A Computationally Efficient Musculoskeletal Model of the Lower Limb for the Control of Rehabilitation Robots: Assumptions and Validation

Round 1

Reviewer 1 Report

Introduction - the last section (lines 103-116) of the introduction could be shortened and only information related to the aims and goals of the study should be presented, as now this last section repeats the abstract and part of the results/methodology. 

Methodology - lines 254-261: various measurement equipment was used, how it was synchronized, and more information about the sampling frequency of the data acquisition is needed. Was data sampled/downsampled to 120 fps? What normalization procedure of EMG recordings was used if any? 

Table 1: STD for max extension angular velocity is missing ("Data" instead of number value). 

 

Author Response

Reviewer 1

Reviewer’s comment 1:

Introduction - the last section (lines 103-116) of the introduction could be shortened and only information related to the aims and goals of the study should be presented, as now this last section repeats the abstract and part of the results/methodology. 

Authors' response:

Following the recommendations of Reviewer 1 and Reviewer 2, we have substantially modified the Introduction. In particular, the last part has been simplified to avoid the redundancies pointed out by the reviewer. 

 

Reviewer’s comment 2:

Methodology - lines 254-261: various measurement equipment was used, how it was synchronized, and more information about the sampling frequency of the data acquisition is needed. Was data sampled/downsampled to 120 fps? What normalization procedure of EMG recordings was used if any?

Authors' response:

We have used a Kinescan human movements analysis system. It consists of a video photogrammetry system and force platforms. Both devices are hardware synchronized and can be sampled at the same rate. In this study, movements and forces were sampled at 120 Hz. The system can connect other measurement systems, EMG, for example, to which it sends a capture trigger signal. EMG recordings were made with Noraxon Myotrace400 equipment, which provides the RMS value of the EMG signal. The RMS signal was recorded with a sampling frequency of 1000 Hz. The RMS signal was subsampled at 100 Hz and then smoothed and interpolated at instants measured by the motion analysis system. For this, we used a smoothing procedure based on cubic local regression.

We used EMG signal data measured in mV and not normalized. We did not normalize the EMG signal since the measurement scale in each subject is irrelevant in the two analyses realized. On the one hand, in the Force -EMG signal correlation, only the monotonic dependence relations between both signals are involved (Spearman's correlation); the result would not change with a scale change associated with normalization. We also used EMG signal to calibrate the model (from the records of the exercises with a load of 0 kg and a load of 12 kg), thus obtaining a relationship between the force (N) and the EMG signal (mV). This relationship is used to estimate the forces in the exercises with a load of 6 kg.

An explanation of these points has been included in the material and methods section:

“The equipment provides the RMS value (root mean square) of the EMG signal sampled at 1000 Hz. This signal was subsampled at 100 Hz, smoothed, and interpolated to obtain the values at the same time instants as Kinescan.”

 

Reviewer’s comment 3:

Table 1: STD for max extension angular velocity is missing ("Data" instead of number value).

Authors' response:

The missing value is std= 22.4(º/s). It has been corrected in the new version of the paper.

 

The authors greatly appreciate the time and effort in reviewing this paper.

Reviewer 2 Report

The present contribution aims at the improvement of a lower limb model for the application in control algorithms of parallel rehabilitation robots. Therefore, an existing dynamic model is reduced omitting inertial effects in order to allow for real-time calculation of the internal forces. Estimation of the muscle forces is done using Lagrange multipliers substituting dynamic optimization. The model facilitates the analysis of one functional degree of freedom. A case study compares simulation results of the developed static geometric model with the original dynamic model as well as empirical measurements.

Overall, the paper is well written (excluding the introduction). The topic is interesting and relevant.

Nevertheless, there are several limitations:

