Application of Electric Propulsion for Motion Control of ... - Science Direct

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ScienceDirect Procedia Engineering 185 (2017) 291 – 298

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

Application of electric propulsion for motion control of spacecraft which function on non Keplerian orbits Olga L. Starinovaa,*, Irina V. Gorbunovaa, Maxim K. Faina, Roman M. Khabibullina, Andrey Yu. Shornikova a

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

Abstract This article discusses the design-ballistic optimization for spacecraft, which move under the influence of the gravitational fields of complex configuration and electric propulsion. Such conditions exist near asteroids, for example in the systems of two gravitating bodies, in the vicinity of the libration points. In this paper, we propose to use a method of iterative mission optimization for electric propulsion spacecraft, using sequences of the mathematically specified models of movement and the spacecraft design. The previously obtained analytical results, which describes the planar movement of spacecraft with solar electric propulsion, allows the initial approach in the iterative scheme of optimization to be constructed. A method of modeling and optimization of interplanetary missions ballistic schemes is developed, based on a combination of Pontryagin’s maximum principle formalism conditions of transversality and methods of mathematical programming, allowing restriction characteristics for real interplanetary missions to be considered. Recommendations for the choice of the design-ballistic parameters of the interplanetary mission’s spacecraft are obtained taking into account the features of a nuclear and solar power plant for purposes of the delivery of a payload to the required heliocentric orbit, to the Moon and near asteroid. © 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; non Keplerian orbit.

1. Introduction Space exploration programs are very expensive and do not give quick feedback. One of the possible ways to solve this problem is to use the perspective propulsion system with high performance, for instance, using electric propulsion or a solar sail.

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.2017.03.343

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Olga L. Starinova et al. / Procedia Engineering 185 (2017) 291 – 298

Nomenclature

m0 mt0 m pl

=

the mass of the space tug in the parking orbit at the t 0 time

=

the payload mass

m pp

=

the power plant mass

m ps

=

the propulsion system mass

mf1 , m f 2 mt mc D pp , D c

=

the fuel masses of forward and back flights

J ps , J c = K pp , K ps k G ^0, 1`

the specific mass coefficients on thrust of the propulsion system and of the construction

ri j sp

= =

=

the tank mass, including the propellant distribution system

=

the construction mass

=

the specific mass coefficients on power of the power plant and of the construction

=

the power and the thrust efficiency coefficient

= the specific mass coefficient of the tank = the function of thrust switching T = the total mission duration F x = the dependence the specific fuel consumption of the phase vector x x = the phase vector, when consist of the radius-vector r , the velocity vector V and the relative weight of fuel consumed m a = the currently vector acceleration of propulsion a0 = the nominal acceleration of spacecraft e = the trusting direction unit vector = the gravity acceleration g f = the total perturbation acceleration vector P pl = the relative payload mass the radius-vectors of attraction center the specific impulse of engine

The application of electric propulsion is one of the most promising research areas for future space missions. For example, some theoretical and technical aspects of low thrust application are described in the articles [1-4, 10]. The areas of the search for the optimal control laws are shown in the works [2-9]. The problem of integrated design-ballistic optimization of the electric propulsion missions is a non-classical search problem of the criteria’s extremum on parametric variables and unknown functions. The search for the optimal control function and the corresponding trajectory is called the ballistic problem of the mission’s optimization. The search for the spacecraft design parameters is known as the design problem of the mission’s optimization. Unfortunately, these problems cannot be separated from electric propulsion spacecraft, and this fact significantly complicates the solution process. The issue of ballistic optimization of a low thrust mission is reduced to the two-point boundary value problem [3-6, 9]. However, this problem is complicated by the necessity to allow continuous control action, the magnitude of which is comparable with other disturbances [2, 4-6, 8]. Therefore, the motion of spacecraft with low thrust, has to be described in non-Keplerian motion. In recent years, research regarding the designing optimal trajectories of the spacecraft motion in complex gravitational fields has grown significantly These fields surround points of the libration, neighborhoods of the planet-satellite system and regions near asteroids of irregular shape. For example, the optimal interplanetary trajectories and the trajectories of flights to the Moon pass near the libration point L1 Earth-Moon system as shown in works [3, 5-6]. Since an acceleration rate is less than the gravitational acceleration, the optimal control laws

