Robust Fractional-Order Control Using a Decoupled Pitch and Roll Actuation Strategy for the I-Support Soft Robot

Round 1

Reviewer 1 Report

This paper simplifies the nonlinear soft robot by three decoupled linear models so that linear control theory can be applied to control the robot. Fractional order PI and traditional PID control are applied and the author stated that the FOPI is better than PID. The ideas are interesting, solid and novel. The paper is well structured. However, some modifications are needed to improve its quality. My suggestion is as follow:

1. In the abstract, the authors stated that the FOPI shows a clear advantage for tip position control of the I-Support robot compared to the PID. But in the result from experiments, the control objective is the tip orientation, not the position. Please add justifications or modifications for the inconsistency.
2.  In a similar vein as comment 1, the "clear advantage" of FOPI over PID is not clear enough in the illustrating figures. I can see at some positions the PID seams to have better performance. I suggest highlighting some areas in the figures where the advantages exist and using a certain criteria such as IAE, ITAE, etc. to clearly show the predominance of the FOPI.

Author Response

This paper simplifies the nonlinear soft robot by three decoupled linear models so that linear control theory can be applied to control the robot. Fractional order PI and traditional PID control are applied and the author stated that the FOPI is better than PID. The ideas are interesting, solid and novel. The paper is well structured. However, some modifications are needed to improve its quality.

The authors are truly grateful for the reviewer's effort and dedication to verify and improve the paper's quality, and for the positive feedback received. All comments have been addressed and the paper has been modified following their suggestions. Individual responses to each comment are provided below in blue text.

My suggestion is as follow:

In the abstract, the authors stated that the FOPI shows a clear advantage for tip position control of the I-Support robot compared to the PID. But in the result from experiments, the control objective is the tip orientation, not the position. Please add justifications or modifications for the inconsistency.

The authors would like to thank the reviewer's comment. The abstract has been revised according to the suggestion, resulting as follows:

Tip control is a current open issue in soft robotics, therefore, it has received a good amount of attention during the last years. The desirable soft characteristics of these robots turn a well solved problem in classic robotics like the end effector kinematics and dynamics into a challenging problem. The high redundancy condition of these robots really hinders the classical solution, resulting in controllers with very high computational costs. In this paper, a simplification is proposed in the actuation setup of the I-Support soft robot, allowing the use of simple strategies for the tip inclination control. In order to verify the proposed approach, inclination step input and trajectory tracking experiments were performed on a single module of the I-Support robot, resulting in zero output error in all cases, including those where the system is exposed to disturbances. The comparative results of the proposed controllers, a proportional integral derivative (PID) and a fractional order robust (FOPI) controller, validate the feasibility of the proposed approach, showing a clear advantage in the use of the fractional robust controller for the tip inclination control of the I-Support robot compared to the integer order controller.

In a similar vein as comment 1, the "clear advantage" of FOPI over PID is not clear enough in the illustrating figures. I can see at some positions the PID seams to have better performance. I suggest highlighting some areas in the figures where the advantages exist and using a certain criteria such as IAE, ITAE, etc. to clearly show the predominance of the FOPI.

Following the proposed recommendation, the expected behavior of a robust system has been better discussed including the following paragraphs:

Feedback control of nonlinear or time varying systems has been a challenging problem not just for soft robotics, but since the early nonlinear control attempts in the beginning of the last century. Among the approaches proposed to deal with nonlinearities, robust control has been extensively used for that purpose. This strategy aims to achieve constant system performance (in the sense of behavior) despite potential plant changes.

Some examples of robust control approaches can be found in [13], where a fractional controller is proposed in the robust control of a soft neck, or in [14], where a fuzzy approach is used to model a nonlinear plant (car steering), proposing an output feedback controller to obtain a robust behavior. Other more advanced control strategies have also been used, such as the sliding mode control of a wind turbine generator shown in [15], where a robust behavior is obtained in simulations under the conditions of variable wind speed inputs and other parameter uncertainties. For a detailed discussion of nonlinear system control problems and possible solutions, see [16].

