# Treadmill Deck Performance Optimization Design Based on Muscle Activity during Running
## 2 Materials and Methods  
### MSD Model  
Among the crucial roles of the musculoskeletal system, movement stands out as a paramount function. There have been extensive experimental studies and modeling techniques discussed by scholars. While experimental methods hold value in elucidating multiple facets of locomotion, they are subject to certain constraints. For example, not all quantities can be easily measured, experiments cannot study isolated effects of parameters, and experiments might require access to numerous participants and specific testing conditions. [PERSON] and [PERSON] improved the 4-degree-of-freedom active MSD model based on the LN model and provided a detailed description of the model parameters [26]. The model presented in this paper is based on the LNZN model. We added a layer to the model that represents the subsystem of the treadmill deck. The changed model is shown in Figure 2. There are five masses in this model. The upper and lower limbs' rigid mass units are shown as \(m_{1}\) and \(m_{3}\), respectively, which are connected by a set of linear springs and dampers, denoted by \(k_{1}\) and \(c_{1}\), respectively. \(m_{2}\) is the soft tissue mass unit of the lower limb, connected to the rigid mass \(m_{3}\) of the upper limb by a spring with a stiffness coefficient of \(k_{3}\), and connected to the rigid mass \(m_{1}\) of the lower limb by a set of variable stiffness \(k_{2}\) and damping \(c_{2}\). \(m_{4}\) is the soft tissue mass of the upper limb connected to the rigid mass of the upper limb by a spring with a stiffness coefficient of \(k_{5}\) and a set of parallel linear stiffness (\(k_{4}\)) and damping (\(c_{4}\)), where rigid mass represents the mass of the human skeleton and wobbling mass represents the mass of the human body muscle and blood, etc. \(m_{5}\), \(k_{t}\), and \(\zeta_{t}\) represent the mass, stiffness, and damping ratio of the treadmill deck, respectively. Figure 1: Flowchart of the overall structure of the work. The default model parameter values utilized in this paper's numerical simulation are consistent with those provided in the LNZN model, summarized and proposed by [PERSON] and [PERSON] [24], and the specific values are shown in Table 1. The initial displacement of all mass blocks is 0. According to the modified model, the equation governing motion is derived as follows:  
\[ \begin{array}{l}m_{1}\ddot{x}_{1}=m_{1}g+k_{1}(x_{3}-x_{1})+k_{2}(x_{2}-x_{1 })+c_{1}\big{(}\dot{x}_{3}-\dot{x}_{1}\big{)}+c_{2}\big{(}\dot{x}_{2}-\dot{x}_{ 1}\big{)}-fg\\ m_{2}\ddot{x}_{2}=m_{2}g+k_{2}(x_{1}-x_{2})+k_{3}(x_{3}-x_{2})+c_{2}\big{(}\dot{x }_{1}-\dot{x}_{2}\big{)}\\ m_{3}\ddot{x}_{3}=m_{3}g+k_{1}(x_{1}-x_{3})+k_{3}(x_{2}-x_{3})+(k_{4}+k_{5})(x_{4 }-x_{3})+c_{1}\big{(}\dot{x}_{1}-\dot{x}_{3}\big{)}+c_{4}\big{(}\dot{x}_{4}- \dot{x}_{3}\big{)}\\ m_{4}\ddot{x}_{4}=m_{4}g+(k_{4}+k_{5})(x_{3}-x_{4})+c_{4}\big{(}\dot{x}_{3}- \dot{x}_{4}\big{)}\\ m_{5}\ddot{x}_{5}=m_{5}g+fg-k_{t}x_{5}-2\zeta_{t}\sqrt{m_{5}}k_{t}\dot{x}_{5} \end{array}  \tag{1}\]  
where the coefficients \(m\), \(c\), and \(k\), respectively, reflect mass, damping, and stiffness in Figure 2, and \(g\) denotes the gravitational acceleration. Furthermore, following the proposal of [PERSON] and [PERSON] [26], the magnitude of GRF is represented by \(F_{g}\) and we modify the vertical contact force \(F_{g}\) according to the modified model as:  
\[F_{g}= \begin{cases}A_{c}\Big{[}a(x_{1}-x_{5})^{b}+c(x_{1}-x_{5})^{d}(\dot{x}_ {1}-\dot{x}_{5})^{e}\Big{]}&x_{1}>0\\ 0&x_{1}\leq 0\end{cases}  \tag{2}\]  
In effect, this is a nonlinear viscoelastic model of \(F_{g}\), where the parameters \(a\), \(b\), \(c\), \(d\), and \(e\) are shoe-specific parameters. In the LNZN model, [PERSON] and [PERSON] [24] proposed