Multi-objective Evolutionary Optimization of Evasive Maneuvers Including Observability Performance

Multi-objective Evolutionary Optimization of Evasive Maneuvers Including Observability Performance*Yu Dateng, Luo Yazhong ,Jiang Zicheng, Tang GuojinCollege of Aerospace Science and EngineeringNational University of Defense TechnologyChangsha, Chinaluoyz@Abstract—This paper investigates optimal orbital evasion problem with considering observability performance by using a multi-objective optimization approach. The degree of observability is defined as a new performance index, which has a negative correlation with the accuracy degree of relative state estimation. A two-objective optimization model is then formulated and the NSGA-II algorithm is employed to obtain the Pareto-optimal solution set. The numerical results show that the proposed approach can effectively and efficiently demonstrate the relations among the evasive mission characteristic parameters. The proposed approach offers a novel view in solving orbital evasion problem.Keywords—Optimal Evasive maneuver; Evasion Problem; Observability; NSGA-II; Angles-OnlyI.I NTRODUCTIONMany published papers have studied the optimal evasion problem [1-6],which in fact is a path-planning problem. In the problem, a pursuer tries to approach to its target through several maneuvers and the target, namely an evader, expects to escape from the pursuer through its evasive maneuvers. By employing a multi-objective optimization approach, this paper studies the optimal evasion strategy of an evader in which a pursuer uses angles-only navigation.Varieties of evasive maneuver strategies have been proposed. For example, Forte [7] analyzed an equivalent linearized 3-D optimal evasion problem. Shinar [8] obtained a closed-form solution for a switching function and considered the effective navigation gain. The optimal evasive maneuver has also been applied to many aerospace problems. Patera [9-11] firstly proposed an evasive maneuver strategy within the framework of collision probability. Kelly [12] introduced an optimal rendezvous evasive maneuver method using a nonlinear optimization technology. Bombardelli [13] obtained a maneuver direction to maximize the missing distance numerically, which is a function of the arc length separation between the maneuver point and the predicted collision point. In [14], an optimal evasive maneuver was studied for fixed and unfixed maneuver directions, respectively.The above-mentioned studies related to evasive maneuver always assume that ideal measurement could be provided and the influence of navigation performance is hardly considered. In practice, the influence of navigation performance is significant and needs to be analyzed deeply. In this paper, we introduce the observability of a system as a new index by considering accuracy of navigation estimation. As we know, the maneuvers of an evader and pursuer can alter the observability of the system during angles-only navigation [15, 16]. Woffinden [17] obtained an analogic relationship between maneuvers and the system observability in the field of orbit rendezvous. Vallado [18] pointed out that relative motion diversity has a positive correlation with the system observability. Grzymisch and Fichter [19, 20] provided an analytical derivation of the observability conditions to find an optimal maneuver, which could ensure observability and obtain the best possible navigation estimate, simultaneously.Generally, if an evader wants to escape from a pursuer, it is a prerequisite to need an appropriate maneuver direction and sufficient maneuver time. Besides the observability, the relative distance should also be guaranteed during the approaching. Thus, the problem has multiple planning variables and design objectives. The evasive maneuver problem therefore corresponds to a multiobjective optimization problem, and the traditional single objective optimization model (which usually only maximize the relative distance) will encounter great difficulty in solving such an evasive problem.In this work, a novel multi-objective optimization approach is proposed to design evasive maneuvers by introducing an additional new performance index related to system observability. The remainder of this paper is organized as follows. First, the orbital evasion problem is described. Then, the model of observability is established and the degree of observability (DOO) is introduced. A non-dominated sorting genetic algorithm II (NSGA-II) is employed to planning an optimal evasive maneuver, where both DOO and relative distance can be maximized. Finally, an example of optimal evasive maneuvers obtained by multi-objective optimization results is provided and numerical simulations show a desired improvement of DOO and relative distance. In addition, a simulation of the extended Kalman filter (EKF) estimation is performed. The results indicate that the optimal evasive maneuvers can effectively reduce the accuracy of the relative state estimation and weaken the navigation performance.This study was co-supported by the National Natural Science Foundation of China (No. 11272346 and 11402295) and the National Program on Key Basic Research Project (No. 2013CB733100).978-1-4799-7492-4/15/$31.00 © 2015 IEEEII.P ROBLEM O VERVIEWAn orbital evasion problem contains two spacecraft—evader and pursuer. Pursuer’s objective is to approach or capture the evader, and evader’s objective is to escape the pursuer and keep a relative safe distance from it. Assume that a pursuer starts from a rather long distance ( 50 km) with angles-only measurements. In such a distance, the pursuer usually executes two or three impulses to approach the evader, which means the approaching strategy is not time-varying. Therefore, the orbital evasion problem can be regarded as a one-sided optimal control problem. The one-sided optimization problem considers only one player and is equal to a flight path optimization problem.As for a flight path optimization problem, three control variables are needed—the magnitude of the impulse, appropriate maneuver direction and maneuver time. The magnitude of the impulse is constrained by the maneuvering ability of the evader, so the last two are chosen as optimization variables and the first one is set as a constant in this work. Besides the traditional safety index—relative distance, the observability is introduced as an evasive index. The maneuver direction and maneuver time can change the observability of the system, because the observability of the system is related to the relative motion which is influenced by the two optimization variables. Thus the two optimization variables have a coupling relationship with the observability and relative distance.The optimal maneuver is expected to decrease the observability, and maximize the relative distance simultaneously. The evasive maneuver problem becomes an optimization problem that contains two control variables and two objective functions. To solve this multiobjective optimization problem, the quantification model of observability and the multiobjective optimization model is needed.III.O BSERVABILITY QUANTIFICATION In this section, the accuracy degree of relative position and velocity estimation is analyzed. There are different indexes available to describe the observability, such as the distance error metric [22] and condition number of the observability matrix [23]. Different from the previous indexes, a new index to represent the level of state estimation is introduced here and we call it the observability. The covariance matrix of the relative position and velocity estimation is then determined and the DOO of the relative state is analyzed based on the covariance matrix.A.Measurement EquationsIn general, optical camera is the most common measuring equipment, and it provides two angle measurements (azimuth θ and elevation ε) at each instant of time. Figure 1 illustrates the schematic of the geometry of the pursuer-evader system along with the azimuth and elevation angles in the cameraframe, where FPO , FMO, and FEOare the pursuer, camera, andevader coordinate systems, respectively. In the followinganalysis, FMO and FPOare regarded as the same coordinate.The orbital coordinate system is defined as follows: x is directed along the position vector from the center of the earth to the pursuer, y is directed along the inverse direction of the velocity, and z=x⨯y obeys the right-handed coordinate system. Orbital coordinates attached to the evader are similar and are omitted here.Fig. 1.The relationship of relative measurement.Consider an evader in a near-circular orbit. The distance from a pursuer to the evader is fairly smaller than that from the evader to the center of the earth. The Clohessy-Wiltshire equations are therefore applied to describe the relative motion between these two spacecraft.The relative motion of the evader in the orbital coordinates is governed by the dynamics equations as follows:=+X AX BU(1) where A is a state space matrix, B is a control input matrix, and U is acceleration vector. The solution to (1) is(,)(,)dtt t v stt t t sτ=+⎰XΦXΦU(2)where(,)t tΦ is a state transition matrix()(,)e[]rr rvt tr vvr vvΦΦt tΦΦ-⎡⎤===⎢⎥⎣⎦AΦΦΦ(3)If the thrust of pursuer is discrete and an impulse at1kt-isdenoted as1k-∆v, then the relative motion of the pursuer atkt can be presented as1111(,)(,)k k k k v k k kt t t t----=+∆XΦXΦv (4) Equations (2) and (4) give two explicit expressions for the relative state recursion for continuous and discrete thrusts, respectively.If denoting the relative position between the pursuer and evader in the measurement coordinate systemas[]TPEx y z=r, the relationship betweenPEr and