8.3 Multi-Body Simulation-Driven Multi-Muscle Model
9.1.3 Muscle Resulting Elbow Moment
122 Chapter 9: Results
around the olecranon. Hence, for increasing the triceps length, the triceps’ orientation deviates from the orientation of the line by up to 40◦. The biceps’ deflection of the reac-tion force is without contact mostly less than 1 mm, i.e. very small and negligible. With contact, the biceps’ deflection superposes for a long biceps the impact of the contact. For a very extended forearm, the biceps touches the humerus, hence, the deflection increases to approximately 20◦.
For the results of the MTC depicted in Figure 9.19 and 9.20, less artifacts are arising as the muscle reaction forces are not reducing to zero or changing the sign as they do if rigid tendons are assumed.
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elbow angle θ [°]
muscle lever arm [mm]
triceps biceps
Figure 9.21: The lever arms for triceps and biceps brachii for the TDM are de-termined by taking the derivative of the muscle lengths with respect to θ.
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elbow angle [°]
muscle moment [Nmm]
triceps biceps
Figure 9.22: The product of lever arm and muscle force yields the resulting el-bow joint moment which depends on θ and α. The thick curves consider con-tact whereas the thin curves do not. The curve with the lowest absolute value be-longs to α = 0 and raises in ∆α = 0.1 steps up to α = 1to build 11 curves for each muscle.
muscle activation and has an impact on the vector-resulting lever arm. Furthermore, the contact formulation has an influence on the vector-resulting lever arm as it influences the orientation of the reaction force strongly.
For the triceps, see Figure 9.23a and b, the lever arm is, for a flexing elbow, a mono-tonically decreasing surface. Not considering contact, the lever arm is between -8 mm, for large elbow angles, and -30 mm, for small elbow angles. While the muscle activation’s dependency is mostly only weak, it has a strong influence on a fully extended forearm.
Towards a passive muscle, the lever arm drops. Considering contact, the overall muscle activation’s dependency is stronger. Rather significant is the fact, that the lever arm including contact is less linear. Contact causes a concave bend for the elbow angle de-pendency. For an extended elbow, the lever arm including contact even becomes positive.
Furthermore, the distinct kink at the extended elbow towards a passive muscle state is not existing.
For the biceps, see Figure 9.23b and d, the lever arm values are mostly between 10 and 15 mm. For the passive, fully flexed muscle, the muscle reaction force is very close to zero.
Hence, its orientation is not clearly defined anymore and therefore, the resulting biceps reaction force and moment, respectively, are not well defined anymore. When contact is also considered, the slight decrease towards the fully extended elbow vanishes and the biceps lever arm is about 13.5 mm and very close to constant.
As the lever arms are determined using the exact vectorial locations, it is possible now to incorporate the change of the location of the point of action and the different orientations of the muscle reaction force. To point out the impact of the change of location, in Figure 9.24 the difference between the results of Figure 9.23 are subtracted
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(a) triceps, without contact (b) biceps, without contact
(c) triceps, with contact (d) biceps, with contact
Figure 9.23: The VRLA lever arms are a function of the muscle activation, the elbow angle, and the effects of contact. For the few simulations, where the biceps reaction force becomes very small and negative, the algorithm fails to predict a reasonable lever arm.
from the results considering the change of the point of action.
For the triceps, Figure 9.24a shows that the influence of the change of the point of action on the lever arm is not very large. It is smaller than 1 mm. The results of Figure 9.24c show that the behaviour of the lever arm is not considerable influenced by the effects of contact. Only for θ > 120◦, the influence of contact increases and the lever arm raises up to 1 mm. For the biceps (Figure 9.24b), the values are even smaller and within 0.4 mm. The area, in where the biceps reaction forces become very small, the lever arm is indifferent and less significant. The results of the lever arm appear not to be significantly influenced by the effects of contact as the surface in Figure 9.24d is very similar to the one in Figure 9.24b.
In Figure 9.25, the elbow moment resulting from using the tendon-displacement method (TDM) is compared to the results using the vector-resulting lever arm (VRLA) method.
If the moment using the VRLA method is plotted similar to Figure 9.22 using lines, the results would be hardly recognisable as the resulting lines are intersecting each other. To improve the comparability of the results, the following plots are visualised using surface plots and Figure 9.25c and 9.25d repeat the results of Figure 9.22 as surface plots.
In Figure 9.25, the triceps’ elbow moment can be seen on the left and the biceps’ elbow moment on the right. The four figures, Figure 9.25c-f, show the resulting moment for using
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(a) triceps, without contact (b) biceps, without contact
(c) triceps, with contact (d) biceps, with contact
Figure 9.24: Investigates the impact of the change of location of the point of action on the VRLA, i.e. the solution of VRLA is subtracted from VRLA+.
the TDM (second row) and the VRLA (third row). The first and the fourth row show the impact of the effects of contact on the resulting moment using the TDM resulting moment and on the VRLA method resulting moment, respectively. This is done by subtracting the moment including the effects on contact from the moment not including contact.
The second row shows the moment resulting from using the TDM and the third row the moment from using the VRLA method. All four plot include contact.
For the resulting moment using the TDM without contact, the triceps takes its maxi-mum absolute value at |MT(θ = 100◦, α= 1)|=| −25.59|kNmm and|MT(θ = 102◦, α= 1)|=|−26.07|kNmm if contact is considered. For the VRLA method, the triceps induced elbow moment is significantly weaker. The triceps exhibits its largest absolute moment with |MT(θ = 73◦, α = 1)| = | −17.53|kNmm for not taking into account contact and
|MT(θ = 73◦, α = 1)| = | − 16.89|kNmm if contact is considered. Using the VRLA method, the triceps’ moment passes the neutral line for a fully flexed elbow, as the value of the lever arm changes its sign. I.e. for θ > 130◦, the triceps acts as an elbow flexor throughout all levels of activations! As the biceps’ VRLA is roughly 2.5 times smaller than the one determined by the TDM, the resulting elbow moment is also smaller. While the maximum value for the TDM is MB(θ = 10◦, α= 1) = 24.28 kNmm without contact, the absolute maximal value for contact is MB(θ = 10◦, α= 1) is 22.22 kNmm.
126 Chapter 9: Results
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(a) triceps, comparing contact, TDM
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muscle moment [Nmm]
(b) biceps, comparing contact, TDM
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(c) triceps, TDM
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θ [°] α [−]
muscle moment [Nmm]
(d) biceps, TDM
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muscle moment [Nmm]
(e) triceps, VRLA
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muscle moment [Nmm]
(f ) biceps, VRLA
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θ [°]
muscle moment [Nmm]
(g) triceps, comparing contact, VRLA
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θ [°] α [−]
muscle moment [Nmm]
(h) biceps, comparing contact, VRLA
Figure 9.25: The resulting triceps moment is plotted versus the muscle activation and elbow angle. The first two rows are determined using the TDM. The lower two rows are determined using the VRLA in addition to considering a changing point of action. For the upper and the lower row, the moment resulting from considering contact is subtracted from the moment resulting from not considering contact to highlight the influence of contact on the resulting moment.