[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86381-en":3,"doc-seo-86381-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":20,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},86381,8796095360427,"Lucas Martin","https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d",8,"Research & Report","Accelerating Sampling-Based Control via Learned Linear Koopman Dynamics","Efficient model predictive path integral (MPPI) control is developed for systems with complex nonlinear dynamics, addressing the computational bottleneck of repeated nonlinear propagation during trajectory sampling. A learned linear deep Koopman operator (DKO) model replaces the true nonlinear dynamics for rollout, enabling faster rollout and more efficient trajectory sampling. The DKO is learned directly from interaction data without requiring analytic system models. MPPI-DK is tested in simulation on benchmark robotics tasks and validated on hardware with quadruped reference tracking, achieving near-MPPI performance with substantially reduced computational cost for real-time control.","Accelerating Sampling-Based Control via Learned Linear Koopman Dynamics  \nWenjian Hao, Yuxuan Fang, Zehui Lu, and Shaoshuai Mou  \narXiv :2603 .05385v2 [ cs .RO] 13 Jul 2026  \nAbstract—This paper presents an efficient model predictive path integral (MPPI) control framework for systems with complex nonlinear dynamics. To improve the computational efficiency of classic MPPI while preserving control performance, we replace the nonlinear dynamics used for trajectory propagation with a learned linear deep Koopman operator (DKO) model, enabling faster rollout and more efficient trajectory sampling. The DKO dynamics are learned directly from interaction data, eliminating the need for analytical system models. The resulting controller, termed MPPI-DK, is evaluated in simulation on inverted pendulum swing-up and stabilization and surface vehicle navigation tasks, and validated on hardware through reference-tracking experiments on a quadruped robot. Experimental results demonstrate that MPPI-DK achieves control performance close to MPPI using true dynamics while substantially reducing computational cost, enabling efficient real-time control on robotic platforms. These results indicate that MPPI-DK provides an effective surrogate for samplingbased control when nonlinear dynamics propagation limits the achievable control frequency.  \nI. INTRODUCTION  \nThe control of robotic systems with nonlinear, highdimensional dynamics remains challenging, particularly for rapid maneuvers and real-time operation. Although model predictive control (MPC) can explicitly handle state and input constraints while optimizing performance over a receding horizon [1], [2], its use at high control frequencies is often limited by repeated online optimization, nonlinear dynamics propagation, and occasional infeasibility. These computational demands can limit real-time robotic deployment. To improvescalability for highly nonlinear systems, model predictive path integral (MPPI) control approximates optimal control updates through Monte Carlo trajectory sampling rather than deterministic online optimization [3] . This samplingbased formulation accommodates nonlinear dynamics and nonconvex costs, supports parallel computation, and has been successfully applied to autonomous driving [3], aerial robotics [4], and legged locomotion [5] .  \nA key limitation of MPPI is the repeated propagation of nonlinear dynamics during trajectory sampling, which can restrict control frequency and scalability for computationally demanding systems. Data-driven surrogate models, particularly deep neural networks (DNNs) [6], can approximate  \nThis material is based upon work supported by the Defense Advanced Research Projects Agency (DARPA) under of the Learning Introspective Control (LINC) project (grant no. N65236-23-C-8012) Any opinions, findingsand conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the DARPA or the U.S. Government.  \nW. Hao, Y. Fang, and S. Mou are with the School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907, USA. (Email:{hao93, fang394, [mous](mous}@purdue.edu)[}](mous}@purdue.edu)[@purdue.edu](mous}@purdue.edu))  \nZ. Lu is an independent researcher. (Email: [zehuilu789@gmail.com](zehuilu789@gmail.com))  \ncomplex dynamics but may still incur substantial cost when evaluated repeatedly within sampling-based control. An alternative perspective of data-driven dynamics learning is using the Koopman operator theory, which represents nonlinear dynamics as linear evolution in a lifted state space [7], [8] . Extended dynamic mode decomposition (EDMD) implements this idea using manually selected lifting functions [9], whereas deep Koopman operator (DKO) methods learn these lifting functions directly from data using DNNs [10]–[12] . DKO models have been integrated with MPC and model-based reinforcement learning [13], [14] . Their linear structure enables efficient state pro","cbCainUSq2jzVMqL","https://ap.wps.com/l/cbCainUSq2jzVMqL","pdf",785684,6,1,"English","en",105,"# Introduction\n## MPPI control challenges and sampling-based motivation\n## Koopman operator and deep Koopman operator overview\n# Problem Formulation and Preliminaries\n## Notation\n# Proposed Framework\n## DKO-accelerated MPPI concept\n## Simulation and hardware validation","[{\"question\":\"What problem does the paper address in sampling-based MPPI control?\",\"answer\":\"The paper targets the computational cost of repeatedly propagating nonlinear dynamics during MPPI trajectory sampling, which can limit control frequency and scalability for complex systems.\"},{\"question\":\"How does MPPI-DK accelerate MPPI without sacrificing performance?\",\"answer\":\"MPPI-DK replaces nonlinear dynamics used in rollouts with a learned linear deep Koopman operator, enabling faster state propagation in the lifted space while keeping compatibility with stochastic sampling and nonconvex costs.\"},{\"question\":\"What evidence is provided for MPPI-DK effectiveness?\",\"answer\":\"The method is evaluated in simulation on tasks such as inverted pendulum swing-up and surface vehicle navigation, and validated on hardware using reference tracking experiments on a quadruped robot, showing near-MPPI performance with reduced computational cost.\"}]",1784211366,15,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"accelerating-sampling-based-control-via-learned-linear-koopman-dynamics","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/accelerating-sampling-based-control-via-learned-linear-koopman-dynamics/86381/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address in sampling-based MPPI control?","Question",{"text":75,"@type":76},"The paper targets the computational cost of repeatedly propagating nonlinear dynamics during MPPI trajectory sampling, which can limit control frequency and scalability for complex systems.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does MPPI-DK accelerate MPPI without sacrificing performance?",{"text":80,"@type":76},"MPPI-DK replaces nonlinear dynamics used in rollouts with a learned linear deep Koopman operator, enabling faster state propagation in the lifted space while keeping compatibility with stochastic sampling and nonconvex costs.",{"name":82,"@type":73,"acceptedAnswer":83},"What evidence is provided for MPPI-DK effectiveness?",{"text":84,"@type":76},"The method is evaluated in simulation on tasks such as inverted pendulum swing-up and surface vehicle navigation, and validated on hardware using reference tracking experiments on a quadruped robot, showing near-MPPI 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