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ScienceDirect Procedia Engineering 185 (2017) 396 – 403

6th Russian-German Conference on Electric Propulsion and Their Application

Special refill spacecraft debris collector, equipped with electro rocket engine of low-thrust, design parameters optimization

Sergey A. Ishkova, Gregory A. Filippova* a

Samara University, 34, Moskovskoye shosse, Samara, 443086, Russia

Abstract The problem of near Earth space debris is studied in this article. A special spacecraft debris collector, equipped with electro rocket engine of low-thrust, for large space debris disposal is introduced. The mass model of spacecraft debris collector, one-off or reusable, is obtained. Ballistic scheme, spacecraft debris collector orbital transfer from parking orbit to space debris orbit, its disposal in Earth atmosphere and reusable spacecraft return to the parking orbit are studied. For the introduced criteria of transport transfer efficiency and assumption about consistency of acceleration from thrust, analytical relations for spacecraft design parameters calculation are obtained. The design parameters calculation results are shown in a generalized form. © 2017 2017The TheAuthors. Authors.Published Published Elsevier B.V. © byby Elsevier Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility of the organizing committee of RGCEP – 2016. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 6th Russian-German Conference on Electric Propulsion and Their Application Keywords: space debris; special spacecraft debris collector; criteria function; characteristic velocity gains; design parameters; total resource

1. Introduction Fragments of space debris (FSD) – are all man-made objects and their parts in space, which are uncontrollable, and cannot be used for any purposes, but pose a danger for active spacecraft. FSD can be divided into categories. First, FSD of diameter less than 1 cm. Second, FSD with a diameter of 1 – 10 cm and, third, FSG diameter with a diameter greater than 10 cm. Many of this objects are observable and information about its orbit storage in catalogs. Orbits of FSD can be classified. First, low Earth orbits of 400 – 1000 km altitude. Second, FSD at geostationary orbit and, third, FSD at altitudes about 1000 – 5000 km [1]. Methods of space debris disposal can be divided to two categories: passive (preemptive) and active methods. Active methods propose special spacecraft (SDC), which transfer FSD to low disposal orbit (LDO), that perigee height is about 200 km, and from that FSD enter to Earth atmosphere. An especially important problem for SDC is disposal FSD near the International Space Station – the most expensive object in space. The most dangerous is large FSD – uncontrolled spacecraft, booster stages and other. These objects cannot completely burn in atmosphere and, during landing, can damage Earth objects. Furthermore, point of landing forecast, for these objects, during its self-locking in Earth – poses a difficult technical problem. The FSD dissent and landing in predetermined area of Earth surface is an important and current problem, and SDC can solve this problem. * Corresponding author. Tel.: +7-960-814-6648. E-mail address: [email protected]

1877-7058 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 6th Russian-German Conference on Electric Propulsion and Their Application

doi:10.1016/j.proeng.2017.03.321

397

Sergey A. Ishkov and Gregory A. Filippov / Procedia Engineering 185 (2017) 396 – 403

SDC equipped with electro rocket engines (ERE) of low-thrust are justified [2] as they are characterized by high specific impulse and allow low mass spacecraft with long operational time to be created. ERE can be used as sustainer engines and also as engines in orientation and docking systems. Nowadays, there are two types of SDC. The first type are SDC of one time in use and and the second one is a reusable refill SDC. The ballistic schemes of SDC operations are introduced next. SDC stay at a parking orbit with an altitude of 400 km. At the first stage, other services determine the FSD, that will dispose, and transfer data to SDC. The motion control system of SDC calculates the control program and SDC makes orbital transfer to the FSD, makes rendezvous transfer and fixed FSD on-board. Next, if possible, the SDC moves to another FSD and fixes it it on-board. At the second stage, FSD makes orbital transfer to LDO and stays there, until sufficient conditions for FSD disposal to a predetermined area of Earth surface. Next, SDC makes disposal of FSD, it enters into the Earth’s atmosphere and FSD burns in atmosphere or lands in predetermined area. Reusable SDC, after unfixed FSD, at a third stage, moves to a parking orbit, refills as necessary, and is ready for the next operation. For reusable FSD, this cycle repeats. Modern methods of spacecraft design assume iterations, during design and ballistic parameters optimization. In this article, we consider “zero” iteration, to determine area of design and ballistic parameters, based on simple mass models. 2. Mass model of one-time use SDC Let us study mass model of a one-time use SDC: M0

