The majority of human body mass is due to its soft tissue,^{1} with that soft tissue having the potential to move relative to the underlying skeletal structures during locomotion.^{2} However, researchers typically assume each human body segment is rigid, thus ignoring intrasegment soft tissue motion when performing inverse dynamics analyses.^{3} Some simulation models of human locomotion have included a wobbling mass component. The inclusion of this model component has been necessary for the accurate estimation of ground reaction forces,^{4,5} but has not been exploited to examine the potential influence on the resultant joint moments.

Soft tissue motion occurs during human movement and yet our understanding of its role when performing inverse dynamics analyses of human motion is not clear, in particular for the analyses of movements which include an impact.^{6} It has been shown that soft tissue motion has a profound effect on intrasegmental dynamics just following foot–ground impact during a drop landing.^{7} The human body is made up of different types of soft tissue (eg, adipose tissue and muscle tissue), with varying elastic properties.^{8} The amount of the different types of soft tissue varies between individuals.^{1} The effects of the inertial and elastic properties of these soft tissue components on the dynamics of human motion are still largely unknown.

Forward dynamics models of running have included wobbling mass components,^{4,5} with the aim of more accurately reproducing ground reaction forces. Given that experimental studies have shown that walking energetics^{9,10} and running energetics^{11,12} are influenced by soft tissue motion, the question arises as to what influence does this soft tissue motion have on the dynamics of human running? Therefore, it was the purpose of this study to investigate the effects in running gait of the wobbling mass motion on resultant joint moments and powers.

## Methods

A hybrid inverse dynamics model including experimentally driven kinematics and simulated wobbling motion to account for the motions of soft tissue during the stance phase of running is presented. Experimental data are collected on one subject to quantify the potential effects of soft tissue motion on knee and hip joint dynamics. The details of the experimental methods, procedure for the performing of the inverse dynamics, and how the required soft tissue elastic properties were determined are presented.

### Experimental Methods

One healthy male adult participated in this study. All procedures were approved by the Penn State Institutional Review Board, and informed consent was obtained. The participant was 27 years old, 168 cm tall, had a body mass of 70 kg, and was 24.0% body fat. Body fat mass was determined from segment lengths, widths, circumferences, and skinfold measurements.^{13} The participant ran down a 20-m runway at 4.0 m·s^{−1}. The participant was a rearfoot striker and ran using their regular running shoes. Passive reflective markers (9 mm in diameter) were placed on the feet, shank, thigh, and pelvis and tracked at 300 Hz using a motion capture system (Motion Analysis Corporation, Santa Rosa, CA). There were 4 markers per segment to determine segment position and orientation, and an additional 15 markers on both the shank and thigh to quantify soft tissue motion. With a force plate (Kistler, Novi, MI) at the center of the runway ground reaction forces were measured for one step at 1200 Hz. Segment coordinate systems, segment angles, and joint angles were defined according to the International Society of Biomechanics standards.^{14–16} Marker trajectories were used to determine joint and segment positions and orientations based on a least squared method.^{17} All marker trajectories that were utilized to calculate the segment pose were low-pass filtered at 10 Hz, while force plate signals were low-pass filtered at 30 Hz using a second-order Butterworth filter applied in forward and backward directions. Segment inertial parameters were determined by the methods outlined in de Leva.^{18}

### Calculation of Joint Moments

Custom MATLAB code (Mathworks, Inc, Natick, MA) was used to calculate sagittal plane resultant joint forces, moments, and powers at the ankle, knee, and hip joints during the stance phase (Figure 1). Two separate models were examined, a rigid model and a model containing wobbling masses. The rigid model included 4 segments: the foot, shank, thigh, and pelvis segments. The wobbling mass model had 2 additional segments where the shank and thigh included wobbling masses, each connected to its rigid counterpart via a linear spring-damper system. Each spring-damper system included a translational and torsional spring damper to constrain the motion of the wobbling mass. The model rigid and wobbling components were connected at their corresponding center-of-mass locations.

The motion of the wobbling masses was determined by a forward dynamic simulation. The kinematics of the rigid components were determined by the inverse kinematic analysis outlined in the previous subsection. Solution of a series of ordinary differential equations using the rigid components kinematics gave wobbling mass motion. The ordinary differential equations that dictate the motion of the wobbling component of each segment, S, are,

where *g* is the gravitational constant (9.81 m·s^{−1}). The determination of the elastic parameters is discussed in the following subsection.

