In this paper, we extend ergodic search to any surface that can be approximated by a triangle mesh. The basis functions are approximated through finite element methods on a triangle mesh of the domain. We formally prove that this approximation converges to the continuous case as the mesh approximation converges to the true domain. We demonstrate that on domains where analytical basis functions are available (plane, sphere), the proposed method obtains equivalent results, and while on other domains (torus, bunny, wind turbine), the approach is versatile enough to still search effectively. Lastly, we also compare with an existing ergodic search technique that can handle complex domains and show that our method results in a higher quality exploration.

We present a novel formulation of ergodic trajectory optimization that can be specified over general domains using kernel maximum mean discrepancy. Ergodic trajectory optimization is an effective approach that generates coverage paths for problems related to robotic inspection, information gathering problems, and search and rescue. These optimization schemes compel the robot to spend time in a region proportional to the expected utility of visiting that region. Current methods for ergodic trajectory optimization rely on domain-specific knowledge, e.g., a defined utility map, and well-defined spatial basis functions to produce ergodic trajectories. Here, we present a generalization of ergodic trajectory optimization based on maximum mean discrepancy that requires only samples from the search domain. We demonstrate the ability of our approach to produce coverage trajectories on a variety of problem domains including robotic inspection of objects with differential kinematics constraints and on Lie groups without having access to domain specific knowledge. Furthermore, we show favorable computational scaling compared to existing state-of-the-art methods for ergodic trajectory optimization with a trade-off between domain specific knowledge and computational scaling, thus extending the versatility of ergodic coverage on a wider application domain.

In this work, we consider the problem of multiagent informative path planning (IPP) for robots whose sensor visibility continuously changes as a consequence of a time-varying natural phenomenon. We leverage ergodic trajectory optimization (ETO), which generates paths such that the amount of time an agent spends in an area is proportional to the expected information in that area. We focus specifically on the problem of multi-agent drone search of a wildfire, where we use the time-varying environmental process of smoke diffusion to construct a sensor visibility model. This sensor visibility model is used to repeatedly calculate an expected information distribution (EID) to be used in the ETO algorithm. Our experiments show that our exploration method achieves improved information gathering over both baseline search methods and naive ergodic search formulations.

Effective robotic exploration in continuous domains requires planning trajectories that maximize coverage over a predefined region. A recent development, Stein Variational Ergodic Search (SVES), proposed parallel ergodic exploration (a key approach within the field of robotic exploration), via Stein variational inference that computes a set of candidate trajectories approximating the posterior distribution over the solution space trajectories. While this approach leverages GPU parallelism well, the trajectories in the set might not be distinct enough, leading to a suboptimal set. In this paper, we propose two key methods to diversify the solution set of this approach. First, we leverage the signature kernel within the SVES framework, introducing a pathwise, sequence-sensitive interaction that preserves the Markovian structure of the trajectories and naturally spreads paths across distinct regions of the search space. Second, we propose a derivative-free evolution-strategy interpretation of SVES that exploits batched, GPU-friendly fitness evaluations and can be paired with approximate gradients whenever analytic gradients of the kernel are unavailable or computationally intractable. The resulting method both retains SVES’s advantages while diversifying the solution set and extending its reach to black-box objectives. Across planar forest search, 3D quadrotor coverage, and model-predictive control benchmarks, our approach consistently reduces ergodic cost and produces markedly richer trajectory sets than SVES without significant extra tuning effort.

In this work, we propose a computationally efficient algorithm for visual policy learning that leverages differentiable simulation and first-order analytical policy gradients. Our approach decouple the rendering process from the computation graph, enabling seamless integration with existing differentiable simulation ecosystems without the need for specialized differentiable rendering software. This decoupling not only reduces computational and memory overhead but also effectively attenuates the policy gradient norm, leading to more stable and smoother optimization. We evaluate our method on standard visual control benchmarks using modern GPU-accelerated simulation. Experiments show that our approach significantly reduces wall-clock training time and consistently outperforms all baseline methods in terms of final returns. Notably, on complex tasks such as humanoid locomotion, our method achieves a $4\times$ improvement in final return, and successfully learns a humanoid running policy within 4 hours on a single GPU. Videos and code are available on https://haoxiangyou.github.io/Dva_website

This systems paper presents the implementation and design of RB5, a wheeled robot for autonomous long-term exploration with fewer and cheaper sensors. Requiring just an RGB-D camera and low-power computing hardware, the system consists of an experimental platform with rocker-bogie suspension. It operates in unknown and GPS-denied environments and on indoor and outdoor terrains. The exploration consists of a methodology that extends frontier- and sampling-based exploration with a path-following vector field and a state-of-the-art SLAM algorithm. The methodology allows the robot to explore its surroundings at lower update frequencies, enabling the use of lower-performing and lower-cost hardware while still retaining good autonomous performance. The approach further consists of a methodology to interact with a remotely located human operator based on an inexpensive long-range and low-power communication technology from the internet-of-things domain (i.e., LoRa) and a customized communication protocol. The results and the feasibility analysis show the possible applications and limitations of the approach.Code—The open-source software stack is made available on the project repository webpage†.

