Optimal Rendezvous Trajectories as a Function of ...

AIAA 2010-7663

AIAA Guidance, Navigation, and Control Conference 2 - 5 August 2010, Toronto, Ontario Canada

Optimal Rendezvous Trajectories as a Function of Thrust-to-Weight Ratio Yuri Ulybyshev * Rocket-Space Corporation "Energia", Korolev, Moscow Region, 141070, Russia An optimal rendezvous trajectory analysis of long-range guidance (or phasing and transfer to a target orbit) related with design of low-thrust propulsion system is considered. Typical rendezvous missions to low Earth orbit space stations for an active spacecraft with different thrust-to-weight ratios (or thrust acceleration) are studied. The method of pseudoimpulse sets is used for the computations. This approach combines large-scale linear programming algorithms with the well-known discretization of the trajectories on small segments and uses discrete pseudo-impulse sets which are considered independently for each segment. Such method is very suitable for spacecraft trajectory optimization with arbitrary thrust from high to low. Existence of low-thrust optimal solutions is briefly discussed. An analysis of thrust-to-weight ratio factor for phasing orbit is presented. As for the high thrust case, for low-thrust trajectories there also are optimal phase solutions, i.e. there is a range of initial phase angles for which required characteristic velocity is almost equal to optimal orbit transfer between initial and target orbit. Possible rendezvous trajectory types in a wide ranges of thrust acceleration values from high to low (including a continuous multirevolution burn) and phase angles are described.

Nomenclature

a, a

=

thrust acceleration vector and its magnitude

A Ae

=

matrix of inequality constraints

=

matrix of equality constraints

e, er, en, eb

=

thrust direction unit vector and its components in local vertical/local horizontal coordinate system - radial, along-track, cross-track

i

=

segment number

j

=

pseudo-impulse number

J k

=

performance index

=

quantity of pseudo-impulses at each segment

m

=

number of boundary conditions

n

=

quantity of segments

q

=

weight coefficient vector

ρ

=

vector of relative coordinates

NR

=

specified number of revolutions for rendezvous mission

P

=

boundary condition vector

*

Head of Space Ballistic Department, Senior Member AIAA, Email: [email protected], [email protected]

Copyright © 2010 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

2

rω.

=

mean radius of circular reference orbit

t

=

time

ΔtAN

=

relative time from ascending node

V

=

vector of relative velocity

T

=

orbit period

X

=

vector of decision variables

Δti

=

duration of i-th segment

ΔVi(j)

=

characteristic velocity of j-th pseudo-impulse at i-th segment

ΔVx

=

characteristic velocity

μ

=

gravitational parameter for the Earth

ϑ

=

pitch angle , Fig. 4

ϕ

=

phase angle, Fig. 1

ψ

=

yaw angle, Fig. 4

Ф, Фρρ, ФρV, ФVρ, ФVV

=

transition matrix and its sub-matrices

ω

=

mean angular velocity of orbit motion

S

I. Introduction

PACECRAFT with high thrust such as the “Soyuz”, the “Progress”, the “Space Shuttle”, and the ATV was usually used for the rendezvous missions to the International Space Station (ISS) or other space stations (SSs). The potential of solar–electrical propulsion system for future space missions has been well recognized. This propulsion system produces low thrust with high specific impulse, greatly reducing the initial spacecraft mass. This mass reduction makes possible a new generation of small spacecraft for rendezvous missions and maneuvers in a vicinity of the ISS or other space stations. Low-thrust rendezvous trajectory optimization has been extensively analyzed in the published literature [1-7]. However, most of these analyses has focused on numerical or analytical methods for solutions of the problem. An important part of most practical rendezvous missions is a phasing strategy. The target objective of the phasing segment is to reduce the phase angle between an active spacecraft (AS) and a SS, based on a difference of their orbital periods. Such issues for high-thrust rendezvous missions was detailed studied [8-12]. In the case of continuous medium- or low-thrust spacecraft the phasing strategy is closely related with available thrust-to-weigth ratio. The major objective of this paper is to present an optimal rendezvous trajectory analysis related with this interrelation. Typical rendezvous missions to the ISS for extended range of possible orbital parameters and different thrust accelerations (or thrust-to-weight ratios) are considered. For the computations we use method of pseudo-impulse sets [6, 13-14] that based on discretization of the trajectory into segments and introducing of a set for each segment. On the one hand, this substantially increases the problem dimensionality, but, on the other hand, allows one to formulate the problem into a large-scale linear programming form. Present-day linear programming methods use interior-point algorithms to solve such problems. The method is very suitable for spacecraft trajectory optimization with arbitrary thrust from high to low. The paper will be included two major parts. First, it is an analysis of thrust-to-weight ratio factor for phasing orbit. Possible types of optimal rendezvous trajectories with continuous thrust will be described in the second part. 2 American Institute of Aeronautics and Astronautics

3

II. High-Thrust Rendezvous Missions for Space Stations A typical rendezvous mission included the following flight phases: - launch of active spacecraft at a near coplanar to the SS orbit; - long-range guidance (phasing and transfer to a vicinity SS or an aim point) based on the separated orbit determination for each spacecraft using the ground segment and/or space navigation systems GLONASS or GPS; - short-range guidance based on relative navigation using relative measurements of the range and direction between the AS and SS and docking. In the following only the long-range guidance phase will be considered. At the initial time after launch the AS to an initial orbit, both spacecraft are placed in near-coplanar, near-circular orbits with different altitudes. Usually, the AS orbit period (and respectively mean altitude) is below than the SS orbit period, i.e. Т0 < Тf. The geocentric angle between the AS and SS ϕ is named as the phase angle (Fig.1). The angle is defined an along-track displacement between them at the initial time. For a high-thrust AS, such as the “Soyuz”, the long range guidance is performed as an orbit transfer at an aim point with a near zero phase angle at a specified final time tf [8-9, 12]. Currently, for twoday rendezvous missions [12], the first two burns at 3th - 4th revolutions formed a phasing orbit that must be eliminated the phase displacement. The next two phasing burns are performed at 28th - 30th revolutions with transfer to the aim point. Usually, a midcourse burn at 17th revolution is used to correct errors accumulated during the first day of flight.

Figure 1. Phase angle geometry. For impulsive solutions (i.e. for high thrust) there is a range of initial phase angles for which required characteristic velocity for the trajectories is almost equal and near to the optimal orbit transfer between the orbits. Therefore the phasing performed without of an additional characteristic velocity (in a realistic case, there can be small differences for very low-altitude phasing orbits due to the atmospheric perturbation). For these cases, the phasing orbits are placed inside the ring between the initial and target orbits (see Fig. 1). The solutions are named as optimal phase solutions. The phases without the optimal range are required phasing orbits without the ring (below of the initial or higher of the target orbit) and respectively more high characteristic velocity.

III. Optimization of Continuous Thrust Rendezvous Trajectories A. Optimization Methods Based on Discrete Sets of Pseudo-Impulses The methods use discretization of the spacecraft trajectory on segments and sets of pseudo-impulses for each segment [6, 13-14]. Suppose that the final time of the rendezvous trajectory tf is specified. Introduce a set of n segments as the partition [t0, t1, t2, …,, tn], with t0=0 and tn=tf. The mesh points ti are referred to as nodes, the 3 American Institute of Aeronautics and Astronautics

4 intervals Δti=[ti+1, ti] are referred to as trajectory segments. The simplest case of the control space for the thrust vector is a plane. As an example, it can be a local horizontal or orbit plane. We consider an i-th segment independent of all the other segments. Suppose that the thrust direction in the plane is arbitrary. All of the possible thrust directions can be present as a set of pseudo-impulses

ΔVi ( j ) e i( j ) within the unit circle with a small angle of

Δϕ =2π/k between them (Fig. 2a). Suppose that there is an optimal impulse ΔViopt for the i-th segment. inadmissible approximation

Δϕ ΔVi opt

admissible approximation

b)

a)

Figure 2. Set of pseudo-impulses in a plane. Thus we can present the optimal impulse by the sum k

ΔVi opt = ∑ ΔVi ( j ) e i( j )

(1)

1

with a constraint for the characteristic velocities of the pseudo-impulses (Fig. 2b): k

∑ ΔVi ( j ) ≤ 1

(2)

1

It is evident that the optimal vector approximation of the optimal impulse by the pseudo-impulses is a sum of the two nearest neighbor pseudo-impulses. In a similar way, it can be consider a three-dimensional case for the possible thrust directions. A set of pseudo-impulses can be constructed using a uniform distribution on the unit sphere. An example of such set on the full sphere is depicted in Fig. 3. Similarly to the planar case, for each segment, the sum of characteristic velocities of the pseudo-impulses should be constrained by the inequality (2). The best approximation of an optimal impulse ΔVi opt is also a sum of nearest neighbor pseudo-impulses.

1

0.5

0

-0.5

-1 -1

-1 0

0 1

1

Figure 3. Near uniform point distributions in 3D space. 4 American Institute of Aeronautics and Astronautics

5

Define a (n×k)-dimension vector of decision variables

X T = [ ΔV1(1) , ΔV1( 2 ) ,...ΔV1( k ) , ΔV2(1) , ΔV2( 2 ) ,...ΔV2( k ) ,...ΔVn( k ) ]

(3)

It should be noted that all the vector components must be nonnegative. For the vector, according to the previous statements, the following linear inequality can be written

AX≤b ,

(4)

where A is a n×(n×k)-dimension matrix of the following form ( all of the unspecified elements equal to zero)

⎫ 1 1 1 ...1 ⎡ 

⎤⎪ ⎢ k ⎥⎪ 1 1 1 ... 1 ⎢ ⎥ ⎪⎪

k ⎥⎬ n A=⎢ ....... ⎢ ⎥⎪ ⎢ ⎥ 1 1 1 ...

1⎥ ⎪⎪ ⎢⎣ k   

⎦ ⎪ ⎭ n ×k

(5a)

and a n-dimension vector

b T = [ 1, 1, 1 .....1 , 1 ].

(5b)

For the decision variable vector X, the boundary conditions can be expressed as

ΔP f = P f − P*f = A e X ,

(6)

where ΔP f is a target vector, Ае is a m×(n×k)-dimension matrix of partial derivatives

⎡ ∂P1 ⎢ ∂V ( j ) ⎢ 1 A e = ⎢... ∂P ⎢ (sj ) ⎢⎣ ∂V1 where

∂Pq ∂Vi ( j )

∂P1 ∂V1( 2 ) ... ∂Ps ∂V1( 2 )

... ... ...

∂P1 ∂V2( j ) ... ∂Ps ∂V2( j )

∂P1 ∂V2( 2 ) ... ∂Ps ∂V2( 2 )

... ... ...

∂P1 ⎤ ∂Vn( k ) ⎥ ⎥ ... ∂Ps ⎥ ⎥ ∂Vn( k ) ⎥⎦

(7)

is a partial derivative which can be computed using analytical relations or numerically. T

Introduce a (n×k)-dimension vector of weight coefficients as q =[1 1…1 1] for the equal segments. Then, a performance index corresponding to the minimum characteristic velocity for the trajectory can be written as

J = min ( q T ⋅ X

)

(8) As the result, we have a classical linear programming problem with constraints of a linear inequality and equality given by Eqs. (4) and (6), respectively. The elements of the decision variable vector X must be nonnegative and constrained

0 ≤ ΔVi ( j ) ≤ 1 . 5 American Institute of Aeronautics and Astronautics

(9)

6 The segments in the linear programming form are formally considered independent of each other. Therefore, additional post-processing and validation are required for the linear programming solutions. It is necessary to find all of the segments corresponding to the non-zero decision variables. The adjacent segments among these should be joined in burns. More detailed description of the methods using discrete sets of pseudo-impulses is given in [6, 1314]. B. Mathematical Model for Rendezvous Trajectories The motion of active and passive spacecraft is considered in the neighborhood of a certain reference circular orbit of radius rω . The well-known equations of the relative motion [15-16] can be written in the following form:

where

ρ T = [r , n, b]

and

⎡ ρ ⎤ ⎡ 0 I ⎤ ⎡ ρ ⎤ ⎡0⎤ ⎢V ⎥=⎢ ⎥⎢ ⎥ + ⎢ ⎥ , ⎣  ⎦ ⎣ A B ⎦ ⎣ V ⎦ ⎣a ⎦ V T = r, n , b are the vectors of

[

]

(10) relative coordinates and velocities in the local

vertical/local horizontal coordinate system (Fig. 4), I is the identity matrix, 0 is zero matrix, and a is the thrust acceleration vector,

⎡3 0 0 ⎤ ⎡ 0 2 0⎤ A = ⎢0 0 0 ⎥, B = ⎢ − 2 0 0⎥. ⎢ ⎥ ⎢ ⎥ 0 0 1 0 0 0 − ⎥⎦ ⎣⎢ ⎣⎢ ⎦⎥

(11)

The analytical solution of Eq.(10) for a passive trajectory is known [16] and can be represented in the form:

⎡ ρ0 ⎤ ⎡Φ ρρ (t ) Φ ρV (t )⎤ ⎡ ρ0 ⎤ ⎡ ρ(t ) ⎤ t = P(t ) = ⎢ Φ ( ) ⎢ V ⎥ = ⎢Φ (t ) Φ (t ) ⎥ ⎢ V ⎥ , ⎥ t V ( ) VV ⎣ ⎦ ⎣ 0 ⎦ ⎣ Vρ ⎦⎣ 0 ⎦

(12)

where Ф, Фρρ, ФρV, ФVρ, and ФVV are the transition matrix and its sub-matrices, respectively. The expressions for these matrices are given in the Appendix.

Figure 4. Local vertical/local horizontal coordinate system.

6 American Institute of Aeronautics and Astronautics

7 Let us suppose that the sets of the segments and pseudo-impulses are defined. The number of boundary conditions for the rendezvous problem is m=6. The partial derivatives of the boundary conditions for an i-th segment and j-th pseudo-impulse can be written as

⎡ Δρ f ⎤ ⎡Ф ρV (t f − ti )⎤ ( j) ΔP i = ⎢ ΔVi = Q(t f − ti )e i( j ) ΔVi ( j ) , =⎢ ⎥ ⎥ ⎣ ΔV f ⎦ ⎣ФVV (t f − ti ) ⎦

ΔVi ( j )

(13)

e i( j ) is the unit vector of the pseudo-impulse. For the segments with a finite thrust the obvious expression is used ΔVi = ai ⋅ Δti , where аi is the thrust acceleration on whose maximum values the following constraint is imposed 0≤аi≤аimax. where

is the velocity pseudo-impulse,

C. Typical Boundary Conditions for Rendezvous Trajectories Rendezvous trajectories were considered for various types of an AS placed at an initial low near-circular orbit (altitude of ~250 km) approaching a space station at a near-circular orbit of ~500 km altitude according to a twoday rendezvous mission (in NR ~30 revolutions) [6]. The spacecraft orbits were assumed to be non-coplanar with a difference in inclination Δi = 0.35°. Such trajectories are close to mission profiles used in practice for rendezvous of spacecraft to the ISS, but with an enhanced difference of altitudes and inclinations. Without loss of generality, let T

T

the vector of the boundary conditions for the aim point be Рf =0 . The specified final time is

tf=NRT=(45h15m30s) and the target vector is ΔРfT=[250 km, 118 km − ϕ ⋅ rω , 38 km, 0.5 m/s, -433 m/s, -14.3 m/s], where ϕ is the phase angle. Radius of the reference orbit is rω = 6753 km. The rendezvous trajectories was discretized into 36 segments at each revolution with a uniform distribution in the argument of latitude, i.e. n=30х36=1080, and used a uniform distribution of the pseudo-impulses on a unit sphere with k=500 points. Therefore, the number of the decision variables is nxk=540,000.

IV. Optimal Rendezvous Trajectories with Continuous Thrust A. Existence of Optimal Solutions Suppose that all of the boundary conditions and final time are specified. An optimal rendezvous trajectory solution for impulsive thrust is always exists (perhaps, with the exception of a singular case). Such solution can be physically meaningless. As an example, the trajectory pass “through the Earth”, but in mathematical sense, the solution is exists. By contrast to the impulsive thrust case, optimal rendezvous trajectory solutions for continuous thrust with a thrust-to-weigth ratio are not necessarily. If a solution has been found, then other solutions for phase angles near to the phase angle of the solution can be found, etc. By this way, a set of optimal solutions can obtained. In the general case, the set must be contained two subsets. Similarly to the high-thrust case, the first subset is optimal phase solutions with a range of initial phase angles for which required characteristic velocity for the trajectories is almost equal and near to the optimal orbit transfer between initial and target orbits. Second, it is the solutions without of the range with more high characteristic velocities. B. Phasing for Rendezvous Trajectories with Continuous Thrust The optimal phase range depends substantially on the thrust-to-weight ratio. Optimal phase ranges for different values of the thrust acceleration are shown in Fig. 5. For comparison, the optimal phase range for optimal impulsive solutions is also depicted (dashed red line). It should be noted that there is a minimum thrust acceleration for that the phase range is degenerated to a point or a phase angle value. For an example, see below. 7 American Institute of Aeronautics and Astronautics

8

320

a=0.005 m/s

2

300

x

ΔV , m/s

280

a=0.002 m/s

2

260 a=0.01 m/s

240 a=0.0015 m/s

220 200 a=0.0012 m/s

2

a=0.001 m/s

2

2

2

Impulsive solution

180 a=0.0011 m/s 2 160 0

100

200

300

400

500

600

Phase angle, deg

Figure 5. Required ΔVx for different thrust accelerations. By contrast to the high-thrust case, the phasing orbit for a medium- or low-thrust optimal phase trajectory is a spinning spiral with two burns, usually, at each revolution (Fig. 6).

Intersection line between orbit planes

Burns

Figure 6. Spiral phasing trajectory.

V. Brief Qualitative Analysis of Optimal Trajectories with Continuous Thrust A. Possible Trajectory Types For optimal phase solutions, there are the following trajectory types (in a wide range of thrust acceleration values from high to low): 1. near to impulsive rendezvous trajectory (medium thrust); 2. two burns at an quantity of revolutions (medium and low thrust); 3. two burns at all of the revolutions (low thrust); 4. near to continuous multi-revolution burn (very low thrust). It should be noted that the bounds between the types are very relative. In a sense, it is a continuous set of the trajectories as a function of the thrust-to-weight ratio. An example of third mode solution (phase angle ~380°) is presented in Fig. 7, where burn distribution as sequences of adjacent segments for each revolution is presented. Each segment is depicted as a color-filled rectangle. The colors of the rectangles correspond to the required thrust pitch and yaw angles in compliance with the color axis scaling (the right side of each part of the figure). Each full colored segment is mean that it is the maximum acceleration or thrust (a little number of very small open space is related

8 American Institute of Aeronautics and Astronautics

9 with numerical solution tolerance). The exceptions are only the first and last segments in burns. Note that different color axes are used for the pitch and yaw angles. The thrust directions are near to the local horizontal plane and yaw angles to two values of ±20°. a) Pitch angle (a=0.002 m/s 2; Δ Vx=155.1 m/s, Phase angle= 380 deg) 90

Pitch angle, deg

Continuous burn

80

80 60 70 40

Δt

AN

, min

60 20 50 0 40

Segment

-20

30 -40 20 -60 10 -80 0 0

5

10

15

20

25

30

Revolution number

90

b) Yaw angle (a=0.002 m/s 2 ; Δ Vx=155.1 m/s, Phase angle= 380 deg)

Yaw angle, deg

150

Δt

AN

, min

80 70

100

60

50

50 0 40 -50

30 20

-100

10 0 0

-150 5

10

15

20

25

30

Revolution number Figure 7. Example of burn distribution and thrust direction for third type. There is a minimum thrust acceleration for that the solution of the rendezvous problem for the specified boundary conditions and final time exists. The solution is corresponded to the continuous thrust at the transfer time interval, i.e. it is one multi-revolution burn with continuous change of the thrust direction (Fig. 8). In the case of a thrust acceleration decreasing there is absence of the solutions, but some solutions can be exist for more higher values of the specified final time tf .