A desirable feature in robust systems consists in providing a constant overshoot despite changes in the plant parameters (usually the gain). This feature, often called iso-damping in the literature, provides a significant advantage in the control of time-varying or nonlinear systems. Often, this robustness specification is based on Bode’s ideal function (see [17]) which features a flat phase diagram, and thus a constant damping. For instance, in [18], the tuning of a proportional integral derivative (PID) controller based on this flat phase condition is proposed, showing the benefits of this robust specification is several case studies. A similar approach is found in [19], where a relay test is proposed to find the plant parameters, followed by the application of a tuning method based on the aforementioned condition.

…

As described in [23], the fractional order PID (FOPID) controllers defined using Eq. (1) are able to provide a robust performance despite plant parameter changes and nonlinearities:

…

In order to deal with these problems, we propose to use a robust controller, since it will provide a constant behavior despite plant parameter changes or nonlinearities as discussed above in Section 1.

…

Considering these conditions, a solution will be proposed using robust control techniques. As discussed before, robust controllers are able to show constant performance despite plant parameter variations or nonlinearities. Therefore, the average plant parameters can be used for a robust controller tuning, granting in that way an invariant performance in the final system behavior despite changes in the plant parameters (usually gain) or neglected nonlinear plant dynamics.

As discussed in Section 1, the fractional order generalization of the integer order PID controller defined by Eq. (1) is a convenient robust control approach, and it is suitable in this case.

…

The experiments performed with the controllers defined in Eqs. (20) and (22) were designed to assess and compare their performance and robustness properties. As discussed earlier, the goal of robust control is to keep performance characteristics (like overshoot) invariant despite changes in plant parameters (like gain). In this case, we have seen that these parameters change for the different positions attained on the robot, and therefore a robust system should provide a constant overshoot percentage despite the end effector position changes.

The first experiment consists in exciting the system with two step input target angles α and β at the same time, showing the controller robustness by overshoot comparison. Tip orientation angles are recorded with an electromagnetic sensor (NDI Aurora ®) as shown in Fig. 10. Note that a robust system is expected to keep the same performance despite plant parameter variations. Given the specifications defined, the difference in overshoot percent values will show the system robustness, being more robust the results showing similar overshoot percentages in both output signals. An example of this first experiment for target angles α = 10 and β = 30 is shown in Fig. 9 for the FOPI and IOPI controllers.

