V.G. Petukhov Khrunichev State Research and Production Space Center

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

INTRODUCTION V.G. Petukhov. Low Thrust Trajectory Optimization 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. 3

INTRODUCTION 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. 4

INTRODUCTION 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 convergence domain 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. 5

INTRODUCTION 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). 6

Problem: to solve non-linear system (1) with respect to vector z Let z 0 - initial approximation of solution. Then, (2) where b - residuals when z = z 0. 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)). 1. CONTINUATION METHOD V.G. Petukhov. Low Thrust Trajectory Optimization 7

CONTINUATION METHOD V.G. Petukhov. Low Thrust Trajectory Optimization Application of continuation method to optimal control boundary 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. and perturbation equations for residuals and sensitivity matrix calculation: Extended initial conditions: 8

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

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

2.1. TRAJECTORY OPTIMIZATION PROBLEM Cost function: (constant power, nuclear electric propulsion) (variable power, solar electric propulsion) Equations of motion:d 2 x/dt 2 =  x +a Initial conditions:x(0)=x 0 (t 0 ), v(0)=v 0 (t 0 )+V  e  Boundary conditions 1) rendezvous:x(T)=x k (t 0 +T), v(T)=v k (t 0 +T) 2) flyby:x(T)=x k (t 0 +T) where x, v - SC position and velocity vectors,  - gravity field force function, a - thrust acceleration vector, x 0, v 0 - departure planet position and velocity vectors, x k, v k - 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. OPTIMAL PLANETARY TRANSFER V.G. Petukhov. Low Thrust Trajectory Optimization 11

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

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

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

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

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

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

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

2.7. METHOD OF CONTINUATION WITH RESPECT TO GRAVITY PARAMETER Sequence of trajectory calculation using basic continuation method Sequence of trajectory calculation using 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 OPTIMAL PLANETARY TRANSFER V.G. Petukhov. Low Thrust Trajectory Optimization 19

Let x 0 (0), x 0 (T) - departure planet position when t=0 and t=T; x k - 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 x 0 and projection of x k into the initial orbit plane). The solution of Kepler equation gives corresponding values of mean anomalies M 0 and M k (M=E-e  sinE, where E=2  arctg{[(1-e)/(1+e)] 0.5 tg( /2)} is eccentric anomaly). Mean anomaly is linear function of time at the keplerian orbit: M=M 0 +n  (t-t 0 ), where n=(  0 /a 3 ) 0.5 is mean motion. Therefore, the condition of angular distance invarianct is M k +2  N rev =nT+M 0, where N rev is number of complete orbits. So initial value of the primary gravity parameter is  0 =[( M k +2  N rev - M 0 )/T] 2 a 3, and current one is  (  )=  0 +(1-  0 ) . The shape and size of orbits should be invariance witn respect to , therefore v(t,  )=  (  ) 0.5 v(t, 1). OPTIMAL PLANETARY TRANSFER V.G. Petukhov. Low Thrust Trajectory Optimization 20

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

Numerical example: Mercury rendezvous Constant power, launch date January 1, 2001, transfer duration 1200 days All solutions are obtained using coasting along the Earth orbit as initial approximation Basic version of continuation method Continuation with respect to gravity parameter 5 complete orbits7 complete orbits OPTIMAL PLANETARY TRANSFER V.G. Petukhov. Low Thrust Trajectory Optimization 22

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

3. OPTIMAL TRANSFER TO LUNAR ORBIT VARIABLE 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) L 2 -rendezvous trajectory; 3) Trajectory from the point L 2 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. V.G. Petukhov. Low Thrust Trajectory Optimization 24

VALIDATION OF TRAJECTORY DIVIDING INTO ARCS 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 opening width ~60000 km Hill’s sphere Region of satellite motion Moon to Earth 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/s Opening width in the L 2 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/s 2 ) SC relative velocity should be not greater ~10 m/s when distance from L 2 is less ~30000 km. OPTIMAL TRANSFER TO LUNAR ORBIT V.G. Petukhov. Low Thrust Trajectory Optimization 25

EARTH_MOON L 2 RENDEZVOUS Model problem of transfer from circular Earth orbit (altitude км, inclination 63°, right ascension of ascending node 12°, lattitude argument 0°; launch date January 5, 2001) a, mm/s t, days t, days t, days t, days 4 complete orbits5 complete orbits 6 complete orbits7 complete orbits OPTIMAL TRANSFER TO LUNAR ORBIT V.G. Petukhov. Low Thrust Trajectory Optimization 26

