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Orbital rendezvous mission planning using mixed integer nonlinear programming
Acta Astronautica 68 (2011) 1070–1078
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Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Orbital rendezvous mission planning using mixed integer nonlinear programming$ Jin Zhang n, Guo-jin Tang, Ya-Zhong Luo, Hai-yang Li College of Aerospace and Materials Engineering, National University of Defense Technology, Changsha, China
a r t i c l e i n f o
abstract
Article history: Received 9 June 2010 Received in revised form 19 September 2010 Accepted 21 September 2010 Available online 13 October 2010
The rendezvous and docking mission is usually divided into several phases, and the mission planning is performed phase by phase. A new planning method using mixed integer nonlinear programming, which investigates single phase parameters and phase connecting parameters simultaneously, is proposed to improve the rendezvous mission’s overall performance. The design variables are composed of integers and continuous-valued numbers. The integer part consists of the parameters for station-keeping and sensorswitching, the number of maneuvers in each rendezvous phase and the number of repeating periods to start the rendezvous mission. The continuous part consists of the orbital transfer time and the station-keeping duration. The objective function is a combination of the propellant consumed, the sun angle which represents the power available, and the terminal precision of each rendezvous phase. The operational requirements for the spacecraft–ground communication, sun illumination and the sensor transition are considered. The simple genetic algorithm, which is a combination of the integer-coded and real-coded genetic algorithm, is chosen to obtain the optimal solution. A practical rendezvous mission planning problem is solved by the proposed method. The results show that the method proposed can solve the integral rendezvous mission planning problem effectively, and the solution obtained can satisfy the operational constraints and has a good overall performance. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Orbital rendezvous Mission planning Mixed integer nonlinear programming Genetic algorithm
1. Introduction In the rendezvous and docking (RVD) mission, for acquiring the desired docking condition, the chaser must execute many complicated operations such as orbital maneuvers, rendezvous navigation sensor switches and space–ground communications. Several station-keeping points are deployed in the rendezvous trajectory to obtain a stable orbital profile [1], and then the rendezvous process is divided into several phases such as homing, closing and approaching. Fehse [2] provided the common considerations for every rendezvous phase, and the planning of rendezvous mission is usually performed phase by phase.
$
This paper was presented during the 60th IAC in Daejeon. Corresponding author. Tel.: + 86 731 457 6316; fax: + 86 731 451 2301. E-mail address: [email protected] (J. Zhang). n
0094-5765/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2010.09.024
To improve the rendezvous mission’s overall performance, besides the parameters in each single phase, the parameters at the interfaces of different phases should also be investigated. These interface parameters include the ranges of station-keeping points, the ranges of the sensor-switching points, the number of maneuvers of different rendezvous phases, and so on. Now the design variables are composed of integers and continuous-valued numbers, and the rendezvous mission planning inevitably becomes a complicated mixed integer nonlinear programming (MINLP) problem. Recently, MINLP has been applied to the space mission planning: Ross and D’Souza [3] proposed a hybrid optimal control framework for space mission planning and the discrete variables are solved in integer programming subproblem; Luo et al. [4] proposed a hybrid strategy to optimize the rendezvous phasing trajectory, and the discrete variables are solved by integer-coded genetic algorithm. It is difficult to solve MINLP problem due to the combinatorial nature and the potential multiple local minimum points [5].
J. Zhang et al. / Acta Astronautica 68 (2011) 1070–1078
t0nrep
Nomenclature C K0 Ki Mi Sj X Xdesign duri m ni nrep nrKi nrSj rKi rSj t00
tik
covariance matrix initial aiming point station-keeping point penalty coefficient of the constraint sensor-switching point relative position and velocity design variables station-keeping duration mass of the chaser number of maneuvers number of repeating periods serial number of the station-keeping point candidate serial number of the sensor-switching point candidate relative range of the station-keeping point relative range of the sensor-switching point initial time of the whole mission design
Dti Dtinterval,j Dtprep,j U Uv b dX dDv
e Z li
o
The automated rendezvous mission, which is controlled onboard, cannot be initiated until the chaser is first guided by the ground mission control center to the initial aiming point K0. Based on the information obtained by the relative navigation sensors, the chaser’s actions are automatically controlled by the onboard system after K0. Fig. 1 shows the process of the rendezvous mission considered in this paper: the chaser transfers from K0 to K3 using three sets of maneuvers with station-keeping at Ki(i= 1,2,3); the maneuvers are controlled onboard using the information obtained by Sensor 1, Sensor 2 and Sensor 3, with sensor-switching at S1 and S2; the chaser communicates with the ground stations at K0 and Ki. The sun
2. Rendezvous mission planning considerations When we study orbital rendezvous, the chaser’s movement is usually described in the target centered orbital coordinate system o xyz, which is defined as [2]: the z-axis, also called R-bar, is along the position vector from the target spacecraft to the earth;
K2
initial time to start the mission at nrepth repeating period burning time of the orbital maneuver orbital transfer time of a rendezvous phase time interval between the sensor transition and the next maneuver preparation time for the sensor to obtain steady filter information state transition matrix of the relative position and velocity state transition matrix of the impulse orbital sun angle navigation error burning error spacecraft elevation angle seen from the ground station sun’s line of sight angle weight factor in the objective function mean orbital angular rate of the target
the y-axis, also called H-bar, is along the opposite direction of the orbit normal; the x-axis, also called V-bar, is toward the direction of the velocity and completes the right-handed system.
Previous studies mainly focused on the rendezvous trajectory planning for a single phase [6], especially on the fuel optimal rendezvous maneuver problem, and few studies dealt with the rendezvous mission planning which investigates single phase parameters and phase connecting parameters simultaneously. The goal of this paper is to propose a rendezvous mission planning method using MINLP, which can improve the overall performance of several rendezvous phases.
K3
1071
K1
O
S2
S1
K0
Fig. 1. The process of a rendezvous mission.
