Geodesic
Discussion on angular asymmetry in the solutions of SLIP model
Mathematica Applicanda, Tome 51 (2023) no. 2, pp. 223-237.

Voir la notice de l'article provenant de la source Annales Societatis Mathematicae Polonae Series

We consider a spring-mass model of human running which is built upon an inverted elastic pendulum. The model itself consists of two sets of differential equations - one set describes the motion of the centre of mass of a runner in contact with the ground (support phase), and the second set describes the phase of no contact with the ground (flight phase). In our previous approach, we assumed that periodic solutions in the support phase are symmetrical with respect to the touch-down and take-off angles for the large spring constant (or small angle of attack). Based on proposed solutions, we introduce analytical approximations of an asymmetrical boundary value problem, which brings our model closer to real running. By appropriately concatenating asymptotic solutions for the two gait phases, we are able to reduce the dynamics to a one-dimensional apex to apex return map and then to investigate the existence and stability of periodic solutions. Unlike in the symmetrical version, we could not find sufficient conditions for this map to have a unique stable fixed point. Extending the model with the possibility of taking off with the angle other than during landing, the aforementioned asymmetry, is necessary in the context of real run considerations. Thanks to this, our work could be enriched by experimental results. In this paper, we will present the possible reasons for the instability of asymmetric solutions in conjunction with conclusions from the observation of real runs.
DOI : 10.14708/ma.v51i2.7194
Classification : 37N20, 70K20, 92C10
Mots-clés : spring-mass model, running, approximate solution, fixed point 
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Piotr Kowalczyk; Łukasz Płociniczak; Zofia Wróblewska. Discussion on angular asymmetry in the solutions of SLIP model. Mathematica Applicanda, Tome 51 (2023) no. 2, pp.  223-237. doi : 10.14708/ma.v51i2.7194. http://geodesic.mathdoc.fr/articles/10.14708/ma.v51i2.7194/

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