Todd D. Murphey

Applying micro-patterns to surfaces has been shown to impart useful physical properties such as drag reduction and hydrophobicity. However, current manufacturing techniques cannot produce micro-patterned surfaces at scale due to high-cost machinery and inefficient coverage techniques such as raster-scanning. In this work, we use multiple robots, each equipped with a patterning tool, to manufacture these surfaces. To allow these robots to coordinate during the patterning task, we use the ergodic control algorithm, which specifies coverage objectives using distributions. We demonstrate that robots can divide complicated coverage objectives by communicating compressed representations of their trajectory history both in simulations and experimental trials. Further, we show that robot-produced patterning can lower the coefficient of friction of metallic surfaces. This work demonstrates that distributed multi-robot systems can coordinate to manufacture products that were previously unrealizable at scale.

Applying micro-patterns to surfaces has been shown to impart useful physical properties such as drag reduction and hydrophobicity. However, current manufacturing techniques cannot produce micro-patterned surfaces at scale due to high-cost machinery and inefficient coverage techniques such as raster-scanning. In this work, we use multiple robots, each equipped with a patterning tool, to manufacture these surfaces. To allow these robots to coordinate during the patterning task, we use the ergodic control algorithm, which specifies coverage objectives using distributions. We demonstrate that robots can divide complicated coverage objectives by communicating compressed representations of their trajectory history both in simulations and experimental trials. Further, we show that robot-produced patterning can lower the coefficient of friction of metallic surfaces. This work demonstrates that distributed multi-robot systems can coordinate to manufacture products that were previously unrealizable at scale.

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 at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/sunnyshi0310/KoopmanRobo/tree/main</uri>.

Generative model-based imitation learning methods have recently achieved strong results in learning high-complexity motor skills from human demonstrations. However, imitation learning of interactive policies that coordinate with humans in shared spaces without explicit communication remains challenging, due to the significantly higher behavioral complexity in multi-agent interactions compared to non-interactive tasks. In this work, we introduce a structured imitation learning framework for interactive policies by combining generative single-agent policy learning with a flexible yet expressive game-theoretic structure. Our method explicitly separates learning into two steps: first, we learn individual behavioral patterns from multi-agent demonstrations using standard imitation learning; then, we structurally learn inter-agent dependencies by solving an inverse game problem. Preliminary results in a synthetic 5-agent social navigation task show that our method significantly improves non-interactive policies and performs comparably to the ground truth interactive policy using only 50 demonstrations. These results highlight the potential of structured imitation learning in interactive settings.

Coverage motion planning is essential to a wide range of robotic tasks. Unlike conventional motion planning problems, which reason over temporal sequences of states, coverage motion planning requires reasoning over the spatial distribution of entire trajectories, making standard motion planning methods limited in computational efficiency and less amenable to modern parallelization frameworks. In this work, we formulate the coverage motion planning problem as a statistical inference problem from the perspective of flow matching, a generative modeling technique that has gained significant attention in recent years. The proposed formulation unifies commonly used statistical discrepancy measures, such as Kullback-Leibler divergence and Sinkhorn divergence, with a standard linear quadratic regulator problem. More importantly, it decouples the generation of trajectory gradients for coverage from the synthesis of control under nonlinear system dynamics, enabling significant acceleration through parallelization on modern computational architectures, particularly Graphics Processing Units (GPUs). This paper focuses on the advantages of this formulation in terms of scalability through parallelization, highlighting its computational benefits compared to conventional methods based on waypoint tracking.

Ergodic control synthesizes optimal coverage behaviors over spatial distributions for nonlinear systems. However, existing formulations model the robot as a non-volumetric point, whereas in practice a robot interacts with the environment through its body and sensors with physical volume. In this work, we introduce a new ergodic control formulation that optimizes spatial coverage using a volumetric state representation. Our method preserves the asymptotic coverage guarantees of ergodic control, adds minimal computational overhead for real-time control, and supports arbitrary sample-based volumetric models. We evaluate our method across search and manipulation tasks -- with multiple robot dynamics and end-effector geometries or sensor models -- and show that it improves coverage efficiency by more than a factor of two while maintaining a 100% task completion rate across all experiments, outperforming the standard ergodic control method. Finally, we demonstrate the effectiveness of our method on a robot arm performing mechanical erasing tasks. Project website: https://murpheylab.github.io/vec/