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illumination condition and the effects of errors also play important roles in rendezvous mission. The details of the main considerations are described as follows. 2.1. Considerations for each single phase 2.1.1. Orbital transfer The chaser’s orbital transfer from Ki 1(i=1,2,3) to Ki is calculated onboard based on the well-known Clohessy– Wiltshire (C–W) equations [2] 8 F > € > oz_ ¼ x > x2 > m > > < Fy y€ þ o2 y ¼ ð1Þ > m > > > F > > _ : z€ þ2ox3 o2 z ¼ z m where o is the mean orbital angular rate of the target, m is the mass of the chaser, and Fx, Fy and Fz are the thrust components of the chaser’s engine. The transfer time from Ki 1 to Ki is þ Dti ¼ ti ti1
ð2Þ
ti
þ is the time when the chaser arrives at Ki and ti1 where is the time when the chaser leaves Ki 1. The station-keeping duration at Ki is
duri ¼ tiþ ti
ð3Þ
It is assumed that the time intervals between each two successive maneuvers within the same rendezvous phase is equal to each other, so the burning time of the kth (k= 1,2,...,ni) maneuver in ith rendezvous phase is þ t ti1 þ ðk1Þ þ ti1 tik ¼ i ni 1
Let þ þ DXi ¼ Xðti ÞUðti ,ti1 ÞXðti1 Þ
ð6Þ
DVi ¼ ½ðDvi1 ÞT ðDvi2 ÞT ðDvini ÞT T
ð7Þ
Fi ¼ ½Uv ðti ,ti1 ÞUv ðti ,ti2 Þ Uv ðti ,tini Þ
ð8Þ
The minimum norm solution [7] is
DVi ¼ Fi T ðFi Fi T Þ1 DXi
ð9Þ
If ni = 2, the minimum norm solution is the only solution. If ni 4 2, the fuel optimal solution can be obtained using optimizing algorithms with the minimum norm solution as the initial guess. Many methods for solving the fuel optimal rendezvous problem can be found from the references of Ref. [6]. 2.1.2. Errors Different errors can affect the rendezvous mission in the process of navigating, targeting and burning. Targeting errors are mainly caused by the linear dynamic model used, but their effects decrease when the chaser approaches the target. Navigating errors and burning errors are the main errors considered in this paper. The linear covariance (LinCov) [8] method is used to propagate errors, and the interaction between the errors which affect the rendezvous mission at different time is ignored. The covariance matrix at the terminal of each rendezvous phase is þ þ T CdXi ¼ Uðti ,ti1 ÞCdXi0 Uðti ,ti1 Þ þ
ni X
Uv ðti ,tik ÞCdDvik Uv ðti ,tik ÞT
k¼1
ð4Þ
ð10Þ
where ni( Z2) is the number of maneuvers from Ki 1 to Ki and tik is the burning time. Using the impulse maneuver approximation, the orbital transfer from Ki 1 to Ki is expressed as
where CdXi0 is the covariance matrix of the initial navigation errors, CdDvik is the covariance matrix of kth maneuver’s burning errors. For reducing the effects of errors, each maneuver is recomputed before burning. The effects of the errors before the last two maneuvers of each phase can be reduced effectively by the recomputation, and they cannot affect X i directly. Let dXiðni 1Þ and dDviðni 1Þ be the navigation error and burning error at (ni 1)th maneuver, respectively. Let dXini and dDvini be the navigation error and burning error at nith maneuver, i.e. the last maneuver of ith rendezvous phase, respectively. Considering the correcting ability of each maneuver’s recomputation, rXi , the position part of X i , is mainly affected by dXiðni 1Þ and dDviðni 1Þ . While vXi , the velocity part of X i , is mainly affected by dXini and dDvini . The errors’ coupling effects, which appeared at the non-diagonal part of the covariance matrix, are ignored. So, the covariance matrix at the terminal of each rendezvous phase is h CdXi ¼ E1 Uðti ,tiðni 1Þ ÞCdXiðn 1Þ Uðti ,tiðni 1Þ ÞT i i þ Uv ðti ,tiðni 1Þ ÞCdDviðn 1Þ Uv ðti ,tiðni 1Þ ÞT
þ þ ÞXðti1 Þþ Xðti Þ ¼ Uðti ,ti1
ni X
Uv ðti ,tik Þ Dvik
ð5Þ
k¼1
_ y, _ z_ ÞT , where X ¼ ðx,y,z, x, ½oðt2 t1 Þ, 0
B1 B B B0 B B Uðt2 , t1 Þ ¼ B B0 B B B0 B B @0 0
2 6 6 6 6 6 6 Uv ðt2 , t1 Þ ¼ 6 6 6 6 6 6 6 4
s ¼ sin½oðt2 t1 Þ, 4s3oðt2 t1 Þ
c ¼ cos 2ð1cÞ
0
6½oðt2 t1 Þs
c
0
0
43c
0
6oð1cÞ
4c3
0
2s
os 0
0
0 2s
c 0
0 c
o 0 2ðc1Þ
o 4c3
s
0
o
2ðc1Þ
0
o
3os
4s3oðt2 t1 Þ
o
0 s
o 0
2ð1cÞ
o 0 s
o
0
2s
0
c
0
2s
0
c
0
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5
o 0 s
o
1 C C C C C C C C, C C C C C A
i
þE2 ½Uðti ,tini ÞCdXin Uðti ,tini ÞT i
þ Uv ðti ,tini ÞCdDvin Uv ðti ,tini ÞT i
ð11Þ
J. Zhang et al. / Acta Astronautica 68 (2011) 1070–1078
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where E1 ¼ diagð1,1,1,0,0,0Þ, E2 ¼ diagð0,0,0,1,1,1Þ, and diag represents the diagonal matrix. 2.1.3. Sun illumination For obtaining more solar energy, the chaser’s solar arrays are pointed toward the sun to gain the necessary supply of power [2], and the angle between the sunlight and the normal of the solar arrays plane is kept as small as possible. The chaser is assumed to hold an earth oriented attitude, and the smallest angle theoretically equals to b-angle [2], i.e. the angle of the sun relative to the orbital plane. With a smaller b, the spacecraft has the ability to gain more solar energy, and then b partly represents the power ability of the spacecraft. The optical sensors, which are usually used for rendezvous navigation, also have requirements for sun illumination. As shown in Fig. 2, the sun directly appears in the sensor’s field of view may greatly disturb the sensor, where Z is the sun’s line of sight angle in the sensor’s view and Z0 is the half cone angle of the sensor’s field of view. Then, the constraints for the sun illumination condition of optical sensors are given by
Z 4 Z0
ð12Þ
2.2. Considerations for the phase interface 2.2.1. Station-keeping At the station-keeping point Ki (i= 1,2,3), the relative velocity is small and the relative position changes slowly. For obtaining a stable orbital profile, the mission is checked both onboard and on ground. From K3 to docking, the chaser’s position, velocity, attitude and angular rate are controlled simultaneously in a closed-loop mode. rKi , i.e. the relative range of Ki, is a connecting parameter of ith phase and (i 1)th phase, and several candidates of rKi need the designer to make a choice. Let lrKi be the number of rKi candidates and nrKi 2 f1,2, . . . ,lrKi g be the serial number of the chosen candidate. 2.2.2. Communication Although the rendezvous process is automatically controlled from K0, the direct space–ground communication ability is also preferred at K0 and every stationkeeping point Ki(i=1–3), so that the ground mission control center has the ability to abort the rendezvous mission when some failures happen and are identified on ground but not found onboard.
ε0
Fig. 3. The geometry of the communication condition.
As shown in Fig. 3, when the spacecraft appears above the ground station’s local horizon, the communication between the spacecraft and the ground station is available [2]. Considering the disturbance of the atmosphere and the terrain, the available communication condition is given by [9]
e 4 e0
ð13Þ
where e is the spacecraft elevation angle seen from the ground station, and e0 is a small positive angle. Generally, the target runs in a repeating-ground-track orbit. If the target arrives at K0 with some communication condition at the time t00, then it will obtain a similar communication condition after several repeating periods at the time t0nrep ¼ t00 þ nrep T
ð14Þ
where T is the target’s repeating period, and nrep is the number of repeating periods. 2.2.3. Sensor-switching For obtaining high navigation precision, three different relative navigation sensors are utilized according to their different performance at different relative range. The choice of rSj ðj ¼ 1,2Þ, i.e. the relative range of the sensor transition point Sj, also affect the overall performance. Let n o lrSj be the number of rSj candidates and nrSj 2 1,2,:::,lrSj be the serial number of the chosen candidate. After the chaser switches sensors at Sj, some preparation time Dtprep,j is required to obtain steady sensor filter information that a high navigation precision can be used in the next targeting calculation. A following orbital maneuver performed within Dtprep,j is not permitted
Dtinterval,j Z Dtprep,j
ð15Þ
where Dtinterval,j is the time interval between the sensor transition at Sj and the next maneuver. 3. MINLP problem
η
0
η
3.1. Design variables The design variables are composed of integers and continuous-valued numbers:
Fig. 2. The geometry of the sun illumination constraint of optical sensors.
