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ScienceDirect Procedia Engineering 185 (2017) 411 – 417

6th Russian-German Conference on Electric Propulsion and Their Application Boundary problem solution algorithm for the task of controlled spacecraft motion in irregular gravitational field of an asteroid

Andrey Shornikova, Olga Starinovaa* a

Samara University, 443086,34 Moskovskoye Shosse, Samara, Russia

Abstract The problem of controlled spacecraft motion in vicinity of asteroids is an important flight dynamics task. Solution of the problem consists of gravitational field simulations and control trajectory scheme defininition. We propose to use the model of single gravity points for motion simulation of a spacecraft in an irregular gravitational field of the Eros 433 asteroid. The equations of spacecraft motion are the corresponding equations of the n-body problem. The motion trajectory can be split into a number of from-point-topoint motion parts. Each part can be considered as a two-point Boundary task. The Authors developed a concept of Boundary problem solution based on the Pontryagin’s principle and modified Newton’s step-by-step approximation method. The Authors developed a special software for the Boundary problem solutions. A boundary task of the controlled spacecraft’s transfer between circular orbits from 200 km to 100 km is presented in the paper. © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2017 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of RGCEP – 2016. Peer-review under responsibility of the scientific committee of the 6th Russian-German Conference on Electric Propulsion and Their Application Keywords: spacecraft; electric propulsion engine; electrospray engine; irregular gravitational field; n-body task; boundary task; asteroid.

1.

Introduction

Over the last years, asteroid and comet explorations have become more popular. The reasons for this are: asteroid and comet investigations provide valuable insights into the history of the Universe, asteroids are considered as an important source of resources for commercial and exploration needs and, moreover, as objects that provide operational experience for future space explorations [1]. The spacecraft is usually equipped with an electric propulsion engine that allows manoeuvring in vicinity of asteroids in the long term. While designing spacecraft missions researchers face a problem of controlled motion near asteroids. Control schemes are supposed to be effective in the context of time restriction and fuel consumption. The optimization of low-thrust trajectories is a well-known subject [2], and is performed by using numerical procedures. There are a number of approaches for optimizing a flight trajectory. However, the task becomes complicated because orbital motion about asteroids is highly nonlinear due to inhomogeneities in the irregular gravitational field. Classical theories of motion close to spheroidal bodies cannot be applied as for inhomogeneous bodies the Keplerian forces do not provide a good approximation of the system dynamics [3]. *

Corresponding author. Tel.: +7 902 379 47 04 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.2017.03.323

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Nomenclature control spacecraft’s angles barycentric rectangular coordinate system specific impulse of electrospray engine gravity constant mass of gravitate point system mass of spacecraft propulsive force vector radius vector of spacecraft’s position radius vector of Sun’s position radius vector of Sun’s position with respect to a spacecraft’s position Boolean function (0 or 1) fuel consumption gravity parameter gravitational potential The paper considers the spacecraft’s trajectory as a superposition of separate regions. Each region is a trajectory segment that can be defined as a two-point Boundary task: position of start point and destination point are known, power plant characteristics are also known, the control program is supposed to be identified. Authors developed the optimization scheme of controlled spacecraft’s motion based on the Pontryagin’s principle and Newton’s approximation method. It is proposed considering the task of the transfer between circular orbits for the motion near asteroid Eros in the paper. The choice of the cosmic body is explained by the shape of the asteroid. The bulk of the asteroid is a dumbbell that creates a non-spherical and unstable in time gravitational field. 2.

