About

Todd Murphey is a Professor of Mechanical Engineering at Northwestern University and the Director of the Center for Robotics and Biosystems. His research focuses on computational methods in data-driven control, information theory in physical systems, and embodied intelligence. His projects include robotic exploration using mechanical contact, human-in-the-loop control, and shared control. Professor Murphey has received significant recognition such as the National Science Foundation CAREER award in 2006 and has served as an editor for IEEE Transactions on Robotics from 2014 to 2018. He is also a member of the Air Force Scientific Advisory Board since 2019. His academic background includes a Ph.D. in Control and Dynamical Systems from the California Institute of Technology and a B.S. in Mathematics from the University of Arizona. In addition to his research, he has developed courses focusing on systems analysis, machine dynamics, and numerical methods in optimal control, emphasizing project-based learning.

Research topics

• Computer Science
• Artificial Intelligence
• Engineering
• Geometry
• Mathematics
• Physics
• Medicine
• Acoustics
• Control engineering
• Algorithm
• Electrical engineering
• Mathematical analysis
• Physical medicine and rehabilitation

Selected publications

• Manufacturing Micro-Patterned Surfaces with Multi-Robot Systems

  ArXiv.org · 2026-03-18

  articleOpen accessSenior author

  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.

  PublisherOA PDF

• Manufacturing Micro-Patterned Surfaces with Multi-Robot Systems

  arXiv (Cornell University) · 2026-03-18

  preprintOpen accessSenior author

  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.

  PublisherDOI

• New <i>IEEE Transactions on Robot Learning</i>

  IEEE Robotics & Automation Magazine · 2026-03-01

  article
  PublisherDOI

• Koopman Operators in Robot Learning

  IEEE Transactions on Robotics · 2026-01-01 · 1 citations

  article

  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>.

  PublisherDOI

• Fast Ergodic Search With Kernel Functions

  IEEE Transactions on Robotics · 2025-01-01 · 2 citations

  articleSenior author

  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.

  PublisherDOI

• Scalable Coverage Trajectory Synthesis on GPUs as Statistical Inference

  ArXiv.org · 2025-11-14

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

• Flow Matching Ergodic Coverage

  ArXiv.org · 2025-04-24

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

• Sample-Efficient Online Control Policy Learning with Real-Time Recursive Model Updates

  ArXiv.org · 2025-09-10

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

• Flow Matching Ergodic Coverage

  2025-06-21

  articleOpen accessSenior author

  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.

  PublisherDOI

• Inverse Mixed Strategy Games with Generative Trajectory Models

  ArXiv.org · 2025-02-05

  preprintOpen accessSenior author

  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.

  PublisherOA PDFDOI

Recent grants

• CAREER: Planning and Control for Overconstrained Mechanisms

  NSF · $425k · 2006–2009

• CPS: Synergy: Collaborative Research: Mutually Stabilized Correction in Physical Demonstration

  NSF · $700k · 2013–2018

• Collaborative Research: Major: Puppet Choreography and Automated Marionettes

  NSF · $374k · 2008–2013

• RI: Small: Collaborative Research: Information-driven Autonomous Exploration in Uncertain Underwater Environments

  NSF · $235k · 2017–2021

• Stability and Optimality Properties of Sequential Action Control for Nonlinear and Hybrid Systems

  NSF · $375k · 2017–2021

Frequent coauthors

• Giuseppe Carbone

  77 shared

• Jian S. Dai

  Genesis HealthCare

  73 shared

• Yu Zhou

  Institute of Computing Technology

  72 shared

• Howie Choset

  44 shared

• Gregory S. Chirikjian

  National University of Singapore

  43 shared

• Michael Kassler

  43 shared

• I‐Ming Chen

  Nanyang Technological University

  42 shared

• Ian Abraham

  42 shared

Awards & honors

• National Science Foundation CAREER award (2006)
• Defense Science Study Group 2014-2015