Faster-Than-Natural Satellite Circumnavigation via Continuous Control^†

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

One of the interesting points in the article is the consideration of the relative motion with a angular velocity different from the natural one. But there is no explanation of this problem statement and how it is related to applied problems. 

First 10 pages of the article are devoted to general things related to the equations of motion of satellite in the gravitational field of the Earth. As a result, it is shown how the J2 affects the relative motion. Both the equations of motion and the influence of the J2 are well known. Than relative motion equations are considered and their solutions were obtained using the Laplace domain. It is not very clear to me why it is necessary to discuss the solutions of these equations because they are again well known. The article then discusses several control laws, none of which are new. Also according to the figures, ΔV consumption required is quite high to implement these control laws. To sum up, the article lacks novelty, which is a significant problem and therefore it is a reject. 

Author Response

Comment 1: One of the interesting points in the article is the consideration of the relative motion with a angular velocity different from the natural one. But there is no explanation of this problem statement and how it is related to applied problems. 

Response 1:   A justification  of the need for fast circumnavigation has been expanded in the  third paragraph of the introduction:

“The vast majority of research on the control of relative trajectories addresses the case of natural motion, since it requires significantly less fuel expenditure.   However, in time-critical situations, for instance in military operations regarding the loss or potential loss of a critical space asset, the benefit of responsiveness may outweigh the high fuel cost.  Other constraints may also necessitate a faster-than-natural relative trajectory, such as the relatively short observation time available for a given point on the Earth for remote sensing missions.”

 

Comment 2:  First 10 pages of the article are devoted to general things related to the equations of motion of satellite in the gravitational field of the Earth. As a result, it is shown how the J2 affects the relative motion. Both the equations of motion and the influence of the J2 are well known. Than relative motion equations are considered and their solutions were obtained using the Laplace domain. It is not very clear to me why it is necessary to discuss the solutions of these equations because they are again well known. The article then discusses several control laws, none of which are new. Also according to the figures, ΔV consumption required is quite high to implement these control laws. To sum up, the article lacks novelty, which is a significant problem and therefore it is a reject. 

Response 2:  The reviewer is absolutely correct in pointing out well-known material that could be eliminated.  Much of the EOM development (including J2), Laplace solution to the linear EOMS, LQT theory and other unnecessary or redundant equations have been eliminated.   The influence of J2 is still included in Section 2.3 to give some baseline of the errors that exist without the controller as is discussed in the first paragraph of Section 2.3 on p. 8.  The reviewer is also correct that the feedback control is not new, but this feedforward control does not exist in the literature as now stated in the second to last paragraph of introduction on p.4:

“This paper presents a continuous control design for faster-than-natural relative motion that uses a newly developed feedforward control which significantly mitigates the deleterious impact that noise can have on previously developed architectures.”

 Also, a control architecture for fast circumnavigation that deals effectively with noise has not been demonstrated in the literature as now stated in the abstract:

“One challenge of continuous control for faster-than-natural relative motion is its susceptibility to noise due to the large nominal thrust that is required.  The deleterious effect of noise has not been addressed in the literature for this problem. “

 The reviewer is correct in pointing out that the DeltaV requirements are unreasonably high.   We have decreased the size and angular velocity of the baseline case in order to reduce the acceleration by more than an order of magnitude which is potentially achievable by current technology as now addressed at the bottom of p. 12 and top of p.13:

“The maximum magnitude of the control effort in Figure 4b is 4.0E-4 m/s^2 which is larger by about a factor of two compared to typical current day continuous-thrust spacecraft [20,21].  For example, Ref. [21] cites a maximum acceleration of 2.3E-4 m/s^2 for the SpaceX Starlink  V2.0mini satellite.   However, none of the spacecraft surveyed were optimized for this particular mission, since no such satellite has been acknowledged in the open literature.   So, the levels of thrust dictated here are likely to be possible with current or very near term technology.  As discussed below, for relative orbits smaller than the baseline of this work, the thrust requirements can be significantly reduced.   Thus, advances in relative navigation also improve the practicality of this mission concept.”

 

Reviewer 2 Report

Comments and Suggestions for Authors

Please see the attached file.

Comments for author File: Comments.pdf

Author Response

Comment 1: The contributions of this paper are unclear, making it difficult to identify its originality. The authors should explicitly state their contributions. For example, they should clarify how this work has been revised and expanded from their previous work presented at SciTech in early 2024. Additionally, they should specify which control systems have been newly constructed based on their insights, identify the main idea if any of the control systems were developed by the authors themselves, and clearly articulate the ultimate goal of exploring the three control systems.