Gravitational field model

The asteroid Eros 433 is considered in the paper with the physical properties [4]: x size is 34.4х11.2х11.2 km, x mean diameter is 16.8 km, x mass is | 6.69 ˜1015 kg, x rotation period is 5.27 hours. Authors propose the barycentric model of the gravitational potential as the model of the gravitational potential of Eros 433. Therefore, it was decided to consider the gravitational field of the asteroid as a superposition of Ngravitate points. A number of the points and theirs positions determine accuracy of the model [5]: n

mt

¦ m o U(r i

i 1

SV

n

)

¦ U (r ) i

i 1

i

n

G˜

¦ i 1

mi ( x  xi )  ( y  yi ) 2  ( z  zi ) 2 2

(1)

The model of Eros 433 gravitational field is the superposition of two mass points: 4.356 ˜1015 kg, 2.334 ˜1015 kg, which are rotating around the single barycentre with an angular velocity 5.6 ˜10 4 rad/sec. Precise values of masses and the distance between two gravity points were calculated by the direct search method (Fig.1).

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Andrey Shornikov and Olga Starinova / Procedia Engineering 185 (2017) 411 – 417

Fig 1. Gravitational field model of Eros 433

3.

Model of controlled motion

It was decided to use the barycentric model of gravitational potential to simulate the motion of a spacecraft equipped with an electric power plant. A coordinate system is the rectangular barycentric system BsXYZ. The following assumptions are made: the spacecraft is a mass point, a spacecraft has no impact on mass point motions (restricted circular three body problem); the Sun has impact on the spacecraft and the asteroid (Fig. 2).

Fig 2. Asteroid Eros 433 and the n-body problem scheme

Eros and the spacecraft appertain to the plane BsXY. The BsZ axis is the axis of the asteroid`s rotation. The BsX axis directs to the vernal equinox J . Vector form of the motion equation of the disturbed spacecraft is: ˜˜

r SV



P

1

AST

 P 2 AST §Δ r · rSV  P SUN ¨ 3  3SUN ¸  P 3 rSV © ' r SUN ¹

It was decided to use angle D (between thrust vector and BsY axis) as control angles (Fig.3).

P

and BsX axis) and angle

(2)

E (between thrust vector P

Fig 3. Control angles of the spacecraft’s power plant

4.

Two-point Boundary problem formalization

It is proposed to determine the solution for two boundary problem. As was considered earlier, the whole trajectory can be split into parts. The proposed solution scheme can be used for defining optimal control program for each particular part of the whole trajectory. In this case, the task of two control circular orbit transfer is considered. As a criterion of optimal trajectory, the trajectory complies with Pontryagin’s maximum principle. A trajectory that

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Andrey Shornikov and Olga Starinova / Procedia Engineering 185 (2017) 411 – 417

complies with the Pontryagin’s maximum principle is the optimal one [6]. Authors consider the criterion of optimality – maximum speed of operation. The control program that guarantees minimal time of flight is defined: T

T

³ dt o min

(3)

0

According to the Pontryagin’s maximum principle, it is relevant to assign conjugate variables vector ψ (t ) to all elements of the main radius vector rSV in order to form a Hamiltonian:

ψt

ψ t , r

ψ V t , \ m

T

(4)

First of all, it is necessary to determine the Hamiltonian and, after that, to define partial derivatives and define control angles equations [7]: ψV V  ψ r r  ψ mWG

H

(5)

The system of conjugate variables: dψ dt



wH wrSV

(6)

The optimal control program is considered as a vector [8]:

{D opt (t ),

u(t )

E opt (t ), G (t )}

(7)

After all manipulations, computations and simplifications, equations for optimal control angles are as follows: D

ª \ Vx arccos« « \ 2 \ 2 \ 2 Vx Vy Vz ¬

º » » ¼

E

ª \ Vy arccos« « \ 2 \ 2 \ 2 Vx Vy Vz ¬

º » » ¼

(8)

Consequently, 14 differential equations in the barycentric multidimensional set (barycentric coordinate system) were obtained: x 3 equations of spacecraft’s position: r ( x, y , z )T ;

5.

x

3 equations of spacecraft’s velocities: V

x

3 equations of conjugate variables for spacecraft’s position: ψr

x

3 equations of conjugate variables for spacecraft’s Velocities: ψ V

x

1 equation of spacecraft’s fuel consumption: mSV ;

x

1 equation of conjugate variable for spacecraft’s fuel consumption: \ m .

