Todd D. Murphey

Aleksandra Kalinowska, Elnaz Davoodi, Florian Strub et al. · deepmind

While it is known that communication facilitates cooperation in multi-agent settings, it is unclear how to design artificial agents that can learn to effectively and efficiently communicate with each other. Much research on communication emergence uses reinforcement learning (RL) and explores unsituated communication in one-step referential tasks -- the tasks are not temporally interactive and lack time pressures typically present in natural communication. In these settings, agents may successfully learn to communicate, but they do not learn to exchange information concisely -- they tend towards over-communication and an inefficient encoding. Here, we explore situated communication in a multi-step task, where the acting agent has to forgo an environmental action to communicate. Thus, we impose an opportunity cost on communication and mimic the real-world pressure of passing time. We compare communication emergence under this pressure against learning to communicate with a cost on articulation effort, implemented as a per-message penalty (fixed and progressively increasing). We find that while all tested pressures can disincentivise over-communication, situated communication does it most effectively and, unlike the cost on effort, does not negatively impact emergence. Implementing an opportunity cost on communication in a temporally extended environment is a step towards embodiment, and might be a pre-condition for incentivising efficient, human-like communication.

Thomas A. Berrueta, Allison Pinosky, Todd D. Murphey

Robots and animals both experience the world through their bodies and senses. Their embodiment constrains their experiences, ensuring 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.

Jake Ketchum, Sophia Schiffer, Muchen Sun et al.

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.

Muchen Sun, Allison Pinosky, Ian Abraham et al.

Functional registration algorithms represent point clouds as functions (e.g. spacial occupancy field) avoiding unreliable correspondence estimation in conventional least-squares registration algorithms. However, existing functional registration algorithms are computationally expensive. Furthermore, the capability of registration with unknown scale is necessary in tasks such as CAD model-based object localization, yet no such support exists in functional registration. In this work, we propose a scale-invariant, linear time complexity functional registration algorithm. We achieve linear time complexity through an efficient approximation of L2-distance between functions using orthonormal basis functions. The use of orthonormal basis functions leads to a formulation that is compatible with least-squares registration. Benefited from the least-square formulation, we use the theory of translation-rotation-invariant measurement to decouple scale estimation and therefore achieve scale-invariant registration. We evaluate the proposed algorithm, named FLS (functional least-squares), on standard 3D registration benchmarks, showing FLS is an order of magnitude faster than state-of-the-art functional registration algorithm without compromising accuracy and robustness. FLS also outperforms state-of-the-art correspondence-based least-squares registration algorithm on accuracy and robustness, with known and unknown scale. Finally, we demonstrate applying FLS to register point clouds with varying densities and partial overlaps, point clouds from different objects within the same category, and point clouds from real world objects with noisy RGB-D measurements.

Annalisa T. Taylor, Malachi Landis, Ping Guo et al.

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.

Taosha Fan, Hanlin Wang, Michael Rubenstein et al.

In this paper, we consider the problem of planar graph-based simultaneous localization and mapping (SLAM) that involves both poses of the autonomous agent and positions of observed landmarks. We present CPL-SLAM, an efficient and certifiably correct algorithm to solve planar graph-based SLAM using the complex number representation. We formulate and simplify planar graph-based SLAM as the maximum likelihood estimation (MLE) on the product of unit complex numbers, and relax this nonconvex quadratic complex optimization problem to convex complex semidefinite programming (SDP). Furthermore, we simplify the corresponding complex semidefinite programming to Riemannian staircase optimization (RSO) on the complex oblique manifold that can be solved with the Riemannian trust region (RTR) method. In addition, we prove that the SDP relaxation and RSO simplification are tight as long as the noise magnitude is below a certain threshold. The efficacy of this work is validated through applications of CPL-SLAM and comparisons with existing state-of-the-art methods on planar graph-based SLAM, which indicates that our proposed algorithm is capable of solving planar graph-based SLAM certifiably, and is more efficient in numerical computation and more robust to measurement noise than existing state-of-the-art methods. The C++ code for CPL-SLAM is available at https://github.com/MurpheyLab/CPL-SLAM.

Alexander Broad, Todd Murphey, Brenna Argall

We present a shared control paradigm that improves a user's ability to operate complex, dynamic systems in potentially dangerous environments without a priori knowledge of the user's objective. In this paradigm, the role of the autonomous partner is to improve the general safety of the system without constraining the user's ability to achieve unspecified behaviors. Our approach relies on a data-driven, model-based representation of the joint human-machine system to evaluate, in parallel, a significant number of potential inputs that the user may wish to provide. These samples are used to (1) predict the safety of the system over a receding horizon, and (2) minimize the influence of the autonomous partner. The resulting shared control algorithm maximizes the authority allocated to the human partner to improve their sense of agency, while improving safety. We evaluate the efficacy of our shared control algorithm with a human subjects study (n=20) conducted in two simulated environments: a balance bot and a race car. During the experiment, users are free to operate each system however they would like (i.e., there is no specified task) and are only asked to try to avoid unsafe regions of the state space. Using modern computational resources (i.e., GPUs) our approach is able to consider more than 10,000 potential trajectories at each time step in a control loop running at 100Hz for the balance bot and 60Hz for the race car. The results of the study show that our shared control paradigm improves system safety without knowledge of the user's goal, while maintaining high-levels of user satisfaction and low-levels of frustration. Our code is available online at https://github.com/asbroad/mpmi_shared_control.

