Compensation Control Strategy for Orbital Pursuit-Evasion Problem with Imperfect Information

Round 1

Reviewer 1 Report

Nice article. It is written in a simple and clear language for engineers. All scientific results are confirmed by extensive numerical modeling. The article will undoubtedly be of interest to both university students and engineers.

Comments. 1) Figure 6 mistakenly repeats Figure 5.
2) According to the reviewer, two references to primary sources should be added to the list of cited literature:

W.H. Clohessy and R.S. Wiltshire, Journal of Aerospace Sciences, Vol. 27, No. 9, 1960, pp. 653–658.

S.W. Shepperd, Journal of Guidance, Control, and Dynamics, Vol. 14, No. 6, 1991, pp. 1318–1322.

Author Response

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Reviewer 2 Report

In general, the paper deals with an interesting topic, but it lacks of a good structure and of a systematic approach: different topics are dealt in succession and it is not easy for the reader to understand the connections between them.

Specific remarks are:

• Avoid repetitions in the text, e.g. lines 70-74 with lines 139 to 143.
• Variables or figures are often not properly defined or presented, e.g.
  • tgo is not used in eq. 10.
  • rho is undefined in eq. 10.
  • omega is undefined at line 185.
  • There is no explanation of what the functions and the 4 cases are in Fig. 2.
• Two spacecrafts in proximity will experience the same gravity conditions. What does gravity difference (line 73) mean?
• Is it correct to speak of pursuit-evasion games (even in the title), when the proposed Guaranteed Cost Strategy is only yielding sub-optimal solutions (see also the last remark of this list)?
• Does the method in Sec. 3.2.2 need to be repeated at every time step, in the case of uncertainty (e.g. noisy measurements)? If yes, how practical is it?
• Even though the assumptions at lines 256-257 do not hold for the orbital case, they do not seem to constitute a novelty on their own, in my opinion.
• It is incorrect to inject noise directly on the position states in the UKF formulation. Noise should be injected only as the result of external dynamics on the spacecraft, i.e. accelerations/torques. Then, through the process noise matrix Q, this noise will be propagated to all the states.
• In example 1, if the evader can escape in 207 sec and the pursuer can intercept in 212 sec. why is the pursuer winnning? Have you arbitrarily decided that the guaranteed miss of the pursuer was smaller than that of the evader? Or is the evader not playing optimally? Can this be considered a fail of the proposed Guaranteed Cost Strategy?

 

 

Author Response

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Round 2

Reviewer 2 Report

The authors have mostly answered to my comments. The only comment I would like to give is to their response 5. I believe that the proposed strategy for computing t_go might not be adequate in some cases, for example in the presence of bearings-only measurements, which is the case of many optical-based sensors (e.g. cameras) that can be used on board of a satellite (range sensors are much more expensive and problematic in terms of accuracy and pointing).

In the case of bearings-only measurements in a pursuit-evasion context, though, the t_go is unobservable, as demonstrated in many literature papers, as for example:

- Oshman, Y.; Davidson, P. Optimization of observer trajectories for bearings-only target lo-calization.IEEE Trans. Aerosp. Electron. Syst.1999,35, 892–902.
- Battistini, S.; Shima, T. Differential games missile guidance with bearings-only measure-ments.IEEE Trans. Aerosp. Electron. Syst.2014,50, 2906–2915

I believe it might be worth to declare that the proposed analysis is limited by some hypothesis.

Author Response

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This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.

Round 1

Reviewer 1 Report

This paper deals with orbital pursuit-evasion game with imperfect information. The control problem is formulated on the basis of Clohessy-Wiltshire equations of circular orbital motion that comes from linearization of standard orbital motion equations. The control problem consists in minimizing final distance between the pursuer and evader. This is another article about pursuit-evasion problem and its novelty and contribution should be clearly demonstrated especially with reference to [7] and many other papers in the filed. 

I have the following comments.

