*2.2. Experiments*

In a gait laboratory, subjects walked at their preferred speed while wearing 16 retroreflective markers, as shown in Figure 2. The 3D trajectories are collected at 100 Hz by a 12 camera optical capture system (Vicon MX, OML, UK). The GRF was collected at 1000 Hz by three force plates (AMTI, 40060, Advanced Mechanical Technology, Inc., Watertown, MA, USA). Anthropometric parameters including height, mass, and leg length of each subject were measured and recorded. All the subjects were asked to walk barefoot at their preferred walking cadence. The distance of the walking track was about 7 m and had 3 force plates embedded in it. For all subjects, 15 trials of data were recorded for each subject.

**Figure 2.** Measurement of lower limb displacement during level ground walking. (**a**) The experimental setup; (**b**) the reflective markers on the front side; and (**c**) the reflective markers on the back side.

#### *2.3. Multi-Mass-Spring Model of the Lower Limbs*

A simple model that can characterize the dynamic behaviors of the lower limbs during walking is the foundation for understanding human motion. To describe the kinematics and kinetics in the vertical direction of both the left and right lower limbs, a multi-mass-spring model that includes both the knee and hip joints of the lower limbs is proposed as shown in Figure 3. The trunk and upper limbs are assumed to be concentrated mass points; moreover, the thigh and shank are both characterized as mass points.

**Figure 3.** The dynamic model of the human body with ground reaction force.

The analytical formula can then be written as:


000 0 0 0 0 0000 0 0 0 male female = 0.6028\* = 0.5824\* ; male female = 0.1416\* = 0.1478\* ; male female = 0.6028\* = 0.5824\* ; male female = 0.1416\* = 0.1478\* ; where *m* is the mass of the trunk, upper limbs, and head in total, *m<sup>t</sup>* and *m<sup>s</sup>* are the masses of the thighs and shanks, respectively, based on the relationship of the segment mass to body mass '*M*' given by Leva [36], *<sup>m</sup>*male<sup>=</sup> 0.6028 <sup>∗</sup> *<sup>M</sup>*; *<sup>m</sup>*female<sup>=</sup> 0.5824 <sup>∗</sup> *<sup>M</sup>*, *m* male *<sup>t</sup>* = 0.1416∗*M*; *m* female *<sup>t</sup>* = 0.1478∗*M*, the foot is neglected in the model and its mass is included in the shank, *m*male *<sup>s</sup>* <sup>=</sup> 0.057∗*M*; *<sup>m</sup>*female *<sup>s</sup>* = 0.061∗*M*; *xlt* and *xrt* denote the vertical displacements of the left and right thigh, respectively; *xls* and *xrs* refer to the vertical displacements of the left and right shanks, respectively; *F<sup>l</sup>* and *F<sup>r</sup>* are the left and right GRF in vertical, respectively; *klh* and *krh* indicate the vertical stiffness of left and right

hip, respectively; and *klk* and *krk* correspond to the vertical stiffness of left and right knee, respectively. ( ) 

( )

Then the vertical stiffness of the hip and knee are derived as follows: 

( )

 

male female = 0.057\* ; = 0.061\*

$$\begin{cases}(k\_{lh}+k\_{rh})\mathbf{x}-k\_{lh}\mathbf{x}\_{lt}-k\_{rh}\mathbf{x}\_{rt}=m\ddot{\mathbf{g}}-m\ddot{\mathbf{x}}\\-k\_{lh}\mathbf{x}+(k\_{lh}+k\_{lk})\mathbf{x}\_{lt}-k\_{lk}\mathbf{x}\_{ls}=m\_{l}\ddot{\mathbf{g}}-m\_{l}\ddot{\mathbf{x}}\_{lt}\\-k\_{rh}\mathbf{x}+(k\_{rh}+k\_{rk})\mathbf{x}\_{rt}-k\_{rk}\mathbf{x}\_{rs}=m\_{l}\ddot{\mathbf{g}}-m\_{l}\ddot{\mathbf{x}}\_{rs}\\-k\_{lh}\mathbf{x}\_{lt}+k\_{lk}\mathbf{x}\_{ls}=m\_{s}\ddot{\mathbf{g}}-\ddot{\mathbf{f}}\_{l}-m\_{s}\ddot{\mathbf{x}}\_{ls}\\-k\_{rk}\mathbf{x}\_{rt}+k\_{rk}\mathbf{x}\_{rs}=m\_{s}\ddot{\mathbf{g}}-\ddot{\mathbf{F}}\_{r}-m\_{s}\ddot{\mathbf{x}}\_{rs}\end{cases}\tag{2}$$