The kinematics of the wobbling components just prior to touchdown are assumed to be the same as that of their corresponding rigid components. For the stance phase, wobbling mass kinematics of each segment were calculated by numerically solving Equations 1, 2, and 3 using the MATLAB ordinary differential equation solver ode113.^{19} The resultant spring-damper forces and moments were utilized while performing the inverse dynamics analysis.^{20} The resultant spring-damper forces of the thigh and shank acted at the center of mass of the rigid thigh and shank segments, respectively, and were treated as external forces for purposes of inverse dynamics. For example, the wobbling shank spring-damper forces and moments were included in the inverse dynamic calculation of the resultant reaction forces and moments at the knee.

### Elastic Parameter Determination

^{21}

**, at each frame,**

*F**i*, as follows,

*n*is the number of markers,

**x**

_{j}is a 3 × 1 vector describing the position of the

*j*th marker during the running trial,

*i*,

**X**

_{j}is a 3 × 1 vector describing the position of the

*j*th marker in the reference configuration determined during a static trial, and

^{22}A continuous wavelet transform with a Morse mother wavelet was utilized to visually inspect the time–frequency profile of the volume ratio and to determine the damped natural frequency,

^{23}The natural frequency (

^{24}

*ξ*is the damping ratio and

*m*is the mass of the wobbling mass. To calculate the torsional spring-damper parameters, the mass

To investigate the effects of varying the elastic properties of the wobbling mass, given that these are not known precisely, the natural frequency, damping ratio, and wobbling mass were systematically changed. The natural frequency, see Equation 5, was varied as 67%, 100%, and 133% of the experimentally determined values. Human soft tissue oscillates between one and 3 times before vibrations decay^{8}; therefore, the damping ratio was given values of 0.1, 0.2, and 0.3 to correspond to this elastic behavior. The mass and moment of inertia of the wobbling component was varied as their proportion of total segment mass and moment of inertia, using values of 20%, 40%, and 60%. This resulted in a total of 27 spring mass-damper combinations.

The Pearson correlation coefficient,^{25} a measure of linear correlation, was used to calculate the association between each wobbling parameter on the net negative mechanical work, net positive mechanical work, and total mechanical work produced at each joint.

## Results

The change in volume ratio indicated by the deformation gradient tensors of the thigh and shank indicate large volume changes during stance compared with the swing phase (Figure 2A and 2B). The corresponding continuous wavelet transforms shows that the soft tissue of the shank and thigh deformed at a damped frequency of approximately 18 and 16 Hz, respectively (Figure 2C and 2D). This set the range for the damped natural frequency of the wobbling masses at 12 to 24 Hz and 10 to 20 Hz for the shank and thigh, respectively.

—(A) The change in volume ratio as calculated from the thigh markers. (B) The change in volume ratio as calculated from the shank markers. (C) The continuous wavelet transform of the thigh volume ratio. (D) The continuous wavelet transform of the shanks volume ratio. In all 4 figures, the pair of vertical lines indicate heel strike and toe-off in the gait cycle. The color bars in (C) and (D) indicate the normalized amplitude of the volume changes.

Citation: Journal of Applied Biomechanics 38, 2; 10.1123/jab.2021-0051

—(A) The change in volume ratio as calculated from the thigh markers. (B) The change in volume ratio as calculated from the shank markers. (C) The continuous wavelet transform of the thigh volume ratio. (D) The continuous wavelet transform of the shanks volume ratio. In all 4 figures, the pair of vertical lines indicate heel strike and toe-off in the gait cycle. The color bars in (C) and (D) indicate the normalized amplitude of the volume changes.

Citation: Journal of Applied Biomechanics 38, 2; 10.1123/jab.2021-0051

—(A) The change in volume ratio as calculated from the thigh markers. (B) The change in volume ratio as calculated from the shank markers. (C) The continuous wavelet transform of the thigh volume ratio. (D) The continuous wavelet transform of the shanks volume ratio. In all 4 figures, the pair of vertical lines indicate heel strike and toe-off in the gait cycle. The color bars in (C) and (D) indicate the normalized amplitude of the volume changes.

Citation: Journal of Applied Biomechanics 38, 2; 10.1123/jab.2021-0051

The maximum simulated wobbling displacement relative to the underlying rigid segments occurred at the lowest frequency (10–12 Hz) and highest wobbling component inertial properties (75% of total segment mass and moment of inertia) combinations (Table 1). The maximum wobbling displacement of the shank and thigh segments relative to the underlying rigid segments was <1.5 and 2.0 cm, respectively. The maximum wobbling angular displacement of the shank and thigh segments relative to the underlying rigid segments was <5° and 6°, respectively.

