###### Abstract
The models based on differential equations and the models based on operators are dominant phenomenology-based hysteresis models. For example, the Bouc-Wen model, as a represent  
Figure 1: PAM-driven soft manipulators. (**a**) OCTARM continuum manipulator; (**b**) BIONIC HANDLING ASSISTANT; (**c**) a cylindrical manipulator; (**d**) continuum robot with parallel cable and pneumatic artificial muscles; (**e**) seven parallel PAMs-based manipulator with variable stiffness; (**f**) four parallel PAMs-based manipulators with variable stiffness. based models, adopts nonlinear differential equations to calculate the PAM hysteresis [24]. Krasnosel'skii-Pokrovskii (KP) models [25; 26], modified Prandtl-Ishlinskii (MPI) models [27], Preisach models [28; 29], generalized Prandtl-Ishlinskii (GPI) [30; 31] models, and some other operator-based models fit hysteresis by weighting and summing multiple hysteresis operators. The above existing hysteresis models are mostly concerned with the inherent hysteresis (compared with coupled hysteresis effects) in PAMs, and different PAM couplings are rarely concerned in hysteresis models. For compensation control of parallel PAMs-based manipulators, mechanical couplings are usually regarded as unmodeled disturbances, and they are solved by introducing feedback controllers parallel with individual inverse hysteresis compensators, complicating the controller design. As an example, reference [32] employed the Preisach model to characterize the hysteresis behavior of PAMs and integrated a feedback controller with the inverse Preisach compensator to address the interference among different PAMs. Although reference [33] proposed a lumped model considering mechanical coupling and simple friction with constant curvature assumption for a pneumatic manipulator, it did not accurately describe the hysteresis behavior of the soft manipulator. Particularly, the model calculation is too complicated to achieve control; hence, it needs to be linearized, and a closed-loop controller is employed simultaneously to enhance control accuracy. To avoid time-consuming calculation and model simplification, reference [34] derived a recurrent neural network-based model for a pneumatic manipulator, focusing on learning the coupled relationship between actuator length variations and the manipulator's posture. However, this model lacks sufficient calculations for hysteresis effects between pressure and the actuator length responses, and is not suitable for low-level controller design. Based on the above discussions, the research work on modeling of the coupled hysteresis effect in the multiple PAMs-based manipulator is still open, and successful modeling methods can greatly promote the further development of pneumatic manipulator control. This paper conducts research on modeling and compensation of the coupled hysteresis behavior of the soft manipulator with variable stiffness. The main contributions of this paper are illustrated as follows:  
1. This paper firstly presents a series of experiments to characterize the coupled hysteresis behavior of the soft manipulator with variable stiffness based on PAMs. 2. According to existing literature reviews, now, available hysteresis models mostly concentrate on modeling individual hysteresis effects (compared with coupled hysteresis effects) in the PAM, and few hysteresis models have been established involving hysteresis couplings of the PAM-based manipulator, especially with variable stiffness. Based on the input-output behavior characteristics, a coupled hysteresis model that comprehensively considers stiffness influence is proposed. The newly proposed model is constructed of three serial parts, including an inherent hysteresis part, an actuator coupling part, and a stiffness-dependent part. To our knowledge, the model in this paper innovatively predicts the hysteresis coupling of the parallel PAMs-based robot with variable stiffness, exhibiting outstanding generalization ability and precision.