(Vx,Vy,Vz )T ;

(\ x ,\ y ,\ z )T ; (\ Vx ,\ Vy ,\ Vz )T ;

Solution algorithm of the two-point Boundary problem 0

The spacecraft assigns with the radius vector rSV in the first time moment (start position) and it is required to K

obtain the position rSV (destination position) within the shortest possible time. The destination position is assumed to be reached if the following conditions have been fulfilled:

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Andrey Shornikov and Olga Starinova / Procedia Engineering 185 (2017) 411 – 417

x x

an obtained position matches the target value; destination velocity values are not fixed;

rSV o\ m 1; rSV o\ m 0 . As previously noted, conjugate variables are placed in correspondence with the radius vector, velocity vector and 0 K mass values. The values rSV , rSV and electric engine parameters ( P P0 and Isp Isp 0 ) are defined for the particular task. Therefore, the boundary conditions are represented for the case when iteration process convergence is reached when \ m 0 : x

0

0

K

rSV

0

rSV

K

k

x , y , z ,Vx ,Vy ,Vz , m ,\ 0

0

0

0

z

0

0

x , y , z ,Vx ,Vy ,Vz ,\ k

k

k

k

k

k

x0

,\ y0 ,\ z0 ,\ Vx0 ,\ Vy0 ,\ Vz0 ,\ m0



1

T

(9)

0

T

mk

The peak closing error vector γ peak and integration step are also defined. These parameters identify an accuracy of calculations. The closing error vector equation: γ

r  r K 2

0 2

SV

(10)

SV

It is noteworthy that approximation convergence depends mainly on the initial values choice. Practically, minimal initial values variance have a significant impact on the approximation convergence. \ m 0 (practically, calculations can be After that, 14 differential equation’s system are integrated until lim t of stopped if \ m d 1˜1010 is reached). The closing error is calculated for the obtained spacecraft’s position

r  r K1 2

γ1

it

ψ1

SV

is

>\

K0 2

SV

necessary x0

to

. If γ 1 d γ peak then calculation processes can be finished – the goal is reached. In other case, increase

each

conjugate

variable

by

@

the

first-step

increment

value

T1 :

 T1 ,\ y0  T1 ,\ z0  T1 ,\ Vx0  T1 ,\ Vy0  T1 ,\ Vz0  T1 . Then, 14 differential equation’s system is integrated 6

times for every conjugate variable alteration. After that, matrix equation is defined Ax B (system of linear equations):

A

ª x1\ x T  x1k 0 « « \ x TT 1 « y1  yk 0 « T « z\ x T  z 1 k0 « 1 T « « Vx1\ x T  Vxk1 0 « « \ x TT 1 «Vy1  Vyk 0 « T « Vz\ x T  Vz1 k0 « 1 T ¬«

\ T

x1 y  x1k 0

T

\ T

y1 y  yk1 0

T

\ T

z1 y  z 1k 0

T

\ y T

\

T

x1\ z T  xk1 0

x1\ Vx T  xk1 0

x1 Vy  x1k 0

y1\ z T  yk1 0

y1\ Vx T  yk1 0

y1 Vy  yk1 0

z1\ z T  z 1k 0

z1\ Vx T  z 1k 0

z1 Vy  z 1k 0

T

T

T

T

T

T

T

\

T

\

T

T

T

\ Vy T

 Vxk1 0

Vx1\ z T  Vxk1 0

Vx1\ Vx T  Vxk1 0

Vx1

Vy1 y  Vyk1 0

Vy1\ z T  Vyk1 0

Vy1\ Vx T  Vyk1 0

Vy1 Vy  Vyk1 0

Vz1\ z T  Vz1k 0

Vz1\ z T  Vz1k 0

Vz1\ Vx T  Vz1k 0

Vz1 Vy  Vz1k 0

Vx1

T

\ T

T

T

B

T

T

T

ª xk 1  xk 0 º « y y » k0 » « k1 « z k1  z k 0 » » « «Vxk 1  Vxk 0 » «Vyk 1  Vyk 0 » » « ¬«Vz k 1  Vz k 0 ¼»

