Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 1 (June 2018) 1–5___________________________________________________________________________________________________________Optimal Hohmann-Type Impulsive Ellipse-to-Ellipse Coplanar Rendezvous
Xiwen Tian, Yingmin Jia

The Seventh Research Division and the Center for Information and Control, School of Automation Science andElectrical Engineering, Beihang University (BUAA), 37 Xueyuan Road, Haidian District
Beijing, 100191, China
E-mail: tianxiwen123@163.com; ymjia@buaa.edu.cn
www.buaa.edu.cn

Abstract
This paper devotes to the problem of ellipse-to-ellipse coplanar rendezvous, where the solution and distribution ofHohmann-type optimal impulsive rendezvous are investigated. The analytical relation between the initial states andrendezvous time are derived for Hohmann-type, and the optimal impulse amplitudes are given thereupon. The
distribution boundary of Hohmann-type model is obtained according to the Hohmann transfer and Hohmann withcoasts. Simulations are demonstrated to analyze the influences of the solution and distribution.
Keywords: Optimal impulsive rendezvous, Hohmann-type rendezvous, ellipse-to-ellipse, optimal distribution.

classification, distribution and guidance precision. For
the case of elliptic orbit rendezvous, Wang6 used thestate transition matrix given by Yamanaka7 to calculate
the optimal solution of four-impulse model, but theanalytical solution and the distribution are difficult to beachieved. Chen8,9 studied the ellipse-to-circle coplanar
rendezvous based on his results on the dynamicalequations for elliptic orbit rendezvous in low
eccentricity, and provided the solutions anddistributions of all types optimal models. Motivated bywhich, our previous work10 considered the ellipse-toellipse coplanar rendezvous and obtained the analyticalsolution and distribution of four-impulse model. In thispaper, we will further investigate the Hohmann-typemodel for optimal impulsive ellipse-to-ellipse coplanar
rendezvous.

1. Introduction
Optimal impulsive rendezvous is aimed at obtaining
minimum-fuel guidance strategy for spacecraft
rendezvous, which has attracted considerable attention.
Despite that Lawden’s necessary conditions1 for optimal
impulsive trajectories and Lion’s improving methods2
for non-optimal trajectories have provided someguidelines to solve the optimal problem where the initialstates and rendezvous time are specified, thedistributions of optimal models cannot be obtained
clearly in these way. So far, only Prussing’s theory3 of
optimal impulsive rendezvous on close circular orbits iscomplete in its theoretical system, which derives thesolutions and distributions of optimal impulsive models
by solving the primer vector equations and boundary
value problem. A reference frame in mean velocity orbit
was built by Frank4, and showed better performance indescribing the impulse locations and magnitudes thanthe mean radius orbit in Prussing’s results. Xie5 focusedon the selection of reference frame for optimal
impulsive rendezvous, and investigated the effect on the2. Dynamics Description

Copyright © 2018, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
1

Journal of Robotics, Networking and Artificial Life, Vol. 5 No. 1 (June 2018) 1–5___________________________________________________________________________________________________________The relative motion between two spacecrafts in elliptic
orbits was derived in our previous work10, which is stillused in this paper and given as follow:r 3 r 2(1)2 r

3.1. Solution of Hohmann transfer
The impulse direction can be obtained from the solution(6), while the impulse time and magnitudes needed becalculated according to the following boundary value
problem:
2 1 C0V1
4S 3 F 0(8)
x F
V2
2S0

The initial and terminal states of system (1) arex0

x01 , x02 , x03 , x04

xF

0, 0, 0, 0

where

1kc 1 ec cos f cx01

T(2)

T

3 4C

kt 1 et cos ft

1kc 2 ec sin f cx03x04

kc

32(3)

1kt 2 et sin ft

1 ec cos f c

232kt

where,
magnitudes of two impulse, andtime. From (4) and (8), we haved3 sin F
V12S

1x02

1 et cos ft

2x2d3 sind3 sinx03 , d 2d3d12

d 22 , d 4

2 x01

x04 ,

arcsin

d1d3

B sin

22 B cos

2 A sin

2C3C

D

2d4 sin F

(10)
(11)

d1

Fx01
201x

2
1dd1,sin201x

2
1d(12)On the other hand, from the second row of (8), it has
3d4 2d3 sin F
2d3 sin(13)
V14S 3 F(5)Combining (5), (9) and (13), it has

3 F x01C d1S
21 C

2d1

fcft

(14)

The appropriate initial states and rendezvous time which
satisfy the necessary conditions of Hohmann transfer
can be obtained by solving (11) and (14) together, andthen, the second impulse
can be obtained as

The solution to primer vector equations corresponding
to system (1) can be given in the following form:

A cos

F

where

3. Optimal Hohmann-Type Rendezvous

1sin

F2x01
d12 sincos

3 x01 2 x04(9)

Substituting (5) into (10), it can be obtained that

whered1

d 3 cos
2 1 C

andare theis the rendezvousthen(4)d3 sin3d 4 2d3 cos

2d 4V1is the difference of phase angle between twoand
are their eccentricities andspacecrafts;
true anomalies, respectively.
The states at phase angle was also deduced[10]:2d 4 d3 cos
3d 4 2d3 sin

1(6)

V2

Hohmann-type model is a special case of optimal twoimpulse rendezvous, where the coefficients of (6) are(7)A B C 0, D1d4

d3 sin F
2S(15)

3.2. Distribution of Hohmann-type model

then. It can be verified that thenecessary conditions of optimal impulsive rendezvous
are satisfied for any phase angle .

