{"id":2248,"date":"2024-09-24T13:59:27","date_gmt":"2024-09-24T17:59:27","guid":{"rendered":"https:\/\/cecas.clemson.edu\/ugvaidya\/?page_id=2248"},"modified":"2024-10-17T21:20:30","modified_gmt":"2024-10-18T01:20:30","slug":"publication-demo","status":"publish","type":"page","link":"https:\/\/cecas.clemson.edu\/ugvaidya\/publication-demo\/","title":{"rendered":"Publications"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-page\" data-elementor-id=\"2248\" class=\"elementor elementor-2248\">\n\t\t\t\t<div class=\"elementor-element elementor-element-cfcef2c e-flex e-con-boxed e-con e-parent\" data-id=\"cfcef2c\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-887d918 elementor-widget__width-auto exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-n-accordion\" data-id=\"887d918\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;default_state&quot;:&quot;expanded&quot;,&quot;max_items_expended&quot;:&quot;one&quot;,&quot;n_accordion_animation_duration&quot;:{&quot;unit&quot;:&quot;ms&quot;,&quot;size&quot;:400,&quot;sizes&quot;:[]}}\" data-widget_type=\"nested-accordion.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"e-n-accordion\" aria-label=\"Accordion. Open links with Enter or Space, close with Escape, and navigate with Arrow Keys\">\n\t\t\t\t\t\t<details id=\"e-n-accordion-item-1430\" class=\"e-n-accordion-item\" open>\n\t\t\t\t<summary class=\"e-n-accordion-item-title\" data-accordion-index=\"1\" tabindex=\"0\" aria-expanded=\"true\" aria-controls=\"e-n-accordion-item-1430\" >\n\t\t\t\t\t<span class='e-n-accordion-item-title-header'><div class=\"e-n-accordion-item-title-text\"> Under Review <\/div><\/span>\n\t\t\t\t\t\t\t<span class='e-n-accordion-item-title-icon'>\n\t\t\t<span class='e-opened' ><svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t<span class='e-closed'><svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-fd077f2 e-con-full e-flex e-con e-child\" data-id=\"fd077f2\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-2431987 e-con-full e-flex e-con e-child\" data-id=\"2431987\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-e4b0bde exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"e4b0bde\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Ajinkya Joglekar, Chinmay Samak, Tanmay Samak, Venkat Krovi, and Umesh Vaidya,\u00a0<a href=\"https:\/\/www.researchgate.net\/profile\/Ajinkya-Joglekar\/publication\/380152547_Expanding_Autonomous_Ground_Vehicle_Navigation_Capabilities_through_a_Multi-Model_Parameterized_Koopman_Framework\/links\/662fbdf906ea3d0b7419695b\/Expanding-Autonomous-Ground-Vehicle-Navigation-Capabilities-through-a-Multi-Model-Parameterized-Koopman-Framework.pdf\" target=\"_blank\" rel=\"noopener\">Expanding Autonomous Ground Vehicle Navigation Capabilities through a Multi-Model Parameterized Koopman Framework, Sub<\/a>mitted for publication in Journal.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-07fde9e e-con-full e-flex e-con e-child\" data-id=\"07fde9e\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-462ce7f premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"462ce7f\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-462ce7f\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-462ce7f\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>We introduce the Multi-Model Parameterized Koopman (MMPK) framework, a novel end-2-end data-driven modeling and control pipeline for enabling autonomous navigation in Uncrewed Ground Vehicles. MMPK builds upon the Koopman Extended Dynamic Mode Decomposition (KEDMD) algorithm, offering a flexible model- and controladaptation in the presence of time-varying uncertainties with both ego-vehicle and operational-environment parameters. Unlike traditional methods, MMPK addresses challenges such as overfitting and reliance on a singular global model by adopting a set of pose-agnostic representations of positional data and curvature-parameterized Koopman models, thereby effectively mitigating data bias. The end-2-end unified pipeline encompasses an offline MMPK modeling and an online outer-loop control design consisting of model-based trajectory planning and linear Model Predictive Control adapted to switched Koopman dynamics. The performance of the proposed pipeline is verified via simulation and experimental testing using a 1\/5 th scale Ackermann-steered ground vehicle platform (AgileX Hunter SE) and benchmark driving profiles. Comparative evaluations demonstrate MMPK\u2019s superior path-tracking capabilities and the effectiveness of its local planning strategy in bridging the Model-Sim-Real gap.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-e0b61a0 e-con-full e-flex e-con e-child\" data-id=\"e0b61a0\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-a2aa9cd exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"a2aa9cd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>C Vilas Samak, T Vilas Samak, A Joglekar, U Vaidya, V Krovi, <a class=\"gsc_a_at\" href=\"https:\/\/scholar.google.com\/citations?view_op=view_citation&amp;hl=en&amp;user=bzUVrKIAAAAJ&amp;sortby=pubdate&amp;citation_for_view=bzUVrKIAAAAJ:6ZxmRoH8BuwC\">Digital Twins Meet the Koopman Operator: Data-Driven Learning for Robust Autonomy<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-f0db2a2 e-con-full e-flex e-con e-child\" data-id=\"f0db2a2\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-669c6dd premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"669c6dd\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-669c6dd\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-669c6dd\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>Contrary to on-road autonomous navigation, off-road autonomy is complicated by various factors ranging from sensing challenges to terrain variability. In such a milieu, data-driven approaches have been commonly employed to capture intricate vehicle-environment interactions effectively. However, the success of data-driven methods depends crucially on the quality and quantity of data, which can be compromised by large variability in off-road environments. To address these concerns, we present a novel workflow to recreate the exact vehicle and its target operating conditions digitally for domain-specific data generation. This enables us to effectively model off-road vehicle dynamics from simulation data using the Koopman operator theory, and employ the obtained models for local motion planning and optimal vehicle control. The capabilities of the proposed methodology are demonstrated through an autonomous navigation problem of a 1:5 scale vehicle, where a terrain-informed planner is employed for global mission planning. Results indicate a substantial improvement in off-road navigation performance with the proposed algorithm (5.84x) and underscore the efficacy of digital twinning in terms of improving the sample efficiency (3.2x) and reducing the sim2real gap (5.2%).<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-a21932e e-con-full e-flex e-con e-child\" data-id=\"a21932e\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-8f5f6fc exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"8f5f6fc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Joseph Moyalan, Sriram S. K. S Narayanan, and Umesh Vaidya,\u00a0<a href=\"https:\/\/arxiv.org\/abs\/2407.05133\" target=\"_blank\" rel=\"noopener\">Control Density Function for Robust Safety and Convergence<\/a>, Submitted for publication in Journal.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-1694fdf e-con-full e-flex e-con e-child\" data-id=\"1694fdf\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0c0c3c7 premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"0c0c3c7\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-0c0c3c7\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-0c0c3c7\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>We introduce a novel approach for safe control design based on the density function. A control density function (CDF) is introduced to synthesize a safe controller for a nonlinear dynamic system. The CDF can be viewed as a dual to the control barrier function (CBF), a popular approach used for safe control design. While the safety certificate using the barrier function is based on the notion of invariance, the dual certificate involving the density function has a physical interpretation of occupancy. This occupancy-based physical interpretation is instrumental in providing an analytical construction of density function used for safe control synthesis. The safe control design problem is formulated using the density function as a quadratic programming (QP) problem. In contrast to the QP proposed for control synthesis using CBF, the proposed CDF-based QP can combine both the safety and convergence conditions to target state into single constraints. Further, we consider robustness against uncertainty in system dynamics and the initial condition and provide theoretical results for robust navigation using the CDF. Finally, we present simulation results for safe navigation with single integrator and double-gyre fluid flow-field examples, followed by robust navigation using the bicycle model and autonomous lane-keeping examples.