{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T14:47:49Z","timestamp":1787496469522,"version":"build-2736575974"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T00:00:00Z","timestamp":1675209600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["2021YFE0203400"],"award-info":[{"award-number":["2021YFE0203400"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key Research and Development Program of China","doi-asserted-by":"publisher","award":["MHP\/043\/20"],"award-info":[{"award-number":["MHP\/043\/20"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Innovation and Technology Commission under the Mainland-Hong Kong Joint Funding Scheme (MHKJFS), and the Hong Kong Special Administrative Region, China","award":["2021YFE0203400"],"award-info":[{"award-number":["2021YFE0203400"]}]},{"name":"Innovation and Technology Commission under the Mainland-Hong Kong Joint Funding Scheme (MHKJFS), and the Hong Kong Special Administrative Region, China","award":["MHP\/043\/20"],"award-info":[{"award-number":["MHP\/043\/20"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Bioengineering"],"abstract":"<jats:p>The stiffness of lower limb joints is a critical characteristic of walking. To investigate the potential of establishing a simple and universal model to describe the characteristics related to vertical vibration during human walking, vertical stiffness is introduced at the knee and hip. A multi-mass-spring model of the human body is established in the vertical direction. In the Fourier form, results of experiments on 14 healthy adults show that the vertical displacements of joints are a function of the leg length and walking cadence, while the ground reaction force is a function of the body weight and walking cadence. The obtained universal equations of vertical displacement and ground reaction force are employed as the input parameters to the proposed multi-mass-spring model. Thus, the vertical stiffness in the knee and hip can then be estimated simultaneously by the subject\u2019s weight, leg length, and walking cadence. The variation of vertical stiffness shows different time-varying trends in different gait phases across the entire gait cycle. Finally, the proposed model for vertical stiffness estimation is validated by the vertical oscillation of the pelvis. The average error across three gait cycles for all subjects is 20.48%, with a standard deviation of 5.44%. These results display that the vertical stiffness of knee and hip across the entire gait cycle can be directly estimated by individual parameters that are easy to measure. It provides a different view of human walking analysis and may be applied in future pathological gait recognition, bipedal robots, and lower limb exoskeletons.<\/jats:p>","DOI":"10.3390\/bioengineering10020187","type":"journal-article","created":{"date-parts":[[2023,2,1]],"date-time":"2023-02-01T06:49:36Z","timestamp":1675234176000},"page":"187","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Simultaneous Estimation of the Vertical Stiffness in the Knee and Hip for Healthy Human Subjects during Walking"],"prefix":"10.3390","volume":"10","author":[{"given":"Huan","family":"Zhao","sequence":"first","affiliation":[{"name":"Key Laboratory of Education Ministry for Modern Design and Rotor Bearing System, Xi\u2019an Jiaotong University, 28 Xianning West Road, Xi\u2019an 710049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Junyi","family":"Cao","sequence":"additional","affiliation":[{"name":"Key Laboratory of Education Ministry for Modern Design and Rotor Bearing System, Xi\u2019an Jiaotong University, 28 Xianning West Road, Xi\u2019an 710049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7221-5906","authenticated-orcid":false,"given":"Wei-Hsin","family":"Liao","sequence":"additional","affiliation":[{"name":"Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong 999077, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1082","DOI":"10.1126\/science.1107799","article-title":"Efficient Bipedal Robots Based on Passive-Dynamic Walkers","volume":"307","author":"Collins","year":"2005","journal-title":"Science"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1789","DOI":"10.1016\/j.istruc.2021.05.048","article-title":"Reproducing vertical human walking loads on rigid level surfaces with a damped bipedal inverted pendulum","volume":"33","author":"Lin","year":"2021","journal-title":"Structures"},{"key":"ref_3","first-page":"4","article-title":"Biologically inspired design and development of a variable stiffness powered ankle-foot prosthesis","volume":"11","author":"Wei","year":"2019","journal-title":"J. Mech. Robot"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1186\/s12984-019-0517-9","article-title":"Compliant lower limb exoskeletons: A comprehensive review on mechanical design principles","volume":"16","author":"Torricelli","year":"2019","journal-title":"J. Neuroeng. Rehabil"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"12042","DOI":"10.1088\/1742-6596\/2096\/1\/012042","article-title":"Development of the lower extremity exoskeleton dynamics model using in the task of the patient verticalization","volume":"2096","author":"Saypulaev","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7451683","DOI":"10.1155\/2020\/7451683","article-title":"Knee joint biomechanics in physiological conditions and how pathologies can affect it: A systematic review","volume":"2020","author":"Zhang","year":"2020","journal-title":"Appl. Bionics Biomech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3040","DOI":"10.1109\/TMECH.2021.3051724","article-title":"Reconstructing walking dynamics from two shank-mounted inertial measurement units","volume":"26","author":"Li","year":"2021","journal-title":"IEEE\/ASME Trans. Mechatron."