Closed-loop control remains an open challenge in soft robotics. The nonlinear responses of soft actuators under dynamic loading conditions limit the use of analytic models for soft robot control. Traditional methods of controlling soft robots underutilize their configuration spaces to avoid nonlinearity, hysteresis, large deformations, and the risk of actuator damage. Furthermore, episodic data-driven control approaches such as reinforcement learning (RL) are traditionally limited by sample efficiency and inconsistency across initializations. In this work, we demonstrate RL for reliably learning control policies for dynamic balancing tasks in real-time single-shot hardware deployments. We use a deformable Stewart platform constructed using parallel, 3D-printed soft actuators based on motorized handed shearing auxetic (HSA) structures. By introducing a curriculum learning approach based on expanding neighborhoods of a known equilibrium, we achieve reliable single-deployment balancing at arbitrary coordinates. In addition to benchmarking the performance of model-based and model-free methods, we demonstrate that in a single deployment, Maximum Diffusion RL is capable of learning dynamic balancing after half of the actuators are effectively disabled, by inducing buckling and by breaking actuators with bolt cutters. Training occurs with no prior data, in as fast as 15 minutes, with performance nearly identical to the fully-intact platform. Single-shot learning on hardware facilitates soft robotic systems reliably learning in the real world and will enable more diverse and capable soft robots.

The absence of sensitive tools for quantifying movement dysfunction hinders our ability to study the underlying causes of motor impairments and makes it difficult to demonstrate the effectiveness of therapeutic approaches. Consequently, it slows down progress in developing novel treatment protocols, including personalized impairment-targeted interventions. While there exist well-established metrics of static and quasi-static motion, such as reaching range, little emphasis has been placed on quantifying dynamic response-controlled and timing-sensitive movements where the continuous modulation of motor activity is required to respond to real-time stimuli. In this study, we employ robot-assisted virtual tasks that require dynamic motion in the upper limb, and develop metrics that assess dynamic capabilities by quantifying the frequency spectra of movement during these tasks. We assess chronic survivors of hemiparetic stroke across three dynamic tasks (n=13 for the first two tasks and n=48 for the third task). We find that hemiparetic stroke causes a significant decline in dynamic response at frequencies above 1.5Hz in the paretic upper limb, with the degree of functional loss dependent on the clinically assessed severity of motor impairment. Frequency-based metrics of dynamic motion could be used to assess performance during everyday activities, such as brushing teeth, cooking, or cleaning. A versatile quantitative assessment of dynamic motion can accelerate progress in understanding and rehabilitating functional dynamic motion in individuals with neuromotor disorders.

Closed-loop control remains an open challenge in soft robotics. The nonlinear responses of soft actuators under dynamic loading conditions limit the use of analytic models for soft robot control. Traditional methods of controlling soft robots underutilize their configuration spaces to avoid nonlinearity, hysteresis, large deformations, and the risk of actuator damage. Furthermore, episodic data-driven control approaches such as reinforcement learning (RL) are traditionally limited by sample efficiency and inconsistency across initializations. In this work, we demonstrate RL for reliably learning control policies for dynamic balancing tasks in real-time single-shot hardware deployments. We use a deformable Stewart platform constructed using parallel, 3D-printed soft actuators based on motorized handed shearing auxetic (HSA) structures. By introducing a curriculum learning approach based on expanding neighborhoods of a known equilibrium, we achieve reliable single-deployment balancing at arbitrary coordinates. In addition to benchmarking the performance of model-based and model-free methods, we demonstrate that in a single deployment, Maximum Diffusion RL is capable of learning dynamic balancing after half of the actuators are effectively disabled, by inducing buckling and by breaking actuators with bolt cutters. Training occurs with no prior data, in as fast as 15 minutes, with performance nearly identical to the fully-intact platform. Single-shot learning on hardware facilitates soft robotic systems reliably learning in the real world and will enable more diverse and capable soft robots.

Data-driven control methods need to be sample-efficient and lightweight, especially when data acquisition and computational resources are limited -- such as during learning on hardware. Most modern data-driven methods require large datasets and struggle with real-time updates of models, limiting their performance in dynamic environments. Koopman theory formally represents nonlinear systems as linear models over observables, and Koopman representations can be determined from data in an optimization-friendly setting with potentially rapid model updates. In this paper, we present a highly sample-efficient, Koopman-based learning pipeline: Recursive Koopman Learning (RKL). We identify sufficient conditions for model convergence and provide formal algorithmic analysis supporting our claim that RKL is lightweight and fast, with complexity independent of dataset size. We validate our method on a simulated planar two-link arm and a hybrid nonlinear hardware system with soft actuators, showing that real-time recursive Koopman model updates improve the sample efficiency and stability of data-driven controller synthesis -- requiring only &lt;10% of the data compared to benchmarks. The high-performance C++ codebase is open-sourced. Website: https://www.zixinatom990.com/home/robotics/corl-2025-recursive-koopman-learning.