The distribution of optimal Hohmann-type rendezvous
is to illustrate the existence of feasible solution. To
investigate the distribution, rendezvous time is chosen
as the X-coordinate and the special phase angle defined
below as Y-coordinate:2

Journal of Robotics, Networking and Artificial Life, Vol. 5 No. 1 (June 2018) 1–5___________________________________________________________________________________________________________1.5d4 F d3 sinFd3 sinF(16)be rendezvous time solved by (11) and (14),is the corresponding special phase angle. Ifand, then it is just the Hohmanntransfer. The two impulses are implemented atand. However, when the real rendezvous time islonger than, the coasts are needed to save the fuel.
Ifand, it is a Hohmann modelwith terminal coast. The two impulses are implemented
atand, and the residual time
is for terminal coast. The special phase angle
andrendezvous time should satisfy the following relation
andLetFh

1.5d 4d3 sinFhFFhIfsatisfies

21, FFFig 1 Distribution of Hohmann-type model

Fhd3 sin
FhF

4. Simulations(17)

In this section, simulation examples are presented toshow the guidance performance and distribution ofHohamman-type impulsive rendezvous.

,and the special phase angle
Fh1.5d 4d3 sinFFh, F

FhF

Fh21F

4.1. Hohmann ellipse-to-ellipse rendezvous

Fhd3 sin(18)

It is assumed the semi-major axis and eccentricities of
the target orbit and chaser orbit are initially(km),(km),and, respectively. Letand(rad) be theappropriate rendezvous time and initial difference ofphase angle, respectively, which satisfy (11) and (14).
And denote (m) as the optimal radius of reference
frame, (m) as the initial relative distance, (m),and (rad) as the guidance errors.

,then, after the initial coast for time
, the special
phase angle will become exactly. This case is aHohmann model with initial coast, and the impulses areimplemented atand.
Let
1.5d 4 F Fh
Fh1
Fh

Fh 2

d3 sinFhFh

1.5d 4d3 sin

Fd3 sin
FFhF(19)

Table 1. Results of Hohmann impulsive rendezvous

Fhd3 sin

Ifand, then there exists
a Hohmann model with both initial coast and terminal
coast. As shown in Fig.1., this case is illustrated in themiddle of the curves expressed by (17) and (18), that is
the shadow part. Denoteas the intersection
point of the curves determined by (16) and (18), then
the two impulses are implemented atand. The initial and terminal coast last for timeand, respectively.
From the above, the optimal Hohmann-type
impulsive rendezvous has four models, all of whose
impulse magnitudes are determined by (9) and (15), andimpulse direction is along the tangential direction.

12

34

3.62
7.12e-03
6.70e+06
4.99e+04
6.80e+01
3.38e-05
5.82e-06
1.29e+02

3.44
9.68e-03
7.92e+06
7.05e+04
3.38e+04
4.51e-03
2.87e-04
5.14e+03

2.80
6.04e-03
7.81e+06
4.83e+04
3.09e+04
4.22e-03
3.50e-05
5.71e+02

2.68
3.46e-03
6.71e+06
2.71e+04
2.49e+01
8.92e-06
3.80e-06
5.08e+01

Simulation results of Hohmann transfer for ellipseto-ellipse rendezvous are demonstrated in Table 1,
which shows that: (1) with different true anomalies,
even if the other initial states are the same, the3

Journal of Robotics, Networking and Artificial Life, Vol. 5 No. 1 (June 2018) 1–5___________________________________________________________________________________________________________rendezvous time and initial difference of phase angle
which satisfy (11) and (14) varies much; (2) the optimal
radius of reference frame also changes with the trueanomaly; (3) the guidance precision is high when thechaser initially stays around the perigee.
4.2. Distribution of Hohmann-type model
To investigate the distribution of Hohmann-type ellipseto-ellipse rendezvous, we take rendezvous timeasthe X-coordinate andas Y-coordinate. Fig.2
shows the distribution of Hohmann-type model withdifferent true anomalies and eccentricities.

(d)

(a)

(f)

Fig 2 Distributions with different true anomalies andeccentricities
5. Conclusion
This paper extends our previous work10 to theHohmann-type optimal impulsive rendezvous. By
defining the special phase angle, we derived theanalytical solution for Hohmann transfer, and obtainedthat the optimal Hohmann-type impulsive rendezvous
has four models, i.e. Hohmann transfer, Hohmann withinitial coast, Hohmann with terminal coast andHohmann with both coasts. In further research, we will
integrate all optimal models in one map, including four-(c)4Journal of Robotics, Networking and Artificial Life, Vol. 5 No. 1 (June 2018) 1–5___________________________________________________________________________________________________________

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J. Guid. Contr. Dynam., 25(1) (2002) 60-66.
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impulse, three-impulse, three-impulse with coasts, twoimpulse, two-impulse with coasts, and Hohmann-type.
Acknowledgements
This work was supported by NSFC (61327807,61521091, 61520106010, 61134005) and the National
Basic Research Program of China (973 Program:2012CB821200, 2012CB821201)
References
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2. P.M. Lion and M. Handelsman, Primer vector on fixedtime impulsive trajectories, AIAA J., 6(1) (1968) 127-132.
3. J. E. Prussing, Optimal Multi-Impulse Orbital rendezvous,
Sc.D. Thesis. (Cambridge: Dept. of Aeronautics andAstronautics, Massachusetts Institute of Technology,1967).

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