Xdesign ¼ ðnrK1 ,nrK2 ,nrK3 ,nrS1 ,nrS2 ,n1 ,n2 ,n3 ,nrep ,dur1 ,dur2 ,dur3 , Dt1 , Dt2 , Dt3 Þ
ð16Þ
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Among the variables, nrep is the parameter representing when to start the rendezvous mission, ni and Dti are the parameters of each rendezvous phase, and nrKi , duri and nrSj are the parameters connecting different phases. Now, the parameters within different rendezvous phases and the parameters at the interfaces are designed at the same time. 3.2. Objective function The objective function is a combination of the total velocity increment, the mean sun angle b (corresponding to the power available) and the weighted sum of terminal covariance matrixes f¼
ni 3 X X
9Dvik 9 þ l0 9b9 þ
i¼1k¼1
3 X
3 X
qffiffiffiffiffiffiffiffiffiffiffiffi CdXi jj
j¼1
rKi
0
li @
i¼1
qffiffiffiffiffiffiffiffiffiffiffiffi1 CdXi jj A þ o rKi j¼4 6 X
ð17Þ where the unit of b is radian, li(i= 0,1,2,3) is the weight factor, CdXi jj is the jth diagonal element of CdXi , rKi is the relative range of the station-keeping point Ki. 3.3. Constraints The requirements for communicating at station-keeping points are given by gi ¼ e0 max½em ðtKi Þ r0 ði ¼ 1,2,3Þ
ð18Þ
where tKi is the time when the station-keeping at Ki happens, em(t)(m= 1,2,y,ngr) is the chaser’s elevation angle seen from the mth ground station at time t, and ngr is the number of ground stations. The requirements for gaining steady navigation information after switching sensors are ( g4 ¼ Dtprep,1 Dtinterval,1 r0 ð19Þ g5 ¼ Dtprep,2 Dtinterval,2 r0 It is assumed that only the third sensor is an optical one. The lighting requirement is given by g6 ¼ Z0 minðZðtÞÞ r0ðtS2 r t r t3 Þ
ð20Þ
where tS2 is the time when the second sensor transition happens. 4. Optimization method 4.1. Treatment of constraints The penalty function is used to deal with the constraints, and then the total objective function is F ¼fþ
6 X
fMi max½gi ,0g
ð21Þ
i¼1
have been widely used in different optimization problems. Hybrid algorithms combining genetic algorithm with local search method are usually preferred for solving MINLP problem [5], but hybrid algorithms are difficult to operate well. Here, a mix-coded GA, which is a modification of the real-coded GA to include the integer variables, is employed. Actually, it is just a simple combination of the integer-coded GA and the real-coded GA. The design variables, which consist of integers and real numbers, are regarded as the chromosome of the individual. The fitness function is chosen as Fit ¼ CF
ð22Þ
where C is a large constant. The arithmetical crossover, uniform mutation and tournament selection are applied. Details of the GA can be found in Ref. [10].
4.3. Approximation in calculation Because the GA evaluates the objective function many times, the small computation cost of each evaluation is preferred. In each evaluation of the GA: (1) the fuel optimal solution for each rendezvous phase’s orbital transfer is approximated by the minimum norm solution, which is calculated analytically by Eq. (9); (2) the chaser’s orbital parameters, em, b and Z at the given time t, are obtained by interpolation, based on the target trajectory information. Before the GA is involved, the target trajectory is simulated using numerical integration for the whole design time interval. The target’s em, b and Z are calculated at each integral step, then we get the time sequences of em, b and Z. 8 t ¼ ðt00 ,t00 þ step,. . .,t00 þ l step,. . .,t00 þ lmax stepÞT > > > > > < em ¼ ðem0 , em1 ,. . ., eml ,. . ., emlmax ÞT
b ¼ ðbm0 , bm1 ,. . ., bml ,. . ., bmlmax ÞT > > > > > : g ¼ ðZm0 , Zm1 ,. . ., Zml ,. . ., Zmlmax ÞT
ðl ¼ 0,1,. . .,lmax Þ
ð23Þ where step is the integration step, lmax is the integral number of times and lmax step denotes the whole design time interval. For the time t, let lt be the maximum integer which is less than or equal to t/step. In each evaluation of the GA, em(t), b(t) and Z(t) are calculated with eml, bml, and Zml (l =lt 1,lt,lt +1) by three points polynomial interpolation, respectively [11]. From K0 to K3, the chaser is close to the target’s orbit in the earth centered coordinate system, so the interpolation makes a good approximation.
where Mi(i= 1,2,y,6) is the penalty coefficient. 4.2. Mix-coded genetic algorithm
5. Results
Genetic algorithms (GAs) are search algorithms based on the biologic evolution mechanism. Versions of GAs
In this section, the method proposed above is applied to a practical rendezvous mission.
J. Zhang et al. / Acta Astronautica 68 (2011) 1070–1078
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5.1. Computation condition
The covariance matrixes of the navigation errors for each sensor are shown, respectively:
The Gregorian universal coordinated time (UTCG) of t00 is 21 March 2010 12:00:00.000. The target runs in a 2-day repeating-ground-track orbit and its initial state (semimajor axis, eccentricity, inclination, RAAN, argument of periapsis, true anomaly) is
diag½ð2 mÞ2 ,ð2 mÞ2 ,ð2 mÞ2 ,ð0:03 m=sÞ2 ,ð0:03 m=sÞ2 ,ð0:03 m=sÞ2
diag½ð0:2 m þ 1:0 103 rÞ2 ,ð0:2m þ 1:0 103 rÞ2 , ð0:2 m þ1:0 103 rÞ2 ,ð0:01 m=sÞ2 ,ð0:01 m=sÞ2 ,ð0:01 m=sÞ2 diag½ð0:2 m þ 1:0 103 rÞ2 ,ð0:2m þ 1:0 103 rÞ2 ,
Etar0 ¼ ð6714:2692km,0,42:41,98:7656811,01,01Þ
ð0:2 m þ1:0 103 rÞ2 ,ð0:01 m=sÞ2 ,ð0:01 m=sÞ2 ,ð0:01 m=sÞ2
Only the J2 term of the earth nonspherical perturbation is considered, and the other orbital perturbations are ignored. The repeating period T is about 169,441.345 s, and the search space of nrep is {0,1,y,15}. The parameters of ground stations are shown in Table 1. The initial state of the chaser relative to the target is
where r is the relative range of the measure point. The covariance matrix of burning errors is
X0 ¼ ð70,000 m,0 m,20,000 m,33m=s,0 m=s,0 m=sÞ The maximum measurable ranges of the three sensors are about 100 km, 15 km and 500 m, respectively. The relative ranges of candidate station-keeping points and candidate sensor transition points are shown in Table 2. The search space of ni, Dti and duri are shown in Table 3. The half cone angle of the third sensor’s field of view is Z0 = 101. The required preparation time are Dtprep,1 = 300 s and Dtprep,2 =120 s, respectively.
Table 1 Parameters of ground stations. Station name Station Station Station Station Station Station
1 2 3 4 5 6
Latitude (deg)
Longitude (deg)
Altitude (m)
e0
30 0 30 30 0 30
130 100 50 50 100 120
0 0 0 0 0 0
5 5 5 5 5 5
(deg)
Table 2 Parameters of candidate station-keeping points and candidate sensor transition points. Candidate serial number
rK1 (m)
1 2 3 4 5
6000 5000 4000 3000 2000
rK2 (m)
600 500 400 300 200
rK3 (m)
60 50 40 30 20
rS1 (m)
14,000 13,000 12,000 11,000 10,000
CdDvik ¼ diag½ð0:02 m=s þ 0:01Dvikx Þ2 , ð0:02 m=s þ0:01Dviky Þ2 ,ð0:02 m=s þ 0:01Dvikz Þ2 where Dvikx, Dviky and Dvikz are the three components of the impulse vector Dvik. The coefficients of the objective function are shown in Table 4. The GA parameters are shown in Table 5. 5.2. A group of solutions with detailed results The target’s orbit is simulated with the information provided in Section 5.1. Fig. 4 shows the time history of b in a month from t00. Fig. 5 shows the target’s ground track of three orbital periods from t0nrep . The MINLP problem for rendezvous mission planning is solved using the mix-coded GA. The algorithm ends at the maximum number of generations. On a personal computer with CPU 3.0 GHz, six trials with the same computation condition are performed and each requires about 2 min. The design variables and the objectives are shown in Table 6. The best solution obtained within the six trials is Solution4, which has been labeled in bold. For Solution 4, the burning time and impulse components are provided in Table 7, 9b9 is about 3.91, and the elements on the diagonal of the error covariance matrixes at the terminal of different rendezvous phases are provided in Table 8. Table 4 Coefficients in the objective function. Parameter
Value
Parameter
Value
l0 l1 l2 l3
5 5 5 5 10,000
M1 M2 M2 M3 M4 M5
5000 5000 5000 5 5 5000
rS2 (m)
500 450 400 350 300
Table 3 Search space of the maneuver number of times, transfer time and station-keeping duration. Phase number
ni
Dti (s)
duri (s)
1 2 3
{2,3,4} {2,3,4} {2,3,4}
[500,8000] [500,8000] [500,8000]
[60,300] [60,300] [60,300]
C
Table 5 GA parameters. Parameter
Value
Population size Maximum number of generations Scale of tournament selection Probability of crossover Probability of mutation
500 100 3 0.5 0.3
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From the time history of max(em) which is shown in Fig. 6, we find that the station-keeping at K1 can be monitored by Station 3, the station-keeping at K2 can be monitored by Station 6 and the station-keeping at K3 can be monitored by Station 2. The constraints g1, g2 and g3 are satisfied. The first sensor transition occurred at 2062.52 s, it is between the second and the third maneuver of the first
Table 7 Burning time and velocity increment. Phase number
tik (s)
1
0 1771.275 3542.55
4.89858 3.70772 4.29560
0 0 0
0.75961 0.55202 1.31039
2
3661.83 4999.595 6337.36
0.00861 0 0.00861
0 0 0
0.44352 0.06601 0.44352
3
6493.115 7081.788 7670.462 8259.135
0.06290 0.06396 0.06396 0.06290
0 0 0 0
0.21893 0.03531 0.03531 0.21893
60
β(deg)
40
Dvikx (m/s)
Dviky
Dvikz
(m/s)
(m/s)