Although reinforcement learning (RL) algorithms have demonstrated impressive capabilities in simulation, their transition into the real-world often reveals a performance gap. A key challenge of real-world deployment is ensuring robustness to complex or unmodeled physical phenomena, such as anisotropic friction during locomotion. This discrepancy between simulated and real-world performance underscores the need for hardware benchmarks to rigorously evaluate and improve RL algorithms for physical deployment. In this work, we present NoodleBot—a low-cost, untethered three-link swimmer robot—as a hardware benchmark for RL algorithms. This benchmark is intended to complement embodied learning approaches, where simulations provide confidence that algorithms are able to learn from scratch in challenging environments. We demonstrate three algorithms learning on the platform and compare the results to learning with a simulated swimmer, highlighting the importance of effective state exploration to agent performance. We also show the ability of one algorithm to learn in single-shot hardware deployments. Design, firmware, and software are available open source at https://github.com/MurpheyLab/NoodleBot.

Ergodic coverage effectively generates exploratory behaviors for embodied agents by aligning the spatial distribution of the agent's trajectory with a target distribution, where the difference between these two distributions is measured by the ergodic metric.However, existing ergodic coverage methods are constrained by the limited set of ergodic metrics available for control synthesis, fundamentally limiting their performance.In this work, we propose an alternative approach to ergodic coverage based on flow matching, a technique widely used in generative inference for efficient and scalable sampling.We formally derive the flow matching problem for ergodic coverage and show that it is equivalent to a linear quadratic regulator problem with a closed-form solution.Our formulation enables alternative ergodic metrics from generative inference that overcome the limitations of existing ones.These metrics were previously infeasible for control synthesis but can now be supported with no computational overhead.Specifically, flow matching with the Stein variational gradient flow enables control synthesis directly over the score function of the target distribution, improving robustness to the unnormalized distributions; on the other hand, flow matching with the Sinkhorn divergence flow enables an optimal transportbased ergodic metric, improving coverage performance on nonsmooth distributions with irregular supports.We validate the improved performance and competitive computational efficiency of our method through comprehensive numerical benchmarks and across different nonlinear dynamics.We further demonstrate the practicality of our method through a series of drawing and erasing tasks on a Franka robot.

Game-theoretic models are effective tools for modeling multi-agent interactions, especially when robots need to coordinate with humans. However, applying these models requires inferring their specifications from observed behaviors—a challenging task known as the inverse game problem. Existing inverse game approaches often struggle to account for behavioral uncertainty and measurement noise, and leverage both offline and online data. To address these limitations, we propose an inverse game method that integrates a generative trajectory model into a differentiable mixed-strategy game framework. By representing the mixed strategy with a conditional variational autoencoder (CVAE), our method can infer high-dimensional, multi-modal behavior distributions from noisy measurements while adapting in real-time to new observations. We extensively evaluate our method in a simulated navigation benchmark, where the observations are generated by an unknown game model. Despite the model mismatch, our method can infer Nash-optimal actions comparable to those of the ground-truth model and the oracle inverse game baseline, even in the presence of uncertain agent objectives and noisy measurements.

Current methods based on Neural Radiance Fields fail in the low data limit, particularly when training on incomplete scene data. Prior works augment training data only in next-best-view applications, which lead to hallucinations and model collapse with sparse data. In contrast, we propose adding a set of views during training by rejection sampling from a posterior uncertainty distribution, generated by combining a volumetric uncertainty estimator with spatial coverage. We validate our results on partially observed scenes; on average, our method performs 39.9% better with 87.5% less variability across established scene reconstruction benchmarks, as compared to state of the art baselines. We further demonstrate that augmenting the training set by sampling from any distribution leads to better, more consistent scene reconstruction in sparse environments. This work is foundational for robotic tasks where augmenting a dataset with informative data is critical in resource-constrained, a priori unknown environments. Videos and source code are available at https://murpheylab.github.iollow-data-nerfl

Ergodic coverage effectively generates exploratory behaviors for embodied agents by aligning the spatial distribution of the agent's trajectory with a target distribution, where the difference between these two distributions is measured by the ergodic metric. However, existing ergodic coverage methods are constrained by the limited set of ergodic metrics available for control synthesis, fundamentally limiting their performance. In this work, we propose an alternative approach to ergodic coverage based on flow matching, a technique widely used in generative inference for efficient and scalable sampling. We formally derive the flow matching problem for ergodic coverage and show that it is equivalent to a linear quadratic regulator problem with a closed-form solution. Our formulation enables alternative ergodic metrics from generative inference that overcome the limitations of existing ones. These metrics were previously infeasible for control synthesis but can now be supported with no computational overhead. Specifically, flow matching with the Stein variational gradient flow enables control synthesis directly over the score function of the target distribution, improving robustness to the unnormalized distributions; on the other hand, flow matching with the Sinkhorn divergence flow enables an optimal transport-based ergodic metric, improving coverage performance on non-smooth distributions with irregular supports. We validate the improved performance and competitive computational efficiency of our method through comprehensive numerical benchmarks and across different nonlinear dynamics. We further demonstrate the practicality of our method through a series of drawing and erasing tasks on a Franka robot.

