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# Using Torque-Angle and Torque- Velocity Models to Characterize Elbow Mechanical Function: Modeling and Applied Aspects

Diane Haering, Charles Pontonnier, Nicolas Bideau, Guillaume Nicolas, Georges Dumont

► To cite this version:

Diane Haering, Charles Pontonnier, Nicolas Bideau, Guillaume Nicolas, Georges Dumont. Using Torque-Angle and Torque-Velocity Models to Characterize Elbow Mechanical Function: Modeling and Applied Aspects. Journal of Biomechanical Engineering, 2019, 141 (8), pp.084501. 10.1115/1.4043447. hal-02074561

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Submitted on 5 Apr 2019

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USING TORQUE-ANGLE AND TORQUE-
VELOCITY MODELS TO CHARACTERIZE
ELBOW MECHANICAL FUNCTION:
MODELING AND APPLIED ASPECTS

Haering, Diane<fnref n="1" />

IBHGC, ENSAM ParisTech, F-75014 Paris, France

diane.haering@gmail.com

Pontonnier, Charles

Univ Rennes, CNRS, INRIA, IRISA, UMR6074, F-35000 Rennes.

charles.pontonnier@ens-rennes.fr

Bideau, Nicolas

Univ Rennes, M2S, EA1274, F-35000 Rennes.

nicolas.bideau@univ-rennes2.fr

Nicolas, Guillaume

Univ Rennes, M2S, EA1274, F-35000 Rennes.

guillaume.nicolas@univ-rennes2.fr

Dumont, Georges

Univ Rennes, CNRS, INRIA, IRISA, UMR6074, F-35000 Rennes.

georges.dumont@ens-rennes.fr

<footnote n="1">¹ Corresponding author</footnote>
ABSTRACT

Characterization of muscle mechanism through the torque-angle and torque-velocity relationships is critical
for human movement evaluation and simulation. In-vivo determination of these relationships through
dynamometric measurements and modelling is based on physiological and mathematical aspects. However,
no investigation regarding the effects of the mathematical model and the physiological parameters
underneath these models was found. The purpose of the current study was to compare the capacity of
various torque-angle and torque-velocity models to fit experimental dynamometric measurement of the
elbow and provide meaningful mechanical and physiological information. Therefore, varying mathematical
function and physiological muscle parameters from the literature were tested. While a quadratic torque-
angle model seemed to increase predicted to measured elbow torque fitting, a new power-based torque-
velocity parametric model gave meaningful physiological values to interpret with similar fitting results to a
classical torque-velocity model. This model is of interest to extract modeling and clinical knowledge
characterizing the mechanical behavior of such a joint.

1 **Keywords:** maximal joint torque - isokinetic dynamometer – torque-angle-velocity

2 relationship, maximal power velocity, muscle mechanics.

3
4 **INTRODUCTION**

5 Joint strength models are valuable representations of the torque generation
6 capacities of a human, useful in direct assessment as well as in musculoskeletal modeling
7 and analyses of human body. These models assume that muscles are viscoelastic
8 actuators [1-3], resulting at the joint level in Joint Torque-Angle and Torque-Velocity
9 Relationships (JTAR and JTVR respectively, and their coupling JTAVR). Fitting such models
10 to specific subjects while keeping their physiological meaning relevant remains an issue.
11 Basically, models are fitted to isokinetic measurements of joint torques in different angle
12 and angular velocity conditions [1-3].

13 At sarcomere scale, force-length relationship is asymmetrical piecewise linear due
14 to actin and myosin cross-bridge dynamics [4]. At muscle scale, the inter-fiber variability
15 blurs the transient states [5]. At joint scale, muscle-specific non-linear moment arms [6]
16 bring additional transformation into the torque-angle relationship. These observations
17 make it difficult to choose between various JTAR models, and no consensus exists in the
18 literature: normal curve [7,8], quadratic spline [9,10], cubic spline [11], cosinus wave [12],
19 or sinus exponential wave [13]. In a previous study, differences between those models in
20 fitting experimental isokinetic measures have been observed [14], particularly in the
21 eccentric portion of JTAR, evidencing the interaction between angle and velocity in such
22 models. Meanwhile, JTVR is mostly represented with hyperbolic functions [15,16],
23 although it might not cover all the joint velocity range [17].

24 JTAR and JTVR integrate parameters supposed to be physiologically meaningful.
25 At joint scale, parameters reflect partially the muscle physiology, even if the joint reflects
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the interaction between multiple muscles. For JTAR, maximal isometric torque Γ<sub>max</sub>,
optimal isometric angle α<sub>0</sub>, and maximal range of motion RoM, are recurrent parameters.

For JTVR, Yeadon et al. 2006 [18] introduced maximal eccentric torque, maximal concentric velocity and technical parameters. Anderson, et al. 2007 [12] added eccentric to concentric force ratio and velocities at 75% and 50% of maximal isometric force within the physiological range. In this last model, the two concentric parameters are dependent; and the model lacks derivative continuity between concentric and eccentric portions that can lead to unrealistic JTAVR fitting to data, particularly exhibiting continuity jumps. As proposed in [19], maximal range of velocity can be useful for JTVR extrapolation in high velocity regions.