ME MT MC,

(1)

where M 0 initial (start) SDC mass, M E SDС mass, determined by effective power of power plant, M T SDC mass, determined by active operational time (working fluid mass), M C fixed and constant mass of SDC (mass of constructions, navigation and control systems, FSD docking and fixed system and other constant masses). Mass components M E and M T are written in the in the formula:

ME MT

PC , 2 §P · t ¨ mC ¸, ©C ¹ J

(2)

where P – total sustainer engines thrust, C – exhaust velocity of ERE, t – total ERE operation time, mC working fluid flow rate (in one second) for SDC orientation supporting, J specific (dimension less) mass of power plant, taking into account mass of solar panel and its degradation with time. The next assumptions are introduced : 1. ERE exhaust velocity C and thrust P stay constant during all SDC resource. 2. Acceleration from thrust, during every stage of SDC operation, is constant and calculated for average SDC mass. Total operation duration is written in the next formula:

t t1t 2

Vx1 Vx2 , w1 w2

(3)

where V x1 and V x2 gains of characteristic velocity at first and second stage, w 1 and w 2 average acceleration from thrust, which, in accordance with assumptions, are determined as:

398


w0 , k1

P

w1

M1 M0 T 2

(4)

P

w1

M 0 M T1

M T2 2

M FSD

w0 , k2

(5)

where M T1 and M T2 working fluid consumption at stages, M FSD FSD mass, w0 initial acceleration from thrust, k1 and k 2 correction coefficients, determined by the following formulas:

M T1 , 2 M0

k1

1

k2

M M1 M T2 1 T FSD . M0 2 M0 M0

(6)

The design parameters problem optimization: for determine values of M 0 , M FSD , M C , V x1 , Vx2 and J , obtain parameters C and P from criteria of minimal operation duration:

Сopt , Popt

arg min t .

(7)

In accordance with iteration process of design parameters optimization, we assume, that during problem (7) solving, correction coefficients k1 and k 2 and gains of working fluid for SDC orientation supporting t mC does not depend on design parameters C and P. White balance mass and time equation: J

M0

T

PC t P MC, 2 C

Vx1 k1 Vx2 k 2 w0

Vx* . w0

(8)

(9)

In eq. (8), gains of working fluid for SDC orientation supporting t mC included in FSD constant mass

M C , and Vx* Vx1 k1 Vx2 k2 specific characteristic velocity gains during operation. After dividing left and right side of eq. (8) to M 0 , we obtain w0 : 1 Pc , C t J 2 C

w0

where P c

(10)

MC . M0

(11)

Substitute (10) in (9), express explicitly time t and, after transformation, obtain:

t

J

C 2 Vx*

2 C 1 Pc Vx*

.

(12)

399


Determine derivative from t by C, and equating it to zero, we obtain solution of problem (7):

Vx* . 1 Pc By substituting solution (12) in (9) and (10), we obtain: Copt

2

(13)

0 wopt

1 Pc 2 ,

(14)

2 J Vx*

J V x*

t min

2

Popt

M0

2

1 P c 2

(15)

,

1 P c 2 .

(16)