Many soft robots struggle to produce dynamic motions with fast, large displacements. We develop a parallel 6 degree-of-freedom (DoF) Stewart-Gough mechanism using Handed Shearing Auxetic (HSA) actuators. By using soft actuators, we are able to use one third as many mechatronic components as a rigid Stewart platform, while retaining a working payload of 2kg and an open-loop bandwidth greater than 16Hz. We show that the platform is capable of both precise tracing and dynamic disturbance rejection when controlling a ball and sliding puck using a Proportional Integral Derivative (PID) controller. We develop a machine-learning-based kinematics model and demonstrate a functional workspace of roughly 10cm in each translation direction and 28 degrees in each orientation. This 6DoF device has many of the characteristics associated with rigid components—power, speed, and total workspace— while capturing the advantages of soft mechanisms.

Current methods based on Neural Radiance Fields fail in the low data limit, particularly when training on incomplete scene data. Prior works augment training data only in next-best-view applications, which lead to hallucinations and model collapse with sparse data. In contrast, we propose adding a set of views during training by rejection sampling from a posterior uncertainty distribution, generated by combining a volumetric uncertainty estimator with spatial coverage. We validate our results on partially observed scenes; on average, our method performs 39.9% better with 87.5% less variability across established scene reconstruction benchmarks, as compared to state of the art baselines. We further demonstrate that augmenting the training set by sampling from any distribution leads to better, more consistent scene reconstruction in sparse environments. This work is foundational for robotic tasks where augmenting a dataset with informative data is critical in resource-constrained, a priori unknown environments. Videos and source code are available at https://murpheylab.github.io/low-data-nerf/.

Scaling to arbitrarily large bundle adjustment problems requires data and compute to be distributed across multiple devices. Centralized methods in prior works are only able to solve small or medium size problems due to overhead in computation and communication. In this paper, we present a fully decentralized method that alleviates computation and communication bottlenecks to solve arbitrarily large bundle adjustment problems. We achieve this by reformulating the reprojection error and deriving a novel surrogate function that decouples optimization variables from different devices. This function makes it possible to use majorization minimization techniques and reduces bundle adjustment to independent optimization subproblems that can be solved in parallel. Moreover, an efficient closed-form warm start strategy has been presented that always improves bundle adjustment estimates. We further apply Nesterov’s acceleration and adaptive restart to improve convergence while maintaining its theoretical guarantees. Despite limited peer-to-peer communication, our method has provable convergence to first-order critical points under mild conditions. On extensive benchmarks with public datasets, our method converges much faster than decentralized baselines with similar memory usage and communication load. Compared to centralized baselines using a single device, our method, while being decentralized, yields more accurate solutions with significant speedups of up to 953.7x over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="sans-serif">C</mml:mi> <mml:mi mathvariant="sans-serif">e</mml:mi> <mml:mi mathvariant="sans-serif">r</mml:mi> <mml:mi mathvariant="sans-serif">e</mml:mi> <mml:mi mathvariant="sans-serif">s</mml:mi> </mml:math> and 174.6x over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi mathvariant="sans-serif">D</mml:mi> <mml:mi mathvariant="sans-serif">e</mml:mi> <mml:mi mathvariant="sans-serif">e</mml:mi> <mml:mi mathvariant="sans-serif">p</mml:mi> <mml:mi mathvariant="sans-serif">L</mml:mi> <mml:mi mathvariant="sans-serif">M</mml:mi> </mml:math> . Code: https://github.com/facebookresearch/ DABA .

Game-theoretic models are effective tools for modeling multi-agent interactions, especially when robots need to coordinate with humans. However, applying these models requires inferring their specifications from observed behaviors -- a challenging task known as the inverse game problem. Existing inverse game approaches often struggle to account for behavioral uncertainty and measurement noise, and leverage both offline and online data. To address these limitations, we propose an inverse game method that integrates a generative trajectory model into a differentiable mixed-strategy game framework. By representing the mixed strategy with a conditional variational autoencoder (CVAE), our method can infer high-dimensional, multi-modal behavior distributions from noisy measurements while adapting in real-time to new observations. We extensively evaluate our method in a simulated navigation benchmark, where the observations are generated by an unknown game model. Despite the model mismatch, our method can infer Nash-optimal actions comparable to those of the ground-truth model and the oracle inverse game baseline, even in the presence of uncertain agent objectives and noisy measurements.

Eye gaze tracking is increasingly popular due to improved technology and availability. In the domain of assistive device control, however, eye gaze tracking is often used in discrete ways (e.g., activating buttons on a screen), and does not harness the full potential of the gaze signal. In this article, we present a method for collecting both reactionary and controlled eye gaze signals, via screen-based tasks designed to isolate various types of eye movements. The resulting data allows us to build an individualized characterization for eye gaze interface use. Results from a study conducted with participants with motor impairments are presented, offering insights into maximizing the potential of eye gaze for assistive device control. Importantly, we demonstrate the potential for incorporating direct continuous eye gaze inputs into gaze-based interface designs; generally seen as intractable due to the 'Midas touch' problem of differentiating between gaze movements for perception versus for interface operation. Our key insight is to make use of an individualized measure of smooth pursuit characteristics to differentiate between gaze for control and gaze for environment scanning. We also present results relating to gaze-based metrics for mental workload and show the potential for the concurrent use of eye gaze for control input as well as assessing a user's mental workload both offline and in real-time. These findings might inform the development of continuous control paradigms using eye gaze, as well as the use of eye tracking as the sole input modality to systems that share control between human-generated and autonomy-generated inputs.