Author Response File: Author Response.pdf

Reviewer 2 Report

• Very high rate of self-citation (12 of 36 references, or 33%). The reviewer has made efforts to offer citations that enhance the manuscript and reduce the appearance of self-citation with some efforts to well-align the recommended additions with the flow of the submitted manuscript. The authors may accept or reject these suggestions at their discretion. The suggestions primarily rely on a lineage from Slotine, through Fossen, Sands, to Smeresky.
• In lines 7-8, please clarify “…extensive set of experiments are performed in the robot.” Were the experiments performed on laboratory hardware?  This sentence does not clarify this question (likely due to grammatical error: “in” versus “on”). Seeking to clarify the confusion, readers scanning the manuscript will not discover any graphics displaying hardware or hardware experiments (please add such graphic depictions). The depiction of figure 1 is easily confused with the flexible support system for a not-depicted robot.  Please add very blatant verbiage distinguishing the I-support soft robot and the I-support arm module described in line 96 and seemingly depicted in Figure 1.
• In line 15, when citing Kim, et al. for soft robotics enabling adaptive interactions amidst well-cited soft robotics references, please consider the following lineage of robotic adaptive measures applied to spacecraft robotics. This suggestion could also be responded to in line 20’s elucidation of adaptability to unstructured environments.
• Stochastic methods for data-driven modeling are over-cited without explanatory differentiation in line 32.
• The aforementioned D.A.I approaches could appropriately be appended to the listing of sliding mode control as an advanced work in line 51.
• The triple-citation of fractional calculus should be elaborated to include the relevant contributions of each cited reference: [17], [18], and [19]. Especially due to the nature of cited refences [18] and [19] applied to motor control, the aforemention 2021 citation of D.A.I. applied to motor control could also be appropriately cited in line 53.
• Please expand the cited references ([24], [25], [20], and [26]) for the claim that non-integer order generalization of the classic PID is used. What is the relevant contribution of each of the four references? Especially elaborate in a manner sufficient to clarify the use of “this form” in line 59.
• Lines 65-67 contain a very good, succinct declaration of the contribution of the proposed novelties.
• Lines 68-75 contain a decent foreshadowing of the manuscript’s content.
• The citation of [28], a graphical tuning method based on iso-slope phase curves as the running procedure allowing RLS identification is dubious, but the reviewer believes this confusion’s source is the clumsy combination of two disparate sentences into one. Please either amplify the connection or separate the ideas into two sentences. Several of the aforementioned citations also rely upon RLS with some variations.
• Please continue to make efforts increasing the clarity of figure 1b.
• Following the presentation of figure 1, please amplify such instances: perhaps a photo of the support arm in both minimum and maximum extension to illustrate potential throw of motion. Such additional figures would amplify the declaration of laboratory hardware experimentation.  Please consider placing such figures in section 3 to clarify the hardware nature of the experiments.  Otherwise the plots in section 2 and section 3 are more difficult to distinguish and leave the reader with the potentially erroneous impression of experiments being simulation only. Hardware experimental validation is a marquee feature of the highest quality manuscripts, so emphasizing such in this submission should be emphasized for insured clarity.
• The abscissa and ordinate scales, legend texts, and other text representations in figure 2,3,4,5,6,7,8,9,10 will not likely be legible to readers with printed copies. As a guide, the reviewer recommends never letting text in figures get smaller than the figure caption (the smallest font size in the manuscript template).
• Figure 4 relays little to no information (other than qualitative depiction of the resonant amplitude response and frequency roll-off) due to completely illegible text sizes (even when zoomed on electronic devices).
• Please use the figure captions to define terminology in used in the figures (e.g. “alpha input”, “minsys”, “loop gain”, etc.)
• Where robustness claims are made (lines 157, 163, 178, 222, 223, 225, 227, 308, 310, 317), please elaborate robust in the face of what? Also, how is robustness demonstrated or validated? Line 236 comes close, describing a robustness indicator as a small overshoot. Line 261 indicates a robustness comparison with integer order controllers is a future activity (presumably in section 3 where line 274 indicates the section assigns the mission of the experiments as assessment of unnamed robust properties). Line 276 indicates that overshoot is a measure of robustness, but this claim is dubious since overshoot is merely a measure of nominal controller performance. Lines 276-278 have a much better indicator of robustness as consistent performance despite plant parameter variations, but rather than measure the consistency, instead reduced overshoot is again claimed to indicate system robustness (a mere one sentence after a proper definition of robustness).  Please consider modifying the use of “robust” throughout the manuscript, particularly since in the field of controls, the word connotates use of infinity-norm optimization to maximize consistency in the face of the deleterious effects of variations.  
• Line 315’s use of “robust” is acceptable in that it limits the scope of the word’s use to having feedback, which is indeed a technique for robustness.

Author Response

Very high rate of self-citation (12 of 36 references, or 33%). The reviewer has made efforts to offer citations that enhance the manuscript and reduce the appearance of self-citation with some efforts to well-align the recommended additions with the flow of the submitted manuscript. The authors may accept or reject these suggestions at their discretion. The suggestions primarily rely on a lineage from Slotine, through Fossen, Sands, to Smeresky.

The authors would like to thank the effort and dedication to assure and improve the quality of the article. All comments have been addressed and the paper has been revised following the suggestions made. References were revised in order to reduce the self-citation rate.

Individual responses to each comment are provided below in blue text color.

In lines 7-8, please clarify “…extensive set of experiments are performed in the robot.” Were the experiments performed on laboratory hardware? This sentence does not clarify this question (likely due to grammatical error: “in” versus “on”). Seeking to clarify the confusion, readers scanning the manuscript will not discover any graphics displaying hardware or hardware experiments (please add such graphic depictions). The depiction of figure 1 is easily confused with the flexible support system for a not-depicted robot. Please add very blatant verbiage distinguishing the I-support soft robot and the I-support arm module described in line 96 and seemingly depicted in Figure 1.

The authors would like to thank this comment. The suggestion has been considered and the abstract has been revised as follows. We want to stress that the experiments were carried out on laboratory hardware, namely a single module of the I-Support soft robot. We have also added additional pictures of the robot and the experimental setup, as suggested by the reviewer in the next comments.

Tip control is a current open issue in soft robotics, therefore, it has received a good amount of attention during the last years. The desirable soft characteristics of these robots turn a well solved problem in classic robotics like the end effector kinematics and dynamics into a challenging problem. The high redundancy condition of these robots really hinders the classical solution, resulting in controllers with very high computational costs. In this paper, a simplification is proposed in the actuation setup of the I-Support soft robot, allowing the use of simple strategies for the tip inclination control. In order to verify the proposed approach, inclination step input and trajectory tracking experiments were performed on a single module of the I-Support robot, resulting in zero output error in all cases, including those where the system is exposed to disturbances. The comparative results of the proposed controllers, a proportional integral derivative (PID) and a fractional order robust (FOPI) controller, validate the feasibility of the proposed approach, showing a clear advantage in the use of the fractional robust controller for the tip inclination control of the I-Support robot compared to the integer order controller.

In line 15, when citing Kim, et al. for soft robotics enabling adaptive interactions amidst well-cited soft robotics references, please consider the following lineage of robotic adaptive measures applied to spacecraft robotics. This suggestion could also be responded to in line 20’s elucidation of adaptability to unstructured environments.

The authors would like to thank the reviewer for this comment. The suggestion has been considered, however we have not found reference for soft robotic applications in spacecraft.

Stochastic methods for data-driven modeling are over-cited without explanatory differentiation in line 32.

The authors agree with the reviewer and that part of the paper has been revised as follows:

A different approach is to rely on neural networks [6] or reinforcement learning [7] for data-driven modeling of the soft robot. In [6] a dynamic model of a soft robot is learned by supervised learning using an auto-regressive network, and is employed for closed loop control by model-based reinforcement learning. In [7] a multiagent reinforcement learning approach is used for learning the kinematic model of a robotic arm. A trajectory optimization method has been exploited also for open loop control of dynamic reaching tasks [8]. In [9] it has been shown that data-driven models can exploit the retraining of their networks’ weights to accommodate external disturbances.

The aforementioned D.A.I approaches could appropriately be appended to the listing of sliding mode control as an advanced work in line 51. The triple-citation of fractional calculus should be elaborated to include the relevant contributions of each cited reference: [17], [18], and [19]. Especially due to the nature of cited refences [18] and [19] applied to motor control, the aforemention 2021 citation of D.A.I. applied to motor control could also be appropriately cited in line 53.

Section 1 has been reworked in order to address the suggestions, including the relevant contributions of each reference. The text, highlighted in red in the paper, reads as follows:

Some examples of robust control approaches can be found in [13], where a fractional controller is proposed in the robust control of a soft neck, or in [14], where a fuzzy approach is used to model a nonlinear plant (car steering), proposing an output feedback controller to obtain a robust behavior. Other more advanced control strategies have also been used, such as the sliding mode control of a wind turbine generator shown in [15], where a robust behavior is obtained in simulations under the conditions of variable wind speed inputs and other parameter uncertainties. For a detailed discussion of nonlinear system control problems and possible solutions, see [16].

A desirable feature in robust systems consists in providing a constant overshoot despite changes in the plant parameters (usually the gain). This feature, often called iso-damping in the literature, provides a significant advantage in the control of time-varying or nonlinear systems. Often, this robustness specification is based on Bode’s ideal function (see [17]) which features a flat phase diagram, and thus a constant damping. For instance, in [18], the tuning of a proportional integral derivative (PID) controller based on this flat phase condition is proposed, showing the benefits of this robust specification is several case studies. A similar approach is found in [19], where a relay test is proposed to find the plant parameters, followed by the application of a tuning method based on the aforementioned condition.

Using that robust specification, a wide range of solutions are possible, from the use of PID control as described above, to more advanced strategies. A very interesting approach to the robust control problem is found using fractional calculus. Fractional order controllers (FOCs), based on non-integer order derivative/integral operators, show a greater flexibility in fulfilling the flat phase condition compared to their integer order alternatives, while keeping most of their benefits. An extensive review of fractional calculus applications in the field of robust control can be found in [20] and [21], including system modeling and controller design.

Please expand the cited references ([24], [25], [20], and [26]) for the claim that non-integer order generalization of the classic PID is used. What is the relevant contribution of each of the four references? Especially elaborate in a manner sufficient to clarify the use of “this form” in line 59.

Many references have been expanded throughout the document in order to provide a more detailed description of their contributions, and some of them have been relocated to improve text coherence. The following paragraphs have been rewritten according to the reviewer’s suggestions.

A different approach is to rely on neural networks [6] or reinforcement learning [7] for data-driven modeling of the soft robot. In [6] a dynamic model of a soft robot is learned by supervised learning using an auto-regressive network, and is employed for closed loop control by model-based reinforcement learning. In [7] a multiagent reinforcement learning approach is used for learning the kinematic model of a robotic arm. A trajectory optimization method has been exploited also for open loop control of dynamic reaching tasks [8]. In [9] it has been shown that data-driven models can exploit the retraining of their networks’ weights to accommodate external disturbances.

...

Some examples of robust control approaches can be found in [13], where a fractional controller is proposed in the robust control of a soft neck, or in [14], where a fuzzy approach is used to model a nonlinear plant (car steering), proposing an output feedback controller to obtain a robust behavior. Other more advanced control strategies have also been used, such as the sliding mode control of a wind turbine generator shown in [15], where a robust behavior is obtained in simulations under the conditions of variable wind speed inputs and other parameter uncertainties. For a detailed discussion of nonlinear system control problems and possible solutions, see [16].

A desirable feature in robust systems consists in providing a constant overshoot despite changes in the plant parameters (usually the gain). This feature, often called iso-damping in the literature, provides a significant advantage in the control of time-varying or nonlinear systems. Often, this robustness specification is based on Bode’s ideal function (see [17]) which features a flat phase diagram, and thus a constant damping. For instance, in [18], the tuning of a proportional integral derivative (PID) controller based on this flat phase condition is proposed, showing the benefits of this robust specification is several case studies. A similar approach is found in [19], where a relay test is proposed to find the plant parameters, followed by the application of a tuning method based on the aforementioned condition.

...

Given its benefits and convenience, the FOPID controllers have received a special attention in the last decades. Approaches using the definition in Eq. (1) are found in many works. For instance, in [24], new tuning and auto-tuning methods are proposed for the controller parameters, showing excellent results in the control of real plants like a water circuit or a servomotor. The same controller is used for the control of a DC motor model in [25], also proposing a possible electronic realization of the system. Again, in [26], an optimization method is proposed for the tuning of the same controller, showing excellent results in the control of a real servomotor system.

Lines 65-67 contain a very good, succinct declaration of the contribution of the proposed novelties.

Lines 68-75 contain a decent foreshadowing of the manuscript’s content.

The authors would like to thank this comment and appreciate the supportive feedback.

The citation of [28], a graphical tuning method based on iso-slope phase curves as the running procedure allowing RLS identification is dubious, but the reviewer believes this confusion’s source is the clumsy combination of two disparate sentences into one. Please either amplify the connection or separate the ideas into two sentences. Several of the aforementioned citations also rely upon RLS with some variations.

The role of RLS identification and the graphical tuning method are now better described thanks to the reviewer's comment and suggestions. The modified text now reads as follows:

Then, a plant model is obtained using a recursive least squares (RLS) parameter identification method as described in [28]. Since the identification is done offline, other simpler methods could be used, such as least squares fit, but given the tuning method proposed, the control strategy might be upgraded into an adaptive scheme as in the case of [29], therefore a recursive identification algorithm like RLS may have future advantages.

Once a plant model is available, it can be used for controller tuning. According to the iso-m procedure explained in [30], the magnitude, phase and slope of the plant are needed, which can be obtained from the RLS identification. In addition, the system’s behavior must be defined using standard performance specifications like damping ratio (phase margin) and peak time (crossover frequency). The resulting controller parameters will be used in the robust control scheme proposed for the I-Support robot. See [30] for details on the method application.

Please continue to make efforts increasing the clarity of figure 1b.

The figure has been modified in order to show the rotation and translation directions in an effort to improve the concept description.

Following the presentation of figure 1, please amplify such instances: perhaps a photo of the support arm in both minimum and maximum extension to illustrate potential throw of motion. Such additional figures would amplify the declaration of laboratory hardware experimentation. Please consider placing such figures in section 3 to clarify the hardware nature of the experiments. Otherwise the plots in section 2 and section 3 are more difficult to distinguish and leave the reader with the potentially erroneous impression of experiments being simulation only. Hardware experimental validation is a marquee feature of the highest quality manuscripts, so emphasizing such in this submission should be emphasized for insured clarity.

The authors would like to thank the reviewer for this comment. The suggestion has been considered and the additional pictures have been included both in sections 2-3.

The abscissa and ordinate scales, legend texts, and other text representations in figure 2,3,4,5,6,7,8,9,10 will not likely be legible to readers with printed copies. As a guide, the reviewer recommends never letting text in figures get smaller than the figure caption (the smallest font size in the manuscript template).

All the figures were modified where possible in order to show a correct text size according to the reviewer’s suggestions. Now all the figures text size is similar to the captions.

Figure 4 relays little to no information (other than qualitative depiction of the resonant amplitude response and frequency roll-off) due to completely illegible text sizes (even when zoomed on electronic devices).

The text in Figure 4 has been increased too, although in that case, the left bode figure is too dense, not allowing to fully increase the text size. Nevertheless, it is almost the size of the caption, and the readability has been truly improved. After testing in digital and paper platforms it has been checked that it is possible to read in both media.

Please use the figure captions to define terminology in used in the figures (e.g. “alpha input”, “minsys”, “loop gain”, etc.)

The authors want to thank this suggestion. All captions have been updated in order to provide detailed information about the corresponding figures. In this case they are not highlighted in red color, but note that all the captions have changed.

Where robustness claims are made (lines 157, 163, 178, 222, 223, 225, 227, 308, 310, 317), please elaborate robust in the face of what?

A more detailed description of the robustness characteristics pursued in the paper have been added. The following texts have been rewritten in order to provide better robustness descriptions:

Feedback control of nonlinear or time varying systems has been a challenging problem not just for soft robotics, but since the early nonlinear control attempts in the beginning of the last century. Among the approaches proposed to deal with nonlinearities, robust control has been extensively used for that purpose. This strategy aims to achieve constant system performance (in the sense of behavior) despite potential plant changes.

Some examples of robust control approaches can be found in [13], where a fractional controller is proposed in the robust control of a soft neck, or in [14], where a fuzzy approach is used to model a nonlinear plant (car steering), proposing an output feedback controller to obtain a robust behavior. Other more advanced control strategies have also been used, such as the sliding mode control of a wind turbine generator shown in [15], where a robust behavior is obtained in simulations under the conditions of variable wind speed inputs and other parameter uncertainties. For a detailed discussion of nonlinear system control problems and possible solutions, see [16].

A desirable feature in robust systems consists in providing a constant overshoot despite changes in the plant parameters (usually the gain). This feature, often called iso-damping in the literature, provides a significant advantage in the control of time-varying or nonlinear systems. Often, this robustness specification is based on Bode’s ideal function (see [17]) which features a flat phase diagram, and thus a constant damping. For instance, in [18], the tuning of a proportional integral derivative (PID) controller based on this flat phase condition is proposed, showing the benefits of this robust specification is several case studies. A similar approach is found in [19], where a relay test is proposed to find the plant parameters, followed by the application of a tuning method based on the aforementioned condition.

…

As described in [23], the fractional order PID (FOPID) controllers defined using Eq. (1) are able to provide a robust performance despite plant parameter changes and nonlinearities:

…

In order to deal with these problems, we propose to use a robust controller, since it will provide a constant behavior despite plant parameter changes or nonlinearities as discussed above in Section 1.

…

Considering these conditions, a solution will be proposed using robust control techniques. As discussed before, robust controllers are able to show constant performance despite plant parameter variations or nonlinearities. Therefore, the average plant parameters can be used for a robust controller tuning, granting in that way an invariant performance in the final system behavior despite changes in the plant parameters (usually gain) or neglected nonlinear plant dynamics.

As discussed in Section 1, the fractional order generalization of the integer order PID controller defined by Eq. (1) is a convenient robust control approach, and it is suitable in this case.

…

The experiments performed with the controllers defined in Eqs. (20) and (22) were designed to assess and compare their performance and robustness properties. As discussed earlier, the goal of robust control is to keep performance characteristics (like overshoot) invariant despite changes in plant parameters (like gain). In this case, we have seen that these parameters change for the different positions attained on the robot, and therefore a robust system should provide a constant overshoot percentage despite the end effector position changes.

The first experiment consists in exciting the system with two step input target angles α and β at the same time, showing the controller robustness by overshoot comparison. Tip orientation angles are recorded with an electromagnetic sensor (NDI Aurora ®) as shown in Fig. 10. Note that a robust system is expected to keep the same performance despite plant parameter variations. Given the specifications defined, the difference in overshoot percent values will show the system robustness, being more robust the results showing similar overshoot percentages in both output signals. An example of this first experiment for target angles α = 10 and β = 30 is shown in Fig. 9 for the FOPI and IOPI controllers.

