PRECISION ZEM/ZEV FEEDBACK GUIDANCE ALGORITHM UTILIZING VINTI S ANALYTIC SOLUTION OF PERTURBED KEPLER PROBLEM

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1 AAS PRECISION ZEM/ZEV FEEDBACK GUIDANCE ALGORITHM UTILIZING VINTI S ANALYTIC SOLUTION OF PERTURBED KEPLER PROBLEM Jaemyung Ahn, * Yanning Guo, and Bong Wie A new implementation o a zero-eort-miss/zero-eort-velocity (ZEM/ZEV) eedback guidance scheme, which utilizes the computational eiciency and accuracy o analytical orbit propagators, is proposed. Kepler s propagator and Vinti s algorithm or the solution o perturbed Kepler problem are used to predict the ZEM and ZEV state vectors and determine the acceleration command, which are oten obtained by numerical integration o the equations o motion in the conventional approach. A procedure or optimal time-to-go estimation that can be used with the new guidance algorithm or intercept o moving target is introduced. The eectiveness o the proposed implementation is demonstrated by the perormance comparison with the open-loop optimization approach through several case studies. INTRODUCTION Guidance o missiles and space vehicles is one o most critical technologies required or their successul mission operations. Various study results on this subject as applied to orbital intercept, rendezvous, and planetary landing problems can be ound in the literature. The mathematical methodologies employed by such studies are diverse as well including the optimal control approaches [1-6], the classical proportional navigation (PN) guidance and its variants [7], and other approaches. Among these methodologies, the zero-eort-miss/zero-eort-velocity (ZEM/ZEV) eedback guidance and its variants is a practical guidance algorithm that has been extensively examined in the past [8-12]. The generalized ZEM/ZEV guidance scheme [8,9] is physically easy to understand and simple to implement due to its conceptual simplicity. It is a eedback guidance law that can deal with perturbations and/or uncertainties in a robust way in particular compared with open-loop, optimization-based guidance algorithms. The perormance o the ZEM/ZEV algorithm is satisactory or various practical applications as well. With an assumption o a uniorm gravity ield, the algorithm provides a true optimal so- * Associate Proessor, Dept. o Aerospace Engineering, KAIST, South Korea. jaemyung.ahn@kaist.ac.kr Assistant Proessor, Dept. o Control Science and Engineering, Harbin Institute o Technology, China. guoyn@hit.edu.cn Vance Coman Endowed Chair Proessor, Asteroid Delection Research Center, Department o Aerospace Engineering, Iowa State University, 2271 Howe Hall, Ames, IA , USA. bongwie@iastate.edu 1

2 lution. For some ballistic missile intercept and asteroid intercept scenarios with spherical gravity, it was demonstrated that the ZEM/ZEV approach can even compete with corresponding openloop optimal solutions, while its eedback characteristics make it more suitable or realistic situations with uncertainties [8-12]. For long-range missions that require the consideration o position-dependent gravity model, however, there are some challenges in using the proposed ZEM/ZEV algorithm. The irst challenge is the ast and accurate prediction o the ZEM/ZEV inormation. Under the assumption o spherical gravitational ield (with or without additional perturbations), when the states (position and velocity) o an object at an initial time is speciied, it is not possible to express its states at a given time as a closed orm in general. For example, or a missile intercept problem presented by Guo et al. [8], the ZEM was computed using a numerical integration o both the interceptor and target missiles. An actual on-board implementation o this numerical integration-based algorithm can be diicult due to the high computational burden. The second challenge is the determination o terminal time t or time-to-go t go or ree terminal time problems. Guo et al. [9] proposed a ormulation to determine the time-to-go or a Mars landing problem using ZEM/ZEV guidance with an assumption that the direction o gravity vector is constant. Newman [11] developed a series o polynomial equations to determine the terminal time associated with various assumptions on gravity (zero, constant, linear, quadratic, and linearized inverse square) in the context o missile intercept using ZEM (without considering ZEV) guidance. But the authors could not ind any previous studies on terminal time (or time-togo) determination problems or ZEM/ZEV guidance with spherical (or more complex) gravity model. This paper addresses the aorementioned two challenges by 1) introducing analytic propagators or ZEM/ZEV prediction and 2) developing a ormulation to determine the terminal time associated with minimum control-energy guidance problem. The position-dependent gravity model and two analytic propagators the Kepler propagator (or spherical gravity model) and the Vinti propagator (with additional perturbed gravity model including J 2, J 3, and J 4 terms) [13] are used. A general ormulation to determine the terminal time subject to inal position/velocity constraint or a moving target is proposed. Various cases (minimum energy missile intercept, head-on missile intercept, and elliptical orbital rendezvous) will be examined in this paper. The solutions obtained by using the ZEM/ZEV guidance algorithm with terminal time determined using the proposed ormula will be compared with the open-loop optimization results to demonstrate its practical eectiveness. OPTIMAL MISSILE/SPACECRAFT GUIDANCE PROBLEM Consider the dynamics o a missile or spacecrat, moving in Earth-Centered-Inertial (ECI) coordinate rame, described by (1) where r and v are the position and velocity o the missile, a is the acceleration o the missile applied by its propulsion system, and g is the gravitational acceleration. Vector g is expressed as the unction o the position vector r = [x, y, z] T as ollows: (2) 2

