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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 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.