3.1. Precision of Kinematic Reconstruction
We first evaluate the quality of the reconstructions by computing the root-mean-square error (RMSE) of reconstructed and reference data and comparing torques and forces with (literature) reference values. The RMSE is computed by
with
and
denoting the reconstructed and reference generalized positions of the degree of freedom (DOF)
k. Instead of assessing each DOF individually, we compute the RMSE for translational and rotational DOF groups, denoted by the set of indices
. The RMSE is calculated over all multiple shooting nodes and normalized by the number of DOFs
and the number of multiple shooting nodes
m. The results are:
cm and for non-amputee athlete 1,
cm and for non-amputee athlete 2,
cm and for non-amputee athlete 3,
cm and for the amputee athlete.
As the RMSEs are smaller than cm for translational and (≈0.6) for rotational DOFs, the solution of the least squares OCP follows the reference movement quite closely. Most of the deviations between the reconstructed and the reference motion are due to small variances in the exact contact point and the rather simple foot model, which approximates the contact by a single contact point. In reality, though, this contact point moves a few centimeters along the forefoot during contact. It should be noted at this point that the computed RMSEs only account for deviations between the reconstructed and reference data, however not for measurement errors and errors in the transfer of the measured marker positions to model joint angles. Especially considering the coordinates’ range of motion, the RMSEs are small enough to accept the quality of the reconstruction.
3.2. Validation of Ground Reaction Forces
To demonstrate that our approach generates reasonable ground reaction forces and joint torques, we compare the reconstructed ground reaction forces with the filtered measurements from force plates (
Figure 6).
Figure 6 shows the anterior-posterior (TX), mediolateral (TY) and vertical (TZ) components of both the measured filtered and reconstructed ground reaction forces for all four athletes investigated. Apart from minor deviations, the reconstructed and measured forces fit together well. The anterior-posterior force (TX) is composed of a braking and propulsive component and the time of the zero transition is appropriate, too. For the vertical forces (TZ), the reconstructed and measured data show a parabolic course, in accordance with literature on sprint dynamics [
21,
22]. However, there are minor deviations in each of the three force components, which we will discuss briefly. In the measured anterior-posterior force (TX), there is a peak at the beginning of the contact which is not reconstructed. In addition, the propulsive component is over- or underestimated, especially during the second contact. This is particularly noticeable in the amputee athlete’s second contact. Apart from the possibility of residual artifacts in the measured data, two causes in particular appear plausible: First, we use a rather simple foot-contact model which is a strong approximation of reality (but as the overall good results show, it is certainly justified). On the other hand, especially for the second observation, it seems probable that the formulation of the optimal control problem allows fulfilling the regularization term of the objective function (
10a) more strongly during the second contact phase. Here it would be interesting to analyze a longer step sequence. Regarding the mediolateral ground reaction forces (TY), it is noticeable that the reconstructed forces follow the measured ones rather roughly. We attribute this to the simplified foot model, since even small displacements of the contact point in mediolateral direction have a large influence. Finally, we consider the vertical force (TZ): Here, the reconstructed forces are very good, only for the non-amputee athlete 2 is the rapid increase in the measured data at the beginning of the contact phase not present.
3.3. Validation and Analysis of Joint Torques
As a third and final validation step, we compare the reconstructed joint torques with literature values for non-amputee [
23,
24,
25] and double transtibial sprinting [
6]. To our knowledge, no joint torques for unilateral amputee sprinting have been published yet.
Figure 7 shows the mean and standard deviation of the joint torques of the non-amputee athletes (blue solid line and shaded regions) as well as the joint torque curves of the amputee athlete (red lines where dashed red lines show the curves of the unaffected side joints if applicable). Intuitively, one would expect that non-amputee athletes would run symmetrically, i.e., that the movements of joints that are present twice would show similar curves, if one takes into account the phase shift and possible sign changes between the sides. Accordingly, and because it is not clear why the right side of the amputee athlete should be compared with the right side of the non-amputee athlete, we consider for the non-amputee athletes the mean values of both steps of the three sprinters. In the subsequent section, we will examine in more detail how symmetrically the non-amputee athletes actually run.
