V.G. Petukhov E-mail: petukhov@mtu-net.ru Khrunichev State Research and Production Space Center 2 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 3 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. V.G. Petukhov. Low Thrust Trajectory Optimization INTRODUCTION 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. V.G. Petukhov. Low Thrust Trajectory Optimization 4 INTRODUCTION 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. V.G. Petukhov. Low Thrust Trajectory Optimization 5 INTRODUCTION 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). V.G. Petukhov. Low Thrust Trajectory Optimization 6 7 1. CONTINUATION METHOD Problem: to solve non-linear system f (z ) 0 (1) with respect to vector z Let z0 - initial approximation of solution. Then f ( z0 ) b, (2) where b - residuals when z = z0. Let consider z(), where is a scalar parameter and equation f ( z) (1 )b (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: dz f z1 ( z)b, z(0) z0 d 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 (4) CONTINUATION METHOD 8 Application of continuation method to optimal control boundary value problem Optimal motion equations (after principle maximum application): dx Hp , dt dp H x dt Boundary conditions (an example): x(0) x 0 , x(T ) x k Boundary value problem parameters and residuals: z p(0), f x(T ) x k Sensitivity matrix: fz Associated system of optimal motion o.d.e. and perturbation equations for residuals and sensitivity matrix calculation: x(T ) z dx Hp , dt dp H x , dt d x x p H pp , H px dt z z z d p x p H xp H xx dt z z z Extended initial conditions: V.G. Petukhov. Low Thrust Trajectory Optimization x(0) x 0 , x(T ) x k , x p 0, I z z CONTINUATION METHOD 9 Using continuation method for low-thrust trajectory optimization problem Optimal control problem reduction to the boundary value problem by maximum principle CONTINUATION METHOD Initial approximation z0 1st version of o.d.e. right parts calculation Initial residuals b calculation by optimal motion o.d.e. integrating for given initial approximation z0 of boundary value problem parameters Continuation method’s o.d.e. integrating dz f z1 ( z)b, z(0) z0 d 2nd version of o.d.e. right parts calculation 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 finitedifference Solution z(1) V.G. Petukhov. Low Thrust Trajectory Optimization 10 2. OPTIMAL PLANETARY TRANSFER VARIABLE SPECIFIC IMPULSE PROBLEM V.G. Petukhov. Low Thrust Trajectory Optimization OPTIMAL PLANETARY TRANSFER 11 2.1. TRAJECTORY OPTIMIZATION PROBLEM 1 T 2 Cost function: a dt (constant power, nuclear electric propulsion) 0 2 2 1 T a dt (variable power, solar electric propulsion) 2 0 N (x , t ) Equations of motion: d2x/dt2=x+a Initial conditions: x(0)=x0(t0), v(0)=v0(t0)+Ve 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 OPTIMAL PLANETARY TRANSFER 12 2.2. OPTIMAL MOTION EQUATIONS (CONSTANT POWER) 1 dx H a Ta pTx pTv x pTv a 2 dt a pv Hamiltonian: Optimal control: Optimal motion equations: dx ~ 1 H p Tv p v p Tx p Tv x 2 dt d 2x p , x v dt 2 d 2p v 2 xx p v . dt Residuals: x(T ; a 0 , a 0 ) x k f (rendezvous) v(T ; a 0 , a 0 ) v k Optimal Hamiltonian: x(T ; a 0 , a 0 ) x k (flyby) f p v (T ; a 0 , a 0 ) Boundary value problem