Max Muchen Sun, Ayush Gaggar, Peter Trautman et al.

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.

Max Muchen Sun, Francesca Baldini, Katie Hughes et al.

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.

Jueun Kwon, Max M. Sun, Todd Murphey

Ergodic control synthesizes optimal coverage behaviors over spatial distributions for nonlinear systems. However, existing formulations model the robot as a non-volumetric point, but 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.

Max M. Sun, Todd Murphey

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.

Max Muchen Sun, Allison Pinosky, Todd Murphey

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.

Ayush Gaggar, Todd D. Murphey

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/.

Taosha Fan, Joseph Ortiz, Ming Hsiao et al.

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. 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 Ceres and 174.6x over DeepLM. Code: https://joeaortiz.github.io/daba.

Taosha Fan, Todd Murphey

We consider the problem of distributed pose graph optimization (PGO) that has important applications in multi-robot simultaneous localization and mapping (SLAM). We propose the majorization minimization (MM) method for distributed PGO ($\mathsf{MM-PGO}$) that applies to a broad class of robust loss kernels. The $\mathsf{MM-PGO}$ method is guaranteed to converge to first-order critical points under mild conditions. Furthermore, noting that the $\mathsf{MM-PGO}$ 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 ($\mathsf{AMM-PGO}^*$) and without ($\mathsf{AMM-PGO}^{\#}$) -- have faster convergence in contrast to the $\mathsf{AMM-PGO}$ method without sacrificing theoretical guarantees. In particular, the $\mathsf{AMM-PGO}^{\#}$ 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 $\mathsf{AMM-PGO}^*$ method using a master node to aggregate information from all the other nodes. The efficacy of this work is validated through extensive applications to 2D and 3D 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.

Annalisa T. Taylor, Thomas A. Berrueta, Todd D. Murphey

Active learning is a decision-making process. In both abstract and physical settings, active learning demands both analysis and action. This is a review of active learning in robotics, focusing on methods amenable to the demands of embodied learning systems. Robots must be able to learn efficiently and flexibly through continuous online deployment. This poses a distinct set of control-oriented challenges -- one must choose suitable measures as objectives, synthesize real-time control, and produce analyses that guarantee performance and safety with limited knowledge of the environment or robot itself. In this work, we survey the fundamental components of robotic active learning systems. We discuss classes of learning tasks that robots typically encounter, measures with which they gauge the information content of observations, and algorithms for generating action plans. Moreover, we provide a variety of examples -- from environmental mapping to nonparametric shape estimation -- that highlight the qualitative differences between learning tasks, information measures, and control techniques. We conclude with a discussion of control-oriented open challenges, including safety-constrained learning and distributed learning.

Muchen Sun, Francesca Baldini, Peter Trautman et al.

Cooperatively avoiding collision is a critical functionality for robots navigating in dense human crowds, failure of which could lead to either overaggressive or overcautious behavior. A necessary condition for cooperative collision avoidance is to couple the prediction of the agents' trajectories with the planning of the robot's trajectory. However, it is unclear that trajectory based cooperative collision avoidance captures the correct agent attributes. In this work we migrate from trajectory based coupling to a formalism that couples agent preference distributions. In particular, we show that preference distributions (probability density functions representing agents' intentions) can capture higher order statistics of agent behaviors, such as willingness to cooperate. Thus, coupling in distribution space exploits more information about inter-agent cooperation than coupling in trajectory space. We thus introduce a general objective for coupled prediction and planning in distribution space, and propose an iterative best response optimization method based on variational analysis with guaranteed sufficient decrease. Based on this analysis, we develop a sampling-based motion planning framework called DistNav that runs in real time on a laptop CPU. We evaluate our approach on challenging scenarios from both real world datasets and simulation environments, and benchmark against a wide variety of model based and machine learning based approaches. The safety and efficiency statistics of our approach outperform all other models. Finally, we find that DistNav is competitive with human safety and efficiency performance.

Taosha Fan, Kalyan Vasudev Alwala, Donglai Xiang et al.

