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V.G. Petukhov E-mail: petukhov@mtu-net.ru

V.G. Petukhov E-mail: petukhov@mtu-net.ru. Khrunichev State Research and Production Space Center. LOW THRUST TRAJECTORY OPTIMIZATION. 2. V.G. Petukhov. Low Thrust Trajectory Optimization. CONTENTS. INTRODUCTION 1. CONTINUATION METHOD

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V.G. Petukhov E-mail: petukhov@mtu-net.ru

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  1. V.G. PetukhovE-mail: petukhov@mtu-net.ru Khrunichev State Research and Production Space Center LOW THRUST TRAJECTORY OPTIMIZATION

  2. 2 V.G. Petukhov. Low Thrust Trajectory Optimization CONTENTS INTRODUCTION 1. CONTINUATION METHOD 2. OPTIMAL PLANETARY TRANSFERVARIABLE SPECIFIC IMPULSE PROBLEM 3. OPTIMAL TRANSFER TO LUNAR ORBITVARIABLE SPECIFIC IMPULSE PROBLEM 4. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS CONSTANT SPECIFIC IMPULSE PROBLEM CONCLUSION

  3. 3 V.G. Petukhov. Low Thrust Trajectory Optimization INTRODUCTION It is presented common methodical approach to computation different problems of low thrust trajectory optimization. This approach basis is formal reduction of maximum principle’s two points boundary value problem to the initial value problem. This reduction is realized by continuation method.

  4. INTRODUCTION 4 V.G. Petukhov. Low Thrust Trajectory Optimization Low-thrust trajectory optimization: T.M. Eneev, V.A. Egorov, V.V. Beletsky, G.B. Efimov,M.S. Konstantinov, G.G. Fedotov, Yu.A. Zakharov, Yu.N. Ivanov, V.V. Tokarev, V.N. Lebedev,V.V. Salmin, S.A. Ishkov, V.V. Vasiliev, T.N. Edelbaum, F.W. Gobetz, J.P. Marec, N.X. Vinh, K.D. Mease, C.G. Sauer,C. Kluever, V. Coverstone-Carroll, S.N. Williams,M. Hechler, etc. Continuation method: M. Kubicek, T.Y. Na, etc.

  5. INTRODUCTION 5 V.G. Petukhov. Low Thrust Trajectory Optimization • Conventional numerical optimization methods shortcomings • small region of convergence; • computational unstability; • neessity to select initial approximation when it is absent any a-priori information concerning solution. • These problems partially are connected with optimization problem nature (problems of optimal solution stability, existance, and bifurcation). But most of numerical methods introduce own restrictions which are not directly connected with the mathematical problem properties. So the convergencedomain of practically all numerical methods is essential smaller in comparison with the extremal point attraction domain in the space of unknown boundary value problem parameters. • Methodical shortcomings are connected with the computational unstability, the convergence domain boundedness, and (in case of direct methods) the big problem dimensionality.

  6. INTRODUCTION 6 V.G. Petukhov. Low Thrust Trajectory Optimization Purpose of new continuation method “Regularization” of numerical trajectory optimization, i.e. elimination (if possible) the methodical deffects of numerical optimization. Particularly, the was stated and solved problem of trajectory optimization using trivial initial approximation (the coasting along the initial orbit for example). Applied trajectory optimization problems under consideration 1. Planetary low thrust trajectory optimization (the variable specific impulse problem); 2. Lunar low thrust trajectory optimization within the frame of restricted problem of three bodies (the variable specific impulse problem); 3. Optimal low thrust trajectories between non-coplanar elliptical orbits (the constant specific impulse problem).

  7. 7 1. CONTINUATION METHOD Problem: to solve non-linear system (1) with respect to vector z Let z0 - initial approximation of solution. Then , (2) where b - residuals when z = z0. Let consider z(), where  is a scalar parameter and equation (3) with respect to z(). Obviously, z(1is solution of eq. (1). Let differentiate eq. (2) on  and solve it with respect to dz/d: (4) Just after integrating eq. (4) from 0 to 1 we have solution of eq. (1).Equation (4) is the differential equation of continuation method(the formal reduction of non-linear system (1) into initial value problem (4)). V.G. Petukhov. Low Thrust Trajectory Optimization