the measurement parameters of the optical camera isarctan(/)()arctan(Ey xzεθ⎡⎤⎡⎤==⎢⎥⎢⎥-⎢⎥⎣⎦⎣⎦h X (5)where ε and θ are the elevation and azimuth, respectively. If denoting the distances from the center of the earth to the evader and to the pursuer as E r and P r , respectively, the evader and pursuer are in nearly the same orbital plane and with a phase angle of ϕ during the approaching procedure [21], and they have the following relationship2222cos PEE PE P rr r r r ϕ=+- (6)whereE r(7) In general, E r is prescribed to the pursuer before approaching. From (6) and (7), the transition matrix PE Q from EO F to PO Fis obtained as0001PP PE P P r r r r ⎤⎥⎢⎥=⎢⎥⎢⎥⎣⎦Q (8) Substituting P PE PE =r Q r into (5) leads to the measurement equation()X =+Z h υ(9)where T []εθυυ=υ is the measurement noise of the opticalcamera, ευ and θυ are Gaussian and white with zero mean. From (9), ()X h is a nonlinear matrix function of X , and the measurement sensitive matrix for the EKF can be obtained as follows:()11121323T212223h h h h h h ⨯∂⎡⎤==⎢⎥∂⎣⎦h X H X 0 (10) According to the EKF, (1), (3), (9) and (10) allow us to estimate the relative position and velocity for angles-only relative navigation. From the above, it is not difficult to find that the orbital maneuver from pursuer or evader affects the relative navigation and measurement, and the observability of the system.B. Covariance of Relative State EstimationExpanding (9) at |1ˆk k -X as Taylor series to the first orderapproximation, we get()()()()()|1|1|1Tˆ|1|1ˆˆˆˆk k k k k kk k k k k k k k ---=--∂=+-∂=+-X X h X Z h X XX X h X H X X (11)where |111ˆˆ(,)k k k k k t t ---=X ΦX . Thus we rewrite (11) as()|1|1ˆˆk k k k k k k ---⎡⎤=+-⎣⎦X X H Z h X (12)where H -stands for an arbitrarily generalized inverse matrixof H . Owing to11111ˆ[()]k N k N k N k N k N --+-+-+-+-+=+-X X H Z h X(13)where N =1, 2, 3,…., it can further be rewritten as11111ˆˆ(,)(,)[()]k N k k k N k k k N k N k N t t t t --+-+-+-+-+=+-ΦX ΦX H Z h X (14)or()11111ˆˆ(,)k k k N k k N k N k N t t ---+-+-+-+⎡⎤=+-⎣⎦X X ΦH Z h X (15) where k X is the relative position and velocity estimation at K t from the observation of Z at 1K N t -+. Using theobservations of {}12,,...,K N K N K Z Z Z -+-+ in connection with (12) and (15), we get the expression for the relative state estimation at K t()()11ˆˆ(,)ˆkk ki k i i ii k N k t t --=-+-⎡⎤=+-⎣⎦=+-∑X X ΦH Z h X X ΓZ h (16) where[]T1122(,)(,)k N k N k k N k N k k t t t t -+-+-+-+=ΓH ΦH ΦHTTT T 12k N k N k -+-+⎡⎤=⎣⎦Z Z Z Z()()()TT T T 1122ˆˆˆk N k N k N k N k k-+-+-+-+⎡⎤=⎣⎦h h X h X h X From (16), the accuracy of the relative state estimation is related to the matrix Γ. Assume that k X in (16) is a true description of the relative states, the measurement noise Z h -on the error of the relative state estimation |ˆk k k N --X X is reflected by the characteristics of 1Γ-. The covariance matrixof N measurements of noise is[]12diag N =R R R R(17)where 22diag , 1,2,,i i N θεσσ⎡⎤==⎣⎦R . The covariance matrix of relative state estimation cov P is then obtained as [24]()[][]11T11cov 1(,)(,)k T i i k i i i k i k N t t t t ----=-+⎧⎫==⎨⎬⎩⎭∑P ΓV ΓH ΦR H Φ(18)C. Introduction of Degree of ObservabilityEquation (18) for k N = means that all of the N measurements are used to estimate the relative state in k . Consequently, the covariance matrix of the relative state estimation is[][]1T11cov 1(,)(,)N i i N i i i N i t t t t ---=⎧⎫==⎨⎬⎩⎭∑P H ΦR H ΦFIM (19)where the matrix FIM is the Fisher information matrix [25] of the measurements, which contains the information of the system relative states acquired by the measurement sensor. It should be mentioned that if the matrix 0P (the covariance of the initial relative state estimation) is invertible, (19) is further expressed as [24]11cov 0(--=P P +FIM)(20) From the basic property of the covariance matrix, (19) represents the accuracy of the relative state estimation. Consequently, cov P allows us to define the DOO of the relative states. That is, the DOO of the relative position DOO R and the DOO of the relative velocity DOO Vare defined belowDOO R =(21) DOO V =(22)respectively, where cov ijP represents the i th row and j th column element of cov P . The definition defined above considers the estimated accuracy of all directions, which is more reasonable than other indices. Obviously, the DOO represents the accuracy of the relative state estimation of one spacecraft to the other. The larger the values of DOO R and DOO V are, the lower the accuracy of the relative state estimation is. Moreover, if the observation abilities of two spacecraft are equal, the reciprocal DOO of the pursuer and evader is the same.D. DOO Calculation Based on Fictitious OrbitAs we know, an evasive maneuver changes the coordinate system of the evader, which means that the calculation of the relative states needs to be corrected, otherwise the calculated FIM is incorrect. In order to solve this problem, a fictitious orbit that represents the orbit of a non-maneuvering evader is shown in Fig. 2.k X -,k E Pursuer orbitEvader orbit aftermaneuverFig. 