Besides models fitting issues, physiological significance is useful for interpretation.

For example, maximal strength and muscle compositions are useful in ergonomics [20,21]. In sports, specific velocity at maximal power can be a training focus [22]. In clinics, eccentric to concentric strength ratio is an indicator for pathology [23]. Last, musculoskeletal analysis needs such parameters to calibrate models to subjects [24,25].

A direct in-vivo estimation of these parameters remains an issue, since it requires cadaveric, invasive or expensive measurements [24–27]. Joint strength models are therefore useful to get these values indirectly [12,24,28].

The purpose of the current study is to investigate the effects of mathematical
models and muscle parameters on JTAR and JTVR from modeling and applied points of
views. We assumed that:

* [H1] an asymmetrical JTAR can reduce torque prediction errors;
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* [H2] a new power-based JTVR can improve the physiological relevance of such models.

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# METHODS

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## Ethics and Participants

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Under INRIA national ethics committee agreement (COERLE #2017-002), twenty-two male participants (33±6 years; 1.81±0.07 m; 78±9 kg) gave their informed consent to participate in the study.

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## Isokinetic measurement

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Participants seated upright with the arm alongside on a Con-Trex MJ® isokinetic dynamometer (CMV AG, Dübendorf, Switzerland) according the manual guidelines (fig. 1). The dynamometer axis was aligned with the elbow flexion axis at 90° for maximum precision. Maximal distance between handle and arm brace without hampering elbow flexion was chosen to minimize elbow displacement away from the dynamometer axis.

Similarly, tight straps were used to immobilize thorax. Range of motion was adjusted to the subject.

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Goniometric measurement was used to calibrate angular values. Three passive flexion-extension trials at 60°.s⁻¹, 120°.s⁻¹, or 180°.s⁻¹ were recorded for gravity and passive components compensation.

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After a ten minutes sub-maximal warm-up, five voluntary flexion or extension hold for five seconds at angles evenly spaced throughout the range of motion were recorded as isometric trials. Then, three repetitions of concentric-passive cycles at 60°.s⁻¹, 120°.s⁻¹, or 180°.s⁻¹ or eccentric-passive trials at 60°.s⁻¹, 120°.s⁻¹, or 180°.s⁻¹ in flexion and extension
70 were recorded as isokinetic trials. Within subject ranges of motion, reaching isokinetic
71 state limited the maximal velocity to 180°.s⁻¹. During 180°.s⁻¹ trials, subjects were asked
72 to anticipate their effort at the end of the previous passive cycle to decrease delay and
73 ensure a sufficient maximal contraction time during trials. To decrease fatigue effects,
74 trials order was randomized within subjects, flexion and extension trials were alternated,
75 and a 45 seconds rest was respected between each trial. All measurements were collected
76 by the same experimenter. Angle, angular velocity and torque were recorded at 256 Hz.

77 Only data corresponding to isokinetic states were used for analysis, discarding the first
78 milliseconds of trials (215 ms to 355 ms) during which muscle activation is not maximal
79 [18]. For each condition, the repetition with the largest average torque was selected.

80

81 INSERT FIGURE 1 HERE

82

83 JTAR models

84 Five JTAR mostly encountered in the literature were implemented as Normal [3,10],
85 Quadratic [11,12], Cosinus [13], Cubic [14] and Sinus-exponential [15] models.
86 Parameters of all these models are the maximal isometric torque Γ<sub>max</sub>, the optimal joint
87 angle α<sub>0</sub> and the maximal range of isometric force production RoM. These models are
88 extensively described in the supplementary material and presented in figure 2.

89

90 INSERT FIGURE 2 HERE

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## 92 JTVR models

Two JTVR models were compared. The Anderson-based model is an adapted version of [12]. The power-based model is a new polynomial function of the maximal power velocity and other physiological parameters selected from the literature.

The Anderson-based model introduces 3 parameters: $\omega_{\Gamma_{.75}}$ - Velocity at 75% of maximal isometric torque, $E$ - Eccentric to concentric torque index and an additional $\omega_{\Gamma_{.5}}/\omega_{\Gamma_{.75}}$ ratio as an optimization constraint rather than an arbitrary value as in the original version.

The power-based model depends on the concentric velocity at maximal power, $\omega_{Pmax}$, because of its unique correlation with muscle composition [29,30]. $\omega_{Pmax}$ is also used as an inflexion constraint for the concentric part of the JTAR. The model also depends on $\omega_{min}/\omega_{max}$ - Max. eccentric to concentric velocity ratio, $\Gamma_{ECC}/\Gamma_{CON}$ - Max. eccentric to concentric torque ratio and $\omega_{max}$ - Maximum concentric velocity at which muscle sustains no more tension [31].

Both models are extensively described in the supplementary material and presented in figure 3.

## 109 Fitting models to data

A least-square-curve-fitting method (trust-region algorithm, Matlab® Optimization Toolbox™) minimized the quadratic distance between modelled and measured torques by optimizing models parameters.
114 Isometric, concentric and eccentric parameters were optimized in successive steps as
115 recommended by [12]. The five JTAR models were tested in the isometric step and were
116 combined with both JTVR models in the 2nd and 3rd steps.

