Hurmuzlu Y, Basdogan C. On the measurement of dynamic stability of human locomotion. J Biomech Eng. 1994;116(1):30–6. pmid:8189711 with being the average time of one cycle within the one-minute interval Δ t. δ h is the phase, which within a simulation is chosen randomly being any number between zero and 2 π. h specifies the number of harmonics contributing, with m being the highest one. The maximal harmonic is identified from the Fourier transform of a subject’s movement. t T denotes the time for the transient effect decreasing to e −1. The transient effect averaged over the n th minute is Let t {\displaystyle t} represent time and let f ( t , ⋅ ) {\displaystyle f(t,\cdot )} be a function which specifies the dynamics of the system. That is, if a {\displaystyle a} is a point in an n {\displaystyle n} -dimensional phase space, representing the initial state of the system, then f ( 0 , a ) = a {\displaystyle f(0,a)=a} and, for a positive value of t {\displaystyle t} , f ( t , a ) {\displaystyle f(t,a)} is the result of the evolution of this state after t {\displaystyle t} units of time. For example, if the system describes the evolution of a free particle in one dimension then the phase space is the plane R 2 {\displaystyle \mathbb {R}

As known from previous studies [ 21, 22] the influence of morphing and transient effect is small compared with the differences between individuals. While morphing is present in all trials, the transient effect is not observable in all cases (20 of 25 cases for running, 8 of 25 cases for biking). For biking, the transient effect is less prominent compared to running. We suspect the fixation of the legs with the foot connected to the pedal and the hip very much fixed onto the saddle, there is limited freedom in movement variation. The tibia position and its associated acceleration is often settled onto the attractor from the start onwards. A different situation is seen in running, where the kinematic chain is unfixed near the location of the accelerometer at the distal end of the tibia. Here the probability to start a movement close to the subject’s attractor, resulting in no visible transient effect, is small. Interestingly, the most experienced runners show the least transient effect. at attractor point j. Here b is the controlling constant and σ k( j) the attractor’s standard deviation, which is divided by the average of the attractor’s deviation 〈 σ k〉. This takes care of the changing width of the acceleration bundle. The correction term, being activated at time t b, is modeled as The transient effect is a temporary oscillation around the attractor at the beginning of a cyclic movement. The starting value of the oscillation might be very individual, specific to the subject, and having a part of the starting value occurring by sheer chance. We model the deviation as the solution of a damped harmonic oscillator, where the transient term can be viewed as the departure from the morphed attractor

Method

Discover a faster, simpler path to publishing in a high-quality journal. PLOS ONE promises fair, rigorous peer review, A dynamical system is generally described by one or more differential or difference equations. The equations of a given dynamical system specify its behavior over any given short period of time. To determine the system's behavior for a longer period, it is often necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computers. 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. The purpose of this paper was to find a quantitative description of cyclic motion with the capacity to simulate individuals’ characteristic movement. A model was proposed consisting of six contributing parts. Individual attractor, morphing, short time fluctuation, transient effect, control mechanism and sensor noise. Simulations based on this model showed the same distinctive variations as the measured data. In all cases the similarity analysis of same subjects produced higher results— and —compared with different subject combinations— and . Measurements of the respective simulations are clearly identifiable, confirming the model’s suitability for describing cyclic motion. The nine constants together with the subject’s attractor approximations are characteristic for a person’s movement and the influence of the recording sensors. A time-dependent individual attractor morphing is described as the attractor change from start t S to end t E minute by minute. The equation is of heuristic nature. It must be capable of describing the changes of a given attractor and its development to the final attractor as a function of time. We take care of this process by taking attractor approximations at beginning and end and describe the morphing of the two attractor approximations, introducing the three dimensionless constants a 0, a 1, a 2, by

The comparison of a subject’s attractors of a 1-hour measurement to an independent super attractor allows approximation of the magnitude of morphing. The maximal difference between attractors from independent measurements of one subject is restricted by the maximal possible morphing. Morphing can deform an attractor in many different ways, which most probably results in δM’s of comparable values. Therefore, results as shown in Figs 7 and 8 might represent good approximations of typical morphing magnitudes. Still, the determination of the attractor remains a challenging issue. In mathematical systems, like the famous “Lorenz map”, the attractor is reached after the transient effect subsided. There, either a stable regular attractor is reached or a strange one is seen. Here, although data of the cyclic motion never completely reaches regularity, neither is the behavior completely chaotic. The regularity is, as mentioned before, good enough to discriminate between individuals.

Conclusion

Nashner LM. Balance adjustments of humans perturbed while walking. J Neurophysiol. 1980;44(4):650–64. pmid:7431045

Broscheid KC, Dettmers C, Vieten M. Is the Limit-Cycle-Attractor an (almost) invariable characteristic in human walking? Gait Posture. 2018;63:242–7. pmid:29778064 Fluctuation in the form of a “random walk”. These are changes around a morphed attractor described with the iteration 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]

References

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. 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 Fig 1. Schematic two-dimensional depiction of the three-dimensional recognition horizon (red) and compared attractor (blue).