We present a novel approach to shared control of human-machine systems. Our method assumes no a priori knowledge of the system dynamics. Instead, we learn both the dynamics and information about the user's interaction from observation through the use of the Koopman operator. Using the learned model, we define an optimization problem to compute the optimal policy for a given task, and compare the user input to the optimal input. We demonstrate the efficacy of our approach with a user study. We also analyze the individual nature of the learned models by comparing the effectiveness of our approach when the demonstration data comes from a user's own interactions, from the interactions of a group of users and from a domain expert. Positive results include statistically significant improvements on task metrics when comparing a user-only control paradigm with our shared control paradigm. Surprising results include findings that suggest that individualizing the model based on a user's own data does not effect the ability to learn a useful dynamic system. We explore this tension as it relates to developing human-in-the-loop systems further in the discussion.
The Hilbert space transformation is defined by a basis function, @math , which transforms the standard state definition into a Hilbert space. In practice, this basis function is often a non-linear combination of the initial state representation and is used to approximate the Koopman operator. This basis function can be significantly larger in dimensionality than the initial state space. Common choices for the basis function include Hermite polynomials and radial basis functions @cite_6 . By acting on the Hilbert state representation, the Koopman operator is able to capture the complex, nonlinear dynamics described by the evolution operator.
@cite_6
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The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator. We will show that this approach is, in effect, an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (Mezic in Nonlinear Dynamics 41(1–3): 309–325, 2005). Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions.