Astrodynamics of Interplanetary Cubesats

Transcription

1 Astrodynamics of Interplanetary Cubesats F. Topputo Politecnico di Milano, Italy 06 F. Topputo, Politecnico di Milano. All rights reserved. icubesat 06, 5th Interplanetary CubeSat Workshop 4-5 May 06, Oxford, United Kingdom

2 Outline Interplanetary trajectory design Conventional spacecraft Cubesats Main challenges Case study: Cubesat mission to Mars Proposed strategy Ballistic capture Dual propulsion Critical analysis

3 Interplanetary trajectory design for standard spacecraft Aim: To find best path for a conventional spacecraft Fuel-optimal, time-optimal, energy-optimal, etc. Manoeuvres accomplished through on-board propulsion Small errors in nominal trajectory zeroed with TCM 6 DoF control usually available (RCS) Control authority is not an issue S/C designed to cope with off-nominal conditions, unless catastrophic events occur S/C over-actuated 3

4 Trajectory design for interplanetary cubesats Aim: To find best solution under much tighter constraints Power generated Propellant stored Thrust exerted Interplanetary cubesats have much less control authority Capability of executing al manoeuvres strongly limited These features set new challenges in astrodynamics Arrival: How to acquire a final, closed about a planet? Cruise: How to accomplish interplanetary transfer? Departure: How to leave Earth? 4

5 Case study: Cubesat mission to Mars Devised strategy involves Arrival: Performing ballistic capture upon Mars arrival Cruise: Using on-board micro-propulsion Departure: Using hybrid propulsion to leave Earth Case study: A cubesat mission to Mars 5

6 Ballistic capture (in a nutshell) A massless particle is (temporarily) ballistically captured by a primary if (along ) its Kepler energy (H) goes from positive to negative The two-body state changes from hyperbolic to elliptic Requires n-body dynamics, with n 3 Permanent capture require dissipation The opposite behavior is ballistic escape Planet Temporary ballistic capture 6

7 Why ballistic capture Saves propellant Reduces hyperbolic excess velocity upon arrival Lowers magnitude of arrival maneuver, or Thus saving propellant, m p /m 0 = exp[ v/(i sp g 0 )] Widens launch windows Target is a point in space, not planet Increases safety Avoids single-point injection failures v Hyperbolic arrival 7

8 Why ballistic capture Saves propellant Reduces hyperbolic excess velocity upon arrival Lowers magnitude of arrival maneuver, or Thus saving propellant, m p /m 0 = exp[ v/(i sp g 0 )] Widens launch windows Target is a point in space, not planet Increases safety Avoids single-point injection failures v Hyperbolic arrival 7

9 Why ballistic capture Saves propellant Reduces hyperbolic excess velocity upon arrival Lowers magnitude of arrival maneuver, or Thus saving propellant, m p /m 0 = exp[ v/(i sp g 0 )] Widens launch windows Target is a point in space, not planet Increases safety Avoids single-point injection failures v Hyperbolic arrival 7

10 Why ballistic capture Saves propellant Reduces hyperbolic excess velocity upon arrival Lowers magnitude of arrival maneuver, or Thus saving propellant, m p /m 0 = exp[ v/(i sp g 0 )] Widens launch windows Target is a point in space, not planet Increases safety Avoids single-point injection failures v Hyperbolic arrival Low-energy arrival 7

11 Why ballistic capture Saves propellant Reduces hyperbolic excess velocity upon arrival Lowers magnitude of arrival maneuver, or Thus saving propellant, m p /m 0 = exp[ v/(i sp g 0 )] Widens launch windows Target is a point in space, not planet Increases safety Avoids single-point injection failures v Hyperbolic arrival Low-energy arrival 7

12 Why ballistic capture Saves propellant Reduces hyperbolic excess velocity upon arrival Lowers magnitude of arrival maneuver, or Thus saving propellant, m p /m 0 = exp[ v/(i sp g 0 )] Widens launch windows Target is a point in space, not planet Increases safety Avoids single-point injection failures v Hyperbolic arrival Low-energy arrival 7