117 Since range of motion and acquisition rate were constant for all trial, duration and frame
118 number varied for each velocity. To guarantee equal weight of all velocities on fitting,
119 samples of equal number of frames were selected within the algorithm.

120 Models comparison

121 First, models were compared in terms of ability to fit data. Adjusted correlation and linear
122 regression coefficients were compared between all combinations of JTAR and JTVR
123 models. Additionally, a one-way repeated measures Anova was performed to test the
124 effects of JTAR models on isometric torque prediction errors; and a two-way repeated
125 measures Anova was performed to test the effects of JTAR and JTVR models on isokinetic
126 (concentric + eccentric) torque prediction errors. Mauchly normality and sphericity test
127 was performed. Then, the Distribution Cumulative Differences (Matlab® Statistics &
128 Machine Learning Toolbox™) was performed for the Anovas. Results are presented with
129 significance level set to p ≤ .05 and significance power F.

130 Second, the optimized parameters of all models were compared and confronted to
131 literature values.

132 RESULTS

133 The significant effects of JTAR and JTVR were not different between flexion or extension
134 motions. Results for both motions are presented together in this section.
Torque predicted by cosinus, quadratic, and cubic models displayed larger correlation with experimental data than normal and sinus-exponential models (table 1). Highest correlations were obtained for isometric data. Correlation for concentric data was higher with the power-based model than with the Anderson-based model but lower for eccentric data.

INSERT TABLE 1 HERE

Sphericity was verified for all data ($p < 0.01$). ANOVA revealed that JTAR had significant effects on isometric prediction errors ($p < 0.01$, $F = 3.86$). Post-hoc tests attributed the smallest errors to the quadratic model ($p < 0.001$, Fig 4a). Error increased significantly of 0.19, 0.53, 2.67, and 4.25 N.m between cosinus, cubic, sinus-exponential and normal models respectively.

The ANOVA showed an ordinal interaction between these models on average error ($p < 0.01$, $F = 18.36$), plus significant effects of JTAR ($p < 0.01$, $F = 13.15$) and JTVR ($p < 0.05$, $F = 4.66$). Normal and sinus-exponential models still displayed the largest errors - 10% larger than other models - in combination with both JTVR models (Fig. 4b). The power-based model displayed larger overall error than Anderson's model only when combined with sinus-exponential and normal torque-angle models.

INSERT FIGURE 4 HERE
Average isometric parameters obtained with each model are presented in table 2. In flexion, $Γ_{max}$ varied between 63 N.m and 69 N.m, $RoM$ varied between 155° and 175°, and $α_0$ varied between 59° and 102°. In extension, $Γ_{max}$ between 60 N.m and 66 N.m, $RoM$ between 164° and 179°, and $α_0$ between 56° and 99° were obtained in extension. Significant effects of the model were found for $Γ_{max}$ and $α_0$ only in extension ($p < 0.05$, $F = 2.93$, and $p < 0.01$, $F = 3.90$ respectively). Average $Γ_{max}$ obtained with cubic and sinus-exponential models, and average $α_0$ obtained with the normal model, differed by more than 10% from values found in the literature.

INSERT TABLE 2 HERE

Optimal $ω_{max}$, $ω_{min}$, and $ω_{Pmax}$ obtained from the new model combined to each isometric model are presented in table 3. No statistical effect of the model was found for these parameters.

INSERT TABLE 3 HERE

Optimal $Γ_{ECC}/Γ_{CON}$ ratios obtained with both JTVR are presented in table 4. The ANOVA showed an effect of the JTAR, but no effect of the JTVR and an interaction between JTAR and JTVR. Differences identified through the post-hoc tests were about 1% of the average ratio values.
179 INSERT TABLE 4 HERE

180

181 Discussion

182 The purpose of the current study was to investigate the capacity of JTAR and JTVR models

183 to represent the elbow mechanical function from modeling and applied points of views.

184 We first compared their ability to fit the experimental data and then interpreted their
185 physiological parameters in comparison to the literature.

186 Differences in fitting experimental data were observed. Both choices of JTAR and JTVR

187 showed statistical effects and interaction on maximal torque prediction. For JTAR, [H1]

188 was not verified, since the asymmetrical models displayed significantly larger errors than

189 two of the symmetrical models (quadratic and cosinus). This may be explained by

190 interactions between maximal forces and moment arms of multiple muscles crossing the

191 elbow [6]. For the elbow joint, the quadratic JTAR appears as the most accurate and

192 adaptable model on a large cohort. Those results might be joint-specific. As reported in

193 our preliminary study [14], average prediction error is increased by the JTVR. Associated

194 with one of the two best JTAR (i.e. quadratic, cosinus, cubic), the power-based and

195 Anderson-based models gave similar prediction levels. Although the new physiological

196 parameters may improve the model meaningfulness, it did not improve its compliance to

197 fit measured data. Especially, the additional constraint added to ensure derivative

198 continuity between concentric and eccentric parts of the model seemed to decrease

199 eccentric fitting efficiency of the model. However, it ensured that the torque envelope