2. Illustration of relative states calculation.Before calculating the DOO, we firstly determine the relative states ,k E X of an evader after maneuver with a fictitious orbit and the relative states ,k P X of a pursuer with the fictitious orbit in k measurements, separately. Accordingly, the relative states of the evader after a maneuver with the pursuer after k measurements can be obtained to be,,k k E k P XX X =- (23) Under such a circumstance, the DOO of the evader and pursuer can be determined.IV. M ULTI -OBJECTIVE O PTIMIZATION M ODELA. Optimization VariablesIn order to optimize the direction of evasive maneuver, an impulse is assumed to be fixed in the following analysis, and azimuth θ, elevation ε, and maneuver time t are selected as three control parameters to be optimized, viz.,T [,,]t εθ=G (24)Taking into account the spatial geometric relationship between the two spacecraft, the constraint conditions for an evasive maneuver problem read322220m t Tππθππε⎧-≤≤-⎪⎪⎨-≤≤⎪⎪≤<⎩ (25) where T is total observation time and m t is maneuver time.In order to simplify our problem further, here we make some observations on the range of the optimization variables. For two moving objects in space, it is well known from [18] that an increased difference of the system relative motion gives rise to an increase of system observability (e.g., the relative motion feature of two coplanar spacecraft in a circle orbit is more unobvious than that in an elliptic orbit). Therefore, the observability of two coplanar spacecraft is weaker than that of two non-coplanar spacecraft, which is equivalent to that the DOO is higher in the present analysis. Due to this reason, the evader prefers to choose an evasive maneuver in a plane, so the values of elevation can be constrained in a range near zero. As for the maneuver time, we constrain it in a constant time κ, the reason for which is that the evasive maneuver has little effect on increasing the DOO if the evasive maneuver is too late. Under such circumstance, new constraint conditions are stated as32200m t ππθεκ⎧-≤≤-⎪⎪⎨=⎪⎪≤<⎩ (26) Thus, two control parameters are left to be optimizedT [,]t θ=G (27)B. Objective FunctionsTo solve a multiobjective optimization problem, a reasonable evaluation index is required to be determined firstly. For the evasive maneuver problem under consideration, the variation of DOO during evasion plays a key role. Also, the minimal relative distance as a traditional safety evaluation index ensures the safety of an evader during its evasion. Based on these considerations, in what follows we choose two indexes —minimal relative distance L and DOO to evaluate the optimization results.During evasion, the evader expects the DOO of the system to be larger. At the same time, the relative distance between thetwo spacecraft also becomes larger. Therefore, we choose an objective function as()()1DOO2max max f R f L ⎧=⎪⎨=⎪⎩G G (28) A multiobjective problem is formulated as a nonlinear constrained problem as follows:12Maximize[(),()]F F G Gwhere 1()F G refers to the DOO function of velocity and 2()F G the relative distance function of velocity. a. DOOSince the DOOs of the relative position DOO R and relative velocity DOO V can be determined by the system state Fisher information matrix in k measurements, from (19) one gets[][]T11FIM (,)(,)kiiki i i k i k N t t t t -=-+=∑H ΦR H Φ (29)Hence the FIM is a function of the relative states depending on the pursuer and evader. For convenience of analysis, it is assumed that the impulse of the pursuer does not change. Consequently, one has 1F expressed by1()(FIM())F F =G G (30)Here DOO R or DOO V can be chosen as an optimization function. In the following, as an example we only take DOO R as an optimization function.b. Minimal Relative DistanceIn order to guarantee the safety of evader, minimal relative distance during the approaching should be considered. Recalling (4), one has the relationship between the relative states and impulse2()(())k F F =G X G (31)Now subtracting the state vectors of one spacecraft from that of the other at k moment, from (22) one obtains the relative states,,k k E k P =-XX X (32) The minimal relative distance ismin((1:3)),1,2,...i L Xi == (33) where i Xis a 6-dimension state function, (1:3)i X is the relative position state vector, and the norm means the 2-norm of a vector.C. NSGA-II AlgorithmIn this study, one of the most well-known multi-objective evolutionary algorithms, the NSGA-II, is employed to solve the multi-objective optimization problem, which was proposed by Deb et al. [26]. This algorithm uses the idea of transforming the multi-objectives to a single fitness