ACT Global Optimization Competition Workshop

Transcription

1 ACT Global Otimization Cometition Worksho Mikhail S.KONSTANTINO Gennadiy G. FEDOTO Moscow Aiation Institute iachesla G. PETUKHO Khruniche State Research and Production Sace Center

2 OBJECTIE FUNCTION: PRELIMINARY CONSIDERATION J m f (- ast ) T ast Imact oint in the icinity of asteroid s erihelion 1) Retrograde orbit ) Imact in the icinity of S/C erihelion 3) S/C ahelion distance should be maimized Minimum roellant Launch date and transfer duration should be otimized Giant lanets swingbys or low-thrust should be considered 1) Swingbys are refered in comarison with thrusting ) Transfer duration tends to maimal

3 THEREFORE Thanks ACT & Dario Izzo for an interesting roblem

4 FLYBY CONSTRAINTS: ASYMPTOTIC ELOCITY ROTATION Maimal angles flyby sequences E JSA or E JSJA are refered Ma betta deg Mercury enus Earth Mars Juiter Saturn - heliocentric arabola inf km/s

5 LOW-THRUST TRAJECTORY: BOUNDARY CONDITIONS ( ) ( ) ( ) ( ) ( ). m m Initial oint: Final oint: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin sin cos sin cos y z y z y y y y z y y l l M M γ β γ β β Swingby: where µ π β min ma 1 1 arccos r ( ) ( ) ( ) ( ) ( ) ( ). 1 ) ( sgn T T T m T T T m ast ast ast ast

6 ERSIONS OF TRAJECTORY PROFILE AND COMPUTATIONAL TECHNIQUES 1. Flyby sequence in the inner Solar system (ahelion increasing asidal line ositioning) 1.1. E E or E coast flyby sequence 1.. EE E E or trajectory using low-thrust arc. Retrograde trajectory shaing in the outer Solar system using Giants flybys and/or low thrust EJSJA EJSA ESJA ESA or EJA flyby sequence using low-thrust arcs if necessary -or - EA trajectory using low-thrust arc(s). Lambert soler Conic-atched interlanetary trajectory calculation based on the Keler equation TPBP soler (numerical integration targeting to the net lanet arying the trajectory arc duration and dearture asymtotic elocity direction). Trajectory arc can include one thrusting arc using arametric thrust steering Power-limited (LP) roblem soler Constant ejection elocity (LP-to-CE) roblem soler

7 PROBLEM COMPLEXITY 1. Large number of routes under consideration: Swingby 1 st lanet nd lanet M th lanet 1 orbit orbits 1 st half of orbit nd half of orbit N orbits Descending node Ascending node. Local otimization of low-thrust arcs: bifurcations numerical stability and conergence roblems Global otimization assumes using of reliable local otimization techniques. Local otimization technique of multily swingbys low-thrust trajectory is insufficiently elaborated. So some simlifications are ineitable. Due to roblem comleity researcher s intuition and eerience should be used along with otimization techniques using

8 SOFTWARE EPOCH LP-roblem soler E-ProTO multi-orbits trajectory otimization CE-roblem GALTT Graity-Assisted Low-Thrust Trajectory Ealuation PL CE-roblem soler Other software: LP CE continuation (frozen trajectory structure) Lambert solers Conic-atched interlanetary trajectory calculation based on the Keler equation Legend: - reiously deeloed and eloited software - modified software - new-deeloed software

9 TRAJECTORY DESCRIPTION: INITIAL FLYBY SEQUENCE 1. E days. EE days ericenter radius km 3. EE days ericenter radius km 4. EEE days ericenter radius km

10 TRAJECTORY DESCRIPTION: EARTH-to-EARTH TRAJECTORY ARC 1. Tangential thrust (braking). LP-trajectory 3. CE-trajectory 1 Boundary conditions continuation wrt. graity arameter Pitch deg. Continuation LP CE transfer duration 5. EEEE 1. days ericenter radius km thrusting days MJD