20 Table 8 Elements on the diagonal of the error covariance matrixes.
0
-20
-40 0
5
10
15 20 time (day)
25
Fig. 4. Time history of target’s orbital sun angle.
30
j
qffiffiffiffiffiffiffiffiffiffiffiffi CdX1 jj
qffiffiffiffiffiffiffiffiffiffiffiffi CdX2 jj
qffiffiffiffiffiffiffiffiffiffiffiffi CdX3 jj
1 2 3 4 5 6
173.31 28.14 165.68 0.07 0.036 0.045
64.48 31.40 68.84 0.036 0.036 0.039
20.59 19.70 24.41 0.036 0.036 0.037
Fig. 5. Ground track of the target.
Table 6 Results of six trials. Solution number
Design variables
ni 3 P P Dvik
f
i¼1k¼1
1 2 3 4 5 6
(5,2,1,3,3,3,4,4, 8,159.232,133.165,167.661,3541.24,2575.01,1897.87) (5,2,1,2,2,4,3,4,8,176.047,199.398,140.175,3532.94,2536.41,1895.36) (5,3,1,4,2,3,4,4, 8, 162.866,175.978,150.715,3523.45,2630.9,1816.26) (5,2,1,2,4,3,3,4,8,119.28,155.755,135.355,3542.55,2675.53,1766.02) (5,2,1,2,4,3,3,4,8,166.713,148.826,145.775,3549.02,2556.81,1925.49) (3,3,2,4,1,3,4,4,134.116,182.291,142.211,3586.14,4591.48,690.525,8)
14.89 14.81 14.85 14.75 14.74 15.00
29.21 29.19 29.00 28.87 29.29 28.88
J. Zhang et al. / Acta Astronautica 68 (2011) 1070–1078
rendezvous phase, and Dtinterval,1 = 1480.030 s. The second sensor transition occurred at 7238.162 s, it is between the second and the third maneuver of the third rendezvous phase, and Dtinterval,2 =432.30 s. Then, the constraints g4 and g5 are satisfied. From Fig. 7, which shows the time history of Z, we find that Z is always bigger than Z0 after the second sensor transition and the constraint g6 is satisfied.
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So all the constraints considered are satisfied. From Table 6, we obtain that (1) Dt2 and Dt3 of Solution 4 are quite different from Solution 6, while the objective functions of the two solutions have similar values. The integral rendezvous mission planning problem has multiple local minimum points. (2) Solution 1–Solution 5 have similar variables and objective functions. This can be explained by the GA’s bad local search ability [5].
100
max (εm) (deg)
80
5.3. Numerical experiments for changing computing condition
first station-keeping
60
second station-keeping third station-keeping
40 20 0 -20 0
2000
4000 6000 time (s)
8000
Fig. 6. Time history of chaser’s maximum elevation angle seen from ground stations.
200
5.3.1. Results for changing communication condition Station 2, Station 3 and Station 6 are the ground stations which are used for communication at the stationkeeping points of Solution 4. When one of the three ground stations is unavailable, it will cause the increase of the objective function as shown in Table 9. 5.3.2. Results for changing lighting condition If the second sensor is also a optical one, i.e. Z o Z0 is both required for the second and the third sensor, the best solution obtained is Xdesign ¼ ð4,2,1,3,2,3,3,4,9,140:335,148:733,125:5, 3461:56,2651:55,1957:91Þ The total velocity increment is 15.41 m/s. The objective function is 30.73, which is a little more than the objective function of Solution 4. 9b9 ¼ 12:11, and is bigger than Z0. Actually, when 9b9 4 Z0, the sun would never appear in the field view of the second or third sensor during the whole rendezvous mission.
150 η(deg)
6. Conclusion
100 first sensor-translation
50
second sensor-translation
0 0
2000
4000 time(s)
6000
8000
Fig. 7. Time history of sun’s light of sight angle in sensors’ view.
A rendezvous mission planning method, which treats several rendezvous phases as an entirety, has been proposed using MINLP. The method is applied to a practical rendezvous mission planning problem. The results with detailed information show that the method proposed can solve the integral rendezvous mission planning problem effectively, and the solution obtained can satisfy the operational constraints and has a good overall performance. The numerical experiments for changing computing condition show that with the proposed planning framework, the considerations in different phases and the considerations at the interfaces
Table 9 Results of numerical experiments for changing communication condition. Unavailable ground station
Design variables
Station 2 Station 3 Station 6
(5,3,1,3,3,3,3,4,,8,216.249,145.527,131.059,3399.68,2722.19,2677.05) (3,2,2,2,4,2,3,4,8,155.433,225.607,115.387,2620.05,3382.86,1207.62) (4,1,2,3,3,3,3,4,8,211.886,169.757,113.317,3371.02,4687.72,821.105)
ni 3 P P
9Dvik 9
f
i¼1k¼1
14.77 20.85 15.65
31.75 36.30 30.61
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can trade with each other to improve the mission’s overall performance. The proposed method is valuable for operational rendezvous mission planning. References [1] J.L. Goodman, History of space shuttle rendezvous and proximity operations, Journal of Spacecraft and Rockets 43 (5) (2006) 944–959. [2] W. Fehse, Automated Rendezvous and Docking of Spacecraft, Cambridge University Press, London, 2003 pp. 31–32, 40–41, 113–170. [3] I.M. Ross, C.N. D’Souza, Hybrid optimal control framework for mission planning, Journal of Guidance, Control and Dynamics 28 (4) (2005) 686–697. [4] Y.Z. Luo, H.Y. Li, G.J. Tang, Hybrid approach to optimize a rendezvous phasing strategy, Journal of Guidance, Control and Dynamics 30 (1) (2007) 185–191.
[5] V.B. Gantovnik, Z. Gurdal, et al., Genetic algorithm for mixed integer nonlinear programming problems using separate constraint approximations, AIAA Journal 43 (8) (2005) 1844–1849. [6] D.J. Jezewski, J.P Brazzel, et al., A survey of rendezvous trajectory planning, Advances in the Astronautical Sciences 76 (1992) 1373–1396. [7] Guo-jin Tang, Ya-zhong Luo, Hai-yang Li, Optimal robust linearized impulse rendezvous, Aerospace Science and Technology 11 (7–8) (2007) 563–569. [8] D.K. Geller, Linear covariance techniques for orbital rendezvous analysis and autonomous onboard mission planning, Journal of Guidance, Control, and Dynamics 29 (6) (2006) 1404–1414. [9] W.J. Larson, J.R. Wertz, Space Mission Analysis and Design, third ed, Microcosm Press, El Segundo, 1999 pp. 110–120. [10] D.E. Goldberg, B. Korb, K. Deb, Messy genetic algorithms: motivation, analysis, and first results, Complex Systems 3 (5) (1989) 493–530. [11] W.H. Press, S.A. Teukolsky, et al., Numerical Recipes, the Art of Scientific Computing, third ed., Cambridge University Press, London, 2007 pp. 110–120.
