 Journal of Robotics, Networking and Artificial Life

Volume 5, Issue 1, June 2018, Pages 1 - 5

Optimal Hohmann-Type Impulsive Ellipse-to-Ellipse Coplanar Rendezvous

Authors
Xiwen Tiantianxiwen123@163.com, Yingmin Jiaymjia@buaa.edu.cn
The Seventh Research Division and the Center for Information and Control, School of Automation Science and Electrical Engineering, Beihang University (BUAA), 37 Xueyuan Road, Haidian District, Beijing, 100191, China
Available Online 30 June 2018.
DOI
https://doi.org/10.2991/jrnal.2018.5.1.1How to use a DOI?
Keywords
Optimal impulsive rendezvous; Hohmann-type rendezvous; ellipse-to-ellipse; optimal distribution
Abstract

This paper devotes to the problem of ellipse-to-ellipse coplanar rendezvous, where the solution and distribution of Hohmann-type optimal impulsive rendezvous are investigated. The analytical relation between the initial states and rendezvous 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 with coasts. Simulations are demonstrated to analyze the influences of the solution and distribution.

Open Access
This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

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 some guidelines to solve the optimal problem where the initial states and rendezvous time are specified, the distributions of optimal models cannot be obtained clearly in these way. So far, only Prussing’s theory3 of optimal impulsive rendezvous on close circular orbits is complete in its theoretical system, which derives the solutions 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 in describing the impulse locations and magnitudes than the mean radius orbit in Prussing’s results. Xie5 focused on the selection of reference frame for optimal impulsive rendezvous, and investigated the effect on the classification, distribution and guidance precision. For the case of elliptic orbit rendezvous, Wang6 used the state transition matrix given by Yamanaka7 to calculate the optimal solution of four-impulse model, but the analytical solution and the distribution are difficult to be achieved. Chen8,9 studied the ellipse-to-circle coplanar rendezvous based on his results on the dynamical equations for elliptic orbit rendezvous in low eccentricity, and provided the solutions and distributions of all types optimal models. Motivated by which, our previous work10 considered the ellipse-to-ellipse coplanar rendezvous and obtained the analytical solution and distribution of four-impulse model. In this paper, we will further investigate the Hohmann-type model for optimal impulsive ellipse-to-ellipse coplanar rendezvous.

2. Dynamics Description

The relative motion between two spacecrafts in elliptic orbits was derived in our previous work10, which is still used in this paper and given as follow:

{δr¨=3δr+2δθδθ¨=2δr˙

The initial and terminal states of system (1) are

x0=[x01,x02,x03,x04]TxF=[0,0,0,0]T
where
x01=kc(1+eccosfc)1kt(1+etcosft)1x02=βx03=kc12ecsinfckt12etsinftx04=kc32(1+eccosfc)2kt32(1+etcosft)2
β is the difference of phase angle between two spacecrafts; ec, et and fc, ft are their eccentricities and true anomalies, respectively.

The states at phase angle τ was also deduced:

x(τ)=[2d4d3cos(τ+φ)β3d4τ+2d3sin(τ+φ)2d3sinφd3sin(τ+φ)3d4+2d3cos(τ+φ)]
where
d1=x03,d2=3x01+2x04d3=d12+d22,d4=2x01+x04,φ=arcsin(d1d3)

3. Optimal Hohmann-Type Rendezvous

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

λ1=Acosτ+Bsinτ+2Cλ2=2Bcosτ2Asinτ3Cτ+D
Hohmann-type model is a special case of optimal two-impulse rendezvous, where the coefficients of (6) are
A=B=C=0,D=±1
then λ1 = 0, |λ2| = 1. It can be verified that the necessary conditions of optimal impulsive rendezvous are satisfied for any phase angle τ.

3.1. Solution of Hohmann transfer

The impulse direction can be obtained from the solution (6), while the impulse time and magnitudes needed be calculated according to the following boundary value problem:

[2(1Cτ)04Sτ3τF02Sτ03+4Cτ1][ΔV1ΔV2]=x(τF)
where Sτ = sin τ, Cτ = cos τ, ΔV1 and ΔV2 are the magnitudes of two impulse, and τF is the rendezvous time. From (4) and (8), we have
ΔV1=d3sin(τF+φ)2SτΔV1=2d4+d3cos(τF+φ)2(1Cτ)
then
d3(sin(τF+φ)sinφ)=2d4sinτF
Substituting (5) into (10), it can be obtained that
x012+d12sin(τF+θ)=d1
where
cosθ=x01x012+d12,sinθ=dx012+d12
On the other hand, from the second row of (8), it has
ΔV1=β+3d4τ2d3sin(τF+φ)+2d3sinφ(4Sτ3τF)
Combining (5), (9) and (13), it has
3τF(x01Cτ+d1Sτ)2(1Cτ)+2d1=fcft
The appropriate initial states and rendezvous time which satisfy the necessary conditions of Hohmann transfer can be obtained by solving (11) and (14) together, and then, the second impulse ΔV2 can be obtained as
ΔV2=d4+d3sin(τF+φ)2Sτ

3.2. Distribution of Hohmann-type model

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:

δθF=β+1.5d4τFd3sin(τF+φ)+d3sinφ

Let τFh be rendezvous time solved by (11) and (14), and δθFh is the corresponding special phase angle. If τF = τFh and δθF = δθFh, then it is just the Hohmann transfer. The two impulses are implemented at τ1 = 0 and τ2 = τF. However, when the real rendezvous time is longer than τFh, the coasts are needed to save the fuel.