  8. CONTINUATION METHOD 8 Application of continuation method to optimal controlboundary value problem Optimal motion equations(after principle maximum application): Boundary conditions (an example): Boundary value problem parameters and residuals: Sensitivity matrix: Associated system of optimal motion o.d.e. andperturbation equations for residuals and sensitivitymatrix calculation: Extended initial conditions: V.G. Petukhov. Low Thrust Trajectory Optimization

  9. CONTINUATION METHOD 9 Optimal control problem reductionto the boundary value problemby maximum principle CONTINUATION METHOD Initial residuals b calculationby optimal motion o.d.e. integratingfor given initial approximation z0 of boundary value problem parameters Initialapproximation z0 2nd versionof o.d.e. right partscalculation 1st versionof o.d.e. right parts calculation Continuation method’s o.d.e. integrating with respect to  from 0 to 1 Associated integrating of optimal motion equations and perturbations equations for current z() to calculate current residuals f(z,) and sensitivity matrix fz(z,) Integrating of optimal motion equations for current z() to calculate current residuals f(z,) and for pertubed z() to calculate fz(z,) by finite-difference Solutionz(1) V.G. Petukhov. Low Thrust Trajectory Optimization Using continuation methodfor low-thrust trajectory optimization problem

  10. 10 V.G. Petukhov. Low Thrust Trajectory Optimization 2. OPTIMAL PLANETARY TRANSFERVARIABLE SPECIFIC IMPULSE PROBLEM

  11. OPTIMAL PLANETARY TRANSFER 11 Cost function:(constant power, nuclear electric propulsion) (variable power, solar electric propulsion) Equations of motion:d2x/dt2=x+a Initial conditions:x(0)=x0(t0), v(0)=v0(t0)+Ve Boundary conditions 1) rendezvous: x(T)=xk(t0+T), v(T)=vk(t0+T) 2) flyby: x(T)=xk(t0+T) where x, v - SC position and velocity vectors,  - gravity field force function,a - thrust acceleration vector, x0, v0 - departure planet position and velocity vectors,xk, vk - arrival planet position and velocity vectors, V - initial hyperbolic excess of SC velocity, e -direction of V, N(x,t) - the current power to the initial one ratio. V.G. Petukhov. Low Thrust Trajectory Optimization 2.1. TRAJECTORY OPTIMIZATION PROBLEM

  12. OPTIMAL PLANETARY TRANSFER 12 Hamiltonian: Optimal control: Optimal Hamiltonian: Optimal motion equations: Residuals: Boundary value problem parametersand initial residuals vectors: (rendezvous) (flyby) V.G. Petukhov. Low Thrust Trajectory Optimization 2.2. OPTIMAL MOTION EQUATIONS(CONSTANT POWER)

  13. OPTIMAL PLANETARY TRANSFER 13 2.3. EQUATIONS OF CONTINUATION METHOD Boundary value problem immersioninto the one-parametric family: Boundary value problem parametersinitial value and solution: Differential equations of continuationmethod: Differential equations for calculation right parts of continuation method’s differential equations: V.G. Petukhov. Low Thrust Trajectory Optimization

  14. OPTIMAL PLANETARY TRANSFER 14 2.4. TRAJECTORY SEQUENCEWHICH IS CALCULATED BY CONTINUATION METHODUSING COASTING AS INITIAL APPROXIMATION 5 Earth-to-Mars, rendezvous,launch date June 1, 2000, V= 0 m/s,T=300 days 1 - coast trajectory (1= 0)2-4 - intermediate trajectories (0< 2 < 3 < 4 < 1) 5 - final (optimal) trajectory (5= 1) 4 3 2 1 V.G. Petukhov. Low Thrust Trajectory Optimization

  15. OPTIMAL PLANETARY TRANSFER 15 2.5. NUMERICAL EXAMPLESOPTIMAL TRAJECTORIES TO MERCURY AND NEAR-EARTH ASTEROIDS V.G. Petukhov. Low Thrust Trajectory Optimization

  16. OPTIMAL PLANETARY TRANSFER 16 OPTIMAL ORBITAL PLANE ROTATION EXAMPLES Optimal 90°-rotationof orbital plane Optimal 120°-rotationof orbital plane V.G. Petukhov. Low Thrust Trajectory Optimization

