LOW THRUST ORBIT TRANSFER UNDER SOLAR ECLIPSE CONSTRAINT

Transcription

1 Asher Space Research Institute Technion, Israel Institute of Technology LOW THRUST ORBIT TRANSFER UNDER SOLAR ECLIPSE CONSTRAINT M. Guelman, A. Gipsman, A. Kogan, A. Kazarian 1

2 Electric Propulsion Electric propulsion system is a set of components arranged so as to convert electrical power from the spacecraft power system into the kinetic energy of a propellant jet. There are three types of electric propulsion systems: ELECTROTHERMAL Propellant is heated electrically and then expanded through nozzle ELECTROSTATIC Propellant is ionized and then accelerated by a large electrostatic field ELECTROMAGNETIC Propellant is accelerated by the interaction of the current driven through the gas with the magnetic field 2

3 Chronology The earliest known record in which the idea of electric propulsion system for rocket vehicles appears - the notebook of rocket pioneer R. Goddard K. Tsiolkovskii pointed out at a possibility of using the electrical power for acceleration of propellant H. Oberth included a chapter on electric propulsion in his classic book on rocketry and space travel Late 1950-ies First experiments on electric propulsion started July First electric propulsion system, SERT I, equipped with ion engines, launched to orbit by NASA November Soviet Zond 2, which carried six Pulsed Plasma Thrusters (PPT) launched to Mars 3

4 Electric vs. Chemical Propulsion Chemical propulsion Energy stored in chemical bonds of propellant creates thrust - energy limited ADVANTAGES: No separate energy source is required High thrust to mass ratio DISADVANTAGES: Specific impulse is limited to several hundreds seconds (typically ) Electric propulsion Electrical energy is used to accelerate propellant and create thrust power limited ADVANTAGES: Specific impulse in the range of thousands seconds Low, highly controllable thrust - precise pointing is possible DISADVANTAGES: Separate energy source is required Low thrust to mass ratio 4

5 Application of Electric Propulsion High specific impulse makes it advantageous option for space missions with large delta V requirement Low propellant consumption allows to launch more payload, increase the mission duration or to reduce the launch cost Growth of satellite operation lifetime and available onboard power stimulates the increase of electric propulsion application onboard different spacecraft At present electric propulsion is used mostly for GEO satellite stationkeeping. Recent scientific missions that used electric propulsion: Deep Space- 1 (NASA), Smart -1 (ESA), Hayabusa (JAXA) 5

6 Hall Thruster Operation Principle H a ll th r u s te r is a n e le c tr ic th r u s te r w ith c lo s e d e le c tr o n d r ift ( H a ll c u r r e n t) 6

7 H a l l T h r u s t e r s 7

8 Hall Thrusters: Some Facts Power range 50 W 70 kw Typical specific impulse sec Used in space onboard Russian spacecraft since 1972 in orbit correction propulsion systems From the 1990-ies research and development of HT started in US, Europe, Israel etc. ESA Smart-1 uses Hall thruster as primary propulsion system 8

9 Satellites Equipped with Hall Thrusters GEO telecommunication satellites Express A Express AM SESAT 9

10 ESA SMART-1 10

11 ORBIT TRANSFER Orbit transfer using low thrust is analyzed for solar electric propulsion systems. The continuous-thrust solutions are extended to the case in which an eclipse shadow arc restricts thrusting in sunlight only, so that thrusting is now intermittent during each orbit. The magnitude of the thrust is assumed to be constant and the direction to be controllable. A numerical solution to the problem of optimal solar electric propulsion orbital transfer is obtained under the general action of Earth oblateness, drag effects and Earth shadowing. 11

12 Shadowing Function General lighting conditions for a satellite orbiting the Earth. The umbra, or shadow cone, is the conical shaded region opposite the direction of the Sun in which the entire disk of the Sun is hidden by the disk of the Earth. An eclipse occurs when the spacecraft enters the umbra. In the penumbra, the disk of the Sun is partially, but not wholly, obstructed by the central body disk. Satellite Orbit Sun Sun Sun Earth x* x * Umbra Penumbra Penumbra z * y * 12

13 Regions of total and partial solar eclipses z * Γ 3 Γ 32 Γ 1 y * Earth Disk x * Γ 3 (ρ,ϕ ) Sun Disk Earth Disk Γ 2 Γ 1 Γ 2 z * Γ 3 ρ* 1 13

