*3.4. Validation of the Model*

It is reasonable to validate the model by evaluating the pelvis displacement estimated by the model because the vertical stiffness has been difficult, if not impossible, to measure during walking until now. Errors between the model solution and the measured displacement of the pelvis are calculated. The measured displacement was collected in the experiment in Section 2.2. It contains three strides, and each stride has differences in oscillation. Therefore, the variable of time '*t*' in Equation (8) is set to 3.5 s in order to include three strides. With the empirical coefficients and phases in Table 1, the vertical displacement of the thigh and shank is expressed and substituted into Equation (4). The obtained vertical displacement of the pelvis for one subject is then compared to the measured displacement as presented in Figure 10.

**Figure 10.** The pelvis trajectory calculated by the model and real measurement.

\* measure solutiom measure

1

1 \*

It can be observed that the model solution of the pelvis displacement is approximately consistent with the measured displacement. This proved that the proposed model could characterize walking characteristics such as vertical stiffness and pelvis oscillation by leg length, body weight, and walking cadence. Moreover, to illustrate the universality and stability of the identification process, the model error is calculated from the solved pelvis displacement and the measured pelvis displacement as follows:

$$E = \frac{1}{t \ast F\_{\text{s}}} \sum\_{nt=1}^{t \ast F\_{\text{S}}} \frac{\varkappa\_{nt}^{\text{measure}} - \varkappa\_{nt}^{\text{solution}}}{\varkappa\_{nt}^{\text{measure}}},\tag{13}$$

where *t* equals 3.5 s as mentioned before, *F<sup>s</sup>* equals the sampling frequency of the motion capture system, which is 100, and subscript *nt* refers to the number of time points. The errors for all the subjects are shown in Table 2. For different individuals, the errors range from 11.94 to 29.14%, and the mean error is 20.48% while the standard deviation is 5.44%.

**Table 2.** The error of the estimated pelvis displacement by model.


#### **4. Discussion**

The primary aim of this study is to estimate the vertical stiffness of the knee and hip using individual parameters that are easy to measure during walking. To achieve this aim, a multi-mass-spring model was established. Furthermore, the Fourier series was used to fit the vertical displacements of lower limb segments and VGRF required in the established model with individual parameters such as leg length and walking cadence. According to the established model, the vertical stiffness of the knee and hip was estimated by leg length, body weight, and walking cadence across the entire gait cycle. Furthermore, the established lower limb model was validated by its solution of pelvic displacement and real measurement.

Firstly, our results implied that the established multi-spring model is effective at characterizing walking characteristics. There were different dynamic models for stiffness calculation, as shown in Table 3. A typical human gait model using a nonlinear angular spring and dash pot at each point was established to find the optimum joint stiffness of the hip and ankle in the stance phase [31]. It also found that stiffness variation was affected by gait pattern and cadence. An actuated dissipative model combining the optimization-based minimal biped model and the spring-loaded inverted-pendulum model was established for the stance phase, and 2 <sup>×</sup> 104 N/m (5 <sup>×</sup> <sup>10</sup><sup>3</sup> to 1 <sup>×</sup> <sup>10</sup><sup>5</sup> N/m) of the leg stiffness achieved the closest GRF profile [32]. This supported our finding that the vertical stiffness of the hip in the stance phase is sometimes varied at a high value level, as displayed in Figure 9. The quasi-stiffness of the knee and ankle was predicted using statistical models based on subject weight and height [18,20]. They provided the foundation for the idea that immeasurable characteristics can be predicted by measurable parameters. A point mass with two massless springs was also established as a dynamic model to calculate the leg stiffness in the stance phase [39] and to predict the trajectory of the center of mass. Compared to these dynamic models for stiffness estimation in the stance phase, the multi-mass spring model established in this study can estimate stiffness across the entire gait cycle, and its solution of pelvis displacement has been validated.


**Table 3.** The comparison between the typical and proposed dynamic models for stiffness calculation. . . . . .

Aside from the ability of the proposed model in our study to be consistent across the entire gait cycle, the vertical stiffness of joints in our study was a different concept from traditional joint stiffness. Traditionally, the stiffness of the knee and hip was calculated as the quotient of the moment and joint angle change in the sagittal plane, and the moment was calculated by the trajectory data and the GRF [40]. This joint stiffness illustrated the relationship between the angle and the corresponding moment applied to the joints during walking. While the vertical stiffness of the joints investigated in this study shows a link between vertical oscillation of lower limb segments and VGRF.