Ergodic search enables optimal exploration of an information distribution with guaranteed asymptotic coverage of the search space. However, current methods typically have exponential computational complexity and are limited to Euclidean space. We introduce a computationally efficient ergodic search method. Our contributions are two-fold as follows: First, we develop a kernel-based ergodic metric, generalizing it from Euclidean space to Lie groups. We prove this metric is consistent with the exact ergodic metric and ensures linear complexity. Second, we derive an iterative optimal control algorithm for trajectory optimization with the kernel metric. Numerical benchmarks show our method is two orders of magnitude faster than the state-of-the-art method. Finally, we demonstrate the proposed algorithm with a peg-in-hole insertion task. We formulate the problem as a coverage task in the space of SE(3) and use a 30-s-long human demonstration as the prior distribution for ergodic coverage. Ergodicity guarantees the asymptotic solution of the peg-in-hole problem so long as the solution resides within the prior information distribution, which is seen in the 100% success rate.

Robots navigating in crowded areas should negotiate free space with humans rather than fully controlling collision avoidance, as this can lead to freezing behavior. Game theory provides a framework for the robot to reason about potential cooperation from humans for collision avoidance during path planning. In particular, the mixed strategy Nash equilibrium captures the negotiation behavior under uncertainty, making it well suited for crowd navigation. However, computing the mixed strategy Nash equilibrium is often prohibitively expensive for real-time decision-making. In this paper, we propose an iterative Bayesian update scheme over probability distributions of trajectories. The algorithm simultaneously generates a stochastic plan for the robot and probabilistic predictions of other pedestrians’ paths. We prove that the proposed algorithm is equivalent to solving a mixed strategy game for crowd navigation, and the algorithm guarantees the recovery of the global Nash equilibrium of the game. We name our algorithm Bayesian Recursive Nash Equilibrium (BRNE) and develop a real-time model prediction crowd navigation framework. Since BRNE is not solving a general-purpose mixed strategy Nash equilibrium but a tailored formula specifically for crowd navigation, it can compute the solution in real-time on a low-power embedded computer. We evaluate BRNE in both simulated environments and real-world pedestrian datasets. BRNE consistently outperforms non-learning and learning-based methods regarding safety and navigation efficiency. It also reaches human-level crowd navigation performance in the pedestrian dataset benchmark. Lastly, we demonstrate the practicality of our algorithm with real humans on an untethered quadruped robot with fully onboard perception and computation.

Micro-structured surfaces possess useful properties such as friction modification, anti-fouling, and hydrophobicity. However, manufacturing these surfaces in an affordable, scalable, and efficient manner remains challenging. Standard coverage methods for surface patterning require precise placement of micro-scale features over meter-scale surfaces with expensive tooling for support. In this work, we address the scalability challenge in surface patterning by designing a mobile robot with a credit-card-sized footprint to generate micro-scale divots using a modulated tool tip. We provide a control architecture with a target feature density to specify surface coverage, eliminating the dependence on individual indentation locations. Our robot produces high-fidelity surface patterns and achieves automatic coverage of a surface from sophisticated target images. We validate an exemplary application of such micro-structured surfaces by controlling the friction coefficients at different locations according to the density of indentations. These results show the potential for compact robots to perform scalable manufacturing of functional surfaces, switching the focus from precision machines to small-footprint devices tasked with matching only the density of features.

When a robot encounters a novel object, how should it respond$\unicode{x2014}$what data should it collect$\unicode{x2014}$so that it can find the object in the future? In this work, we present a method for learning image features of an unknown number of novel objects. To do this, we use active coverage with respect to latent uncertainties of the novel descriptions. We apply ergodic stability and PAC-Bayes theory to extend statistical guarantees for VAEs to embodied agents. We demonstrate the method in hardware with a robotic arm; the pipeline is also implemented in a simulated environment. Algorithms and simulation are available open source, see http://sites.google.com/u.northwestern.edu/embodied-learning-hardware .

Operating multi-robot teams of diverse agents is an ongoing challenge for emergency deployments, where inter-agent connectivity is rare and environments are unpredictable. Heterogeneous systems must be capable of adapting autonomously while maintaining safety. Here, we develop an algorithm for heterogeneous decentralized multi-robot systems to independently manage safety constraints with provable guarantees for safety and communication in a coverage task. We demonstrate this algorithm in settings where up to 100 agents navigate a simulated cluttered environment with safety constraints that change as agents observe hazards. Further, we show that the performance of a system with a largely disconnected network is equivalent to a fully connected communication network, suggesting that treating connectivity as a constraint may be unnecessary with an appropriate control strategy.