T

T

T

T

\

T

\

T

T

T

 Vxk1 0

x1\ Vz T  xk1 0 º » T » y1\ Vz T  yk1 0 » » T z1\ Vz T  z 1k 0 » » T » Vx1\ Vz T  Vxk1 0 » » T » Vy1\ Vz T  Vyk1 0 » » T Vz1\ Vz T  Vz1k 0 » » T ¼»

(11)

(12)

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Andrey Shornikov and Olga Starinova / Procedia Engineering 185 (2017) 411 – 417

X

ª 'x º «' » « y» « 'z » « » « ' Vx » «' Vy » « » ¬« ' Vz ¼»

(13)

This linear system can be solved by the Gaussian elimination technique [9]. The vector of conjugate variables increments X is gained. Consequently, the next conjugate variables vector and spacecraft’s radius vector:

\

ψ2 rSV

2

x0

 ' x ,\ y0  ' y ,\ z0  ' z ,\ Vx0  'Vx ,\ Vy0  'Vy ,\ Vz0  'Vz ,\ m0

x , y , z ,Vx ,Vy ,Vz , m , ψ

T

0

0

0

0

0

z

0



1

T

(14)

2

Next step: 14 differential equation’s system is integrated. The closing error is calculated for the obtained spacecraft’s position γ 2

r  r K2 2

SV

K1 2

SV

. If γ 1 t γ 2 then there is a need to increase each conjugate variable by

the first-step increment value T 2 , integrate the system 6 times, define matrix equation etc. If γ 1 d γ 2 then X is supposed to be divided in Newton’s method coefficient ] . Usually, ] 2 [10], however there is an opportunity to change the coefficient manually in the developed software in the calculation process. It allows increasing approximation convergence substantially: if the researcher notices that boundary problem solution “digs” itself, he is able to change the process by changing Newton’s method coefficient manually. The whole calculation loop terminates until γ n d γ peak : last closing error vector is less than or equal to the destination closing error. One more reason for loop termination is insolubility of linear equation system (matrix equation). Practically, it means that initial data (start position vector, destination position vector etc.) has no physical sense: the solution for the particular task does not exist. It is also noteworthy that power plant has two work patterns: 1 – running engine, 0 – nonrunning engine. Boolean function G calculates every iterational step. As maximal operational speed boundary task is considered, algorithm convergence can be improved by setting G equal to 1 at every step. In addition, all calculations can be done in dimension values and dimensionless values as well. Using dimensionless values allows algorithm convergence to be improved [11]. 6.

Effectiveness of Boundary problem’s solution algorithm

The authors propose to identify the effectiveness of the algorithm by calculating the most common control scheme – orbital transfer. For this purpose, the authors developed the software to simulate spacecraft motion in irregular gravitational fields. It was decided to consider the transfer between two orbits – 200 km and 100 km from the barycenter of the system. All requirements that were defined earlier hold for this case. The weight of the spacecraft model is 1200 kg (that is similar to Dawn mission [12] and Rosetta mission [13]). The initial commencing speed in every simulation case was chosen close to the circular velocity of Eros as the body with only one gravity point. The launch date is 30.05.2018, the starting position is (200 0 0) km, the starting velocity vector is (0 0.0012 0) km/sec, the thrust is 1 mkN, specific impulse is 2000 sec. Integration step is 300 sec: integration scheme is the Runge-Kutta method (forth-order accuracy). Fig. 4 represents the obtained trajectory and the optimal program for the control angles. The resulted closing error is 1.456. Flight time is 3.836 days. Fuel consumption is 2.2 ˜ 10 5 k g .