We propose a novel sparse constrained formulation and from it derive a real-time optimization method for 3D human pose and shape estimation. Our optimization method, SCOPE (Sparse Constrained Optimization for 3D human Pose and shapE estimation), is orders of magnitude faster (avg. 4 ms convergence) than existing optimization methods, while being mathematically equivalent to their dense unconstrained formulation under mild assumptions. We achieve this by exploiting the underlying sparsity and constraints of our formulation to efficiently compute the Gauss-Newton direction. We show that this computation scales linearly with the number of joints and measurements of a complex 3D human model, in contrast to prior work where it scales cubically due to their dense unconstrained formulation. Based on our optimization method, we present a real-time motion capture framework that estimates 3D human poses and shapes from a single image at over 30 FPS. In benchmarks against state-of-the-art methods on multiple public datasets, our framework outperforms other optimization methods and achieves competitive accuracy against regression methods. Project page with code and videos: https://sites.google.com/view/scope-human/.

Aleksandra Kalinowska, Ahalya Prabhakar, Kathleen Fitzsimons et al.

With growing access to versatile robotics, it is beneficial for end users to be able to teach robots tasks without needing to code a control policy. One possibility is to teach the robot through successful task executions. However, near-optimal demonstrations of a task can be difficult to provide and even successful demonstrations can fail to capture task aspects key to robust skill replication. Here, we propose a learning from demonstration (LfD) approach that enables learning of robust task definitions without the need for near-optimal demonstrations. We present a novel algorithmic framework for learning tasks based on the ergodic metric -- a measure of information content in motion. Moreover, we make use of negative demonstrations -- demonstrations of what not to do -- and show that they can help compensate for imperfect demonstrations, reduce the number of demonstrations needed, and highlight crucial task elements improving robot performance. In a proof-of-concept example of cart-pole inversion, we show that negative demonstrations alone can be sufficient to successfully learn and recreate a skill. Through a human subject study with 24 participants, we show that consistently more information about a task can be captured from combined positive and negative (posneg) demonstrations than from the same amount of just positive demonstrations. Finally, we demonstrate our learning approach on simulated tasks of target reaching and table cleaning with a 7-DoF Franka arm. Our results point towards a future with robust, data-efficient LfD for novice users.

Pavel Chvykov, Thomas A. Berrueta, Akash Vardhan et al.

Self-organization is frequently observed in active collectives, from ant rafts to molecular motor assemblies. General principles describing self-organization away from equilibrium have been challenging to identify. We offer a unifying framework that models the behavior of complex systems as largely random, while capturing their configuration-dependent response to external forcing. This allows derivation of a Boltzmann-like principle for understanding and manipulating driven self-organization. We validate our predictions experimentally in shape-changing robotic active matter, and outline a methodology for controlling collective behavior. Our findings highlight how emergent order depends sensitively on the matching between external patterns of forcing and internal dynamical response properties, pointing towards future approaches for design and control of active particle mixtures and metamaterials.

Taosha Fan, Todd Murphey

In this paper, we generalize proximal methods that were originally designed for convex optimization on normed vector space to non-convex pose graph optimization (PGO) on special Euclidean groups, and show that our proposed generalized proximal methods for PGO converge to first-order critical points. Furthermore, we propose methods that significantly accelerate the rates of convergence almost without loss of any theoretical guarantees. In addition, our proposed methods can be easily distributed and parallelized with no compromise of efficiency. The efficacy of this work is validated through implementation on simultaneous localization and mapping (SLAM) and distributed 3D sensor network localization, which indicate that our proposed methods are a lot faster than existing techniques to converge to sufficient accuracy for practical use.

Wanxin Jin, Todd D. Murphey, Zehui Lu et al.

This paper proposes a novel approach that enables a robot to learn an objective function incrementally from human directional corrections. Existing methods learn from human magnitude corrections; since a human needs to carefully choose the magnitude of each correction, those methods can easily lead to over-corrections and learning inefficiency. The proposed method only requires human directional corrections -- corrections that only indicate the direction of an input change without indicating its magnitude. We only assume that each correction, regardless of its magnitude, points in a direction that improves the robot's current motion relative to an unknown objective function. The allowable corrections satisfying this assumption account for half of the input space, as opposed to the magnitude corrections which have to lie in a shrinking level set. For each directional correction, the proposed method updates the estimate of the objective function based on a cutting plane method, which has a geometric interpretation. We have established theoretical results to show the convergence of the learning process. The proposed method has been tested in numerical examples, a user study on two human-robot games, and a real-world quadrotor experiment. The results confirm the convergence of the proposed method and further show that the method is significantly more effective (higher success rate), efficient/effortless (less human corrections needed), and potentially more accessible (fewer early wasted trials) than the state-of-the-art robot learning frameworks.

Maurice Rahme, Ian Abraham, Matthew L. Elwin et al.