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Orbital rendezvous mission planning using mixed integer nonlinear programming$ Jin Zhang n, Guo-jin Tang, Ya-Zhong Luo, Hai-yang Li College of Aerospace and Materials Engineering, National University of Defense Technology, Changsha, China
a r t i c l e i n f o
abstract
Article history: Received 9 June 2010 Received in revised form 19 September 2010 Accepted 21 September 2010 Available online 13 October 2010
The rendezvous and docking mission is usually divided into several phases, and the mission planning is performed phase by phase. A new planning method using mixed integer nonlinear programming, which investigates single phase parameters and phase connecting parameters simultaneously, is proposed to improve the rendezvous mission’s overall performance. The design variables are composed of integers and continuous-valued numbers. The integer part consists of the parameters for station-keeping and sensorswitching, the number of maneuvers in each rendezvous phase and the number of repeating periods to start the rendezvous mission. The continuous part consists of the orbital transfer time and the station-keeping duration. The objective function is a combination of the propellant consumed, the sun angle which represents the power available, and the terminal precision of each rendezvous phase. The operational requirements for the spacecraft–ground communication, sun illumination and the sensor transition are considered. The simple genetic algorithm, which is a combination of the integer-coded and real-coded genetic algorithm, is chosen to obtain the optimal solution. A practical rendezvous mission planning problem is solved by the proposed method. The results show that the method proposed can solve the integral rendezvous mission planning problem effectively, and the solution obtained can satisfy the operational constraints and has a good overall performance. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Orbital rendezvous Mission planning Mixed integer nonlinear programming Genetic algorithm
1. Introduction In the rendezvous and docking (RVD) mission, for acquiring the desired docking condition, the chaser must execute many complicated operations such as orbital maneuvers, rendezvous navigation sensor switches and space–ground communications. Several station-keeping points are deployed in the rendezvous trajectory to obtain a stable orbital profile [1], and then the rendezvous process is divided into several phases such as homing, closing and approaching. Fehse [2] provided the common considerations for every rendezvous phase, and the planning of rendezvous mission is usually performed phase by phase.
$
This paper was presented during the 60th IAC in Daejeon. Corresponding author. Tel.: + 86 731 457 6316; fax: + 86 731 451 2301. E-mail address: [email protected] (J. Zhang). n
0094-5765/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2010.09.024
To improve the rendezvous mission’s overall performance, besides the parameters in each single phase, the parameters at the interfaces of different phases should also be investigated. These interface parameters include the ranges of station-keeping points, the ranges of the sensor-switching points, the number of maneuvers of different rendezvous phases, and so on. Now the design variables are composed of integers and continuous-valued numbers, and the rendezvous mission planning inevitably becomes a complicated mixed integer nonlinear programming (MINLP) problem. Recently, MINLP has been applied to the space mission planning: Ross and D’Souza [3] proposed a hybrid optimal control framework for space mission planning and the discrete variables are solved in integer programming subproblem; Luo et al. [4] proposed a hybrid strategy to optimize the rendezvous phasing trajectory, and the discrete variables are solved by integer-coded genetic algorithm. It is difficult to solve MINLP problem due to the combinatorial nature and the potential multiple local minimum points [5].
J. Zhang et al. / Acta Astronautica 68 (2011) 1070–1078
t0nrep
Nomenclature C K0 Ki Mi Sj X Xdesign duri m ni nrep nrKi nrSj rKi rSj t00
tik
covariance matrix initial aiming point station-keeping point penalty coefficient of the constraint sensor-switching point relative position and velocity design variables station-keeping duration mass of the chaser number of maneuvers number of repeating periods serial number of the station-keeping point candidate serial number of the sensor-switching point candidate relative range of the station-keeping point relative range of the sensor-switching point initial time of the whole mission design
Dti Dtinterval,j Dtprep,j U Uv b dX dDv
e Z li
o
The automated rendezvous mission, which is controlled onboard, cannot be initiated until the chaser is first guided by the ground mission control center to the initial aiming point K0. Based on the information obtained by the relative navigation sensors, the chaser’s actions are automatically controlled by the onboard system after K0. Fig. 1 shows the process of the rendezvous mission considered in this paper: the chaser transfers from K0 to K3 using three sets of maneuvers with station-keeping at Ki(i= 1,2,3); the maneuvers are controlled onboard using the information obtained by Sensor 1, Sensor 2 and Sensor 3, with sensor-switching at S1 and S2; the chaser communicates with the ground stations at K0 and Ki. The sun
2. Rendezvous mission planning considerations When we study orbital rendezvous, the chaser’s movement is usually described in the target centered orbital coordinate system o xyz, which is defined as [2]: the z-axis, also called R-bar, is along the position vector from the target spacecraft to the earth;
K2
initial time to start the mission at nrepth repeating period burning time of the orbital maneuver orbital transfer time of a rendezvous phase time interval between the sensor transition and the next maneuver preparation time for the sensor to obtain steady filter information state transition matrix of the relative position and velocity state transition matrix of the impulse orbital sun angle navigation error burning error spacecraft elevation angle seen from the ground station sun’s line of sight angle weight factor in the objective function mean orbital angular rate of the target
the y-axis, also called H-bar, is along the opposite direction of the orbit normal; the x-axis, also called V-bar, is toward the direction of the velocity and completes the right-handed system.
Previous studies mainly focused on the rendezvous trajectory planning for a single phase [6], especially on the fuel optimal rendezvous maneuver problem, and few studies dealt with the rendezvous mission planning which investigates single phase parameters and phase connecting parameters simultaneously. The goal of this paper is to propose a rendezvous mission planning method using MINLP, which can improve the overall performance of several rendezvous phases.
K3
1071
K1
O
S2
S1
K0
Fig. 1. The process of a rendezvous mission.