Koopman operator theory offers a rigorous treatment of dynamics and has been emerging as an alternative modeling and learning-based control method across various robotics sub-domains. Due to its ability to represent nonlinear dynamics as a linear (but higher-dimensional) operator, Koopman theory offers a fresh lens through which to understand and tackle the modeling and control of complex robotic systems. Moreover, it enables incremental updates and is computationally inexpensive, thus making it particularly appealing for real-time applications and online active learning. This review delves deeply into the foundations of Koopman operator theory and systematically builds a bridge from theoretical principles to practical robotic applications. We begin by explaining the mathematical underpinnings of the Koopman framework and discussing approximation approaches for incorporating inputs into Koopman-based modeling. Foundational considerations, such as data collection strategies as well as the design of lifting functions for effective system embedding, are also discussed. We then explore how Koopman-based models serve as a unifying tool for a range of robotics tasks, including model-based control, real-time state estimation, and motion planning. The review proceeds to a survey of cutting-edge research that demonstrates the versatility and growing impact of Koopman methods across diverse robotics sub-domains: from aerial and legged platforms to manipulators, soft-bodied systems, and multi-agent networks. A presentation of more advanced theoretical topics, necessary to push forward the overall framework, is included. Finally, we reflect on some key open challenges that remain and articulate future research directions that will shape the next phase of Koopman-inspired robotics. To support practical adoption, we provide a hands-on tutorial with executable code at https://shorturl.at/ouE59.

Tactile perception is important for robotic systems that interact with the world through touch. Touch is an active sense in which tactile measurements depend on the contact properties of an interaction--e.g., velocity, force, acceleration--as well as properties of the sensor and object under test. These dependencies make training tactile perceptual models challenging. Additionally, the effects of limited sensor life and the near-field nature of tactile sensors preclude the practical collection of exhaustive data sets even for fairly simple objects. Active learning provides a mechanism for focusing on only the most informative aspects of an object during data collection. Here we employ an active learning approach that uses a data-driven model's entropy as an uncertainty measure and explore relative to that entropy conditioned on the sensor state variables. Using a coverage-based ergodic controller, we train perceptual models in near-real time. We demonstrate our approach using a biomimentic sensor, exploring "tactile scenes" composed of shapes, textures, and objects. Each learned representation provides a perceptual sensor model for a particular tactile scene. Models trained on actively collected data outperform their randomly collected counterparts in real-time training tests. Additionally, we find that the resulting network entropy maps can be used to identify high salience portions of a tactile scene.

Tactile perception is important for robotic systems that interact with the world through touch. Touch is an active sense in which tactile measurements depend on the contact properties of an interaction—e.g., velocity, force, acceleration— as well as properties of the sensor and object under test. These dependencies make training tactile perceptual models challenging. Additionally, the effects of limited sensor life and the near-field nature of tactile sensors preclude the practical collection of exhaustive data sets even for fairly simple objects. Active learning provides a mechanism for focusing on only the most informative aspects of an object during data collection. Here we employ an active learning approach that uses a data-driven model’s entropy as an uncertainty measure and explore relative to that entropy conditioned on the sensor state variables. Using a coverage-based ergodic controller, we train perceptual models in near-real time. We demonstrate our approach using a biomimentic sensor, exploring "tactile scenes" composed of shapes, textures, and objects. Each learned representation provides a perceptual sensor model for a particular tactile scene. Models trained on actively collected data outperform their randomly collected counterparts in real-time training tests. Additionally, we find that the resulting network entropy maps can be used to identify high salience portions of a tactile scene.

Robots and animals both experience the world through their bodies and senses. Their embodiment constrains their experiences, ensuring that they unfold continuously in space and time. As a result, the experiences of embodied agents are intrinsically correlated. Correlations create fundamental challenges for machine learning, as most techniques rely on the assumption that data are independent and identically distributed. In reinforcement learning, where data are directly collected from an agent’s sequential experiences, violations of this assumption are often unavoidable. Here we derive a method that overcomes this issue by exploiting the statistical mechanics of ergodic processes, which we term maximum diffusion reinforcement learning. By decorrelating agent experiences, our approach provably enables single-shot learning in continuous deployments over the course of individual task attempts. Moreover, we prove our approach generalizes well-known maximum entropy techniques and robustly exceeds state-of-the-art performance across popular benchmarks. Our results at the nexus of physics, learning and control form a foundation for transparent and reliable decision-making in embodied reinforcement learning agents. The central assumption in machine learning that data are independent and identically distributed does not hold in many reinforcement learning settings, as experiences of reinforcement learning agents are sequential and intrinsically correlated in time. Berrueta and colleagues use the mathematical theory of ergodic processes to develop a reinforcement framework that can decorrelate agent experiences and is capable of learning in single-shot deployments.