As described in detail in
Section 2, we approximate the action of all muscles at a related joint (i.e., antagonist and agonist muscles) by net torque actuators. All four mentioned publications report sagittal plane moments for (part of) the leg joints based on measurements and standard inverse dynamics techniques. For the sagittal plane hip joint torques during the stance phase, the trend from external flexion to external extension torques is consistent between the reconstructed and the literature data. The orders of magnitude match, too. However, Bezodis and colleagues [
23] found a peak at the beginning of the contact phase which our non-amputee models do not generate. In accordance with [
24], they computed maximal extension and flexion torques of
N
−1 for the knee joints which corresponds to the reconstructed values of 1 N
−1 for the maximal extension and 3 N
−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
−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 (
)
/
−1 which fit the reconstructed torques.
Brüggemann and co-workers [
6] investigated double transtibial amputee sprinting. While the order of magnitude of the reported hip moments match the ones from our reconstruction of a unilateral transtibial amputee sprinter, there are significant differences in the curves. They probably stem from differences between one-sided and double amputee athletes concerning leg stiffness of the biological versus the prosthetic leg and mass differences between the two. The deviations in the peak flexion torque of the prosthetic ankle might be partly due to the same reason. In addition, they are probably a result of the different prosthesis models – they differ both in reality and in the computer model.
As already mentioned, we examined the results of the reconstruction in the sagittal plane (2D), among others also the joint torques, in a previous publication [
13]. In the present study, we focus on a comprehensive analysis of the (a)symmetry between steps and the angular momentum about the center of mass (CoM). Nevertheless, we will now briefly review the reconstructed joint torques and discuss especially those in the frontal and transversal planes: Without looking more closely at any of the subdiagrams, clear differences between the non-amputee and amputee athletes are apparent from the very first look. In addition, even when comparing the two sides of the amputee’s body (solid line: body side of the amputated leg, dashed line: side of the biological leg), differences can be seen directly. In the sagittal plane, the active joint torques in the affected leg are significantly lower, except for the hip joint during the contact phase. The latter remains completely within the flexion region during the contact phase, while a transition from flexion to extension torque can be seen in non-amputee athletes. Presumably, the low knee flexion moment in the sagittal plane and the strong hip rotation in the transversal plane are compensated here. Especially in the hip rotation moments of the frontal and transversal planes, the influence of the different geometry between affected and non-affected leg is clearly visible. On the one hand, the reduced torque in the knee joint of the affected leg could indicate that this joint is likewise affected by the amputation. On the other hand, there are indicators that the conscious stiffening of the knee joint could be a strategy to make optimal use of the prosthesis [
26,
27]. At first glance, it seems appropriate in this context that the passive torque generated in the prosthetic ankle is almost twice as high as in the biological ankle. However, a closer look reveals it as an artefact of the geometry of the prosthetic device as the large joint torque is generated at the very posterior prosthetic ankle joint and the ground reaction forces are of comparable magnitude for both non-amputee and amputee athletes. In the sagittal plane, the torques in the biological leg of the amputee athlete are on average greater than those of the non-amputee athletes, especially in the knee joint during ground contact. To run at a comparable speed to that of the non-amputee athletes, the amputee must therefore generate significantly more torque with the non-amputee leg. The progression of the hip torque during the contact phase is also interesting, since the torque is in the extension range during the entire phase. In the frontal and sagittal plane, the torques of the biological leg are also higher on average than those of the non-amputee athletes. In addition, they fluctuate more, which indicates that more permanent readjustment is necessary to achieve a stable sprint movement. The observation of the arm torques shows that especially the arm that is on the opposite side to the prosthetic leg has significantly higher torques in all planes. The amputee must therefore use this arm more to compensate for the inter-limb asymmetry. However, the amputee’s other arm also has slightly greater torque compared to the non-amputee’s arm. This also applies to the spinal joints: the torques are significantly higher on average than in the non-amputee control group.
To be able to make another quantitative statement, we calculated the average of the absolute joint torques for each joint and normalized them with respect to the individual body mass
M and covered distance
D by:
The total duration of the motion
is individual for each athlete. The computed values are given in
Table 2.