measure by the creation of a number of fronts, sorted according to nondomination. During the fitness assignment, the first front is created as theset of solutions that is not dominated by any solutions in the population. These solutions are given the highest fitness and temporarily removed from the population, then a second nondominated front consisting of the solutions that are now nondominated is built and assigned the second-highest fitness, and so forth. This is repeated until each of the solutions has been assigned a fitness. After each front has been created, its members are assigned crowding distances normalized distance to closest neighbors in the front in the objective space) later to be used for niching. The NSGA-II has been successfully applied in spacecraft trajectory optimization, for example, in designing a three-objective impulse rendezvous problem [27] and a two-objective robust rendezvous problem with considering uncertainty [28].V.N UMERICAL S IMULATIONConsider an illustrative example. For simplicity, it is assumed that the orbit control of the pursuer is based on the CW equations. Assume that after pursuer executes an approaching maneuver(00t =), initial relative states 0X at 0t and expected terminal relative states of the pursuer at 2000 s f t = areT 0[41933.5 m,-27232.0 m,-50.0 m,-49.94 m/s,10.18 m/s,0.0]=XT [3000 m,0,0 ,0 m/s,0 ,0.0]f =XThe standard fourth-order Runge-Kutta method is applied in numerical simulation. An optical camera located on the chaser is employed for relative measurements, and the azimuth angle θand the elevation angle ε with standard deviations of 0.001θσ= rad and 0.001εσ=rad are obtained, respectively. An initial range from the pursuer to the evader is taken as 50 km, an initial covariance matrix and a covariance matrix of measurement noise are respectively2222220diag[(2000m),(2000m),(2000m),(5m/s),(2m/s),(1m/s)]=P ,22diag[(0.001rad),(0.001rad)]i =R , i =1, 2, …, NIn addition, the measurement frequency of the optical camera is 1 Hz. The evasive impulse of the evader is fixed to 3m/s and 6m/s in magnitude. All the data mentioned above are acquired based on the project we have ever done.A. Multiobjective Optimization ResultsThe parameters of NSGA-II are as follows: population size, 150; maximum number of generations, 80; probability of crossover, 0.90; and probability of mutation, 0.08. Constraints are chosen the same as those in section IV, and κ=50s is taken. NSGA-II was used to optimize the evasive trajectories for the different configurations. It should be noted that 10independent runs for each test case were completed by considering the stochastic characteristic of the NSGA-II, and all of the Pareto-optimal solutions obtained in the 10 runs were compared and revised by deleting the inferior and repeating solutions. The revised Pareto-optimal solutions of the 10 runs were then regarded as the final Pareto solution set.In order to analyze the influence of the impulse magnitude on the Pareto solutions, different Pareto-optimal fronts were compared. Figure 3 compares the Pareto-optimal fronts withdifferent magnitude of impulse, in which the converged Pareto-optimal solutions are indicated by symbols ―o,‖ ―*‖.4DOO of Relative Position(m)M i n i m a l R e l a t i v e D i s t a n c e (m )Fig. 3. Pareto fronts with different impulse magnitude.Each one of scattered data in Fig. 3 depends on its two optimal control parameters T [,]t θ=G , namely the maneuver direction and maneuver time. It is known from Fig. 3 that the magnitude of impulse has a more obvious effect on minimal relative distance than DOO of relative position. It can be seen that insome specific zones, the solutions become discontinuous. It is easier to understand this from the relationship of relative position and estimation. Both relative distance and measurements influence the value of DOO of relative position. A larger relative distance with good measurements (which means that the relative trajectory is more far away from the trajectory without maneuver) and a smaller relative distance with bad measurements can obtain similar DOO, thus discontinuity generates. These can be seen in Sec V.C and Woffinden obtained similar results in [17]. For different safety requirements of minimal relative distance and DOO of relative states, it is easy to choose a suitable solution from optimal solutions in Fig. 3, so the evader can acquire a desired minimal relative distance and terminal relative state estimation.B. Confirmation of pareto optimalityFig. 