9 American Institute of Aeronautics and Astronautics

10

90

a) Pitch angle (a=0.00095 m/s 2; Δ Vx=160.8 m/s, Phase angle= 305 deg)

Pitch angle, deg

80 80 60 70 40

Δt

AN

, min

60 20 50 0 40 -20 30 -40 20 -60 10 -80 0 0

5

10

15

20

25

30

Revolution number

90

b) Yaw angle (a=0.00095 m/s 2; Δ Vx=160.8 m/s, Phase angle= 305 deg)

Yaw angle, deg

150

Δt

AN

, min

80 70

100

60

50

50 0 40 -50

30 20

-100

10 0 0

-150 5

10

15

20

25

30

Revolution number Figure 8. Solution of minimum thrust-to-weight ratio for optimal phase. It should be noted that for an inverse-square gravity field, the primer vector that defined optimal thrust direction [17] is a function of the eccentric anomaly with periodic and secular terms. Therefore the thrust direction rate in the burns must be of the same order as the orbital angular velocity, and a high-frequency chattering should be lacking. As an example, pitch and yaw angle histories at two revolutions for the previous minimum thrust acceleration trajectory are presented in Fig. 9. As shown this direction rate is the same order as the orbital angular velocity that is very compliance with qualitative properties for optimal space trajectories [13, 17].

10 American Institute of Aeronautics and Astronautics

11

50

Pitch and yaw angles, deg

40 30

Pitch angle

20

28-th revolution

10 0 -10

6-th revolution

-20 -30

Yaw angle

-40 -50 0

10

20

30

40

Δt

50

60

70

80

90

, min

AN

Figure 9. Examples of pitch and yaw angles histories for 6-th and 28-th revolutions. For the phase angles without the optimal range there are another trajectory types. An example is shown in Fig. 10. The phase angle is ~520°. In this case, we have continuous multi-revolution burns with thrust directions are also near to the local horizontal plane and final burns with negative transversal direction. In fact, it is two-burn rendezvous trajectory. 2

90

a) Pitch angle (a=0.002 m/s ; Δ Vx=212.4 m/s, Phase angle= 520 deg)

Pitch angle, deg

80 80 60 70 40

Δt

AN

, min

60 20 50 0 40 -20 30 -40 20 -60 10 -80 0 0

5

10

15

20

25

30

Revolution number 2

90

b) Yaw angle (a=0.002 m/s ; Δ Vx=212.4 m/s, Phase angle= 520 deg)

, min

AN

150

Multi-revolution continuous burn

80

Δt

Yaw angle, deg

70

100

60

50

50 0 40 Burns with negative transversal direction

30

-50

20

-100

10 0 0

-150 5

10

15

20

25

30

Revolution number

Figure 10. Example of burn distribution for phase angle without optimal range.

11 American Institute of Aeronautics and Astronautics

12 B. Interrelation Between Phasing and Thrust-to-Weigth Ratio A schematic diagram for dependence of optimal solutions vs the phase angle and thrust acceleration is presented in Fig. 11. Since for the trajectories, the pitch angles are near to zero, only the yaw angle distributions are depicted. The solutions without optimal phase range corresponded to spiral phasing orbits without the ring between the initial and target orbits (see Fig. 1). 2

Acceleration, m/sec ( or thrust-to-weight ratio) 0.005

50 0

50

Δ V x= 153.83

0

10 Δ V = 158.54

20

30

0

0

x

0.002

50 0

0

10

20

30

30

0

10

0

20

30

0

10

20

30

Δ V x=157.58

50

0

10 20 Δ V = 155.71

30

0

0

10 20 Δ V = 205.68

30

10

30

x

50

0

0

Δ V x= 153.74

x

50 0

10 20 Δ V x= 154.53

50

Δ V x= 222.26

0.0015

50

Δ V x= 153.27

50

0

10

20

30

0

0

20

300 deg 450 deg 150 deg Figure 11. Changes of solutions for different phase angles and thrust-to-weight ratios(yaw angles distributions).

