1. Introduction
The knowledge of the maximum forces that an individual can exert on the environment allows one to assess his ability to perform a given task without exceeding physiological limits. In relation to the required force level, it is an important factor to consider when evaluating a workplace using an ergonomics assessment method such as RULA [
1,
2], REBA [
3] or OCRA [
4]. Maximum force can be used as an objective indicator for defining criteria for discomfort evaluation [
5]; and in the framework of digital human modeling, it is needed for tuning muscle fatigue models’ parameters [
6]. Furthermore, it can contribute to a more efficient human–robot collaboration in manufacturing [
7,
8] or in rehabilitation contexts [
9]. Indeed, if a robotic system is aware of the limits of the human operator or patient, it can evaluate more efficiently the assistance it needs to provide. Under a static condition, the distribution of the maximum external forces that the upper-limb can apply to the outside world, defined as the force feasible set (FFS), is known to be anisotropic and posture-dependent [
10,
11]. The dependence of the FFS on the posture is due to the muscle length, which determines the muscle’s isometric force; muscular moment arms, which contribute to the joint torques; and the Jacobian matrix of the upper limb, which links these joint torques to the external force at the hand. Additionally, due to the anisotropy of the FFS, external force amplitude can be very different according to the direction of these forces. Therefore, the knowledge of a maximal value for a limited set of directions, as is usually found in the literature [
12,
13,
14,
15], may not be sufficient to globally represent the force capabilities of an individual. It is believed that existing formalisms used in robotics could be interesting for evaluating the FFS of the hand in a given upper-limb posture because they have the ability to characterize the FFS in every Cartesian direction given the posture and assumptions of maximum joint torques. Two variants are available: the force ellipsoid (FE) [
16] and the force polytope (FP) [
17]. In the biomechanics field, the FE and FP can be adapted by considering measured maximum joint torques [
11,
18,
19]. To apply these formalisms to human limbs, it must be noted that the maximum joint torques change as a function of joint angle and direction of rotation [
20,
21,
22], and also depend on the other joint posture because of the presence of multi-articular muscles. The models based on measured joint torques are called the scaled force ellipsoid (SFE) and scaled force polytope (SFP) [
11,
19]. Considering the human upper-limb, few studies that were only conducted in the horizontal plane compared the SFE and SFP with measured forces [
10,
11,
18]. For different elbow flexion values, a comparison between FFS predictions with FE, SFE, FP and SFP was done, but no ground truth measures of maximal forces exerted at the hand was proposed [
23]. Additionally, some preliminary results on one subject with both joint torques and 3D force measurements were provided in [
24]. In this framework, one purpose of the present study was to evaluate the four models of FFS (FE, SFE, FP and SFP) from recordings of human upper-limb posture and maximum joint torques with a dedicated dynamometer. The second purpose was to compare the modeled FFSs to a measured force polytope (MFP) constructed from the measurement of maximal forces in a set of directions obtained from a force sensor. Different parameters were considered in order to compare modeled FFSs and the MFP. They characterize the force amplitudes (maximum predicted force and volume which produces an overall evaluation) but also the shapes of the FFSs (more or less elongated) and their orientations (angle between the main axes of the FFS). In addition, the RMS error was calculated for all points on the surface of the modeled FFS and the MFP. It was hypothesized that SFE would underestimate the MFP, and the SFP would overestimate it thanks to their assumptions. It was also hypothesized that shape and orientation would be correctly predicted by the models.
3. Results
In this section, the results of the comparison between the parameters of MFP and those of FE, FP, SFE and SFP are presented.
The measured joint torques are summarized in
Table 2. The measured joint angles during the joint torques and force measurements were relatively close despite two significant differences (
Table 3). The mean(SD) of joint angles displayed on the first line of
Table 3 correspond to that of the torques measurement on the dynamometer for the considered degree of freedom. The second line provides the mean(SD) of the joint angles of the complete upper-limb dofs during the force measurements.
The ANOVA on isotropy showed significant main effect (F(4, 24) = 8.40,
p < 0.05) and the post hoc test revealed lower value of isotropy for the MFP compared with the FE, SFE, FP and SFP (FE: 0.969 ± 0.006, SFE: 0.970 ± 0.010, FP: 0.971 ± 0.007, SFP: 0.966 ± 0.015 vs. MFP: 0.926 ± 0.034,
p < 0.05 for each comparison) (
Figure 3).
The FE and FP volumes and maximal forces were not included in the statistical analysis because of the unitary joint torques assumption. The ANOVA on volume showed significant main effect (F(2, 12) = 10.081,
p < 0.05) and post hoc test indicated that the SFE volume was lower than that of MFP (SFE:
±
N
vs. MFP:
±
N
,
p < 0.05) (
Figure 4).
The ANOVA on maximal forces showed a significant main effect (F(2, 12) = 27.298,
p < 0.05). The Dunnett post hoc test indicated that the MFP maximal force (
Figure 5) was lower than that of SFP and higher than that of SFE (SFE: 329.2 ± 79.5 N vs. MFP: 518.1 ± 134.4 N,
p < 0.05; SFP: 619.0 ± 160.3 N vs. MFP,
p < 0.05) (
Figure 5).
Results concerning the angles between the main axis showed a significant main effect (F(3, 18) = 14.192,
p < 0.05). Tukey’s post-hoc indicated that both
SFE/MFP and
SFP/MFP are significantly smaller than
FE/MFP and
FP/MFP (
p < 0.05) (
Table 4).
As an example, the sphere of RMS errors between MFP, SFE and SFP is given for a subject in
Figure 6 and RMS errors data are provided in
Table 5.
4. Discussion
The objective of this study was to evaluate four prediction models of force feasible sets (FE, SFE, FP and SFP) against force measurements (MFP). Comparisons were made on various representative parameters such as global orientation, volume, isotropy and maximum force. In addition, detailed prediction errors were evaluated. These results are original and constitute the first comparison of 3D modeling of FFS with both hand force and joint torque measurements on the upper limb. As such, these results significantly complement and enrich previous studies that involved only a very small number of subjects [
10,
11,
18] or only considered maximal force [
40,
41]. We will now comment on the differences observed in light of the data in the literature, while taking a critical look at the hypotheses retained for the models. In addition, our results allow us to discuss the relative importance of the postural component, i.e., that related to the Jacobian matrix, compared to that related to the joint torques.
Data for measured isometric maximum joint torques were consistent with literature data for similar postures and in the same population [
20,
22,
42,
43]. For example, shoulder joint torque values for extension and abduction for a healthy population aged 20–29 years are 91.9 (19.7) [
42] N·m and 60.2 (14.0) N·m [
22] compared to 87.5 (25.7) N·m and 68.7 (20.0) N·m in this study, respectively.
The study of the literature enabled us to highlight the anisotropy of the FFSs, i.e., the difference in the amplitude of the forces according to the direction of application [
10,
11]. All the proposed FFS models had this property. In fact, we can see that the isotropy values were higher than 0.93, characteristic of an elongated shape. Consequently, a preferential direction of the force was obtained, generally oriented in the antero-posterior and downward direction (
Figure 7). The isotropy of the MFP was statistically lower than that of the four models. The more elongated shape of the SFE compared to the MFP was due to the fact that there was an underestimation of the forces along the smaller axes of the ellipsoid compared to the preferential axis. For the SFP, the opposite trend was observed with a correct estimate along the smaller axes and an overestimation along the major axis (
Figure 8).
The volume of the SFE significantly underestimated the overall measured force production capacities (MFP) whereas no statistical difference was found between the MFP and the SFP. For the latter, a significant inter-individual variability was observed and no clear trend appeared with respect to the MFP with, depending on the subjects, lower or greater values. It thus appeared that the assumptions chosen for the type of joint torques norm had a strong influence on the volume of the FFS models.
Moreover, compared to the MFP, there was a significant underestimation of the maximum force by more than 36.5% by the SFE and an overestimation of 19.5% by the SFP. Once again, the assumptions used to construct the models significantly affected the observed results. Indeed, the definition of the ellipsoids was based on the use of an Euclidian-type norm in order to define the joint torques limits. Thus, when a joint torque was maximum at the level of one dof, the others were all equal to zero, which was not very representative of an individual’s motor control. In the musculoskeletal system, when a maximum joint torque is produced at the shoulder, it is possible to generate a joint torque at the elbow. Conversely, for polytopes, the infinite type norm allowed all torques to be at their maximum values simultaneously, which also seemed physiologically unrealistic. Thus, as expected, there was an underestimation of the FFS with SFE and an overestimation with SFP. The fact that a maximum torque on one dof leads to a null value on the others limits the force production evaluation. Similarly, the possibility that all torques are maximum does not take into account the coupling between the different degrees of freedom. The latter is due to the presence of bi and pluri-articular muscles of the musculoskeletal system.
The parameter that was best evaluated by the models was the direction of the main axis of the MFP. In the case of ellipsoids, the maximum forces were exerted according to the latter. For polytopes, the SVD allowed one to find the principal axis taking into account globally all its vertices. Thus, this axis may not correspond exactly to the maximum forces. In the present case, the angle between the principal axis of the FE, the SFE, the FP and the SFP with that of the MFP was between 7.4 ± 3.3° and 14.8 ± 3.0° (absolute value: 2.9 to 19.6°) (
Table 4). In terms of overall orientation, the biomechanical models (SFE and SFP) provided the best results compared to the MFP.