1. There is an error in Eq (1), it should be 3\omega^2 x_i+2\omega \dot{y}_i.
2. Section 4.1 was borrowed from ref. [7], but neither in [7] nor in the article there is no proof of optimality of strategy (10). What is necessary condition of optimality? It should be also pointed out that the problem is in fact minimax problem where (6) is first minimized wrt. U_p, which gives the cost dependent on U_p, and then maximized wrt. U_e.
3. J(t) in (9) is not correctly defined, is it the same as in (8)? Is t_f fixed or not? What is a domain of J in (8) and in (9)? Please explain it.
4. Some comments on the convergence of homotopy method would be valuable.
5. The sign in (27) should depend on i.
6. The authors consider additive sensor/input noise and the Extended Kalman Filter is used to improve quality of control. The input noise w in (28) is not correctly defined. In fact, (28) should be treated as stochastic equation. Discretization of (28) is unnecessary. It is possible to construct continuous-discrete filter, that produce state estimate also between the measurements.
7. Matrix \Gama (T) in (30) is incorrectly defined, see e.g. https://en.wikipedia.org/wiki/Discretization or any book on Kalman Filtering.
8. In my opinion, the EKF is not very good solution. It is often unstable, involve very good initial guess and and it has large mean squred error. It would be better to use Unscented Kalman Filter or even try to find an optimal filter. This should not be very difficult, because the state equation is linear.

Finally I conclude, that the problem considered is interesting, especially in the area of uncertainty and disturbance compensation, but the contribution and novelty should be clearly described and the above errors must be removed. 

Author Response

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Reviewer 2 Report

The paper is on a topic of orbital pursuit-evasion game, which is important to the space community. The game has imperfect information owing to measurement noise and time delay. The authors use C-W equations to model the dynamics, homotopy method combined with Newton's method to calculate time to go for the closed-loop solution, compensation for time delay, and an extended Kalman filter to estimate the states. 

1. While the approach seems appropriate, the contributions of the paper to the literature are unclear. The introduction make it clear what new insight does the paper provide, and how does it advance the literature.  

2. What is the novelty in the methodology employed in the paper? Please clearly write that in the introduction.   

3. I do not think that the following statement in the Introduction section is not quite true: "Most existing studies assume the game has complete information." There are several studies that do not consider complete information. In fact, beyond the studies considering perfect information, the literature can be classified into studies on incomplete information, imperfect information, as well as uncertain information. Please see Table 1 in the recent paper "Linear Regression Models Applied to Spacecraft Imperfect Information Pursuit-Evasion Differential Games" by Linville and Hess. 

4.  It is not clear how the time delay Delta t is accounted for in the compensation strategy - in fact, it does not appear in the equation(s). 

5. The paper is easy to follow, but it has grammatical issues that need to be corrected. 

6. The conclusions should reflect the insight provided by the simulations conducted in the paper.  

Author Response

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Reviewer 3 Report

The paper presents an interesting solution for the compensation of the time-delay in orbital pursuit-evasion games. The main thing to revise, in my opinion, is how the results are showcased. A lot of different cases are presented, but I believe that many examples do not add a lot to the discussion and they rather make reading heavier, e.g. the cases where there is no time-delay. On the other hand, the random part of the simulations is not very well described. There is no information on the noise figures and on the initial estimation errors. Furthermore, in the presence of random variables in the simulations, it is preferable to adopt a Monte Carlo simulation approach, rather than performing a single run. Definitely, the authors could simplify the numerical simulations session, reducing the number of cases and increasing the number of runs in a Monte Carlo fashion.

Minor issues are:

• Lines 158-161 are a repetition of the introduction (lines 78-82)
• The arc length constraint on line 189 is not defined
• Could you give a definition of what the uncertainty set represents?
• It would be better to plot the ZEM rather than the three position coordinates (e.g. fig. 4)

Author Response

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Round 2

Reviewer 1 Report

Dear Authors

Detailed comments

1. Eq. (1) still contain an errors.
2. Matrix A in (4) does not correspond to (1) and contain an error.
3. There is no guarantee that the denominators in (10) are positive.
4. There is no proof that Z(t) tends to zero under the strategy (10).
5. There is an implicit assumption that the evader strategy given in (10) is optimal. But we don't know whether or not U_E is optimal. Maybe there exist different evader's strategy that is better than (10)? Hence there is no evidence that (10) is correct.
6. It follows from (15), that ||Z(t_f)||<0 for some \rho_E, \rho_P and t_f. It is rather impossible.
7. Statement "From Equation (15), it can be seen that when the pursuer adopts UP* , the cost will not exceed a fixed value." is trivial. Since the final time is fixed then the cost is ALWAYS bounded by some value, but this value depend on final time and can grow.

8. Stochastic model (34) is not properly formulated. It has been assumed in advance in (34), that the state X of the system is perfectly known which is not true. As a consequence (35) is not proper discrete time model of the system. Matrix \Gamma_k in (36) is incorrect (see my previous review).

    

    Finally I conclude that my previous comments has not been taken into account, and there is still no proof of the stability of the proposed strategy. That's why I recommend that this article should be rejected.

    

    

    

    

Reviewer 3 Report

Thanks for adequately addressing my previous comments.