SSCAK-MPC: Steady-State Constrained Adaptive Koopman Model Predictive Control for Soft Manipulators

Abstract

Soft manipulators face challenges in modeling and real-time control due to nonlinear dynamics, model mismatch and drift, as well as inadequate physical consistency of data-driven models. This paper proposes a Steady-State Constrained Adaptive Koopman Model Predictive Control (SSCAK-MPC) framework to tackle these challenges. The framework consists of two stages: (Stage I) offline nominal Koopman modeling and (Stage II) online adaptive control. In the offline stage, steady-state data are incorporated into the Koopman identification process through a regularization term to improve prediction accuracy and steady-state consistency. In the online stage, an adaptive Koopman model is constructed by adding residual matrices to the nominal model. These residual matrices are updated online from sliding window data to compensate for time-varying model discrepancies. By integrating the adaptive Koopman model into a linear MPC framework, the optimal control problem is simplified to a computationally efficient quadratic program (QP). Under the assumption of bounded composite disturbances, recursive feasibility of the nominal MPC problem and practical input-to-state stability (ISS) of the disturbed nominal closed-loop system are established. Experiments on a pneumatic soft manipulator (PSM) show that the proposed SSCAK-MPC framework outperforms baseline methods in tracking accuracy and trajectory recovery under the tested scenarios. Note to Practitioners—This work is motivated by practical challenges in controlling soft manipulators, which exhibit strong nonlinearities, time-varying dynamics, and model discrepancies caused by unmodeled dynamics, material aging, or load changes. Conventional model-based controllers often struggle to achieve high-precision tracking under such conditions. In many applications, repeatedly shutting down the robot for model re-identification is undesirable. The proposed SSCAK-MPC framework addresses this issue through a two-stage strategy: a steady-state constrained nominal Koopman model is identified offline, and only the residual model discrepancy is updated online during operation. This design reduces the online adaptation burden and avoids repeated full-model retraining. Consequently, the framework is particularly suitable for soft robotic systems that require real-time adaptation without system shutdown or full-model re-identification. Experimental results on a pneumatic soft manipulator (PSM) demonstrate improved tracking accuracy and trajectory recovery, indicating its potential for safe manipulation and human-centered robotic applications.