Super Attractor: Methods for Manifesting a Life beyond Your Wildest Dreams

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Sehle A, Mündermann A, Starrost K, Sailer S, Becher I, Dettmers C, et al. Objective assessment of motor fatigue in multiple sclerosis using kinematic gait analysis: a pilot study. Journal of NeuroEngineering and Rehabilitation. 2011;8(1):59. Hausdorff JM, Zemany L, Peng C, Goldberger AL. Maturation of gait dynamics: stride-to-stride variability and its temporal organization in children. J Appl Physiol (1985). 1999;86(3):1040–7.

Dingwell JB, Kang HG. Differences between local and orbital dynamic stability during human walking. J Biomech Eng. 2007;129(4):586–93. pmid:17655480 A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor, forward in time. The trajectory may be periodic or chaotic. If a set of points is periodic or chaotic, but the flow in the neighborhood is away from the set, the set is not an attractor, but instead is called a repeller (or repellor).In finite-dimensional systems, the evolving variable may be represented algebraically as an n-dimensional vector. The attractor is a region in n-dimensional space. In physical systems, the n dimensions may be, for example, two or three positional coordinates for each of one or more physical entities; in economic systems, they may be separate variables such as the inflation rate and the unemployment rate. [ not verified in body] Strogatz SH. Nonlinear dynamics and Chaos: with applications to physics, biology, chemistry, and engineering. Reading, Mass.: Addison-Wesley Pub.; 1994. xi, 498 p. p. Visual representation of a strange attractor. [1] Another visualization of the same 3D attractor is this video. Code capable of rendering this is available. Hurmuzlu Y. Dynamics of bipedal gait: Part II—Stability analysis of a planar five-link biped. Journal of Applied Mechanics. 1993;60(2):337–43.

When simulating the kinematics and comparing it with real life data, we need to include the measurement error–noise —caused by the sensor characteristics. It can be obtained directly from measuring the output signals of the sensors at rest. The signal of an accelerometer is, subtracting the values caused by the earth’s gravitational field, modeled as white noise. (13) In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, [2] for a wide variety of starting conditions of the system. System values that get close enough to the attractor values remain close even if slightly disturbed. Clermont CA, Benson LC, Osis ST, Kobsar D, Ferber R. Running patterns for male and female competitive and recreational runners based on accelerometer data. Journal of sports sciences. 2019;37(2):204–11. pmid:29920155 Loske S, Nuesch C, Byrnes KS, Fiebig O, Scharen S, Mundermann A, et al. Decompression surgery improves gait quality in patients with symptomatic lumbar spinal stenosis. Spine J. 2018.Wilson RC, Jones PW. A comparison of the visual analogue scale and modified Borg scale for the measurement of dyspnoea during exercise. Clinical Science. 1989;76(3):277–82. pmid:2924519 For this reason, we propose a mathematical model of the kinematic of the human cyclic motion based on acceleration data. It allows simulation of cyclic movement and comparison with measured data. We illustrate this model as a superposition of six mathematical terms covering the motion as a (1) limit-cycle attractor, (2) individual attractor morphing, (3) short time random fluctuation in form of “random walk”, (4) the transient effect describing initial oscillations around the attractor at the onset of the activity subsiding with increasing time, (5) a control process being activated when stride variations tend to exceed the morphed attractors’ boundaries, and (6) the influence of noise generated by the measurement device—accelerometers. Thus, this model allows extension of earlier findings specifically about the variability of subjects’ cyclic movement with its fixed and random components. Fig 8. All 5 bike trials of all 5 subjects compared to their personal, but independent super attractor for minutes 11 to 60.

Data Availability: All data files are available from zenodo.org under the direct link http://doi.org/10.5281/zenodo.3518415.Bipedal gait, especially walking, has been the most decisive development of homo sapiens to surpass their ancestors and relatives [ 1]. In the past centuries further cyclic motions like swimming, cycling, rowing or skiing came along, to overcome natural obstacles, to facilitate traveling and then as leisure activities. Recently, cyclic motion descriptions have served as biological templates for developments in robotics together with developments in artificial intelligence [ 2]. Although cyclic movements are performed a thousand-fold each day in everyday life, their underlying composition and structure is not fully understood.