3 In Eq. (3), g c and g p respectively denote the spherical (Keplerian) and perturbed gravitational acceleration terms [14-15] described by (3) (4) (5) In Eqs. (4) and (5), is the Earth s gravitational parameter, ( J 2, J, J ) are the zonal harmonics terms, and R is the equatorial radius o the Earth. Also r is the magnitude o position vector r. 3 4 A missile intercept problem can be deined as a dynamic optimization problem or determining the acceleration history o the missile, a, subject to the dierential equations describing the interceptor missile s dynamics with a speciied objective unction and initial/terminal conditions. The objective unction J is expressed as (6) where is the terminal cost unction (set up as zero in this problem). Initial and terminal conditions o the missile are assumed as r(0) = r M,0 (7) v(0) = v M,0 (8) Note that, or the intercept or rendezvous o a moving target, the inal position o the target can be calculated rom the target dynamics. It is assumed that the target is moving subject to the gravity only (i.e. ree-all) whose equations o motion are described as (11) where r T and v T are the position and velocity o the target, respectively. For an intercept problem, equals r () t and the terminal missile velocity is unspeciied. For a rendezvous problem, on T the other hand, we have to speciy both o terminal position and velocities as: and. (9) (10) (12) (13) 3

4 Now the optimal control problem is completely deined by Eq. (6) (objective unction), Eqs. (1)-(2) (dynamic constraints), and Eqs. (7)-(10) (initial and terminal constraints) [16]. To simpliy the problem ormulation, we may assume that the gravity vector g is a unction o time only ( gr () = g(), t gr ( ) = g () t ). The Hamiltonian or this problem is then deined as T T where p and p are co-state vectors associated with Eqs. (1) and (2), respectively. r v The dierential equations or the co-states are (14) (15) From Eq. (15), we can conclude that p r is a constant vector. Using this properties, p v is determined as (17) (16) where p = p ( t ) is value o the co-state vector p at terminal time., v v v The optimality condition or this problem is given as H 0= = a+ p a By combining Eqs. (17) and (18), we obtain the optimal acceleration vector as v (18) (19) where is the time-to-go or the intercept. Equation (19) indicates that the optimal control law or this intercept/rendezvous problem is determined by inding two vectors: p and r. It should be noted that the solution is true optimal under the assumption:. OPTIMAL FEEDBACK GUIDANCE USING ZEM/ZEV This section presents a procedure to implement Eq. (19) as a eedback orm the unction o current position and velocity. Assume that that the position and velocity o the missile at current time t are respectively r ( t ) and v ( t ). From Eq. (2), the velocity o the missile at time t is expressed as (20) where is the ree-all velocity o the missile at t. By applying Eq. (19) to Eq. (20), we obtain 4