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"42099","DOI":"10.1088\/1742-6596\/1865\/4\/042099","article-title":"Long time prediction of human lower limb movement based on IPSO-BPNN","volume":"1865","author":"Jin","year":"2021","journal-title":"J. Phys. Conf. Ser."},{"key":"ref_9","first-page":"011001","article-title":"What is an artificial muscle? A comparison of soft actuators to biological muscles","volume":"17","author":"Nishikawa","year":"2021","journal-title":"Bioinspir. Biomim."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"116727","DOI":"10.1016\/j.jsv.2021.116727","article-title":"A spring-loaded inverted pendulum model for analysis of human-structure interaction on vibrating surfaces","volume":"522","author":"Yang","year":"2022","journal-title":"J. Sound Vib."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1016\/0167-9457(93)90010-M","article-title":"Joint stiffness: Myth or reality","volume":"12","author":"Latash","year":"1993","journal-title":"Hum. Mov. Sci."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"9914278","DOI":"10.1155\/2021\/9914278","article-title":"Application of leg, vertical, and joint stiffness in running performance: A literature overview","volume":"2021","author":"Struzik","year":"2021","journal-title":"Appl. Bionics Biomech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"644582","DOI":"10.3389\/fbioe.2021.644582","article-title":"The impact of locomotor speed on the human metatarsophalangeal joint kinematics","volume":"9","author":"Wang","year":"2021","journal-title":"Front. Bioeng. Biotechnol."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"110803","DOI":"10.1016\/j.jbiomech.2021.110803","article-title":"Effects of sex and walking speed on the dynamic stiffness of lower limb joints","volume":"129","author":"Santos","year":"2021","journal-title":"J. Biomech."},{"key":"ref_15","unstructured":"van der Kooij, H., Fricke, S.S., Veld, R.C., Prieto, A.V., Keemink, A.Q., Schouten, A.C., and van Asseldonk, E.H. (2021). A device and method to identify hip, knee and ankle joint impedance during walking. arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2563","DOI":"10.1109\/LRA.2021.3062355","article-title":"Stiffness modulation in a humanoid robotic leg and knee","volume":"6","author":"Russell","year":"2021","journal-title":"IEEE Robot. Autom. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1109\/TCDS.2020.2968845","article-title":"Exoskeleton online learning and estimation of human walking intention based on dynamical movement primitives","volume":"13","author":"Qiu","year":"2021","journal-title":"IEEE Trans. Cogn. Dev. Syst."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Shamaei, K., Sawicki, G., and Dollar, A.M. (2013). Estimation of quasi-stiffness of the human knee in the stance phase of walking. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0059993"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1016\/j.bjpt.2021.02.001","article-title":"Hip passive stiffness is associated with midfoot passive stiffness","volume":"25","author":"Fajardo","year":"2021","journal-title":"Braz. J. Phys. Ther."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Shamaei, K., Sawicki, G., and Dollar, A.M. (2013). Estimation of Quasi-Stiffness and propulsive work of the human ankle in the stance phase of walking. PLoS ONE, 8.","DOI":"10.1371\/journal.pone.0059935"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"562","DOI":"10.1109\/TBME.2012.2230261","article-title":"The difference between stiffness and quasi-stiffness in the context of biomechanical modeling","volume":"60","author":"Rouse","year":"2013","journal-title":"IEEE T. Bio.-Med. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1016\/0966-6362(95)01045-9","article-title":"Gait characterization via dynamic joint stiffness","volume":"4","author":"Davis","year":"1996","journal-title":"Gait Posture"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1002\/acr.20012","article-title":"Knee joint stiffness during walking in knee osteoarthritis","volume":"62","author":"Dixon","year":"2010","journal-title":"Arthrit. Care Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1080\/02640414.2012.710755","article-title":"A review of models of vertical, leg, and knee stiffness in adults for running, jumping or hopping tasks","volume":"30","author":"Serpell","year":"2012","journal-title":"J. Sports Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1016\/j.humov.2015.01.005","article-title":"Leg and vertical stiffness (a)symmetry between dominant and non-dominant legs in young male runners","volume":"40","author":"Pappas","year":"2015","journal-title":"Hum. Mov. