**1. Introduction**

Walking is one of the most common daily activities of humans, and a large number of engineered locomotion systems are designed to emulate human walking, such as bipedal walkers [1,2], biologically inspired prosthetic limbs [3], and lower limb exoskeletons [4,5]. Research in these fields requires knowledge of the stiffness of lower limbs [6–8] since lower limbs act as supports and actuators in walking [9,10]. As stiffness is a multifactorial expression of the musculoskeletal system [11–14], stiffness in the lower limbs has been studied a lot [15–17]. There are several types of 'stiffness' such as leg stiffness, joint stiffness, and vertical stiffness [18,19]. Leg stiffness is the quotient of ground reaction force (GRF) and the change in leg length. The joint stiffness is the torsional stiffness, which is calculated as the quotient of the moment and joint angle for passive walking. Furthermore, the instantaneous slope of the joint's torque-angle profile is described and defined as quasistiffness [20,21]. In addition, joint stiffness at the ankle, knee, and hip is typically defined as the ratio of the change in muscle moment to joint angular displacement [22,23]. Vertical stiffness is generally used to describe the linear movements that occur in the vertical direction, such as hopping and jumping [24]. It was defined as the quotient of vertical ground reaction force (VGRF) and the center of mass displacement [25].

**Citation:** Zhao, H.; Cao, J.; Liao, W.-H. Simultaneous Estimation of the Vertical Stiffness in the Knee and Hip for Healthy Human Subjects during Walking. *Bioengineering* **2023**, *10*, 187. https://doi.org/10.3390/ bioengineering10020187

Academic Editors: Christina Zong-Hao Ma, Zhengrong Li and Chen He

Received: 28 December 2022 Revised: 25 January 2023 Accepted: 28 January 2023 Published: 1 February 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

At times, vertical stiffness and leg stiffness were used interchangeably for jumping activities, but it is actually vertical stiffness rather than leg stiffness [26]. Moreover, the relationships between the leg stiffness, vertical stiffness, and joint stiffness of the stance phase in running are compared [27]. It also illustrated that joint stiffness was associated with limb stiffness (vertical stiffness and leg stiffness). As for walking, leg stiffness is calculated as the resultant GRF in the direction of the connection between the center of pressure and hip joint center, and symmetry in bilateral leg stiffness and stiffness sharing proved useful for a more complete gait assessment in children with diplegic cerebral palsy [28].

As reviewed in [29], loading, motion, and cycle all influenced the mechanical characteristics components of walking. Walking metrics such as vertical oscillation, cadence, speed, and step length can be employed to estimate the GRF during walking by a deep learning network regression algorithm [30]. To analyze the inner relationship between stiffness and walking characteristics, some dynamic models have been established. A human gait model in two degrees of freedom was developed to calculate the time-varying stiffness of the joint, and the stiffness is found to be affected by gait pattern and cadence [31]. To realize human-like GRF patterns, an actuated dissipative spring-mass model was also proposed by introducing spring-damping units to the optimization-based minimal biped model [32]. Results illustrated that stiffness and objective weight affect the number and size of peaks in the VGRF and stance time. The vertical movement of the center of mass was related to the stabilization strategies of the double support phase and the single support phase, and the difference was also reflected in the GRF [33]. In addition, the alterations of VGRF during walking were also associated with the appearance of neurodegenerative diseases [34,35].

From the above studies, it can be summarized that the vertical characteristic is crucial for assessing the walking ability of humans. However, the vertical characteristic of joints has not been studied since the reported 'vertical stiffness' was at the whole-body level and the joint stiffness was focused on the moment and angle applied to them. In addition, all kinds of the mentioned 'stiffness' were calculated only at the stance phase and based on the measured GRF, displacement, angle, and moment, which are expensive to measure.

Therefore, the objective of this study is twofold: (i) to establish a universal gait dynamic model that can estimate both the immeasurable stiffness and measurable displacement, and (ii) to estimate the vertical stiffness of the knee and hip during walking by the individual parameters. Based on these concepts, the vertical stiffness of lower limb joints is hypothesized to be directly estimated by individual parameters like leg length, body weight, and walking cadence.

#### **2. Materials and Methods**

To evaluate the vertical stiffness of lower limb joints continuously and completely, a multi-mass-spring model of the lower limbs is established. Then the vertical displacements of the lower limb during walking are collected and summarized into a uniform equation. Moreover, the vertical stiffness of the hip and knee is derived. The entire process is displayed in Figure 1.

**Figure 1.** The process of the estimation of vertical stiffness by a dynamic model.