5 The terminal velocity o the missile is expressed as (21) The terminal position o the missile r ( t ) can be obtained by substituting Eq. (21) into Eq. (1) as By evaluating the integral in Eq. (23), we obtain the ollowing result: (22) (23) (24) where is the ree-all position o the missile. Note that the expressions FF FF or v and r not exact because the gravitational acceleration is the unction o its position that is aected by the control acceleration a. In this paper, we assumed that g is the unction o time and used these expressions. Let us deine the zero-eort-miss (ZEM) and the zero-eort-velocity (ZEV) as (25) Equations (22) and (24) are combined with the terminal conditions, Eqs. (9) and (10), to determine the two vectors p r and p. Two types o problem with dierent terminal conditions are v, considered. In this section, we assume that the inal time t is given. (26) Type 1: Terminal position and velocity on a curve ( ) From Eqs. (22) and (24), we have the two set o linear equations or p and p, as ollows: r, v (27) By solving these equations, we can determine p and p as r, v (28) (29) When Eqs. (29) and (30) are applied to Eq. (19), we have the optimal control law or Type 1 o the orm: (30) 5

6 (31) Type 2: Terminal position speciied on a curve, terminal velocity ree ( ) I the terminal velocity is ree, the terminal co-state associated with velocity can be determined as (32) From Eq. (24), p is determined as r Finally, we can obtain the optimal control law or Type 2 as (33) (34) Calculation o ZEM and ZEV It should be noted that the optimal control laws or both cases use the ZEM/ZEV inormation, FF FF which require the ree-all position and velocity vectors ( r ( t ), v ( t )). Dierent methodologies can be used to obtain these ZEM/ZEV values. As the simplest (and computationally least FF FF expensive) approach, we can derive the analytic expressions or r ( t ) and v ( t ) under the assumption o uniorm gravity: gr ()= g = a constant vector. In this case, the ZEM and ZEV are expressed as ollows: (35) (36) While this approach with constant gravity assumption works ine or relatively short-range missions, its eectiveness is reduced or missions with longer ranges in which the changes in the direction o gravity is not negligible. A straightorward modiication o Eqs. (35)-(36) would be the method o direct integration used in [8]. In this approach, the ree-all position and velocity are obtained by numerically integrating Eqs. (1)-(2) until the inal time t with a= 0. Similarly, the target s position and velocity are obtained by numerical integration o Eqs. (11)-(12). The advantage o this approach is its lexibility the eects o non-spherical gravity terms (and even atmospheric drag) can be included. However, the high computational resource consumption required or the numerical integration can make it prohibitively diicult to be implemented onboard. 6

7 Analytic propagators can be considered as the third alternative, which is used in this paper. We consider two dierent propagators: Kepler s propagator and Vinti s propagator. The Kepler s propagator computes the ZEM and ZEV by solving Kepler s initial value problem (with spherical gravity assumption) or both o the missile and target. The solution o the Kepler s problem is expressed as (37) where, g,, and can be obtained by various methods such as iteration and series solution [14-15]. The Vinti s propagator applies the Vinti Spheroidal Method to reine the solution o Kepler s initial value problem so that the eects o zonal harmonics ( J 2, J, and J 3 4 terms) are included [13]. DETERMINATION OF TERMINAL TIME The ZEM/ZEV guidance laws expressed in Eqs. (31) and (34) are derived with an assumption that the terminal time t is speciied. This section introduces a ormula to determine t or terminal time ree problems. Note that the problems discussed in the previous section have terminal boundary conditions that a subset o terminal state x s are speciied on a curve. Following [16], the terminal boundary conditions or the optimal control problem with this situation are described as (38) (39) where is the terminal cost unction, p is the co-state vector associated with x, and s s is the co-state vector associated with unspeciied states at t. These equations are customized or Type 1 and Type 2 to derive the relationships that should be satisied at terminal time, which is used to determine t. (40) p c s Type 1: Terminal position and velocity on a curve ( ). The terminal cost unc- In this case, vector x s is composed o the whole states as tion is zero ( ) or this problem, and Eq. (39) becomes (41) By applying Eq. (14) to Eq. (41), we obtain the ollowing relationship: 7