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-90c7efa e-con-full e-flex e-con e-child\" data-id=\"90c7efa\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-4688589 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"4688589\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Umesh Vaidya, When Koopman meets Hamilton and Jacobi, Submitted for Publication in Journal.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-6abc050 e-con-full e-flex e-con e-child\" data-id=\"6abc050\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5ceee8b premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"5ceee8b\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-5ceee8b\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-5ceee8b\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>In this paper, we establish a connection between the spectral theory of the Koopman operator and the solution of the Hamilton Jacobi (HJ) equation. The HJ equation occupies a central place in systems theory, and its solution is of interest in various control problems, including optimal control, robust control, and input-output analysis. One of the main contributions of this paper is to show that the Lagrangian submanifolds, which are fundamental objects for solving the HJ equation, can be obtained using the spectral analysis of the Koopman operator. We present two different procedures for the approximation of the HJ solution. We utilize the spectral properties of the Koopman operator associated with the uncontrolled dynamical system and Hamiltonian systems that arise from the HJ equation to approximate the HJ solution. We present a convex optimization-based computational framework with convergence analysis for approximating the Koopman eigenfunctions and the Lagrangian submanifolds. <br \/>Our solution approach to the HJ equation using Koopman theory provides for a natural extension of results from linear systems to nonlinear systems. We demonstrate the application of this work for solving the optimal control problem. Finally, we present simulation results to validate the paper&#8217;s main findings and compare them against linear quadratic regulator and Taylor series based approximation controllers.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-6879b1e e-con-full e-flex e-con e-child\" data-id=\"6879b1e\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-fe615eb exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"fe615eb\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>U. Vaidya,\u00a0<a href=\"https:\/\/arxiv.org\/abs\/1503.04438\">Stochastic Stability Analysis of Discrete-time System Using Lyapunov Measure<\/a>, Submitted for Publication in Journal.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-b1cc283 e-con-full e-flex e-con e-child\" data-id=\"b1cc283\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-25cdae5 premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"25cdae5\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-25cdae5\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-25cdae5\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>In this paper, we study the stability problem of a stochastic, nonlinear, discrete-time system. We introduce a linear transfer operator-based Lyapunov measure as a new tool for stability verification of stochastic systems. Weaker set-theoretic notion of almost everywhere stochastic stability is introduced and verified, using Lyapunov measure-based stochastic stability theorems. Furthermore, connection between Lyapunov functions, a popular tool for stochastic stability verification, and Lyapunov measures is established. Using the duality property between the linear transfer Perron-Frobenius and Koopman operators, we show the Lyapunov measure and Lyapunov function used for the verification of stochastic stability are dual to each other. Set-oriented numerical methods are proposed for the finite dimensional approximation of the Perron-Frobenius operator; hence, Lyapunov measure is proposed. Stability results in finite dimensional approximation space are also presented. Finite dimensional approximation is shown to introduce further weaker notion of stability referred to as coarse stochastic stability. The results in this paper extend our earlier work on the use of Lyapunov measures for almost everywhere stability verification of deterministic dynamical systems (&#8220;Lyapunov Measure for Almost Everywhere Stability&#8221;, {\\it IEEE Trans. on Automatic Control}, Vol. 53, No. 1, Feb. 2008).<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-ef8f701 e-con-full e-flex e-con e-child\" data-id=\"ef8f701\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-8fb40cc exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"8fb40cc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>K. Ebrahimi, N. Elia, and U. Vaidya, Distributed Robust Optimization via Continuous-Time Dynamics, Submitted for publication in Journal.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1430\" class=\"elementor-element elementor-element-279782f e-con-full e-flex e-con e-child\" data-id=\"279782f\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-e539a31 premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"e539a31\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-e539a31\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-e539a31\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>Abstract not available<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-c39fa29 e-flex e-con-boxed e-con e-parent\" data-id=\"c39fa29\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t<div class=\"elementor-element elementor-element-63b0286 elementor-widget__width-auto exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-n-accordion\" data-id=\"63b0286\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;default_state&quot;:&quot;expanded&quot;,&quot;max_items_expended&quot;:&quot;one&quot;,&quot;n_accordion_animation_duration&quot;:{&quot;unit&quot;:&quot;ms&quot;,&quot;size&quot;:400,&quot;sizes&quot;:[]}}\" data-widget_type=\"nested-accordion.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t<div class=\"e-n-accordion\" aria-label=\"Accordion. Open links with Enter or Space, close with Escape, and navigate with Arrow Keys\">\n\t\t\t\t\t\t<details id=\"e-n-accordion-item-1040\" class=\"e-n-accordion-item\" open>\n\t\t\t\t<summary class=\"e-n-accordion-item-title\" data-accordion-index=\"1\" tabindex=\"0\" aria-expanded=\"true\" aria-controls=\"e-n-accordion-item-1040\" >\n\t\t\t\t\t<span class='e-n-accordion-item-title-header'><div class=\"e-n-accordion-item-title-text\"> Year 2024 <\/div><\/span>\n\t\t\t\t\t\t\t<span class='e-n-accordion-item-title-icon'>\n\t\t\t<span class='e-opened' ><svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t<span class='e-closed'><svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-20f9e10 e-con-full e-flex e-con e-child\" data-id=\"20f9e10\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-0ed146f e-con-full e-flex e-con e-child\" data-id=\"0ed146f\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0ac70ea exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"0ac70ea\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Joseph Moyalan, Sriram S.K.S Narayanan, Andrew Zheng, and Umesh Vaidya,\u00a0<a href=\"https:\/\/ieeexplore.ieee.org\/abstract\/document\/10644714\" target=\"_blank\" rel=\"noopener\">Synthesizing controller for safe navigation using control density function<\/a>, American Control Conference.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-d6564aa e-con-full e-flex e-con e-child\" data-id=\"d6564aa\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-29b0cea premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"29b0cea\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-29b0cea\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-29b0cea\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>We consider the problem of navigating a nonlinear dynamical system from some initial set to some target set while avoiding collision with an unsafe set. We extend the concept of density function to control density function (CDF) for solving navigation problems with safety constraints. The occupancy-based interpretation of the measure associated with the density function is instrumental in imposing the safety constraints. The navigation problem with safety constraints is formulated as a quadratic program (QP) using CDF. The existing approach using the control barrier function (CBF) also formulates the navigation problem with safety constraints as QP. One of the main advantages of the proposed QP using CDF compared to QP formulated using CBF is that both the convergence\/stability and safety can be combined and imposed using the CDF. Simulation results involving the Duffing oscillator and safe navigation of Dubin car models are provided to verify the main findings of the paper.