• There should be no citation in the abstract.
• Line 41/42: Sentence makes no sense. Why does the knee joint model directly affect the arm muscle tendon?
• Line 45: Sentence makes no sense - „need in-vivo a complex personalization process“.
• Line 48: Error - „the direction the of axis of rotation“
• Line 49: What was carried out „in-vitro“? Do you mean that the respective joint parameters were determined „in-vitro“? But what should that mean? How was it done?
• Sentence in line 55-56 does not really connect to the sentence before. Furthermore, more detail is necessary for comprehensibility.
• What means „inherent diversity of muscle interactions“? When comparing to the remainder of the sentence, perhaps just a higher number of muscle compartments in the models could be meant. The wording is very unusual.
• Line 66-68: Sentence is confusing. There is written, that solving the redundancy problem either implies the use of complex optimization or the incorporation of Hill-type muscle models. This indicates, that optimization is not necessary as soon as Hill-type muscles are used. This is not the case. That are two totally different aspects.
• Line 82/83: What is meant by in vitro validation in this case?
• Line 83/84: Sentence is not complete.
• Line 87: What does MME mean? Please introduce all abbreviations.
• Line 87-90: Sentence makes no sense. More details would be helpful.
• The introduction is very jumpy and inconsistent. It should be substantially improved and made more consistent from the beginning until line 90.
• Line 129: HJC means hip joint center? Please introduce the abbreviations.
• Line 136: Please describe the functional calibration method more detailed. How are the joint locations determined? How are the inertial parameters determined? Why are the inertial parameters important for the quasi-static model? Later this is even neglected in this paper.
• Are the maximum isometric muscle forces also subject-specifically adjusted?
• Equation 3: It seems, that the pelvis does not introduce a gravitational force. More details regarding the necessity of the gravitational body forces of the upper body/torso including the pelvis would be helpful.
• Equation 6: Where does the physiological cross-sectional area of the muscles come from? Are these subject-specifically?
• Please explain, what in the context of this contribution is meant by real-time. Please compare the necessary calculation speed with standard static optimization procedures as this would also be really fast for such a simplified model. Using static optimization would then even not be dependent on the narrow band around the observed kinematics.
• Line 312: What does a negative knee moment mean? Is it the joint moment generated by the muscles? If yes, how can it be negative?
• Table 2: Why was the 99th percentile chosen and not the maximum value? Have there been extreme outliers? How is it interpreted at that point?
• Figure 4: It is a misconception, that the forces due to gravity are much smaller in a slow movement than the external ground reaction force. The ground reaction force instead counteracts the gravitational forces in order to prevent the body to free fall with gravitational acceleration towards the earth’s center. This effect is probably only because the other masses were neglected in the model. Please also compare comment further above regarding the masses in the model.
• Line 360: Please elaborate on the Force-EMG calibration. It remains unclear how this was done.

To conclude, the paper should be revised according to the mentioned aspects. Thereafter, resubmission is recommended.

Author Response

Reviewer 2

Reviewer’s comment 1:

The present contribution aims at the improvement of a lower limb model for the application in control algorithms of parallel rehabilitation robots. Therefore, an existing dynamic model is reduced omitting inertial effects in order to allow for real-time calculation of the internal forces. Estimation of the muscle forces is done using Lagrange multipliers substituting dynamic optimization. The model facilitates the analysis of one functional degree of freedom. A case study compares simulation results of the developed static geometric model with the original dynamic model as well as empirical measurements. Overall, the paper is well written (excluding the introduction). The topic is interesting and relevant.

The topic is interesting and relevant. Nevertheless, there are several limitations. There should be no citation in the abstract.

Authors' response:

We greatly appreciate this comment. Related to the Introduction section, following reviewer recommendations, we have rewritten the Introduction, from the beginning to line 90, clarifying the points indicated, as detailed below.

On the other hand, the citation has been removed from the abstract. We have changed the sentence to the following:

“Finally, we have developed a force calculation process based on Lagrange multipliers that provides a closed-form solution; in this way, the problem of dynamic indeterminacy is solved without the need to use an iterative process”

 

Reviewer’s comment 2:

Line 41/42: Sentence makes no sense. Why does the knee joint model directly affect the arm muscle tendon?

Authors' response:

The knee joint model determines the relative movements between relevant anatomical points, such as tendon insertion points or tendon contact at the patella. This can affect the lever arms and forces estimated by the model. We have modified the wording of the sentence as follows:

“In effect, the knee kinematics model defines the estimated relative motion between relevant anatomical points, affecting the muscles' lever arms and the calculated forces. Hence the importance of using models that can be adapted to different people and ranges of movement”.

 

Reviewer’s comment 3:

Line 45: Sentence makes no sense - „ in-vivo a complex personalization process“.

Authors' response:

Thanks for the suggestion. We have modified the wording of the sentence as follows:

“Leardini et al. conducted an exhaustive review of the different joint models used in MSM [4]. Simple knee models using ta revolute pair do not adequately represent its kinematics [5]. In contrast, other complex models, such as those presented by [6], provide good movement estimates but require anatomical information that can only be personalized using medical imaging techniques.”

