IJCAS

This paper presents a data-driven Koopman modeling framework for globally linearizing highly nonlinear dynamical systems in lifted infinite-dimensional state space. In this framework, three data-driven models are proposed and identified to approximate the infinite-dimensional linear Koopman operator through a method called extended dynamic mode decomposition (EDMD). The implementation of EDMD requires a data set of snap pairs and a dictionary of scalar observables, which affects the accuracy of data-driven modeling. Five basis functions are compared and discussed to illustrate their suitable application scenarios. To verify the Koopman data-driven modeling framework, we apply it to the modeling of soft robotic systems, which is thought to be an extremely difficult task due to the large and continuous deformation of soft materials. Results demonstrate that Koopman linear, bilinear, and nonlinear models for both two-dimensional (2D) and three-dimensional (3D) soft robots are superior to the existing state-space modeling approach by achieving less normalized root mean square error (NRMSE). Among the three Koopman models, the constructed nonlinear model has higher performance than the bilinear model, followed by the linear one. Furthermore, the Monomial and Hermite basis functions are the optimal choices for constructing the Koopman linear, bilinear, and nonlinear models for the investigated 2D and 3D soft robots as they have the same structure when the degree of basis function is chosen less than four. Although the Fourier basis function is expected to outperform the Hermite and Gaussian basis functions in modeling systems with oscillatory motion, it is not a preferable selection in the high dimensional soft robotic modeling in our case due to its computational complexity and tendency to be non-convergent. The Sparse Fourier basis function cannot be regarded as a good choice for modeling soft robots, as it is only suitable for generating models with sparse data. It is worth noting that our findings can lay a solid foundation for the dynamics analysis and precise control of highly nonlinear dynamical systems, like soft robots in the future.

Keywords: Data-driven modeling, identification method, Koopman operator, nonlinear systems, soft robots.