parameters and initial residuals vectors: a 0 z : a 0 z z0 V.G. Petukhov. Low Thrust Trajectory Optimization f ( z0 ) b OPTIMAL PLANETARY TRANSFER 2.3. EQUATIONS OF CONTINUATION METHOD Boundary value problem immersion into the one-parametric family: f ( z) (1 )b Boundary value problem parameters initial value and solution: z 0 z0 , z 1 ~z Differential equations of continuation method: dz f z1 ( z)b, z(0) z0 d Differential equations for calculation right parts of continuation method’s differential equations: p v d 2 p v x xx , p p v o d t 2 p v o x xx v p v o p v d 2 x x , xx p v o p v o d t 2 p v o 2 p v d p v x xx . p p v o d t 2 p v o x xx v p v o d2 x x p v , dt2 d2 pv xx p v , dt2 p v d 2 x x , xx 2 p v o p v o d t p v o x(0) x 0 (0), p dx(0) v 0 (0) V v , dt pv p v p Tv x (0) x(0) x(0) x (0) V 0, 0, I 2 , 0, p v0 p v0 p v0 pv pv p v0 p v0 (0) p (0) p (0) p (0) I, v0 0, v0 0, v0 I p v0 p v0 p v0 p v0 V.G. Petukhov. Low Thrust Trajectory Optimization 13 OPTIMAL PLANETARY TRANSFER 2.4. TRAJECTORY SEQUENCE WHICH IS CALCULATED BY CONTINUATION METHOD USING COASTING AS INITIAL APPROXIMATION 5 4 Earth-to-Mars, rendezvous, launch date June 1, 2000, V= 0 m/s, T=300 days 1- 3 1 2 coast trajectory (1= 0) 2-4 - intermediate trajectories (0 < 2 < 3 < 4 < 1) 5 - final (optimal) trajectory (5= 1) V.G. Petukhov. Low Thrust Trajectory Optimization 14 OPTIMAL PLANETARY TRANSFER 2.5. NUMERICAL EXAMPLES OPTIMAL TRAJECTORIES TO MERCURY AND NEAR-EARTH ASTEROIDS V.G. Petukhov. Low Thrust Trajectory Optimization 15 OPTIMAL PLANETARY TRANSFER OPTIMAL ORBITAL PLANE ROTATION EXAMPLES Optimal 90°-rotation of orbital plane Optimal 120°-rotation of orbital plane V.G. Petukhov. Low Thrust Trajectory Optimization 16 OPTIMAL PLANETARY TRANSFER EXAMPLE: INITIAL HYPERBOLIC EXCESS OF VELOCITY IMPACT V.G. Petukhov. Low Thrust Trajectory Optimization 17 OPTIMAL PLANETARY TRANSFER EXAMPLE: NUCLEAR (RIGHT) AND SOLAR (LEFT) ELECTRIC PROPULSION V.G. Petukhov. Low Thrust Trajectory Optimization 18 OPTIMAL PLANETARY TRANSFER 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 using continuation with respect to gravity parameter V.G. Petukhov. Low Thrust Trajectory Optimization 19 OPTIMAL PLANETARY TRANSFER 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-esinE, where E=2arctg{[(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, )=()0.5 v(t, 1). V.G. Petukhov. Low Thrust Trajectory Optimization 20 OPTIMAL PLANETARY TRANSFER 21 Equations of motion: ()x p v , p v ()xx p v x Boundary conditions: x(0) x 0 , x (0) 1/ 2 ( ) v 0 , x(T ) x k , x (T ) 1/ 2 ( ) v k . Residuals: x (T ) x k f 1/ 2 x ( T ) ( ) v k Boundary value problem parameters: z = (pv(0), dpv(0)/dt)T = p v o , p v o Equation of continuation method: dz f fz1 (z) b , z(0) z 0 d T where d 2 x x p v , ( ) xx 2 z z dt z b= f(z0) x (T ) p v o f z d x(T ) p v o dt d x(T ) p v o dt x (T ) p v o x f 1 x vk 2 1/ 2 ( ) p v d 2 p v x , ( ) xx p v xx 2 z z dt z x d 2 x x p v x ( ) xx , 2 z z dt p v d 2 p v x xx p v ( ) xx p v xx , 2 z z dt x x (0) x (0) x (0) x (0) 1 0, v0 , 1/ 2 z z 2 ( ) p v (0) p v (0) p v (0) p v (0) p v (0) p v (0) E, 0. p v0 p v0 p v0 p v0 V.G. Petukhov. Low Thrust Trajectory Optimization OPTIMAL PLANETARY TRANSFER 22 