  17. OPTIMAL PLANETARY TRANSFER 17 V.G. Petukhov. Low Thrust Trajectory Optimization EXAMPLE: INITIAL HYPERBOLIC EXCESS OF VELOCITY IMPACT

  18. OPTIMAL PLANETARY TRANSFER 18 V.G. Petukhov. Low Thrust Trajectory Optimization EXAMPLE: NUCLEAR (RIGHT) AND SOLAR (LEFT) ELECTRIC PROPULSION

  19. OPTIMAL PLANETARY TRANSFER 19 2.7. METHOD OF CONTINUATION WITH RESPECT TO GRAVITY PARAMETER Reasons of continuation method failure: sensitivity matrix degeneration (bifurcation of optimal solutions) Mostly bifurcations of optimal planetary trajectories are connected with different number of complete orbits If angular distance will remain constant during continuation, the continuation way in the parametric space will not cross boundaries of different kinds of optimal trajectories. So, the sensitivity matrix will not degenerate The purpose of method modification - to fix angular distance of transfer during continuation Sequence of trajectory calculation using basic continuation method Sequence of trajectory calculation usingcontinuation with respect to gravity parameter V.G. Petukhov. Low Thrust Trajectory Optimization

  20. OPTIMAL PLANETARY TRANSFER 20 Let x0(0), x0(T) - departure planet position when t=0 and t=T; xk - target planet position when t=T. Let suppose primary gravity parameter to be linear function of , and let choose initial value of this gravity parameter 0 using following condition:1) angular distances of transfer are equal when =0 and =1; 2) When =1 primary gravity parameter equals to its real value (1 for dimesionless equations) The initial approximation is SC coast motion along departure planet orbit. Let the initial true anomaly equals to 0 at the start point S, and the final one equals tok=0+ at the final point K ( is angle between x0 and projection of xk into the initial orbit plane). The solution of Kepler equation gives corresponding values of mean anomalies M0 and Mk (M=E-esinE, where E=2arctg{[(1-e)/(1+e)]0.5tg(/2)} is eccentric anomaly). Mean anomaly is linear function of time at the keplerian orbit: M=M0+n(t-t0), where n=(0/a3)0.5 is mean motion. Therefore, the condition of angular distance invarianct is Mk+2Nrev=nT+M0, where Nrev is number of complete orbits. So initial value of the primary gravity parameter is 0=[( Mk+2Nrev - M0)/T]2a3, and current one is ()=0+(1-0) . The shape and size of orbits should be invariance witn respect to , therefore v(t, )=()0.5v(t, 1). V.G. Petukhov. Low Thrust Trajectory Optimization

  21. OPTIMAL PLANETARY TRANSFER 21 Equations of motion: Boundary conditions: Residuals: Boundary value problem parameters: Equation of continuation method: where z = (pv(0), dpv(0)/dt)T = b = f(z0) V.G. Petukhov. Low Thrust Trajectory Optimization

  22. OPTIMAL PLANETARY TRANSFER 22 Numerical example: Mercury rendezvousConstant power, launch date January 1, 2001, transfer duration 1200 daysAll solutions are obtained using coasting along the Earth orbit as initial approximation Basic versionof continuation method Continuation with respect to gravity parameter 5 complete orbits 7 complete orbits V.G. Petukhov. Low Thrust Trajectory Optimization

  23. OPTIMAL PLANETARY TRANSFER 23 V.G. Petukhov. Low Thrust Trajectory Optimization EXAMPLES: OPTIMAL TRAJECTORIES TO MAJOR PLANETS OF SOLAR SYSTEM

  24. 24 V.G. Petukhov. Low Thrust Trajectory Optimization 3. OPTIMAL TRANSFER TO LUNAR ORBITVARIABLE SPECIFIC IMPULSE PROBLEM It is considered the transfer of SC using variable specific impulse thruster from a geocentric orbit into an orbit around the Moon.The SC trajectory is divided into the 4 arcs: 1) Geocentric spiral untwisting from an initial orbit up to a geocentric intermediate orbit; 2) L2-rendezvous trajectory; 3) Trajectory from the point L2 of Earth-Moon system to a selenocentric intermediate orbit; 4) Selenocentric twisting down to a final orbit. The 1st and 4th arcs can be eliminated if initial and final orbits have high altitude. Trajectories of 2nd and 3rd arcs are defined by continuation method.

