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Adaptive robust control of soft bending actuators: an empirical nonlinear model-based approach



Soft robotics, compared with their rigid counterparts, are able to adapt to uncharted environments, are superior in safe human-robot interactions, and have low cost, owing to the native compliance of the soft materials. However, customized complex structures, as well as the nonlinear and viscoelastic soft materials, pose a great challenge to accurate modeling and control of soft robotics, and impose restrictions on further applications. In this study, a unified modeling strategy is proposed to establish a complete dynamic model of the most widely used pneumatic soft bending actuator. First, a novel empirical nonlinear model with parametric and nonlinear uncertainties is identified to describe the nonlinear behaviors of pneumatic soft bending actuators. Second, an inner pressure dynamic model of a pneumatic soft bending actuator is established by introducing a modified valve flow rate model of the unbalanced pneumatic proportional valves. Third, an adaptive robust controller is designed using a backstepping method to handle and update the nonlinear and uncertain system. Finally, the experimental results of comparative trajectory tracking control indicate the validity of the proposed modeling and control method.


目 的

为软体机器人系统提供统一且完整的动力学建模方法, 并且基于建立的模型设计控制器, 以实现软体机器人的精确位置控制.


1. 提出了一种经验非线性模型及其辨识方法, 提高了气动软体机器人建模的精度; 2. 建立了不平衡气动比例阀的准静态流量模型, 实现了气动系统的动力学建模; 3. 基于模型, 设计了自适应鲁棒控制器, 实现了软体机器人的精确位置控制.

方 法

1. 将传统线性模型的参数设置为位置的函数, 使用泰勒展开、 系统滤波和最小二乘方法, 实现任意阶次的经验非线性模型辨识; 2. 对不平衡气动比例阀进行阀芯受力分析, 推导阀芯位置的准静态方程, 进而推导准静态流量模型; 3. 通过轨迹跟踪对比实验, 验证所提出的模型和控制器的有效性.

结 论

1. 实验结果表明, 仅使用滑模控制器, 就可以实现较高精度的轨迹跟踪, 这证明了所提建模方法的有效性; 2. 使用自适应鲁棒控制器, 并在传统滑模控制器的基础上在线更新参数, 可以有效提高轨迹跟踪精度.

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  1. Abbasi P, Nekoui MA, Zareinejad M, et al., 2020. Position and force control of a soft pneumatic actuator. Soft Robotics, 7(5):550–563.

    Article  Google Scholar 

  2. Al-Ibadi A, Nefti-Meziani S, Davis S, 2018. Active soft end effectors for efficient grasping and safe handling. IEEE Access, 6:23591–23601.

    Article  Google Scholar 

  3. Bieze TM, Largilliere F, Kruszewski A, et al., 2018. Finite element method-based kinematics and closed-loop control of soft, continuum manipulators. Soft Robotics, 5(3): 348–364.

    Article  Google Scholar 

  4. Blumenschein LH, Gan LT, Fan JA, et al., 2018. A tip-extending soft robot enables reconfigurable and deployable antennas. IEEE Robotics and Automation Letters, 3(2):949–956.

    Article  Google Scholar 

  5. Boyraz P, Runge G, Raatz A, 2018. An overview of novel actuators for soft robotics. Actuators, 7(3):48.

    Article  Google Scholar 

  6. Bruder D, Gillespie B, Remy CD, et al., 2019. Modeling and control of soft robots using the Koopman operator and model predictive control. Proceedings of Robotics: Science and Systems.

  7. Chen C, Tang W, Hu Y, et al., 2020. Fiber-reinforced soft bending actuator control utilizing on/off valves. IEEE Robotics and Automation Letters, 5(4):6732–6739.

    Article  Google Scholar 

  8. Chen WB, Xiong CH, Liu CL, et al., 2019. Fabrication and dynamic modeling of bidirectional bending soft actuator integrated with optical waveguide curvature sensor. Soft Robotics, 6(4):495–506.

    Article  Google Scholar 

  9. Deimel R, Brock O, 2013. A compliant hand based on a novel pneumatic actuator. Proceedings of the IEEE International Conference on Robotics and Automation, p.2047–2053.

  10. Deimel R, Radke M, Brock O, 2016. Mass control of pneumatic soft continuum actuators with commodity components. Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, p.774–779.

  11. Falkenhahn V, Hildebrandt A, Neumann R, et al., 2017. Dynamic control of the bionic handling assistant. IEEE/ASME Transactions on Mechatronics, 22(1):6–17.