Response 1:  The abstract, introduction and conclusions have been completely rewritten and some text in the body of the manuscript has been revised in order to address all of these valid critiques.  The contributions, goal of the paper, explanation of which control systems are newly developed  are now more explicitly stated, e.g. in the Abstract:

“One challenge of continuous control for faster-than-natural relative motion is its susceptibility to noise due to the large nominal thrust that is required.  The deleterious effect of noise has not been addressed in the literature for this problem.   This work develops a feedforward control that helps to eliminate unnecessary control effort and jitter in the actuation due to noise.   When implemented in conjunction with existing feedback control schemes, the resulting control system is shown to perform well and shows promise as an implementable option with current technology.”

And in the second to last paragraph of the Introduction on p.4:

“This paper presents a continuous control design for faster-than-natural relative motion that uses a newly developed feedforward control which significantly mitigates the deleterious impact that noise can have on previously developed architectures. The feedback part of the controller could use the controllers from Refs. [15] or [16] or many of the controllers developed for natural motion.  Here, a linear quadratic tracker (a modification of the linear quadratic regulator for nonzero nominal motion) is used for its good performance, ease of design, ease of implementation and ubiquitous nature.   A combined control architecture, with feedforwad and feedback elements, is shown to perform well for a variety of relative orbit sizes and periods.”

Regarding the SciTech paper, we have also added at the end of the Introduction on p. 2: “The main addition to that paper has been the addition of noise to the state measurement data.”  However, the vast majority of the content of this manuscript is the same as the SciTech paper, which we assume to not be a violation of originality since that is a non-referred conference paper while this is an archival journal article.

Note that the description of the investigation as three different control systems has been modified to more clearly show the contributions of this work.

 

Comment 2: It is recommended to focus on the main content in the paper. The section on dynamic models is quite extensive, which may cause readers to lose focus on the primary subject - control systems.
- On pages 6 and 7, the dynamic model can be understood without equations 1 to 3.
- On pages 11 and 12, the derivation of the homogeneous solution is not central to the main subject of this paper.
- On pages 13 and 18, equations 18, 19, and 29 have already been explained earlier using the general form.
- On pages 17 and 18, the readers can refer to numerous previous studies that have addressed how to solve the LQR problem.

Response 2:  Thank you for these excellent suggestions which point out the pitfall of using a thesis as the starting point for a journal article!   Equations 1-3,  the solution to the linear EOMs on pp. 11 and 12, the redundant EOMs (18,19,29) and the LQT development on pp. 17 and 18 have all been removed and the text appropriately modified.

 

Comment 3: On page 9 of Section 3, the nominal relative orbit is obtained by propagating with the J2 perturbation, whereas the nominal relative orbit on page 10 of Section 4 is derived from the CW equations, which assume a circular reference orbit. In other words, the latter nominal relative orbit does not account for the J2 perturbation. To avoid confusion, please distinguish between the two nominal relative orbits by using different terminology or adding an explanation. Additionally, please specify which nominal reference orbit is used to calculate the relative position error.

Response 3:  We have made multiple changes to avoid confusion on the baseline:

• We have eliminated the incorrect sentence the end of the section on p. 10 of the original manuscript: “These error plots for the perturbed natural motion provides a baseline in which the control schemes presented in this paper are compared against.”   (By “baseline” we really meant some metric by which the performance of the controlled motion can be compared rather than the nominal trajectory.)
• We have put the uncontrolled J2 results a new subsection titled “Uncontrolled Natural Motion” on p. 8 and explicitly stated the reasons for this section:
  “The unforced, natural relative motion of the two satellites defined above is discussed in this section for two reasons.   First, the unperturbed version of the resulting relative ellipse is used for the baseline relative motion geometry of the faster-than-natural circumnavigation (only the period is changed).   Second, the error between the perfect ellipse and the perturbed motion for this uncontrolled  case provides some metric by which the performance of the controlled motion can be compared.”
• We have also added on p. 11 after the actual baseline nominal motion, given by Equation 7:
  “All of the error plots in the sections below are with respect to Equation 7.”

 

Comment 4:  On page 12 of Section 4, the authors mention, “the set of equations defined in equation 14 result in no relative thrust if substituted into equation 7.” However, the set of equations is derived with the assumption of zero external force from the outset. Please clarify the causality more carefully.

Response 4:  As the reviewer states, the causality is that the thrust is assumed to be zero in the derivation of the motion – not the other way around as this poorly written sentence indicates.  Since the sentence is redundant with the more accurate text already present before Equation 7,  we have eliminated it. 