Model Parameters and Hip Joint Work

Wobbling mass/moment of inertia | ξ | Negative hip work (% diff.) | Positive hip work (% diff.) | Total hip work (% diff.) | ||
---|---|---|---|---|---|---|

%Total | Hz | Hz | None | J (%) | J (%) | J (%) |

— | — | — | — | −4.6 (—) | 17.0 (—) | 12.4 (—) |

20 | 12 | 10 | 0.1 | −6.5 (−41.8) | 20.6 (21.1) | 14.0 (13.4) |

20 | 12 | 10 | 0.2 | −5.9 (−28.6) | 19.7 (16.0) | 13.8 (11.4) |

20 | 12 | 10 | 0.3 | −5.6 (−21.7) | 19.2 (13.0) | 13.6 (9.8) |

20 | 18 | 15 | 0.1 | −5.1 (−11.8) | 17.3 (2.1) | 12.2 (−1.6) |

20 | 18 | 15 | 0.2 | −5.0 (−9.6) | 17.4 (2.4) | 12.3 (−0.3) |

20 | 18 | 15 | 0.3 | −4.9 (−7.5) | 17.4 (2.5) | 12.5 (0.7) |

20 | 24 | 20 | 0.1 | −4.6 (−0.6) | 17.1 (0.9) | 12.5 (1.1) |

20 | 24 | 20 | 0.2 | −4.7 (−2.5) | 17.2 (1.3) | 12.5 (0.8) |

20 | 24 | 20 | 0.3 | −4.7 (−2.8) | 17.2 (1.3) | 12.5 (0.7) |

40 | 12 | 10 | 0.1 | −8.7 (−89.3) | 24.4 (43.8) | 15.7 (26.8) |

40 | 12 | 10 | 0.2 | −7.3 (−59.5) | 22.5 (32.7) | 15.2 (22.7) |

40 | 12 | 10 | 0.3 | −6.7 (−45.1) | 21.5 (26.6) | 14.8 (19.7) |

40 | 18 | 15 | 0.1 | −5.7 (−24.1) | 17.7 (4.2) | 12.0 (−3.2) |

40 | 18 | 15 | 0.2 | −5.5 (−20.2) | 17.8 (5.0) | 12.3 (−0.6) |

40 | 18 | 15 | 0.3 | −5.3 (−15.9) | 17.9 (5.3) | 12.5 (1.4) |

40 | 24 | 20 | 0.1 | −4.7 (−1.8) | 17.3 (2.0) | 12.6 (2.1) |

40 | 24 | 20 | 0.2 | −4.9 (−5.8) | 17.4 (2.7) | 12.6 (1.5) |

40 | 24 | 20 | 0.3 | −4.9 (−6.1) | 17.4 (2.7) | 12.6 (1.5) |

60 | 12 | 10 | 0.1 | −11.1 (−141.7) | 28.5 (67.7) | 17.3 (40.2) |

60 | 12 | 10 | 0.2 | −8.9 (−93.1) | 25.5 (50.1) | 16.6 (34.1) |

60 | 12 | 10 | 0.3 | −7.8 (−69.8) | 23.8 (40.4) | 16.0 (29.5) |

60 | 18 | 15 | 0.1 | −6.3 (−36.8) | 18.1 (6.5) | 11.8 (−4.7) |

60 | 18 | 15 | 0.2 | −6.0 (−31.1) | 18.3 (7.8) | 12.3 (−0.9) |

60 | 18 | 15 | 0.3 | −5.7 (−24.7) | 18.4 (8.2) | 12.6 (2.1) |

60 | 24 | 20 | 0.1 | −4.8 (−3.7) | 17.5 (3.3) | 12.8 (3.2) |

60 | 24 | 20 | 0.2 | −5.0 (−9.1) | 17.7 (4.1) | 12.7 (2.3) |

60 | 24 | 20 | 0.3 | −5.0 (−9.5) | 17.7 (4.2) | 12.6 (2.2) |

Note: All 27 variations of the wobbling mass model parameters are described. The net negative, net positive, and total mechanical work at the hip joint are reported, with the percentage differences between hip joint work between each wobbling model with respect to the rigid model are in parentheses.

Knee joint kinetics were nearly identical between the rigid and wobbling mass model for all wobbling model parameters (Figure 3). The negative mechanical work and positive mechanical work at the knee joint for the rigid model was −60.7 and 16.4 J, respectively. For all wobbling mass parameter combinations, the negative mechanical work and positive mechanical work at the knee joint was within 1% of that of the rigid model. Due to the near identical knee joint kinetics, the remainder of the results with regards to the joint kinetics will focus on the hip joint.

