AdapJ: An Adaptive Extended Jacobian Controller for Soft Manipulators

TL;DR

The paper tackles soft-manipulator control under strong nonlinearity and hysteresis by introducing AdapJ, an adaptive extended Jacobian controller that uses independent extended inverse Jacobian matrices and online Gauss-Newton updates to approximate inverse dynamics with minimal training data. AdapJ preserves the compact Jacobian-like structure while decoupling parameter dependencies, enabling online adaptation to changing stiffness, damping, control frequencies, and disturbances. Initialization relies on motor babbling and batch optimization, followed by online updates that refine A_* and B_* to reflect current dynamics; the method can degenerate to the traditional inverse Jacobian controller if certain relationships hold. Across extensive simulation and real-world experiments, AdapJ outperforms classical Jacobian, MPC, RNN-based, and iterative feedback controllers in trajectory tracking accuracy, speed, and adaptability, demonstrating high data efficiency and robustness for soft-robot applications.

Abstract

The nonlinearity and hysteresis of soft robot motions present challenges for control. To solve these issues, the Jacobian controller has been applied to approximate the nonlinear behaviors in a linear format. Accurate controllers like neural networks can handle delayed and nonlinear motions, but they require large datasets and exhibit low adaptability. Based on a novel analysis on these controllers, we propose an adaptive extended Jacobian controller, AdapJ, for soft manipulators. This controller retains the concise format of the Jacobian controller but introduces independent parameters. Similar to neural networks, its initialization and updating mechanism leverages the inverse model without building the corresponding forward model. In the experiments, we first compare the performance of the Jacobian controller, model predictive controller, neural network controller, iterative feedback controller, and AdapJ in simulation. We further analyze how AdapJ parameters adapt in response to the physical property change. Then, real-world experiments have validated that AdapJ outperforms the neural network controller, model predictive controller, and iterative feedback controller with fewer training samples and adapts robustly to varying conditions, including different control frequencies, material softness, and external disturbances. Future work may include online adjustment of the controller format and adaptability validation in more scenarios.

Figures (11)

• Figure 1: Qualitative comparison of soft robot controllers. The proposed controller maintains a low structural complexity and computational cost, only slightly greater than that of the Jacobian controller, while outperforming the Jacobian controller 14MY, model predictive controller 19ZT, RNN controller 23ZCc, and iterative feedback controller 24XL in terms of control performance considering accuracy and adaptability.
• Figure 2: Diagram of the standard Jacobian controller. Following initialization of the Jacobian matrix, the Jacobian controller computes the actuation $a_{j,t}$ (red) based on the previous robot state, actuation $s_t, a_{t-1}$ (green), and desired robot state $sd_{t+1}$ (blue). Then the Jacobian matrix updates according to the difference in robot state and actuation $\triangle s_{t+1}, \triangle a_t$ (green).
• Figure 3: The diagram of the Jacobian principle. The nonlinear function $z=-(x^2+1)/6$ is approximated by (A) 3, (B) 5, and (C) 7 linear functions. The blue dotted lines represent the linear approximation functions, and the red lines represent the nonlinear function.
• Figure 4: The diagrams of approximating a multivariable nonlinear function using linear approximation with (A) coupling and (B) independent variables. The grey curved surfaces represent the multivariable nonlinear function $z=-(x^2+y^2)/6$, the red lines represent the nonlinear function $z=-(x^2+1)/6$, and the blue planes represent the linear approximation functions.
• Figure 5: Diagram of our adaptive extended Jacobian controller. The extended inverse Jacobian matrices $A_*, B_*$ are initialized with motor babbling and batch optimization. Then, the actuation $a_{aj,t}$ (red) is computed based on these matrices, previous robot states, actuations $s_*, a_*$ (green), and the desired robot state $sd_{t+1}$ (blue). After execution, the matrices $A_*, B_*$ are updated, and the control loop proceeds to the next iteration.