 

Comment 5: On page 12 of Section 4, the authors state that the simulation conditions are defined to use the simplified solution (equation 14) instead of the general solution (equation 13). As a result, the open-loop control system is not applicable to other simulation conditions, meaning that the three control systems can only be analyzed under limited conditions. Please ensure consistency in the control system properties by using other simulation conditions.

Response 5:  The reviewer is exactly correct that the feedforward control of (old) Equation 17 is only applicable to the specific baseline motion given by (old) Equations 16.  We have generalized the solution to the linear equations which is now given in Equation 5 on p. 10.     This includes general in-plane and out-of-plane amplitudes and phase angles and an offset along the y-axis, but does not include an offset along the x-axis since this does not result in a closed path which is the focus of this concept.  The baseline motion in Equation 7 and feedforward solution in Equation 8 have been appropriately updated.  We have also run simulations for different relative ellipses and found the behavior to follow the same behavior of the specific cases given (e.g. that shorter paths, such as D=0, require less control effort) which is now stated before the conclusions on p. 20:

“Finally, variations in the relative baseline motion exhibit the same type of behavior shown in the examples above, with the salient feature that the control effort decreases with decreased path length of the relative ellipse.”

Reviewer 3 Report

Comments and Suggestions for Authors

The paper discusses various approaches for controlling the relative motion between two satellites for fast satellite circumnavigation missions. It considers three different control approaches to maintain a relative trajectory of the secondary satellite within the required bound with respect to the primary satellite. The authors applied open-loop, feed-back and combined controllers to a defined initial conditions of the primary satellite in inertial frame and the relative position and velocity of the secondary satellite in relative coordinate system.

The result shows that open loop controller was not able to keep the relative position error of the satellite within the bound. Using feed-back controller based on linear quadratic tracking (LQT) (or regulator as it is most commonly called), the state error was maintained within the bound. Then, a noise was introduced to the system. Applying the LQT controller with the noise results in an increase in the control thrust by factor of 2. Therefore, a combined controller using both open-loop and feedback LQT was tested, resulting in bounded position error and reasonable control thrust with less noise compared to using only LQT.

 

The particular remarks are the following:

- I think introduction chapter should be 1., Nomenclature should not be numbered.

- in page 2, F and a, it should be stated that they are thrust vector acceleration. it might confuse the reader with thrust force.

- page 6: three-element, add hyphen.

- page 6: J2 should not have a unit.

- page 7: EOMs was not introduced. 

- page 7: why rotation matrix is Z? I think better representation will be good.

- page 8: unit of inclination is wrong in Table 1 it shows km, and it shows degrees for eccentricity.

- page 8: "Additionally, certain functions native to the simulation

software (MATLAB™) were implemented and configured to suit the scope of this research" not clear what it means?

- page 8: space between the number and the unit: 7000km, 500m, 200m

- page 8: RMS is not defined. 

- page 8: it is not clear how the initial condition of the secondary satellite is defined given 500 m relative orbit ? better to explain how the initial relative position and velocity are obtained.

- page 10: why Nominal is capital letter?

- page 11: how C can be relative orbital radius?

- page 12: in equation 14 how can all the three equations have C where c(t) is out of plane relative motion! I guess it should have another constant. 

- page 12: in equation 14, if we follow the given initial conditions, we have y0 which should represent shift in orbit so i(t) should have + constant which is (y0 - 2x0_dot / n)

- page 13: equation 17a: the first term should be theta_dot not squared

- page 13: equation 17b: second term should be cos not sin

 

 

- It is proposed to consider how relative orbit size might affect the control performance as the system was tested for a little relative orbits.

 

- Could authors elaborate more on the fast circumnavigation orbits that were used? How big is the difference from the natural orbit and how does it affect the choice of control algorithm and its tuning? The algorithms that were used in the study are pretty well known and the novelty is to apply it for non-natural relative orbits. So, it is important to emphasize on the difference with standard relative orbits control.

 

- "The Lyapunov function utilized in this paper is based on Schaub’s work [8], which results in a stable control law for which the maneuvering satellite utilizes to track to a reference orbit. The developed control law however, does not exhibit quick convergence times nor result in optimal fuel usage."

 

 Please consider adding "asymptotical stable control law". How do you whether the convergence speed fast or slow? Do you mean that this is not an exponentially stable? Although, by tuning weight matrices the performance can be improved.

 

- Table 2. What is the reason to consider that high precision of the initial conditions?

