Synthesis of two-dimensional human walking: a test of the λ-model

Abstract

To test the lambda-model version of the equilibrium point hypothesis both for feasibility and validity with respect to the control of terrestrial locomotion, we developed a two-dimensional, eleven-segment musculoskeletal model of the human body including 14 muscle-tendon complexes per leg, three-segment feet, and a physiologically based model of foot-ground interaction. Human walking was synthesized by numerical integration of the coupled muscle-tendon and rigid body dynamics. To this end a control algorithm based on the lambda-model was implemented in the model providing muscle stimulation patterns that guaranteed dynamically stable walking including a balanced trunk. Thus, the timing of the movement is not preset by a central pattern generator but emerges from the interaction of the musculoskeletal system with the control algorithm. The control parameters were found in a trial-and-error approach. The feedforward part of the control scheme consists of just two target configurations each of which is composed of a set of one nominal length per muscle (lambda-model). Variation of gravity reveals that (1) the synthesized walking patterns are close to ballistic walking and (2) this muscularly induced natural walking can only be initiated and maintained in the range between about a tenth and three times earth-bound gravity. Our walking patterns are robust both against parameter variations and shuffling of the swing leg. We discuss our model with respect to gravity scaling, speed control, feedback delay, and the terms "equilibrium point hypothesis" and "central pattern generator."

Robots in human biomechanics—a study on ankle push-off in walking

“…For ground interaction, a modified version of the model described in [32] was applied, generating contact forces at the heel and toe of each foot based on viscoelastic elements. Parameters for the motors' electromechanical properties were experimentally identified and integrated [33] in a separate motor model for each motor type.…”

A macroscopic ansatz to deduce the Hill relation

“…For example, it might constitute a thermodynamical state equation or a fairly complex visco-elastic force law. In numerical neuro-musculo-skeletal model simulations Hill-type models are usually used in the force law sense (Günther and Ruder, 2003;Günther et al, 2007;Houdijk et al, 2006;Kistemaker et al, 2006;van Soest and Bobbert, 1993) interpreting it uni-directionally, with the force given and the respective velocity constituting the model response.…”

What does head movement tell about the minimum number of mechanical degrees of freedom in quiet human stance?

“…Experimentally, here we found mean absolute vertical amplitudes of ankle markers in quiet stance of about 0.07 mm and typical maximum peak-to-peak vertical ground reaction force fluctuations of about ±5 N underneath each single foot and ±1 N for the sum of both feet. These data can be perfectly reproduced by computer simulations of quiet human stance using a more complex two-legged eleven-body neuro-musculo-skeletal model of the human [18]. This latter model incorporates all the foot properties mentioned above.…”

“…This latter model incorporates all the foot properties mentioned above. For example, the fatty pads in the foot are modelled according to physiological data in which the resistive force is typically related about cubically to the compressive deformation [11,12,18,36]. Thus, the eleven-body model including a such parameterised foot is consistent with the measured vertical ankle displacements and force fluctuations.…”

“…This choice enabled comparison of simulation results amongst both models and the experimental data; we provided practically the same value of mean horizontal COM displacement amplitudes (bottom of Table 1: almost 1 mm). The models were simulated with in-house software "simsys" [18,19,23], coded in "C" language, calling routines "dgefa" and "dgesl" (adopted from the "LINPACK" package), and using the variable order, variable step size integrator "de" [37] with 10 −6 relative and absolute error tolerances. "simsys" generates the code of any model's equations of motion in two dimensions (here, in the sagittal plane) and in terms of the Newton-Euler formalism.…”