200 generated by the fitting method was continuous, that may not be the case with the
Anderson-based one. Globally, correlations between measured and predicted JTAVR were weaker in the current study than in other studies in the literature that focused on lower limbs [14,21]. This result may be due to the larger anatomical variability of the upper limb. Moreover, larger misalignment between elbow and dynamometer axes may arise during motion because of the equipment, since the arm position cannot be as controlled as the thigh on the dynamometer. Quantification and correction of this problem could improve the fitting quality and the subsequent parameter estimation. Specific joint strengths models dedicated to fitting may also have been of interest to be tested here [5,36]

For applied perspectives, physiological parameters obtained through optimization of the models seem coherent with the literature. For a group of young healthy men, maximal isometric torque $Γ_{max}$ between 63 N.m and 69 N.m in flexion [2,32], and balanced flexion-extension ratios between .95 and .97 are similar to literature [2,33]. Range of motion $RoM$ obtained with all models is larger than the anatomical reference [34], probably due to an extrapolation of muscle strength beyond realistic elbow configuration due to bony limits. Average optimal angle, $α_0$, obtained for elbow flexion and extension with all models are consistent with the literature [35–37]. For both flexion and extension, $α_0$ found for normal (79°,76°), cosinus (77°,72°), and quadratic (77°,72°) models was close to classically observed average angles [38,39], while cubic (59°,56°) and sinus-exponential (112°,99°) models values were at boundaries. Only $Γ_{max}$ obtained with cubic and sinus-exponential
models, and $\alpha_0$ obtained with the normal model differed by more than 10% from
literature values [32,38].

Concerning concentric and eccentric parameters, concentric velocity at maximal power, $\omega_{Pmax}$, was our focus. Due to its relationship with muscle composition [30], linking mechanical and physiological muscle functions, the implementation of this parameter in the model seemed interesting for applications in sports, rehabilitation or ergonomics [40]. Optimized $\omega_{Pmax}$ values, between 404°.s⁻¹, and 561°.s⁻¹, are about two times larger than $\omega_{Pmax}$ values only based on isokinetic dynamometer for thigh muscles [29]. Previous study showed that measured $\omega_{Pmax}$ correlated better (r=.55) with fiber composition when corrected with a Hill-type model as in our study [41] and could have values between 215°.s⁻¹ and 539°.s⁻¹. Although correlations are seen, only 51.8% of the fiber composition variance was explained by $\omega_{Pmax}$ [42]. For further work, a combination of $\omega_{Pmax}$ measurements with Hill-model correction and electromyography could be investigated [43] and a validation of muscle composition prediction through this technique should be performed.

For the other concentric and eccentric parameters, maximal concentric velocity, $\omega_{max}$, between 1268°.s⁻¹ and 1531°.s⁻¹ in flexion and, 1368°.s⁻¹ and 1667°.s⁻¹ in extension were found. That lays below values reported for baseball players, middle-aged, and elderly men respectively [43,44]. The normal and sinus-exponential models produced the smallest values. For maximal eccentric velocity, $\omega_{min}$, no conclusive reference data were found. In general, physiological parameters related to velocity were obtained by extrapolation of our model beyond highest velocity measured in this study. Since no direct measurement
244 for much higher velocities was possible with such dynamometers, these values cannot be directly validated.

245 Since the degree of meaningfulness of the power-based JTVR was higher, without decreasing significantly the data fitting, [H2] seemed verified.

248 To conclude, five JTAR and two JTVR were compared when fitting experimental dynamometric measurements from modelling and applied perspectives. While a quadratic torque-angle model fitted best the data, a new proposed JTVR increased physiological transparency and clinical relevance without decreasing significantly the data fitting. The study highlights the needs for improvement of dynamometric measurement accuracy for the upper limb and the importance of the meaningfulness of the physiological parameters to be optimized when fitting these models to data.

256

257 ACKNOWLEDGMENT

258 The authors want to acknowledge INRIA who funded a post-doctoral scholarship on this project.

261

262 CONFLICT OF INTEREST

263

264 The authors encountered no conflict of interest for the current study.

265
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376 [41] Maclntosh, B. R., Herzog, W., Suter, E., Wiley, J. P., and Sokolosky, J., 1993, "Human Skeletal Muscle Fibre Types and Force: Velocity Properties," Eur. J. Appl. Physiol., **67**(6), pp. 499-506.

379 [42] Suter, E., Herzog, W., Sokolosky, J., Wiley, J. P., and Macintosh, B. R., 1993, "Muscle Fiber Type Distribution as Estimated by Cybex Testing and by Muscle Biopsy.," Med. Sci. Sports Exerc., **25**(3), pp. 363-370.

382 [43] Kim, C., Gao, Q., Kim, W., and Kim, W., 1994, "Muscle Fiber Type Distribution Estimated by Non-Invasive Technique: Based on Isometric Force and Integrated Electromyography," Clin. Sci., **87**(s1), p. 107.

385 [44] Toji, H., and Kaneko, M., 2007, "Effects of Aging on Force, Velocity, and Power in the Elbow Flexors of Males," J. Physiol. Anthropol., **26**(6), pp. 587-592.

387
Figure Captions List

388
389

Fig. 1 Experimental set up. The participant is seated and attached to the ConTrex dynamometer in upright position with the arm along his side. The axis of the dynamometer is aligned with the epicondylitis axis with the elbow flexed at 90°.

