# Research on Optimal Control of Treadmill Shock Absorption Based on Ground Reaction Force Constraint
## 2 Establishment of the Mathematical Model  
### Biomechanical Motion Model  
Currently, there is extensive experimental research and equivalent function modeling in human motion biomechanics. Scholars have explored various modeling techniques, such as the four-degree-of-freedom MSD model proposed by [PERSON] and [PERSON] [12], known as the LN model. This model integrates the rigid and swinging mass of the upper and lower limbs for the first time. However, as a passive model, the LN model cannot account for changes in the stiffness and damping of soft tissues in the legs. [PERSON] and [PERSON] [13] proposed a new active model--the LNZN model, which is an improvement over the LN model. In this model, the stiffness and damping coefficients of the lower limb muscles are adjustable, and the parameters are described in detail. This paper presents a new model based on the improvement of the LNZN model, adding an additional layer to represent the subsystem of the treadmill platform, as shown in Figure 1a for human running on a treadmill. The modified model is shown in Figure 1b. The model has seven particles. m represents the mass of the system (units: kg); c denotes the damping coefficient of the damper (units: N*s/m); and k indicates the spring constant (units: N/m). The stiffness and mass of the upper and lower limbs are represented by m\({}_{1}\) and m\({}_{3}\), respectively. These are linked by a set of springs and dampers, denoted as k\({}_{1}\) and c\({}_{1}\). The mass of the lower limb soft tissues is represented by m\({}_{2}\), which is connected to the stiffness mass m\({}_{3}\) via a spring with a stiffness coefficient k\({}_{3}\). It is also connected to the lower tissue stiffness mass through a variable stiffness spring-damper system k\({}_{2}\) and c\({}_{2}\). The upper limb stiffness m\({}_{3}\) is directly connected to the lower limb stiffness mass via k\({}_{1}\) and c\({}_{1}\). The mass of the upper limb soft tissues is represented by m\({}_{4}\), which is connected to the upper limb stiffness mass by k\({}_{5}\), k\({}_{4}\), and c\({}_{4}\). In this model, the rigid body masses represent the skeletal mass of the human body, such as m\({}_{1}\) and m\({}_{3}\), while the swinging mass represents the soft tissue mass, including musclesand blood. m\({}_{5}\) represents the stiffness mass of the treadmill platform; kt\({}_{5}\) and ct\({}_{5}\) denote the equivalent damping of the athletic shoes worn during the experiment. m\({}_{6}\) and m\({}_{7}\) are the equivalent masses of the shock-absorbing airbags; k\({}_{6}\) and k\({}_{7}\) are the equivalent spring stiffness of the treadmill platform; c\({}_{6}\) and c\({}_{7}\) are the equivalent damping of the airbags. kt\({}_{6}\) and kt\({}_{7}\) represent the equivalent damping spring stiffness of the treadmill support frame. Additionally, fg is the external force absorbed by the human body (units: N), while fg\({}_{1}\) and fg\({}_{2}\) are the external forces absorbed by the two shock-absorbing airbags (units: N). According to the modified model, the equation of motion can be obtained as follows:  
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