# Asymmetry in Three-Dimensional Sprinting with and without Running-Specific Prostheses

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Modeling of Sprinting Motions with and without Running-Specific Prostheses

#### 2.1.1. Three-Dimensional Subject-Specific Modeling of Unilateral Transtibial Amputee and Non-Amputee Athletes

#### 2.1.2. Mathematical Description of Sprinting Motions

#### 2.2. Least-Squares Optimal Control Problem for Dynamics Reconstruction

## 3. Results and Discussion

#### 3.1. Precision of Kinematic Reconstruction

- $RMS{E}_{\mathrm{trans}}=0.358$ cm and $RMS{E}_{\mathrm{rot}}=0.010\mathrm{rad}\approx 0.57{}^{\circ}$ for non-amputee athlete 1,
- $RMS{E}_{\mathrm{trans}}=0.487$ cm and $RMS{E}_{\mathrm{rot}}=0.004\mathrm{rad}\approx 0.23{}^{\circ}$ for non-amputee athlete 2,
- $RMS{E}_{\mathrm{trans}}=0.421$ cm and $RMS{E}_{\mathrm{rot}}=0.006\mathrm{rad}\approx 0.34{}^{\circ}$ for non-amputee athlete 3,
- $RMS{E}_{\mathrm{trans}}=0.774$ cm and $RMS{E}_{\mathrm{rot}}=0.010\mathrm{rad}\approx 0.57{}^{\circ}$ for the amputee athlete.

#### 3.2. Validation of Ground Reaction Forces

#### 3.3. Validation and Analysis of Joint Torques

^{−1}for the knee joints which corresponds to the reconstructed values of 1 N $\mathrm{m}$ $\mathrm{k}\mathrm{g}$

^{−1}for the maximal extension and 3 N $\mathrm{m}$ $\mathrm{k}\mathrm{g}$

^{−1}for the maximal flexion torques. For the ankle torques, too, the both the course and the magnitude of the reconstructed torques fit well with the literature values with a peak external flexion torque of circa 5 N $\mathrm{m}$ $\mathrm{k}\mathrm{g}$

^{−1}. Transversal and frontal moments for knee and ankle joints are reported by Stafilidis and Arampatzis [24]. However, our models have only the RZ-DOF in the ankle which corresponds with the reported data. As the additional torques are rather small, the simplifications introduced in our models by diminishing the number of DOFs seems justifiable. The torques given by Schache and colleagues [25] correspond well during the stance phase. In addition, they show torques during the swing phase of sprinting at ($8.95\pm 0.70$) $\mathrm{m}$/$\mathrm{s}$

^{−1}which fit the reconstructed torques.

^{−1}and $0.37$ $\mathrm{m}$ $\mathrm{s}$

^{−1}slower on average, respectively (see Table 3). However, the effects due to the different velocities are already reflected to some extent by the normalization with respect to total time and covered distance. We first compare the amputee athlete with the mean value of the non-amputee athletes: it is immediately noticeable that the sum over all joints for the amputee athlete is smaller than for the non-amputee athletes (98.8%), but still quite clearly within the standard deviation. Furthermore, it is noticeable that the values in some joints are lower than those of the non-amputee athletes: in the affected leg (hip: 90.3%, knee: 41.7% compared with the mean of the non-amputee athletes), in the biological ankle (76.0%), and in the right elbow (94.6%). However, except for the affected knee, all values are within the standard deviation regions. In all other joints (except the neck, where the values are practically the same), the values are significantly larger and also lie at the margin (right shoulder) or significantly above the standard deviation regions (unaffected hip, unaffected knee, left shoulder, left elbow, lumbar, thorax). The deviations from the mean ranged from 13.3% to 53.3%. Compared to the mean value of the non-amputee athletes, it is therefore clear that the amputee athlete must apply significantly greater torques in the majority of the joints, which also increases the risk of fatigue and injury in these joints. If we now draw the comparison only to the non-amputee athlete 1, who runs at a comparable speed, the result becomes a little less clear, but remains broadly the same: In the same joints as before, the torque values of the amputee athlete are lower than those of the non-amputee athlete 1 (affected hip: 80.3%, affected knee: 35.7%, biological ankle: 70.4%, right elbow: 88.1% compared to the non-amputee athlete 1). In the joints where the values for the amputee were greater in the previous comparison, the deviations are a little less significant, ranging between 3.7% and 35.3%. Considering the sum over all joints, the value of the amputee athlete is 89.7%. Nevertheless, it is also clear that this comparatively smaller value results in particular from lower torques in the affected leg. Due to the asymmetry of the amputee athlete, there is a clear imbalance in the load on the individual joints, which particularly affects the back and arms. Overall, it can be seen that the amputee must use his upper body more and in some joints in a completely different way to compensate for the asymmetry caused by the prosthesis.