Fig. 2 Normalized torque-angle relationship as defined by the five mathematical models: normal, cosinus, and quadratic models are symmetrical; cubic and sinus-exponential are asymmetrical.

Fig. 3 Normalized torque-velocity models and constraint parameters: in the Anderson-based model (A), we find one derivative constraint at $\omega_{max}$, two independent constraints at $-\omega_{max}$ and $\omega_0$, and two dependent constraints at $\omega_{T.5}$ and $\omega_{T.75}$; in our power-based model (B), we defined three derivative constraints and three independent constraints at $\omega_{max}$, $\omega_{max}$, and $\omega_0$, and an additional derivative constraints at $\omega_{Pmax}$ on the power-velocity relationship.

Fig. 4 Effects of torque-angle models (A) and interaction between the torque-angle and torque-velocity models (B) on average prediction errors. Colored stars represent individuals (one color = one subject). Black dots, blue dots and red diamonds represent the average of all individuals. Asterisks indicate significant difference between means.

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391
Table Caption List

392
393

Table 1 Adjusted correlation coefficients between measured and predicted maximal torque for all models in elbow flexion and extension for each type of data: isometric, concentric and eccentric.

Table 2 Optimal elbow torque-angle parameters obtained with the five isometric models.

Table 3 Optimal elbow torque-velocity parameters of the new model obtained with the five isometric models.

Table 4 Optimal elbow eccentric to concentric ratios obtained with Anderson-based model and the new model combined with each of the five isometric models.

394
395 Figure 1– Experimental set up. The participant is seated and attached to the ConTrex

396 dynamometer in upright position with the arm along his side. The axis of the

397 dynamometer is aligned with the epicondylitis axis with the elbow flexed at 90°.

<figure><img src="image_2.png" /></figure>
Figure 2 – Normalized torque-angle relationship as defined by the five mathematical models: normal, cosine, and quadratic models are symmetrical; cubic and sinus-exponential are asymmetrical.

<figure><img src="image_3.png" /></figure>

403
404
Figure 3 - Normalized torque-velocity models and constraint parameters: in the Anderson-based model (A), we find one derivative constraint at $\omega_{max}$, two independent constraints at $-\omega_{max}$ and $\omega_0$, and two dependent constraints at $\omega_{R.5}$ and $\omega_{R.75}$; in our power-based model (B), we defined three derivative constraints and three independent constraints at $\omega_{max}$, $\omega_{max}$, and $\omega_0$, and an additional derivative constraint at $\omega_{Pmax}$ on the power-velocity relationship.

<figure><img src="image_4.png" /></figure>
Figure 4 – Effects of torque-angle models (A) and interaction between the torque-angle and torque-velocity models (B) on mean errors computed as the average difference between maximal torque measured on the dynamometer and maximal torque predicted by the models. Colored stars represent individuals (one color = one subject). Black dots, blue dots and red diamonds represent the average of all individuals. Asterisks indicate significant difference between means.

418

<figure><img src="image_5.png" /></figure>
Table 1. Adjusted correlation coefficients between measured and predicted maximal torque for all models in elbow flexion and extension for each type of data: isometric, concentric and eccentric.

<table><thead><tr><th colspan="7">R<sup>2</sup></th></tr><tr><th></th><th></th><th></th><th>Normal</th><th>Cosinus</th><th>Quadratic</th><th>Cubic</th><th>Sinus-exponential</th></tr></thead><tbody><tr><td rowspan="12">Anderson-based model</td><td rowspan="3">flexion</td><td>Isometric</td><td>.62</td><td>.86</td><td>.87</td><td>.84</td><td>.74</td></tr><tr><td>Concentric</td><td>.58</td><td>.62</td><td>.62</td><td>.61</td><td>.61</td></tr><tr><td>Eccentric</td><td>.75</td><td>.75</td><td>.75</td><td>.74</td><td>.77</td></tr><tr><td rowspan="3">extension</td><td>Isometric</td><td>.66</td><td>.88</td><td>.89</td><td>.87</td><td>.75</td></tr><tr><td>Concentric</td><td>.74</td><td>.75</td><td>.75</td><td>.76</td><td>.74</td></tr><tr><td>Eccentric</td><td>.71</td><td>.77</td><td>.77</td><td>.76</td><td>.74</td></tr><tr><td rowspan="12">Power-based model</td><td rowspan="3">flexion</td><td>Isometric</td><td>.62</td><td>.86</td><td>.87</td><td>.84</td><td>.74</td></tr><tr><td>Concentric</td><td>.64</td><td>.70</td><td>.70</td><td>.68</td><td>.68</td></tr><tr><td>Eccentric</td><td>.60</td><td>.64</td><td>.64</td><td>.62</td><td>.64</td></tr><tr><td rowspan="3">extension</td><td>Isometric</td><td>.66</td><td>.88</td><td>.89</td><td>.87</td><td>.75</td></tr><tr><td>Concentric</td><td>.81</td><td>.85</td><td>.85</td><td>.85</td><td>.82</td></tr><tr><td>Eccentric</td><td>.64</td><td>.70</td><td>.70</td><td>.70</td><td>.67</td></tr></tbody></table>

424
425 Table 2. Optimal elbow torque-angle parameters obtained with the five isometric models.