Andrey Shornikov and Olga Starinova / Procedia Engineering 185 (2017) 411 – 417 a)

b)

Fig 4. The transfer between circular orbits: (a) - Trajectory of controlled spacecraft motion; (b) - control program of optimal flight in an orbit

7.

Conclusion

The task of spacecraft’s motion simulation in vicinity of Eros 433 was considered. All algorithm parameters were gained under the assumption of the considered task of an orbital transfer. Authors defined irregular gravitational field models and controlled spacecraft’s motion models. Presented mathematical models are in the barycentric threedimensional coordinate system. The proposed approach can be used for real motion simulation tasks. The paper describes the algorithm of two-point boundary problem solution. The criterion of optimality is a maximum speed of operation. The boundary problem solution is based on Pontryagin’s maximum principle. The Hamiltonian of the system and, as a result, control angles were determined. It was decided to consider the Newton’s method of a step-by-step approximation to solve the two-point boundary problem. Special software that was developed by the authors allows users to change Newton’s approximation coefficient in the process of calculations that improves algorithm convergence. The presented algorithm has a loop structure. It includes a number of steps and a closed number of iterations. To illustrate the proposed algorithm – it was decided to consider the transfer between two orbits – 200 km and 100 km from the barycenter of Eros 433. The resulting closing error can be decreased by decreasing iterational step. To summarize, the developed algorithm of two-point boundary problems can be used for tasks of spacecraft motion equipped with an electric propulsion engine near asteroids. References [1] C. Moore, “Technology development for NASA’s asteroid redirect.” [2] Starinova, Olga L. "Optimization methods of laws control of electric propulsion spacecraft in the restricted three-body task." 10TH INTERNATIONAL CONFERENCE ON MATHEMATICAL PROBLEMS IN ENGINEERING, AEROSPACE AND SCIENCES: ICNPAA 2014. Vol. 1637. No. 1. AIP Publishing, 2014. [3] Shornikov A., Starinova O. Simulation of controlled motion in an irregular gravitational field for an electric propulsion spacecraft //Recent Advances in Space Technologies (RAST), 2015 7th International Conference on. – IEEE, 2015. – С. 771-776. [4] Asteroids interner base: URL: http://space.frieger.com/asteroids/ [5] V. Szebehely, “Theory of orbits: the restricted problem of three bodies.” Yale univ New Haven CT, 1967, pp. 10-25. [6] Kopp, Richard E. "Pontryagin maximum principle." Mathematics in Science and Engineering 5 (1962): 255-279. [7] Leitmann, George, ed. Optimization techniques: with applications to aerospace systems. Vol. 5. Academic Press, 1962. [8] Salmin, V. V., and O. L. Starinova. "Optimization of interplanetary flights of spacecraft with low-thrust engines taking into account the ellipticity and noncoplanarity of planetary orbits." Cosmic Research 39.1 (2001): 46-54. [9] Duff, Iain S., Albert Maurice Erisman, and John Ker Reid. Direct methods for sparse matrices. Oxford: Clarendon press, 1986. [10] Ascher, Uri M., Robert MM Mattheij, and Robert D. Russell. Numerical solution of boundary value problems for ordinary differential equations. Vol. 13. Siam, 1994. [11] Filatov A.V., Tkachenko I.S., Tyugashev A.A., Sopchenko E.V. Structure and algorithms of motion control system's software of the small spacecraft // Information Technology and Nanotechnology (ITNT-2015), 2015. Vol. 1490. P. 246-251 [12] Rayman, Marc D., et al. "Dawn: A mission in development for exploration of main belt asteroids Vesta and Ceres." Acta Astronautica 58.11 (2006): 605-616. [13] Glassmeier, Karl-Heinz, et al. "The Rosetta mission: flying towards the origin of the solar system." Space Science Reviews 128.1-4 (2007): 1-21.

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