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-340d434 e-con-full e-flex e-con e-child\" data-id=\"340d434\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0de4005 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"0de4005\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Amar Ramapuram Matavalam, Boya Hou, Hyungjin Choi, Subhonmesh Bose, Umesh Vaidya, Data-Driven Transient Stability Analysis Using the Koopman Operator, Accepted for Publication in International Journal of Electrical Power and Energy Systems.\u00a0\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-6706adc e-con-full e-flex e-con e-child\" data-id=\"6706adc\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-5871d9d exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"5871d9d\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Bhagyashree Umathe and Umesh Vaidya,\u00a0<a href=\"https:\/\/ieeexplore.ieee.org\/abstract\/document\/10374195\" target=\"_blank\" rel=\"noopener\">Spectral Koopman Approach for Identifying Stability Boundary<\/a>, IEEE Control Systems Letters and American Control Conference.\u00a0<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-859a4e6 e-con-full e-flex e-con e-child\" data-id=\"859a4e6\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0b99f13 premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"0b99f13\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-0b99f13\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-0b99f13\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>We consider the problem of navigating a nonlinear dynamical system from some initial set to some target set while avoiding collision with an unsafe set. We extend the concept of density function to control density function (CDF) for solving navigation problems with safety constraints. The occupancy-based interpretation of the measure associated with the density function is instrumental in imposing the safety constraints. The navigation problem with safety constraints is formulated as a quadratic program (QP) using CDF. The existing approach using the control barrier function (CBF) also formulates the navigation problem with safety constraints as QP. One of the main advantages of the proposed QP using CDF compared to QP formulated using CBF is that both the convergence\/stability and safety can be combined and imposed using the CDF. Simulation results involving the Duffing oscillator and safe navigation of Dubin car models are provided to verify the main findings of the paper.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-6af8b75 e-con-full e-flex e-con e-child\" data-id=\"6af8b75\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-90ae2dd exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"90ae2dd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>MS Singh, R Pasumarthy, U Vaidya, S Leonhardt,\u00a0<a href=\"https:\/\/www.biorxiv.org\/content\/10.1101\/2024.06.17.599263v1.full.pdf\" target=\"_blank\" rel=\"noopener\">Functional Control of Network Dynamical Systems: An Information Theoretic Approach,<\/a>\u00a0.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-87574be e-con-full e-flex e-con e-child\" data-id=\"87574be\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-b0b1efc premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"b0b1efc\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-b0b1efc\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-b0b1efc\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>In neurological networks, the emergence of various causal interactions and information flows among nodes is governed by the structural connectivity in conjunction with the node dynamics. The information flow describes the direction and the magnitude of an excitatory neuron\u2019s influence to the neighbouring neurons. However, the intricate relationship between network dynamics and information flows is not well understood. Here, we address this challenge by first identifying a generic mechanism that defines the evolution of various information routing patterns in response to modifications in the underlying network dynamics. Moreover, with emerging techniques in brain stimulation, designing optimal stimulation directed towards a target region with an acceptable magnitude remains an ongoing and significant challenge. In this work, we also introduce techniques for computing optimal inputs that follow a desired stimulation routing path towards the target brain region. This optimization problem can be efficiently resolved using non-linear programming tools and permits the simultaneous assignment of multiple desired patterns at different instances. We establish the algebraic and graph-theoretic conditions necessary to ensure the feasibility and stability of information routing patterns (IRPs). We illustrate the routing mechanisms and control methods for attaining desired patterns in biological oscillatory dynamics.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-b3039af e-con-full e-flex e-con e-child\" data-id=\"b3039af\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-eb0c73e exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"eb0c73e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Shankar A. Deka and Umesh Vaidya,\u00a0<a href=\"https:\/\/ieeexplore.ieee.org\/abstract\/document\/10384288\" target=\"_blank\" rel=\"noopener\">Extensions of the Path-integral formula for computation of Koopman eigenfunctions<\/a>, IEEE Control and Decision Conference.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-ee4f65a e-con-full e-flex e-con e-child\" data-id=\"ee4f65a\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-231295f premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"231295f\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-231295f\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-231295f\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>The paper is about the computation of the principal spectrum of the Koopman operator (i.e., eigenvalues and eigenfunctions). The principal eigenfunctions of the Koopman operator are the ones with the corresponding eigenvalues equal to the eigenvalues of the linearization of the nonlinear system at an equilibrium point. The main contribution of this paper is to provide a novel approach for computing the principal eigenfunctions using a path-integral formula. Furthermore, we provide conditions based on the stability property of the dynamical system and the eigenvalues of the linearization towards computing the principal eigenfunction using the path-integral formula. Further, we provide a Deep Neural Network framework that utilizes our proposed path-integral approach for eigenfunction computation in high-dimension systems. Finally, we present simulation results for the computation of principal eigenfunction and demonstrate their application for determining the stable and unstable manifolds and constructing the Lyapunov function.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-951d3f5 e-con-full e-flex e-con e-child\" data-id=\"951d3f5\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-a6b1aac exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"a6b1aac\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>B Hou, A. Matavalam, S. Bose, U. Vaidya,&nbsp;<a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0167278924001192\" target=\"_blank\">Propagating uncertainty through system dynamics in reproducing kernel Hilbert space<\/a>, Physica D: Nonlinear Phenomena 463, 2024<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1040\" class=\"elementor-element elementor-element-eb94526 e-con-full e-flex e-con e-child\" data-id=\"eb94526\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-46f466c premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"46f466c\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-46f466c\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-46f466c\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>We present a data-driven approach to propagate uncertainty in initial conditions through the dynamics of an unknown system in a reproducing kernel\u00a0<a href=\"https:\/\/www.sciencedirect.com\/topics\/physics-and-astronomy\/hilbert-spaces\">Hilbert space<\/a>\u00a0(RKHS). The uncertainty in initial conditions is represented through its kernel mean embedding (KME) in the RKHS. For a discrete-time Markovian\u00a0<a href=\"https:\/\/www.sciencedirect.com\/topics\/physics-and-astronomy\/dynamical-system\">dynamical system<\/a>, we utilize the conditional mean embedding (CME) operator to encode the underlying dynamics. Learning in RKHS often incurs prohibitive\u00a0<a href=\"https:\/\/www.sciencedirect.com\/topics\/physics-and-astronomy\/data-storage\">data storage<\/a>\u00a0requirements. To circumvent said limitation, we propose an algorithm to propagate uncertainty via a learned\u00a0<em>sparse<\/em>\u00a0CME operator, and provide theoretical guarantees on the approximation error for the embedded distribution with time. We empirically study our algorithm over illustrative dynamical systems and\u00a0<a href=\"https:\/\/www.sciencedirect.com\/topics\/engineering\/power-engineering\">power systems<\/a>.