426 Values displayed in the table correspond to: means ± standard deviations.

<table><thead><tr><th></th><th></th><th>Normal</th><th>Cosinus</th><th>Quadratic</th><th>Cubic</th><th>Sinus-exp</th></tr></thead><tbody><tr><td rowspan="3">FLEXION</td><td>Γ<sub>max</sub> [N.m]</td><td>69 ± 13</td><td>63 ± 11</td><td>63 ± 11</td><td>64 ± 11</td><td>66 ± 12</td></tr><tr><td>RoM [°]</td><td>175 ± 17</td><td>160 ± 33</td><td>155 ± 34</td><td>167 ± 28</td><td>173 ± 24</td></tr><tr><td>α<sub>0</sub> [°]</td><td>79 ± 10</td><td>77 ± 13</td><td>77 ± 14</td><td>102 ± 12</td><td>59 ± 10</td></tr><tr><td rowspan="3">EXTENSION</td><td>Γ<sub>max</sub> [N.m]</td><td>66 ± 18</td><td>60 ± 16</td><td>60 ± 16</td><td>61 ± 17</td><td>64 ± 18</td></tr><tr><td>RoM [°]</td><td>179 ± 4</td><td>167 ± 27</td><td>164 ± 30</td><td>169 ± 23</td><td>173 ± 20</td></tr><tr><td>α<sub>0</sub> [°]</td><td>76 ± 9</td><td>72 ± 11</td><td>72 ± 11</td><td>99 ± 10</td><td>56 ± 9</td></tr></tbody></table>

427
428
Table 3. Optimal elbow torque-velocity parameters of the new model obtained with the five isometric models. Values displayed in the table correspond to: means ± standard deviations.

<table><thead><tr><th colspan="2">Torque-angle model</th><th>Normal</th><th>Cosinus</th><th>Quadratic</th><th>Cubic</th><th>Sinus-exp</th></tr></thead><tbody><tr><td rowspan="3">FLEXION</td><td>ω<sub>max</sub> [°.s<sup>-1</sup>]</td><td>1268 ± 514</td><td>1517 ± 544</td><td>1531 ± 548</td><td>1492 ± 567</td><td>1331 ± 507</td></tr><tr><td>ω<sub>min</sub> [°.s<sup>-1</sup>]</td><td>-173 ± 87</td><td>-301 ± 276</td><td>-369 ± 500</td><td>-330 ± 477</td><td>-238 ± 241</td></tr><tr><td>ω<sub>Pmax</sub> [°.s<sup>-1</sup>]</td><td>404 ± 231</td><td>495 ± 242</td><td>502 ± 500</td><td>490 ± 255</td><td>426 ± 225</td></tr><tr><td rowspan="3">EXTENSION</td><td>ω<sub>max</sub> [°.s<sup>-1</sup>]</td><td>1368 ± 647</td><td>1636 ± 747</td><td>1667 ± 754</td><td>1605 ± 738</td><td>1458 ± 683</td></tr><tr><td>ω<sub>min</sub> [°.s<sup>-1</sup>]</td><td>-297 ± 329</td><td>-509 ± 464</td><td>-531 ± 482</td><td>-536 ± 530</td><td>-391 ± 410</td></tr><tr><td>ω<sub>Pmax</sub> [°.s<sup>-1</sup>]</td><td>432 ± 291</td><td>551 ± 343</td><td>563 ± 347</td><td>537 ± 337</td><td>473 ± 305</td></tr></tbody></table>

432

433
Table 4. Optimal elbow eccentric to concentric ratios obtained with Anderson-based model and the new model combined with each of the five isometric models. Values displayed in the table correspond to: means ± standard deviations.

<table><thead><tr><th></th><th></th><th></th><th>Normal</th><th>Cosinus</th><th>Quadratic</th><th>Cubic</th><th>Sinus-exp</th></tr></thead><tbody><tr><td rowspan="2">FLEXION</td><td rowspan="2">Γ<sub>ECC</sub>/Γ<sub>CON</sub></td><td>Anderson-based</td><td>1.17 ±0.07</td><td>1.18 ±0.08</td><td>1.18 ±0.08</td><td>1.18 ±0.08</td><td>1.18 ±0.08</td></tr><tr><td>ω<sub>max</sub>-based</td><td>1.16 ±0.15</td><td>1.20 ±0.21</td><td>1.20 ±0.22</td><td>1.20 ±0.21</td><td>1.17 ±0.16</td></tr><tr><td rowspan="2">EXTENSION</td><td rowspan="2">Γ<sub>ECC</sub>/Γ<sub>CON</sub></td><td>Anderson-based</td><td>1.21 ±0.10</td><td>1.22 ±0.10</td><td>1.22 ±0.10</td><td>1.22 ±0.10</td><td>1.21 ±0.10</td></tr><tr><td>ω<sub>max</sub>-based</td><td>1.17 ±0.15</td><td>1.34 ±0.30</td><td>1.32 ±0.30</td><td>1.31 ±0.29</td><td>1.21 ±0.20</td></tr></tbody></table>

437
438
SUPPLEMENTARY MATERIAL

USING TORQUE-ANGLE AND TORQUE-VELOCITY MODELS TO
CHARACTERIZE ELBOW MECHANICAL FUNCTION: MODELING
AND APPLIED ASPECTS

Diane Haering, Charles Pontonnier, Nicolas Bideau, Guillaume Nicolas, Georges

Dumont

submitted to Journal of Biomechanical Engineering.

