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Research Papers

# Local State Space Temporal Fluctuations: A Methodology to Reveal Changes During a Fatiguing Repetitive Task

[+] Author and Article Information

Department of Biomedical Engineering, Amirkabir University of Technology, 15875–4413, Tehran, Iransanjarima@alum.sharif.edu

Department of Biomedical Engineering, Amirkabir University of Technology, 15875–4413, Tehran, Iranarshi@aut.ac.ir

Department of Mechanical Engineering, Sharif University of Technology, 11155–9567, Tehran, Iran; Department of Industrial and Management Engineering, Hanyang University, Ansan Gyeonggi-do, 426–791, Republic of Koreaparnianpour@sharif.edu

Saeedeh Seyed-Mohseni

Biomechanics Laboratory, Rehabilitation Research Center, Faculty of Rehabilitation, Iran University of Medical Sciences, 15459–13487, Tehran, Iranseyedmohseni@iums.ac.ir

1

Corresponding author.

J Biomech Eng 132(10), 101002 (Sep 27, 2010) (9 pages) doi:10.1115/1.4002373 History: Received January 09, 2010; Revised August 09, 2010; Posted August 16, 2010; Published September 27, 2010; Online September 27, 2010

## Abstract

The effect of muscular fatigue on temporal and spectral features of muscle activities and motor performance, i.e., kinematics and kinetics, has been studied. It is of value to quantify fatigue related kinematic changes in biomechanics and sport sciences using simple measurements of joint angles. In this work, a new approach was introduced to extract kinematic changes from 2D phase portraits to study the fatigue adaptation patterns of subjects performing elbow repetitive movement. This new methodology was used to test the effect of load and repetition rate on the temporal changes of an elbow phase portrait during a dynamic iso-inertial fatiguing task. The local flow variation concept, which quantifies the trajectory shifts in the state space, was used to track the kinematic changes of an elbow repetitive fatiguing task in four conditions (two loads and two repetition rates). Temporal kinematic changes due to muscular fatigue were measured as regional curves for various regions of the phase portrait and were also expressed as a single curve to describe the total drift behavior of trajectories due to fatigue. Finally, the effect of load and repetition rate on the complexity of kinematic changes, measured by permutation entropy, was tested using analysis of variance with repeated measure design. Statistical analysis showed that kinematic changes fluctuated more (showed more complexity) under higher loads $(p=0.014)$, but did not differ under high and low repetition rates $(p=0.583)$. Using the proposed method, new features for complexity of kinematic changes could be obtained from phase portraits. The local changes of trajectories in epochs of time reflected the temporal kinematic changes in various regions of the phase portrait, which can be used for qualitative and quantitative assessment of fatigue adaptation of subjects and evaluation of the influence of task conditions (e.g., load and repetition rate) on kinematic changes.

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## Figures

Figure 1

Varying fatigue strategies evident in phase space. (a) The subject mainly altered her velocity to comply with the metronome when becoming fatigued. (b) The subject mainly altered her ROM. (c) Alteration in both velocity and ROM occurred. Test condition 3 corresponds to low load and high repetition rate while in test condition 4, both load and repetition rate were low.

Figure 3

(a) Changes in expected values of the epochwise trajectory for selected regions of Fig. 2, (b) plot of the first four SOVs, and (c) the first two SOCs corresponding to the SOVs. The first SOC (SOC1) was used for further analyses.

Figure 4

Plot of SOC1 in all four test conditions for all 15 subjects. These curves show the drift trend of phase space trajectories due to fatigue. Subject 3 was later omitted from statistical analysis because the second test condition was missing.

Figure 5

((a)–(c) and (e)–(g)) Epochwise plot of phase portrait trajectories of two subjects to show gradual drifts due to fatigue. The fresh epoch is shown in gray and is identical in all panels. Some selected subsequent epochs are shown for comparison with the fresh state. The first row of plots corresponds to subject 4, test condition 3 (low load and high repetition rate) as in Fig. 2. The subject in this condition showed a gradual decline of both ROM and peak velocities, so the fluctuation of SOC1 (d) was smooth, resulting in a low PEnt value compared with panel (h) of the other subject. The second row of plots corresponds to subject 13, test condition 4 (low load and low repetition rate) in which the subject showed a more complex behavior having nonmonotonic shrinkage/expansions of trajectories. Therefore, SOC1 had more fluctuations (h) resulting in a higher PEnt value compared with (d). The line style of panels (d) and (h) complies with Fig. 4. Refer to the Appendix for more details on computation of PEnt.

Figure 6

Interaction plot of the effect of load and repetition rate on endurance times or test durations (Tlim) for 14 subjects. Load had a significant effect on Tlim but the post hoc analysis showed that repetition rate affected the Tlim only at low load conditions (p=0.049) not at high load condition (p=0.152). For details of statistical results, see Table 1.

Figure 7

Negative slope of MDF of muscles showing fatigued state. The higher load resulted at more negative slope of MDF for all three muscles (p<0.004) while the repetition rate did not affect the slope of MDF of muscles.

Figure 8

PEnt of SOC1 grouped for load and repetition rate for all subjects. PEnt was significantly higher (p=0.014) for high load.

Figure 2

Phase portrait partitioning to track local kinematic changes. The epochwise analysis of trajectories allows measurement of temporal changes in each region in a discrete time scale. The dash-dotted trajectory is the fresh (first) epoch and the black solid one is the last fatigued epoch. White circles indicate expected values of the fresh trajectory in each region. Black circles are the expected values of trajectory data for subsequent epochs. Connecting black circles in each region constitutes regional curves (Fig. 3), which show the drift of the expected values from the fresh epoch to the last fatigued epoch. Regional curves were subject to SOD analysis to find dominant SOCs.

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