We present a sim-to-real framework that uses dynamics and domain randomized offline reinforcement learning to enhance open-loop gaits for legged robots, allowing them to traverse uneven terrain without sensing foot impacts. Our approach, D$^2$-Randomized Gait Modulation with Bezier Curves (D$^2$-GMBC), uses augmented random search with randomized dynamics and terrain to train, in simulation, a policy that modifies the parameters and output of an open-loop Bezier curve gait generator for quadrupedal robots. The policy, using only inertial measurements, enables the robot to traverse unknown rough terrain, even when the robot's physical parameters do not match the open-loop model. We compare the resulting policy to hand-tuned Bezier Curve gaits and to policies trained without randomization, both in simulation and on a real quadrupedal robot. With D$^2$-GMBC, across a variety of experiments on unobserved and unknown uneven terrain, the robot walks significantly farther than with either hand-tuned gaits or gaits learned without domain randomization. Additionally, using D$^2$-GMBC, the robot can walk laterally and rotate while on the rough terrain, even though it was trained only for forward walking.

Giorgos Mamakoukas, Maria L. Castano, Xiaobo Tan et al.

This paper presents a generalizable methodology for data-driven identification of nonlinear dynamics that bounds the model error in terms of the prediction horizon and the magnitude of the derivatives of the system states. Using higher-order derivatives of general nonlinear dynamics that need not be known, we construct a Koopman operator-based linear representation and utilize Taylor series accuracy analysis to derive an error bound. The resulting error formula is used to choose the order of derivatives in the basis functions and obtain a data-driven Koopman model using a closed-form expression that can be computed in real time. Using the inverted pendulum system, we illustrate the robustness of the error bounds given noisy measurements of unknown dynamics, where the derivatives are estimated numerically. When combined with control, the Koopman representation of the nonlinear system has marginally better performance than competing nonlinear modeling methods, such as SINDy and NARX. In addition, as a linear model, the Koopman approach lends itself readily to efficient control design tools, such as LQR, whereas the other modeling approaches require nonlinear control methods. The efficacy of the approach is further demonstrated with simulation and experimental results on the control of a tail-actuated robotic fish. Experimental results show that the proposed data-driven control approach outperforms a tuned PID (Proportional Integral Derivative) controller and that updating the data-driven model online significantly improves performance in the presence of unmodeled fluid disturbance. This paper is complemented with a video: https://youtu.be/9_wx0tdDta0.

Wanxin Jin, Todd D. Murphey, Dana Kulić et al.

This paper develops the method of Continuous Pontryagin Differentiable Programming (Continuous PDP), which enables a robot to learn an objective function from a few sparsely demonstrated keyframes. The keyframes, labeled with some time stamps, are the desired task-space outputs, which a robot is expected to follow sequentially. The time stamps of the keyframes can be different from the time of the robot's actual execution. The method jointly finds an objective function and a time-warping function such that the robot's resulting trajectory sequentially follows the keyframes with minimal discrepancy loss. The Continuous PDP minimizes the discrepancy loss using projected gradient descent, by efficiently solving the gradient of the robot trajectory with respect to the unknown parameters. The method is first evaluated on a simulated robot arm and then applied to a 6-DoF quadrotor to learn an objective function for motion planning in unmodeled environments. The results show the efficiency of the method, its ability to handle time misalignment between keyframes and robot execution, and the generalization of objective learning into unseen motion conditions.

Alexandra Q. Nilles, Ana Pervan, Thomas A. Berrueta et al.

We present a task-centered formal analysis of the relative power of several robot designs, inspired by the unique properties and constraints of micro-scale robotic systems. Our task of interest is object manipulation because it is a fundamental prerequisite for more complex applications such as micro-scale assembly or cell manipulation. Motivated by the difficulty in observing and controlling agents at the micro-scale, we focus on the design of boundary interactions: the robot's motion strategy when it collides with objects or the environment boundary, otherwise known as a bounce rule. We present minimal conditions on the sensing, memory, and actuation requirements of periodic ``bouncing'' robot trajectories that move an object in a desired direction through the incidental forces arising from robot-object collisions. Using an information space framework and a hierarchical controller, we compare several robot designs, emphasizing the information requirements of goal completion under different initial conditions, as well as what is required to recognize irreparable task failure. Finally, we present a physically-motivated model of boundary interactions, and analyze the robustness and dynamical properties of resulting trajectories.

Hadas Kress-Gazit, Kerstin Eder, Guy Hoffman et al.

Robot capabilities are maturing across domains, from self-driving cars, to bipeds and drones. As a result, robots will soon no longer be confined to safety-controlled industrial settings; instead, they will directly interact with the general public. The growing field of Human-Robot Interaction (HRI) studies various aspects of this scenario - from social norms to joint action to human-robot teams and more. Researchers in HRI have made great strides in developing models, methods, and algorithms for robots acting with and around humans, but these "computational HRI" models and algorithms generally do not come with formal guarantees and constraints on their operation. To enable human-interactive robots to move from the lab to real-world deployments, we must address this gap. This article provides an overview of verification, validation and synthesis techniques used to create demonstrably trustworthy systems, describes several HRI domains that could benefit from such techniques, and provides a roadmap for the challenges and the research needed to create formalized and guaranteed human-robot interaction.

