###### Abstract  
The parallel driving soft manipulator with multiple extensors and contractile pneumatic artificial muscles (PAMs) is able to operate continuously and has varying stiffness, achieving smooth movements and a fundamental trade-off between flexibility and stiffness. Owing to the hysteresis of PAMs and actuator couplings, the manipulator outputs display coupled hysteresis behaviors with stiffness dependence, causing significant positioning errors. For precise positioning control, this paper takes the lead in proposing a comprehensive model aimed at accurately predicting the coupled hysteresis behavior with the stiffness dependence of the soft manipulator. The model consists of an inherent hysteresis submodule, an actuator coupling submodule, and a stiffness-dependent submodule in series. The asymmetrical hysteresis nonlinearity of the PAM is established by the generalized Prandtl-Ishlinskii model in the inherent hysteresis submodule. The serial actuator coupling submodule is dedicated to modeling the actuator couplings, and the stiffness-dependent submodule is implemented with a fuzzy neural network to characterize the stiffness dependence and other system nonlinearities. In addition, an inverse compensator on the basis of the proposed model is conducted. Experiments demonstrate that this model possesses high accuracy and good generalization, and its compensator is effective in decoupling and mitigating hysteresis coupling of the manipulator. The proposed model and control methods significantly improve the positioning accuracy of the pneumatic soft manipulator. p +
Footnote †: journal: applied sciences  
total of nine bellow-based pneumatic actuators are arranged and connected in parallel. Reference [12] designed a cylindrical manipulator (Figure 1c) with each section having three radially symmetric pneumatic chambers. In addition, a dual-plane constraint design (Figure 1d) for the backbone structure was proposed in reference [13] to integrate the parallel cables and pneumatic artificial muscles into the continuum robot. A soft manipulator controlled by antagonistically arranged extensor and contractile PAMs was proposed to enhance their anti-interference capability and load-bearing capacity in reference [14] (Figure 1e), and our previous work also designed a soft manipulator driven by three contractile PAMs and one extensor PAM in parallel [15; 16], as shown in Figure 1f. Based on opposing configurations of extensor PAMs and contractile PAMs, the soft manipulator can output different stiffnesses varying independently from its position. Despite the fact that PAM possesses excellent application prospects in soft robots, because of friction loss, rubber tube elasticity, and the unique geometric structure, it presents a severe asymmetric hysteresis phenomenon, preventing PAM-driven soft manipulators from positioning accurately [17; 18; 19]. Further complicating matters, parallel PAMs contained in the soft manipulator are constrained to interact with each other; thus, the system is negatively affected by the coupled hysteresis among different PAMs, increasing difficulties in precise modeling and control of the PAM-driven soft manipulators to a higher degree. It is noted that the hysteresis coupling effect almost exists in any multiple PAMs-actuated parallel system. In order to compensate and decouple the hysteresis coupling effect, an effective modeling and control strategy should be investigated. Currently, hysteresis modeling and compensation control for smart materials have significantly attracted the attention of researchers [20; 21; 22]. Now, available hysteresis models can be broadly classified into physics-based models and phenomenology-based models. Physics-based models, such as the Maxwell-slip model, are derived from the working principles of systems and material properties [23], meeting difficulties in establishing hysteresis models with high precision for some simplifications. Therefore, phenomenology-based hysteresis models appear, and plenty of work has been conducted.