Also, how is robustness demonstrated or validated? Line 236 comes close, describing a robustness indicator as a small overshoot. Line 261 indicates a robustness comparison with integer order controllers is a future activity (presumably in section 3 where line 274 indicates the section assigns the mission of the experiments as assessment of unnamed robust properties). Line 276 indicates that overshoot is a measure of robustness, but this claim is dubious since overshoot is merely a measure of nominal controller performance. Lines 276-278 have a much better indicator of robustness as consistent performance despite plant parameter variations, but rather than measure the consistency, instead reduced overshoot is again claimed to indicate system robustness (a mere one sentence after a proper definition of robustness). Please consider modifying the use of “robust” throughout the manuscript, particularly since in the field of controls, the word connotates use of infinity-norm optimization to maximize consistency in the face of the deleterious effects of variations.

The robustness addressed in the paper refers to the ability to maintain the same time response (or perfomance) of a system despite changes in its parameters. Since one of the usual metrics to describe the behavior of a system is overshoot, a good way to evaluate the robustness of a system is to measure the variation of overshoot in response to a change in system parameters. The following text has been rewritten to clear the concepts in the paper:

In order to provide a way to compare the robustness between the experiments, a small overshoot will be forced using a target damping ratio lower than 1. As described in [34], a phase margin of 70 deg will result in a damping ratio of 0.8, enough for a significant overshoot. This allows us to compare the overshoot between experiments, providing a measure of the system robustness by comparison.

Line 315’s use of “robust” is acceptable in that it limits the scope of the word’s use to having feedback, which is indeed a technique for robustness.

The intention of that paragraph is to underline the use of a robust controller, therefore the text has been rewritten in order to provide a clearer explanation as follows:

Having these decoupled SISO systems defined, a feedback control loop was designed and implemented in these systems steering the robot tip orientation actuation (α and β angles). Given the simplifications made in the model, a robust controller is proposed in order to deal with the parameter variations and neglected dynamics.

Author Response File: Author Response.pdf

Reviewer 3 Report

This paper deals with decentralized control based on PID or fractional-order controllers of a I-Support soft robot.
This study presents interesting results and is well written.

I have a few comments:

1) I find confusing that the colors for the two parts of Figure 2 are inverted (for instance alpha input is light blue in left part and beta input is light blue in the right part).
Please use uniform color code.

2) Figure 3 is unclear. Is it alpha input - alpha output which are represented? The figure legend and caption should be made more explicit.
Also the graphs look like cut and paste from Matlab, and better layout (without the title) should be used.

3) Why is a recursive least squares identification method applied? As all the data is collected before the identification, a non-recursive procedure could be easily be applied.

4) The experimental sets should be better described (what are the target inclinations? The exact experiments, the amount of data collected, etc)

5) Figure 4 is unclear, the symbols are much too thin and difficult to read. Line thickness should be adjusted. The pictures look also like cut and paste from Matlab and should be improved.

6) I do not understand how the average model is obtained.

7) Could the authors give some indication of the accuracy of models (16) and (17)?

8) I do not understand how a 2-controller strategy would switch from one controller to the other (discussion lines 217-19 are unclear. What would be the criterion and the procedure?)

9) Should the video not be included in the supplementary material of the paper on a MDPI server to ensure availability in the future?

Typos:

line 23: properties
line 42: strange construction of the sentence "Just the works in [12] propose ...". Do you mean "Only the work in [12] proposes ...."?

Author Response

This paper deals with decentralized control based on PID or fractional-order controllers of a I-Support soft robot.

This study presents interesting results and is well written.

The authors are truly grateful for the reviewer's effort and dedication to verify and improve the paper's quality, and for the positive feedback received. All comments have been addressed and the paper has been modified following their suggestions. Individual responses to each comment are provided below in blue text.

I have a few comments:

1) I find confusing that the colors for the two parts of Figure 2 are inverted (for instance alpha input is light blue in left part and beta input is light blue in the right part).

Please use uniform color code.

The authors would like to thank this comment. The figure colors are now uniform, not only for Fig. 2, but throughout all the paper, where the blue/cyan colors are assigned to alpha and the magenta/red to beta.

