Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control (1902.02827v2)

Abstract: Controlling soft robots with precision is a challenge due in large part to the difficulty of constructing models that are amenable to model-based control design techniques. Koopman Operator Theory offers a way to construct explicit linear dynamical models of soft robots and to control them using established model-based linear control methods. This method is data-driven, yet unlike other data-driven models such as neural networks, it yields an explicit control-oriented linear model rather than just a "black-box" input-output mapping. This work describes this Koopman-based system identification method and its application to model predictive controller design. A model and MPC controller of a pneumatic soft robot arm was constructed via the method, and its performance was evaluated over several trajectory following tasks in the real-world. On all of the tasks, the Koopman-based MPC controller outperformed a benchmark MPC controller based on a linear state-space model of the same system.

• The paper introduces a Koopman-based method that transforms nonlinear soft robot dynamics into linear models for effective MPC control.
• The approach leverages EDMD and L¹ regularization to develop a sparse matrix representation that balances complexity with computational efficiency.
• Numerical results demonstrate improved trajectory tracking with the Koopman MPC reducing average error from 2.45 cm to 1.26 cm.

Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Control

The paper addresses the challenge of precise control in soft robots, arising from the intrinsic difficulties of modeling their expansive degrees of freedom and compliant materials. Traditional modeling approaches often come up short because they require simplifying assumptions that limit their applicability. Alternatively, data-driven methods, while capturing behaviors more adaptively, have been criticized for yielding models that are not amenable to traditional control techniques due to their black-box nature. This paper proposes using the Koopman operator theory in conjunction with Model Predictive Control (MPC) as an approach to create explicit linear dynamic models of such nonlinear systems, overcoming significant challenges associated with both traditional and data-driven modeling.

The authors present a method leveraging the Extended Dynamic Mode Decomposition (EDMD) to approximate the Koopman operator, projecting nonlinear state dynamics into a higher-dimensional space where linear control methodologies can be applied efficiently. This process yields a linear model which permits real-time control strategies, such as MPC, to operate effectively even when the underlying system dynamics are inherently nonlinear. Beyond the theoretical underpinnings, a practical implementation is demonstrated using a pneumatic soft robotic arm. The performance of the proposed Koopman-based MPC is rigorously evaluated against a state-space model MPC as a benchmark. It is observed that the Koopman-based MPC demonstrates superior performance across various trajectory following tasks.

Numerical results are emphasized, showcasing how the Koopman-based MPC consistently outperforms its linear state-space counterpart. For instance, in trajectory following experiments, the Koopman-based MPC shows an average tracking error of 1.26 cm compared to an error of 2.45 cm for the linear model-based controller. These results are compelling, considering that the ultimate success of predictive models in soft robotics hinges on their ability to adaptively cope with the system's flexibility and unpredictability.

One of the key contributions of this work is its method for developing sparse matrix models through $L^1$ regularization. By doing so, the resulting Koopman operator representation achieves compelling sparsity, which is a crucial aspect for efficient computation and implementation. While sparsity enhances computational feasibility, the paper explores this trade-off with prediction accuracy, providing insights that balance model complexity with computational efficiency. Its implications for the physical implementation of controllers in real-time scenarios make this approach particularly relevant in complex robotic systems.

From a theoretical perspective, this research lays important groundwork by demonstrating how the Koopman framework can be pragmatically applied to soft robotic systems. The structural challenges associated with high-dimensional calculations are ably handled through the presented methods. Practically, the approach facilitates improved control accuracy with broader implications for soft robotics applications where safe interaction with humans or delicate objects is paramount.

Future developments could explore extensions of the presented methodology to handle more complex systems with additional environmental interactions, such as contact forces. Additionally, automating the choice of basis functions for more efficient Koopman operator computation and enhancing the model's adaptability to external perturbations remain areas ripe for further research. Consequently, this work stands as a notable contribution towards the advancement of adaptive control methodologies in soft robotics, promoting seamless integration between theoretical frameworks and practical control design.