8 (42) From Eq. (17), the terminal acceleration and is expressed as (43) We obtain the terminal velocity v rom Eq. (22). Also note that the time-to-go t go becomes the terminal time t i it is evaluated at t = 0. Then, Eq. (42) is reormulated as where p and p are expressed using ZEM, ZEV, and t r v, by replacing t go in Eqs. (29)-(30) by t. The optimal terminal time can be obtained by solving Eq. (44) or t. The Newton-Raphson method can be used to iterate on t as (44) (45) where t k denotes the predicted value o t at k th iteration. The derivative is expressed as (46) where the derivatives are described by (47) (48) (49) (50) 8

9 (51) Note that the terminal position and velocity constraints, r =Qt ( ) and, are used to evaluate g ( t ) and in Eqs. (44) and (46). Type 2: Terminal position speciied on a curve, terminal velocity ree ( ) When the terminal velocity is ree, vector x is deined as x = r. Similar to Eq. (41), we can set s s up an equation to obtain the terminal time associated with the boundary condition as (52) In addition, because the terminal velocity is ree, rom Eq. (40), the terminal co-state associated with velocity becomes zero as This leads to the zero terminal acceleration ( a( t ) =- p = v, 0, rom Eq. (17)). Then, Eq. (52) is rearranged as (54) I we evaluate this equation at t = 0 (53) FF, p and v r are expressed using ZEM, t, and v ( t ) by replacing t go in Eqs. (22) and (33) by t. Then Eq. (54) becomes the ollowing root-inding problem or t : The analytic derivative used or the Newton-Raphson iteration to solve Eq. (55) is expressed as (55) (56) where d( ZEM )/ dt is deined in Eq. (50). The Newton-Raphson iteration to solve Eqs. (44) and (55) using the analytic derivative expressed in Eqs. (46)-(51) and (56) converges very ast. This procedure can be eectively used to determine the terminal time at the initial phase o the intercept, or even during midcourse guidance procedure to update the terminal time as the outer loop o the main ZEM guidance loop. 9

10 CASE STUDIES This section presents our case studies (Cases A, B, C, and D) or missile and spacecrat intercept/rendezvous problems using the proposed ZEM/ZEV guidance law and/or optimal t selection procedure. The irst two cases are t ree, minimum energy, missile intercept problems where the inal position constraint is imposed. In Case A, the terminal velocity constraint is not applied (ZEM guidance). A special case o direct intercept condition (i.e. the sum o velocity vectors o the interceptor and target is zero) is imposed as the terminal velocity constraint o Case B (ZEM/ZEV guidance). The next two cases are satellite intercept/rendezvous problems. Case C solved a t ree, minimum energy, satellite intercept problem (ZEM guidance), and inally a series o t ixed, minimum energy, satellite rendezvous problems (ZEM/ZEV guidance) with various t values are dealt with in Case D. For each case study, the ZEM/ZEV guidance simulation using the proposed methodology was compared with the open-loop trajectory optimization result. MATLAB was used or the implementation o the ZEM/ZEV guidance simulation, and the open-loop trajectory optimization was conducted using GPOPS-II sotware that operates based on the Gauss pseudo-spectral method [17]. Analytic orbital propagators were implemented using MATLAB translations o the FORTRAN sources provided by Dr. Gim Der o Der Astrodynamics [18]. Case A: Terminal position speciied (Minimum Energy Intercept) The subject o the irst case study is selected as a ballistic missile intercept problem presented in [8]. The dynamics o the intercept missile and the target missile are expressed as Eqs. (1)-(2) and Eqs. (11)-(12), respectively. In this case, only the terminal position constraint is imposed and the constraining curve is deined as the position o the target missile ( ). The initial conditions or the interceptor and target missiles are presented in Table 1. Table 1: Initial Conditions o Interceptor and Target Missiles (Cases A and B) Interceptor Target Initial Position (km, km, km) Initial Velocity (km/s, km/s, km/s) r = [0.0, , ] T r = [0.0, 0.0, ] T 0 M 0 T v = [0.0, 2.006, 5.954] T v = [0.0, 6.785, 2.880] T 0 M 0 T Figure 1 shows the trajectories o the interceptor and target missiles or this case obtained by the ZEM guidance simulation and the open-loop optimization. It can be seen that the two trajectories almost coincide, which suggests that the ZEM guidance produces almost the same result as the open-loop optimization. The inal time t or the ZEM guidance was determined as s (Kepler) / s (Vinti) by solving the root inding problem t ( ) = 0 expressed in Eq. (55). The unction ( t ) is presented in the top plot o Figure 2. The inal time obtained rom the optimization was s., which is almost the same as the value or ZEM guidance. The objective unction and the acceleration proiles rom the two approaches are exhibited in the bottom plots o Figure 2. The curves show almost no dierence as well, which indicates the perormance o the two approaches or this case are almost identical. The perormance summary or the two methods 10

