# A Method for Precise Tracking Control of Pneumatic Artificial-Muscle-Driven Exoskeletal Robot
## 2 Pneumatic-Muscle-Driven Upper Limb Exoskeletal Robot System
### PAM Driver
#### 2.1.1 PAM Output Force Model  
The following model was introduced by [PERSON] [21]. This mathematical model, as shown in equation (1) below, is currently a simplified mechanism of PAM's output force. Nonetheless, this model is based on the following hypotheses: (a) strands cannot be extended; (b) there are frictional forces between the chamber and the strands enveloping it; (c) the outreach and adduction of muscle are always in an ideal state; no relative motion will take place between the two aforementioned parts; (d) PAM can be approximated as a cylinder by disregarding the structural changes at the muscle end seal. The geometrical relationship of PAM is shown in Figure 2.  
\[F(\varepsilon,p)=\pi r_{0}^{2}p\Big{[}a(1-\varepsilon)^{2}-b\Big{]} \tag{1}\]  
In the formula, \(a=3/\tan^{2}a_{0}\); \(b=1/\sin^{2}a_{0}\); \(a_{0}\) is PAM's initial braid angle; \(r_{0}\) is the initial radius; \(\varepsilon=(l_{0}-l)/l_{0}\); \(l_{0}\) is the initial length; \(l\) is PAM's length; and \(p\) is PAM's inner pressure. Figure 2: Structure of pneumatic muscle. In order to accurately model the output force of PAM, the given output force model is obviously insufficient. Elastic forces between weaved strands and the rubber bladder, frictional forces between strands and the rubber bladder, and frictional forces between strands are of necessity when constructing the mathematical model of PAM. The inner strand of PAM is made from nylon, giving it higher structural strength. This enables the muscle to transform the power provided by air into output force. That is to say, the axial adduction force is actually provided by the strand mesh. Great air pressure is put on the strand during PAM's axial outreach, causing extortion, and thus, storing energy. This portion of energy can be calculated by measuring the extortion of the strand. As shown in Figure 3, the tractional force on the mesh can be deduced by balancing the axial force. The elastic force generated by the strand can be inferred by calculating the elastic energy's derivative in the displacement direction, as shown in Formula (2). \[F_{\text{braid}}\ =\frac{dW_{\text{braid}}}{dL}\ =Nb_{0}A_{s_{0}}\frac{dV_{ \epsilon}}{dL}=\frac{N^{2}b_{0}^{3}A_{s_{0}}^{2}E^{2}p^{2}\pi nL}{\left(\pi n NEA _{s_{0}}-pLb_{0}\right)^{3}} \tag{2}\]  
In the above formula, \(A_{s_{0}}\) is the nominal cross-sectional area of the strand, \(E\) is the strand's elastic modulus, \(\mu_{i}\) represents the property parameter of rubber, N is the number of strands, \(W_{\text{braid}}\) is the stored elastic potential energy within, \(V_{\epsilon}\) is the strain energy density, and b is the length of a single strand. The inner expandable chamber of PAM is commonly made of rubber. This is due to the elasticity of the rubber bladder's inner cavity and the prevention of accurate modeling during the inflation process by the outer enveloping strand of the bladder. The elastic energy of the bladder can be deduced according to the virtual work principle. This energy can be represented indirectly by the elastic energy density \(W\):  
\[F_{\text{bladder}}\ =V_{r}\frac{dW}{dL} \tag{3}\]  
In this equation, \(V_{r}\) is the bladder volume. Taking \(t_{0}\) and t as the bladder's inherent thickness and its thickness while inflating, \(L_{0}\) and L are PAM's length under no external influence and during inflation, respectively. The volume of the bladder is:  
\[V_{r} =\pi L_{0}\Big{[}\Big{(}\frac{D_{0}}{2}\Big{)}^{2}-\Big{(}\frac{D _{0}}{2}-t_{0}\Big{)}^{2}\Big{]} \tag{4}\] \[=\pi L_{0}t_{0}(D_{0}-t_{0})\]  
For rubber material, the elastic energy density \(W\) can be represented by the Mooney-Rivlin model:  
\[W=C_{10}(I_{1}-3)+C_{01}(I_{2}-3) \tag{5}\]  
Figure 3: Analysis of force on strand mesh. where \(I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}\), \(I_{2}=\lambda_{1}^{2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}+\lambda_{1}^{2 }\lambda_{3}^{2}\), \(C_{ij}\) is the material constant, and I\({}_{1}\) and I\({}_{2}\) are the Green strain invariant. \(\lambda_{i}\) is the main elongation.