Jacobian Matrix

The Jacobian methods use the Jacobian matrix to infer the relationship between the actuation and position velocities by model linearization. For modeling, the Jacobian matrix is estimated by an optimization problem. The other optimization problem with the Jacobian matrix as a constraint can be employed as the control method, and some works apply the inverse Jacobian matrix directly.

The Jacobian model is applied in Prof. Michael Yip's works for position and force control. Due to its oversimplification, there are some adjustments, like smoothing and sensing fusion, in most cases. Even so, most labs only apply the Jacobian model as the initial solution and turn to the other models in their following works.

In summary, the Jacobian model is a concise and simple model for soft robot modeling and control. This model only uses one matrix for modeling and only requires actuation and position variables in the last steps. The Jacobian model is employed as modeling, and the inverse matrix can be utilized for control. Such a model is updated online with a high frequency in most cases, which can be achieved due to the simple structure. Meanwhile, the linear relationship assumption necessitates online updating and high control frequency. Only position control can be achieved directly, and the other tasks, e.g., force control, require specific adjustments.

Paper List:

14MY: Yip M C, Camarillo D B. Model-less feedback control of continuum manipulators in constrained environments[J]. IEEE Transactions on Robotics, 2014, 30(4): 880-889. 

This is the first paper applying the Jacobian method for soft robot control. One optimization problem is proposed for modeling and Jacobian matrix estimation, while the other optimization problem is proposed for control and actuation variable estimation with the estimated Jacobian matrix as a constraint. The Jacobian matrix is normalized for adjustment. The 2D soft robot can follow trajectories with unknown obstacles thanks to this controller. The control frequency is 1kHz.

16MY: Yip M C, Camarillo D B. Model-less hybrid position/force control: a minimalist approach for continuum manipulators in unknown, constrained environments[J]. IEEE Robotics and Automation Letters, 2016, 1(2): 844-851. 

Based on 14MY, this paper introduces a simple force-displacement model and includes the force control inside the second optimization problem. The control frequency is over 2kHz.

17KLb: Lee K H, Leong M C W, Chow M C K, et al. FEM-based soft robotic control framework for intracavitary navigation[C]//2017 IEEE International Conference on Real-time Computing and Robotics (RCAR). IEEE, 2017: 11-16. 

Compared with MY’s works, this paper employs a FEM model for Jacobian matrix optimization. Moreover, the end positions for the Jacobian matrix are mapped on an image plane.

17MY: Yip M C, Sganga J A, Camarillo D B. Autonomous control of continuum robot manipulators for complex cardiac ablation tasks[J]. Journal of Medical Robotics Research, 2017, 2(01): 1750002. 

This paper exploits 16MY’s method for force and position control and performs complex cardiac ablation tasks. To reject disturbance, a low-pass filter is used. Also, this paper estimates the Jacobian matrix with data from several previous steps, not only the single past step.

17YJ: Jin Y, Wang Y, Chen X, et al. Model-less feedback control for soft manipulators[C]//2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2017: 2916-2922. 

This work employs a complex modular soft robot with 20 chambers working in 3D space. For the Jacobian matrix estimation, this paper makes simplifications based on the physical structure. Moreover, a safe Jacobian matrix is introduced to keep the control stable.

18XW: Wang X, Lee K H, Fu D K C, et al. Experimental validation of robot-assisted cardiovascular catheterization: model-based versus model-free control[J]. International journal of computer assisted radiology and surgery, 2018, 13: 797-804. 

This work compares the position tracking performance of model-based and model-free control (Jacobian method.) The inverse Jacobian matrix is leveraged for control instead of an optimization problem.

19MV: Verghese M, Richter F, Gunn A, et al. Model-free visual control for continuum robot manipulators via orientation adaptation[C]//The International Symposium of Robotics Research. Cham: Springer International Publishing, 2019: 959-970. 

This is one paper from MY’s lab. To obtain an accurate Jacobian matrix estimation, the authors employ orientation adaptation and Kalman filter.

20XW: Wang X, Fang G, Wang K, et al. Eye-in-hand visual servoing enhanced with sparse strain measurement for soft continuum robots[J]. IEEE Robotics and Automation Letters, 2020, 5(2): 2161-2168. 

This paper combines the sensing signals from a camera and fiber bragg gratings for robot state estimation. The Jacobian matrix and inverse Jacobian matrix are involved for modeling and control, respectively.

20YW: Wu Y Y, Tan N. Model-less feedback control for soft manipulators with Jacobian adaptation[C]//2020 International Symposium on Autonomous Systems (ISAS). IEEE, 2020: 217-222. 

This paper smooth the estimated Jacobian matrix change and actuation change to achieve a stable control.