Ana Pervan, Todd D. Murphey

In nature, biological organisms jointly evolve both their morphology and their neurological capabilities to improve their chances for survival. Consequently, task information is encoded in both their brains and their bodies. In robotics, the development of complex control and planning algorithms often bears sole responsibility for improving task performance. This dependence on centralized control can be problematic for systems with computational limitations, such as mechanical systems and robots on the microscale. In these cases we need to be able to offload complex computation onto the physical morphology of the system. To this end, we introduce a methodology for algorithmically arranging sensing and actuation components into a robot design while maintaining a low level of design complexity (quantified using a measure of graph entropy), and a high level of task embodiment (evaluated by analyzing the Kullback-Leibler divergence between physical executions of the robot and those of an idealized system). This approach computes an idealized, unconstrained control policy which is projected onto a limited selection of sensors and actuators in a given library, resulting in intelligence that is distributed away from a central processor and instead embodied in the physical body of a robot. The method is demonstrated by computationally optimizing a simulated synthetic cell.

Ahalya Prabhakar, Ian Abraham, Annalisa Taylor et al.

This paper presents a formulation for swarm control and high-level task planning that is dynamically responsive to user commands and adaptable to environmental changes. We design an end-to-end pipeline from a tactile tablet interface for user commands to onboard control of robotic agents based on decentralized ergodic coverage. Our approach demonstrates reliable and dynamic control of a swarm collective through the use of ergodic specifications for planning and executing agent trajectories as well as responding to user and external inputs. We validate our approach in a virtual reality simulation environment and in real-world experiments at the DARPA OFFSET Urban Swarm Challenge FX3 field tests with a robotic swarm where user-based control of the swarm and mission-based tasks require a dynamic and flexible response to changing conditions and objectives in real-time.

Giorgos Mamakoukas, Orest Xherija, T. D. Murphey

Learning a stable Linear Dynamical System (LDS) from data involves creating models that both minimize reconstruction error and enforce stability of the learned representation. We propose a novel algorithm for learning stable LDSs. Using a recent characterization of stable matrices, we present an optimization method that ensures stability at every step and iteratively improves the reconstruction error using gradient directions derived in this paper. When applied to LDSs with inputs, our approach---in contrast to current methods for learning stable LDSs---updates both the state and control matrices, expanding the solution space and allowing for models with lower reconstruction error. We apply our algorithm in simulations and experiments to a variety of problems, including learning dynamic textures from image sequences and controlling a robotic manipulator. Compared to existing approaches, our proposed method achieves an orders-of-magnitude improvement in reconstruction error and superior results in terms of control performance. In addition, it is provably more memory-efficient, with an O(n^2) space complexity compared to O(n^4) of competing alternatives, thus scaling to higher-dimensional systems when the other methods fail.

Ian Abraham, Alexander Broad, Allison Pinosky et al.

We develop a hybrid control approach for robot learning based on combining learned predictive models with experience-based state-action policy mappings to improve the learning capabilities of robotic systems. Predictive models provide an understanding of the task and the physics (which improves sample-efficiency), while experience-based policy mappings are treated as "muscle memory" that encode favorable actions as experiences that override planned actions. Hybrid control tools are used to create an algorithmic approach for combining learned predictive models with experience-based learning. Hybrid learning is presented as a method for efficiently learning motor skills by systematically combining and improving the performance of predictive models and experience-based policies. A deterministic variation of hybrid learning is derived and extended into a stochastic implementation that relaxes some of the key assumptions in the original derivation. Each variation is tested on experience-based learning methods (where the robot interacts with the environment to gain experience) as well as imitation learning methods (where experience is provided through demonstrations and tested in the environment). The results show that our method is capable of improving the performance and sample-efficiency of learning motor skills in a variety of experimental domains.

Aleksandra Kalinowska, Kyra Rudy, Millicent Schlafly et al.