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t\t<details id=\"e-n-accordion-item-1041\" class=\"e-n-accordion-item\" >\n\t\t\t\t<summary class=\"e-n-accordion-item-title\" data-accordion-index=\"2\" tabindex=\"-1\" aria-expanded=\"false\" aria-controls=\"e-n-accordion-item-1041\" >\n\t\t\t\t\t<span class='e-n-accordion-item-title-header'><div class=\"e-n-accordion-item-title-text\"> Year 2023 and old Papers <\/div><\/span>\n\t\t\t\t\t\t\t<span class='e-n-accordion-item-title-icon'>\n\t\t\t<span class='e-opened' ><svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-minus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h384c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t\t<span class='e-closed'><svg aria-hidden=\"true\" class=\"e-font-icon-svg e-fas-plus\" viewBox=\"0 0 448 512\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\"><path d=\"M416 208H272V64c0-17.67-14.33-32-32-32h-32c-17.67 0-32 14.33-32 32v144H32c-17.67 0-32 14.33-32 32v32c0 17.67 14.33 32 32 32h144v144c0 17.67 14.33 32 32 32h32c17.67 0 32-14.33 32-32V304h144c17.67 0 32-14.33 32-32v-32c0-17.67-14.33-32-32-32z\"><\/path><\/svg><\/span>\n\t\t<\/span>\n\n\t\t\t\t\t\t<\/summary>\n\t\t\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-198d2d7 e-con-full e-flex e-con e-child\" data-id=\"198d2d7\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-ef85bf2 e-con-full e-flex e-con e-child\" data-id=\"ef85bf2\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-16bef7c exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"16bef7c\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Rejitha Raveendran, Arun D. Mahindrakar, and Umesh Vaidya,\u00a0<a href=\"https:\/\/ieeexplore.ieee.org\/abstract\/document\/10323541\" target=\"_blank\" rel=\"noopener\">Dynamical System Approach for Time-Varying Constrained Convex Optimization Problems<\/a>, Accepted for publication in IEEE Transacations of Automatic Control, 2023.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-3a2caf8 e-con-full e-flex e-con e-child\" data-id=\"3a2caf8\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-60065e9 premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"60065e9\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-60065e9\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-60065e9\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>Optimization problems emerging in most of the real-world applications are dynamic, where either the objective function or the constraints change continuously over time. This article proposes projected primal\u2013dual dynamical system approaches to track the primal and dual optimizer trajectories of an inequality constrained time-varying (TV) convex optimization problem with a strongly convex objective function. First, we present a dynamical system that asymptotically tracks the optimizer trajectory of an inequality constrained TV optimization problem. Later, we modify the proposed dynamics to achieve the convergence to the optimizer trajectory within a fixed time. The asymptotic and fixed-time convergence of the proposed dynamical systems to the optimizer trajectory is shown via the Lyapunov-based analysis. Finally, we consider the TV extended Fermat\u2013Torricelli problem of minimizing the sum-of-squared distances to a finite number of nonempty, closed, and convex TV sets, to illustrate the applicability of the projected dynamical systems proposed in this article.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-417d583 e-con-full e-flex e-con e-child\" data-id=\"417d583\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-b7d7e92 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"b7d7e92\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Andrew Zheng, and Sriram S.K.S Narayanan, and Umesh Vaidya,\u00a0<a href=\"https:\/\/arxiv.org\/pdf\/2306.15830.pdf\">Safe navigation using density functions,<\/a>\u00a0IEEE Robotics and Automation Letters.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-bced549 e-con-full e-flex e-con e-child\" data-id=\"bced549\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-dde931a premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"dde931a\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-dde931a\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-dde931a\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>This paper presents a novel approach for safe control synthesis using the dual formulation of the navigation problem. The main contribution of this paper is in the analytical construction of density functions for almost everywhere navigation with safety constraints. In contrast to the existing approaches, where density functions are used for the analysis of navigation problems, we use density functions for the synthesis of safe controllers. We provide convergence proof using the proposed density functions for navigation with safety. Further, we use these density functions to design feedback controllers capable of navigating in cluttered environments and high-dimensional configuration spaces. The proposed analytical construction of density functions overcomes the problem associated with navigation functions, which are known to exist but challenging to construct, and potential functions, which suffer from local minima. Application of the developed framework is demonstrated on simple integrator dynamics and fully actuated robotic systems. Our project page with implementation is available at https:\/\/github.com\/ clemson-dira\/density_feedback_control<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-18e65c2 e-con-full e-flex e-con e-child\" data-id=\"18e65c2\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-6a9c9cd exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"6a9c9cd\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Alok Kumar, Bhagyashree Umathe, Umesh Vaidya, Atul Kelkar,\u00a0<a href=\"https:\/\/asmedigitalcollection.asme.org\/lettersdynsys\/article\/doi\/10.1115\/1.4063479\/1166904\/Safe-Operating-Limits-of-Vehicle-Dynamics-under\">Safe Operating Limits of Vehicle Dynamics under Parameteric Uncertainty using Koopman Spectrum<\/a>, ASME Dynamics Systems and Control Letters. (<strong>Winner of MECC Best Paper Award<\/strong>)<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-a2a0f3a e-con-full e-flex e-con e-child\" data-id=\"a2a0f3a\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0326899 premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"0326899\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-0326899\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-0326899\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>Ground vehicles operate under different driving conditions, which require the analysis of varying parameter values. It is essential to ensure the vehicle\u2019s safe operation under all these conditions of the parameter variation. In this paper, we investigate the safe operating limits of a ground vehicle by performing the reachability analysis for varying parameters using the Koopman spectrum approach. The reachable set is computed using the Koopman principal eigenfunctions obtained from a convex optimization formulation for different values of the parameter. We consider the two degrees-of-freedom nonlinear quarter-car model to simulate the vehicle\u2019s dynamics. Based on the obtained reachable sets, we provide the mean and variance computation framework with parametric uncertainty. The results show that the reachable set for each value provides valuable information regarding the safe operating limits of the vehicle and can assist in developing safe driving strategies.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-9fae0e7 e-con-full e-flex e-con e-child\" data-id=\"9fae0e7\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-aca6cc6 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"aca6cc6\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Shankar Deka, Sriram Subhramanian, Umesh Vaidya,\u00a0<a href=\"https:\/\/arxiv.org\/abs\/2307.06805\">Path-Integral Formula for Computing Koopman Eigenfunction<\/a>s, IEEE Control and Decision Conference.