School of Kinesiology, University of Michigan, 1402 Washington Heights, Ann Arbor, MI 48109-2013, USA
Department of Psychology, University of Michigan, 004 East Hall, 530 Church Street, Ann Arbor, MI 48109-1043, USA.
J Exp Biol. 2021 Mar 18;224(Pt 6):jeb232850. doi: 10.1242/jeb.232850.
Runners are commonly modeled as spring-mass systems, but the traditional calculations of these models rely on discrete observations during the gait cycle (e.g. maximal vertical force) and simplifying assumptions (e.g. leg length), challenging the predicative capacity and generalizability of observations. We present a method to model runners as spring-mass systems using nonlinear regression (NLR) and the full vertical ground reaction force (vGRF) time series without additional inputs and fewer traditional parameter assumptions. We derived and validated a time-dependent vGRF function characterized by four spring-mass parameters - stiffness, touchdown angle, leg length and contact time - using a sinusoidal approximation. Next, we compared the NLR-estimated spring-mass parameters with traditional calculations in runners. The mixed-effect NLR method (ME NLR) modeled the observed vGRF best (RMSE:155 N) compared with a conventional sinusoid approximation (RMSE: 230 N). Against the conventional methods, its estimations provided similar stiffness approximations (-0.2±0.6 kN m) with moderately steeper angles (1.2±0.7 deg), longer legs (+4.2±2.3 cm) and shorter effective contact times (-12±4 ms). Together, these vGRF-driven system parameters more closely approximated the observed vertical impulses (observed: 214.8 N s; ME NLR: 209.0 N s; traditional: 223.6 N s). Finally, we generated spring-mass simulations from traditional and ME NLR parameter estimates to assess the predicative capacity of each method to model stable running systems. In 6/7 subjects, ME NLR parameters generated models that ran with equal or greater stability than traditional estimates. ME NLR modeling of the vGRF in running is therefore a useful tool to assess runners holistically as spring-mass systems with fewer measurement sources or anthropometric assumptions. Furthermore, its utility as statistical framework lends itself to more complex mixed-effects modeling to explore research questions in running.
跑步者通常被建模为弹簧质量系统，但这些模型的传统计算依赖于步态周期中的离散观察（例如最大垂直力）和简化假设（例如腿长），这挑战了观察结果的预测能力和通用性。我们提出了一种使用非线性回归 (NLR) 和完整垂直地面反作用力 (vGRF) 时间序列来对跑步者进行建模的方法，而无需额外的输入和更少的传统参数假设。我们使用正弦逼近推导出并验证了一个时变 vGRF 函数，该函数由四个弹簧质量参数 - 刚度、触地角度、腿长和接触时间 - 来描述。接下来，我们将 NLR 估计的弹簧质量参数与跑步者的传统计算进行了比较。混合效应 NLR 方法 (ME NLR) 与传统正弦逼近方法相比，对观察到的 vGRF 进行了最佳建模（RMSE：155 N）。与传统方法相比，其估计值提供了相似的刚度近似值（-0.2±0.6 kN m），但角度略陡峭（1.2±0.7 度），腿更长（+4.2±2.3 cm），有效接触时间更短（-12±4 ms）。总的来说，这些由 vGRF 驱动的系统参数更接近观察到的垂直冲量（观察：214.8 N s；ME NLR：209.0 N s；传统：223.6 N s）。最后，我们根据传统和 ME NLR 参数估计生成了弹簧质量模拟，以评估每种方法模拟稳定跑步系统的预测能力。在 6/7 名受试者中，ME NLR 参数生成的模型与传统估计相比具有同等或更高的稳定性。因此，ME NLR 对跑步者 vGRF 的建模是一种有用的工具，可以更全面地将跑步者作为弹簧质量系统进行评估，所需的测量源或人体测量学假设更少。此外，它作为统计框架的实用性使其可以应用于更复杂的混合效应建模，以探索跑步中的研究问题。