Individuals post stroke experience motor impairments, such as loss of independent joint control, leading to an overall reduction in arm function. Their motion becomes slower and more discoordinated, making it difficult to complete timing-sensitive tasks, such as balancing a glass of water or carrying a bowl with a ball inside it. Understanding how the stroke-induced motor impairments interact with each other can help design assisted training regimens for improved recovery. In this study, we investigate the effects of abnormal joint coupling patterns induced by flexion synergy on timing-sensitive motor coordination in the paretic upper limb. We design a virtual ball-in-bowl task that requires fast movements for optimal performance and implement it on a robotic system, capable of providing varying levels of abduction loading at the shoulder. We recruit 12 participants (6 individuals with chronic stroke and 6 unimpaired controls) and assess their skill at the task at 3 levels of loading, defined by the vertical force applied at the robot end-effector. Our results show that, for individuals with stroke, loading has a significant effect on their ability to generate quick coordinated motion. With increases in loading, their overall task performance decreases and they are less able to compensate for ball dynamics---frequency analysis of their motion indicates that abduction loading weakens their ability to generate movements at the resonant frequency of the dynamic task. This effect is likely due to an increased reliance on lower resolution indirect motor pathways in individuals post stroke. Given the inter-dependency of loading and dynamic task performance, we can create targeted robot-aided training protocols focused on improving timing-sensitive motor control, similar to existing progressive loading therapies, which have shown efficacy for expanding reachable workspace post stroke.

Ian Abraham, Ahalya Prabhakar, Todd D. Murphey

This paper develops KL-Ergodic Exploration from Equilibrium ($\text{KL-E}^3$), 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 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.

Giorgos Mamakoukas, Ian Abraham, Todd D. Murphey

This paper demonstrates the benefits of imposing stability on data-driven Koopman operators. The data-driven identification of stable Koopman operators (DISKO) is implemented using an algorithm \cite{mamakoukas_stableLDS2020} that computes the nearest \textit{stable} matrix 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. Lastly, 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: \url{https://sites.google.com/view/learning-stable-koopman}.

Ana Pervan, Todd Murphey

We consider the problem of designing synthetic cells to achieve a complex goal (e.g., mimicking the immune system by seeking invaders) in a complex environment (e.g., the circulatory system), where they might have to change their control policy, communicate with each other, and deal with stochasticity including false positives and negatives---all with minimal capabilities and only a few bits of memory. We simulate the immune response using cyclic, maze-like environments and use targets at unknown locations to represent invading cells. Using only a few bits of memory, the synthetic cells are programmed to perform a reinforcement learning-type algorithm with which they update their control policy based on randomized encounters with other cells. As the synthetic cells work together to find the target, their interactions as an ensemble function as a physical implementation of a Bayesian update. That is, the particles act as a particle filter. This result provides formal properties about the behavior of the synthetic cell ensemble that can be used to ensure robustness and safety. This method of simplified reinforcement learning is evaluated in simulations, and applied to an actual model of the human circulatory system.

Kathleen Fitzsimons, Aleksandra Kalinowska, Julius P. A. Dewald et al.

Despite the fact that robotic platforms can provide both consistent practice and objective assessments of users over the course of their training, there are relatively few instances where physical human robot interaction has been significantly more effective than unassisted practice or human-mediated training. This paper describes a hybrid shared control robot, which enhances task learning through kinesthetic feedback. The assistance assesses user actions using a task-specific evaluation criterion and selectively accepts or rejects them at each time instant. Through two human subject studies (total n=68), we show that this hybrid approach of switching between full transparency and full rejection of user inputs leads to increased skill acquisition and short-term retention compared to unassisted practice. Moreover, we show that the shared control paradigm exhibits features previously shown to promote successful training. It avoids user passivity by only rejecting user actions and allowing failure at the task. It improves performance during assistance, providing meaningful task-specific feedback. It is sensitive to initial skill of the user and behaves as an `assist-as-needed' control scheme---adapting its engagement in real time based on the performance and needs of the user. Unlike other successful algorithms, it does not require explicit modulation of the level of impedance or error amplification during training and it is permissive to a range of strategies because of its evaluation criterion. We demonstrate that the proposed hybrid shared control paradigm with a task-based minimal intervention criterion significantly enhances task-specific training.

Ian Abraham, Todd D. Murphey

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.

Alexander Broad, Todd Murphey, Brenna Argall

We describe a shared control methodology that can, without knowledge of the task, be used to improve a human's control of a dynamic system, be used as a training mechanism, and be used in conjunction with Imitation Learning to generate autonomous policies that recreate novel behaviors. Our algorithm introduces autonomy that assists the human partner by enforcing safety and stability constraints. The autonomous agent has no a priori knowledge of the desired task and therefore only adds control information when there is concern for the safety of the system. We evaluate the efficacy of our approach with a human subjects study consisting of 20 participants. We find that our shared control algorithm significantly improves the rate at which users are able to successfully execute novel behaviors. Experimental results suggest that the benefits of our safety-aware shared control algorithm also extend to the human partner's understanding of the system and their control skill. Finally, we demonstrate how a combination of our safety-aware shared control algorithm and Imitation Learning can be used to autonomously recreate the demonstrated behaviors.

