The Evasive Art of Subjective Time Measurement Some Methodological Dilemmas
This newspaper researches an optimal problem of orbital evasion with because space geometry by using an analytical approach. Firstly, an angles-but relative navigation model is built and the definition of completely nonobservable maneuver is proposed. After algebraic analysis of relative space geometry, it is proved that the completely nonobservable maneuver is nonexistent. Based on this, the angle measurements of orbit without evasion are gear up as reference measurements and an analytical solution is derived to find the minimum difference betwixt measurements and the reference measurements in a abiding measuring fourth dimension. Then, an object function using vector multiplication is designed and an optimization model is established and so as to prove the optimality of belittling solution. At concluding, several numerical simulations are performed with different maneuver directions, which verify the effectiveness of the analytical method of this paper for orbital evasion problem. This method offers a new viewpoint for orbital evasion problem.
1. Introduction
Nowadays the satellites face various threats, not only orbit debris but also some noncooperative rendezvous. To increase the survivability of satellites, it is important to take some evasion strategies and perform optimally evasive maneuvers. In this paper, we focus on optimal evasion strategies for an evading satellite against a noncooperative rendezvous spacecraft. It should be mentioned that both of the relevant spacecrafts are agile-spacecraft, and the debris is out of consideration in this newspaper.
The optimal evasion problem has been studied for many years [1–6]. In this problem, a pursuer tries to arroyo its target (namely, an evader) through several evasive maneuvers, at the same time the evader expects to escape from the pursuer through some optimal evasive maneuvers. Varieties of evasion strategies had been proposed. Shinar and Steinberg [7] proposed a airtight form expression for a switching equation with a new navigation gain considered. Forte et al. [8] analyzed an equivalent linearization of the three-dimensional optimal abstention problem.
The optimal evasive strategies have been applied on many aerospace problems. Kelly and Picciotto [nine] proposed an optimal rendezvous evasive method by using a nonlinear optimization engineering. Patera [10–12] firstly introduced an optimal evasive strategy in consideration of collision probability. Bombardelli [13] obtained an optimal maneuver method to numerically maximize the miss distance, which described the arc length separation between the maneuvering rear point and the predicted collision point. Recently, the study of evasive maneuvers has been done with different emphasis, Lee et al. [14, 15] used genetic algorithm to find a solution of minimum fuel consumption and to decide delta-V maneuvers in LEO and GEO. de Jesus and de Sousa [16] investigated the being of symmetry in determining the initial conditions of collisions among objects. The evasion trouble studies mentioned above did not consider the navigation performance influence. In addition, the budgeted objects were always considered to be debris or failure vehicles in well-nigh of the orbital evasion research, which means that the evasion strategy may be useless when the object is changed to noncooperative spacecraft.
In fact, the navigation operation is actually a significant view to analyze evasion strategies. In this paper, we introduced infinite geometry of the two spacecraft in an orbital evasion problem to characterize the measurements as a new index. As known, the system observability can be contradistinct by the maneuvers of evader and pursuer during angles-only navigation [17, xviii]. Vallado [xix] found that the diversity of relative movement had a positive nonlinear correlation with the system observability. Woffinden and Geller [20] derived an analogical correlation betwixt system observability and maneuvers in the orbit rendezvous field. Grzymisch and Fichter [21, 22] establish an optimal maneuver method for rendezvous through analyzing the observability weather condition. Dateng et al. [23] used a multiobjective optimization approach to investigate orbital evasion problem with the consideration of organisation observability. The fundamental reason why the arrangement observability can be altered by the maneuvers of evader and pursuer is that these maneuvers change the relative space geometry and the measurements that the pursuer and the evader tin larn. This is the viewpoint which this paper tried to focus on.
In this work, a novel belittling evasion strategy is proposed to notice optimal evasive maneuvers by considering the variation of relative infinite geometry and measurements. The remainder of this newspaper is organized as follows. First, a problem overview most the orbital evasion is described. And then, an angles-merely relative navigation model is established and the definition of completely nonobservable maneuver is proposed. Afterwards algebraic analysis of relative space geometry, information technology is proved that the completely nonobservable maneuver is nonexistent. Based on this, an analytical solution is derived to find the optimal evasive maneuver. Then, an optimization model is established and numerical solution using genetic algorithm (GA) is introduced so equally to bear witness the optimality of belittling solution. At concluding, a numerical simulation is performed to verify the validity of the method. The results signal that the analytical method proposed in this paper can reach the expected effect.
ii. Problem Overview
In that location are commonly two spacecrafts in an orbital evasion problem (a pursuer and an evader). The objective of pursuer is to capture or approach the evader, and the objective of evader is to find some evasion strategies and escape by the pursuer. This paper researches the optimal evasion strategy when the initial relative distance is about 100 km and the initial orbits are coplanar and HEO (Highly Elliptical Orbit).