Continuous exploration without interruption is important in scenarios such as search and rescue and precision agriculture, where consistent presence is needed to detect events over large areas. Ergodic search already derives continuous trajectories in these scenarios so that a robot spends more time in areas with high information density. However, existing literature on ergodic search does not consider the robot's energy constraints, limiting how long a robot can explore. In fact, if the robots are battery-powered, it is physically not possible to continuously explore on a single battery charge. Our paper tackles this challenge, integrating ergodic search methods with energy-aware coverage. We trade off battery usage and coverage quality, maintaining uninterrupted exploration by at least one agent. Our approach derives an abstract battery model for future state-of-charge estimation and extends canonical ergodic search to ergodic search under battery constraints. Empirical data from simulations and real-world experiments demonstrate the effectiveness of our energy-aware ergodic search, which ensures continuous exploration and guarantees spatial coverage.

In this paper, we address the problem of safe trajectory planning for autonomous search and exploration in constrained, cluttered environments. Guaranteeing safe (collision-free) trajectories is a challenging problem that has garnered significant due to its importance in the successful utilization of robots in search and exploration tasks. This work contributes a method that generates guaranteed safety-critical search trajectories in a cluttered environment. Our approach integrates safety-critical constraints using discrete control barrier functions (DCBFs) with ergodic trajectory optimization to enable safe exploration. Ergodic trajectory optimization plans continuous exploratory trajectories that guarantee complete coverage of a space. We demonstrate through simulated and experimental results on a drone that our approach is able to generate trajectories that enable safe and effective exploration. Furthermore, we show the efficacy of our approach for safe exploration using real-world single- and multi- drone platforms.

Search and rescue applications often need multiple agents to complete a set of conflicting tasks. This paper studies a Multi-Agent Multi-Objective Ergodic Search (MA-MO-ES) approach to this problem where each objective or task is to cover a domain subject to an information map. The goal is to allocate coverage tasks to agents so that all maps are explored ergodically. The combinatorial nature of task allocation makes it computationally expensive to solve for optimal allocation using brute force. Apart from a large number of possible allocations, computing the cost of a task allocation is itself an expensive planning problem. To mitigate the computational challenge, we present a branch and bound-based algorithm with pruning techniques that reduce the number of allocations to be searched to find optimal coverage task allocation. We also present an approach to leverage the similarity between information maps to further reduce computation. Extensive testing on 147 randomly generated test cases shows an order of magnitude improvement in runtime compared to an exhaustive brute force approach.

We present a method for image-guided exploration for mobile robotic systems. Our approach extends ergodic exploration methods, a recent exploration approach that prioritizes complete coverage of a space, with the use of a learned image classifier that automatically detects objects and updates an information map to guide further exploration and localization of objects. Additionally, to improve outcomes of the information collected by our robot's visual sensor, we present a decomposition of the ergodic optimization problem as bi-level coarse and fine solvers, which act respectively on the robot's body and the robot's visual sensor. Our approach is applied to geological survey and localization of rock formations for Mars rovers, with real images from Mars rovers used to train the image classifier. Results demonstrate 1) improved localization of rock formations compared to naive approaches while 2) minimizing the path length of the exploration through the bi-level exploration.

In this article, we demonstrate the benefits of imposing stability on data-driven Koopman operators. The data-driven identification of stable Koopman operators (DISKO) is implemented using an algorithm [1] that computes the neareststablematrix solution to a least-squares reconstruction error. As a first result, we derive a formula that describes the prediction error of Koopman representations for an arbitrary number of time steps, and which shows that stability constraints can improve the predictive accuracy over long horizons. As a second result, we determine formal conditions on basis functions of Koopman operators needed to satisfy the stability properties of an underlying nonlinear system. As a third result, we derive formal conditions for constructing Lyapunov functions for nonlinear systems out of stable data-driven Koopman operators, which we use to verify stabilizing control from data. Finally, we demonstrate the benefits of DISKO in prediction and control with simulations using a pendulum and a quadrotor and experiments with a pusher-slider system. The paper is complemented with a video:https://sites.google.com/view/learning-stable-koopman.

Our work is motivated by humanitarian assistant and disaster relief (HADR) where often it is critical to find signs of life in the presence of conflicting criteria, objectives, and information. We believe ergodic search can provide a framework for exploiting available information as well as exploring for new information in applications, such as HADR. Existing ergodic search methods typically consider search using only a single information map. However, one can readily envision many scenarios where multiple information maps that encode different types of relevant information are used. Ergodic search methods currently do not possess the ability to simultaneously search multiple information maps, nor do they have a way to balance which information gets priority. This leads us to formulate a multiobjective ergodic search (MO-ES) problem, which aims to find the so-called Pareto-optimal solutions, for the purpose of providing human decision makers various solutions that trade off among conflicting criteria. To efficiently solve MO-ES, we develop a framework called sequential local ergodic search (SL-ES), which leverages the recent advances in ergodic search methods as well as the idea of local optimization to efficiently compute Pareto-optimal solutions. Our numerical results show that SL-ES computes solutions of better quality and runs faster than the baselines.