Taosha Fan, Jarvis Schultz, Todd Murphey

This paper addresses the problem of efficiently computing higher-order variational integrators in simulation and trajectory optimization of mechanical systems as those often found in robotic applications. We develop $O(n)$ algorithms to evaluate the discrete Euler-Lagrange (DEL) equations and compute the Newton direction for solving the DEL equations, which results in linear-time variational integrators of arbitrarily high order. To our knowledge, no linear-time higher-order variational or even implicit integrators have been developed before. Moreover, an $O(n^2)$ algorithm to linearize the DEL equations is presented, which is useful for trajectory optimization. These proposed algorithms eliminate the bottleneck of implementing higher-order variational integrators in simulation and trajectory optimization of complex robotic systems. The efficacy of this paper is validated through comparison with existing methods, and implementation on various robotic systems---including trajectory optimization of the Spring Flamingo robot, the LittleDog robot and the Atlas robot. The results illustrate that the same integrator can be used for simulation and trajectory optimization in robotics, preserving mechanical properties while achieving good scalability and accuracy.

Aleksandra Kalinowska, Thomas A. Berrueta, Adam Zoss et al.

Hybrid systems, such as bipedal walkers, are challenging to control because of discontinuities in their nonlinear dynamics. Little can be predicted about the systems' evolution without modeling the guard conditions that govern transitions between hybrid modes, so even systems with reliable state sensing can be difficult to control. We propose an algorithm that allows for determining the hybrid mode of a system in real-time using data-driven analysis. The algorithm is used with data-driven dynamics identification to enable model predictive control based entirely on data. Two examples---a simulated hopper and experimental data from a bipedal walker---are used. In the context of the first example, we are able to closely approximate the dynamics of a hybrid SLIP model and then successfully use them for control in simulation. In the second example, we demonstrate gait partitioning of human walking data, accurately differentiating between stance and swing, as well as selected subphases of swing. We identify contact events, such as heel strike and toe-off, without a contact sensor using only kinematics data from the knee and hip joints, which could be particularly useful in providing online assistance during walking. Our algorithm does not assume a predefined gait structure or gait phase transitions, lending itself to segmentation of both healthy and pathological gaits. With this flexibility, impairment-specific rehabilitation strategies or assistance could be designed.

Ian Abraham, Ahalya Prabhakar, Todd D. Murphey

This paper develops a method for robots to integrate stability into actively seeking out informative measurements through coverage. We derive a controller using hybrid systems theory that allows us to consider safe equilibrium policies during active data collection. We show that our method is able to maintain Lyapunov attractiveness while still actively seeking out data. Using incremental sparse Gaussian processes, we define distributions which allow a robot to actively seek out informative measurements. We illustrate our methods for shape estimation using a cart double pendulum, dynamic model learning of a hovering quadrotor, and generating galloping gaits starting from stationary equilibrium by learning a dynamics model for the half-cheetah system from the Roboschool environment.

Alexander Broad, Todd Murphey, Brenna Argall

We present a novel approach to shared control of human-machine systems. Our method assumes no a priori knowledge of the system dynamics. Instead, we learn both the dynamics and information about the user's interaction from observation through the use of the Koopman operator. Using the learned model, we define an optimization problem to compute the optimal policy for a given task, and compare the user input to the optimal input. We demonstrate the efficacy of our approach with a user study. We also analyze the individual nature of the learned models by comparing the effectiveness of our approach when the demonstration data comes from a user's own interactions, from the interactions of a group of users and from a domain expert. Positive results include statistically significant improvements on task metrics when comparing a user-only control paradigm with our shared control paradigm. Surprising results include findings that suggest that individualizing the model based on a user's own data does not effect the ability to learn a useful dynamic system. We explore this tension as it relates to developing human-in-the-loop systems further in the discussion.

Alexander Broad, Ian Abraham, Todd Murphey et al.

We present a structured neural network architecture that is inspired by linear time-varying dynamical systems. The network is designed to mimic the properties of linear dynamical systems which makes analysis and control simple. The architecture facilitates the integration of learned system models with gradient-based model predictive control algorithms, and removes the requirement of computing potentially costly derivatives online. We demonstrate the efficacy of this modeling technique in computing autonomous control policies through evaluation in a variety of standard continuous control domains.

Ian Abraham, Anastasia Mavrommati, Todd D. Murphey

We develop an algorithm to explore an environment to generate a measurement model for use in future localization tasks. Ergodic exploration with respect to the likelihood of a particular class of measurement (e.g., a contact detection measurement in tactile sensing) enables construction of the measurement model. Exploration with respect to the information density based on the data-driven measurement model enables localization. We test the two-stage approach in simulations of tactile sensing, illustrating that the algorithm is capable of identifying and localizing objects based on sparsely distributed binary contacts. Comparisons with our method show that visiting low probability regions lead to acquisition of new information rather than increasing the likelihood of known information. Experiments with the Sphero SPRK robot validate the efficacy of this method for collision-based estimation and localization of the environment.