 

- Equation 9. The form of equation is typically called as the amplitude-phase form

 

- Could authors please add a couple sentences on the way to select Q, R weight matrices for both cases?

Following the importance of the control weights selection authors could refer to a paper where it was optimization based on different objectives:
Biktimirov, Shamil, Gleb Belyj, and Dmitry Pritykin. "Satellite formation flying for space advertising: From technically feasible to economically viable." Aerospace 9.8 (2022): 419.

 

The paper investigates an important research problem and is well written in general, however it has multiple mistakes and typos and requires additional analysis. Therefore, it requires a major revision to be published in Aerospace journal.

 

Author Response

Response to initial list of minor corrections:  The authors wish to thank Reviewer 3 for the very thorough and careful review, including the correction of several format issues, incorrect units, unclear statements, undefined acronyms and even a couple of errors in our feedforward equations.  Every one of the hyphenated minor corrections listed in the review have been made.  More significant issues/errors have additional explanation following the “>>” symbols.

- I think introduction chapter should be 1., Nomenclature should not be numbered.

- in page 2, F and a, it should be stated that they are thrust vector acceleration. it might confuse the reader with thrust force.

- page 6: three-element, add hyphen.

- page 6: J2 should not have a unit.

- page 7: EOMs was not introduced. 

- page 8: unit of inclination is wrong in Table 1 it shows km, and it shows degrees for eccentricity.

- page 8: "Additionally, certain functions native to the simulation software (MATLAB™) were implemented and configured to suit the scope of this research" not clear what it means?

>>This referred to built-in functions like ‘ode45’ and ‘lqr’, the latter of which is now described in the first sentence of the last  paragraph on p. 14.  The original unclear sentence has been removed.

- page 8: space between the number and the unit: 7000km, 500m, 200m

- page 8: RMS is not defined. 

- page 8: it is not clear how the initial condition of the secondary satellite is defined given 500 m relative orbit ? better to explain how the initial relative position and velocity are obtained.

>>Since this is well established in the literature for relative motion we have (on p. 7 above Table 2) simply added a reference (Analytical Mechanics of Space Systems by Schaub and Junkins) that contains the required calculations.   If the editor or reviewer feel that this should  be explicitly added to the paper, we are happy to do so.

- page 10: why Nominal is capital letter?

- page 11: how C can be relative orbital radius?

>>We now refer to C more explicitly and accurately as the “semi-minor axis of the relative in-plane ellipse”, e.g. on p. 10 after Eqn. 5.

- page 12: in equation 14 how can all the three equations have C where c(t) is out of plane relative motion! I guess it should have another constant. 

>>A different symbol was most definitely required.   We have changed the relative coordinates from r,i,c to x,y,z.

- page 12: in equation 14, if we follow the given initial conditions, we have y0 which should represent shift in orbit so i(t) should have + constant which is (y0 - 2x0_dot / n)

>>There would indeed be an offset in the center of the relative ellipse of (y0 - 2x0_dot / n), but x0_dot is chosen so that this term is exactly zero.  (Note that the value of x0_dot in Table 2 was in error; that has been changed in the revised manuscript.)

- page 13: equation 17a: the first term should be theta_dot not squared

- page 13: equation 17b: second term should be cos not sin

 

 

Comment 1:  It is proposed to consider how relative orbit size might affect the control performance as the system was tested for a little relative orbits.

Response 1:  This is indeed a worthwhile consideration so we have added additional numerical results and analysis regarding the size of the relative trajectory in the first paragraph on p. 19.   On p. 11 afer Equation 8 we have also added:

“Note that the feedforward control is proportional to the size of the ellipse.”

 

Comment 2:   Could authors elaborate more on the fast circumnavigation orbits that were used? How big is the difference from the natural orbit and how does it affect the choice of control algorithm and its tuning? The algorithms that were used in the study are pretty well known and the novelty is to apply it for non-natural relative orbits. So, it is important to emphasize on the difference with standard relative orbits control.

Response 2:  The impact of the fast circumnavigation on the choice of control has now been made much more clear and explicit.   For instance in the abstract:  

“One challenge of continuous control for faster-than-natural relative motion is its susceptibility to noise due to the large nominal thrust that is required.  The deleterious effect of noise has not been addressed in the literature for this problem.   This work develops a feedforward control that helps to eliminate unnecessary control effort and jitter in the actuation due to noise.”

The impact that fast circumnavigation has on tuning is described after the Q and R matricies on p. 15:

“The resulting gain matrix needed to accurately track the relative trajectory only through feedback on state errors is quite large and therefore susceptible to noise as described below.   This is the main weakness of using feedback control for faster-than-natural motion.”