 

Reviewer’s comment 4:

Line 48: Error - „the direction the of axis of rotation“

Authors' response:

Please, see the response to comment 6

 

Reviewer’s comment 5:

Line 49: What was carried out „in-vitro“? Do you mean that the respective joint parameters were determined „in-vitro“? But what should that mean? How was it done?

Authors' response:

Please, see the response to comment 6

 

Reviewer’s comment 6:

Sentence in line 55-56 does not really connect to the sentence before. Furthermore, more detail is necessary for comprehensibility.

Authors' response:

We greatly appreciate this observation. We have rewritten the sentences of comments #4, #5, and #6 as follows:

“However, the most widely used models represent the knee joint using a mobile axis of rotation, whose position and orientation are related to knee flexion angle. This relationship is obtained experimentally in in-vitro studies, which limits its realism and customization capacity [7], mainly when applied to people with a medical condition [8].”

Reviewer’s comment 7:

What means „inherent diversity of muscle interactions“? When comparing to the remainder of the sentence, perhaps just a higher number of muscle compartments in the models could be meant. The wording is very unusual.

Authors' response:

Thank you for your comment. We have rewritten the sentence as follows:

“Second, the models must offer an adequate representation of the contribution of muscle groups to joint movement, which is usually associated with considering a higher number of muscles.”

 

Reviewer’s comment 8:

Line 66-68: Sentence is confusing. There is written, that solving the redundancy problem either implies the use of complex optimization or the incorporation of Hill-type muscle models. This indicates, that optimization is not necessary as soon as Hill-type muscles are used. This is not the case. That are two totally different aspects.

Authors' response:

Following the reviewer comment, we have rewritten this paragraph as follows:

“Generally, it is assumed that increasing the complexity of the model will improve its validity [16]-[17]. However, increasing the number of muscles requires more information about their anatomical and physiological characteristics to calibrate the model [18]. In addition, the resolution of the problem of the redundancy of muscular actions becomes more difficult, requiring complex optimization algorithms with a high computational cost [19]-[20]. For this reason, in many works, adjacent muscles are grouped, or only a single set of muscles is considered when solving the redundancy problem [21].”

 

Reviewer’s comment 9:

Line 82/83: What is meant by in vitro validation in this case?

Authors' response:

Please, see the response to comment 10

Reviewer’s comment 10:

Line 83/84: Sentence is not complete.

Authors' response:

By in vitro validation, we mean the validation of models through experiments with cadaveric specimens, where movement is simulated and forces are measured to compare them with the model estimates, as in the indicated reference [26]. These studies focus on particular aspects, in this example, the representation of the patellofemoral joint to predict its kinematics and quadriceps forces. The results of these studies are difficult to extrapolate to clinical applications, so other procedures based on in vivo measurements of internal forces should be sought, as based on the instrumented prosthesis.

Following the reviewer’s comment, we have modified the wording of lines 82/83/84 to highlight this idea.

“The works with direct validation, i.e., the magnitudes estimated by the model are compared directly with experimental measurements, are scarce, and are often limited to in vitro validations [25]. The results of these studies are difficult to extrapolate to clinical applications, hence the interest in in-vivo validation based on joint contact forces, measured from instrumented knee prostheses [16, 26]. Although it is not a complete validation –only the contact forces are measured, not the muscular ones– the instrumented prostheses are the source of experimental data in vivo with measurements closest to the reality of the internal actions in the lower limb.”

 

Reviewer’s comment 11: 

Line 87: What does MME mean? Please introduce all abbreviations.

Authors' response:

We greatly appreciate this observation. It's a typo; we meant MSM, musculoskeletal models.

 

 

 

 

 

 

Reviewer’s comment 12:

Line 87-90: Sentence makes no sense. More details would be helpful.

Authors' response:

We greatly appreciate this observation. In the new version of the paper, we have modified the wording of the paragraph to highlight the importance of MSM in the operation of rehabilitation robots.

“One of the newest and most interesting applications of MSM is their inclusion within the control system of robotic systems devoted to rehabilitation. The knowledge of the muscular and ligamentous forces during movement allows i) to increase the patient's safety during the process, ii) to monitor the rehabilitation process over time, thus quantifying its progress, and iii) to establish a priori what force levels are to be achieved during movement in specific muscles [27].

Within the mechanical systems that constitute the core of robotic rehabilitation systems, two main types can be distinguished: open kinematic chains, mainly exoskeletons [28-30], and closed kinematic chains, mainly parallel robots.”