11 TYPICAL POWER-LIMITED PROBLEM T 1 Purose: To minimize J a T d a for a dynamical system obeying the differential equations a Ω and haing following boundary conditions: t : () d()/ m() m t T: (T) f d(t)/ f.. Here T is final time a is thrust acceleration and are ectors of sacecraft osition and elocity resectiely Ω is graity otential. Let aly Pontryagin s maimum rincile to reduce this OCP to the TPBP. Hamiltonian of this dynamical system is H -a T a/ T T Ω T a. So otimal control is a and equations of otimal motion become following: d Ω d Ω The boundary conditions for rendezous mission has form: t : () d()/ t T: (T) f d(t)/ f. In fact it is necessary to sole equation f(z) where unknown TPBP arameters -d /. T f f ( ( ) () z) is ector of residuals is ector of ( T ) f z ()

12 ALGORITHMS FOR POWER-LIMITED PROBLEM Continuation (homotoy) technique is used for LP-roblem soling. Purose of continuation method Regularization of numerical trajectory otimization i.e. elimination (if ossible) the methodical deffects of numerical otimization. Particularly there was stated and soled roblem of trajectory otimization using triial initial aroimation (the coasting along the initial orbit for eamle).

13 CONTINUATION TECHNUQUE Continuation (Homotoy) Technique Alication to Otimal Control Problem (OCP) Problem: to sole non-linear system f ( z) with resect to ector z (1) Otimal motion equations (after alying the maimum rincile): d H d H Let z - initial aroimation of solution. Then f( z ) b where b - residuals when z z. Let consider immersion z(τ) where τ is a scalar arameter and equation f( z) ( 1 τ) b (3) with resect to z(τ). Obiously z(1) is solution of eq. (1). Let differentiate eq. () on τ and resole it with resect to dz/dτ: () Boundary conditions (an eamle): Boundary alue roblem arameters and residuals: Sensitiity matri: ( ) ( T ) k z () f ( T ) k f z (T ) z dz 1 fz z b z z dτ ( ) ( ) Just after integrating eq. (4) from to 1 we hae solution of eq. (1). Equation (4) is the differential equation of continuation algorithm (the formal reduction of non-linear system (1) into initial alue roblem (4)). Differential equations (4) are integrated using a high-order Runge-Kutta method with adatie ste size control. (4) Associated system of otimal motion o.d.e. and erturbation equations for residuals and sensitiity matri calculation: d H d H d H H z z d H H z z z z Etended initial conditions: ( ) ( T ) k I z z

14 OCP-SOLER BASED ON CONTINUATION ALGORITHM Reduction of otimal control roblem to the boundary alue roblem by maimum rincile Initial aroimation z Initial residuals b calculation by otimal motion o.d.e. integrating for gien guess alue z of boundary alue roblem arameters CONTINUATION METHOD 1st ersion of o.d.e. righthand side calculation Continuation method s o.d.e. integrating dz 1 fz z b z z dτ ( ) ( ) with resect to τ from to 1 nd ersion of o.d.e. right-hand side calculation Associated integrating of otimal motion equations and erturbation equations for current z(τ) to calculate current residuals f(zτ) and sensitiity matri f z (zτ) Integrating of otimal motion equations for current z(τ) to calculate current residuals f(zτ) and for ertubed z(τ) to calculate f z (zτ) by finitedifference technique Solution z(1)

15 CONTINUATION WITH RESPECT TO GRAITY PARAMETER (1/3) Occasional reasons of continuation algorithm failure: sensitiity matri degeneration (bifurcation of otimal solutions) Mostly bifurcations of otimal lanetary trajectories are connected with different number of comlete orbits. If angular distance will remain constant during continuation the continuation way in the arametric sace will not cross boundaries of different kinds of otimal trajectories. So the sensitiity matri will not degenerate. The urose of the technique modification - to fi angular distance of transfer during continuation Earth-to-Mars rendezous launch date October 1 1 m/s T days 1 - Earth at launch - Earth at arrial 3 - Mars at arrial 4 - intermediate trajectories (τ < 1) 5 - otimal trajectory (τ 1) Sequence of trajectory calculations using basic continuation method Sequence of trajectory calculations using continuation with resect to graity arameter