    Article  Google Scholar 

  12. Fan JZ, Du QL, Yu QG, et al., 2020. Biologically inspired swimming robotic frog based on pneumatic soft actuators. Bioinspiration & Biomimetics, 15(4):046006.

    Article  Google Scholar 

  13. Fang G, Wang XM, Wang K, et al., 2019. Vision-based online learning kinematic control for soft robots using local Gaussian process regression. IEEE Robotics and Automation Letters, 4(2):1194–1201.

    Article  Google Scholar 

  14. Finnemore EJ, Franzini JB, 2002. Fluid Mechanics with Engineering Applications. McGraw-Hill, New York, USA, p.597.

    Google Scholar 

  15. Franco E, Garriga-Casanovas A, 2021. Energy-shaping control of soft continuum manipulators with in-plane disturbances. The International Journal of Robotics Research, 40(1):236–255.

    Article  Google Scholar 

  16. Gerboni G, Diodato A, Ciuti G, et al., 2017. Feedback control of soft robot actuators via commercial flex bend sensors. IEEE/ASME Transactions on Mechatronics, 22(4):1881–1888.

    Article  Google Scholar 

  17. Hamidi A, Almubarak Y, Tadesse Y, 2019. Multidirectional 3D-printed functionally graded modular joint actuated by TCP FL muscles for soft robots. Bio-Design and Manufacturing, 2(4):256–268.

    Article  Google Scholar 

  18. Hyatt P, Kraus D, Sherrod V, et al., 2019a. Configuration estimation for accurate position control of large-scale soft robots. IEEE/ASME Transactions on Mechatronics, 24(1):88–99.

    Article  Google Scholar 

  19. Hyatt P, Wingate D, Killpack MD, 2019b. Model-based control of soft actuators using learned non-linear discrete-time models. Frontiers in Robotics and AI, 6:22.

    Article  Google Scholar 

  20. Ibrahim S, Krause JC, Raatz A, 2019. Linear and nonlinear low level control of a soft pneumatic actuator. Proceedings of the 2nd IEEE International Conference on Soft Robotics, p.434–440.

  21. Jung J, Park M, Kim D, et al., 2020. Optically sensorized elastomer air chamber for proprioceptive sensing of soft pneumatic actuators. IEEE Robotics and Automation Letters, 5(2):2333–2340.

    Article  Google Scholar 

  22. Katzschmann RK, 2018. Building and Controlling Fluidically Actuated Soft Robots: from Open Loop to Model-based Control. PhD Thesis, Massachusetts Institute of Technology, Massachusetts, USA.

    Google Scholar 

  23. Khan AH, Li S, 2020. Sliding mode control with PID sliding surface for active vibration damping of pneumatically actuated soft robots. IEEE Access, 8:88793–88800.

    Article  Google Scholar 

  24. Khan AH, Shao ZL, Li S, et al., 2020. Which is the best PID variant for pneumatic soft robots? An experimental study. IEEE/CAA Journal of Automatica Sinica, 7(2): 451–460.

    Article  Google Scholar 

  25. Kim S, Laschi C, Trimmer B, 2013. Soft robotics: a bioinspired evolution in robotics. Trends in Biotechnology, 31(5):287–294.

    Article  Google Scholar 

  26. Kwon J, Yoon SJ, Park YL, 2020. Flat inflatable artificial muscles with large stroke and adjustable force-length relations. IEEE Transactions on Robotics, 36(3):743–756.

    Article  Google Scholar 

  27. Laschi C, Mazzolai B, Cianchetti M, 2016. Soft robotics: technologies and systems pushing the boundaries of robot abilities. Science Robotics, 1(1):eaah3690.

    Article  Google Scholar 

  28. Li MH, Kang RJ, Branson DT, et al., 2018. Model-free control for continuum robots based on an adaptive Kalman filter. IEEE/ASME Transactions on Mechatronics, 23(1): 286–297.

    Article  Google Scholar 

  29. Liu S, Yao B, 2008. Coordinate control of energy saving programmable valves. IEEE Transactions on Control Systems Technology, 16(1):34–45.

    MathSciNet  Article  Google Scholar 

  30. Luo C, Wang K, Li GY, et al., 2019. Development of active soft robotic manipulators for stable grasping under slippery conditions. IEEE Access, 7:97604–97613.

    Article  Google Scholar 

  31. Marchese AD, Tedrake R, Rus D, 2016. Dynamics and trajectory optimization for a soft spatial fluidic elastomer manipulator. The International Journal of Robotics Research, 35(8):1000–1019.