If τF > τFh and δθF = δθFh, it is a Hohmann model with terminal coast. The two impulses are implemented at τ1 = 0 and τ2 = τFh, and the residual time τFτFh is for terminal coast. The special phase angle δθF and rendezvous time τF should satisfy the following relation

δθF=δθFh+1.5d4(τFτFh)+d3sin(τFh+φ)d3sin(τF+φ)τFh=τ2τ1,τF[τFh,+)

If τF > τFh and the special phase angle δθF satisfies

δθF=δθFh1.5d4(τFτFh)+d3sin(τFτFh+φ)d3sinφτFh=τ2τ1,τF[τFh,+)
then, after the initial coast for time τFτFh, the special phase angle will become exactly δθFh. This case is a Hohmann model with initial coast, and the impulses are implemented at τ1 = τFτFh and τ2 = τF.

Let

δθFh1=δθFh+1.5d4(τFτFh)+d3sin(τFh+φ)d3sin(τF+φ)δθFh2=δθFh1.5d4(τFτFh)+d3sin(τFτFh+φ)d3sinφ
If τF > τFh and δθFh1 < δθF < δθFh2, then there exists a Hohmann model with both initial coast and terminal coast. As shown in Fig. 1., this case is illustrated in the middle of the curves expressed by (17) and (18), that is the shadow part. Denote (τF0, δθF0) as the intersection point of the curves determined by (16) and (18), then the two impulses are implemented at τ1 = τF0τFh and τ2 = τF0. The initial and terminal coast last for time τF0τFh and τFτF0, respectively.

From the above, the optimal Hohmann-type impulsive rendezvous has four models, all of whose impulse magnitudes are determined by (9) and (15), and impulse direction is along the tangential direction.

4. Simulations

In this section, simulation examples are presented to show the guidance performance and distribution of Hohamman-type impulsive rendezvous.

4.1. Hohmann ellipse-to-ellipse rendezvous

It is assumed the semi-major axis and eccentricities of the target orbit and chaser orbit are initially at = 6730 (km), ac = 6750 (km), et = 0.0005 and ec = 0.0004, respectively. Let τF and β (rad) be the appropriate rendezvous time and initial difference of phase angle, respectively, which satisfy (11) and (14). And denote Rr(m) as the optimal radius of reference frame, R(m) as the initial relative distance, Δa(m), Δe and Δθ(rad) as the guidance errors.

Simulation results of Hohmann transfer for ellipse-to-ellipse rendezvous are demonstrated in Table 1, which shows that: (1) with different true anomalies, even if the other initial states are the same, the 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 true anomaly; (3) the guidance precision is high when the chaser initially stays around the perigee.

1 2 3 4
fc 30° 150° 210° 330°
τF 3.62 3.44 2.80 2.68
β 7.12e-03 9.68e-03 6.04e-03 3.46e-03
Rr 6.70e+06 7.92e+06 7.81e+06 6.71e+06
R 4.99e+04 7.05e+04 4.83e+04 2.71e+04
Δa 6.80e+01 3.38e+04 3.09e+04 2.49e+01
Δe 3.38e-05 4.51e-03 4.22e-03 8.92e-06
Δθ 5.82e-06 2.87e-04 3.50e-05 3.80e-06
ΔR 1.29e+02 5.14e+03 5.71e+02 5.08e+01
Table 1.

Results of Hohmann impulsive rendezvous

4.2. Distribution of Hohmann-type model

To investigate the distribution of Hohmann-type ellipse-to-ellipse rendezvous, we take rendezvous time τF as the X-coordinate and δθF/d4 as Y-coordinate. Fig. 2 shows the distribution of Hohmann-type model with different true anomalies and eccentricities.

5. Conclusion

This paper extends our previous work10 to the Hohmann-type optimal impulsive rendezvous. By defining the special phase angle, we derived the analytical solution for Hohmann transfer, and obtained that the optimal Hohmann-type impulsive rendezvous has four models, i.e. Hohmann transfer, Hohmann with initial coast, Hohmann with terminal coast and Hohmann with both coasts. In further research, we will integrate all optimal models in one map, including four-impulse, three-impulse, three-impulse with coasts, two- impulse, 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)