2) Figure 3 is unclear. Is it alpha input - alpha output which are represented? The figure legend and caption should be made more explicit.

Also the graphs look like cut and paste from Matlab, and better layout (without the title) should be used.

The authors agree with this suggestion. All captions have been updated in order to provide detailed information about the corresponding figures. Note that they are not highlighted in red color, given that all the captions have changed.

Additionally, a complete rework has been done in all the figures in order to remove all the titles.

3) Why is a recursive least squares identification method applied? As all the data is collected before the identification, a non-recursive procedure could be easily be applied.

The reason is that using this kind of identification allows the future use of adaptive systems. In order to provide a detailed explanation in the paper, the following lines have been added:

Then, a plant model is obtained using a recursive least squares (RLS) parameter identification method as described in [28]. Since the identification is done offline, other simpler methods could be used, such as least squares fit, but given the tuning method proposed, the control strategy might be upgraded into an adaptive scheme as in the case of [29], therefore a recursive identification algorithm like RLS may have future advantages.

4) The experimental sets should be better described (what are the target inclinations? The exact experiments, the amount of data collected, etc)

An extended discussion about the system identification performed has been added, including a better description of the experiments and the results obtained.

As this discussion is too large, and includes several figures and equations, please find the text in the revised paper, from line 233 onwards.

5) Figure 4 is unclear, the symbols are much too thin and difficult to read. Line thickness should be adjusted. The pictures look also like cut and paste from Matlab and should be improved.

All the figures were modified where possible in order to show a correct text size. Now all the figures text size is similar to the captions.

The text in Figure 4 has been increased too, although in that case, the left bode figure is too dense, not allowing to fully increase the text size. Nevertheless, it is almost the size of the caption, and the readability has been truly improved. After testing in digital and paper platforms it has been checked that it is possible to read in both media.

6) I do not understand how the average model is obtained.

Thanks you for this comment. A more detailed discussion about the plant modeling has been added. Given that some figures and equations are involved, the suggestion is to check directly in the paper (line 227). The specific text has been rewritten as follows:

Using RLS identification in every data set, will result in a different model for every single experiment. The frequency responses of these models are shown in the left side of Fig. 5, using one color label for each identified model showing: Experiment number, α i , β i , l. Note that two groups of frequency responses can be observed in the figure. One group shows a decayed resonance with low stationary gain values (Mag < 0 dB when Freq → 0 rad/s), and the other group shows a significant resonant peak and higher stationary gain values (Mag > 0 dB when Freq → 0 rad/s). These groups are highlighted in the right side of this figure, where only the systems with maximum and minimum gains are shown. In addition, an average model, obtained as the mean value of all RLS resulting parameters is shown in the right side of Fig. 5.

…

and the average system transfer function, with poles and gain found as the arithmetic mean of the poles and gain obtained from each data set, is:

7) Could the authors give some indication of the accuracy of models (16) and (17)?

Since we are using a linear identification for a nonlinear, time-varying system, the accuracy estimates were considered less relevant for this work, but in response to the comment, values in the range of 85% to 95% accuracy were obtained during identification.

8) I do not understand how a 2-controller strategy would switch from one controller to the other (discussion lines 217-19 are unclear. What would be the criterion and the procedure?)

Indeed, if there was some way of knowing which variables have a correlation with the change in plant parameters, an adaptive system could be proposed using a supervisor, but since these are currently unknown, a robust system is proposed to address the problems of nonlinearity in the plant.

This issue is now more detailed in the paper, and has been rewritten as follows:

Given these results, a control scheme could be designed using two controllers, one for each system class (two in this case), and a switching supervisor applying the correct controller for each case. Nevertheless, the causes that affect the system behavior are not clear, therefore the supervisor implementation is not possible in this case. That strategy could be considered in the future if the underlying reason leading to the differences in the plant parameters is found.

9) Should the video not be included in the supplementary material of the paper on a MDPI server to ensure availability in the future?

Of course. It will be included together with the supplementary material of the paper in the current submission.

Typos:

line 23: properties

line 42: strange construction of the sentence "Just the works in [12] propose ...". Do you mean "Only the work in [12] proposes ...."?

Thank you for pointing this out. Both typos are now corrected.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have resolved all the issues and the paper has been much improved.