—The (A) knee joint moment and (B) knee joint power. The thick gray line and black dashed line denote the rigid model and the wobbling model, respectively. The wobbling model parameters are as follows: damping ratio at 0.17, wobbling mass/inertia at 50% of total segment mass/inertia, and wobbling frequency at the midfrequency range of 15 to 18 Hz.

Citation: Journal of Applied Biomechanics 38, 2; 10.1123/jab.2021-0051

—The (A) knee joint moment and (B) knee joint power. The thick gray line and black dashed line denote the rigid model and the wobbling model, respectively. The wobbling model parameters are as follows: damping ratio at 0.17, wobbling mass/inertia at 50% of total segment mass/inertia, and wobbling frequency at the midfrequency range of 15 to 18 Hz.

Citation: Journal of Applied Biomechanics 38, 2; 10.1123/jab.2021-0051

—The (A) knee joint moment and (B) knee joint power. The thick gray line and black dashed line denote the rigid model and the wobbling model, respectively. The wobbling model parameters are as follows: damping ratio at 0.17, wobbling mass/inertia at 50% of total segment mass/inertia, and wobbling frequency at the midfrequency range of 15 to 18 Hz.

Citation: Journal of Applied Biomechanics 38, 2; 10.1123/jab.2021-0051

Hip joint kinetics were affected by including a wobbling model component (Figure 4). The wobbling frequency had the largest effect on the hip joint kinetics. The linear correlation coefficients between the wobbling frequency and the net negative work, net positive work, and total work calculated at the hip joint were .76 (*P* < .01), –.78 (*P* < .01), and –.75 (*P* < .01), respectively. The inertial parameters of the wobbling component had the second largest effect on the hip joint kinetics. The linear correlation coefficients between the inertial properties of the wobbling component and the net negative work, net positive work, and total work calculated at the hip joint were –.40 (*P* = .04), .34 (*P* = .08), and .27 (*P* = .18), respectively. The damping ratio had little effect on the hip joint kinetics. The linear correlation coefficients between the damping ratio and the net negative work, net positive work, and total work at the hip joint were .19 (*P* = .32), –.12 (*P* = .54), and .04 (*P* = .86), respectively. The negative mechanical work and positive mechanical work at the hip joint for the rigid model was −4.6 and 17.0 J, respectively. At the experimentally determined wobbling frequencies (ie, the midfrequency wobbling condition), the magnitude of the negative hip joint work increased by 7.5% to 36.8%, depending on the other wobbling parameters (Table 1). The magnitude of the positive hip joint work increased by 2.1% to 8.2%.

—The hip joint moments at (A) low-, (B) mid-, and (C) high-frequency wobbling mass parameters. The hip joint power at (D) low-, (E) mid-, and (F) high-frequency wobbling mass parameters. The gray thick, black dotted, black dashed, and black thin lines denote the rigid model, 20% wobbling mass/inertia model, 40% wobbling mass/inertia model, and 60% wobbling mass/inertia model, respectively. The damping ratio for all plots is 0.2. The low-, mid-, and high-frequency ranges for the wobbling parameters is 10 to 12 Hz, 15 to 18 Hz, and 20 to 24 Hz, respectively.

Citation: Journal of Applied Biomechanics 38, 2; 10.1123/jab.2021-0051

—The hip joint moments at (A) low-, (B) mid-, and (C) high-frequency wobbling mass parameters. The hip joint power at (D) low-, (E) mid-, and (F) high-frequency wobbling mass parameters. The gray thick, black dotted, black dashed, and black thin lines denote the rigid model, 20% wobbling mass/inertia model, 40% wobbling mass/inertia model, and 60% wobbling mass/inertia model, respectively. The damping ratio for all plots is 0.2. The low-, mid-, and high-frequency ranges for the wobbling parameters is 10 to 12 Hz, 15 to 18 Hz, and 20 to 24 Hz, respectively.

Citation: Journal of Applied Biomechanics 38, 2; 10.1123/jab.2021-0051

—The hip joint moments at (A) low-, (B) mid-, and (C) high-frequency wobbling mass parameters. The hip joint power at (D) low-, (E) mid-, and (F) high-frequency wobbling mass parameters. The gray thick, black dotted, black dashed, and black thin lines denote the rigid model, 20% wobbling mass/inertia model, 40% wobbling mass/inertia model, and 60% wobbling mass/inertia model, respectively. The damping ratio for all plots is 0.2. The low-, mid-, and high-frequency ranges for the wobbling parameters is 10 to 12 Hz, 15 to 18 Hz, and 20 to 24 Hz, respectively.

