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Grab bars have been widely used for assisting elderly people with mobility and providing support for daily activities. This work aims to expand the notion of grab bars beyond fixed installations by the use of a mobile robot that can place a handlebar at any point in space, to optimally support postural transitions. A survey of elderly people and care professionals indicated that such a device must be sturdy, providing secure support without sliding or tipping over, yet also have a compact footprint to be maneuverable within confined spaces. Here, we propose a novel two-body robot structure, consisting of two small-footprint mobile bases connected by a four bar linkage where handlebars are mounted. Each base measures only 29.2 cm wide, making the robot likely the slimmest ever developed for mobile postural assistance. Through kinematic analysis, it is shown that the two-body structure can bear the entire weight of a human body, meeting required load bearing specifications as a handlebar. A control plan is proposed that is generalizable to all robots with two nonholonomic mobile bases connected by a coupling mechanism. This consists of a leader-follower scheme, in which the bases are connected by a virtual spring, as well as various enhancements to waypoint tracking and dead reckoning that allow the robot to smoothly and accurately follow a series of waypoints. A prototype robot is constructed, and its performance is validated experimentally. 

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.