In addition to the values for the amputee athlete and the mean of the non-amputee athletes, we focus on the values for the non-amputee athlete 1 for the comparison of amputee and non-amputee sprinting since the average velocities of the non-amputee athlete 1 and the amputee athlete are of comparable order. The non-amputee athletes 2 and 3 run
−1 and
−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
The diagrams of
Figure 7 show the mean values over both steps in the case of the non-amputee athletes. This is reasonable due to the fact that it is not a priori evident why the we should compare the right/left side of the amputee athlete with the corresponding side of the non-amputee athletes, without further knowledge of which is the “stronger side”. Nevertheless, there are some suggestions that a certain asymmetry is also present in non-amputee elite sprinters and might be beneficial for performance. We therefore now take a closer look at the (a)symmetry in the generalized positions and joint torques of both the non-amputee and unilateral transtibial amputee athletes.
Figure 8 and
Figure 9 show the absolute symmetry values between the two steps with values closer to zero indicating that the two steps are more symmetric. These differences are computed by subtraction of the second step from the first with respecting the necessary phase shifts in leg and arm joints. Hence, their time horizon is only one step consisting of one airborne and one contact phase. In all diagrams (of
Figure 8 and all following figures), the red solid lines represent the data of the amputee athlete. The blue solid lines show the mean curves of the non-amputee athletes with shaded regions indicating the standard deviations. In addition, we show the individual curves of the three non-amputee athletes by dashed lines in shades of blue. All phase durations are scaled to allow a better comparison of the curves at the same relative time points in phases. Individual phase durations are given in
Table 4. The joint torques are normalized by the individual body masses. As the prosthetic device has to be passive, no active torque produced by a torque actuator is present. For the diagrams, we computed the torque generated by the prosthesis via Equation (
1). Passively generated torques are drawn as green solid lines. We refer to the most posterior point of the prosthesis when using the term ‘prosthetic ankle’. At this point, we would like to refer in addition to
Figure 5, which shows the animated sequences of the movements, from which many of the observations described below can also be graphically understood.
Considering both figures together, we immediately notice that the values for the amputee athlete are much larger on average than those of the non-amputee athletes (for both the mean and the individual curves). Hence, non-amputee sprinting is much more symmetric than unilateral transtibial amputee sprinting. This meets our expectations as the body of the amputee athlete is highly asymmetric, while those of the non-amputee athletes are a lot more symmetric. Hence, our above symmetry assumption for the comparison of the joint torques seems justified.
We start the analysis with a description and discussion of the absolute asymmetries in the generalized positions (i.e., overall position in space and joint angles). On average, the differences between the two steps for the translational DOFs are about three and a half times greater in the amputee athlete than in the non-amputee control group. The significantly large difference in the distance travelled in the forward direction plays a particularly important role here: during the second step (following the prosthetic contact phase) the amputee’s center of mass moves about 30 cm further than during the first step. This step takes also about 8 ms longer (see
Table 4). Interestingly, the difference in the step lengths of these two steps is only about 15 cm. It follows that the amputee positions his pelvis significantly differently with respect to the contact point during the two contact phases. For the rotational DOFs, the differences between the two steps are more than twice as great when the amputee is compared with the control group. As expected, the greatest differences are found here in the leg joints and individual joints of the arms. In particular, hip movement in the sagittal plane and knee movement during contact are very asymmetrical. In the arm joints, the asymmetry is particularly evident on the side of the body opposite the amputation (‘swing-side’). It appears that the amputee attempts to compensate for the weight asymmetry of his or her legs with the arm movements.
Another interesting fact is that the asymmetry between the two steps in the amputee case is significantly greater for almost all joints than in the non-amputee athletes; on average almost three times greater for all joints.Here it is particularly clear that the amputee has a completely different actuation strategy firstly for both legs, but also between the steps in which the prosthesis and the biological foot are in contact with the ground. The question arises whether such an actuation strategy, which has to generate very different torques, is disadvantageous overall or in relation to individual joints, e.g., in terms of increased abrasion or fatigue and the associated increased risk of injury.
Although we are particularly interested in the differences in sprinting behavior of the amputee compared to the average of non-amputee athletes, we would like to briefly discuss the aspect of asymmetry in non-amputee sprinting. In both figures we also drew the symmetry values for the individual athletes (dashed lines). It is noticeable that even the non-amputee athletes do not run perfectly symmetrically. Averaged over all diagrams, the non-amputee athlete 3 runs most symmetrically and the non-amputee athlete 2 runs most asymmetrically. Interestingly, the non-amputee athlete 1 is exactly in between in terms of symmetry, although his average speed shows the greatest difference between the two steps (see
Table 3). Interestingly, the athlete which runs most asymmetrically is at the same time the athlete with the lowest average velocity, the shortest contact phase durations and the biggest peak vertical ground reaction forces in this study. Hence, it seems that smaller asymmetries such as in non-amputee athletes 1 and 3 might have little influence on or even be beneficial for sprint performance. However, more pronounced asymmetries (non-amputee athlete) might hinder sprint performance.