16 CONTINUATION WITH RESPECT TO GRAITY PARAMETER (/3) Let () (T) - dearture lanet osition when t and tt; k - target lanet osition when tt. Let suose rimary graity arameter to be linear function of τ and let choose initial alue of this graity arameter µ using following condition: 1) angular distances of transfer are equal when τ and τ1; ) When τ1 rimary graity arameter equals to its real alue (1 for dimensionless equations) The initial aroimation is SC coast motion along dearture lanet orbit. Let the initial true anomaly equals to ν at the start oint S and the final one equals to ν k ν ϕ at the final oint K (ϕ is angle between and rojection of k into the initial orbit lane). The solution of Keler equation gies corresonding alues of mean anomalies M and M k (ME-e sine where E arctg{[(1-e)/(1e)].5 tg(ν/)} is eccentric anomaly). Mean anomaly is linear function of time at the kelerian orbit: MM n (t-t ) where n(µ /a 3 ).5 is mean motion. Therefore the condition of angular distance inariance is M k π N re ntm where N re is number of comlete orbits. So initial alue of the rimary graity arameter is µ [( M k π N re - M )/T] a 3 and current one is µ(τ)µ (1-µ ) τ. The shae and size of orbits should be inariance with resect to τ therefore (t τ)µ(τ).5 (t 1).

17 CONTINUATION WITH RESPECT TO GRAITY PARAMETER (3/3) f f τ Equations of motion: Boundary conditions: Residuals: Boundary alue roblem arameters: Equation of continuation method: where b f(z ) z ( T) o ( T) & d d o d t d t ( ( T) ) ( ( T) & ) τ & 1 / τ µ 1 ( τ) µ τ k o o && µ ( τ ) Ω && µ ( τ) Ω 1 / ( ) &( ) µ ( τ) 1 / ( T) &( T) µ ( τ). f d d d d µτ ( ) Ω z z z µτ ( ) ( ) z Ω Ω z z µ Ω µτ ( ) Ω τ τ z z µ Ω µτ ( ) ( ) τ τ Ω Ω z z ( ) &( ) ( ) &( ) 1 µ 1 / z z τ τ µ ( τ) τ ( ) & ( ) ( ) & ( ) ( ) & ( ) E. & & τ τ k ( T) k 1 / ( T ) µ ( τ) z ( () d ()/) T ( & ) dz f dτ 1 z k f ( z) b τ k o z() z o T

18 Purose: To minimize for a dynamical system obeying the differential equations T w P J δ Ω w P dm m P d δ δ e and haing following boundary conditions: t : () d()/ m() m t T: (T) f d(t)/ f. Here δ is ste-like thrusting function P thrust w ehaust elocity m sacecraft mass. Ω Ω m m P d d w P dm m P d δ δ δ t : () d()/ m() m t T: (T) f d(t)/ f m (T) where ste-like thrusting function > if if 1 s s ψ ψ δ and switching function w m m s 1 ψ In fact it is necessary to sole equation f(z) where ) ( ) ( ) ( ) ( T T T m f f z f is ector of residuals and () () () m z Pontryagin s maimum rincile reduces this OCP to the following TPBP: TYPICAL CE-PROBLEM WITH THRUST SWITCHINGS

19 LP-to-CE CONTINUATION USING FROZEN TRAJECTORY STRUCTURE 1. Sequence of thrusting and coasting arcs is assumed fied. Equations of otimal motion are following: d Ω δ (the equations corresond to LP-roblem d if τ and to CE-roblem if τ1) Ω dm P δ w dm P δ. m 1 m. Switching function: ψ m w 3. Initial conditions: () () m() m. ( 1τ ) P τ m ψ(t 1 ) ψ(t ) ψ(t) Thrusting 4. Final conditions: f τ msgn m ti T i k 1 1 M k ast ( t ) ψ ( t ) T ( ) ast ( t ) ψ ( t ) N arc 1 N arc 1 t 1-1 t 1 t T t t 1 t t 3 k( t i ) ε ε t i Multilier roiding minimal arc length ε

20 EARTH-to-ASTEROID POWER-LIMITED TRAJECTORY Continuation wrt. graity arameter Continuation wrt. boundary conditions

21 TRAJECTORY DESCRIPTION: FINAL (CE) EARTH-to-ASTEROID TRAJECTORY 6. EEEEA 613. days ericenter radius 76. km thrusting days Pitch deg Pitch Yaw deg Yaw MJD MJD

22 FINAL (CE) EARTH-to-ASTEROID TRAJECTORY Inclination deg MJD 1.95 Eccentricity e ma MJD

23 FINAL (CE) EARTH-to-ASTEROID TRAJECTORY 3.E8 Pericenter radius km.5e8.e8 1.5E8 1.E8 5.E7 r min km.e MJD.4E9.E9 Aocenter radius km.e9 1.98E9 1.96E9 1.94E9 1.9E9 1.9E9 1.88E MJD