At such altitude, the pursuer commonly merely has two angle measurements, because the evader is noncooperative and the LiDAR (Low-cal Detection and Ranging) cannot get distance measurements in such a range. Assuming that evader and pursuer both know the initial state of each other, the pursuer will larn the futurity state of evader through bending measurements with the assist of some filters. The bending measurements of orbit without evasion are gear up as reference measurements. Generally, the angle measurements will have a not bad alter when the evader starts an evasive maneuver, meaning that pursuer tin can take hold of the maneuver through filtering immediately. When pursuer gains the evasive maneuver, it can replan its approaching to brand the evasive maneuver invalid.
However, if the evasive maneuver causes minor or fifty-fifty no departure in bending measurements compared with the reference measurements, the accuracy of filtering will exist declined or even not convergent. Thus, the accuracy of navigation has a human relationship with the relative space geometry which is influenced by the evasive maneuver. In this way, the evasive maneuver tin change the accuracy of navigation and the system observability.
The optimal evasive maneuver is expected to minimize the difference of angle measurements pursuer caused between evasion and no evasion, and so the difficulty of filter tracking is increased. When the magnitude of evasive maneuver is given, the evasion trouble changes to an optimal optimization problem of two control variables. In the following, nosotros will introduce both belittling and numerical solutions to solve this problem.
3. The Relationship of Infinite Geometry
In this section, the relationship of relative space geometry and evasive maneuver is analyzed. Many different indexes have been used to describe the observability, such equally the status number of the observability matrix [24] and the distance mistake [25]. A new index that represents the level of land estimation is introduced here which is unlike from the previous indexes, and we call information technology measurement observability.
three.i. State Transfer Matrix for HEO
The relative Local Vertical Local Horizontal (LVLH) coordinate system is used as reference frame to describe orbital relative movement. Since pursuer and evader are in elliptical orbit, the Tschauner-Hempel (TH) equation [26] is introduced to depict the relative motion. The homogeneous solution is Yamanaka-Ankersen land transfer matrix [27]: where and are true anomaly of the reference spacecraft at and , respectively. The expressions of and are every bit follows: where , , , , . is eccentricity of the reference orbit, is gravitational coefficient of the globe, and are the first derivative of and with respect to .
In the following, we gear up and . Assume a spacecraft executes a maneuver at ; so the relative land at with respect to the reference orbit is
three.2. Measurement Equations
In the actual projects, optical camera is one of the most mutual measuring devices and can provide two independent angle measurements (contains superlative and azimuth ) at every moment. Figure 1 shows the geometric schematic of the orbital pursuit-evasion system along with the elevation and azimuth in the frame of photographic camera, where , , and are the evader, pursuer, and photographic camera coordinate systems, respectively. and are regarded as the same coordinate system in the following analysis. The definition of the orbital coordinate system is as follows: is along the position vector which is from the center of the earth to the pursuer, x is perpendicular to in the orbital plane and at the same side of the velocity direction, and obeys the right-handed coordinate arrangement. The definition of the orbital coordinate attached to the evader is omitted here, since the definition is similar.
The relative position between the evader and pursuer in measuring coordinate system is denoted as ; the relationship of and measuring parameters obtained from the optical photographic camera is where and are the elevation and azimuth angles, respectively.
Through linear inverse transformation, (four) is transferred to where is position component of the relative country and is as follows:
According to the EKF, the relative position and velocity tin be estimated using the equations above during the angles-merely relative navigation. It is easy to find that the maneuver of pursuer or evader will affect the relative navigation, measurement, and the observability of the arrangement.
3.3. Space Geometry Analysis
Definition 1. Assume evader executes a nonzero evasive maneuver at ; if the angle measurements pursuer acquired after the evasive maneuver stays the aforementioned compared with those of no maneuver, so the evasive maneuver can be chosen completely unobservable maneuver.