The long-term goal of this work is to enable agents with low-information sensors to perform tasks usually restricted to ones with more sophisticated, high-information sensing capabilities. Our approach is to regulate the motion of these low-information agents to obtain “high-information” results. As a first step, we consider a multi-agent system tasked with locating and tracking a moving target using only noisy binary sensors that measure the presence (or lack thereof) of a target in the sensor's field of view. To generate effective paths for these agents, we use ergodic trajectory optimization with a novel mutual information map that is fast to compute and can handle the discontinuous measurement models often associated with low-information sensing. We compare our approach with existing motion planning methods in multiple simulated experiments. Our experiments show that agents using our method outperform purely coverage-based approaches as well as naive ergodic approaches.

The availability of inexpensive 3D-printed quadrupedal robots motivates the development of learning-based methods compatible with low-cost embedded processors and position-controlled hobby servos. In this work, we show that a linear policy is sufficient to modulate an open-loop trajectory generator, enabling a quadruped to walk over rough, unknown terrain, with limited sensing. The policy is trained in simulation using randomized terrain and dynamics and directly deployed on the robot. We show that the resulting controller can be implemented on resource-constrained systems. We demonstrate the results by deploying the policy on the OpenQuadruped, an open-source 3D-printed robot equipped with hobby servos and an embedded microprocessor.

This article develops KL-ergodic exploration from equilibrium (KL-E3), a method for robotic systems to integrate stability into actively generating informative measurements through ergodic exploration. Ergodic exploration enables robotic systems to indirectly sample from informative spatial distributions globally, avoiding local optima, and without the need to evaluate the derivatives of the distribution against the robot dynamics. Using a hybrid systems theory, we derive a controller that allows a robot to exploit equilibrium policies (i.e., policies that solve a task) while allowing the robot to explore and generate informative data using an ergodic measure that can extend to high-dimensional states. We show that our method is able to maintain Lyapunov attractiveness with respect to the equilibrium task while actively generating data for learning tasks such, as Bayesian optimization, model learning, and off-policy reinforcement learning. In each example, we show that our proposed method is capable of generating an informative distribution of data while synthesizing smooth control signals. We illustrate these examples using simulated systems and provide simplification of our method for real-time online learning in robotic systems.Note to Practitioners—Robotic systems need to adapt to sensor measurements and learn to exploit an understanding of the world around them such that they can truly begin to experiment in the real world. Standard learning methods do not have any restrictions on how the robot can explore and learn, making the robot dynamically volatile. Those that do are often too restrictive in terms of the stability of the robot, resulting in a lack of improved learning due to poor data collection. Applying our method would allow robotic systems to be able to adapt online without the need for human intervention. We show that considering both the dynamics of the robot and the statistics of where the robot has been, we are able to naturally encode where the robot needs to explore and collect measurements for efficient learning that is dynamically safe. With our method, we are able to effectively learn while being energetically efficient compared with state-of-the-art active learning methods. Our approach accomplishes such tasks in a single execution of the robotic system, i.e., the robot does not need human intervention to reset it. Future work will consider multiagent robotic systems that actively learn and explore in a team of collaborative robots.

This paper presents an active learning strategy for robotic systems that takes into account task information, enables fast learning, and allows control to be readily synthesized by taking advantage of the Koopman operator representation. We first motivate the use of representing nonlinear systems as linear Koopman operator systems by illustrating the improved model-based control performance with an actuated Van der Pol system. Information-theoretic methods are then applied to the Koopman operator formulation of dynamical systems where we derive a controller for active learning of robot dynamics. The active learning controller is shown to increase the rate of information about the Koopman operator. In addition, our active learning controller can readily incorporate policies built on the Koopman dynamics, enabling the benefits of fast active learning and improved control. Results using a quadcopter illustrate single-execution active learning and stabilization capabilities during free fall. The results for active learning are extended for automating Koopman observables and we implement our method on real robotic systems.

Although a number of solutions exist for the problems of coverage, search, and target localization-commonly addressed separately-whether there exists a unified strategy that addresses these objectives in a coherent manner without being application specific remains a largely open research question. In this paper, we develop a receding-horizon ergodic control approach, based on hybrid systems theory, that has the potential to fill this gap. The nonlinear model-predictive control algorithm plans real-time motions that optimally improve ergodicity with respect to a distribution defined by the expected information density across the sensing domain. We establish a theoretical framework for global stability guarantees with respect to a distribution. Moreover, the approach is distributable across multiple agents so that each agent can independently compute its own control while sharing statistics of its coverage across a communication network. We demonstrate the method in both simulation and in experiment in the context of target localization, illustrating that the algorithm is independent of the number of targets being tracked and can be run in real time on computationally limited hardware platforms.