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-f571cf8 e-con-full e-flex e-con e-child\" data-id=\"f571cf8\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-9b95498 premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"9b95498\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-9b95498\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-9b95498\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>The paper is about the computation of the principal spectrum of the Koopman operator (i.e., eigenvalues and eigenfunctions). The principal eigenfunctions of the Koopman operator are the ones with the corresponding eigenvalues equal to the eigenvalues of the linearization of the nonlinear system at an equilibrium point. The main contribution of this paper is to provide a novel approach for computing the principal eigenfunctions using a path-integral formula. Furthermore, we provide conditions based on the stability property of the dynamical system and the eigenvalues of the linearization towards computing the principal eigenfunction using the path-integral formula. Further, we provide a Deep Neural Network framework that utilizes our proposed path-integral approach for eigenfunction computation in high-dimension systems. Finally, we present simulation results for the computation of principal eigenfunction and demonstrate their application for determining the stable and unstable manifolds and constructing the Lyapunov function.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-8107fdd e-con-full e-flex e-con e-child\" data-id=\"8107fdd\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0b51602 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"0b51602\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Ajinkya Joglekar, Sarang Sutavani, Chinmay Samak, Tanmay Samak,Krishna Kosaraju, Jonathon Smereka, David Gorsich, Umesh Vaidya, and Venkat Krovi,\u00a0<a href=\"https:\/\/ieeexplore.ieee.org\/abstract\/document\/10341797\">Data-Driven Modeling and Experimental Validation of Autonomous Vehicles using Koopman Operator<\/a><a href=\"https:\/\/ieeexplore.ieee.org\/abstract\/document\/10341797\" target=\"_blank\" rel=\"noopener\">,<\/a>\u00a0International Conference on Intelligent Robots and Systems (IROS).<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-94db943 e-con-full e-flex e-con e-child\" data-id=\"94db943\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-017ba33 premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"017ba33\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-017ba33\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-017ba33\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>This paper presents a data-driven framework to discover underlying dynamics on a scaled F1TENTH vehicle using the Koopman operator linear predictor. Traditionally, a range of white, gray, or black-box models are used to develop controllers for vehicle path tracking. However, these models are constrained to either linearized operational domains, unable to handle significant variability or lose explainability through end-2-end operational settings. The Koopman Extended Dynamic Mode Decomposition (EDMD) linear predictor seeks to utilize data-driven model learning whilst providing benefits like explainability, model analysis and the ability to utilize linear model-based control techniques. Consider a trajectory-tracking problem for our scaled vehicle platform. We collect pose measurements of our F1TENTH car undergoing standard vehicle dynamics benchmark maneuvers with an OptiTrack indoor localization system. Utilizing these uniformly spaced temporal snapshots of the states and control inputs, a data-driven Koopman EDMD model is identified. This model serves as a linear predictor for state propagation, upon which an MPC feedback law is designed to enable trajectory tracking. The prediction and control capabilities of our framework are highlighted through real-time deployment on our scaled vehicle.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-48dce9b e-con-full e-flex e-con e-child\" data-id=\"48dce9b\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-769fae7 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"769fae7\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Joseph Moyalan, Andrew Zheng, Sriram S.K.S Narayanan, Umesh Vaidya,\u00a0<a href=\"https:\/\/arxiv.org\/abs\/2305.02938\">Off-Road Navigation of Legged Robots Using Linear Transfer Operators<\/a>, Modeling, Estimation, Control Conference, (<strong>DSCD Robotics TC Best paper Award<\/strong>)<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-9cc7874 e-con-full e-flex e-con e-child\" data-id=\"9cc7874\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-4873ff4 premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"4873ff4\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-4873ff4\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-4873ff4\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>This paper presents the implementation of off-road navigation on legged robots using convex optimization through linear transfer operators. Given a traversability measure that captures the off-road environment, we lift the navigation problem into the density space using the Perron-Frobenius (P-F) operator. This allows the problem formulation to be represented as a convex optimization. Due to the operator acting on an infinite-dimensional density space, we use data collected from the terrain to get a finite-dimension approximation of the convex optimization. Results of the optimal trajectory for off-road navigation are compared with a standard iterative planner, where we show how our convex optimization generates a more traversable path for the legged robot compared to the suboptimal iterative planner.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-49ed833 e-con-full e-flex e-con e-child\" data-id=\"49ed833\" data-element_type=\"container\" data-e-type=\"container\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t\t<div class=\"elementor-element elementor-element-142b162 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"142b162\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Ajinkya Joglekar, Chinmay Samak, Tanmay Samak, Krishna Chaitanya Kosaraju, Venkat Krovi, Umesh Vaidya,\u00a0<a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S2405896323024266\" target=\"_blank\" rel=\"noopener\">Analytical Construction of Koopman EDMD Candidate Functions for Optimal Control of Ackermann-Steered Vehicles<\/a>, Modeling, Estimation, Control Conference.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-c8f79f2 e-con-full e-flex e-con e-child\" data-id=\"c8f79f2\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-02908db premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"02908db\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-02908db\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-02908db\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>The path-tracking control performance of an autonomous vehicle (AV) is crucially dependent upon modeling choices and subsequent system-identification updates. Traditionally, automotive engineering has built upon increasing fidelity of white- and gray-box models coupled with system identification. While these models offer explainability, they suffer from modeling inaccuracies, non-linearities, and parameter variation. On the other end, end-to-end black-box methods like behavior cloning and reinforcement learning provide increased adaptability but at the expense of explainability, generalizability, and the sim2real gap. In this regard, hybrid data-driven techniques like Koopman Extended Dynamic Mode Decomposition (KEDMD) can achieve linear embedding of non-linear dynamics through a selection of \u201clifting functions\u201d. However, the success of this method is primarily predicated on the choice of lifting function(s) and optimization parameters. In this study, we present an analytical approach to construct these lifting functions using the iterative Lie bracket vector fields considering holonomic and non-holonomic constraints on the configuration manifold of our Ackermann-steered autonomous mobile robot. The prediction and control capabilities of the obtained linear KEDMD model are showcased using trajectory tracking of standard vehicle dynamics maneuvers and along a closed-loop racetrack.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-caef5ce e-con-full e-flex e-con e-child\" data-id=\"caef5ce\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-241da28 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"241da28\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Sarang Sutavani, Andrew Zheng, Ajinkya Joglekar, Jonathon Smereka, David Gorsich, Venkat Krovi, Umesh Vaidya,\u00a0<a href=\"https:\/\/ndia-mich.org\/images\/events\/gvsets\/2023\/papers\/aair\/1%2010PM%20Artificial%20Neural%20Network%20Based%20Terraiin%20Reconstruction%20for%20Off%20Road%20Autonomous%20Vehicles%20Using%20LIDAR.pdf\">Artificial neural network based terrain reconstruction for off-road autonomous vehicle using LIDAR<\/a>.