Dynamic Contrastive Koopman Operator for Tactile Servo of Deformable Linear Objects

Strong nonlinearities and infinite degrees of freedom pose intractable challenges for the manipulation of deformable linear objects (DLOs). This article proposes a novel modeling method based on a deep Koopman operator for nonlinear systems with high-dimensional inputs, enabling precise tactile servo performance. Initially, a dynamic contrastive learning algorithm is proposed to approximate the Koopman operator for the unknown dynamic system in the embedded space. During the training phase, a dynamic negative sampling strategy based on a task-oriented state distance measurement is employed to ensure consistency between the distance metrics in the embedded and the original spaces. Furthermore, a Koopman-based model predictive controller is developed to compensate for modeling errors, with stability conditions explicitly outlined. Extensive experiments demonstrate that the proposed method outperforms the representative deep Koopman algorithms in modeling performance, with 17% and 7% improvements in correlation distance measures. In the downstream DLO manipulation task relying on tracking the desired in-hand tactile state, our framework achieves the lowest average tracking error, a 100% following rate, and reduces the inference time to 1/22 of the state-of-the-art method, highlighting its superior prediction and control capabilities in the high- dimensional tactile servoing task.