3.5. (Symmetry) Analysis of Angular Momentum Around Center of Mass in all Three Planes
Figure 10,
Figure 11 and
Figure 12 show the angular momenta for the sagittal, frontal and transversal planes, respectively. The values are normalized by body mass and body height squared. The diagrams distinguish the contribution of the legs, arms and torso segment to the total angular momentum. At first sight, clear differences in the angular momentum curves between the amputee and the non-amputee athletes are visible. To begin with, we take a look at the total angular momentum of the non-amputee athlete: They follow similar curves as those observed by Hinrichs[
28], though absolute numbers vary because of differences in running velocity. Arms and trunk move around an angular momentum of zero. However, the angular momentum of the two legs is in opposite directions, with the contact leg always having a greater angular momentum. Accordingly, the total angular momentum does not move around zero. The reason for this is the circular motion of the legs, especially the feet (see Hinrichs [
28]). A comparison with the curves of the amputated athlete now shows a similar curve for the biological leg, which consequently follows a comparable circular motion. In contrast, the angular momentum of the prosthetic leg during the contact phase is significantly lower and the curve deviates. We suspect the cause to be the asymmetry of the legs, which is especially caused by the lighter prosthesis. Similar observations can be made for the leg angular momenta in the frontal and transversal planes: For the biological leg, the curves of the amputee athlete are comparable to those of the non-amputee reference group, whereas the curves for the leg affected by the amputation differ more. For rotations around the forward axis (RX), the curve of the affected leg’s angular momentum changes more frequently indicating that the amputee athlete has to adjust this leg several times during the motion. For rotations around the vertical axis (RZ), the prosthetic leg rotates clearly stronger than the biological legs which is probably due to the fact that the prosthetic device is a fixed device which has only one rotational DOF, i.e., the one around the frontal axis.
Coming back to the sagittal plane angular momentum, the differences in the total angular momentum of the amputee athlete versus the control group of non-amputees cannot be completely explained by differences in the leg movement. Overall, the angular momentum of the arm segments is comparable between amputee and non-amputee athletes; there are small differences in the left arm angular momentum during the flight phase following contact with the prosthetic leg. Interestingly, in the frontal and transversal plane, too, is mainly the left arm (i.e., the arm on the opposite body side to the one where the amputation has been) angular momentum that has a different course in comparison to the non-amputee reference group. The left arm angular momentum values are significantly larger, especially during the contact phase with the prosthetic leg and the subsequent flight phase. This indicates that the amputee athlete has to make great use of his left arm for compensation of the weight asymmetry in the legs. The angular momenta of the trunk (spinal) segments are larger in the frontal and sagittal planes as well, in the frontal plane again especially at the end of the prosthetic contact and the subsequent flight phase. Hence, the spinal segments as well account for balancing the lift-off of the relatively light prosthetic leg.
Considering the total angular momenta, we notice that they are larger for the amputee than for the non-amputee reference group in the frontal and transversal planes, i.e., the planes which are generally considered less interesting in linear sprint motions. Interestingly, the total angular momentum of the amputee is relatively symmetrically distributed around the zero line. The rotations in these two planes are therefore overall very symmetrical for the two steps, but show clear asymmetries when the individual components are considered. In the sagittal plane, the total angular momentum is comparable during the first flight phase which is the airborne phase following contact with the biological leg. During the flight phase following prosthetic contact, the angular momentum of the amputee athlete is close to zero, indicating very little rotation around the frontal axis.
In summary, the amputee’s angular momentum values, particularly in the frontal and lateral planes, are on average significantly higher than those of the non-amputee athlete. Regarding the legs, the inter-limb asymmetry mainly affects the rotations in the sagittal plane. However, the amputee athlete generates completely different angular momenta in his arms and spine to achieve stability despite the present asymmetries.