293


depend on the gravitational effects of two or more attraction centers. Formulation of this problem severely complicates the search of an optimal spacecraft control. The methods of the compound design-ballistic optimization of such missions using low thrust are described in this paper. Typically, these problems are solved by either direct or indirect optimization methods using numerical integration [3, 7] of the motion equations. This approach results in significant computational difficulties and does not allow the various design and ballistic parameters of the mission to be analyzed. In this paper, we are developing iterative methods of missions’ optimization for electric propulsion spacecraft, using sequences of the mathematically specified models of the movement and the spacecraft design shape . In addition, this paper shows some of the results of optimal transfers in the Earth-Moon system and near asteroids of a complex shape. 2. Models of system description and methods of solution 2.1. Features of using and characteristics of electric propulsion An electric propulsion system uses electrical energy to electrically expell propellant (reaction mass) at a high speed [1, 10]. Electric thrusters typically use much less propellant than traditional chemical rockets because they have a high specific impulse (1000…10000 1/s).[10-11] Due to limited electric power, the thrust is much weaker compared to chemical rockets, but electric propulsion can provide a little thrust for a long time, so they are traditionally called low thrust propulsion.[3-4] The general trend of electric propulsion development is the increase in power and the propulsion systems efficiency. For example, the high-power electric propulsions thusters (SPD290, TM-50), which are a product of “Experimental Design Bureau FAKEL” and “THNIIMASH”, have power of more than 30 kW. [10-11] Such propulsion systems need developing and designing of the high power energy sources, which commonly have considerable mass. This fact reduces the overall mass efficiency of the spacecraft. Therefore, an accurate design ballistic mission optimization as a whole is required to determine the most advantageous design parameters of energy and propulsion systems. The vector of design parameters must be selected so its coordinates had a significant influence on the criterion. A change in each of these variables produced a different design of the spacecraft. We assume that the spacecraft consists of the following systems: the solar or nuclear power plant, the propulsion system based on electric propulsion; the tanks filled with working fluid, the construction elements, and the other service systems. The design model of the electric propulsion spacecraft can be obtained based on a mass balance equation in the parking orbit. The initial mass of the spacecraft can be represented as the sum of the masses of its general systems, the fuel, and the payload. m0 m pl m pp m ps m f 1 m f 2 mt mc . (1) In a first approximation, the formulas of system masses from the nominal thrust P0 and specific impulse jsp have to form of direct and inverse proportionality. [1, 5-7].

m pp

mt

D pp

P0 j sp g 2K ppK ps

k m f 1 m f 2 , mc

, m ps

Dc

J ps P0 . P0 j sp g

2K ppK ps

J c P0 .

(2) (3)

For space tug with electric propulsion, the output power, and the thrust depend on phase coordinates (distance from spacecraft to the Sun, a rotation angle, a power decreasing due to the Earth radiation belts, and possible shading of solar arrays, work duration of a nuclear reactor, etc.). Therefore, the fuel mass used on the mission depends on the total burning time, the nominal thrust and the specific impulse.

mf

P0 T ³ F x Gdt . j sp g t0

(4)

294


Here F x Px is the function form which depends on the used models of the power plant functioning. This P0

function usually is calculated by the formula F x r k , k | 1,7 2 for spacecraft with solar power point, and by the formula F x

e a x t t0 for spacecraft with nuclear power point. The value

P0 has the meaning of j sp g

T

nominal seconds fuel consumption and the value

³ F x Gdt

has the meaning of the equivalent time of engine

t0

* operation T P . If the function is equal to unity F x { 1 , then the equivalent time of engine operation coincides with

the actual time of engine operation. * Given Eq. (1-4) the spacecraft relative payload mass expressed by formulas if we use notation T P .