#### 3.4. Symmetry Analysis of 3D Kinematics and Dynamics

#### 3.5. (Symmetry) Analysis of Angular Momentum Around Center of Mass in all Three Planes

## 4. Conclusions & Future Work

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The unilateral amputee athlete is a highly asymmetric system due to the prosthetic device.

**Figure 2.**Visualization of the subject-specific models of three non-amputee and one amputee athlete and the running-specific prosthesis.

**Figure 3.**Phase description of sprint motions: Each step consists of an airborne phase, a completely inelastic touchdown and a single-support contact phase. Two full steps are shown in the graphics.

**Figure 4.**Animated sequences of the sprint motions of three non-amputee and one amputee athlete. For an animated version of this figure, see the supplementary video.

**Figure 5.**Animated sequences of the sprint motions of one non-amputee and one amputee athlete from different perspectives to highlight differences in upper body movement. For an animated version of this figure, see the supplementary video.

**Figure 6.**Reconstructed and measured filtered ground reaction forces of three non-amputee and one amputee athlete (rows from top to bottom: non-amputee athlete 1, non-amputee athlete 2, non-amputee athlete 3, amputee athlete; columns from left to right: anterior-posterior force (TX), mediolateral force (TY), vertical force (TZ))

**Figure 7.**Comparison of joint torques for non-amputee and amputee springting. Joint torques are normalized by individual body mass, phase durations are scaled per phase. For the non-amputee athletes, the mean values over both steps of all three athletes are shown together with the regions of standard deviation.

**Figure 8.**Amount of asymmetry in generalized positions of the amputee athlete (red solid line, ‘A’), the mean and standard deviation regions of the non-amputee athletes (blue solid line and region, ‘NA mean’ and the three individual non-amputee athletes (blue dotted lines, ‘NA 1’, ‘NA 2’ and ‘NA 3’). The symmetry values have been computed by subtracting the second step from the first step at respective time points in phases taking into account the phase shifts between left and right steps. Phase durations are scaled for better comparability. We show the absolute symmetry values and a value of 0 would indicate that the motions are perfectly symmetric.

**Figure 9.**Amount of asymmetry in joint torques of the amputee athlete (red solid line, ‘A’), the mean and standard deviation regions of the non-amputee athletes (blue solid line and region, ‘NA mean’ and the three individual non-amputee athletes (blue dotted lines, ‘NA 1’, ‘NA 2’ and ‘NA 3’). The symmetry values have been computed by subtracting the second step from the first step at respective time points in phases taking into account the phase shifts between left and right steps. We show the absolute symmetry values and a value of 0 would indicate that the motions are perfectly symmetric. Phase durations are scaled for better comparability and joint torques are normalized by body mass. The joint torques for the ankle joint rotations in the sagittal plane have been computed with the passively generated torque in the prosthetic device, computed based on Equation (1).

**Figure 10.**Angular momentum about CoM of the amputee athlete and the non-amputee reference group in the frontal plane (rotations around x-axis). Phase durations are scaled and angular momentum values are normalized by body mass and body height squared.

**Figure 11.**Angular momentum about CoM of the amputee athlete and the non-amputee reference group in the sagittal plane (rotations around y-axis). Phase durations are scaled and angular momentum values are normalized by body mass and body height squared. Please note the different y-axis scales for the contribution of the arms.

**Figure 12.**Angular momentum about CoM of the amputee athlete and the non-amputee reference group in the transversal plane (rotations around z-axis). Phase durations are scaled and angular momentum values are normalized by body mass and body height squared.

Non-Amputee Athlete | Amputee Athlete | ||
---|---|---|---|

Body Segment | Degrees of Freedom | Body Segment | Degrees of Freedom |

Pelvis (base) | 6: TX, TY, TZ, RY, RX, RZ | Pelvis (base) | 6: TX, TY, TZ, RY, RX, RZ |

Lumbar | 2: RY, RX | Lumbar | 2: RY, RX |

Thorax | 2: RY, RZ | Thorax | 2: RY, RZ |

Head | 1: RY | Head | 1: RY |

Thigh | 3: RY, RX, RZ | Thigh | 3: RY, RX, RZ |

Shank | 1: RY | Shank | 1: RY |

Foot | 2: RY, RZ | Foot (left) | 2: RY, RZ |

Prosthesis | 1: RY | ||

Upper Arm | 3: RY, RX, RZ | Upper Arm | 3: RY, RX, RZ |

Lower Arm | 1: RY | Lower Arm | 1: RY |

**Table 2.**Average over absolute joint torques per joint normalized by covered distance. For joints that appear on both sides of the body, we give the average over both sides in the case of the non-amputee athletes and the individual values in the case of the amputee athlete. All numbers are given in N $\mathrm{k}\mathrm{g}$

^{−1}.