 

Reviewer’s comment 13:

The introduction is very jumpy and inconsistent. It should be substantially improved and made more consistent from the beginning until line 90.

Authors' response:

Following the reviewer comment, we have rewritten the introduction following the reviewer's suggestions to highlight the three aspects that must take into account to have models better adapted to their use in clinical applications: i) the possibility of customizing the joint kinematics to different movements and people, ii) simplifying the number of muscle actions to consider and iii) validate the models.

 

Reviewer’s comment 14:

Line 129: HJC means hip joint center? Please introduce the abbreviations.

Authors' response:

We agree with the reviewer. We modified the paragraph to better explain these abbreviations: 

“The joints are modeled as a three degrees-of-freedom (DOF) spherical joint at the hip joint center (HJC), a 1-DOF four-bar mechanism at the knee, and a 1-DOF revolute joint at the ankle joint center (AJC), leading to a total of 5 independent generalized coordinates.”

Reviewer’s comment 15:

Line 136: Please describe the functional calibration method more detailed. How are the joint locations determined? How are the inertial parameters determined? Why are the inertial parameters important for the quasi-static model? Later this is even neglected in this paper.

Authors' response

As indicated on lines 125-126, the functional kinematic calibration procedure (hip center position and adjustment of the knee 4R mechanism) is explained in reference [12]. However, following the reviewer's recommendations, we have expanded this information by modifying the wording, which is as follows.

“The location of the HJC and the parameters of the four-bar mechanism that best match the actual motion of the knee are obtained in an offline stage using a functional calibration procedure [13]. For the location of the HJC, we use the method described in [35]. The four-bar mechanism that represents knee motion is determined using an optimal synthesis procedure based on the formulation of [36]”

On the other hand, the inertial parameters were obtained by rescaling using the relations of Dumas et al. [37], as indicated in line 148. The inertial parameters include the body segment's mass, the center of mass locations, and the moments and products of inertia. As the reviewer suggests, in a quasi-static model, the moments and products of inertia are irrelevant. Only the masses and positions of the mass centers are necessary to define the gravitational forces.

In any case, we have used all the inertial parameters in the comparison between the FDM (Full Dynamic Model) and the SGM (Static Geometric Model)

 

 

Reviewer’s comment 16:

Are the maximum isometric muscle forces also subject-specifically adjusted?

Authors' response:

No, maximal force isometric exercises have not been performed, as the EMG signal has not been normalized, as described in the response to Reviewer 1. Moreover, we didn’t use the Hill model to estimate muscle forces and we preferred to use the optimization technique that solves the redundancy problem by optimization, because of the reasons mentioned previously. In this case, there will be no need for the isometric muscle forces parameter. Note that muscle forces estimated using this technique conform well with the EMG values

 

 

Reviewer’s comment 17:

Equation 3: It seems, that the pelvis does not introduce a gravitational force. More details regarding the necessity of the gravitational body forces of the upper body/torso including the pelvis would be helpful.

Authors' response:

Most of the studies that model the lower limb treat it in a similar way as a serial robot. Its end effector is the foot with an external force as measured by the platform fixed on the ground. In this case, the equations of motion, or inverse dynamic equations, are constructed starting from the end-effector, moving from segment to segment via the interconnecting joint, until the base, which is the pelvis in our case. For example, to determine the generalized force at the ankle joint, the only necessary variables are the external force on the foot, the inertial parameters of the foot, and its linear and angular accelerations. Analogously for the tibia and the femur and the corresponding interconnecting joints. Note that the external force as measured by the mentioned platform includes almost half of the body weight ± the inertial effects of the body in a common squat exercise. In other words, the gravitational effects of the other segments of the body are indirectly included using this methodology.

 

 

Reviewer’s comment 18:

Equation 6: Where does the physiological cross-sectional area of the muscles come from? Are these subject-specifically?

Authors' response:

We agree with the reviewer. We rewrote the paragraph next to the equation to explain this concept:

“where Ai represents the i-th muscle cross-sectional area as measured in [15]. It appeared here as a result of minimizing the squared sum of muscle stresses as mentioned previously Hence, the optimization process can be omitted from the formulation of the dynamic model.”

 

 

 

 

 

 

 

 

 

 

 

 

Reviewer’s comment 19:

Please explain, what in the context of this contribution is meant by real-time. Please compare the necessary calculation speed with standard static optimization procedures as this would also be really fast for such a simplified model. Using static optimization would then even not be dependent on the narrow band around the observed kinematics.