13 of periapsis radius rp. Thisby ausing great departure Hohmann Figure 6: sample solution constructed in Figure Left: Figure 6:gardless AA sample solution constructed by isusing infrom Figure 4. 4.Left: transfers where cost increases for increasing rp. Sun-centered frame ( black needed target capture capture Sun-centered frame ( black is is needed tototarget point from Earth; red is capture ; blue point departing Earth; red is capture ; blue departing The redfrom dots in Figure 8 are organized into two di erent sets that correis post-capture ). Right: capture (red) and post-capture spond to two branches of capture sets, see Figure 3. is post-capture ). Right: capture (red) and post-capture (blue) rotating Mars-centered frame. (blue) in in rotating Mars-centered frame. Ballistic capture at Mars th Draft [Revision 0] Saturday 8 November, 04 at 4:4 (c) E. Belbruno and F. Topputo rp 0.5 Y (AU) SOI 0 0 xc xc 0 0 y (adim.) Ballistic capture y (adim.) 3.5 Y (AU)!"#$%&'()*+,-./ :;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{ }~ The set of red dots, as a function of rp is seen to have a series of gaps. This is due to structure of stable and unstable sets as defined x 0 x 0.5 in algorithm for weak stability boundary(see Appendix ). As is 4 4 Mar Mar p described in [6, 3], stable and unstable sets alternate on each radial 3 3 line emanating from secondary body (Mars in this case), giving a 0.5 Cantor-like structure. 3 V The results from Figure 8(b) are summarized in Table 3. vision 0] Saturday 8th November, 04 at 4:4 Mars e< x (c) E. BelbrunoInterplanetary and F. Topputo transfer Table 3: Comparison between ballistic capture transfers and Hohmann transfers c Ear th@dep Ear th@dep c smar Mar Figure : Structure of ballistic capture transfers to Mars. xc to xto c rp rp 3 3 for points in Figure 8(b). The saving, S, is computed as S = ( Vc V )/ Vc, where V associated to H3 case is xconsidered. S xis a X (AU) x (adim.) X (AU) (adim.) 0x 0 measure of efficiency of ballistic capture transfers. tc!p is timeinertial frame Rotating frame (a)(a) Inertial frame (b)(b) Rotating frame of-flight needed to go from xc to rp process is much more benign that high velocity capture maneuver at rp that must be done by a Hohmann transfer. From an operational point of view, this 3 3 is advantageous. H H We now describe se steps in detail in following sections. H3 H3 3 H H3 Point Vc (km) V by (km/s) S (%) P (km) solution 7: A rsample obtained targeting a pointtc!p in 6(days) C 6 (0.99, /). Figure /)..5 Figure () 7: A sample solution obtained by targeting a point in C (0.99, rp (A) % Model 6 6 This solution is interesting due facttarget targetpoint pointxcxcis is33 This solution is interesting due to to fact 00 (B) % 433 When our spacecraft, P, is in motion about Mars, from arrival at xc to Mars from Mars. Also, it takes approximately same amount of time reach km km from Mars. approximately amount of43 time toto reach (B) (C)motion (D) of P by planar elliptic restricted (C) Also, 9897it takes same -4.9% ballistic capture at rp, we(a) model Mars periapsis as case in Figure Left: Sun-centered inertial frame.right: Right: () Mars periapsis case in.04 Figure Left: frame. three-body problem, which () rp takes into account Mars eccentricity ep =H (D) as sun-centered -8.% inertial43 H H rp We view.5 mass of P to be zero..5 rotating Mars-centered frame. H4 H4 rotating Mars-centered frame. H4.5 () Vc or V (km/s) Vc or V (km/s) rp The planar elliptic restricted three-body problem studies motion of a massless particle, P, under gravitational field generated by mutual elliptic motion of two primaries, P, P, of masses m, m, respectively. In this motion. paper, Capt. P is Sun, and P is Mars. The equations for Ball. Capt. of P are mann x00 0 y 0 =!x, 0.5 Hohmann y 00 + x0 =!y..5 5 The derivatives of rp (km) 0subscripts in Eq. () are partial (x, y)!(x, y, f ) =, + 0 ep cos f (b) f = /4 where potential function is (x, y) = µ µ (x + y ) µ( r r µ), From this table it can be seen that time for spacecraft to go from xc to rpisis computed on orderbyofassuming a year. This should time should be able to bein decreased Ball. Capt. This spacecraft being already heliocentric This isvery computed by assuming spacecraft as as being already in heliocentric Hohmann by slightly adjusting V so that distance between spacecraft and c () at Earth s SOI; 4) A second maneuver, VcV performed to inject 0.5 cis, is at Earth s SOI; 4) A second maneuver,, performed to inject 0 Mars0.5decreases.5 rapidly..5 more 5 High altitude Mars s easily accessible rp (km) spacecraft into ballistic capture ; between two maneuvers, 0 ; spacecraft into ballistic capture 5) 5) In In between two maneuvers, Cheaper than Hohmann transfer(s) moves heliocentric space from both Earthand andmars, Mars, spacecraft moves in in heliocentric space farfar from both Earth ()spacecraft (c) f = / 0 refore dynamics is that of two-body problem 5 andand refore dynamics is that two-body problem [9].[9]. The No manoeuvre atof arrival needed! parameters of optimization (to be picked and held fixed) are: 8 The parameters of optimization (to be picked and held fixed) are: (3) on of Hohmann bitangential transfers and ballistic capture and r = (x + µ) + y, r = (x + µ ) + y. byequations ()capture sets C(e, fbarycentric, 0.99, f0 = 0, /4, /. are written in a nonuniformly rotating, adimen0 ), e = / /