P pl 1

D

pp

Dc

g

2K ppK ps

a 0 j sp J ps J c a 0

1 k a0 * TP x, a 0 , j sp . g j sp

(5)

The maximum of relative payload mass P pl is usually used as the optimization criterion of the mission, herewith the total mission duration T is fixed. In this study, we chose the specific impulse and the nominal acceleration of the electric propulsion system as design parameters. Special difficulties in design-ballistic mission’s

* optimization cause the fact that the equivalent time of engine operation TP x, a0 , j sp depends on the design

parameters of the power-propulsion system and the trajectory. This fact significantly complicates the solution process. This circumstance does not allow separating the complex problem into the design and the ballistic parts and leads to the need to simultaneously determining the optimal flight’s trajectories, the control functions of the propulsion system and the design and ballistic parameters of the missions. Formally, the optimization problem is determined so. It is necessary to find: - the control function from a possible area ut U and corresponding trajectory xt X satisfying to boundary conditions xt 0 X 0 , xT X T ,

-

the optimal vector of designed parameters p

a

0,

j sp

T

satisfying to restricts p P ,

- the optimal vector of ballistic parameters b satisfying to restricts bB , which supply a maximum of optimal criteria P pl

^ut ,

p, b`opt

arg max P pl xt , ut , p, b T , X 0 , XT

u t U ,pP ,bB

fixe .

(6)

This problem is so complex that it requires a development of unique methods to its decision. This work describes the iterative method of missions’ optimization for electric propulsion spacecraft, using sequences of the mathematically specified models of the movement and the design shape spacecraft. 2.2. Ballistic model sequences definition The task of ballistic missions’ optimization can be solved with different accuracy depending on the selected motion models. Described method is used clarifying the sequence of models M i , j . [9] The first index corresponds

^

`

to some heliocentric model and the second index correspond to some planetocentric model. The symbol Ai , j is labeled model that was obtained by approximation of solutions correspond to model M i , j . Clarification of the description planetocentric motion is achieved by taking into account not the centrality of the gravitational field of the attracting body, the atmospheric effects, the gravity from planet’s satellites and the Sun, the shadow situation in orbit, the control methods and other factors. The heliocentric motion of the spacecraft is complicated and specified by taking into account the ellipticity and the noncoplanarity of the planet’s orbits, the influence of solar radiation, temperature and working duration on the power plant efficiency, constraints on the

295


phase coordinates and the propulsion control system control. The different ballistic models of this sequence can be used, when we are choosing the program control for various space missions. The simplest model allowing an analytical solution that can be employed as the initial approximation to determine the optimized parameters. The more sophisticated and accurate models had to be selected so that the transition between them was carried out using a continuous change of one parameter. In this case, the results of the previous level can be used as an initial approximation for the next level. In many works of our predecessors, the method of continuation on parameters is used to obtain solutions with close values of the required parameters. [1, 7] In this paper, we propose to extend the application of this method and apply it to motion in a sequence of clarifying models. As a result, gradually clarifies the optimality criterion of the mission and optimal values of design-ballistic parameters. The solutions of the optimization problem in Eq. (6) within the n-th model is stored in the database as a set

^P z , ut , z , pz , bz `,

y z ni

pl

n

n

n

n

i

i

i

i

(7)

which contains the value of the optimality criterion for a given vector design-ballistic parameters P pl z i ; the n

and the optimal vector of ballistic

n n optimal control u t , z i , the optimal vector of designed parameters p z i

n

parameters b z i . The finding process of optimal program control and the corresponding trajectory is an iterative process. 2.3. Methods used to search the optimal control The movement of the electric propulsion spacecraft can be described by the system of the differential equations to take into account the following assumptions [3-9, 11]: the spacecraft is subject to the n-bodies gravitation, the gravitational field of all bodies aren't central, the atmosphere of a planet is standard, the orbits of the attraction centers relative to the barycenter are known. °r a g r, r1 , r2 , r3 ,... f r, r1 , r2 , r3 ,... , ® ° ¯m a0 F x, e G jsp g a0 a F x, e G e 1 m

(8) (9)

Here gr, r1 , r2 , r3 ,... is the vector sum of the gravitational accelerations to all attraction centers,

f r, r1 , r2 , r3 ,... is the vector of total perturbation acceleration, m

m f m0 is the relative mass of fuel consumed.