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Struzik, A., Winiarski, S., and Zawadzki, J. (2022). Inter-Limb Asymmetry of Leg Stiffness in National Second-League Basketball Players during Countermovement Jumps. Symmetry, 14.","DOI":"10.3390\/sym14030440"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"441","DOI":"10.3390\/biomechanics2030034","article-title":"Relationship between Joint Stiffness, Limb Stiffness and Whole\u2013Body Center of Mass Mechanical Work across Running Speeds","volume":"2","author":"Jin","year":"2022","journal-title":"Biomechanics"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kuo, C.-C., Huang, H.-P., Lu, H.-Y., Chen, T.-Y., Wang, T.-M., and Lu, T.-W. (2021). Effects of Tendon Release Surgery on Inter-Limb Leg Stiffness Control in Children with Spastic Diplegic Cerebral Palsy during Gait. Appl. Sci., 11.","DOI":"10.3390\/app11104562"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"e12050","DOI":"10.1016\/j.heliyon.2022.e12050","article-title":"Adopted walking condition for computational simulation approach on bearing of hip joint prosthesis: Review over the past 30 years","volume":"8","author":"Jamari","year":"2022","journal-title":"Heliyon"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Davidson, P., Virekunnas, H., Sharma, D., Piche, R., and Cronin, N. (2019). Continuous Analysis of Running Mechanics by Means of an Integrated INS\/GPS Device. Sensors, 19.","DOI":"10.3390\/s19061480"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1016\/S1350-4533(97)00022-2","article-title":"A stiffness-varying model of human gait","volume":"19","author":"Duan","year":"1997","journal-title":"Med. Eng. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"58","DOI":"10.1016\/j.jbiomech.2019.04.028","article-title":"An actuated dissipative spring-mass walking model: Predicting human-like ground reaction forces and the effects of model parameters","volume":"90","author":"Li","year":"2019","journal-title":"J. Biomech."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"110387","DOI":"10.1016\/j.jbiomech.2021.110387","article-title":"Ground reaction forces intersect above the center of mass in single support, but not in double support of human walking","volume":"120","author":"Vielemeyer","year":"2021","journal-title":"J. Biomech."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"116113","DOI":"10.1016\/j.eswa.2021.116113","article-title":"Severity level diagnosis of Parkinson\u2019s disease by ensemble K-nearest neighbor under imbalanced data","volume":"189","author":"Zhao","year":"2022","journal-title":"Expert Syst. Appl."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"102860","DOI":"10.1016\/j.bspc.2021.102860","article-title":"Accurate identification of Parkinson\u2019s disease by distinctive features and ensemble decision trees","volume":"69","author":"Zhao","year":"2021","journal-title":"Biomed. Signal Process."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1223","DOI":"10.1016\/0021-9290(95)00178-6","article-title":"Adjustments to Zatsiorsky-Seluyanov\u2019s segment inertia parameters","volume":"29","year":"1996","journal-title":"J. Biomech."},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Jamari, J., Ammarullah, M., Saad, A., Syahrom, A., Uddin, M., van der Heide, E., and Basri, H. (2021). The Effect of Bottom Profile Dimples on the Femoral Head on Wear in Metal-on-Metal Total Hip Arthroplasty. J. Funct. Biomater., 12.","DOI":"10.3390\/jfb12020038"},{"key":"ref_38","doi-asserted-by":"crossref","unstructured":"Rubio-Peiroten, A., Garcia-Pinillos, F., Jaen-Carrillo, D., Carton-Llorente, A., Abat, F., and Roche-Seruendo, L. (2021). Relationship between Connective Tissue Morphology and Lower-Limb Stiffness in Endurance Runners. A Prospective Study. Int. J. Environ. Res. Public Health, 18.","DOI":"10.3390\/ijerph18168453"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"11","DOI":"10.1016\/j.jtbi.2011.09.021","article-title":"A model-experiment comparison of system dynamics for human walking and running","volume":"292","author":"Lipfert","year":"2012","journal-title":"J. Theor. Biol."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1016\/j.gaitpost.2015.09.025","article-title":"Alterations in walking knee joint stiffness in individuals with knee osteoarthritis and self-reported knee instability","volume":"43","author":"Gustafson","year":"2016","journal-title":"Gait Posture"},{"key":"ref_41","unstructured":"Panero, E., Anastasio, D., Massazza, G., and Gastaldi, L. (2022). The International Conference of IFToMM ITALY, Springer."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1115\/1.2834293","article-title":"The vertical displacement of the center of mass during walking: A comparison of four measurement methods","volume":"120","author":"Saini","year":"1998","journal-title":"J. Biomech. Eng."}],"container-title":["Bioengineering"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2306-5354\/10\/2\/187\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:21:10Z","timestamp":1760120470000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2306-5354\/10\/2\/187"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,1]]},"references-count":42,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["bioengineering10020187"],"URL":"https:\/\/doi.org\/10.3390\/bioengineering10020187","relation":{},"ISSN":["2306-5354"],"issn-type":[{"value":"2306-5354","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,1]]}}}