Giorgos Mamakoukas, Malcolm A. MacIver, Todd D. Murphey

This paper derives nonlinear feedback control synthesis for general control affine systems using second-order actions---the second-order needle variations of optimal control---as the basis for choosing each control response to the current state. A second result of the paper is that the method provably exploits the nonlinear controllability of a system by virtue of an explicit dependence of the second-order needle variation on the Lie bracket between vector fields. As a result, each control decision necessarily decreases the objective when the system is nonlinearly controllable using first-order Lie brackets. Simulation results using a differential drive cart, an underactuated kinematic vehicle in three dimensions, and an underactuated dynamic model of an underwater vehicle demonstrate that the method finds control solutions when the first-order analysis is singular. Lastly, the underactuated dynamic underwater vehicle model demonstrates convergence even in the presence of a velocity field.

Ahalya Prabhakar, Anastasia Mavrommati, Jarvis Schultz et al.

This paper addresses the problem of enabling a robot to represent and recreate visual information through physical motion, focusing on drawing using pens, brushes, or other tools. This work uses ergodicity as a control objective that translates planar visual input to physical motion without preprocessing (e.g., image processing, motion primitives). % or human-generated training data (i.e., machine learning). We achieve comparable results to existing drawing methods, while reducing the algorithmic complexity of the software. We demonstrate that optimal ergodic control algorithms with different time-horizon characteristics (infinitesimal, finite, and receding horizon) can generate qualitatively and stylistically different motions that render a wide range of visual information (e.g., letters, portraits, landscapes). In addition, we show that ergodic control enables the same software design to apply to multiple robotic systems by incorporating their particular dynamics, thereby reducing the dependence on task-specific robots. Finally, we demonstrate physical drawings with the Baxter robot.

Ian Abraham, Ahalya Prabhakar, Mitra J. Z. Hartmann et al.

Current methods to estimate object shape---using either vision or touch---generally depend on high-resolution sensing. Here, we exploit ergodic exploration to demonstrate successful shape estimation when using a low-resolution binary contact sensor. The measurement model is posed as a collision-based tactile measurement, and classification methods are used to discriminate between shape boundary regions in the search space. Posterior likelihood estimates of the measurement model help the system actively seek out regions where the binary sensor is most likely to return informative measurements. Results show successful shape estimation of various objects as well as the ability to identify multiple objects in an environment. Interestingly, it is shown that ergodic exploration utilizes non-contact motion to gather significant information about shape. The algorithm is extended in three dimensions in simulation and we present two dimensional experimental results using the Rethink Baxter robot.

Elliot Johnson, Jarvis Schultz, Todd Murphey

Variational integrators are well-suited for simulation of mechanical systems because they preserve mechanical quantities about a system such as momentum, or its change if external forcing is involved, and holonomic constraints. While they are not energy-preserving they do exhibit long-time stable energy behavior. However, variational integrators often simulate mechanical system dynamics by solving an implicit difference equation at each time step, one that is moreover expressed purely in terms of configurations at different time steps. This paper formulates the first- and second-order linearizations of a variational integrator in a manner that is amenable to control analysis and synthesis, creating a bridge between existing analysis and optimal control tools for discrete dynamic systems and variational integrators for mechanical systems in generalized coordinates with forcing and holonomic constraints. The forced pendulum is used to illustrate the technique. A second example solves the discrete LQR problem to find a locally stabilizing controller for a 40 DOF system with 6 constraints.

Taosha Fan, Todd Murphey

In this paper, we propose an approach to designing online feedback controllers for input-saturated robotic systems evolving on Lie groups by extending the recently developed Sequential Action Control (SAC). In contrast to existing feedback controllers, our approach poses the nonconvex constrained nonlinear optimization problem as the tracking of a desired negative mode insertion gradient on the configuration space of a Lie group. This results in a closed-form feedback control law even with input saturation and thus is well suited for online application. In extending SAC to Lie groups, the associated mode insertion gradient is derived and the switching time optimization on Lie groups is studied. We demonstrate the efficacy and scalability of our approach in the 2D kinematic car on SE(2) and the 3D quadrotor on SE(3). We also implement iLQG on a quadrator model and compare to SAC, demonstrating that SAC is both faster to compute and has a larger basin of attraction.

Lauren M. Miller, Yonatan Silverman, Malcolm A. MacIver et al.

This paper presents an active search trajectory synthesis technique for autonomous mobile robots with nonlinear measurements and dynamics. The presented approach uses the ergodicity of a planned trajectory with respect to an expected information density map to close the loop during search. The ergodic control algorithm does not rely on discretization of the search or action spaces, and is well posed for coverage with respect to the expected information density whether the information is diffuse or localized, thus trading off between exploration and exploitation in a single objective function. As a demonstration, we use a robotic electrolocation platform to estimate location and size parameters describing static targets in an underwater environment. Our results demonstrate that the ergodic exploration of distributed information (EEDI) algorithm outperforms commonly used information-oriented controllers, particularly when distractions are present.