Clinical management of whiplash-associated disorders is challenging and often unsuccessful, with over a third of whiplash injuries progressing to chronic neck pain. Previous imaging studies have identified muscle fat infiltration, indicative of muscle weakness, in the deep cervical extensor muscles (multifidus and semispinalis cervicis). Yet, kinematic and muscle redundancy prevent the direct assessment of individual neck muscle strength, making it difficult to determine the role of these muscles in motor dysfunction. The purpose of this study was to determine the effects of deep cervical extensor muscle weakness on multi-directional neck strength and muscle activation patterns. Maximum isometric forces and associated muscle activation patterns were computed in 25 test directions using a 3-joint, 24-muscle musculoskeletal model of the head and neck. The computational approach accounts for differential torques about the upper and lower cervical spine. To facilitate clinical translation, the test directions were selected based on locations where resistance could realistically be applied to the head during clinical strength assessments. Simulation results reveal that the deep cervical extensor muscles are active and contribute to neck strength in directions with an extension component. Weakness of this muscle group leads to complex compensatory muscle activation patterns characterized primarily by increased activation of the superficial extensors and deep upper cervical flexors, and decreased activation of the deep upper cervical extensors. These results provide a biomechanistic explanation for movement dysfunction that can be used to develop targeted diagnostics and treatments for chronic neck pain in whiplash-associated disorders.

Ergodic search enables optimal exploration of an information distribution while guaranteeing the asymptotic coverage of the search space. However, current methods typically have exponential computation complexity in the search space dimension and are restricted to Euclidean space. We introduce a computationally efficient ergodic search method. Our contributions are two-fold. First, we develop a kernel-based ergodic metric and generalize it from Euclidean space to Lie groups. We formally prove the proposed metric is consistent with the standard ergodic metric while guaranteeing linear complexity in the search space dimension. Secondly, we derive the first-order optimality condition of the kernel ergodic metric for nonlinear systems, which enables efficient trajectory optimization. Comprehensive numerical benchmarks show that the proposed method is at least two orders of magnitude faster than the state-of-the-art algorithm. Finally, we demonstrate the proposed algorithm with a peg-in-hole insertion task. We formulate the problem as a coverage task in the space of SE(3) and use a 30-second-long human demonstration as the prior distribution for ergodic coverage. Ergodicity guarantees the asymptotic solution of the peg-in-hole problem so long as the solution resides within the prior information distribution, which is seen in the 100% success rate.

A widely accepted explanation for robots planning overcautious or overaggressive trajectories alongside human is that the crowd density exceeds a threshold such that all feasible trajectories are considered unsafe -- the freezing robot problem. However, even with low crowd density, the robot's navigation performance could still drop drastically when in close proximity to human. In this work, we argue that a broader cause of suboptimal navigation performance near human is due to the robot's misjudgement for the human's willingness (flexibility) to share space with others, particularly when the robot assumes the human's flexibility holds constant during interaction, a phenomenon of what we call human robot pacing mismatch. We show that the necessary condition for solving pacing mismatch is to model the evolution of both the robot and the human's flexibility during decision making, a strategy called distribution space modeling. We demonstrate the advantage of distribution space coupling through an anecdotal case study and discuss the future directions of solving human robot pacing mismatch.

Robots navigating in crowded areas should negotiate free space with humans rather than fully controlling collision avoidance, as this can lead to freezing behavior. Game theory provides a framework for the robot to reason about potential cooperation from humans for collision avoidance during path planning. In particular, the mixed strategy Nash equilibrium captures the negotiation behavior under uncertainty, making it well suited for crowd navigation. However, computing the mixed strategy Nash equilibrium is often prohibitively expensive for real-time decision-making. In this paper, we propose an iterative Bayesian update scheme over probability distributions of trajectories. The algorithm simultaneously generates a stochastic plan for the robot and probabilistic predictions of other pedestrians' paths. We prove that the proposed algorithm is equivalent to solving a mixed strategy game for crowd navigation, and the algorithm guarantees the recovery of the global Nash equilibrium of the game. We name our algorithm Bayesian Recursive Nash Equilibrium (BRNE) and develop a real-time model prediction crowd navigation framework. Since BRNE is not solving a general-purpose mixed strategy Nash equilibrium but a tailored formula specifically for crowd navigation, it can compute the solution in real-time on a low-power embedded computer. We evaluate BRNE in both simulated environments and real-world pedestrian datasets. BRNE consistently outperforms non-learning and learning-based methods regarding safety and navigation efficiency. It also reaches human-level crowd navigation performance in the pedestrian dataset benchmark. Lastly, we demonstrate the practicality of our algorithm with real humans on an untethered quadruped robot with fully onboard perception and computation.