For the purpose of concision, the mathematical models used in the paper are extensively
presented here.

1. Joint Torque Angle Relationships (JTAR)

$$
\text{Normal [8]} \qquad \Gamma(\bar{\alpha}) = e^{-\frac{1}{2}(6\bar{\alpha})^2} \qquad (1)
$$

$$
\text{Cosinus [12]} \qquad \Gamma(\bar{\alpha}) = \cos(\pi\bar{\alpha}) \qquad (2)
$$

Quadratic [9,15] \qquad $\Gamma(\bar{\alpha}) = -4\bar{\alpha}^2 + 1$ \qquad (3)

Cubic [11] \qquad $\Gamma(\bar{\alpha}) = \frac{27}{4}\bar{\alpha}^3 - \frac{27}{8}\bar{\alpha}^2 - \frac{27}{16}\bar{\alpha} + \frac{27}{32}$ \quad (4)

Sinus-exponential [13] \qquad $\Gamma(\bar{\alpha}) = \frac{1}{2}\sin(1.919\pi e^{\bar{\alpha}}) + \frac{1}{2}$ \quad (5)

In all of these models, the normalized maximal torque ($\bar{\Gamma} = \frac{\Gamma}{\Gamma_{max}}$) depended on the maximal

isometric torque and on the joint angle to optimal joint angle distance normalized by the maximal

range of isometric force production ($\bar{\alpha} = \frac{\alpha - \alpha_0}{RoM}$) as presented in table 1. Those models are shown

in figure 1. All coefficients used for the normal, quadratic, cubic and sinus-exponential models

were obtained by solving the system of equations expressing the following constraints:

$\bar{\Gamma}(\bar{\alpha}_0) = 1,$

$\bar{\Gamma}(\bar{\alpha}_{min}) = 0,$

$\bar{\Gamma}(\bar{\alpha}_{max}) = 0, \text{ and}$

$\bar{\alpha}_{max} - \bar{\alpha}_{min} = 1; \bar{\alpha}_{max}$ and $\bar{\alpha}_{min}$ being normalized values of maximal and minimal angles.
461

<figure><img src="image_6.png" /></figure>

462

463

Figure 1 – Normalized torque-angle relationship as defined by the five mathematical models: normal, cosine, and quadratic models are symmetrical; cubic and sinus-exponential are asymmetrical.

464

465

466

467

<table><caption>Table 1: JTAR parameters</caption><thead><tr><th colspan="2">Parameters</th><th>Limits</th><th>468</th></tr></thead><tbody><tr><td colspan="2"></td><td>469</td><td></td></tr><tr><td colspan="2"></td><td>470</td><td></td></tr><tr><td rowspan="2">Γ<sub>max</sub></td><td rowspan="2">Max. isometric torque</td><td rowspan="2">0.75 Γ<sub>meas</sub></td><td>471</td><td></td></tr><tr><td>472</td><td></td></tr><tr><td rowspan="3">RoM</td><td rowspan="3">Max. isometric range of motion</td><td rowspan="3">0</td><td>474</td><td></td></tr><tr><td>475</td><td></td></tr><tr><td>476</td><td></td></tr><tr><td rowspan="2">α<sub>0</sub></td><td rowspan="2">Isometric optimal angle</td><td rowspan="2">π/6</td><td>5π/67</td><td></td></tr><tr><td>478</td><td></td></tr><tr><td colspan="2"></td><td>479</td><td></td></tr></tbody></table>

480

481
## 2. Joint Torque Velocity Relationships (JTVR)

### 2.1. JTVR parameters

This section presents the parameters exploited in the models presented below.

<table><caption>Table 2. JTVR parameters</caption><thead><tr><th>Model</th><th colspan="3">Parameters</th><th colspan="2">Limits</th></tr></thead><tbody><tr><td rowspan="3">Anderson-based torque-velocity model</td><td>P<sub>1</sub></td><td>ω<sub>R</sub><sub>.75</sub></td><td>Velocity at 75% of maximal isometric torque</td><td>0</td><td>π</td></tr><tr><td>P<sub>2</sub></td><td>ω<sub>R</sub><sub>.5</sub>/ω<sub>R</sub><sub>.75</sub></td><td>Ratio between velocities at 50% and 75% of maximal isometric torque</td><td>1.9</td><td>2.1</td></tr><tr><td>P<sub>3</sub></td><td>E</td><td>Eccentric to concentric torque index</td><td>.1</td><td>.8</td></tr><tr><td rowspan="4">Power-based torque-velocity model</td><td>P<sub>1</sub></td><td>ω<sub>max</sub></td><td>Max. concentric velocity</td><td>π/3</td><td>5π</td></tr><tr><td>P<sub>2</sub></td><td>ω<sub>Pmax</sub></td><td>Velocity at maximal power</td><td>0.25</td><td>0.4</td></tr><tr><td>P<sub>3</sub></td><td>ω<sub>min</sub>/ω<sub>max</sub></td><td>Max. eccentric to concentric velocity ratio</td><td>-1</td><td>-0.1</td></tr><tr><td>P<sub>4</sub></td><td>Γ<sub>ECC</sub>/Γ<sub>CON</sub></td><td>Max. eccentric to concentric torque ratio</td><td>1.1</td><td>1.8</td></tr></tbody></table>