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-fd77257 e-con-full e-flex e-con e-child\" data-id=\"fd77257\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0db0677 premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"0db0677\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-0db0677\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-0db0677\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>Accurate terrain mapping is of paramount importance for motion planning and safe navigation in unstructured terrain. LIDAR sensors provide a modality, in the form of a 3D point cloud, that can be used to estimate the elevation map of the surrounding environment. But, working with the 3D point cloud data turns out to be challenging. This is primarily due to the unstructured nature of the point clouds, relative sparsity of the data points, occlusions due to negative slopes and obstacles, and the high computational burden of traditional point cloud algorithms. We tackle these problems with the help of a learning-based, efficient data processing approach for vehicle-centric terrain reconstruction using a 3D LIDAR. The 3D LIDAR point cloud is projected on the ground plane, which is processed by a generative adversarial network (GAN) architecture in the form of an image to fill in the missing parts of the terrain heightmap. We train the GAN model on artificially generated datasets and show the method\u2019s effectiveness by means of the reconstructed terrains.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-0949bf4 e-con-full e-flex e-con e-child\" data-id=\"0949bf4\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-7a874f0 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"7a874f0\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Sriram S. K. S. Narayanan , Duvan Tellez-Castro , Sarang Sutavani , Umesh Vaidya, SE(3)\u00a0<a href=\"https:\/\/arxiv.org\/abs\/2305.03868\">Koopman-MPC: Data-driven Learning and Control of Quadrotor UAVs, Modeling, Estimation<\/a>, Control Conference.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-61e8815 e-con-full e-flex e-con e-child\" data-id=\"61e8815\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-9c0056c premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"9c0056c\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-9c0056c\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-9c0056c\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>In this paper, we propose a novel data-driven approach for learning and control of quadrotor UAVs based on the Koopman operator and extended dynamic mode decomposition (EDMD). Building observables for EDMD based on conventional methods like Euler angles (to represent orientation) is known to involve singularities. To address this issue, we employ a set of physics-informed observables based on the underlying topology of the nonlinear system. We use rotation matrices to directly represent the orientation dynamics and obtain a lifted linear representation of the nonlinear quadrotor dynamics in the SE(3) manifold. This EDMD model leads to accurate prediction and can be generalized to several validation sets. Further, we design a linear model predictive controller (MPC) based on the proposed EDMD model to track agile reference trajectories. Simulation results show that the proposed MPC controller can run as fast as 100 Hz and is able to track arbitrary reference trajectories with good accuracy. Implementation details can be found in \\url{<a href=\"https:\/\/github.com\/sriram-2502\/KoopmanMPC_Quadrotor\">this https URL<\/a>}.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-d9042a0 e-con-full e-flex e-con e-child\" data-id=\"d9042a0\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-c5dd384 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"c5dd384\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Boya Hou, Sina Sanjari, Nathan Dahlin, Subhonmesh Bose and Umesh Vaidya,\u00a0<a href=\"https:\/\/proceedings.mlr.press\/v202\/hou23c.html\">Sparse Learning of Dynamical Systems in RKHS: An Operator-Theoretic Approach<\/a>, Accepted for publication in International Conference of Machine Learning (ICML).<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-78b1c93 e-con-full e-flex e-con e-child\" data-id=\"78b1c93\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-d5de80d premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"d5de80d\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-d5de80d\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-d5de80d\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>Transfer operators provide a rich framework for representing the dynamics of very general, nonlinear dynamical systems. When interacting with reproducing kernel Hilbert spaces (RKHS), descriptions of dynamics often incur prohibitive data storage requirements, motivating dataset sparsification as a precursory step to computation. Further, in practice, data is available in the form of trajectories, introducing correlation between samples. In this work, we present a method for sparse learning of transfer operators from\u00a0\u03b2\u03b2-mixing stochastic processes, in both discrete and continuous time, and provide sample complexity analysis extending existing theoretical guarantees for learning from non-sparse, i.i.d. data. In addressing continuous-time settings, we develop precise descriptions using covariance-type operators for the infinitesimal generator that aids in the sample complexity analysis. We empirically illustrate the efficacy of our sparse embedding approach through deterministic and stochastic nonlinear system examples.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-fd2010e e-con-full e-flex e-con e-child\" data-id=\"fd2010e\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-ec15a3e exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"ec15a3e\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Joseph Moyalan, Yongxin Chen, and Umesh Vaidya,\u00a0<a href=\"https:\/\/arxiv.org\/abs\/2210.00723\">Data-Driven Convex Approach for Off-road Navigation via Linear Transfer Operator<\/a>, IEEE Robotics and Automation Letters.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-7246c13 e-con-full e-flex e-con e-child\" data-id=\"7246c13\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-c65208d premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"c65208d\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-c65208d\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-c65208d\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>We consider the problem of optimal navigation control design for navigation on off-road terrain. We use traversability measure to characterize the degree of difficulty of navigation on the off-road terrain. The traversability measure captures the property of terrain essential for navigation, such as elevation map, terrain roughness, slope, and terrain texture. The terrain with the presence or absence of obstacles becomes a particular case of the proposed traversability measure. We provide a convex formulation to the off-road navigation problem by lifting the problem to the density space using the linear Perron-Frobenius (P-F) operator. The convex formulation leads to an infinite-dimensional optimal navigation problem for control synthesis. The finite-dimensional approximation of the infinite-dimensional convex problem is constructed using data. We use a computational framework involving the Koopman operator and the duality between the Koopman and P-F operator for the data-driven approximation. This makes our proposed approach data-driven and can be applied in cases where an explicit system model is unavailable. Finally, we demonstrate the application of the developed framework for the navigation of vehicle dynamics with Dubin&#8217;s car model.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-0b191f6 e-con-full e-flex e-con e-child\" data-id=\"0b191f6\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-63a804f exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"63a804f\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Shankar A. Deka, Umesh Vaidya, and Dimos V. Dimarogonas,\u00a0<a href=\"https:\/\/ieeexplore.ieee.org\/document\/10178189\">Navigation in Time-Varying Densities: An Operator Theoretic Approach<\/a>, European Control Conference, 2023.