  25. OPTIMAL TRANSFER TO LUNAR ORBIT 25 VALIDATION OF TRAJECTORY DIVIDING INTO ARCS Curves of zero velocity(contours of Jacoby’s integral) 1. Typical duration of hyperbolic motion within Hill’s sphere of Moon is ~1 days. 2. Typical velocity increment due to thrust acceleration is ~10 m/s for 1 day if thrust acceleration is ~0.1 mm/s2. 3. Opening width in the L2 vicinity is ~60000 km for SC relative velocity 10 m/s on the Hill’s sphere. To capture SC into the Moon orbit using electric propulsion (thrust acceleration ~0.1 mm/s2) SC relative velocity should be not greater ~10 m/s when distance from L2 is less ~30000 km. Hill’s sphere Region of satellite motion to Earth opening width ~60000 km Moon Region of SC motion for critical Jacoby’s constant Region of SC motion for SC relative velocity 10 m/s on the Hill’s sphere V.G. Petukhov. Low Thrust Trajectory Optimization

  26. OPTIMAL TRANSFER TO LUNAR ORBIT 26 EARTH_MOON L2 RENDEZVOUSModel problem of transfer from circular Earth orbit(altitude 250000 км, inclination 63°, right ascension of ascending node 12°,lattitude argument 0°; launch date January 5, 2001) 4 complete orbits 5 complete orbits 6 complete orbits 7 complete orbits 0.5 a,mm/s2 0.0 0.0 95.0 t, days 0.0 95.0 0.0 95.0 0.0 95.0 t, days t, days t, days V.G. Petukhov. Low Thrust Trajectory Optimization

  27. OPTIMAL TRANSFER TO LUNAR ORBIT 27 EARTH-MOON L2 RENDEZVOUS USING MOON GRAVITY ASSISTED MANEUVER 2.5 orbits 7.5 orbits Gravity assisted maneuver Initial Moon position Initial L2 position Initial orbit Moon orbit Final L2 position Final Moon position 1.0 0.5 Thrust acceleration, mm/s2 Thrust acceleration, mm/s2 0.0 0.0 0 Т, days 95 0 Т, days 95 V.G. Petukhov. Low Thrust Trajectory Optimization

  28. OPTIMAL TRANSFER TO LUNAR ORBIT 28 TRANSFER FROM EARTH-MOON L2 INTO CIRCULAR MOON ORBIT Final orbit: r = 20000 km, i = 90.Transfer: 2.5 orbits, T = 20 days Final orbit: r = 30000 km, i = 0.Transfer: 2.5 orbits, T = 15 days Final orbit: r = 30000 km, i = 0.Transfer: 1.5 orbits, T = 10 days Final (intermediate) orbit Final L2 position Initial L2 position Moon 0.5 mm/s2 Thrustacceleration 0 mm/s2 0 Time, d 10 0 Time, d 15 0 Time, d 20 V.G. Petukhov. Low Thrust Trajectory Optimization

  29. OPTIMAL TRANSFER TO LUNAR ORBIT 29 TRANSFER FROM EARTH-MOON L2 INTO ELLIPTICAL MOON ORBIT (i=90°, hp=300 km, ha=10000 km, 10.5 orbits) Initial L2 position Final orbit Moon Final L2 position 1 mm/s2 Thrustacceleration 0 mm/s2 0 Time, d 30 V.G. Petukhov. Low Thrust Trajectory Optimization

  30. OPTIMAL TRANSFER TO LUNAR ORBIT 30 TRANSFER FROM ELLIPTICAL EARTH ORBIT INTO CIRCULAR MOON ORBIT.TRAJECTORY ARCS Geocentric spiral untwisting Earth-Moon L2 rendezvous Transfer from Earth-Moon L2 into equatorial 30000-km circular Moon orbit Moon Earth Earth 0.5 mm/s2 0.5 mm/s2 Thrustacceleration Thrustacceleration 0 mm/s2 0 mm/s2 0 Time, d 95 0 Time, d 95 V.G. Petukhov. Low Thrust Trajectory Optimization

  31. 31 V.G. Petukhov. Low Thrust Trajectory Optimization 4. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITSCONSTANT SPECIFIC IMPULSE PROBLEM