Authors' response:

We greatly appreciate this observation. In our case, the sampling time of the real-time control unit for the robotic system is 10 ms. As a result, we consider that the calculation time for the model must be lower than 5 ms.  

On the other hand, and in addition to the consideration of the effect of a quasi-static model on the computational time, it should be clear that in this research we have introduced the following simplifications that correspond to a high computational cost and prevent its use in real-time control algorithms:

• The optimization process that was previously used to solve the redundancy problem of muscle forces. It was replaced by a direct analytical solution using Lagrange Multipliers. It is known that the optimization process is an iterative one that requires an initial estimation of the solution.
• The number of muscles under consideration is 18 and their modeling corresponds to high computational time steps. It should be taken into consideration that some of them were modeled using a via-points method that requires the relative distances between the insertions at each instant and the determination of the best insertion points within these via-points. Other muscles such as gastrocnemius lateral and medial are modeled at the femur as contacting a cylinder that approximates the posterior part of the lateral and medial condyles. Moreover, these muscles lose contact with femoral condyles for a given flexion angle of the knee. With respect to the extensor muscles connected to the patella, their insertion points next to the patella are not fixed relative to the femur and depend on the current position of the patella. After that, for all of the calculated insertion points of the muscles, it is needed to calculate the corresponding Jacobian matrix to transform the muscle force to the generalized one.

After applying all these improvements to the algorithm, the calculation time dropped from almost 11 ms for each time step to almost 2.5 ms enabling its use in real-time control algorithm.

 

 

 

 

Reviewer’s comment 20:

Line 312: What does a negative knee moment mean? Is it the joint moment generated by the muscles? If yes, how can it be negative?

Authors' response:

According to the ISB standards, the positive direction of the right knee axis of flexion is mediolateral. Therefore, a positive moment represents an extension moment, and negative moments are flexor. The moments in Table 2 and Figure 4 are the moments associated with the gravitational, inertial, and external actions of the FDM. That moment is negative in almost the entire movement due to the body's weight, and it must be balanced with the muscular forces, as indicated in line 313. We have added an explanation in the explanatory note of Table 2

 

Reviewer’s comment 21:

Table 2: Why was the 99th percentile chosen and not the maximum value? Have there been extreme outliers? How is it interpreted at that point?

Authors' response:

No extreme outliers values have appeared, but we believe that representing the extreme values with a 99th percentile is more robust than using the maximum values.

 

Reviewer’s comment 22:

Figure 4: It is a misconception, that the forces due to gravity are much smaller in a slow movement than the external ground reaction force. The ground reaction force instead counteracts the gravitational forces in order to prevent the body to free fall with gravitational acceleration towards the earth’s center. This effect is probably only because the other masses were neglected in the model. Please also compare reviewer’s comment further above regarding the masses in the model.

Authors' response:

We agree with the reviewer. Perhaps we have incompletely explained the meaning of figure 4. We do not represent the forces but the contribution of each force to the moment at the knee (generalized force). The complete model (FDM) includes the platform forces (external), the weight of the foot and the leg (which are the ones used in the upwards model), and the inertia forces (also of the foot and the leg). The solid grey line (External) would only consider the platform forces as input. These forces are the most considerable contribution to the generalized joint force, and they include the body's weight and inertial forces. The grey dashed line refers to the isolated effect of foot and leg weight. It is relatively small. The contribution of inertial forces (from the foot and leg) appears on the black dashed line and is entirely negligible.

The model uses the information from the force platforms that are recording the weight and inertial forces of the whole body. However, when calculating the forces at the knee from the platform upwards, the contribution of inertial forces associated with the foot and leg is negligible.

We have added an explanation in figure 4

 

Reviewer’s comment 23:

Line 360: Please elaborate on the Force-EMG calibration. It remains unclear how this was done.

Authors' response:

The EMG-Force calibration is explained in the Material and Methods section, section 2.5, point 4 (Predictive ability of the model).

A brief explanation has been included in the header of table 5:

The forces estimated by the SGM model for the L1 load condition are compared with the forces calculated from the Force-EMG calibration curves of the L0 and L2 conditions, using the EMG signal of the L1 tests.

 

The authors greatly appreciate the time and effort in reviewing this paper.