24 FINAL (CE) EARTH-to-ASTEROID TRAJECTORY SMA km 1.14E9 1.1E9 1.1E9 1.8E9 1.6E9 1.4E9 1.E9 1.E9 9.8E8 9.6E8 9.4E MJD RAAN deg MJD Argument of ericenter deg MJD

25 FINAL (CE) EARTH-to-ASTEROID TRAJECTORY Heliocentric distance km.e9 1.8E9 1.6E9 1.4E9 1.E9 1.E9 8.E8 6.E8 4.E8.E8.E MJD elocity km/s 4.5E1 4.E1 3.5E1 3.E1.5E1.E1 1.5E1 1.E1 5.E.E min m/s MJD fin m/s

26 COMPLETE TRAJECTORY Objectie function alue: kg km /sec ; Route: EEEEA Launch date: JD (June ) Escae elocity: km/sec Total duration: days Final S/C mass: kg 1. 1 TW9 arrial 8. 5 th swingby (Earth) 9. 3 rd thrusting arc 7. nd thrusting arc 4. 3 rd swingby (enus). 1 st swingby (enus) 3. nd swingby (Earth) 1. Earth dearture 6. 1 st thrusting arc 5. 4 th swingby (Earth)

27 EXAMPLE OF LOCAL OPTIMIZATION COMPLEXITY: DIRECT EARTH-to-ASTEROID TRAJECTORY J89 J33 J351 J59 1. Initial S/C orbit: line of asides along to asteroid s line of asides; ericenter radius equals to earth orbit radius at dearture date; aocenter radius corresonds to asymtotic elocity.5 km/s; inclination equals to.. Final S/C orbit: line of asides along to asteroid s line of asides; ericenter radius equals to asteroid s ericenter radius; inclination and aocenter radius are aried. 3. Problem: minimum-time transfer to the final orbit with constrained minimal heliocentric distance (. AU). The constraint is regulated by number of orbits (continuation wrt. graity arameter) final inclination and final aocenter radius. 4. Solers: a) Aeraged otimal control roblem (maimum rincile continuation technique E-ProTO software). b) Unaeraged otimal control roblem (maimum rincile continuation technique aeraged solution as an intial guess E-ProTO software)

28 DIRECT EARTH-to-ASTEROID TRAJECTORY: RESULTS INITIAL ORBIT: AU i FINAL ORBIT: AU i FINAL MASS: kg MINIMAL HELIOCENTRIC DISTANCE:.958 AU TRANSFER DURATION:.658 (thrusting) (coasting) 5.44 years J kg km /s

29 TECHNIQUES OF MULTIREOLUTIONAL OPTIMIZATION Equinoctial orbital elements are used: e e cos Ω ω e y e sin( Ω ω ) i tan i cos Ω µ h ( ) i y tan i sin Ω F ν ω Ω Here e ω ν i Ω are Kelerian elements µ - rimary graity arameter. It is considered conentional CE-roblem without any constraints on thrust direction. Maimum rincile reduces the roblem into TPBP. The numerical aeraging oer the orbital eriod is used for comutational consumtion reducing and numerical stability increasing. A numerous ersions of boundary conditions were considered. The continuation (homotoy) rocedure was used to sole minimum-time roblem (see 4.1.1). The simle tyical guess alues: h ±1 e ey i iy (initial alues of co-state ariables) T 1 (dimensionless orbital eriod referred to the initial orbit) as a rule roides stable conergence of otimization. Of course initial alues of co-states from the OCP solution haing close boundary conditions roides imroed conergence. Soler of minimum-roellant roblem uses minimum-time solution as initial aroimation. The factored secant udate algorithm is used for minimum-roellant roblem. Both techniques demonstrates their robustness and efficiency and there were used for a numerous alied roblems.

30 CONCLUSION 1. Tolerable objectie function alue was obtained without using Juiter/Saturn flybys. Global otimization should be suorted by reliable methods of local otimization 3. Continuation technique allows to find global minimum among local minimums deending on restricted number of arameters (boundary conditions transfer duration number of orbits) 4. THANK YOU FOR ATTENTION