The characteristics of the proposed FFS models depend on three parameters: the posture of the kinematic chain represented by the Jacobian matrix, the joint torques and the types of norms used to express the limits of the latter. For the considered posture, the results suggest that the isotropy is little affected by the type of model, because the variations observed between them were small (between 0.966 ± 0.015 for the SFP and 0.971 ± 0.007 for the FP: 0.971 ± 0.007). Thus, one could deduce that for this characteristic of the FFSs, the postural component is preponderant. This observation is supported by [
23], which showed that depending on the elbow flexion angle, isotropies vary significantly between nonscaled and scaled models. It was found that the incorporation of measured values of maximum joint torques (SFE and SFP) improved the prediction of the orientation of the FFS models with respect to the MFP. The appeal of such information is twofold [
19,
44,
45]. The FFS provides information on the set of forces that the upper limb can exert on the environment. In particular, some directions allow for greater forces than others because of the anisotropy of the FFS. If for a given task corresponding to a given direction of force (e.g., drilling), the operator’s posture causes the largest axis of the FFS to be aligned with the direction of the applied force; this might suggest that the posture is chosen so as to maximize the possible force in the direction of interest. In other words, it would mean that the operator has chosen his posture in such a way that he can exert the greatest possible force given the capacities of his musculature. Moreover, the greater the force to be applied, the more the possible choices in terms of the combination of muscular forces should be reduced and the main axis of the FFS should be aligned with the force direction related to the task [
19,
45,
46]. Based on this information, it may be possible to modify the operator’s posture in order to maximize the transmission of force and thus limit the joint torques and reaction forces at the origin of potential musculoskeletal disorders [
44]. The exertion of high intensity forces, handling heavy load over long period of time and working in unfavorable postures are main factors contributing to the incidence of musculoskeletal disorders. High joint torques may be associated with high muscular forces and increased joint reaction forces responsible for muscle and articular tissue damage leading to musculoskeletal disorders. While the results concerning FFS orientations seemed satisfactory, those related to the maximum amplitude of forces must be improved because of the significant variability of the SFP and the differences between the SFE and the MFP. The implementations of the FE and the FP are the simplest because they only require knowledge of the Jacobian matrix and therefore of the posture of the upper limb model. However, these two formalisms were the ones for which the orientation errors of the principal axis with respect to the MFP main axis were the most important. Moreover, they do not give any information on the amplitudes of maximum force. Their use could therefore be envisaged if the degree of precision required on orientation is less crucial (of the order of 15°). The SFE and the SFP provided more precise information and should be preferred. If we consider the results relating to volume and RMSE errors (
Table 5), the SFP clearly appeared to be the best performing model, with, in particular, a percentage of points with an RMSE < 50 N significantly higher than that of the SFP (56.2 (12) vs. 39.7 (12.0)). However, the prediction error on maximum force was greater for the SFP compared to that of the SFE (305.6 (85.3) vs. 244.0 (79.1) N). Therefore, overall, the SFP was the model that seemed the most convincing.
This work has several limitations. Indeed, only one posture has been tested and this analysis should be extended to other cases. One of the limitations concerns the value of joint angles during the measurement of joint forces and torques. Indeed, the configuration of the dynamometer did not always allow one to have joint angles of the unmeasured dofs very close to those adopted during the force measurement. For example, during shoulder torque measurements it was not possible to flex the elbow sufficiently. Since the maximum torques may depend on the position of adjacent joints, this could have had an effect on the torque value. However, special care was given to having postures as close as possible for the two types of measures. In addition, it is necessary to validate the predictions on a wider range of subjects, especially those with pathologies following trauma (spinal cord injury) or central nervous system damage (stroke). It will then be possible to test whether these models are relevant for detecting impairments [
46,
47]. The evaluation of MFP could be improved by providing 3D visual feedback of applied forces to allow better control of the direction of force application. Model scaling is an important issue in the field of biomechanics, and the evaluation of joint torques requires a long and tedious experimental protocol. This temporal constraint has significant consequences on the quality and relevance of the data, especially in the case of people with severe deficiencies.
However, solutions can be considered to replace the measures with the use of regression equations [
48] or musculoskeletal models [
9,
47,
49,
50]. One could thus consider determining maximum isometric torques and thus reduce the time required to evaluate several postures or an entire gesture. In addition, it could be interesting to verify the sensitivity of the models to variations in postures and joint couples of certain dofs.
In this context, the use of musculoskeletal models seems relevant to determine the FFS. Another important advantage of this type of formalism is the possibility of evaluating the FFSs regardless of the posture adopted. Indeed, the input data no longer come simply from joint torques, but directly from the isometric force generated by the muscles. Obviously, particular care will have to be taken to obtain an adequate model geometry and a scaling of the muscular forces in accordance with the physical capacities of an individual.