14 Solar Eclipse Function F(ρ*,z*) Let us now define the function F(ρ*,z*), equal zero in the region Γ 1,where a satellite is in total solar eclipse and equal 1 in the region Γ 3. In the region Γ 2, where a satellite sees a partial solar eclipse, we use a convenient polynomial of degree 3 and make the function F(ρ*,z*) continuous together with its first derivative for arbitrary values of the variable ρ*. F(ρ*,z*) 1 0 a 1 (z*) a 2 (z*) ρ 14

15 15 With a 1 and a 2 are defined by, R s is the solar equatorial radius, R e the Earth equatorial radius and s(t) the Earth- Sun distance. It should be noted that the function F explicitly depends on the time t and the satellite Cartesian coordinates x, y, z. ( ) Γ Γ Γ = * * * * * *,,, ρ ρ ρ ρ ρ ρ if if a a a a a a if F ( ) e e s R z t s R R a + = * 1 ( ) e e s R z t s R R a + + = * 2 Solar Eclipse Function Solar Eclipse Function F(ρ*,z*) *,z*)

16 The optimal control problem of orbital transfer with the solar eclipse constraint will be now considered. We will consider more specifically the case of minimum fuel optimal control with the magnitude of the thrust assumed constant and the direction to be controllable. The system equations of motion are defined by Where r = v r v = µ + a 3 r a d = g d Problem Definition + w d d T + M m 0 with g d the disturbance acceleration due to nonsphericity of the Earth, w d the drag acceleration and T the thrust control vector p 16

17 COST FUNCTION In order to include the eclipse effects the following minimum fuel optimization criterion is defined J t f = t 0 m p[ T( t) ] dt 2 F ( t) In this integral expression, F(t) is the shadowing function and ṃ p is the propellant mass flow rate. 17

18 The inclusion of the function F 2 (t) in the denominator of the penalty function integrand allows to obtain the expression for the optimal switch function considering the solar eclipse constraint. The quality criterion is an approximation of total consumed propellant mass. Propellant mass flow rate is defined by 2 T m p = 2ηP Where T(t) is the modulus of the thrust vector T, P is the electric power dissipated by the thruster and η is the power system efficiency. The optimization problem is to find an admissible control T, which minimizes the quality criterion J over a given time interval (t 0,t f ) and satisfies given initial and final boundary conditions. 18

19 Lagrange's planetary equations Let us now define X = [Ω i θ h e n e m ] T as the vector of orbital elements. Ω is the right ascension of the ascending node, i is the inclination, θ is the argument of latitude, h is the angular momentum, e n, e m are the components of the eccentricity vector on the perpendicular axes lying in the orbital plane. The orbital elements evolution can be described in vector form as d X = dt G(t, X) a d a d = (a dr, a dθ, a dh ) T are the non-keplerian perturbations in the radial, circumferential and normal directions, respectively. The radial direction is along the geocentric radius vector of the spacecraft, the circumferential direction is perpendicular to this radius vector measured positive in the direction of orbital motion and the normal direction is positive along the angular momentum vector of the spacecraft orbit. 19

20 The (6x3) matrix G has the form ( ψ ) G,X = 0 G G nr mr G G G hθ nθ mθ G G G G G Ωh 0 ih γh nh mh 20

21 The equations of motion of a controlled spacecraft in terms of the orbital element set are given by Gauss's form of Lagrange's planetary equations: dω dt rsinθ hsin i di dt rcosθ a h dh = dt = a = dh dh radθ de dt de dt m n h cosθ + en = sinθadr + cosθ + adθ + dh µ 1+ en cosθ + em sinθ n m em sinθ a ( 1+ e cos θ + e sin θ ) tan i h sinθ + em = cosθadr + sinθ + adθ dh µ 1+ en cosθ + em sinθ n m en sinθ a ( 1+ e cosθ + e sinθ) tani dθ dt = µ 2 ( 1+ e cosθ + e sinθ) n h 3 m 2 hsinθ µ ( 1+ e cos θ + e sin θ ) n m a tan i dh 21

22 A NEW INDEPENDENT VARIABLE Instead of time t the independent variable ψ is used, defined by the following differential equation dt = K t ( h,e,e,θ) dψ n m Where K t ( h,e,e,θ) n m = µ h 3 1+ e cosθ + e sinθ 2 ( ) 2 n m The use of this independent variable stabilizes the set of ordinary differential equations. 22