  32. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 32 V.G. Petukhov. Low Thrust Trajectory Optimization Equations of SC motion are written in the equinoctial elements which have not singularty when eccentricty or inclination is nullified. The optimal control problem is reduced into the two-point boundary value problem by maximum principle. This boundary value problem is reduced into the initial value problem by continuation method. It is necessary to integrare system of optimal motion o.d.e. (P-system) and to calculate partial derivatives of final state vector of P-system on the initial value of co-state variables to calculate right parts of continuation method’s o.d.e. The right parts of the P-system are numerically averaged over true lattitude during the P-system integration. Partial derivative of final state vector of P-system on the initial value of co-state vector is calculating using finite differences. The boundary value problem residual vector are calculated as result of first integration of P-system. 6 additional integrations of P-system is required to calculate sensitivity matrix using finite differences. As result, the right parts of the continuation method’s o.d.e. are calculated after solving correspoding linear system. System of continuation method’s o.d.e. is numerically integrated on continuation parameter  from 0 to 1. As a result, the optimal solution is calculated.

  33. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 33 4.1. EQUATION OF MOTION Thrust acceleration components in the orbital reference frame:  - thrust switching function, P - thrust, m - SC mass,  - pitch,  - yaw System of equinoctial elements:  - primary gravity parameter; p, e, , , i,  - keplerian elements. Equation of motionin the equinoctialelements: w - exhaust velocity Boundary conditions:t = 0: t = T: V.G. Petukhov. Low Thrust Trajectory Optimization

  34. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 34 4.2. OPTIMAL CONTROL Cost function: Hamiltonian: Optimal control: или   1 Optimal Hamiltonian: Averaged Hamiltonian does not depends on F, so after averaging . So as orbit-to-orbit transfers are considered, the final value F=F(T) is not fixed pF(T)=0 (transversality condition)  it can be missed terms including pF , where V.G. Petukhov. Low Thrust Trajectory Optimization

  35. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 35 4.3. EQUATIONS OF OPTIMAL MOTION (P-SYSTEM) where - state and co-state vectors, V.G. Petukhov. Low Thrust Trajectory Optimization

  36. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 36 V.G. Petukhov. Low Thrust Trajectory Optimization

  37. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 37 4.4. BOUNDARY VALUE PROBLEM Within the fixed-time problem equation of residuals is following: This equation should be solved with respect to unknown initial value of co-state vector p(0), pm(0). Within the minimum time problem  1 and equations for m andpm are eliminated by substitusion expression m = m0 - (P/w) t into other equations. Equation of residuals is following: This equation should be solved with respect to unknown initial value of co-state vector p(0) and transfer duration T. Continuation method’s equation: , where (minimum time) or z=p (fixed time); b=f(z0)- residual vector for initial z (when =0). The boundary value problem is solved by integration of continuation method’s equation on  from 0 to 1. Partial derivatives of residual vector f on vector z and linear system solving for computation right parts of o.d.e. are processed numerically. V.G. Petukhov. Low Thrust Trajectory Optimization

  38. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 38 V.G. Petukhov. Low Thrust Trajectory Optimization 4.5. DETAILS OF BOUNDARY VALUE PROBLEM SOLVING Boundary value problem is solved by continuation method. The averaged with respect to true lattitude equations of optimal motion are used to calculate residuals f. These equations have singularity when co-state vector p=0, so it is impossible to use zero initial co-state vector (coast motion) as initial approximation. Within the minimum time problem the following initial approximation was used: ph(0)=1 if the final semi-major axis greater than the semi-major axis of initial orbit and ph(0)=-1 otherwise. The rest vector p components were picked out equal to 0 and the initial approximation of transfer duration was T|=0=1 (dimensionless). Using this initial approximation there were found the minimum-time transfers to GEO from the elliptical transfer orbits having inclination 0°-75° and apogee altitude 10000-120000 km.If initial apogee altitude was not match with this range, the solution for a transfer from close initial orbit was used as the initial approximation. It is used numerical averaging the equations of optimal motion on the true lattitude F during these equations integration. The partial derivatives of residualsfwith respect to p(0), T, which are necessary for continuation method, are processed numerically using finite differences. So, there are used numerical integration of numerically averaged equations of optimal motion and numerical differentiating of residuals to calculate right parts of continuation method’s o.d.e.