11 is presented in Table 1. The perormance dierence caused by using dierent analytic propagators was very small. Figure 1: Trajectory or Case A - minimum energy missile intercept Figure 2: Plots o ( t ), objective unction, and acceleration or Case A 11

12 Table 2: Perormance summary Open-loop Optimization vs. ZEM (Case A) Open-Loop Optimization ZEM, predicted t (Propagator: Kepler) ZEM, predicted t (Propagator: Vinti) Terminal time ( t ), s Max. Acceleration, m/s Objective unction (J), m 2 /s 3 3,483 3,492 3,493 Case B: Terminal velocity and position speciied (Head-On Collision Intercept) This case represents a special case o the head-on intercept in which the velocity o the interceptor is exactly the opposite o target s velocity at the inal time. The curve or the terminal position constraint and the terminal velocity constraint are given as and, respectively. The initial conditions or Case A are used in this case as well (presented in Table 1). The interceptor and target missile trajectories obtained by the ZEM/ZEV guidance simulation and the open-loop optimization are plotted together in Figure 3. The two trajectories are almost the same as each other. The solution o Eq. (44) yields the inal time or ZEM/ZEV guidance (Kepler: s / Vinti: s), which is very similar to the value obtained rom optimization (580.5 s). The top plot o Figure 4 shows the curve used to determine t or ZEM/ZEV guidance. Plots comparing the objective unction and acceleration proiles or two approached are presented in the bottom plots o Figure 4 and the summary o perormance comparison is provided in Table 3. Both o them indicate that the proposed guidance works almost as eective as the openloop optimal solution. Like Case A, The perormance dierence caused by using dierent analytic propagators was very small. Table 3: Perormance summary Open-loop Optimization vs. ZEM/ZEV (Case B) Open-Loop Optimization ZEM/ZEV, predicted t (Propagator: Kepler) ZEM/ZEV, predicted t (Propagator: Vinti) Terminal time ( t ), s Max. Acceleration, m/s Objective unction (J), m 2 /s 3 25,337 25,356 25,369 12

13 Figure 3: Trajectory or Case B head-on collision intercept J (m 2 /s 3 ) a (m/s 2 ) (t ), km 2 /s 4 Figure 4: Plots o ( t ), objective unction, and acceleration or Case B 13