Figure two shows the projection of relative motility trajectory in airplane and describes the infinite geometry of completely unobservable maneuver. Assuming the evader's orbit without maneuvers as a reference orbit, then trajectories and describe the relative motion of the pursuer and of the evader when evasion maneuvers are exploited. In Figure 2, represents the relative position vector of the pursuer, and the relative position vector of the evader with an evasive maneuver, with respect to the reference orbit. Definition 1 can be illustrated as follows: the relative motion trajectory satisfies the geometry relationship described in Figure 2, namely, , at any time after evasion ( is nonnegative). Thus, the angle measurements pursuer caused afterwards the evasive maneuver stays the same compared with those without evasion. In this situation, pursuer will never observe out whether evader executes an evasive maneuver. When in Figure 2 exits from the scenario of the pursuer, the correspondent evasive maneuver is chosen completely unobservable maneuver.
However, the completely unobservable maneuver is only a hypothesis about space geometry. Subsequently derivation, information technology can be constitute that the completely unobservable maneuver does non exist. We have the following.
Theorem ii. If the initial relative distance of pursuer and evader is nonzero, the evasive maneuver evader executed cannot become completely unobservable maneuver.
Proof. Presume a completely unobservable maneuver exits. Prepare as initial fourth dimension and take the initial two steps and equally example. The relative position of evader after evasion and the reference orbit are as follows: If the relative country of pursuer and the reference orbit is and the position component is nonzero, then where , and are position vectors.
According to (five) and Definition 1, nosotros have namely, According to the property of solution space, it is easy to find out the following relation: where is nonzero existent number.
Therefore, if (eleven) and (12) are proved to be held, the being of completely unobservable maneuver tin be proved.
From (11), we have
Substitute (13) into (12) and have the second step of (12) into consideration; and so where .
Since is linear transformation and (14) exercise agree, the necessary and sufficient condition of the nonzero solution beingness of is (the initial relative position is nonzero). Thus,
In addition, if the nonzero solution of exists, co-ordinate to the beingness theorem of solutions of linear equation, we tin acquire . Expand and 1 can get
Considering , it is easy to know that if and only if the expression can exist held. Equation (fifteen) is revised equally Substitute into (xvi), so
Set . It can be seen from (17) and (xviii) that the value of makes no influence on the consequence of (17). When is arbitrary, the equalities do non ever hold. Furthermore, (17) always holds only when . This condition goes with the antecedent hypothesis of Theorem two; therefore Theorem 2 is proved in this style. Unobservable maneuvers proofs accept previously been developed for the bearings-just navigation problem in circular orbit [21]; hither the related conclusion is extended to elliptical orbit successfully.
Though the completely unobservable maneuver is nonexistent, it is fix nether the situation of ideal measurement. In reality, if the bending measurements pursuer acquired after the evasive maneuver stays quite close to those of no maneuver and the difference approach to the measurement accuracy, pursuer could be unable to place the evasive maneuver. Thus, these evasive maneuvers can be chosen approximate solution of the completely unobservable maneuver. It can be said that the difficulty of maneuver tracking for pursuer is increased when the departure of angle measurements the evasion acquired is decreased. This is a new index to evaluate the superiority of an evasive maneuver.
four. Optimal Evasion Maneuvers Analysis
In this section, an analytical solution is derived based on the conclusion of Section 3 and meanwhile a numerical solution is given to prove the optimality of analytical solution.
four.ane. Belittling Solution
iv.1.1. Quantification of Space Geometry
To find the optimal evasive maneuver, it is necessary to quantify the relationship of bending measurements variation and space geometry of pursuer and evader. In Euclidean infinite, orthogonality between ii vectors can exist defined past using the notion dot product. Thus, two cavalcade unit of measurement vectors and are orthogonal when . On the contrary, if the scalar product of and is i or −one (), so and are parallel.
From Effigy two we know that is the measuring line of sight of pursuer and the reference orbit, is the measuring line of sight of pursuer and evader after evasion. To each measuring time, the norm of and is constant. Therefore, in lodge to decrease the variation of angle measurements caused by evasion, the included bending of and must be every bit modest as possible. Equally an extreme instance, when the included angle of and is 0 or , the evasive maneuver becomes the completely unobservable maneuver.
Since the norm of and is definite value, if the evasive maneuver is optimal, the scalar product of and should exist maximum. It is seen from Figures ii and 3 that the measuring line of sight tin can be expressed as
At fourth dimension, the scalar production of and is
Information technology is easy to know that the value of is constant at each time; thus the scalar product of and is decided by . Therefore, to minimize the variation of angle measurements at time, the optimization at fourth dimension should minimize the post-obit object:
According to the relationship of evasive maneuver and infinite geometry, (21) tin be revised equally
During the approaching, the optimal evasion should minimize the object at every moment; thus the object function of the whole approaching is as follows: where is the number of measurements.