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illumination condition and the effects of errors also play important roles in rendezvous mission. The details of the main considerations are described as follows. 2.1. Considerations for each single phase 2.1.1. Orbital transfer The chaser’s orbital transfer from Ki 1(i=1,2,3) to Ki is calculated onboard based on the well-known Clohessy– Wiltshire (C–W) equations [2] 8 F > € > oz_ ¼ x > x2 > m > > < Fy y€ þ o2 y ¼ ð1Þ > m > > > F > > _ : z€ þ2ox3 o2 z ¼ z m where o is the mean orbital angular rate of the target, m is the mass of the chaser, and Fx, Fy and Fz are the thrust components of the chaser’s engine. The transfer time from Ki 1 to Ki is þ Dti ¼ ti ti1
ð2Þ
ti
þ is the time when the chaser arrives at Ki and ti1 where is the time when the chaser leaves Ki 1. The station-keeping duration at Ki is
duri ¼ tiþ ti
ð3Þ
It is assumed that the time intervals between each two successive maneuvers within the same rendezvous phase is equal to each other, so the burning time of the kth (k= 1,2,...,ni) maneuver in ith rendezvous phase is þ t ti1 þ ðk1Þ þ ti1 tik ¼ i ni 1
Let þ þ DXi ¼ Xðti ÞUðti ,ti1 ÞXðti1 Þ
ð6Þ
DVi ¼ ½ðDvi1 ÞT ðDvi2 ÞT ðDvini ÞT T
ð7Þ
Fi ¼ ½Uv ðti ,ti1 ÞUv ðti ,ti2 Þ Uv ðti ,tini Þ
ð8Þ
The minimum norm solution [7] is
DVi ¼ Fi T ðFi Fi T Þ1 DXi
ð9Þ
If ni = 2, the minimum norm solution is the only solution. If ni 4 2, the fuel optimal solution can be obtained using optimizing algorithms with the minimum norm solution as the initial guess. Many methods for solving the fuel optimal rendezvous problem can be found from the references of Ref. [6]. 2.1.2. Errors Different errors can affect the rendezvous mission in the process of navigating, targeting and burning. Targeting errors are mainly caused by the linear dynamic model used, but their effects decrease when the chaser approaches the target. Navigating errors and burning errors are the main errors considered in this paper. The linear covariance (LinCov) [8] method is used to propagate errors, and the interaction between the errors which affect the rendezvous mission at different time is ignored. The covariance matrix at the terminal of each rendezvous phase is þ þ T CdXi ¼ Uðti ,ti1 ÞCdXi0 Uðti ,ti1 Þ þ
ni X
Uv ðti ,tik ÞCdDvik Uv ðti ,tik ÞT
k¼1
ð4Þ
ð10Þ
where ni( Z2) is the number of maneuvers from Ki 1 to Ki and tik is the burning time. Using the impulse maneuver approximation, the orbital transfer from Ki 1 to Ki is expressed as
where CdXi0 is the covariance matrix of the initial navigation errors, CdDvik is the covariance matrix of kth maneuver’s burning errors. For reducing the effects of errors, each maneuver is recomputed before burning. The effects of the errors before the last two maneuvers of each phase can be reduced effectively by the recomputation, and they cannot affect X i directly. Let dXiðni 1Þ and dDviðni 1Þ be the navigation error and burning error at (ni 1)th maneuver, respectively. Let dXini and dDvini be the navigation error and burning error at nith maneuver, i.e. the last maneuver of ith rendezvous phase, respectively. Considering the correcting ability of each maneuver’s recomputation, rXi , the position part of X i , is mainly affected by dXiðni 1Þ and dDviðni 1Þ . While vXi , the velocity part of X i , is mainly affected by dXini and dDvini . The errors’ coupling effects, which appeared at the non-diagonal part of the covariance matrix, are ignored. So, the covariance matrix at the terminal of each rendezvous phase is h CdXi ¼ E1 Uðti ,tiðni 1Þ ÞCdXiðn 1Þ Uðti ,tiðni 1Þ ÞT i i þ Uv ðti ,tiðni 1Þ ÞCdDviðn 1Þ Uv ðti ,tiðni 1Þ ÞT
þ þ ÞXðti1 Þþ Xðti Þ ¼ Uðti ,ti1
ni X
Uv ðti ,tik Þ Dvik
ð5Þ
k¼1
_ y, _ z_ ÞT , where X ¼ ðx,y,z, x, ½oðt2 t1 Þ, 0
B1 B B B0 B B Uðt2 , t1 Þ ¼ B B0 B B B0 B B @0 0
2 6 6 6 6 6 6 Uv ðt2 , t1 Þ ¼ 6 6 6 6 6 6 6 4
s ¼ sin½oðt2 t1 Þ, 4s3oðt2 t1 Þ
c ¼ cos 2ð1cÞ
0
6½oðt2 t1 Þs
c
0
0
43c
0
6oð1cÞ
4c3
0
2s
os 0
0
0 2s
c 0
0 c
o 0 2ðc1Þ
o 4c3
s
0
o
2ðc1Þ
0
o
3os
4s3oðt2 t1 Þ
o
0 s
o 0
2ð1cÞ
o 0 s
o
0
2s
0
c
0
2s
0
c
0
3 7 7 7 7 7 7 7 7 7 7 7 7 7 5
o 0 s
o
1 C C C C C C C C, C C C C C A
i
þE2 ½Uðti ,tini ÞCdXin Uðti ,tini ÞT i
þ Uv ðti ,tini ÞCdDvin Uv ðti ,tini ÞT i
ð11Þ
J. Zhang et al. / Acta Astronautica 68 (2011) 1070–1078
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where E1 ¼ diagð1,1,1,0,0,0Þ, E2 ¼ diagð0,0,0,1,1,1Þ, and diag represents the diagonal matrix. 2.1.3. Sun illumination For obtaining more solar energy, the chaser’s solar arrays are pointed toward the sun to gain the necessary supply of power [2], and the angle between the sunlight and the normal of the solar arrays plane is kept as small as possible. The chaser is assumed to hold an earth oriented attitude, and the smallest angle theoretically equals to b-angle [2], i.e. the angle of the sun relative to the orbital plane. With a smaller b, the spacecraft has the ability to gain more solar energy, and then b partly represents the power ability of the spacecraft. The optical sensors, which are usually used for rendezvous navigation, also have requirements for sun illumination. As shown in Fig. 2, the sun directly appears in the sensor’s field of view may greatly disturb the sensor, where Z is the sun’s line of sight angle in the sensor’s view and Z0 is the half cone angle of the sensor’s field of view. Then, the constraints for the sun illumination condition of optical sensors are given by
Z 4 Z0
ð12Þ
2.2. Considerations for the phase interface 2.2.1. Station-keeping At the station-keeping point Ki (i= 1,2,3), the relative velocity is small and the relative position changes slowly. For obtaining a stable orbital profile, the mission is checked both onboard and on ground. From K3 to docking, the chaser’s position, velocity, attitude and angular rate are controlled simultaneously in a closed-loop mode. rKi , i.e. the relative range of Ki, is a connecting parameter of ith phase and (i 1)th phase, and several candidates of rKi need the designer to make a choice. Let lrKi be the number of rKi candidates and nrKi 2 f1,2, . . . ,lrKi g be the serial number of the chosen candidate. 2.2.2. Communication Although the rendezvous process is automatically controlled from K0, the direct space–ground communication ability is also preferred at K0 and every stationkeeping point Ki(i=1–3), so that the ground mission control center has the ability to abort the rendezvous mission when some failures happen and are identified on ground but not found onboard.
ε0
Fig. 3. The geometry of the communication condition.
As shown in Fig. 3, when the spacecraft appears above the ground station’s local horizon, the communication between the spacecraft and the ground station is available [2]. Considering the disturbance of the atmosphere and the terrain, the available communication condition is given by [9]
e 4 e0
ð13Þ
where e is the spacecraft elevation angle seen from the ground station, and e0 is a small positive angle. Generally, the target runs in a repeating-ground-track orbit. If the target arrives at K0 with some communication condition at the time t00, then it will obtain a similar communication condition after several repeating periods at the time t0nrep ¼ t00 þ nrep T
ð14Þ
where T is the target’s repeating period, and nrep is the number of repeating periods. 2.2.3. Sensor-switching For obtaining high navigation precision, three different relative navigation sensors are utilized according to their different performance at different relative range. The choice of rSj ðj ¼ 1,2Þ, i.e. the relative range of the sensor transition point Sj, also affect the overall performance. Let n o lrSj be the number of rSj candidates and nrSj 2 1,2,:::,lrSj be the serial number of the chosen candidate. After the chaser switches sensors at Sj, some preparation time Dtprep,j is required to obtain steady sensor filter information that a high navigation precision can be used in the next targeting calculation. A following orbital maneuver performed within Dtprep,j is not permitted
Dtinterval,j Z Dtprep,j
ð15Þ
where Dtinterval,j is the time interval between the sensor transition at Sj and the next maneuver. 3. MINLP problem
η
0
η
3.1. Design variables The design variables are composed of integers and continuous-valued numbers:
Fig. 2. The geometry of the sun illumination constraint of optical sensors.