And reiterated on the bottom of p.18 after Figure 8:

“The addition of the feedforward control architecture allows for a much larger weight on the control term of the cost function, which reduces the optimal feedback gains significantly.   With these smaller feedback gains, the system becomes less sensitive to noise, and the feedforward term  provides the majority of the control needed to accurately track the desired relative trajectory.”

 

Comment 3:  "The Lyapunov function utilized in this paper is based on Schaub’s work [8], which results in a stable control law for which the maneuvering satellite utilizes to track to a reference orbit. The developed control law however, does not exhibit quick convergence times nor result in optimal fuel usage."

Please consider adding "asymptotical stable control law". How do you whether the convergence speed fast or slow? Do you mean that this is not an exponentially stable? Although, by tuning weight matrices the performance can be improved.

Response3: I think a poorly worded sentence in the Introduction led to some confusion:

“The Lyapunov function utilized in this paper is based on Schaub's work, which results in a stable control law for which the maneuvering satellite utilizes to track to a reference orbit.”

The “this paper” in this sentence refers to the paper by Servidia and Espana, not the current manuscript.   These details were for control of a natural motion trajectory and not highly relevant to the current work, so I removed this paragraph as a part of the revisions to the Introduction which is now more comprehensive with respect to the work on faster-than-natural relative motion.

 

Comment 4:   Table 2. What is the reason to consider that high precision of the initial conditions?

Response 4:  Many digits of accuracy are needed to match the period between the two satellites.   This is not important for the cases with feedback control, but it is necessary for the unforced example and the case with only feedforward control.

 

Comment 5:   Equation 9. The form of equation is typically called as the amplitude-phase form

Response 5:  This equation has been eliminated in an effort to keep the content focused on material that is core to this development, versus material that is well-known and characterized in the literature.

 

Comment 6:   Could authors please add a couple sentences on the way to select Q, R weight matrices for both cases?

Following the importance of the control weights selection authors could refer to a paper where it was optimization based on different objectives:
Biktimirov, Shamil, Gleb Belyj, and Dmitry Pritykin. "Satellite formation flying for space advertising: From technically feasible to economically viable." Aerospace 9.8 (2022): 419.

Response 6:  The method of tuning the LQT is now described at the top of p. 15 before the Q and R matricies are given:

“Equal weights are assumed for all states in the Q matrix and the control components in the R matrix.  Since the cost function only depends on the ratio of the Q and R matricies, this leaves only one scalar variable to be tuned.   The scalar multiplier for the R  matrix is determined iteratively in order to satisfy the constraint on the error bound.”

The multi-objective optimization of both time and fuel of the paper by Biktimirov, et al. is not relevant to this problem with a specified time, but this very interesting paper has been added to the introduction as an example of reconfiguration control  for natural motion trajectories.

 

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

I really appreciate the efforts the authors made to take my comments into account.

I have only one question left. I can't understand what are advantages of feedforward controller presented in section 3. I don't see any analytical justification for its stability or optimality in any of the parameters. And the same things I can't understand about combined controller. 

Author Response

Comment 1:  I have only one question left. I can't understand what are advantages of feedforward controller presented in section 3. I don't see any analytical justification for its stability or optimality in any of the parameters. And the same things I can't understand about combined controller. 

 

Response 1:  To more clearly justify the use of the feedforward we have modified the beginning of Section 3 on pp. 9 and 10:

“The feedforward control represents the nominal control that is needed to track the nominal faster-than-natural relative path assuming a linear dynamics model with no disturbances.   It is the exact (optimal) control for the desired path and the simplified model.  It is not necessarily stable or optimal when implemented in a realistic model, but as long as the disturbances are relatively small, then the difference between feedforward and optimal control is also small.”

This is followed by a simple proof of the somewhat intuitive assertion which is on p. 10 (Equations 4-7).  And then after Equation 7 we have added:

“This provides the fundamental justification for the feedforward controller. Since it provides all but some small deviation from the desired control, the feedback control gains needed to account for unmodelled effects can be small.   This in turn results in lower control effort and less sensitivity to noise than a controller that uses only feedback control.”

Reviewer 2 Report

Comments and Suggestions for Authors

Please find the attached document. 

Comments for author File: Comments.pdf

Author Response

Comment 1:   In Section 6, the properties of the proposed control architecture are well organized. However, considering that an effective conclusion should remind the reader of the main purpose, this section could be further improved. Please summarize and state the main issue and the purpose of this paper in the Conclusion. 2.