During the model solution, the vertical oscillations of lower limb segments and VGR were utilized. The vertical oscillation of lower limb segments was fitted by the Fourier series with leg length and walking cadence, while the VGRF was represented with body weight and walking cadence. The amplitude coefficient and initial phases shown in Table 1 contributed to a universal and mathematical expression. These findings were supported by previous findings. Fourier series, for example, had been used to characterize the pelvic trajectory [41].

Since the vertical stiffness of the knee and hip was obtained solely by individual parameters such as body weight, leg length, and walking cadence, which are all easy and cheap to measure, it implies that VGRF, body weight, and vertical oscillation of body segments have inherent relationships. This is similar to the previous research. The body weight influenced the GRF, and the vertical displacement of the body for a given individual was determined by the effective leg length [32]. Furthermore, it was demonstrated that the VGRF estimated the vertical displacement of the body mass [42].

Moreover, when compared to previous studies, our study illustrates the time-varying process of vertical stiffness corresponding to the gait phase across the entire gait cycle. In vertical, knee stiffness is near zero in the midstance and high in the terminal stance and initial swing. These findings are consistent with previous research, which found that the knee stiffness determined by the slope of the knee moment-angle curve is approximately zero at the start of the stance and increases in the late stance [21]. In addition, it is worthy to note that the vertical stiffness of the knee across the entire cycle is varied in the same order of magnitude while being different for the hip.

The contributions of this study are as follows: (1) the uniform equation to depict the vertical oscillations and VGRF of different people is obtained with individual walking cadences as well as leg length and body weight, respectively; (2) the multi-mass-spring model is established to identify the vertical stiffness of hip and knee simultaneously, and this stiffness can be represented by the body weight, leg length, and walking cadences; and (3) the obtained vertical stiffness is validated by the comparison between the estimated displacement and the measured displacement of the pelvis.

There are also some limitations that need to be considered. The main limitation is the size of the subject. Fourteen subjects walked at their preferred speed, obtaining a homogeneous sample. The analyses could be generalized only to the range of age, height, and walking cadence that the statistical significance supports. Similar estimations could be carried out for other groups, such as older adults and children. Another limitation is that several simplifications were employed. Both the mass and length of the left and right lower limbs were regarded as the same, and the ankle and foot were ignored. A more sophisticated model could be considered to take the asymmetrical factors and eliminated terms into account.

#### **5. Conclusions**

In summary, the vertical stiffness of the knee and hip can be simultaneously estimated by a multi-mass-spring model. It has been found that the vertical oscillations of lower limb segments were universally expressed by walking cadence and leg length, while vertical ground reaction force was represented by walking cadence and body weight. Moreover, the vertical stiffness of the knee and hip were finally estimated by the walking cadence, leg length, and body weight. The variation of the estimated vertical stiffness across the entire gait cycle displayed different trends toward different gait phases. Additionally, the proposed model was validated efficiently by the estimated vertical oscillation of the pelvis across three gait cycles for the 14 different subjects. The remarkable results obtained in this study represent a different view for future studies on human walking analysis. In the near future, more sophisticated models that consider ankle and damping will be constructed and extended to more human groups.

**Author Contributions:** Conceptualization, H.Z. and J.C.; methodology, H.Z.; software, H.Z.; validation, H.Z., J.C. and W.-H.L.; formal analysis, H.Z.; investigation, H.Z.; resources, J.C. and W.-H.L.; data curation, H.Z.; writing—original draft preparation, H.Z.; writing—review and editing, H.Z., J.C. and W.-H.L.; visualization, H.Z.; supervision, J.C. and W.-H.L.; project administration, J.C. and W.-H.L.; funding acquisition, J.C. and W.-H.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Key Research and Development Program of China [grant number 2021YFE0203400], the Innovation and Technology Commission under the Mainland-Hong Kong Joint Funding Scheme (MHKJFS), and the Hong Kong Special Administrative Region, China [Project No. MHP/043/20].

**Institutional Review Board Statement:** The study was conducted in accordance with the Declaration of Helsinki and approved by the Institutional Review Board of Xi'an Jiaotong University (protocol code 2020-1330 and date of approval: 26 October 2020).

**Informed Consent Statement:** Informed consent was obtained from all subjects involved in the study.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author.

**Acknowledgments:** We would like to thank some colleagues for their useful discussion.

**Conflicts of Interest:** The authors declare no conflict of interest.