The solution to vertical stiffness in the knee is as follows: ( )

$$\begin{array}{c} k\_{lk} = \frac{\left(m\_s \text{g} - \text{F}\_l - m\_s \bar{\mathbf{x}}\_{lk}\right)}{\left(\mathbf{x}\_{lk} - \mathbf{x}\_{lt}\right)}\\ k\_{rk} = \frac{\left(m\_s \text{g} - \text{F}\_r - m\_s \bar{\mathbf{x}}\_{rk}\right)}{\left(\mathbf{x}\_{rk} - \mathbf{x}\_{rt}\right)} \end{array} \tag{3}$$

2

Moreover, the pelvis displacement can be derived as: = + ( ) 

2

*Ax*<sup>2</sup> + *Bx* + *C*= 0 *A* = −2*mtg* + 2*m<sup>t</sup>* .. *xlt* − *klk*(*xls* − *xlt*) − *mtg* + *m<sup>t</sup>* .. *xrt* − *krk*(*xrs* − *xrt*) *B* = *mtgxrt* − *m<sup>t</sup>* .. *xltxrt* + *klk*(*xls* − *xlt*)*xrt* + *mtgxlt* − *m<sup>t</sup>* .. *xrtxlt* +*krk*(*xrs* − *xrt*)*xlt* + *mtgxlt* − *m<sup>t</sup>* .. *xltxlt* + *klkxlsxlt* − *klkxlt* 2 +*mtgxrt* − *m<sup>t</sup>* .. *xrtxrt* + *krkxrsxrt* − *krkxrt* <sup>2</sup> + (*mt<sup>g</sup>* <sup>−</sup> *<sup>m</sup><sup>t</sup>* .. *xlt*)(*xlt* + *xrt*) *C* = −2*xltxrtmtg* + *mtxltxrt*( .. *xlt* + .. *xrt*) − *klkxltxrtxls* + *klkxlt* <sup>2</sup>*xrt* −*krkxltxrtxrs* + *klkxltxrt* <sup>2</sup> <sup>−</sup> *<sup>x</sup>ltxrt*(*mt<sup>g</sup>* <sup>−</sup> *<sup>m</sup><sup>t</sup>* .. *xlt*) *x* = <sup>−</sup>*B*<sup>±</sup> √ *<sup>B</sup>*2−4*AC* 2*A* . (4) + 2 2 2 2 +( )( ) 2 +() + + ( ) 4 = 2 =( + ( )) / ( ) 

+ ( ) + +

The stiffness of the hip can be described as: =( + ( )) / ( ) 

$$\begin{array}{l}k\_{lh} = (m\_{l}g - m\_{l}\ddot{\mathbf{x}}\_{lt} + k\_{lk}(\mathbf{x}\_{ls} - \mathbf{x}\_{lt}))/(\mathbf{x}\_{lt} - \mathbf{x})\\k\_{rh} = (m\_{l}g - m\_{l}\ddot{\mathbf{x}}\_{rl} + k\_{rk}(\mathbf{x}\_{rs} - \mathbf{x}\_{rt}))/(\mathbf{x}\_{rl} - \mathbf{x})\end{array} \tag{5}$$

therefore, the outputs of the model are hip stiffness, knee stiffness, and the vertical displacement of the pelvis, and they can be calculated from the inputs such as the ground reaction force, mass, and vertical displacement of the thighs and shanks. As for the vertical displacement of both left and right thighs and shanks, they can be represented with anthropometric parameters as conducted in the following section.