Equation (23) shows that the evasive effectiveness is not only dependent on the maneuvers performed by but also dependent on the position where they are executed. Since this equation is airtight form, it allows for simple inclusion inside a global trajectory optimization scheme as an additional objective or independent objective, weighed past other objective such as relative distance or fuel consumption. This would let a global trajectory optimizer to choose maneuvers that will increase the difficulty of maneuver tracking for pursuer. It is worth noting that (23) provides explicit solutions for any arbitrary set of initial weather (nonzero initial relative position).
4.2. Algebraic Optimal Evasive Maneuver
To observe the optimal evasive maneuver for a abiding duration of measurement, the optimization variable of interest in (23) is the evasive maneuver . The initial state is fixed when the evasive maneuver is executed. is the number of measurements.
Since (23) is a linear function for evasive maneuver, it can merely exist minimized with respect to . After the former quantification analysis higher up, the optimal evasive maneuver can be constitute based on (23). It is logical to limit the desired evasive maneuver magnitude in order to observe the optimal evasive maneuver direction. So, this constraint can be mathematically posed equally follows:
It should be noted that additional constraints could be considered here, if information technology is needed. Closed form solutions would be possible by the following steps with constraints mentioned above.
The constrained optimization problem transfers to minimize (23) with respect to the evasive maneuver , under the equality constraint proposed by (24). Since at that place is no inequality constraint in this problem, information technology can be converted to an equivalent unconstrained problem with the Lagrange multiplier technique used in [28]. The trouble is converted to minimize the Lagrangian function where is the Lagrange multiplier respective to the equality constrain in (24).
The first-order optimality atmospheric condition are given past the derivatives of the Lagrangian function with respect to the optimization variables equal to zero, as well as the Lagrange multiplier. Thus, nosotros have
Co-ordinate to (26), we have
Take the 2d-order optimality conditions into consideration in order to place the stationary point corresponding to the minimum of the Lagrangian function.
Co-ordinate to (27) and (28), an algebraic expression for optimal evasive maneuver can exist obtained as follows:
Equation (29) is the analytical expression of optimal evasive maneuver including the initial states and the expectable evasive maneuver magnitude . Till at present, the expression of the analytical solution is obtained, and it provides the optimal evasive maneuver with respect to relative space geometry and the angle measurements. The state transition and input transition were proposed in Section 3.
4.3. Numerical Solution
In order to prove the optimality of belittling solution, an optimization model is established in this section. One of the most well-known evolutionary algorithms, GA, is employed to solve the optimization trouble. The GA has been successfully applied in spacecraft trajectory optimization, for example, in designing low-thrust trajectories [29] and solving two different kinds of bug typical to astrodynamics [xxx].
Optimization Variables. Since the desired evasive maneuver magnitude is express to , in guild to notice the optimal direction of evasive maneuver, azimuth and elevation are selected as ii optimization variables; namely,
In consideration of the space geometric relationship between evader and pursuer, the constraint conditions for an orbital evasion problem read
In order to farther simplify the trouble, hither we focus on the value range of the optimization variables. For 2 satellites in orbit, it is known from [19] that an increased difference of the arrangement relative motion has a positive correlation with organisation observability. Therefore, if the initial orbit is coplanar, the evasive maneuver should better be coplanar, likewise.
Under such circumstance, new constraint conditions are equally stated in the post-obit: Objective Office. Presume that and are the unit vectors of and , separately. According to Section 4.1, it is easy to know that the closer to ane the value of is, the smaller the variation of angle measurements is. During the measurement, when the sum of is max, the evasive maneuver is optimal. Thus, objective office is equally follows: where and are equal to the unit vector of and .