 

Response 1:  The first two sentences of the Conclusion on p. 23 of the revised manuscript have been modified and expanded to:

" This paper demonstrates that continuous control for faster-than-natural satellite circumnavigation is a practical mission design option.   A specific control architecture is proposed consisting of a newly developed feedforward controller that is used in conjunction with an existing feedback control scheme.  The feedforward term constitutes the majority of the control needed to track the desired relative motion, allowing for reduced gains on the feedback control which avoids excessive control effort and sensitivity to noise.”

 

Comment 2:  On page 7, equation 3 does not account for the rotation of the relative frame (ℛ). The control force ?̅ℛ is developed in the relative frame, while ?̅?ℎ???? is represented in the inertial frame. Considering that each axis of the relative frame rotates along with the ?̂-axis direction, the relationship between accelerations in the inertial frame and the relative frame should include centrifugal acceleration, Coriolis acceleration, and Euler acceleration. (In this paper, the Euler acceleration is omitted due to the constant rotation of the relative frame.)

 

Response 2:  This is an observant comment, but the control force ?̅ℛ is an inertial quantity – i.e. it is not an acceleration relative to (or as observed from) the rotating frame, it is a (specific) force expressed/represented in the rotating frame.   Forces (like position vectors) are always inertial – they are physical quantities (as opposed to the mathematical description of acceleration) that do not change with frame.   This is evident, for instance, in the CW equations in Eq 4 (now Eqn 8), were the left hand side is the relative frame components of the inertial acceleration and the right hand side is the inertial specific force (again, in relative frame components).

To avoid this confusion we have changed the nomenclature for the force to use “F” instead of “a” and refer to the control as a “specific force” instead of an “acceleration”.   The former term is a bit more cumbersome, but is definitely more accurate and hopefully more clear.

 

Minor comments response:   All of the minor comments have been implemented as suggested by the reviewer.  

 

Reviewer 3 Report

Comments and Suggestions for Authors

Thank you to paper authors for big work on paper improvement during revision. It was significantly improved. There are still some comments to be addressed to explore the considered problem comprehensively. I believe after the next round of revision it will well fit for further publication.

Questions:

- Page 3 (end of page). There is a reasonable thought about preferring continuous control instead of impulsive because of typical propulsion's specific impulse difference. Nevertheless, the impulsive maneuvers requires less dV in comparison to the LQR controller for example. Then a trade-off appears between dV and Isp difference.

So, the statement could be somehow more supported. Even though, it is not clear whether the particular relative trajectories considered in the study can be maintained via impulsive maneuvers as between the maneuvers the system will follow natural dynamics.

- Page 7, "This set of initial conditions were chosen to model typical satellites operating within the LEO regime, as a semi-major axis of 7000 km and 65°of inclination are both the approximate mean parameters based on a TLE query of all active LEO satellites." 

The statement does not sound rational as LEO region has few typical orbit types that does not really fall into the example. It would be even better to consider a particular mission use case, particular orbit type - SSO or other, or just to consider the example without mentioning the averaging over satellites' TLE data.
Authors could also mention If its mean or osculating orbital elements set as it will slightly affect simulations results.

- A naive question is to elaborate a little more on derivation of equations 8. Is it the second derivative of the equations? Can it be proven that the feedforward controller works properly via simple analysis or numerical simulations? Actually, this could work as a good estimate on control resources requirements for a given fast circumnavigation trajectory. Maybe there can be a limit from the propulsion system itself to maintain certain period circumference trajectories. It would be interesting to consider.  

- Thank you for adding more analysis on combined control architecture. Nevertheless, there is a still missing question of influence of control weights matrices selection for different relative trajectories maintenance.

Can an individual control weight matrices selection for a particular circumnavigation trajectory decrease the propellant consumption after all or the primary source of control need comes from the feedforward part? Can it be rigorously shown as it would be one of the main outcomes of the study?

- It would be great to reflect more studies outcomes in the conclusion section. 

Minor:
- Page 8,  Missing word "Figure", "The resulting relative motion seen in 2"

- Page 5, typo in the expression for secondary satellite position vector - all components appeared to be $Z_s$

- Nomenclature: "Relative orbital period of the secondary satellite about the primary". I would suggest to rephrase it and probably put emphasize that this is "period of the satellite relative orbit for fast circumnavigation"

- "Angular rate of rotation" - is it angular velocity or just a mean motion?