Phase angle

An example of a continuous solution changes for the range of possible phase angles is shown in Fig. 12. The solutions with the limit phase angles are corresponded to the continuous thrust for the transfer time interval, i.e. it is one multi-revolution burn but with one switching of the yaw angle.

Figure 12. Changes of solutions in range of possible phase angles. 12 American Institute of Aeronautics and Astronautics

13

As shown from the results, there is a mean phase angle that corresponds to solutions with almost equal pair burns at each revolution and the minimum thrust acceleration solution. In a sense, the solutions for phasing angles with equal displacements from the mean phase angle (on the right and left) are mirror images of each other. C. Qualitative Aspects of the Rendezvous Trajectories In closing it may be said that as is well-known from primer vector theory [17], nonsingular optimal space trajectories must be formed by intervals of maximum thrust and coast arcs separated by a finite number of switches, and the optimal thrust direction is always aligned with the primer vector. By this is meant that the characteristic velocity for the segments of a burn must correspond to maximum value of

ΔVi max = ai max ⋅ Δti . The exceptions

are only the first and last segments in the burn. A non-compliance with these qualitative properties for a rendezvous trajectory may indicate that it is a singular solution. In spite of independent consideration of the segments in the linear programming form, all of the presented optimal solutions give excellent agreement with this property of optimal space trajectories. And lastly, to our knowledge, for general continuous thrust orbit transfers including rendezvous trajectories, determination of the optimal number of burns is a very difficult and unsolved problem. In the examined analysis the optimal number of the burns is automatically determined in the post-processing of the linear programming solutions. These considerations also apply to borderline cases (as an example – two burns at a revolution joined a continuous burn).

VI. Conclusion In this paper an optimal rendezvous trajectory analysis related with design of low-thrust propulsion system are presented. Qualitative aspects of optimal near-coplanar, near-circular rendezvous trajectories to space stations at an interface between the phasing strategy and thrust-to-weight ratio are considered. Existence of optimal solutions and their possible types are briefly discussed. For continuous medium- and low-thrust thrust spiral trajectories, as for the case of high-thrust, there also is a range of initial phase angles for which required characteristic velocity for the trajectories is almost equal and near to the optimal orbit transfer between the initial and target orbits. This feature is truth up to a minimum possible thrust acceleration for specified boundary conditions and final time. The continuous thrust rendezvous trajectories with the phasing angles without the ranges are needs phasing orbits without the ring between the initial and target orbits. The presented results may also be used as a part of the spacecraft design process for determination of a required spacecraft thrust-to-weight ratio for rendezvous missions.

Appendix Sub-matrices of the transition matrix from Eq.(12) can be represented in the form:

Φ ρρ

0 ⎤ ⎡ 2 − 3 cos ωt 0 0 ⎥ = ⎢6(sin ωt − ωt ) 1 ⎥ ⎢ 0 0 cos ωt ⎥⎦ ⎢⎣

Φ ρV

sin ωt ⎡ ⎢ ωt ⎢ 2 = ⎢ − (1 − cos ωt ) ⎢ ω ⎢ 0 ⎢⎣

,

2(1 − cos ωt ) ωt 2 sin ωt − 3ωt

ω 0

(A1)

⎤ 0 ⎥ ⎥ 0 ⎥ ⎥ sin ωt ⎥ ω ⎥⎦

13 American Institute of Aeronautics and Astronautics

,

(A2)

14

where

ω=

μ

ΦVρ

0 0 ⎤ ⎡ 3ω sin ωt 0 ⎥ = ⎢6ω (1 − cos ωt ) 1 ⎥ ⎢ 0 0 sin ω − t ⎦⎥ ⎣⎢

ΦVV

2 sin ωt 0 ⎤ ⎡ cos ωt 0 ⎥ = ⎢ − 2 cos ωt 4 cos ωt − 3 ⎥ ⎢ 0 0 cos ωt ⎥⎦ ⎢⎣

rω3

,

(A3)