14 Case C: Terminal position speciied (Chasing type intercept, ree terminal time) Case C is similar to Case A in that only the inal position is speciied, but the dierence is that the directions o initial velocity vectors o target and interceptor spacecrat are the same. The target satellite is in an equatorial orbit with perigee/apogee altitudes o 700 km / 1,000 km. The interceptor satellite orbit is a circular orbit with an inclination angle o 15 degrees. The initial conditions and orbital parameters are provided in Table 4. Table 4: Initial Conditions o Interceptor and Target Satellites (Cases C and D) Interceptor Satellite Target Satellite Initial Position (km, km, km) Initial Velocity (km/s, km/s, km/s) r = [ , 0.0, ] T r = [ , 0.0, 0.0] T 0 M 0 T v = [0.0, , 0.0] T v = [0.0, , 0.0] T 0 M 0 T Orbital Parameters Perigee Altitude (km) Apogee Altitude (km) 1, Inclination (deg.) 0 15 The trajectories obtained by the two methods are presented in Figure 5. The solution o the root-inding problem expressed in Eq. (55) is solved to obtain the inal time t = 1,486 seconds. The optimal inal time rom numerical optimization was 1,469 s (Kepler) / 1,465 s (Vinti) approximately 1% smaller than the value ound by the proposed method. The top plot o Figure 6 shows the unction ( t ) expressed in Eq. (55). The bottom plots o Figure 6 compare the objective unction and acceleration proiles or the two approaches. While the proiles are very similar, unlike Cases A and B, we can observe the dierences in two curves or both proiles. The dierence (although small) can be attributed to the longer light time o Case C (~1,500 sec.) compared with the previous two cases ( sec.). The perormance comparison or two approaches are summarized in Table 5. The perormance dierence caused by using dierent analytic propagators was negligible in Case C, too. Table 5: Perormance summary Open-loop Optimization vs. ZEM/ZEV (Case C) Open-loop Optimization ZEM/ZEV, predicted t (Propagator: Kepler) ZEM/ZEV, predicted t (Propagator: Vinti) Terminal time ( t ), s 1,486 1,469 1,465 Max. Acceleration, m/s Objective unction (J), m 2 /s

15 Figure 5: Trajectory or Case C minimum energy intercept J (m 2 /s 3 ) a (m/s 2 ) (t ), km 2 /s Figure 6: Plots o ( t ), objective unction, and acceleration or Case C 15

16 Case D: Terminal velocity and position speciied (rendezvous, ixed terminal time) The last case is concerned with a rendezvous problem between two spacecrat. Initial conditions or the interceptor and target spacecrat are the same as those or Case C. The curves or the inal position and velocity are the position and velocity o the target spacecrat, respectively (, ). Figure 7 presents the plots o ( t ) and. It can be observed that there is no t value that satisies the relationship t ( ) = 0. Two dierent situations can cause this. First, the optimal trajectory or the rendezvous with minimum control energy would require very long operation time and range. This can result in a very large change in the direction o gravity vector, and the calculation o the optimal t using Eq. (44) can ail. Secondly, it is also possible that this rendezvous problem actually does not have a inite optimal t value (i.e. the larger t, the smaller J). Further analysis is required to determine the true cause o ailure in rootinding. The issue can be resolved by adding the terminal-time minimization term to the objective t T unction as J = ò ( G+ a a /2) dt [3]. Note that, to apply the ZEM/ZEV guidance with this 0 objective unction, the transversality condition described in Eqs. (42) and (55) should be modiied as well, which will be conducted as a uture study. Table 6 summarizes the results o ZEM/ZEV guidance simulations and open-loop optimizations or dierent t values (inal time speciied). Perormance indices and maximum accelerations o the ZEM/ZEV guidance simulations or various inal time (500 5,500 sec.) is compared with those rom open-loop optimizations. For missions with relatively short ranges, the ZEM/ZEV eedback guidance perorms almost the same as the open-loop optimization. This can be observed in Figure 8 and the top plots o Figure 10, which shows the rendezvous trajectory, the cost unction, and the acceleration proile or inal time o 1,000 seconds. The dierence becomes signiicant or longer mission times. In addition, while the ZEM/ZEV has its minimum cost around t = 5,000 sec., the objective unction o the open-loop optimization keep decreasing. While the root cause o this phenomenon should be urther investigated, one o possible explanation would be that the inaccuracy in the prediction o ZEM/ZEV due to the eect o a(t) results in the degradation in the perormance. The perormance dierence caused by using dierent analytic propagators was negligible or relatively short missions, but or very long mission like the condition or t = 5,500 s, the dierence got increased. Table 6: Perormance comparison ZEM/ZEV guidance vs. Open-loop optimization t, s 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000 5,500 OPT J, m 2 / s 3 152, ,185 1,847 1,353 1,285 1,246 1, ZEM J (Kepler), m 2 /s 3 152,420 16,444 4,225 1,958 1,604 1,752 1,978 2,025 1,660 1,116 2,652 ZEM J (Vinti), m 2 /s 3 152,420 16,445 4,227 1,962 1,611 1,763 1,991 2,034 1,655 1,116 2,862 OPT a, m/s max ZEM a (Kepler), m/s max ZEM a (Vinti), m/s max 16

