Spring-mass running: simple approximate solution and application to gait stability

Abstract

The planar spring-mass model is frequently used to describe bouncing gaits (running, hopping, trotting, galloping) in animal and human locomotion and robotics. Although this model represents a rather simple mechanical system, an analytical solution predicting the center of mass trajectory during stance remains open. We derive an approximate solution in elementary functions assuming a small angular sweep and a small spring compression during stance. The predictive power and quality of this solution is investigated for model parameters relevant to human locomotion. The analysis shows that (i), for spring compressions of up to 20% (angle of attack X60 ; angular sweep p60 ) the approximate solution describes the stance dynamics of the center of mass within a 1% tolerance of spring compression and 0:6 tolerance of angular motion compared to numerical calculations, and (ii), despite its relative simplicity, the approximate solution accurately predicts stable locomotion well extending into the physiologically reasonable parameter domain. (iii) Furthermore, in a particular case, an explicit parametric dependency required for gait stability can be revealed extending an earlier, empirically found relationship. It is suggested that this approximation of the planar spring-mass dynamics may serve as an analytical tool for application in robotics and further research on legged locomotion. r

Structured variability of steady-speed locomotion in rats

“…Indeed, our statistical analyses revealed few parameters that discriminate between footfall orders. Those parameters include relative limb length as well as relative limb protraction and retraction -parameters that play a predominant role in self-stability during perturbed locomotion in bipeds (Geyer et al, 2005;Daley and Biewener, 2006;Seyfarth et al, 2002). We therefore suggest that these parameters play an overall important role during terrestrial locomotion.…”

“…During perturbed locomotion, animals instead rely on passive dynamic mechanisms that include spring-mass mechanics and intrinsic mechanisms (see also Biewener and Daley, 2007). In humans and birds, simple spring-mass mechanics mitigate sudden changes in terrain height (Daley and Biewener, 2006;Geyer et al, 2005;Grimmer et al, 2008;Seyfarth et al, 2002). Limbs act as springs and help the system to return to the locomotor trajectory between steps.…”

“…Limbs act as springs and help the system to return to the locomotor trajectory between steps. Interestingly, bipeds maintain passive spring-mass dynamics when limb contact angle, effective limb length and limb stiffness exhibit particularly limited ranges (Seyfarth et al, 2002;Geyer et al, 2005;Grimmer et al, 2008). In addition, intrinsic mechanisms, including force-length, force-velocity and history-dependent properties as well as postural effects on joint dynamics at the musculoskeletal level help to reduce the control effort and consequently the complexity of the nervous system (see also Biewener and Daley, 2007).…”

A Dynamical Systems Approach to Learning: A Frequency-Adaptive Hopper Robot

“…Spring-mass systems have been widely used to study fundamental aspects of locomotion [3,11] and several robots based on this concept have been presented [17,8,5]. In [12] the mechanical stability of spring-mass systems is discussed. Recently, robots with legs including elastic elements have been presented [10,13,14].…”

Stride-to-stride energy regulation for robust self-stability of a torque-actuated dissipative spring-mass hopper

“…Our approach is based on recently proposed analytic approximations to the dynamics of a dissipative SLIP model, 24 which can capture the effects of viscous damping in the leg much more accurately than previously available methods in the literature that rely on the conservation of energy. [25][26][27][28] Since trajectories of this dissipative model lack symmetry properties necessary to indirectly deduce stability properties without an explicit return map, 29,30 we use our analytic approximations to generalize previous uses of Poincaré methods 26,[31][32][33][34] to the stability analysis of running with a dissipative spring-mass model.…”