The role of intrinsic muscle properties for stable hopping—stability is achieved by the force–velocity relation

Abstract

A reductionist approach was presented to investigate which level of detail of the physiological muscle is required for stable locomotion. Periodic movements of a simplified one-dimensional hopping model with a Hill-type muscle (one contractile element, neither serial nor parallel elastic elements) were analyzed. Force-length and force-velocity relations of the muscle were varied in three levels of approximation (constant, linear and Hill-shaped nonlinear) resulting in nine different hopping models of different complexity. Stability of these models was evaluated by return map analysis and the performance by the maximum hopping height. The simplest model (constant force-length and constant force-velocity relations) outperformed all others in the maximum hopping height but was unstable. Stable hopping was achieved with linear and Hill-shaped nonlinear characteristic of the force-velocity relation. The characteristics of the force-length relation marginally influenced hopping stability. The results of this approach indicate that the intrinsic properties of the contractile element are responsible for stabilization of periodic movements. This connotes that (a) complex movements like legged locomotion could benefit from stabilizing effects of muscle properties, and (b) technical systems could benefit from the emerging stability when implementing biological characteristics into artificial muscles.

Preparing the leg for ground contact in running: the contribution of feed-forward and visual feedback

“…While the most reduced models of walking and running -compass gait model and spring mass model -showed that stable walking is possible without the need for any adaptation (Blickhan, 1989;Garcia et al, 1998;Geyer et al, 2006;Goswami et al, 1998;Seyfarth et al, 2002;Srinivasan and Ruina, 2006), some of their offspring models emphasize the benefit of feed-forward timed adaptation of leg characteristics (Birn-Jeffery and Daley, 2012;Blum et al, 2010;Ernst et al, 2012;Karssen et al, 2011) and muscle activity (Haeufle et al, 2010) for the stabilization of locomotion patterns. As our results clearly indicate such a time-dependent feed-forward change in muscle activity in the absence of visual feedback (Fig.…”

Evaluating Morphological Computation in Muscle and DC-Motor Driven Models of Hopping Movements

“…More precisely, the higher MC in the MusFib model can be attributed to the force-velocity characteristics of the muscle fibers. Previous hopping simulations showed that these non-linear contraction dynamics reduce the influence of the controller on the actual hopping kinematics (Haeufle et al, 2010(Haeufle et al, , 2012. This means, that the kinematic trajectory is more predetermined by the material properties as compared to the linearized muscle model MusLin, or the motor model DCMot.…”

“…In a reduced model, hopping motions can be described by a one-dimensional differential equation (Haeufle et al, 2010):…”

“…In the MusFib model, the leg force is modeled to incorporate the active muscle fibers' contraction dynamics. The model has been motivated and described in detail elsewhere (Haeufle et al, 2010(Haeufle et al, , 2012(Haeufle et al, , 2014. In a nutshell, the material properties of the muscle fibers are characterized by two terms modulating the leg force The first term q(t) represents the muscle activity.…”

“…Here, we use a maximum isometric muscle force F max = 2.5 kN, an optimal muscle length l opt = 0.9 m, force-length parameters w = 0.45 m and c = 30, and force-velocity parametersl max = −3.5ms −1 , K = 1.5, and N = 1.5 (Haeufle et al, 2010). In this model, periodic hopping is generated with a controller representing a mono-synaptic force-feedback.…”