EARTH-MOON L 2 RENDEZVOUS USING MOON GRAVITY ASSISTED MANEUVER Thrust acceleration, mm/s Т, days 95 Moon orbit Final Moon position Initial Moon position Initial L 2 position Final L 2 position Initial orbit Gravity assisted maneuver Thrust acceleration, mm/s Т, days orbits7.5 orbits OPTIMAL TRANSFER TO LUNAR ORBIT V.G. Petukhov. Low Thrust Trajectory Optimization 27

TRANSFER FROM EARTH-MOON L 2 INTO CIRCULAR MOON ORBIT Final orbit: r = km, i = 0 . Transfer: 1.5 orbits, T = 10 days Final orbit: r = km, i = 0 . Transfer: 2.5 orbits, T = 15 days Final orbit: r = km, i = 90 . Transfer: 2.5 orbits, T = 20 days Thrust acceleration 0.5 mm/s 2 0 mm/s 2 0 Time, d 10 0 Time, d 150 Time, d 20 Moon Final (intermediate) orbit Initial L 2 position Final L 2 position OPTIMAL TRANSFER TO LUNAR ORBIT V.G. Petukhov. Low Thrust Trajectory Optimization 28

TRANSFER FROM EARTH-MOON L 2 INTO ELLIPTICAL MOON ORBIT (i=90°, h p =300 km, h a =10000 km, 10.5 orbits) Thrust acceleration 1 mm/s 2 0 mm/s 2 0 Time, d 30 Moon Final orbit Initial L 2 position Final L 2 position OPTIMAL TRANSFER TO LUNAR ORBIT V.G. Petukhov. Low Thrust Trajectory Optimization 29

TRANSFER FROM ELLIPTICAL EARTH ORBIT INTO CIRCULAR MOON ORBIT. TRAJECTORY ARCS Geocentric spiral untwistingEarth-Moon L 2 rendezvous Transfer from Earth-Moon L 2 into equatorial km circular Moon orbit Moon Earth Thrust acceleration 0.5 mm/s 2 0 mm/s 2 0 Time, d 95 Thrust acceleration 0.5 mm/s 2 0 mm/s 2 0 Time, d 95 OPTIMAL TRANSFER TO LUNAR ORBIT V.G. Petukhov. Low Thrust Trajectory Optimization 30

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

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. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS V.G. Petukhov. Low Thrust Trajectory Optimization 32

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

OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS V.G. Petukhov. Low Thrust Trajectory Optimization 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  p F (T)=0 (transversality condition)   it can be missed terms including p F , where Cost function: Hamiltonian: Optimal control: Optimal Hamiltonian: или   OPTIMAL CONTROL 34

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

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

OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS V.G. Petukhov. Low Thrust Trajectory Optimization Continuation method’s equation:, where (minimum time) or z=p (fixed time); b=f(z 0 ) - 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 BOUNDARY VALUE PROBLEM Within the minimum time problem   1 and equations for m and p m are eliminated by substitusion expression m = m 0 - (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. 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), p m (0). 37

4.5. DETAILS OF BOUNDARY VALUE PROBLEM SOLVING OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS V.G. Petukhov. Low Thrust Trajectory Optimization 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: p h (0)=1 if the final semi- major axis greater than the semi-major axis of initial orbit and p h (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 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 residuals f with 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. 38

4.6. OPTIMAL SOLUTION IN NON-AVERAGED MOTION OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS V.G. Petukhov. Low Thrust Trajectory Optimization 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 of p F was 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 mm/s 2. An optimal thrust steering examples are presented below. 39

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

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

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

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

OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS V.G. Petukhov. Low Thrust Trajectory Optimization Orbital evolution and optimal thrust steering for optimal apogee altitude of initial orbit (h a = км, i = 65°) Perigee & apogee distance and semi-major axis Eccentricity 44

Eccentricity OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS V.G. Petukhov. Low Thrust Trajectory Optimization Perigee & apogee distance and semi-major axis Orbital evolution and optimal thrust steering for superoptimal apogee altitude of initial orbit (h a = км, i = 65°) Braking-acceleration Braking 45

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

V.G. Petukhov. Low Thrust Trajectory Optimization 48 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 L 2 -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