In social club to use GA, the objective function is revised as
Thus, the GA optimization model is established. Through the results comparing of analytical and numerical solutions, the optimality of analytical solution can be proved.
v. Simulation
Simulation results are presented in this section. Consider an illustrative case: evader is in a HEO and semimajor centrality is 45485189 1000, eccentricity is 0.713, inclination of orbit is 1.10187 rad, right ascension of ascending node (RAAN) is 0.84489 rad, statement of perigee is 4.7022 rad, and true anomaly is 1.4856 rad. Presume that the initial fourth dimension is , and initial relative states at is
The pursuer uses optical camera to get relative measurements. The measurement frequency of the optical camera is causeless equally 0.1 Hz. The magnitude of evasive impulse is fixed to 3 chiliad/s (namely, grand/south). The optimization parameters of GA are as follows: 100 for population size; xxx for maximum generations number; 0.90 for crossover probability; and 0.08 for mutation probability. Constraint weather are chosen as those in Section 4.2.
If the evasion is aimed at 101 angle measurements, so the measuring time is 1000 south. Information technology is easy to know that the minimum value of objective part should be −101. Thus, the optimization results are obtained in Figure 4.
It can be seen from Figure 4 that the numerical solution is −100.9998. After 15 generations, the results accomplish convergence. The management of optimal evasive maneuver is . Take correlated parameters into (29), the analytical solution is acquired and the direction of analytical evasive maneuver is .
The measurement effectiveness of the optimal evasive maneuvers is validated by comparing the GA-optimal and analytical-optimal evasive maneuvers to the propagation of 80 dissimilar maneuver directions, covering from to . The angle measurements during the previous 1000 s are shown in Figure five.
Prepare the angle measurements of reference orbit equally reference measurements. The angle measurements with an evasive maneuver at each fourth dimension minus reference measurements get the variation of angle measurements. The variation of bending measurements during the previous thousand south is given in Figure 6.
(a)
(b)
Every bit seen from Effigy 6, the variation of bending measurements caused by the GA-optimal and analytical-optimal solutions are much smaller than other maneuver directions. Although the analytical-optimal variation of bending measurements at the end fourth dimension is bigger than that of GA-optimal, the sum of the variation is optimal for the whole measuring time. It tin be seen that the variation of angle measurements caused by the analytical-optimal solution is no more than 0.001 rad for the previous 800 s, which means that the pursuer will fifty-fifty not find out the evasion if the measurement accuracy is non improve than 0.001 rad.
In social club to show that the proposed belittling and numerical maneuvers provide the nigh guidance fault in the filter gauge, a Kalman filter is used as the navigation filter. The simulation conditions are the same with the simulation in Figures 4–6, and the initial guidance error is assumed to exist nonzero. The simulation step is ready to v s. The guidance mistake of the optimal evasive maneuvers is validated by comparing the GA-optimal and analytical-optimal evasive maneuvers to the propagation of 60 different maneuver directions, covering from to . The guidance errors in the two directions of orbit plane are shown in Figures 7 and 8.
Equally Figures seven and 8 evidence, the analytical and numerical maneuvers provide the well-nigh improvement of guidance error. Though the guidance fault acquired by the analytical maneuvers is better than that of numerical maneuvers, the deviation is minor. The simulation results testify that the analytical analysis in Section 4 is effective. Moreover, since the solution is analytical, it will have a potential application in applied science utilization.
6. Conclusions
An analytical optimal evasion strategy is proposed for an evading satellite against a noncooperative rendezvous spacecraft. This work extends the inquiry objective to not just orbit droppings but also spacecraft with maneuver ability. Through analysis of relative space geometry, the completely unobservable maneuver is defined and proved to be nonexistent. An analytical airtight-form solution is proposed to compute optimal evasive maneuvers for angles-but navigation. Based on this analytical method, an optimal evasive maneuver can be quickly obtained. Since the previous evasion strategies required numerical optimization which a lot of time is needed, the method in this work should exist an effective improvement compared with the previous evasion country of art. Moreover, in order to decline the navigation accurateness of pursuer, the relative infinite geometry is analyzed and quantified. Though the derivation is under the situation of HEO, information technology is also adaptable for orbital evasion problem in circular orbit.
This work proves that the angle measurements and navigation accuracy can be used in orbital evasion problem, which is often neglected in previous inquiry. This enquiry proposes a new inquiry method that means a potential footstep closer to technology utilization.
Conflicts of Interest
The authors declare that they take no conflict of interests.
Acknowledgments
Project is supported past the National Natural Science Foundation of Red china (Grant no. 11572345).
Copyright
Copyright © 2017 Dateng Yu et al. This is an open access commodity distributed nether the Creative Commons Attribution License, which permits unrestricted utilise, distribution, and reproduction in whatever medium, provided the original work is properly cited.
Source: https://www.hindawi.com/journals/ijae/2017/4164260/