### 2.2. Anderson-based model

$$ \begin{cases} \Gamma(\omega) = \frac{2P_1P_2+\omega(P_2-3P_1)}{2P_1P_2+\omega(2P_2-4P_1)}, & \omega \ge 0 \\ \Gamma(\omega) = \left(\frac{2P_1P_2+\omega(P_2-3P_1)}{2P_1P_2+\omega(2P_2-4P_1)}\right)(1-P_3\omega), & \omega < 0 \end{cases} \quad (1) $$

where $P_1$ is the velocity at 75% of maximal isometric torque, $P_2$ is the ratio between velocities at 50% and 75% of maximal isometric torque, and $P_3$ is a maximal eccentric to maximal concentric torque index.
2.3. Power-based torque-velocity model 496

$$
\begin{equation}
\begin{cases}
\Gamma(\bar{\omega}) = 0, & \mathbf{1} \le \bar{\omega} \\
\Gamma(\bar{\omega}) = a_1 \bar{\omega}^3 + b_1 \bar{\omega}^2 + c_1 \bar{\omega} + d_1, & \mathbf{0} \le \bar{\omega} < \mathbf{1} \\
\Gamma(\bar{\omega}) = a_2 \bar{\omega}^3 + b_2 \bar{\omega}^2 + c_2 \bar{\omega} + d_2, & \mathbf{P_3} \le \bar{\omega} < \mathbf{0} \\
\Gamma(\bar{\omega}) = P_4, & \bar{\omega} < \mathbf{P_3}
\end{cases}
\tag{2}
\end{equation}
$$

498 with:

$$
499 \qquad \bar{\omega} = \frac{\omega}{P_1},
$$

$$
500 \qquad a_1 = - \frac{3P_2^2 - 4P_2 + 1}{4P_2^3 - 6P_2^2 + 2P_2},
$$

$$
501 \qquad b_1 = \frac{2P_2^3 - 3P_2 + 1}{2P_2^3 - 3P_2^2 + P_2},
$$

$$
502 \qquad c_1 = \frac{8P_2^3 - 9P_2^2 + 1}{4P_2^3 - 6P_2^2 + 2P_2},
$$

503 \qquad $d_1 = 1,$

504 and

505 \qquad $a_2$

$$
506 \qquad = - \frac{P_3 - 4P_2 - 9P_2^2 P_3 - 12P_2^2 P_4 + 8P_2^3 P_3 + 8P_2^3 P_4 + 12P_2^2 - 8P_2^3 + 4P_2 P_4}{2P_3^3 (2P_2^3 - 3P_2^2 + P_2)},
$$

$$
507 \qquad b_2 = \frac{2P_2^3 - 3P_2 + 1}{2P_2^3 - 3P_2^2 + P_2},
$$

$$
508 \qquad c_2 = \frac{8P_2^3 - 9P_2^2 + 1}{4P_2^3 - 6P_2^2 + 2P_2},
$$

509 \qquad $d_2 = 1,$

510 where $P_1$ is the maximal concentric velocity, $P_2$ is the velocity at maximal power, $P_3$
511 is maximal eccentric velocity to maximal concentric velocity ratio, and $P_4$ is the
512 maximal eccentric to maximal concentric torque ratio.

513

514
515
516
517

**2.4. Model representations**

<figure><img src="image_7.png" /><figcaption>Figure 2 – Normalized torque-velocity models and constraint parameters: in the Anderson-based model (A), we find one derivative constraint at $\omega_{max}$, two independent constraints at $-\omega_{max}$ and $\omega_0$, and two dependant constraints at $\omega_{r.5}$ and $\omega_{r.75}$; in the power-based model (B), we defined three derivative constraints and three independent constraints at $\omega_{max}$, $\omega_{max}$, and $\omega_0$, and an additional derivative constraints at $\omega_{Pmax}$ on the power-velocity relationship.</figcaption></figure>

518
519
520
521
522
523
524
## 3. Visualization of model to measurement fitting

### 3.1. Joint Torque Angle Relationships

<figure><img src="image_8.png" /><figcaption>Figure 3 – Example of comparison between normal, cosines, quadratic, cubic and sinus exponential JTAR models fitting on isometric maximum torque measurements (black squares).</figcaption></figure>
532
533

**3.2. Joint Torque Angle & Joint Torque Velocity Relationships**

<figure><img src="image_9.png" /></figure>

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535
536
537

<caption>Figure 4 – Example of a combined JTAR and JTVR model mesh fitting on isometric maximum (dots) and isokinetic (lines) torque measurements for a. elbow flexion, and b. elbow extension.</caption>