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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. 

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. 

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. 

This is Jasmine Terrones' Master thesis in the department of Mechanical Engineering, MIT. 

This paper presents a computational method, called Bootstrapped Koopman Direct Encoding (B-KDE) that allows us to approximate the Koopman operator with high accuracy by combining Koopman Direct Encoding (KDE) with a deep neural network. Deep learning has been applied to the Koopman operator method for finding an effective set of observable functions. Training the network, however, inevitably faces difficulties such as local minima, unless enormous computational efforts are made. Incorporating KDE can solve or alleviate this problem, producing an order of magnitude more accurate prediction. KDE converts the state transition function of a nonlinear system to a linear model in the lifted space of observables that are generated by deep learning. The combined KDE-deep model achieves higher accuracy than that of the deep learning alone. In B-KDE, the combined model is further trained until it reaches a plateau, and this computation is alternated between the neural network learning and the KDE computation. The result of the MSE loss implies that the neural network may get rid of local minima or at least find a smaller local minimum, and further improve the prediction accuracy. The KDE computation however, entails an effective algorithm for computing the inner products of observables and the nonlinear functions of the governing dynamics. Here, a computational method based on the Quasi-Monte Carlo integration is presented. The method is applied to a three-cable suspension robot, which exhibits complex switched nonlinear dynamics due to slack in each cable. The prediction accuracy is compared against its traditional counterparts. 

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.