Reviewer 3 Report

The article reports a novel biomechanical modeling approach for the control of a rehabilitation robot. The manuscripts is well written and organized, however, some modifications are needed before the acceptance to be published:

• In the abstract section, please report numerical results obtained from this work.
• In the abstraction section, other modeling methods related to wearable robots should be mentioned:

Xu, Ningcun, et al. " Analysis of a Novel 5-DOF Upper Limb Exoskeleton Rehabilitation Robot." 2020 39th Chinese Control Conference (CCC). IEEE, 2020

Shojaei Barjuei, Erfan, Darwin G. Caldwell, and Jesús Ortiz. "Bond graph modeling and kalman filter observer design for an industrial back-support exoskeleton." Designs 4.4 (2020): 53.

Aftab, Zohaib, and Asad Ali. "Modeling and simulation of a lower-body wearable exoskeleton using robotics’ techniques." International Journal of Mechanical Engineering and Robotics Research 7.3 (2018): 313-318.

• In the equation (1), how the elasticity parameter has been considered? Authors should explain this issue explicitly.
• Which software has been used in this work? Is there any program developed for the modeling approach?
• In the conclusion section, what are the challenging of this modeling appraoch?

Author Response

REVIEWER 3

The article reports a novel biomechanical modeling approach for the control of a rehabilitation robot. The manuscript is well written and organized, however, some modifications are needed before the acceptance to be published:

Reviewer’s comment 1:

In the abstract section, please report numerical results obtained from this work.

Authors' response:

We agree with the reviewer. We modified the abstract such that it reveals the obtained numerical results.

 

Reviewer’s comment 2:

In the abstraction section, other modeling methods related to wearable robots should be mentioned:

Xu, Ningcun, et al. " Analysis of a Novel 5-DOF Upper Limb Exoskeleton Rehabilitation Robot." 2020 39th Chinese Control Conference (CCC). IEEE, 2020

Shojaei Barjuei, Erfan, Darwin G. Caldwell, and Jesús Ortiz. "Bond graph modeling and kalman filter observer design for an industrial back-support exoskeleton." Designs 4.4 (2020): 53.

Aftab, Zohaib, and Asad Ali. "Modeling and simulation of a lower-body wearable exoskeleton using robotics’ techniques." International Journal of Mechanical Engineering and Robotics Research 7.3 (2018): 313-318.

 

Authors' response:

Following also a comment of Reviewer 2, we have modified the wording of the Introduction section and we have also included the references (line 100) proposed by the reviewer.  

 

 

 

 

Reviewer’s comment 3:

In the equation (1), how the elasticity parameter has been considered? Authors should explain this issue explicitly.

Authors' response:

We agree with the reviewer. There exist many other aspects that are not modeled in equation (1). In our study, we are trying to simplify as much as we can model to enable its use in real-time control systems without losing its validity. Furthermore, we found that taking away more terms from the equation, such as the inertial part for slow-motion exercises such as squat, will have a trivial effect on the overall generalized forces, when compared to other important forces such as external ones. “the error resulting from neglecting the inertial forces is insignificant (less than 0.1% of the total range)” as can be observed from the table (3). Additionally, including elasticity at the contact between the bones of the model may lead to an extremely complex model that needs special software to work on it with a very high computational time. See for example

Rahman M, Sharifi Renani M, Cil A, Stylianou AP. Musculoskeletal Model Development of the Elbow Joint with an Experimental Evaluation. Bioengineering (Basel). 2018;5(2):31. Published 2018 Apr 20. doi:10.3390/bioengineering5020031

Where a full model was used to model the elbow joint including contact and elasticity using ADAMS software. A simulation of 35 s needed almost 30 min for its resolution.

 

Reviewer’s comment 4:

Which software has been used in this work? Is there any program developed for the modeling approach?

Authors' response:

Initially, for the sake of simplicity and robustness, MATLAB programming language was used to build, extract results and verify the algorithm. After that it was rewritten in C programming language to enable its use in real-time control systems. 

 

 

 

 

Reviewer’s comment 5:

In the conclusion section, what are the challenging of this modeling approach?

Authors' response:

We agree with the reviewer’s comment and we have emphasized the challenging of the model approach including the following paragraph:

“In summary, the proposed model incorporates the essential characteristics that allow an excellent personalization to obtain reasonable estimates of the internal forces through a simple model and with a low computational cost. These are the fundamental characteristics for the use of MSM models for real-time control of rehabilitation robots.”

 

 

The authors greatly appreciate the time and effort in reviewing this paper.

Round 2

Reviewer 3 Report

The article has been improved a lot and now can be accepted for the publication.

Author Response

Thank you very much for your comment.

 

The authors greatly appreciate the time and effort of the editor and the reviewers in reviewing this paper.