The optimal program control is defined with use of Pontryagin’s maximum principle formalism on the trajectory parts. Also we will enter the vector of the costates Ψ Ψ r , ΨV , \ m T . The problem of optimum control design consists in definition of change programs G t , et and delivering a maximum to an optimality criterion and flight boundary conditions providing performance. The thrust change function can be found from conditions of Hamiltonian maximum is:

e

ΨV ,G ΨV

\ ΨV 1, ' ! 0 , where ' m . ® 1 m j sp g ¯0, ' d 0

(10)

Thus, the optimal control for choosing model is determined by the initial values of co-states. The co-state equations are defined by the partial derivative of the Hamiltonian by Eq. (14). Thus, the optimal control problem on transfer boils down to the following boundary problem: it is necessary to find such initial values of co-state variables, which starting and ending conditions are compiled at the optimal trajectory edges, with motion and optimal control laws by Eq. (10). The modifying Newton method and the continuation of parameters method [1] were used to solving of boundary problems. Well-known, if the initial approach of the costate vector is far from the decision then the high dimension the boundary value problems possess bad convergence. The optimization process of the mission associated with the repeated solution of variational problems of optimal hops at different values of the vectors of design and ballistic

296


parameters. The solution to these challenges requires heuristic approaches, qualifications of the contractor and do not allow for the automation of the optimization process. Good results can be obtained by using intelligent information-processing system (IPS) designed for the optimization of interplanetary missions with low thrust. [12] The longer the user is working on optimizing this mission, the more results in form Eq. (7) are accumulated. At the request of the user, the developed system can obtain or clarify the approximation dependences

f j z ni describing the obtained numerical results y j z ni .

3. Results of optimization of testing mission Let us consider the iteration process on an example of test task solution. The start orbit of electric propulsion spacecraft is the heliocentric Earth orbit. The finishing heliocentric orbit has the following parameters: the * * eccentricity is zero, the radius r f equals 0.5 a.u., the inclination i f equals 10 degree and the longitude of accident * node : f is 0 degree. Such task statement is implemented when the spacecraft lives the Earth action sphere by a

chemical upper stage. For instance, the rocket carrier “Sous” and booster “Fregat” allows the output from the Earth *

action sphere the for spacecraft with a mass of about m0 = 2400 kg. In the first approximation, we used the analytical solutions for transfer between planar circular orbits with the constant control angles of thrust direction and without changing the spacecraft mass ( M i ,1 model). In the next iteration, we numerically found the optimal control law without changing the spacecraft mass ( M i , 2 model). The good initial approximation allowed manifold solutions to be obtained for the solution of a range of parameters ( rf >0,1; 5@ , a 0 >0,001; 5@ ). The solutions obtained were approximate to the polynomial dependences ( Ai , 2 model). This process is described in detail in work [9]. Fig. 1 shows these approximate dependences for the different acceleration a 0 and the given radius rf for flights from Earth to a required heliocentric circular orbit: (a) – the flight duration, (b) – the angular flight distance. The use of these dependencies allows the process of solving boundary value problems for more complex models of spacecraft's motion to be automated. As the initial values of the costates the values obtained by known solution of the second model are used. The transition from the model M i , 2 to the model M i ,3 allows navigating to objectives with measurable values of specific impulse (and ultimately the flow of the working fluid). At sufficiently high speed, on expiration of the working fluid the error in calculating the duration and angular range of flight related to the change of spacecraft's mass is from 5 to 15 percent. However, the structure of the optimal control and the associated trajectory change little. For comparison, Fig. 2 shows the trajectory of the flight with infinite specific impulse and the specific impulse of real propulsion systems (2600 1/s).

(a)

(b) (a) – the flight duration, (b) – the angular flight distance

Fig. 1. The approximating dependence for the different acceleration

a0

and the given radius rk are obtained by the

Earth to a required heliocentric circular orbit

Ai , 2

model for flights from


The following refinement of the model is performed to clarify the fuel consumption with the introduction of the passive sections of the trajectory. To do this, a move to the model M i , 4 is made. Fig. 3 shows trajectories with the same boundary conditions as in Fig. 2(b), but with fixed flight durations of T=370 days and T=460 days. The dependence of the relative fuel consumption from the duration of the trip is traditional in nature: the fuel consumption decreases when increasing the flight duration.