Joint | Non-Amputee Athlete 1 | Non-Amputee Athlete 2 | Non-Amputee Athlete 3 | Mean of Non- AmputeeAthletes | Amputee Athlete | |
---|---|---|---|---|---|---|

Right Side | Left Side | |||||

hip | 0.81 | 0.61 | 0.74 | 0.72 0.08 | 0.65 | 0.84 |

knee | 0.28 | 0.21 | 0.25 | 0.24 0.03 | 0.10 | 0.31 |

ankle | 0.27 | 0.23 | 0.24 | 0.25 0.02 | - | 0.19 |

shoulder | 0.17 | 0.15 | 0.13 | 0.15 0.02 | 0.17 | 0.23 |

elbow | 0.059 | 0.053 | 0.052 | 0.055 0.003 | 0.052 | 0.073 |

thorax | 0.29 | 0.23 | 0.28 | 0.27 0.02 | 0.37 | |

lumbar | 0.28 | 0.29 | 0.35 | 0.31 0.03 | 0.37 | |

neck | 0.04 | 0.03 | 0.02 | 0.03 0.01 | 0.03 | |

sum | 3.78 | 3.05 | 3.47 | 3.43 0.30 | 3.39 |

Step 1 | Step 2 | Average of Both Steps | |
---|---|---|---|

(Flight and Right/Affected Contact) | (Flight and Left/Unaffected Contact) | ||

Non-amputee athletes (mean) | ($9.11\pm 0.3$) $\mathrm{m}$ $\mathrm{s}$^{−1} | ($9.17\pm 0.37$) $\mathrm{m}$ $\mathrm{s}$^{−1} | ($9.14\pm 0.30$) $\mathrm{m}$ $\mathrm{s}$^{−1} |

Non-amputee athlete 1 | $9.37$$\mathrm{m}$$\mathrm{s}$^{−1} | $9.55$$\mathrm{m}$$\mathrm{s}$^{−1} | $9.46$$\mathrm{m}$$\mathrm{s}$^{−1} |

Non-amputee athlete 2 | $8.78$$\mathrm{m}$$\mathrm{s}$^{−1} | $8.81$$\mathrm{m}$$\mathrm{s}$^{−1} | $8.80$$\mathrm{m}$$\mathrm{s}$^{−1} |

Non-amputee athlete 3 | $9.18$$\mathrm{m}$$\mathrm{s}$^{−1} | $9.16$$\mathrm{m}$$\mathrm{s}$^{−1} | $9.17$$\mathrm{m}$$\mathrm{s}$^{−1} |

Amputee athlete | $9.03$$\mathrm{m}$$\mathrm{s}$^{−1} | $10.06$$\mathrm{m}$$\mathrm{s}$^{−1} | $9.54$$\mathrm{m}$$\mathrm{s}$^{−1} |

**Table 4.**Phase durations of the amputee athlete and the non-amputee athletes. Phases are numbered as ordered in Figure 3: 1—first flight phase, 2—right contact phase (affected leg), 3—second flight phase, 4—left contact phase (unaffected leg).

Phase | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Non-amputee athletes (mean) | $(0.147\pm 0.012)\mathrm{s}$ | $(0.092\pm 0.002)\mathrm{s}$ | $(0.149\pm 0.012)\mathrm{s}$ | $(0.093\pm 0.002)\mathrm{s}$ |

Non-amputee athlete 1 | $0.124\mathrm{s}$ | $0.092\mathrm{s}$ | $0.124\mathrm{s}$ | $0.098\mathrm{s}$ |

Non-amputee athlete 2 | $0.168\mathrm{s}$ | $0.088\mathrm{s}$ | $0.172\mathrm{s}$ | $0.088\mathrm{s}$ |

Non-amputee athlete 3 | $0.150\mathrm{s}$ | $0.096\mathrm{s}$ | $0.150\mathrm{s}$ | $0.094\mathrm{s}$ |

Amputee athlete | $0.144$$\mathrm{s}$ | $0.080$$\mathrm{s}$ | $0.152$$\mathrm{s}$ | $0.084$$\mathrm{s}$ |

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**MDPI and ACS Style**

Emonds, A.L.; Mombaur, K.
Asymmetry in Three-Dimensional Sprinting with and without Running-Specific Prostheses. *Symmetry* **2021**, *13*, 580.
https://doi.org/10.3390/sym13040580

**AMA Style**

Emonds AL, Mombaur K.
Asymmetry in Three-Dimensional Sprinting with and without Running-Specific Prostheses. *Symmetry*. 2021; 13(4):580.
https://doi.org/10.3390/sym13040580

**Chicago/Turabian Style**

Emonds, Anna Lena, and Katja Mombaur.
2021. "Asymmetry in Three-Dimensional Sprinting with and without Running-Specific Prostheses" *Symmetry* 13, no. 4: 580.
https://doi.org/10.3390/sym13040580