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-98cce11 e-con-full e-flex e-con e-child\" data-id=\"98cce11\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-677286b premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"677286b\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-677286b\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-677286b\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>This paper considers the problem of optimizing robot navigation with respect to a time-varying objective encoded into a navigation density function. We are interested in designing state feedback control laws that lead to an almost everywhere stabilization of the closed-loop system to an equilibrium point while navigating a region optimally and safely (that is, the transient leading to the final equilibrium point is optimal and satisfies safety constraints). Though this problem has been studied in literature within many different communities, it still remains a challenging non-convex control problem. In our approach, under certain assumptions on the time-varying navigation density, we use Koopman and Perron-Frobenius Operator theoretic tools to transform the problem into a convex one in infinite dimensional decision variables. In particular, the cost function and the safety constraints in the transformed formulation become linear in these functional variables. Finally, we present some numerical examples to illustrate our approach, as well as discuss the current limitations and future extensions of our framework to accommodate a wider range of robotics applications.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-8f84cfb e-con-full e-flex e-con e-child\" data-id=\"8f84cfb\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-cc4430a exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"cc4430a\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Joseph Moyalan, Hyungjin Choi, Yongxin Chen, and Umesh Vaidya,\u00a0<a href=\"https:\/\/arxiv.org\/abs\/2202.01856\">Data-Driven Optimal Control via Linear Operator Theory: A Convex Approach<\/a>, Automatica.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-aa55360 e-con-full e-flex e-con e-child\" data-id=\"aa55360\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-fa5bccd premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"fa5bccd\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-fa5bccd\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-fa5bccd\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>This paper is concerned with data-driven optimal control of nonlinear systems. We present a convex formulation to the optimal control problem (OCP) with a discounted cost function. We consider OCP with both positive and negative discount factor. The convex approach relies on lifting nonlinear system dynamics in the space of densities using the linear Perron-Frobenius (P-F) operator. This lifting leads to an infinite-dimensional convex optimization formulation of the optimal control problem. The data-driven approximation of the optimization problem relies on the approximation of the Koopman operator using the polynomial basis function. We write the approximate finite-dimensional optimization problem as a polynomial optimization which is then solved efficiently using a sum-of-squares-based optimization framework. Simulation results are presented to demonstrate the efficacy of the developed data-driven optimal control framework.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-a346b2d e-con-full e-flex e-con e-child\" data-id=\"a346b2d\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-9250473 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"9250473\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>U. Vaidya, and Duvan Tellez,\u00a0<a href=\"https:\/\/arxiv.org\/abs\/2202.01387\">Data-driven Stochastic Optimal Control using linear transfer operators<\/a>, IEEE Transactions of Automatic Control.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-264538c e-con-full e-flex e-con e-child\" data-id=\"264538c\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-c26ce08 premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"c26ce08\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-c26ce08\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-c26ce08\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>We provide a data-driven framework for optimal control of a continuous-time stochastic dynamical system. The proposed framework relies on the linear operator theory involving linear Perron-Frobenius (P-F) and Koopman operators. Our first results involving the P-F operator provide a convex formulation to the optimal control problem in the dual space of densities. This convex formulation of the stochastic optimal control problem leads to an infinite-dimensional convex program. The finite-dimensional approximation of the convex program is obtained using a data-driven approximation of the P-F operator. Our second results demonstrate the use of the Koopman operator, which is dual to the P-F operator, for the stochastic optimal control design. We show that the Hamilton Jacobi Bellman (HJB) equation can be expressed using the Koopman operator. We provide an iterative procedure along the lines of a popular policy iteration algorithm based on the data-driven approximation of the Koopman operator for solving the HJB equation. The two formulations, namely the convex formulation involving P-F operator and Koopman based formulation using HJB equation, can be viewed as dual to each other where the duality follows due to the dual nature of P-F and Koopman operators. Finally, we present several numerical examples to demonstrate the efficacy of the developed framework.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-aa901e1 e-con-full e-flex e-con e-child\" data-id=\"aa901e1\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-fd36177 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"fd36177\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Sarang Sutavani, Bhagysahree Umathe, and Umesh Vaidya,\u00a0<a href=\"https:\/\/ieeexplore.ieee.org\/document\/10156079\">Small Gain Theorem and L2 Gain Computation in Large using Koopman Spectrum<\/a>, American Control Conference.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-5a434a9 e-con-full e-flex e-con e-child\" data-id=\"5a434a9\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-0566aee premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"0566aee\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-0566aee\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-0566aee\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>The paper is about\u00a0L2-gain computation and the small-gain theorem for nonlinear input-output systems. We show that the Koopman operator\u2019s spectrum can provide conditions for\u00a0L2-gain guarantees and small-gain theorem-based stability of interconnection over a large region of the state space. The large region in the state space can be characterized in terms of the region where Koopman eigenfunctions and the solution of the Hamilton Jacobi equation are well defined. The connection of system\u00a0L2-gain to the spectral properties of the Koopman operator has led to a novel approach, based on the approximation of the Koopman spectrum, for the computation of the\u00a0L2-gain and stability verification of the interconnected system. We present simulation results including application of the developed framework to a power system example.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-b976c77 e-con-full e-flex e-con e-child\" data-id=\"b976c77\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-f36b875 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"f36b875\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Moirangthem Sailash Singh, Ramkrishna Pasumarthy, Umesh Vaidya, Steffen Leonhardt,<a href=\"https:\/\/www.nature.com\/articles\/s41598-023-32762-7\">\u00a0On Quantification and Maximization of InformationTransfer in Network Dynamical Systems<\/a>, Scientific Report.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-7120e06 e-con-full e-flex e-con e-child\" data-id=\"7120e06\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-f5748ed premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"f5748ed\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-f5748ed\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-f5748ed\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>Information flow among nodes in a complex network describes the overall cause-effect relationships among the nodes and provides a better understanding of the contributions of these nodes individually or collectively towards the underlying network dynamics. Variations in network topologies result in varying information flows among nodes. We integrate theories from information science with control network theory into a framework that enables us to quantify and control the information flows among the nodes in a complex network. The framework explicates the relationships between the network topology and the functional patterns, such as the information transfers in biological networks, information rerouting in sensor nodes, and influence patterns in social networks. We show that by designing or re-configuring the network topology, we can optimize the information transfer function between two chosen nodes. As a proof of concept, we apply our proposed methods in the context of brain networks, where we reconfigure neural circuits to optimize excitation levels among the excitatory neurons.