Despite theoretical benefits of collaborative robots, disappointing outcomes are well documented by clinical studies, spanning rehabilitation, prostheses, and surgery. Cognitive load theory provides a possible explanation for why humans in the real world are not realizing the benefits of collaborative robots: high cognitive loads may be impeding human performance. Measuring cognitive availability using an electrocardiogram, we ask 25 participants to complete a virtual-reality task alongside an invisible agent that determines optimal performance by iteratively updating the Bellman equation. Three robots assist by providing environmental information relevant to task performance. By enabling the robots to act more autonomously-managing more of their own behavior with fewer instructions from the human-here we show that robots can augment participants' cognitive availability and decision-making. The way in which robots describe and achieve their objective can improve the human's cognitive ability to reason about the task and contribute to human-robot collaboration outcomes. Augmenting human cognition provides a path to improve the efficacy of collaborative robots. By demonstrating how robots can improve human cognition, this work paves the way for improving the cognitive capabilities of first responders, manufacturing workers, surgeons, and other future users of collaborative autonomy systems.

We present a method for controlling a swarm using its spectral decomposition—that is, by describing the set of trajectories of a swarm in terms of a spatial distribution throughout the operational domain—guaranteeing scale invariance with respect to the number of agents both for computation and the operator tasked with controlling the swarm. We use ergodic control, decentralized across the network, for implementation. In the DARPA OFFSET program field setting, we test this interface design for the operator using the swarm tactics and offensive mission planning (STOMP) interface—the same interface used by Raytheon BBN throughout the duration of the OFFSET program. In these tests, we demonstrate that our approach is scale-invariant—the user specification does not depend on the number of agents; it is persistent—the specification remains active until the user specifies a new command; and it is in real time—the user can interact with and interrupt the swarm at any time. Moreover, we show that the spectral/ergodic specification of swarm behavior degrades gracefully as the number of agents goes down, enabling the operator to maintain the same approach as agents become disabled or are added to the network. We demonstrate the scale-invariance and dynamic response of our system in a field-relevant simulator on a variety of tactical scenarios with up to 50 agents. We also demonstrate the dynamic response of our system in the field with a smaller team of agents. Finally, we make the code for our system available.

We consider the problem of distributed pose graph optimization (PGO) that has important applications in multirobot simultaneous localization and mapping (SLAM). We propose the majorization minimization (MM) method for distributed PGO ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {MM\text{--}PGO}$</tex-math></inline-formula> ) that applies to a broad class of robust loss kernels. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {MM\text{--}PGO}$</tex-math></inline-formula> method is guaranteed to converge to first-order critical points under mild conditions. Furthermore, noting that the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {MM\text{--}PGO}$</tex-math></inline-formula> method is reminiscent of proximal methods, we leverage Nesterov's method and adopt adaptive restarts to accelerate convergence. The resulting accelerated MM methods for distributed PGO—both with a master node in the network ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {AMM\text{--}PGO}^*$</tex-math></inline-formula> ) and without ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {AMM\text{--}PGO}^{\#}$</tex-math></inline-formula> )—have faster convergence in contrast to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {MM\text{--}PGO}$</tex-math></inline-formula> method without sacrificing theoretical guarantees. In particular, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {AMM\text{--}PGO}^{\#}$</tex-math></inline-formula> method, which needs no master node and is fully decentralized, features a novel adaptive restart scheme and has a rate of convergence comparable to that of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {AMM\text{--}PGO}^*$</tex-math></inline-formula> method using a master node to aggregate information from all the nodes. The efficacy of this work is validated through extensive applications to 2-D and 3-D SLAM benchmark datasets and comprehensive comparisons against existing state-of-the-art methods, indicating that our MM methods converge faster and result in better solutions to distributed PGO.

Gait generation for soft robots is challenging due to the nonlinear dynamics and high dimensional input spaces of soft actuators. Limitations in soft robotic control and perception force researchers to hand-craft open loop controllers for gait sequences, which is a non-trivial process. Moreover, short soft actuator lifespans and natural variations in actuator behavior limit machine learning techniques to settings that can be learned on the same time scales as robot deployment. Lastly, simulation is not always possible, due to heterogeneity and nonlinearity in soft robotic materials and their dynamics change due to wear. We present a sample-efficient, simulation free, method for self-generating soft robot gaits, using very minimal computation. This technique is demonstrated on a motorized soft robotic quadruped that walks using four legs constructed from 16 “handed shearing auxetic” (HSA) actuators. To manage the dimension of the search space, gaits are composed of two sequential sets of leg motions selected from 7 possible primitives. Pairs of primitives are executed on one leg at a time; we then select the best-performing pair to execute while moving on to subsequent legs. This method-which uses no simulation, sophisticated computation, or user input-consistently generates good translation and rotation gaits in as low as 4 minutes of hardware experimentation, outperforming hand-crafted gaits. This is the first demonstration of completely autonomous gait generation in a soft robot.