23 Pontryagin s Maximum Principle The optimization problem will be solved by the Pontryagin s maximum principle. Since the magnitude of the thrust is constant the variable T can be either T 0 or zero. The components of the thrust vector T r, T θ and T h are constrained by T r + Tθ + Th = T 2 0 Using the maximum principle an optimal switching function S w is defined by, S w = K t ( 2 1 p F )( M m ) p F 2 p 0 Where p m is the co-state variable for the propellant mass and p(p r, p θ, p h ) the vector of co-state variables, F is the solar eclipse function and K t defines the independent variable. p 23

24 OPTIMAL SOLUTION If S w m ( T ) p p 0 T = T0 T0 p0 ( ) m p T0 If S w < T = 0 T 0 24

25 NUMERICAL RESULTS Transfer From sun-synchronous 720 km orbit To sun-synchronous 420 km orbit RAAN Inclination Argument of perigee Eccentricity Mean motion RAAN Inclination Argument of perigee Eccentricity Mean motion Ω=60.0º i=98.269º ω=0º e= n= rev/day Ω=free i=97.103º ω=free; e= n= rev/day 25

26 APOGEE AND PERIGEE ALTITUDE TIME HISTORY 750 Altitude at apogee and perigee, km Flight time, day 26

27 INCLINATION TIME HISTORY Inclination, deg Flight time, day 27

28 THRUST COMPONENTS Thrust components, mn Radial Circumferential Normal Flight time, day 28

29 Consumed Propellant Mass Time History 4.5 Consumed propellant mass, kg Flight time, day 29

30 Venus Project: Orbital Issues 30

31 Venus Mission The VENµS mission is a joint project of the Israeli Space Agency and the Centre National d Etudes Spatiales (CNES). It includes two major experiments: in-flight tests of solar electric propulsion in LEO, and highresolution, multispectral remote sensing of the earth. The first experiment comprises the tests of a propulsion system with Israeli Hall thrusters (IHET) commanded by a completely autonomous onboard control system. It will propel an inter-orbital transfer from the initial LEO 720 km altitude to a final orbit of 420 km altitude and maintain the latter orbit. The mission profile will consist of three stages: 1) Multispectral earth observations from the initial orbit 2) Inter-orbital transfer 3) Multispectral earth observations from the final orbit The Asher Space Research Institute is involved in planning the second stage of the mission and in developing the onboard orbital control algorithms. 31

32 Work Objectives Dynamics and control of electrically propelled inter- LEO transfer The study addresses the two major issues: Preflight orbit design; Its in-flight implementation 32

33 I. Physical Preliminaries 33

34 Dynamic Environment in LEO Natural factors Man-caused factors Newtonian gravity ~10 1 m/s 2 Control thrust ~10-4 m/s 2 Polar oblateness (J 2 ) ~10-2 m/s 2 Thrust exec.error ~10-5 m/s 2 Higher zonals, total <10-5 m/s 2 Xe mass consumption <10-5 m/s 2 Sectorial harmonics ~10-5 m/s 2 Aging TBD Lunar & solar gravity ~10-7 m/s 2 Aerodynamic drag <10-6 m/s 2 Light pressure ~10-8 m/s 2 Underlined items can cause secular perturbations. 34

35 Dynamic Environment in LEO (2) Polar oblateness dominates among natural perturbations. It causes secular (linear in time) evolution of some orbital elements and leaves others unchanged except for small oscillations with the orbital period and its multiples. Higher gravity harmonics in total contribute ~1% to the oscillatory evolution of orbit and to the precession rate. Air drag causes slow decrease of orbital period and eccentricity. 35

36 Nominal Inter-Orbital Transfer From sun-synchronous 720 km orbit (29 revs per 2 days) To sun-synchronous 420 km orbit (31 revs per 2 days); Sun-synchronism to be maintained during the transfer. Maintenance of synchronism with Earth rotation impossible Goal: minimum transfer duration 36