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-23883d0 e-con-full e-flex e-con e-child\" data-id=\"23883d0\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-f9929b9 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"f9929b9\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Hongzhe Yu, Joseph Moyalan, Umesh Vaidya, and Yongxin Chen,\u00a0<a href=\"https:\/\/arxiv.org\/abs\/2306.00596\">Data-driven optimal control under safety constraints using sparse Koopman approximation<\/a>, International Conference on Robotics and Automation (ICRA).<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-03a964c e-con-full e-flex e-con e-child\" data-id=\"03a964c\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-753713a premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"753713a\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-753713a\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-753713a\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>In this work we approach the dual optimal reach-safe control problem using sparse approximations of Koopman operator. Matrix approximation of Koopman operator needs to solve a least-squares (LS) problem in the lifted function space, which is computationally intractable for fine discretizations and high dimensions. The state transitional physical meaning of the Koopman operator leads to a sparse LS problem in this space. Leveraging this sparsity, we propose an efficient method to solve the sparse LS problem where we reduce the problem dimension dramatically by formulating the problem using only the non-zero elements in the approximation matrix with known sparsity pattern. The obtained matrix approximation of the operators is then used in a dual optimal reach-safe problem formulation where a linear program with sparse linear constraints naturally appears. We validate our proposed method on various dynamical systems and show that the computation time for operator approximation is greatly reduced with high precision in the solutions.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-943a962 e-con-full e-flex e-con e-child\" data-id=\"943a962\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-ec34bd4 exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"ec34bd4\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p>Shankar A. Deka, Umesh Vaidya, and Dimos V. Dimarogonas,\u00a0<a href=\"https:\/\/ieeexplore.ieee.org\/document\/10178189\">Navigation in Time-Varying Densities: An Operator Theoretic Approach<\/a>, European Control Conference.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-542848b e-con-full e-flex e-con e-child\" data-id=\"542848b\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-7fe708d premium-modal-dismissible-yes exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-premium-addon-modal-box\" data-id=\"7fe708d\" data-element_type=\"widget\" data-e-type=\"widget\" data-settings=\"{&quot;premium_modal_box_animation&quot;:&quot;fadeInDown&quot;}\" data-widget_type=\"premium-addon-modal-box.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\n\t\t<div class=\"premium-modal-box-container\" data-settings=\"{&quot;trigger&quot;:&quot;text&quot;,&quot;show_on_exit&quot;:false}\">\n\t\t\t<div class=\"premium-modal-trigger-container\">\n\t\t\t\t\t\t\t\t\t<span data-toggle=\"premium-modal\" data-target=\"#premium-modal-7fe708d\" class=\"premium-modal-trigger-text\">\n\t\t\t\t\t\t<div >\n\t\t\t\t\t\t\tRead Abstract\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t\t\t<\/div>\n\n\t\t\t<div id=\"premium-modal-7fe708d\" class=\"premium-modal-box-modal\"\n\t\t\trole=\"dialog\"\n\t\t\tstyle=\"display: none\"\n\t\t\t>\n\t\t\t\t<div class=\"premium-modal-box-modal-dialog\" data-delay-animation=\"\" data-modal-animation=\"fadeInDown animated-fast\">\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-header\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<h3 class=\"premium-modal-box-modal-title\">\n\t\t\t\t\t\t\t\t\tAbstract\t\t\t\t\t\t\t\t<\/h3>\n\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-close-button-container\">\n\t\t\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-close\" data-dismiss=\"premium-modal\">&times;<\/button>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-body\">\n\t\t\t\t\t\t\t\t\t\t\t\t\t<p>This paper considers the problem of optimizing robot navigation with respect to a time-varying objective encoded into a navigation density function. We are interested in designing state feedback control laws that lead to an almost everywhere stabilization of the closed-loop system to an equilibrium point while navigating a region optimally and safely (that is, the transient leading to the final equilibrium point is optimal and satisfies safety constraints). Though this problem has been studied in literature within many different communities, it still remains a challenging non-convex control problem. In our approach, under certain assumptions on the time-varying navigation density, we use Koopman and Perron-Frobenius Operator theoretic tools to transform the problem into a convex one in infinite dimensional decision variables. In particular, the cost function and the safety constraints in the transformed formulation become linear in these functional variables. Finally, we present some numerical examples to illustrate our approach, as well as discuss the current limitations and future extensions of our framework to accommodate a wider range of robotics applications.<\/p>\n\t\t\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t\t\t<div class=\"premium-modal-box-modal-footer\">\n\t\t\t\t\t\t\t<button type=\"button\" class=\"premium-modal-box-modal-lower-close\" data-dismiss=\"premium-modal\">\n\t\t\t\t\t\t\t\tClose\t\t\t\t\t\t\t<\/button>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t\t\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div role=\"region\" aria-labelledby=\"e-n-accordion-item-1041\" class=\"elementor-element elementor-element-7e4a417 e-con-full e-flex e-con e-child\" data-id=\"7e4a417\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t<div class=\"elementor-element elementor-element-ec5a9dc exad-sticky-section-no exad-glass-effect-no elementor-widget elementor-widget-text-editor\" data-id=\"ec5a9dc\" data-element_type=\"widget\" data-e-type=\"widget\" data-widget_type=\"text-editor.default\">\n\t\t\t\t<div class=\"elementor-widget-container\">\n\t\t\t\t\t\t\t\t\t<p><a href=\"https:\/\/cecas.clemson.edu\/ugvaidya\/publications\/\" target=\"_blank\" rel=\"noopener\">Check Old Papers Here<\/a><\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/details>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t<div class=\"elementor-element elementor-element-3225f36 e-flex e-con-boxed e-con e-parent\" data-id=\"3225f36\" data-element_type=\"container\" data-e-type=\"container\">\n\t\t\t\t\t<div class=\"e-con-inner\">\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>Under Review Ajinkya Joglekar, Chinmay Samak, Tanmay Samak, Venkat Krovi, and Umesh Vaidya,\u00a0Expanding Autonomous Ground Vehicle Navigation Capabilities through a Multi-Model Parameterized Koopman Framework, Submitted for publication in Journal. Read Abstract Abstract &times; We introduce the Multi-Model Parameterized Koopman (MMPK) framework, a novel end-2-end data-driven modeling and control pipeline for enabling autonomous navigation in Uncrewed &hellip; <a href=\"https:\/\/cecas.clemson.edu\/ugvaidya\/publication-demo\/\" class=\"more-link\">Continue reading <span class=\"screen-reader-text\">Publications<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":11,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-2248","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/cecas.clemson.edu\/ugvaidya\/wp-json\/wp\/v2\/pages\/2248","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cecas.clemson.edu\/ugvaidya\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/cecas.clemson.edu\/ugvaidya\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/cecas.clemson.edu\/ugvaidya\/wp-json\/wp\/v2\/users\/11"}],"replies":[{"embeddable":true,"href":"https:\/\/cecas.clemson.edu\/ugvaidya\/wp-json\/wp\/v2\/comments?post=2248"}],"version-history":[{"count":182,"href":"https:\/\/cecas.clemson.edu\/ugvaidya\/wp-json\/wp\/v2\/pages\/2248\/revisions"}],"predecessor-version":[{"id":4514,"href":"https:\/\/cecas.clemson.edu\/ugvaidya\/wp-json\/wp\/v2\/pages\/2248\/revisions\/4514"}],"wp:attachment":[{"href":"https:\/\/cecas.clemson.edu\/ugvaidya\/wp-json\/wp\/v2\/media?parent=2248"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}