## 2 Coupled Hysteresis Characteristics of the Soft Manipulator
### Hysteresis Coupling Effects and Stiffness-Dependence  
In order to provide an understanding of the coupled hysteresis behaviors of the parallel PAMs-driven manipulator system, each of three contractile PAMs in the soft manipulator is inflated with \(p*=0.65+0.65\sin(0.04\pi t)\) bar alternately, as well as the other two with zero pressure. Pressure \(p_{0}\) in the extensor PAM keeps 1 bar. Experimental results on the pressure-length variations of contractile PAMs are shown in the first three lines of Figure 4. And the sampling frequency is set to 1000 Hz. Then, all contractile PAMs are inflated together. Figure 4j-l show the relationships between \(p_{1}\) (the pressure of PAM I) and \(y_{1}\) (the displacement of PAM I), \(p_{2}\) (the pressure of PAM II) and \(y_{2}\) (the displacement of PAM II), and \(p_{3}\) (the pressure of PAM III) and \(y_{3}\) (the displacement of PAM III), respectively. As introduced in the above Section 2.1, when the pressure within contractile PAMs remains unchanged, the manipulator stiffness increases as the pressure of the extensor PAM rises [15]. Therefore, this paper employs the pressure \(p_{0}\) within the extensor PAM to represent the manipulator stiffness. To reveal the influence of manipulator stiffness on its output behavior, relationships between \(p_{1}=0.35\sin(0.04\pi t)+0.35\) bar and output variations of PAM I, PAM II, and PAM III corresponding to \(p_{0}=0\), \(0.5\), \(1.0\), \(1.5\), \(2.0\) and \(2.5\) bar and \(p_{2}=p_{3}=0\) bar are investigated, respectively. Figure 5 shows partial results with \(p_{0}=0,\ 0.5,\ 1.0\) bar. In view of Figures 4 and 5, the hysteresis and mechanical coupling effects in the manipulator system can be clearly seen. (1) Hysteresis effects: There are obvious hysteresis characteristics in PAMs, and the hysteresis is asymmetric and possesses a dead zone. The maximum width ratio of the hysteresis loop calculated in the form of \(e_{w}=\left(\left|y_{up}-y_{down}\right|/\left|y_{\max}-y_{\min}\right|\right) \times 100\%\), (\(y^{i}_{up}\) and \(y^{i}_{down}\) are PAM output values corresponding to the middle input value, \(y^{i}_{\max}\) and \(y^{i}_{\min}\) define the maximal and minimal PAM output, respectively) indicates that \(e_{w}\) is within the range of [28.5%, 50.1%]. The hysteresis will cause serious inaccuracy and even oscillation during the manipulator's motion. (2) Actuator coupling effects: It can be seen that there are coupling behaviors among three contractile PAMs. Taking the first line of Figure 4 as an example, if no coupling exists, the displacements of PAM II (\(y_{2}\)) and PAM III (\(y_{3}\)) should be equal to zero, with \(p_{2}=p_{3}=0\), while the outputs of PAM II and PAM III show obvious responses to the input excitation \(p_{1}\). And their hysteresis curves exhibit different appearances although \(p_{2}=p_{3}=0\). This is because there exist handmade errors in the structure parameters of pneumatic artificial muscles, causing more intricate nonlinearity with coupled hysteresis. Figure 3: The pneumatic soft manipulator. (**a**) Actual photograph of the manipulator. (**b**) Front view of the manipulator. (3) Variable stiffness effects: The comprehensive experimental results in Figure 5 suggest that the stiffness variation produces non-negligible effects on coupled hysteresis loops, especially on the outputs of PAMs with low pressure. It is very intuitive to see that the pressure \(p_{0}\) has a nonlinear effect on the outputs of contractile PAMs, which means that the stiffness state acts on the coupled outputs of the soft manipulator system. The output amplitudes of contractile PAM II and PAM III sharply decrease with increasing stiffness, while the output amplitude of the contractile PAM I has a small change compared to PAM II and PAM III. The reason is that pressure in the PAM II and PAM III remains zero, which is much lower than that in the PAM I, and they have weaker anti-interference ability.