14 Ballistic capture in news Topputo & Belbruno, arxiv, 05 Topputo & Belbruno, CMDA, 05 9

15 Reaching Mars with micro propulsion Aim: target a point in deep space, xc, to ensure capture using on-board Ion Propulsion Assumptions: m0 = kg, S/C in parabolic state wrt Earth T(AU) =.3e^(-.6*AU) [mn] Isp(AU) = 3887.*AU^ 384*AU [s] TOF = 79 days (3. years), mp =.79 kg (mass at Mars = 9. kg) Y (AU) c x c Earth@Dep Mars@r p X (AU) Thrust (mn) Thrust angle (rad) T Tmax time (days) time (days) φ 0

16 Escaping Earth Cubesats likely launched as piggy back payloads No control on launch date Released in low-altitude (LEO, GTO) Earth-bounded Escaping with on-board propulsion may be cumbersome Long duration needed to escape - Pointing, operations, costs, etc. strongly affected Much radiation dose accumulate - Solar arrays, shielding, etc. strongly affected Images taken from

17 Dual propulsion idea Both chemical and low-thrust propulsion on-board system How it works: S/C launched as piggy back in LEO Earth escape achieved with chemical, impulsive burn Short duration, less radiation Masses involve in chemical propulsion are thrown away Dual-staged S/C, interfaces, complexity Cruise accomplished with on-board low-thrust propulsion Concept proven in ESA study in 0 Implemented by Lisa Pathfinder (for or reasons)

18 Dual propulsion for cubesats (a) Ballistic escape via impulsive maneuver and lunar (b) Transfer trajectory as viewed in Sun-centered (b) Ballistic capture and low-thrust descending at Cubesat achieves escape with its own chemical propulsion system Cubesat performs Earth- Mars transfer with its own low-thrust propulsion Upon arrival, ballistic capture is performed (and low-altitude achieved) 3

19 Wrap up and conclusions Cubesats have been used successfully for Earth observation/communication Wandering in solar system with extremely lowresources space systems (cubesats) raises a set of completely new challenges in astrodynamics Ideas have been presented to attempt answering se new questions These include Performing ballistic capture Having a dual propulsion system More in-depth analyses needed 4