  39. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 39 V.G. Petukhov. Low Thrust Trajectory Optimization 4.6. OPTIMAL SOLUTION IN NON-AVERAGED MOTION The real and averaged evolutions of orbital motion are close each to other due to the relatively low thrust acceleration level. To check accuracy of optimal averaged solution, the obtained optimal p(0) and T were used for numerical integration of non-averaged equations of motion. The initial value of true lattitude F was chosen arbitrary (the perigee or apogee values mostly). The initial value ofpFwas equals to 0 (see note above). The optimal thrust steering and insertion errors were calculated as result of numerical integration of the non-averaged equations. The relative errors due to averaging did not exceed 0.1% for transfer from an elliptical orbit to GEO when thrust acceleration was 0.1-0.5 mm/s2. An optimal thrust steering examples are presented below.

  40. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 40 4.7. OPTIMAL ORBITAL EVOLUTIONAND OPTIMAL THRUST STEERING(MINIMUM-TIME PROBLEM) Perigee distance Apogee distance Semi-major axis Orbital evolution for suboptimal apogee altitude of initial orbit(ha = 30000 km, i = 75°) Distance, km 1. Average apogee, semi-major axis, and eccentricity have maximum during transfer.2. Perigee distance increases monotonously. Time, days Inclination, deg Time, days Eccentricity Time, days V.G. Petukhov. Low Thrust Trajectory Optimization

  41. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 41 Braking-acceleration Acceleration-braking Acceleration Optimal thrust steering for suboptimal apogee altitude of initial orbit(ha = 30000 km, i = 75°) Yaw, deg Time, days Pitch, deg Time, days Angle of attack, deg Time, days V.G. Petukhov. Low Thrust Trajectory Optimization

  42. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 42 OPTIMAL THRUST STEERING pitch angle of attack path angle Angle, deg Yaw, deg Time, days Time, days pitch angle of attack path angle Angle, deg Yaw, deg Time, days Time, days pitch angle of attack path angle Yaw, deg Angle, deg Time, days Time, days V.G. Petukhov. Low Thrust Trajectory Optimization

  43. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 43 OPTIMAL THRUST STEERING t=141 d t=80 d t=2 d t=141 d t=80 d t=2 d Angle of attack, deg Pitch, deg True anomaly, deg True anomaly, deg t=141 d t=80 d t=2 d Yaw, deg True anomaly, deg V.G. Petukhov. Low Thrust Trajectory Optimization

  44. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 44 Orbital evolutionand optimal thrust steeringfor optimal apogee altitudeof initial orbit(ha = 140000 км, i = 65°) Perigee & apogee distance and semi-major axis Eccentricity V.G. Petukhov. Low Thrust Trajectory Optimization Eccentricity

  45. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 45 Orbital evolutionand optimal thrust steering forsuperoptimal apogee altitudeof initial orbit(ha = 240000 км, i = 65°) Perigee & apogee distance and semi-major axis Braking-acceleration Braking V.G. Petukhov. Low Thrust Trajectory Optimization Eccentricity Eccentricity

  46. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 47 4.8. OPTIMIZATION OF TRANSFER FROM ELLIPTIC ORBIT TO GEO Initial perigee altitude 250 km,SC mass in the GEO 450 kg, thrust 0.166 N, specific impulse 1500 s Initialinclination ° Initial apogee altitude, thousands km i0=75° Transfer duration, days i0=65° i0=51.3° i0=0° Initial apogee altitude, thousands km V.G. Petukhov. Low Thrust Trajectory Optimization

  47. 48 V.G. Petukhov. Low Thrust Trajectory Optimization CONCLUSION The developed continuation method demonstrated extremely effectiveness for variable specific impulse problem. The combination of two continuation versions (basic continuation method and continuation with respect to gravity parameter) allows to process planetary mission analysis fast and exhaustevely. The L2-ended low thrust trajectories were optimized using the continuation method. These solutions were used to construct quasioptimal trajectories between Earth and Moon orbits. The version of continuation method allows to carry out full-scale analysis of the low-thrust mission to GEO from the inclined elliptical transfer orbit. So, the continuation method performances make this method an effective and useful tool for analysis the wide range of electric propulsion mission

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