3 shows a number of Pareto solutions that were obtained by NSGA-II. However, thus far, the Pareto optimality of these solutions has not yet been verified. This section partially shows the Pareto optimality of the solutions bycomparing them with the DOO-optimal and distance-optimal solutions.4DOO of Relative Position(m)M i n i m a l R e l a t i v e D i s t a n c e (m )Fig. 4. Comparisons between Pareto-optimal, DOO-optimal, and Distance-optimal solutions.Assuming the magnitude of impulse is 3m/s, by using an optimization approach based on genetic algorithm, we obtained the DOO-optimal and distance-optimal solutions. The terminal DOO R of DOO-optimal solution is 946.94m and L is 9160m, as listed in Table I. Three Pareto-optimal solutions obtained by NSGA-II are also listed in Table I. Without considering the performance of DOO, only the minimal relative distance L is chosen as the objective function in order to obtain the distance-optimal solution. This solution is obtained with a function value of DOO R 185.94m, and in correspondence with L 11538m, as listed in Table I. The case without maneuver is also listed in Table I for comparison. Three Pareto-optimal solutions obtained by NSGA-II are also listed in Table I. The DOO-optimal, distance-optimal, and Pareto-optimal solutions are plotted in Fig. 4. It can be seen that the DOO-optimal solution is located at the end point of the Pareto-optimal fronts. The distance-optimal solution is near the end point of the Pareto-optimal fronts, which may because the prematurity of the algorithm we used. Therefore, the solutions obtained using NSGA-II are confirmed by the Pareto optimality by comparing them with the DOO-optimal and the distance-optimal solutions.TABLE I. C OMPARISON OF P ARETO -O PTIMAL , DOO-O PTIMAL , AND D ISTANCE -O PTIMAL Index Terminal DOO R (m)Minimal relative distance L(m)θ(rad)t (s) DOO-Optimal 946.94 9160 -3.066 20 Pareto-Optimal 1 941.37 9182 -3.052 15 Pareto-Optimal 2 806.33 9395 -3.086 9 Pareto-Optimal 3 239.24 10241 -3.226 0 Distance-Optimal 185.94 11538 -3.781 0 Without maneuver36.632310From Table I, the terminal DOO with DOO-optimal evasive maneuver is quite higher than that with distance-optimal, meaning that the terminal accuracy of the relative states with optimal evasive maneuver estimation is much lowerthan that without evasive maneuver and that with maneuver of largest relative distance. As shown in Table I, a comparison of Pareto-solution 3 with the distance-optimal solution shows that a 12.67% improvement in distance is obtained with an decrease in the DOO by 53m (22.28%). On the other hand, a comparison of the DOO-optimal solution with Pareto-solution 1 shows that an decrease in DOO by 5.57 m (0.59%) results in an improvement of minimal relative distance L by only 0.24%. Seeing from Table I and Fig. 3, another conclusion is that DOO is more correlated with maneuver direction and maneuver time than the magnitude of maneuver.C. Solution Validation using Trajectory SimulationIn what follows the effectiveness of simulations is validated through relative motion with 3- error ellipses. Figure 5 shows the error ellipses of the in-plane relative position estimation of the angles-only relative navigation with different evasion maneuver and without evasion maneuver. The control parameters in Fig.5 are obtained from Table I. It is seen from Fig. 5 that the size of the error ellipses of both relative motions decreases during approaching, since both the changing azimuth with time and the increasing number of measurements will improve the estimated accuracy. However, when the evader escapes with optimal evasion maneuver, the error ellipses change much more slowly. Moreover, the eccentricity of all the error ellipses of two relative motions is very large, and the major axis of all these error ellipses points to the evader, which indicates that the uncertainty of the relative position is large along the LOS direction.At the simulation time of 2000 s, the major axis length of the error ellipses without maneuver and with distance-optimal are all smaller than that with DOO-optimal evasion maneuver. With the optimal evasive maneuver, it is clear that the evader effectively increases the uncertainty of the relative position estimation along the LOS direction for angles-only relative navigation.x 1044x(m)y (m )Fig. 5. Error ellipses of in-plane relative position estimation of the threetypes of relative motions.VI. C ONCLUSIONSThis paper studied the orbital evasion problem using multi-objective evolutionary algorithms. The quantized model of observability is introduced as a new index, and DOO is introduced as one new objective function. This approach takes the navigation estimation —degree of observability (DOO) andtraditional safety index —minimal relative distance into account. We employed the NSGA-II to design evasive maneuvers. The results show that the proposed approach can effectively and efficiently demonstrate the relations among the mission characteristic parameters, in which it is found that DOO is more correlated with maneuver direction and maneuver time than the magnitude of maneuver. 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