- "Relative angular velocity of the secondary satellite" - does it really mean relative angular velocity? or mean motion of the satellite relative orbit for circumnavigation / relative motion position vector angular velocity

- Eq (1) Earth equatorial radius should be used which is not specified

 

Author Response

Comment 1:   Page 3 (end of page). There is a reasonable thought about preferring continuous control instead of impulsive because of typical propulsion's specific impulse difference. Nevertheless, the impulsive maneuvers requires less dV in comparison to the LQR controller for example. Then a trade-off appears between dV and Isp difference.

So, the statement could be somehow more supported. Even though, it is not clear whether the particular relative trajectories considered in the study can be maintained via impulsive maneuvers as between the maneuvers the system will follow natural dynamics.

Response 1:    The reviewer is exactly correct that the impulsive case requires less dV, while the continuous case has a more favorable Isp.   To quantify this tradeoff and support the assertion that the continuous case can require less propellant, at the end of Section 4 on pp. 21 and 22 we have made a comparison with an impulsive case from Ref 12:

“The control effort can be quantified in a single metric by integrating the absolute value of the specific thrust magnitude to get a total Δ? for the circumnavigation. For the baseline case, using a simple rectangle integration approximation yields a total Δ? of 1.038 m/s for one period of the relative trajectory. Using this metric, the control effort of the continuous control can be compared to the results for impulsive control from Ref. [12]. In Fig. 5 of that study, the Δ? requirements are given for circumnavigation at different distances from the primary satellite with different relative motion periods. For the case with a radial distance of 20 meters, an alongtrack distance of 40 meters, an out-of-plane distance of 20 meters, and a relative frequency of 1.5 times the mean motion of the primary, the Δ? is about 0.1 m/s. If we implement the continuous control of this paper for the equivalent case (C=D=20m and T??? = 1ℎ?) then the computed Δ? is 0.208 m/s. The control for this case is given in 11. While the Δ? requirement is almost twice that of the impulsive case, the specific impulse advantage of the continuous thrust system would typically result in a significantly lower propellant mass."

 

Comment 2:    Page 7, "This set of initial conditions were chosen to model typical satellites operating within the LEO regime, as a semi-major axis of 7000 km and 65°of inclination are both the approximate mean parameters based on a TLE query of all active LEO satellites." 

The statement does not sound rational as LEO region has few typical orbit types that does not really fall into the example. It would be even better to consider a particular mission use case, particular orbit type - SSO or other, or just to consider the example without mentioning the averaging over satellites' TLE data.
Authors could also mention If its mean or osculating orbital elements set as it will slightly affect simulations results.

Response 2:   Yes, the approach of averaging over all active satellites was probably overly complex and not particularly relevant.  Any LEO example serves to demonstrate the efficacy of the control.  Also, while the issue of mean/osculating elements does indeed have only a very slight (unobservable) impact on the results, we have also addressed this for completeness.   The text on p. 7 is now:

“An example LEO satellite is considered with a semi-major axis of 7000 km and 65°of inclination; all other orbital parameters were set to zero.  These orbital parameters were all considered osculating elements in the computation of the initial state.”

 Comment 3:   A naive question is to elaborate a little more on derivation of equations 8. Is it the second derivative of the equations? Can it be proven that the feedforward controller works properly via simple analysis or numerical simulations? Actually, this could work as a good estimate on control resources requirements for a given fast circumnavigation trajectory. Maybe there can be a limit from the propulsion system itself to maintain certain period circumference trajectories. It would be interesting to consider.  

Response 3:  For the first question we have reworded the text before Equations 8 (now Equations 13) on p.12 of the revised manuscript to try to be more clear on their derivation:

“To determine the feedforward control for the nominal motion of Equations 12, the first and second time derivatives of Equations 12 are determined and substituted into Equations 8. The resulting expressions for the feedforward control are:”

As to a proof that the “feedforward controller works properly”, we make no claim that the feedforward is stable or accurate for the nonlinear/perturbed system, which is why the feedback piece is essential.   The only claim we make is that the feedback piece of the control is relatively small compared to the feedforward piece, which is now shown in a simple proof on p. 10.   And, yes, this property means that the feedforward control can be used to estimate control resources for a variety of mission parameters.   While that could be a nice addition to the paper, the authors feel it is beyond the scope and focus of the paper.

 

Comment 4:   Thank you for adding more analysis on combined control architecture. Nevertheless, there is a still missing question of influence of control weights matrices selection for different relative trajectories maintenance.