Xdesign ¼ ðnrK1 ,nrK2 ,nrK3 ,nrS1 ,nrS2 ,n1 ,n2 ,n3 ,nrep ,dur1 ,dur2 ,dur3 , Dt1 , Dt2 , Dt3 Þ
ð16Þ
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Among the variables, nrep is the parameter representing when to start the rendezvous mission, ni and Dti are the parameters of each rendezvous phase, and nrKi , duri and nrSj are the parameters connecting different phases. Now, the parameters within different rendezvous phases and the parameters at the interfaces are designed at the same time. 3.2. Objective function The objective function is a combination of the total velocity increment, the mean sun angle b (corresponding to the power available) and the weighted sum of terminal covariance matrixes f¼
ni 3 X X
9Dvik 9 þ l0 9b9 þ
i¼1k¼1
3 X
3 X
qffiffiffiffiffiffiffiffiffiffiffiffi CdXi jj
j¼1
rKi
0
li @
i¼1
qffiffiffiffiffiffiffiffiffiffiffiffi1 CdXi jj A þ o rKi j¼4 6 X
ð17Þ where the unit of b is radian, li(i= 0,1,2,3) is the weight factor, CdXi jj is the jth diagonal element of CdXi , rKi is the relative range of the station-keeping point Ki. 3.3. Constraints The requirements for communicating at station-keeping points are given by gi ¼ e0 max½em ðtKi Þ r0 ði ¼ 1,2,3Þ
ð18Þ
where tKi is the time when the station-keeping at Ki happens, em(t)(m= 1,2,y,ngr) is the chaser’s elevation angle seen from the mth ground station at time t, and ngr is the number of ground stations. The requirements for gaining steady navigation information after switching sensors are ( g4 ¼ Dtprep,1 Dtinterval,1 r0 ð19Þ g5 ¼ Dtprep,2 Dtinterval,2 r0 It is assumed that only the third sensor is an optical one. The lighting requirement is given by g6 ¼ Z0 minðZðtÞÞ r0ðtS2 r t r t3 Þ
ð20Þ
where tS2 is the time when the second sensor transition happens. 4. Optimization method 4.1. Treatment of constraints The penalty function is used to deal with the constraints, and then the total objective function is F ¼fþ
6 X
fMi max½gi ,0g
ð21Þ
i¼1
have been widely used in different optimization problems. Hybrid algorithms combining genetic algorithm with local search method are usually preferred for solving MINLP problem [5], but hybrid algorithms are difficult to operate well. Here, a mix-coded GA, which is a modification of the real-coded GA to include the integer variables, is employed. Actually, it is just a simple combination of the integer-coded GA and the real-coded GA. The design variables, which consist of integers and real numbers, are regarded as the chromosome of the individual. The fitness function is chosen as Fit ¼ CF
ð22Þ
where C is a large constant. The arithmetical crossover, uniform mutation and tournament selection are applied. Details of the GA can be found in Ref. [10].
4.3. Approximation in calculation Because the GA evaluates the objective function many times, the small computation cost of each evaluation is preferred. In each evaluation of the GA: (1) the fuel optimal solution for each rendezvous phase’s orbital transfer is approximated by the minimum norm solution, which is calculated analytically by Eq. (9); (2) the chaser’s orbital parameters, em, b and Z at the given time t, are obtained by interpolation, based on the target trajectory information. Before the GA is involved, the target trajectory is simulated using numerical integration for the whole design time interval. The target’s em, b and Z are calculated at each integral step, then we get the time sequences of em, b and Z. 8 t ¼ ðt00 ,t00 þ step,. . .,t00 þ l step,. . .,t00 þ lmax stepÞT > > > > > < em ¼ ðem0 , em1 ,. . ., eml ,. . ., emlmax ÞT
b ¼ ðbm0 , bm1 ,. . ., bml ,. . ., bmlmax ÞT > > > > > : g ¼ ðZm0 , Zm1 ,. . ., Zml ,. . ., Zmlmax ÞT
ðl ¼ 0,1,. . .,lmax Þ
ð23Þ where step is the integration step, lmax is the integral number of times and lmax step denotes the whole design time interval. For the time t, let lt be the maximum integer which is less than or equal to t/step. In each evaluation of the GA, em(t), b(t) and Z(t) are calculated with eml, bml, and Zml (l =lt 1,lt,lt +1) by three points polynomial interpolation, respectively [11]. From K0 to K3, the chaser is close to the target’s orbit in the earth centered coordinate system, so the interpolation makes a good approximation.
where Mi(i= 1,2,y,6) is the penalty coefficient. 4.2. Mix-coded genetic algorithm
5. Results
Genetic algorithms (GAs) are search algorithms based on the biologic evolution mechanism. Versions of GAs
In this section, the method proposed above is applied to a practical rendezvous mission.
J. Zhang et al. / Acta Astronautica 68 (2011) 1070–1078
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5.1. Computation condition
The covariance matrixes of the navigation errors for each sensor are shown, respectively:
The Gregorian universal coordinated time (UTCG) of t00 is 21 March 2010 12:00:00.000. The target runs in a 2-day repeating-ground-track orbit and its initial state (semimajor axis, eccentricity, inclination, RAAN, argument of periapsis, true anomaly) is
diag½ð2 mÞ2 ,ð2 mÞ2 ,ð2 mÞ2 ,ð0:03 m=sÞ2 ,ð0:03 m=sÞ2 ,ð0:03 m=sÞ2
diag½ð0:2 m þ 1:0 103 rÞ2 ,ð0:2m þ 1:0 103 rÞ2 , ð0:2 m þ1:0 103 rÞ2 ,ð0:01 m=sÞ2 ,ð0:01 m=sÞ2 ,ð0:01 m=sÞ2 diag½ð0:2 m þ 1:0 103 rÞ2 ,ð0:2m þ 1:0 103 rÞ2 ,
Etar0 ¼ ð6714:2692km,0,42:41,98:7656811,01,01Þ
ð0:2 m þ1:0 103 rÞ2 ,ð0:01 m=sÞ2 ,ð0:01 m=sÞ2 ,ð0:01 m=sÞ2
Only the J2 term of the earth nonspherical perturbation is considered, and the other orbital perturbations are ignored. The repeating period T is about 169,441.345 s, and the search space of nrep is {0,1,y,15}. The parameters of ground stations are shown in Table 1. The initial state of the chaser relative to the target is
where r is the relative range of the measure point. The covariance matrix of burning errors is
X0 ¼ ð70,000 m,0 m,20,000 m,33m=s,0 m=s,0 m=sÞ The maximum measurable ranges of the three sensors are about 100 km, 15 km and 500 m, respectively. The relative ranges of candidate station-keeping points and candidate sensor transition points are shown in Table 2. The search space of ni, Dti and duri are shown in Table 3. The half cone angle of the third sensor’s field of view is Z0 = 101. The required preparation time are Dtprep,1 = 300 s and Dtprep,2 =120 s, respectively.
Table 1 Parameters of ground stations. Station name Station Station Station Station Station Station
1 2 3 4 5 6
Latitude (deg)
Longitude (deg)
Altitude (m)
e0
30 0 30 30 0 30
130 100 50 50 100 120
0 0 0 0 0 0
5 5 5 5 5 5
(deg)
Table 2 Parameters of candidate station-keeping points and candidate sensor transition points. Candidate serial number
rK1 (m)
1 2 3 4 5
6000 5000 4000 3000 2000
rK2 (m)
600 500 400 300 200
rK3 (m)
60 50 40 30 20
rS1 (m)
14,000 13,000 12,000 11,000 10,000
CdDvik ¼ diag½ð0:02 m=s þ 0:01Dvikx Þ2 , ð0:02 m=s þ0:01Dviky Þ2 ,ð0:02 m=s þ 0:01Dvikz Þ2 where Dvikx, Dviky and Dvikz are the three components of the impulse vector Dvik. The coefficients of the objective function are shown in Table 4. The GA parameters are shown in Table 5. 5.2. A group of solutions with detailed results The target’s orbit is simulated with the information provided in Section 5.1. Fig. 4 shows the time history of b in a month from t00. Fig. 5 shows the target’s ground track of three orbital periods from t0nrep . The MINLP problem for rendezvous mission planning is solved using the mix-coded GA. The algorithm ends at the maximum number of generations. On a personal computer with CPU 3.0 GHz, six trials with the same computation condition are performed and each requires about 2 min. The design variables and the objectives are shown in Table 6. The best solution obtained within the six trials is Solution4, which has been labeled in bold. For Solution 4, the burning time and impulse components are provided in Table 7, 9b9 is about 3.91, and the elements on the diagonal of the error covariance matrixes at the terminal of different rendezvous phases are provided in Table 8. Table 4 Coefficients in the objective function. Parameter
Value
Parameter
Value
l0 l1 l2 l3
5 5 5 5 10,000
M1 M2 M2 M3 M4 M5
5000 5000 5000 5 5 5000
rS2 (m)
500 450 400 350 300
Table 3 Search space of the maneuver number of times, transfer time and station-keeping duration. Phase number
ni
Dti (s)
duri (s)
1 2 3
{2,3,4} {2,3,4} {2,3,4}
[500,8000] [500,8000] [500,8000]
[60,300] [60,300] [60,300]
C
Table 5 GA parameters. Parameter
Value
Population size Maximum number of generations Scale of tournament selection Probability of crossover Probability of mutation
500 100 3 0.5 0.3
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From the time history of max(em) which is shown in Fig. 6, we find that the station-keeping at K1 can be monitored by Station 3, the station-keeping at K2 can be monitored by Station 6 and the station-keeping at K3 can be monitored by Station 2. The constraints g1, g2 and g3 are satisfied. The first sensor transition occurred at 2062.52 s, it is between the second and the third maneuver of the first
Table 7 Burning time and velocity increment. Phase number
tik (s)
1
0 1771.275 3542.55
4.89858 3.70772 4.29560
0 0 0
0.75961 0.55202 1.31039
2
3661.83 4999.595 6337.36
0.00861 0 0.00861
0 0 0
0.44352 0.06601 0.44352
3
6493.115 7081.788 7670.462 8259.135
0.06290 0.06396 0.06396 0.06290
0 0 0 0
0.21893 0.03531 0.03531 0.21893
60
β(deg)
40
Dvikx (m/s)
Dviky
Dvikz
(m/s)
(m/s)