,

(A4)

is the angular velocity of orbital motion for the reference orbit, μ is the Earth’s gravitational

parameter.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Edelbaum, T., “Optimum Low-Thrust Rendezvous and Station-Keeping,” AIAA Journal, Vol. 2, No. 7, 1964, pp. 1196-1201. Coverstone-Carroll, V, and Prussing, J., "Optimal Cooperative Power-Limited Rendezvous Between Neighboring Circular Orbits," Journal of Guidance, Control, and Dynamics, Vol. 16, No. 6, 1993, pp. 1045-1054. Carter, T. E., "Optimal Power-Limited Rendezvous for Linearized Equations of Motion," Journal of Guidance, Control, and Dynamics, Vol. 17, No. 5, 1994, pp. 1082-1086. Kechichian, J.A., “Minimum-Time Low-Thrust Rendezvous and Transfer Using Epoch Mean Longitude Formulation,” Journal of Guidance, Control, and Dynamics, Vol.22, No. 3, 1999 , pp.421-432. Guelman, M., and Aleshin M., “Optimal Bounded Low-Thrust Rendezvous with Fixed Terminal-Approach Direction,” Journal Of Guidance, Control, And Dynamics, Vol. 24, No. 2, 2001, pp. 378-385. Ulybyshev, Y., “Optimization of Multi-Mode Rendezvous Trajectories with Constraints,” Cosmic Research, Vol. 46, No. 2, 2008, pp. 133-147. Wall, B. J., and Conway B.A., “Shape-Based Approach to Low-Thrust Rendezvous Trajectory Design,” Journal Of Guidance, Control, And Dynamics, Vol. 32, No. 1, 2009, pp. 95-101. Navigatsionnoe obespechenie poleta orbital’nogo kompleksa “Salyut-6”–“Soyuz”–“Progress” (Navigation Support for a Flight of the Salyut-6–Soyuz–Progress Orbital Complex), Eds. Petrov, B.N. and Bazhinov, I.K., Moscow, Nauka, 1985 (in Russian). Baranov, A. A., “An Algorithm for Calculating Parameters of Multi-Orbit Maneuvers in Remote Guidance,” Cosmic Research, Vol. 28, No. 1, 1990, pp. 61–67 (translation of Kosmicheskie Issledovaniya). Fehse, W., Automated Rendezvous and Docking of Spacecraft, Cambridge Univ. Press, London, 2003, Chap. 2. Luo, Y-.Z, Lei, Y-.J., and Li, H-.Y., “Optimization of Multiple-Impulse, Multiple-Revolution, RendezvousPhasing Maneuvers,” Journal Of Guidance, Control, And Dynamics, Vol. 30, No. 4, 2007, pp. 946-952. Murtazin, R. and Budylov, S., “Short rendezvous missions for advanced Russian human spacecraft,” Acta Astronautica, 2010 (In Press, doi:10.1016/j.actaastro.2010.05.012 ) or 60th International Astronautical Congress 2009, IAC 2009, Volume 4, 2009, pp.3238-3246. Ulybyshev, Y., “Continuous Thrust Orbit Transfer Optimization Using Large-Scale Linear Programming,” Journal of Guidance, Control, and Dynamics, Vol. 30, No. 2, 2007, pp.427-436. Ulybyshev, Y., “Spacecraft Trajectory Optimization Based on Discrete Sets of Pseudo-Impulses,” Journal of Guidance, Control, and Dynamics, Vol. 32, No. 4, 2009, pp.1209-1217 (see also AIAA Paper 20086276). Clohessy, W.H., and Wiltshire, R.S., "Terminal Guidance Systems for Satellite Rendezvous," Journal of the Aerospace Sciences, Vol. 27, Sept. 1960, pp. 653-658, 674. Appazov, R.F. and Sytin, O.G., Metody proektirovaniya traektorii nositelei i sputnikov Zemli (Methods of Designing Trajectories of Launch Vehicles and Satellites of the Earth), Moscow, Nauka, 1987, Chap. 5 (in Russian). Lawden, D.F., Optimal Trajectories for Space Navigation, Batterworths, London, 1963, Chap. 5. 14 American Institute of Aeronautics and Astronautics