Can an individual control weight matrices selection for a particular circumnavigation trajectory decrease the propellant consumption after all or the primary source of control need comes from the feedforward part? Can it be rigorously shown as it would be one of the main outcomes of the study?

Response 4:  As now described more clearly and rigorously in the text and added simple mathematical justification from the bottom of p. 9 through almost the end of p. 10 (including Equations 4-7), the primary source of the control does indeed come from the feedforward part.  So the selection of control weight matricies cannot improve the propellant consumption, but can definitely increase consumption if not properly tuned! 

 

Comment 5:   It would be great to reflect more studies outcomes in the conclusion section. 

Response 5:  We are not sure what is specifically suggested here.   There are very few studies that consider the faster-than-natural motion as described in the introduction.   We now have a comparison with the impulsive study of Ref. 12 at the end of Section 3 on p. 22 which seems a more appropriate location than the conclusions.

 

 

Minor comments response:  All of the minor comments have been addressed.   With regard to the "Relative angular velocity of the secondary satellite"  comment, we do not think that use of “mean motion” would be an appropriate since that has a specific context for absolute orbital motion, so instead we have change the description of thetadot to “Angular velocity of the relative position vector of the secondary satellite.”

 

Round 3

Reviewer 2 Report

Comments and Suggestions for Authors

Please see the attached file. 

Comments for author File: Comments.pdf

Author Response

Introductory Comments:  The authors have made significant efforts to improve this paper within a short period, and in this reviewer’s opinion, many aspects now meet a sufficient level for publication. However, while the authors stated that all minor comments were addressed, this reviewer has found that not all of them were implemented. This reviewer has identified additional minor errors, as well as previously commented issues (some mentioned twice) that remain uncorrected. Although minor mistakes may seem insignificant and have limited direct impact on the quality of the paper, their recurrence can raise concerns about the paper's overall quality. Such oversights may lead readers to question the reliability of the numerical simulations, given the parallels between the textual and methodological accuracy. Now, it is strongly recommended that the authors focus on thoroughly resolving all remaining minor issues in this revision to enhance the paper’s overall quality and precision.

 

Response:  The authors agree that care and precision are the cornerstone of our profession and should be exhibited to the utmost extent in the numerical work as well as the manuscript.   We are both grateful and apologetic that the reviewer’s attention to detail exceeded our own.   We have reread the manuscript and made some corrections in addition to the specific ones requested by the reviewer:

-We now have consistent capitalization in the nomenclature.

-We have made several additions to the nomenclature to account for all variables in the manuscript except for the orbital elements on p. 7 and 8 of the revised manuscript which are not in any equations and are very standard.

-We have corrected some small punctuation errors.

-We have performed minor changes to the text to increase clarity and precision.

-We are now more precise and concise with the notation for the thrust/control.  We have eliminated all use of “F” for the control and redefined the variables for “u”  which now has a superscript to indicate the frame and a subscript to indicate the architecture (feedforward, feedback, combined).

 

Comment 1:  Please ensure consistency and clarity in their definitions. ? (on page 15) and ?̅ in Figure 1 (on page 5) are used and defined in the main content. But they are not introduced in Nomenclature. Additionally, definitions are typically placed immediately before or after their first appearance. Please relocate the definition of ?̅ from page 11 to page 5.

 

Response 1:  Both ? and ?̅ have been added to the Nomenclature.  The definition of ?̅ has been moved from p. 11 to the bottom of p. 6 and top of p. 7 of the revised manuscript where it is closer proximity to Fig. 1 but still made logical sense as far as the development of the text.

 

Comment 2:   The abbreviation LEO is still not defined before used. The abbreviations EO (electrooptical) and RF (radio frequency) are defined, but are never used again.

 

Response2: “Low Earth orbit” now introduces the acronym LEO on the bottom of p. 7 of the revised manuscript and the acronyms RF and EO have been removed.

 

Comment 3:   On page 22, there is typo: The control for this case is given in 11. On page 23, the caption of Figure 11 requires correction.

 

Response 3:  The word “Figure” is added in the third to last line on p. 22 of the revised manuscript and the caption of Figure 11 on p. 23 is now “Control history for comparison to impulsive approach in Ref. 12”

Reviewer 3 Report

Comments and Suggestions for Authors

Dear authors, thank you for addressing most of the comments of the previous round of review. It has been significantly improved.

Although, authors decided to leave out of scope some comments that could bring scientific novelty to the research topic. It is their right and I will not keep pushing to consider it.

Summarizing, I feel that the paper now is of good enough level and suggest to accept it. 

Author Response

Thank you for your time and assistance in improving the paper.