Citation: Journal of Applied Biomechanics 38, 2; 10.1123/jab.2021-0051

The most extreme differences between the rigid and wobbling mass models were found at low-frequency wobbling conditions (Figure 4). The magnitude of the peak hip joint moments was over 50% greater for the wobbling model compared with that of the rigid model. The peak hip power increased by over 50% from 4.0 to 6.7 W·kg^{−1}. Furthermore, the timings of the peak hip joint moments and powers were shifted by up to 15% later in the stance phase. Qualitatively, the profiles of the hip joint moments and powers for the low-frequency wobbling conditions relative to the rigid model from 40% of the stance phase onwards (Figure 4A and 4D).

In general, the hip kinetics at midfrequency wobbling conditions were not affected as much as those at the lower frequency wobbling conditions. The magnitude of the peak hip joint moments and powers were still about 50% greater for the wobbling model compared with that of the rigid model, but the timings of the peaks were nearly the same as the rigid model. The hip kinetics at high-frequency wobbling conditions were most similar to the rigid model compared with that of the low- and midfrequency wobbling conditions. The magnitude of the peak hip joint moments and powers for the wobbling model at high-frequency wobbling conditions were within 15% of that of the rigid model.

## Discussion

In this study, a rigid body model of running was compared with a model with wobbling mass components. The purpose of this study was to investigate the effects of the wobbling mass motion on resultant joint moments and joint power at the knee and hip. The soft tissue motion of the shank and thigh was measured to vibrate at 18 and 16 Hz, respectively. These values fall in the range of the frequencies determined in previous studies.^{8,26} In the simulation, the displacements of the wobbling mass model components were within the range reported by Schmitt and Guenther.^{27}

The frequencies at which the wobbling mass model vibrated were systematically changed to investigate model sensitivity but also served to highlight how joint kinetics could be influenced by a more or less lean participant. The results of this study suggests that joint kinetics derived from inverse dynamics analysis are sensitive to the mechanical properties of the model utilized to perform the analysis. Notably, for model versions with low- and midfrequency wobbling, the hip joint power may be underestimated by as much as 50% at certain instances of the stance phase relative to that of a rigid model (ie, a model without wobbling components). The differences between the rigid model and wobbling model become more extreme if the virtual leanness of the model decreases. That is, as the mass and moment of inertia of the wobbling component increase, the effects of the wobbling component on the joint kinetics becomes more extreme. This suggests that inverse dynamics analyses of less lean persons should take into account the motion of the soft tissue with respect to the underlying bone. Conversely, the outputs of the rigid model were nearly identical to a model with a wobbling mass, representative of a leaner participant. This interpretation mirrors the experimental evidence from the analysis of obese and nonobese subjects walking.^{10}

Resultant joint moments are often used to estimate the forces produced by the muscles that span the joints of interest, for example, using optimization methods.^{27,28} In such methods, the muscle forces are estimated by minimizing an objective function, while simultaneously ensuring the estimated muscle forces produce the measured resultant joint moments. Errors in the estimated resultant joint moments at the hip, due to not accounting for the wobbling mass, can affect muscle force estimation for muscles both at the hip and knee joints due to biarticular muscles that span both joints (eg, rectus femoris muscle and semitendinosus muscle). This in turn may affect the estimation of muscle forces that span the ankle due to biarticular muscles that span both the knee and ankle (eg, gastrocnemius muscle).

The present study has a few limitations. One of the limitations of the study was that the soft tissue motion relative to the underlying bone was not measured directly. Therefore, it is difficult to ascertain the accuracy of the wobbling mass time–displacement profiles (and subsequent intrasegment forces) without a ground truth comparison. Furthermore, the motion of the soft tissue during running affects the accuracy of the bone pose estimation.^{29} Therefore, it may be that the kinematics of the rigid model also includes the motions of the soft tissue, and this may effect the accuracy of the calculated joint kinetics for the rigid model. The present study did not collect sufficient kinematic information to permit a full accounting for the energetics of the running stride,^{30} but such analysis would provide more information about the energetics of the soft tissue motion and augment the analysis undertaken. In addition, the model did not account for the motion of the wobbling masses with respect to the local anatomical reference frames. It has been shown that viscoelastic properties of human soft tissue may differ between motion parallel and perpendicular to the skin surface.^{8} Finally, the study only examined one subject, therefore the results are strictly germane to that subject; although, aspects of the results may generalize, and the methods presented are appropriate to be employed to examine other subjects.

During running, particularly following foot–ground contact, the soft tissues of the leg move relative to the underlying bones. Modeling of intersegmental dynamics, which lumps the bone and surrounding soft tissues into a single component, fails to capture the effects of this soft tissue motion. The motion of soft tissue appears to contribute to the hip joint kinetics. and when possible, should not be ignored during inverse dynamic analyses.

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