Short communicationA mathematical analysis to address the 6 degree-of-freedom segmental power imbalance
Introduction
A segmental power analysis is a useful biomechanical tool (Caldwell and Forrester, 1992), which has been used in analyzing human movement to indicate the source and net rate of energy transfer (flow) between the rigid bodies of biomechanical models (Aleshinsky, 1986, Robertson and Winter, 1980, van Ingen Schenau and Cavanagh, 1999). Segmental power calculations utilize segment endpoint dynamics (kinetic method), but a theoretically equivalent method is to measure changes in the segment’s energy state (kinematic method) (Zajac et al., 2002). Several researchers have used independent measures of segmental power to explain how power flow between segments relates to changes in the energy state of the segments in activities like walking (Aleshinsky, 1986, Caldwell and Forrester, 1992, Robertson and Winter, 1980, Zelik et al., 2015), pedaling (Kautz et al., 1994, Kautz and Neptune, 2002), running (Caldwell and Forrester, 1992), wheelchair propulsion (Guo et al., 2003), lifting (De Looze et al., 1992), and various endurance sports (van Ingen Schenau and Cavanagh, 1999). Researchers have also used this mathematical equivalence to assess the accuracy of specific models (McGibbon and Krebs, 1998) based on how closely powers calculated using the kinetic method match with those using the kinematic method. Several investigators theorized the kinematic method is more accurate as it is based only on motion and anthropometric estimates (Caldwell and Forrester, 1992, Robertson and Winter, 1980).
The models and corresponding model assumptions used to analyze segmental power flow influence how results may be interpreted. A pin-joint model, which fixes segment ends at a coincident point, has been used for two- (i.e. sagittal plane) or three-dimensional gait analyses (De Looze et al., 1992, McGibbon and Krebs, 1998, Robertson and Winter, 1980). However, use of the pin-joint model may require segment lengths and inertial alignment (e.g. segment center of mass position) to change due to a shared joint center with adjacent segments, thus violating rigid body assumptions. Conversely, a 6 degree-of-freedom (6 DOF) model for three-dimensional gait analyses fixes segment characteristics, which can lead to relative displacement between adjacent segment ends, and thus non-equivalent segment endpoint velocities at a joint (Buczek et al., 1994, McGibbon and Krebs, 1998). While both models have limitations, the translational power resulting from the intersegmental joint force and the segment endpoint velocities in a 6 DOF model is valuable to include for a complete mechanical energy analysis of human gait (Buczek et al., 1994, Geil et al., 2000, Zelik et al., 2015).
Independent of chosen model, the kinetic and kinematic methods typically do not provide experimentally equivalent results, leading to a “power imbalance” (PI) (McGibbon and Krebs, 1998). Using a three-dimensional analysis, McGibbon and Krebs reported using the pin-joint model resulted in a mean absolute PI over stance ranging from 9.9–25.6 W for the shank and 6.8–23.4 W for the thigh. The mean absolute PI was reduced when segment lengths were fixed and radial velocities of the distal and proximal ends of the segment relative to the segment’s center of mass were accounted for (1.1–5.0 W and 0.7–4.1 W for the shank and thigh, respectively). However, while fixed segment lengths reduced the PI within a segment, there was a large power discrepancy between segment ends across a joint (e.g. 10.7–37.8 W at the knee), which was considered an “energy well” (McGibbon and Krebs, 1998).
Thus, identifying the source of the PI is important for effectively characterizing energetic measures in the study of human movement. To date, the foot is the only segment whose PI was computationally accounted for by the inclusion of a calculation for distal foot segmental power (Siegel et al., 1996).
The purpose of this study was to determine the source of PI by conducting a mathematical analysis to equate the kinematic and kinetic methods for a 6 DOF model. We theorized accounting for power due to relative displacement between the distal end of a segment and the joint center in the kinetic model (relative displacement power) would reduce the PI. We then experimentally characterized the PI with and without accounting for the relative displacement power.
Section snippets
Computational development
Using Newton-Euler formulas (Siegler and Liu, 1997) in inverse dynamics calculations (Robertson et al., 2013), the general form for the proximal joint intersegmental force () for any segment m, linked by n number of segments, is given by Eq. (1) where , , , and represent the segment mass, segment center of mass acceleration, gravity (9.81 m/s2), and ground reaction force, respectively. Similarly, the proximal net joint moment () is given by Eq. (2) where , , , and
Experimental method
Experimental data were derived from a coded database of nine healthy subjects (34 ± 10 years, 1.69 ± 0.10 m, 75.6 ± 16.2 kg), consented under an IRB approved protocol, walking with standard shoes on an instrumented treadmill (Bertec Corp., Columbus, OH). Kinematic data were collected using a seven-camera motion capture system (Motion Analysis, Santa Rosa, CA). Motion capture and force data were sampled at 240 Hz and 1200 Hz and low-pass filtered at 6 Hz and 25 Hz, respectively, and analyzed in
Results
The experimental segmental powers (Fig. 2) and PI (Fig. 3, Fig. 4) revealed accounted for nearly all . The average absolute segmental PI was reduced from 0.046 ± 0.015 W/kg, 0.034 ± 0.008 W/kg, and 0.023 ± 0.015 W/kg for the shank, thigh and pelvis, respectively, using the anatomical definition to ≤0.001 ± 0.000 W/kg using the joint center definition in the kinetic method. For context, the percent difference between these two measures was 98.4%, 95.7%, and 95.6% for the shank,
Discussion
The mathematical analysis presented explains how the segmental PI between segmental power and rate of energy change is influenced by the definition of the distal translational velocity term. An AR definition of the distal translational velocity ignores the relative displacement of segment ends at a joint, resulting in a PI. A JC definition includes a relative displacement power () to accurately equate segmental power and rate of energy change mathematically.
The term computationally
Conflict of interest
The authors have no financial or personal relationships with individuals or organizations that inappropriately influenced this work. All authors have no conflicts of interest to disclose.
Acknowledgements
This material was based upon work supported by the National Science Foundation (NSF) Graduate Research Fellowship Grant No. 1247394, the Center for Research in Human Movement Variability of the University of Nebraska at Omaha and the National Institute of Health (P20GM109090), the University of Delaware College of Health Sciences, and the Mechanical Engineering Department Helwig Fellowship. It was also supported by the BADER consortium, a Department of Defense Congressionally Directed Medical
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