2 J Vx*

3. Mass model of a refillable SDC Let us study the mass model of reusable SDC. We assume, that operation, during one cycle, carried out during time t M (motor time), and we neglect for waiting time and duration of SDC passive motion. Then, taking into account the assumptions, we can write:

tM

Vx1 Vx2 Vx3 , w1 w2 w3

t1 t 2 t3

(17)

where t 3 and Vx3 time and characteristic velocity gains at stage of SDC transfer to parking orbit (third stage of operation, for refill SDC). Average acceleration from thrust obtained as:

w3

P M 0 M T1 M T2

M T3

w1 and w2

obtained from (4), acceleration from thrust at third stage w3

w0 , k3

(18)

2

where k3 correction coefficient, are determined as: k2

1

M T1 M T2 M T3 . M0 M0 2 M0

(19)

Similar to relation (9), we can write:

tM

Vx1 k1 Vx2 k1 Vx3 k1 w0 w0 w0

Vx* , w0

(20)

where Vx* specific characteristic velocity gains during one cycle of operation. Consider that the total resource of SDC limit in time by value T, which is known. Since, number of cycles, which SDC carried out during resource are determine by:

400


n

T w0 Vx*

(21)

.

As criteria during design parameters of SDC refill optimization, takes into account criterion function J as ration ship of all mass at the orbit (with refill working fluid mass) to total characteristic velocity gains, spent to FSD disposal: M 0 n 1 M T (22) J o min . n Vx* Function (21) will characterize characteristic velocity gains efficiency to FSD, located at various orbits, disposal. Mass components M T and M 0 are rewritten in the formula:

MT

tM P C

M0

J

M 0 Vx* , C

(23)

w0 M 0 C M 0 Vx* M C. 2 C

(24)

The criterion function (21) is rewritten, taking into consideration (22) and (23). After transformation, we obtain:

J

§ T w0 Vx* · ¸ M C ¨1 ¨ C C ¸¹ © o min . *· § ¨1 J w0 C Vx ¸ T w0 ¨ C ¸¹ 2 ©

(25)

The problem of refillable SDC design parameters optimization reduces to initial acceleration from thrust

w0

and ERE exhaust velocity determination, as minimal argument of function (25):

Сopt , w0opt

arg min J .

(26)

All other parameters in (26) are known. Let us study solution of problem (26) for special case. Assume, that in the first case, w0 derivative

d J , and equating it to zero, we obtain quadratic equation, which solution takes the form: dC

T w

* Сopt

const. Determine

0

Vx*

2

2

T T w0 Vx* , J

(27)

* where Сopt optimal value of ERE exhaust velocity for given initial acceleration from thrust.

Let us study the second special case. Assume, that С

const. Determine

d J , and equating it to zero, we d w0

obtain solution:

w0opt

*

C Vx* T

§ · T ¨ 1 2 1¸, 2 ¨ ¸ J C © ¹

(28)

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Sergey A. Ishkov and Gregory A. Filippov / Procedia Engineering 185 (2017) 396 – 403 *

where w0opt optimal value of initial acceleration from thrust for given ERE exhaust velocity. Determine of optimal C and w0 provide calculation of all SDC design parameters. For large value of number of cycles n, optimal exhaust velocity Сopt will weakly depend on V x* . Since, we can approximately write:

~ Vx*6 2 T Vx*6 ,

Copt ~ where T

(29)

T * specific resource, Vx6 J

n Vx* specific total characteristic velocity gains.

Equation for criterion function (25) and function

P c (11), taking into account introduced variables, is written in

the formula:

~ J

J MC

Pс

1

1 V x*6 V x* C , § V x*6 C V x* · * ¸ ¨1 V x6 ~ ¨ C ¸¹ 2 T © 1

V x*6 C V x* . ~ C 2 T

(30)

(31)

From solutions (30) and (31) we can conclude, that SDC design parameters essentially depend on specific characteristic velocity gains V x* required for orbital transfer. We can determine minimal value of specific characteristic velocity gains V x* in accordance with mathematical theory of optimal process [3]. For this purpose we should use Pontrigins maximum principle and solve two-point boundary value problems for an ordinary differential equations system. In celestial mechanics with low-thrust methods can be applied, capable to determine characteristic velocity gains approximately [4]. Consequently, the problem of characteristic velocity gain V x* determine is difficult, is left out of this research and requires special consideration, like [5]. In all numerical calculations, the value of specific characteristic velocity gains V x* will be constant. 4. Numeric calculation Let us calculate design parameters of one time use SDC. At fig. 1 and 2 we plot the main design parameters of one time use SDC in a generalized form. In fig. 1 we plot optimal ERE exhaust velocity Copt (12) for some constant values of coefficient P c (11) as a function of characteristic velocity gains V x* . At fig. 2 we plot the dependance between relation of operation duration tmin (15) and specific (dimension less) mass of power plant J as a function of characteristic velocity gains V x* .

402


Pс

0,8 Pс

0,5 Pс

Fig. 1. Optimal ERE exhaust velocity

Copt

for some constant values of coefficient

Pс

0,3

P c as function of characteristic velocity gains V x*

0,8 Pс

0,5

Pс

0,3

Fig. 2. The dependance between relation of operation duration t min and specific (dimension less) mass of power plant * characteristic velocity gains V x

J

as a function of

Let us calculate the design parameters of refillable SDC. At fig. 3 – 5 we plot calculation results in accordance with (28) – (30) as functions of specific characteristic velocity gains Vx*6 n Vx*. Specific characteristic velocity gains at one cycle of operation is 1 km/s. At fig. 3 we plot optimal ERE exhaust velocity Copt (28) as a function of specific characteristic velocity gains

Vx*6 . Next, at fig.4 we plot coefficient P с (30) as a function of specific characteristic velocity gains Vx*6 . Finally, ~ at fig. 5, we plot criterion function J (29) as a function of specific characteristic velocity gains Vx*6 . Refer to fig. 3 W – 5 plot for different SDC’s specific resource: 50 and 100 year (5 and 10 years of motor time (17) kg respectively).

W ~ T 100 year kg

~ T

50 year

Fig. 3. The optimal ERE exhaust velocity

W kg

Copt

(28) as function of specific characteristic velocity gains

Vx*6


W ~ T 100 year kg ~ T

Fig. 4. The coefficient

50 year

W kg

P с (30) as function of specific characteristic velocity gains Vx*6

~ T

50 year

W kg

W ~ T 100 year kg

Fig. 5. The criterion function

~ J

(29) as function of specific characteristic velocity gains

Vx*6

5. Conclusion In accordance with numerical calculation results, we can conclude: 1. For one time use SDC , we need not ERE with high exhaust velocity, as we can see from fig. 1. 2. For refillable SDC, during specific characteristic velocity gains increasing, optimal ERE exhaust velocity decreases (see fig. 3), due to initial acceleration from thrust increasing with the purpose of the carried out operation with constant resource. 3. Specific criterion function have a flat minimum (see fig. 5), consequently we can determine rational range of specific characteristic velocity gains at an operation cycle from 20 to 60 km/s. During final SDC design parameter optimization, it allows advanced criteria to be taken into account. References [1] S.S. Veniaminov, A.M. Chervonov Space debris – a threat for mankind. ISR RAS, Moscow, 2012. [2] A.P. Alpatov, V.P. Gusynin Space vehicle with electric thruster for gathering fine space debris. Conference Proceedings 59Th International Astronautical Congress 2008 (IAC 2008) Glasgow (United Kingdom) 29 September – 3 October, 2008. [3] L.S. Pontrigin V.G. Boltyansky The mathematical theory of optimal process. Moscow State Univercity, Moscow, 1976. [4] S.A. Ishkov, V.V. Salmin, O.L. Starinova Solution methods for variational problems of low thrust space flight mechanics. European Academy of Natural Science Press, Hanover, 2014. [5] S.A. Ishkov, G.A. Filippov A disposal of the space debris with special spacecraft debris collector using low thrust. Engineering Letters, Is. 23(2) (2015) 98-109.

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