20 Table 8 Elements on the diagonal of the error covariance matrixes.
0
-20
-40 0
5
10
15 20 time (day)
25
Fig. 4. Time history of target’s orbital sun angle.
30
j
qffiffiffiffiffiffiffiffiffiffiffiffi CdX1 jj
qffiffiffiffiffiffiffiffiffiffiffiffi CdX2 jj
qffiffiffiffiffiffiffiffiffiffiffiffi CdX3 jj
1 2 3 4 5 6
173.31 28.14 165.68 0.07 0.036 0.045
64.48 31.40 68.84 0.036 0.036 0.039
20.59 19.70 24.41 0.036 0.036 0.037
Fig. 5. Ground track of the target.
Table 6 Results of six trials. Solution number
Design variables
ni 3 P P Dvik
f
i¼1k¼1
1 2 3 4 5 6
(5,2,1,3,3,3,4,4, 8,159.232,133.165,167.661,3541.24,2575.01,1897.87) (5,2,1,2,2,4,3,4,8,176.047,199.398,140.175,3532.94,2536.41,1895.36) (5,3,1,4,2,3,4,4, 8, 162.866,175.978,150.715,3523.45,2630.9,1816.26) (5,2,1,2,4,3,3,4,8,119.28,155.755,135.355,3542.55,2675.53,1766.02) (5,2,1,2,4,3,3,4,8,166.713,148.826,145.775,3549.02,2556.81,1925.49) (3,3,2,4,1,3,4,4,134.116,182.291,142.211,3586.14,4591.48,690.525,8)
14.89 14.81 14.85 14.75 14.74 15.00
29.21 29.19 29.00 28.87 29.29 28.88
J. Zhang et al. / Acta Astronautica 68 (2011) 1070–1078
rendezvous phase, and Dtinterval,1 = 1480.030 s. The second sensor transition occurred at 7238.162 s, it is between the second and the third maneuver of the third rendezvous phase, and Dtinterval,2 =432.30 s. Then, the constraints g4 and g5 are satisfied. From Fig. 7, which shows the time history of Z, we find that Z is always bigger than Z0 after the second sensor transition and the constraint g6 is satisfied.
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So all the constraints considered are satisfied. From Table 6, we obtain that (1) Dt2 and Dt3 of Solution 4 are quite different from Solution 6, while the objective functions of the two solutions have similar values. The integral rendezvous mission planning problem has multiple local minimum points. (2) Solution 1–Solution 5 have similar variables and objective functions. This can be explained by the GA’s bad local search ability [5].
100
max (εm) (deg)
80
5.3. Numerical experiments for changing computing condition
first station-keeping
60
second station-keeping third station-keeping
40 20 0 -20 0
2000
4000 6000 time (s)
8000
Fig. 6. Time history of chaser’s maximum elevation angle seen from ground stations.
200
5.3.1. Results for changing communication condition Station 2, Station 3 and Station 6 are the ground stations which are used for communication at the stationkeeping points of Solution 4. When one of the three ground stations is unavailable, it will cause the increase of the objective function as shown in Table 9. 5.3.2. Results for changing lighting condition If the second sensor is also a optical one, i.e. Z o Z0 is both required for the second and the third sensor, the best solution obtained is Xdesign ¼ ð4,2,1,3,2,3,3,4,9,140:335,148:733,125:5, 3461:56,2651:55,1957:91Þ The total velocity increment is 15.41 m/s. The objective function is 30.73, which is a little more than the objective function of Solution 4. 9b9 ¼ 12:11, and is bigger than Z0. Actually, when 9b9 4 Z0, the sun would never appear in the field view of the second or third sensor during the whole rendezvous mission.
150 η(deg)
6. Conclusion
100 first sensor-translation
50
second sensor-translation
0 0
2000
4000 time(s)
6000
8000
Fig. 7. Time history of sun’s light of sight angle in sensors’ view.
A rendezvous mission planning method, which treats several rendezvous phases as an entirety, has been proposed using MINLP. The method is applied to a practical rendezvous mission planning problem. The results with detailed information show that the method proposed can solve the integral rendezvous mission planning problem effectively, and the solution obtained can satisfy the operational constraints and has a good overall performance. The numerical experiments for changing computing condition show that with the proposed planning framework, the considerations in different phases and the considerations at the interfaces
Table 9 Results of numerical experiments for changing communication condition. Unavailable ground station
Design variables
Station 2 Station 3 Station 6
(5,3,1,3,3,3,3,4,,8,216.249,145.527,131.059,3399.68,2722.19,2677.05) (3,2,2,2,4,2,3,4,8,155.433,225.607,115.387,2620.05,3382.86,1207.62) (4,1,2,3,3,3,3,4,8,211.886,169.757,113.317,3371.02,4687.72,821.105)
ni 3 P P
9Dvik 9
f
i¼1k¼1
14.77 20.85 15.65
31.75 36.30 30.61
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can trade with each other to improve the mission’s overall performance. The proposed method is valuable for operational rendezvous mission planning. References [1] J.L. Goodman, History of space shuttle rendezvous and proximity operations, Journal of Spacecraft and Rockets 43 (5) (2006) 944–959. [2] W. Fehse, Automated Rendezvous and Docking of Spacecraft, Cambridge University Press, London, 2003 pp. 31–32, 40–41, 113–170. [3] I.M. Ross, C.N. D’Souza, Hybrid optimal control framework for mission planning, Journal of Guidance, Control and Dynamics 28 (4) (2005) 686–697. [4] Y.Z. Luo, H.Y. Li, G.J. Tang, Hybrid approach to optimize a rendezvous phasing strategy, Journal of Guidance, Control and Dynamics 30 (1) (2007) 185–191.
[5] V.B. Gantovnik, Z. Gurdal, et al., Genetic algorithm for mixed integer nonlinear programming problems using separate constraint approximations, AIAA Journal 43 (8) (2005) 1844–1849. [6] D.J. Jezewski, J.P Brazzel, et al., A survey of rendezvous trajectory planning, Advances in the Astronautical Sciences 76 (1992) 1373–1396. [7] Guo-jin Tang, Ya-zhong Luo, Hai-yang Li, Optimal robust linearized impulse rendezvous, Aerospace Science and Technology 11 (7–8) (2007) 563–569. [8] D.K. Geller, Linear covariance techniques for orbital rendezvous analysis and autonomous onboard mission planning, Journal of Guidance, Control, and Dynamics 29 (6) (2006) 1404–1414. [9] W.J. Larson, J.R. Wertz, Space Mission Analysis and Design, third ed, Microcosm Press, El Segundo, 1999 pp. 110–120. [10] D.E. Goldberg, B. Korb, K. Deb, Messy genetic algorithms: motivation, analysis, and first results, Complex Systems 3 (5) (1989) 493–530. [11] W.H. Press, S.A. Teukolsky, et al., Numerical Recipes, the Art of Scientific Computing, third ed., Cambridge University Press, London, 2007 pp. 110–120.