    Article  Google Scholar 

  32. Mohanty A, Yao B, 2011. Indirect adaptive robust control of hydraulic manipulators with accurate parameter estimates. IEEE Transactions on Control Systems Technology, 19(3):567–575.

    Article  Google Scholar 

  33. Müller D, Raisch A, Hildebrandt A, et al., 2020. Nonlinear model based dynamic control of pneumatic driven quasi continuum manipulators. Proceedings of the IEEE/SICE International Symposium on System Integration, p.277–282.

  34. Pang W, Wang JB, Fei YQ, 2018. The structure, design, and closed-loop motion control of a differential drive soft robot. Soft Robotics, 5(1):71–80.

    Article  Google Scholar 

  35. Polygerinos P, Wang Z, Overvelde JTB, et al., 2015. Modeling of soft fiber-reinforced bending actuators. IEEE Transactions on Robotics, 31(3):778–789.

    Article  Google Scholar 

  36. Polygerinos P, Correll N, Morin SA, et al., 2017. Soft robotics: review of fluid-driven intrinsically soft devices; manufacturing, sensing, control, and applications in human-robot interaction. Advanced Engineering Materials, 19(12):1700016.

    Article  Google Scholar 

  37. Skorina EH, Luo M, Ozel S, et al., 2015. Feedforward augmented sliding mode motion control of antagonistic soft pneumatic actuators. Proceedings of the IEEE International Conference on Robotics and Automation, p.2544–2549.

  38. Tang ZQ, Heung HL, Tong KY, et al., 2020. A probabilistic model-based online learning optimal control algorithm for soft pneumatic actuators. IEEE Robotics and Automation Letters, 5(2):1437–1444.

    Article  Google Scholar 

  39. Thuruthel TG, Ansari Y, Falotico E, et al., 2018. Control strategies for soft robotic manipulators: a survey. Soft Robotics, 5(2):149–163.

    Article  Google Scholar 

  40. Wang T, Zhang YC, Chen Z, 2018. Design and verification of model-based nonlinear controller for fluidic soft actuators. Proceedings of the IEEE/ASME International Conference on Advanced Intelligent Mechatronics, p.1178–1183.

  41. Xiang Z, 2010. Research on the Key Technologies of Pneumatic High-speed on-off Valve. PhD Thesis, Zhejiang University, Hangzhou, China (in Chinese).

    Google Scholar 

  42. Yang Y, Li Y, Chen Y, 2018. Principles and methods for stiffness modulation in soft robot design and development. Bio-Design and Manufacturing, 1(1):14–25.

    Article  Google Scholar 

  43. Yao B, 1997. High performance adaptive robust control of nonlinear systems: a general framework and new schemes. Proceedings of the 36th IEEE Conference on Decision and Control, p.2489–2494.

  44. Yao B, Bu FP, Reedy J, et al., 2000. Adaptive robust motion control of single-rod hydraulic actuators: theory and experiments. IEEE/ASME Transactions on Mechatronics, 5(1):79–91.

    Article  Google Scholar 

  45. Zhang C, Zhu P, Lin Y, et al., 2021. Fluid-driven artificial muscles: bio-design, manufacturing, sensing, control, and applications. Bio-Design and Manufacturing, 4(1): 123–145.

    Article  Google Scholar 

  46. Zhang J, Sheng J, O’Neill CT, et al., 2019. Robotic artificial muscles: current progress and future perspectives. IEEE Transactions on Robotics, 35(3):761–781.

    Article  Google Scholar 

  47. Zhou JS, Chen XJ, Chang Y, et al., 2019. A soft-robotic approach to anthropomorphic robotic hand dexterity. IEEE Access, 7:101483–101495.

    Article  Google Scholar 

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Author information




Cong CHEN and Jun ZOU designed the research, processed the data, wrote the first draft of this manuscript, and revised the final version. Jun ZOU provided funding support.

Corresponding author

Correspondence to Jun Zou.

Ethics declarations

Cong CHEN and Jun ZOU declare that they have no conflict of interest.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 51875507, 51821093, and U1908228)

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Cite this article

Chen, C., Zou, J. Adaptive robust control of soft bending actuators: an empirical nonlinear model-based approach. J. Zhejiang Univ. Sci. A 22, 681–694 (2021).

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  • Pneumatic soft bending actuator
  • Empirical nonlinear model identification
  • Unbalanced pneumatic proportional valve modeling
  • Adaptive robust control
  • Trajectory tracking


  • 气动软弯曲执行器
  • 经验非线性模型辨识
  • 不平衡气动比例阀建模
  • 自适应鲁棒控制
  • 轨迹跟踪

CLC number

  • TP242.3