(a)

(b) (a) с=10000 km/sec, (b) с=26 km/sec Fig. 2. Optimal trajectory depending on injection velocity (rk = 0,5 a.u, a0 = 0,4 mm/sec2)

(a) (b) (a) flight duration T = 370 days, (b) flight duration T = 460 days Fig. 3. Optimal trajectory change depending on flight duration increasing:

Fig. 4. The program of optimal control and the optimal trajectory

297

298


Finally, for more accurate results, taking into account the cost of changing the orbital plane, we move to noncomplanar motion model M i ,6 . Fig. 4 shows the program of optimal control and a projection of the optimal trajectory on the Ecliptic plane and the plane perpendicular to the plane of the Ecliptic. This trajectory is the desired optimal trajectory that satisfies all boundary conditions. The obtained data demonstrate that the refinement of the model of motion does not significantly affect the solution, and only requires to choose the sequence of models that would allow to move from one model to another for certain "boundary" conditions without jumps. 4. Conclusion The use of Pontryagin principle of maximum in the optimal control problem for non-Keplerian trajectories allows the optimal steering and nonsmooth motion to be obtained for spacecraft with various design parameters. The iterative optimization method of interplanetary missions with low thrust, using the sequence of movement for specified mathematical models and space vehicle design shape is developed. Based on Pontryagin’s maximum principle, the necessary conditions of a control law optimality for all models of the sequence are received. The new particular analytical decision describing the planar movement of the space vehicle with solar electric propulsion systems is received. It allows the initial approach in the iterative scheme of optimization to be constructed. The obtained results for Earth-Moon mission with results from [3, 4, 10] exploiting other methods were obtained. The results for the time-optimal mission are in good accord with the results of this work. However, the described methodology allows the optimal thrust-on and thrust-off trajectory parts to be obtained automatically. Clearly, when increasing the needed transfer duration, the fuel consumption is redued and for ballistic design of flight, it is necessary to balance between fuel consumption and mission time. So the applied methodology demonstrates its effectiveness for optimization of the compound multiple-turn trajectories of electric propulsion spacecraft in the frame of n-bodies gravity fields. Findings may be used to solve problems of orbit formation and to calculate the needed design-ballistic parameters of spacecraft. Acknowledgments This research was started with the RFBR Grant № 14-08-00559. References [1] H. W. Loeb, D. Feil, G. A. Popov, V. A. Obukhov, V. V. Balashov, A. I. Mogulkin, V. M. Murashkov, A. N. Nesterenko and S. Khartov, “Design of High-Power High-Specific Impulse RF-IonThruster,” 32nd International Electric Propulsion Conference, Wiesbaden, Germany, September 11–15, 2011. [2] V. Szebehely, “Theory of orbits: the restricted problem of three bodies.” Yale univ New Haven CT, 1967, pp. 10-25. J. T. Betts, and S. O. Erb, “Optimal low thrust trajectories to the moon.” SIAM Journal on Applied Dynamical Systems, Vol. 2, No. 2, 2003, pp. 144-170. [3] R. McKay, M. Macdonald, J. Biggs and C. McInnes, “Survey of highly non-Keplerian orbits with low-thrust propulsion,” Journal of Guidance, Control, and Dynamics, Vol. 34, No. 3, 2011, pp. 645-666. [4] O. L. Starinova, D. V. Kurochkin and I. L. Materova, “Optimal control choice of non-Keplerian orbits with low-thrust propulsion,” AIP Conference Proceeding 1493 (American Institute of Physics, Melville, NY, 2012), 964. [5] O. L. Starinova “Optimization methods of laws control of electric propulsion spacecraft in the restricted three-body task,” AIP Conference Proceeding 1637 (American Institute of Physics, Melville, NY, 2014), 1056. [6] P. V. Kazmerchuk, V. V. Malyshev and V. E. Usachev “Method for optimization of trajectories including gravitational maneuvers of a spacecraft with a solar sail”, Journal of Computer and Systems Sciences International. 01/2007; 46(1), pp. 150-161. [7] A. Shornikov and O. L. Starinova “Simulation of controlled motion in an irregular gravitational field for an electric propulsion spacecraft”, Proceedings of 7th International Conference on Recent Advances in Space Technologies, 2015, IEEE, 2015, pp. 771-776. [8] O. L. Starinova, “Optimum driving on speed between circular coplanar orbits”, Izvestia of Samara Research Center of RAN 7.1, 2005, pp. 9298. [9] A. I. Vasin, A. S. Koroteev, A. S. Lovtsov, V. A. Muravlev, A. A. Shagayda and V. N. Shutov “Review of works on Electric propulsion at Keldysh Research Center,” Works of MAI 60, 2012 [10] “Development of a stationary plasma thruster” http://www.fakel-russia.com