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1. Koopman Dynamic Modeling for Global and Unified Representations of Rigid Body Systems Making and Breaking Contact

   O'Neill, Cormac; Asada, Harry (October 2024, IEEE)

   A global modeling methodology based on Koopman operator theory for the dynamics of rigid bodies that make and break contact is presented. Traditionally, robotic systems that contact with their environment are represented as a system comprised of multiple dynamic equations that are switched depending on the contact state. This switching of governing dynamics has been a challenge in both task planning and control. Here, a Koopman lifting linearization approach is presented to subsume multiple dynamics such that no explicit switching is required for examining the dynamic behaviors across diverse contact states. First, it is shown that contact/noncontact transitions are continuous at a microscopic level. This allows for the application of Koopman operator theory to the class of robotic systems that repeat contact/non-contact transitions. Second, an effective method for finding Koopman operator observables for capturing rapid changes to contact forces is presented. The method is applied to the modeling of dynamic peg insertion where a peg collides against and bounces on the chamfer of the hole. Furthermore, the method is applied to the dynamic modeling of a sliding object subject to complex friction and damping properties. Segmented dynamic equations are unified with the Koopman modeling method. 

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2. Global, Unified Representation of Heterogenous Robot Dynamics Using Composition Operators: A Koopman Direct Encoding Method

   https://doi.org/10.1109/TMECH.2023.3253599  

   Asada, H. Harry (May 2023, IEEE/ASME Transactions on Mechatronics)

   The dynamic complexity of robots and mechatronic systems often pertains to the hybrid nature of dynamics, where governing equations consist of heterogenous equations that are switched depending on the state of the system. Legged robots and manipulator robots experience contact-noncontact discrete transitions, causing switching of governing equations. Analysis of these systems have been a challenge due to the lack of a global, unified model that is amenable to analysis of the global behaviors. Composition operator theory has the potential to provide a global, unified representation by converting them to linear dynamical systems in a lifted space. The current work presents a method for encoding nonlinear heterogenous dynamics into a high dimensional space of observables in the form of Koopman operator. First, a new formula is established for representing the Koopman operator in a Hilbert space by using inner products of observable functions and their composition with the governing state transition function. This formula, called Direct Encoding, allows for converting a class of heterogenous systems directly to a global, unified linear model. Unlike prevalent data-driven methods, where results can vary depending on numerical data, the proposed method is globally valid, not requiring numerical simulation of the original dynamics. A simple example validates the theoretical results, and the method is applied to a multi-cable suspension system. 

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3. Control-Coherent Koopman Modeling: A Physical Modeling Approach

   Asada, Harry; Solano--Castellanos, Jose (December 2024, IEEE)

   The modeling of nonlinear dynamics based on Koopman operator theory, originally applicable only to autonomous systems with no control, is extended to nonautonomous control system without approximation of the input matrix. Prevailing methods using a least square estimate of the input matrix may result in an erroneous input matrix, misinforming the controller. Here, a new method for constructing a Koopman model that yields the exact input matrix is presented. A set of state variables are introduced so that the control inputs are linearly involved in the dynamics of actuators. With these variables, a lifted linear model with the exact input matrix, called a Control-Coherent Koopman Model, is constructed by superposing control input terms, which are linear in local actuator dynamics, to the Koopman operator of the associated autonomous nonlinear system. As an example, the proposed method is applied to multi degree-of-freedom robotic arms, which are controlled with Model Predictive Control (MPC). It is demonstrated that the prevailing Dynamic Mode Decomposition with Control (DMDc) using an approximate input matrix does not provide a satisfactory result, while the Control-Coherent Koopman Model performs well with the correct input matrix, even performing better than the bilinear formulation of the Koopman operator. 

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4. Active Learning of Dynamics for Data-Driven Control Using Koopman Operators

   https://doi.org/10.1109/TRO.2019.2923880  

   Abraham, Ian; Murphey, Todd D. (July 2019, IEEE Transactions on Robotics)

   This paper presents an active learning strategy for robotic systems that takes into account task information, enables fast learning, and allows control to be readily synthesized by taking advantage of the Koopman operator representation. We first motivate the use of representing nonlinear systems as linear Koopman operator systems by illustrating the improved model-based control performance with an actuated Van der Pol system. Information-theoretic methods are then applied to the Koopman operator formulation of dynamical systems where we derive a controller for active learning of robot dynamics. The active learning controller is shown to increase the rate of information about the Koopman operator. In addition, our active learning controller can readily incorporate policies built on the Koopman dynamics, enabling the benefits of fast active learning and improved control. Results using a quadcopter illustrate single-execution active learning and stabilization capabilities during free-fall. The results for active learning are extended for automating Koopman observables and we implement our method on real robotic systems. 

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5. Robot Manipulator Control Using a Robust Data-Driven Method

   https://doi.org/10.3390/fractalfract7090692  

   Rahmani, Mehran; Redkar, Sangram (September 2023, Fractal and Fractional)

   Robotic manipulators with diverse structures find widespread use in both industrial and medical applications. Therefore, designing an appropriate controller is of utmost importance when utilizing such robots. In this research, we present a robust data-driven control method for the regulation of a 2-degree-of-freedom (2-DoF) robot manipulator. The nonlinear dynamic model of the 2-DoF robot arm is linearized using Koopman theory. The data mode decomposition (DMD) method is applied to generate the Koopman operator. A fractional sliding mode control (FOSMC) is employed to govern the data-driven linearized dynamic model. We compare the performance of Koopman fractional sliding mode control (KFOSMC) with conventional proportional integral derivative (PID) control and FOSMC prior to linearization by Koopman theory. The results demonstrate that KFOSMC outperforms PID and FOSMC in terms of high tracking performance, low tracking error, and minimal control signals. 

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