17 Figure 7: Plots o ( t ) and or Case D Y (km) d(t )/dt, km 2 /s 5 (t ), km 2 /s 4 Figure 8: Trajectory or Case D orbital rendezvous ( t = 1,000 sec.) 17

18 Figure 9: Trajectory or Case D orbital rendezvous ( t = 3,500 sec.) J (m 2 /s 3 ) a (m/s 2 ) J (m 2 /s 3 ) a (m/s 2 ) Figure 10: Objective Function and Control Input (t = 1,000 / 3,500 sec.) 18

19 SUMMARY AND FUTURE WORK The results o the case studies indicate that the analytic-propagator-based ZEM/ZEV guidance implementation with proposed t prediction produces the control input and state proiles very similar to the solutions obtained through an open-loop o-line optimization or relatively long-range missions (light angle o 90 deg.). The algorithm provides solutions or very-long-range rendezvous problem like Case D (light angle larger than 180 deg.), but its objective unction and input/state proiles were dierent rom those o open-loop optimal solutions. Furthermore, under the assumption o continuous tracking (as was made or case studies), the implementations with dierent analytic propagators (Kepler and Vinti) do not show meaningul perormance dierence. Authors expect, however, that in realistic situations in which target s inormation is not always available during the intercept/rendezvous mission, the advantage o implementation based on Vinti s propagator will be apparent. Comparison o the proposed algorithm with other eedback guidance laws (e.g. PN guidance and its variants) rom the perspectives o the perormance and the computational cost can be considered or uture work. Use o the proposed algorithm or dierent applications (e.g. precision landing on a target rotating with the planet) and use o the waypoint-optimized ZEM/ZEV guidance concept [9] as applied to Case D can be another interesting research subject or urther exploration. CONCLUSIONS This paper has presented a new implementation o the ZEM/ZEV eedback guidance by utilizing the computational eiciency and accuracy o analytic orbital propagators. The problem was ormulated as an optimal control problem that minimizes the integral o the control energy with terminal states speciied on a plane or a curve, which yields the solution o the ZEM/ZEV eedback control orm. A ormula to determine the terminal time or the ZEM/ZEV eedback guidance or t ree problems using the analytic propagator has been developed. The eectiveness o the proposed implementation and optimal terminal-time estimation methodology was demonstrated using missile/spacecrat intercept and rendezvous case studies. ACKNOWLEDGEMENT The authors would like to express sincere thanks to Dr. Gim Der who has provided the FORTRAN version o Vinti s propagator, which was directly translated into MATLAB and used or this paper. The authors would also like to thank Ms. Yao Zhang (Ph.D. graduate student at Harbin Institute o Technology) or conducting oline trajectory optimizations or long-range missions o Case D. This work was prepared under a research grant rom the National Research Foundation o Korea (NRF-2013M1A3A3A ). The authors thank the National Research Foundation o Korea or the support o this research work. REFERENCES [1] Bryson, A., and Y. Ho, Applied Optimal Control, Wiley, New York, NY, 1969, pp [2] Battin, R., An Introduction to the Mathematics and Methods o Astrodynamics, Revised Edition, AIAA, Reston, VA, 1999, pp [3] D Souza, C., An Optimal Guidance Law or Planetary Landing, AIAA Paper ,

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