37 SEP Peculiarities The satellite carries two thrusters but not more than one may operate at the same time Thrusters axes are fixed in the body frame, thrust direction is controlled by rotations of the satellite as a whole. Solar arrays are also fixed in the body frame. Thus, available power correlates with thrust direction. Thruster cannot work at the input power below 250 W. Thrust, specific impulse, and efficiency depend on the available power. Satellite attitudes and therefore thrust directions are subject to constraints. 37

38 VENuS Satellite a u -n =π χ u 38

39 General Approach Traditionally, design of low thrust trajectories is considered as a variational problem that provides global optimum to thrust control program. An alternative approach, closed-loop locally optimal guidance algorithm, is preferable 39

40 Optimal control vs. guidance Optimal control works in open loop. The task is formulated as a 2-point boundary problem. It - provides globally optimal trajectories - is slow and sensitive to dynamic model errors; - is not well adapted to work with control or phase state constraints Computing a transfer as long as ~10 3 rev unaffordable onboard. Guidance works in closed loop with navigation feedback. Thus it is fast and robust with respect to dynamic model errors; - easy adaptation to constraints; the same algorithm can work as the onboard control algorithm. It provides locally optimal guidance. Extra mass and time quite affordable 40

41 II. Implementation of Guidance algorithm 41

42 Guidance Algorithm Features 1. It deals with orbital elements, not with state vectors. 2. Its output is the thrust direction that provides the fastest evolution of the elements towards their destination values. 2. In order to eliminate the control jitter at approaching the destination orbit, mean elements are used instead of osculating ones 42

43 e 3 E 0 2 ρ optimal transfer e 1 Thrust Controller ( ) T E E W( E E); = f f E f e 2 2 ( ρ ) ( ρ ) T G u = umax = u 2 max T G f d E = Gu E Orbital elements dt ( E f E) ( E E) When the initial and destination orbits are close to each other, a proper choice of W assures an optimal thrust program, even though the transfer may count many revolutions. Optimal transfer in the space of orbital elements presents as a straight line. This gives the lead to W optimization. It is done once for all, at preflight preparation phase. 43

44 Attitude constraints avoidance Banned are the attitudes that imply a) Telescope axis getting inside the prohibited cone that encircles the orbital velocity vector; b) Both star trackers having Sun or Earth in their FOV ( with a margin for straylight) c) Insufficient available power (solar arrays >58 away from the sun). Normally, the unconstrained optimal attitude does not violate more than one constraint in a time, and the violation is weak. Then a constraint avoidance algorithm computes optimal corrections to the unconstrained attitudes; otherwise it switches the thrust off. 44

45 Thruster Firing Constraints Firing a thruster should not last less than T min (=10 minutes). At real-time operation, the control loop should be capable of control prediction at least T min ahead. Though easy to enable in principle, it increases the demanded computing resources. An economical algorithm was been developed that uses a simplified dynamic model. 45

46 Mean versus osculating elements Altitude at apogee and perigee, km Flight time, day A sample of osculating apogee and perigee heights External perturbations and control thrust force oscillatory behavior of osculating elements. Oscillations permeate the feedback and cause parasitic self-exciting oscillations in the control loop. Mean elements contain no short-term oscillations and therefore safe. A practical way to evaluate them is averaging the osculating elements over the period prior to the current instant. 46

47 III. Numerical Results 47

48 Nominal Transfer Summary Constraint Avoidance Strategy Transfer time, days Time out of eclipse, hours Total burning time, hours Xe mass, kg (15% for cathode added) IHET1: On/Off cycles Burning time, hours Xe mass, kg (anode only) IHET2: On/Off cycles Burning time, hours Xe mass, kg (anode only) = constraints ignored; 1 = constraints avoidance 48

49 Nominal transfer Sample thrust program 0.20 Control acceleration, mm/s ECLIPSE ECLIPSE ECLIPSE Time, days 49

50 Sample of power history Nominal & Emergency transfers Input power, W 200 nominal case STR1 in failute STR2 in failure IHET1 in failure Time, days 50

51 SUMMARY & CONCLUSIONS 1. An optimal algorithm for orbit transfer with Solar Electric Propulsion was developed that explicitly takes into account eclipse constraints 2. A closed loop guidance algorithm was developed that provides a satisfactory solution to the thrust control and to its autonomous onboard implementation 3. The guidance algorithm is nearly optimal in terms of transfer duration. 4. The algorithm deals with specially chosen mean elements. Algorithms for their evaluation from GPS data has been developed and tested 51

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