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 orbits V.G. Petukhov. Low Thrust Trajectory Optimization 7 complete orbits OPTIMAL PLANETARY TRANSFER EXAMPLES: OPTIMAL TRAJECTORIES TO MAJOR PLANETS OF SOLAR SYSTEM V.G. Petukhov. Low Thrust Trajectory Optimization 23 24 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) 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. V.G. Petukhov. Low Thrust Trajectory Optimization 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. Hill’s sphere Region of satellite motion Moon opening width ~60000 km 3. Opening width in the L2 vicinity is ~60000 km for SC relative velocity 10 m/s on the Hill’s sphere. to Earth 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. 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 OPTIMAL TRANSFER TO LUNAR ORBIT 26 EARTH_MOON L2 RENDEZVOUS Model 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 t, days 95.0 0.0 t, days 95.0 0.0 t, days V.G. Petukhov. Low Thrust Trajectory Optimization 95.0 0.0 t, days 95.0 OPTIMAL TRANSFER TO LUNAR ORBIT 27 EARTH-MOON L2 RENDEZVOUS USING MOON GRAVITY ASSISTED MANEUVER 2.5 orbits Gravity assisted maneuver 7.5 orbits Initial Moon position Initial L2 position Initial orbit Moon orbit Final L2 position Final Moon position 1.0 Thrust acceleration, mm/s2 Thrust acceleration, mm/s2 0.5 0.0 0 Т, days 0.0 95 0 V.G. Petukhov. Low Thrust Trajectory Optimization Т, days 95 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 Thrust acceleration 0.5 mm/s2 0 mm/s2 0 Time, d 10 0 Time, d 15 0 V.G. Petukhov. Low Thrust Trajectory Optimization Time, d 20 OPTIMAL TRANSFER TO LUNAR ORBIT 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 Thrust acceleration 1 mm/s2 0 mm/s2 0 Time, d 30 V.G. Petukhov. Low Thrust Trajectory Optimization 29 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 Thrust acceleration Thrust acceleration 0.5 mm/s2 0 mm/s2 0 mm/s2 0 Time, d 95 V.G. Petukhov. Low Thrust Trajectory Optimization 0 Time, d 95 31 4. OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS CONSTANT SPECIFIC IMPULSE PROBLEM V.G. Petukhov. Low Thrust Trajectory Optimization OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 32 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. V.G. Petukhov. Low Thrust Trajectory Optimization OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 4.1. EQUATION OF MOTION Thrust acceleration components in the orbital reference frame: a P cos cos m ar P sin cos m an - thrust switching function, P - thrust, m - SC mass, - pitch, - yaw P sin m System of equinoctial elements: h p ex e cos e y e sin i i x tan cos 2 i y tan i sin 2 F - primary gravity parameter; p, e, , , i, - keplerian elements. Equation of motion in the equinoctial elements: 1 e x cos F e y sin F i x sin F i y cos F ~ 1 i x2 i y2 w - exhaust velocity dh Ph h cos cos , dt m de x Ph sin F sin cos 1 cos F e x cos cos e y sin , dt m de y Ph cos F sin cos 1 sin F e x cos cos e x sin , dt m di x Ph 1~ cos F sin , dt m 2 di y Ph 1~ sin F sin , dt m 2 2 dF Ph 3 sin , dt h m dm P , dt w Boundary conditions: t = 0: h h0 , ex ex 0 , e y e y 0 , i x i x 0 , i y i y 0 , m m0 t = T: h hk , ex exk , e y e yk , i x i xk , i y i yk V.G. Petukhov. Low Thrust Trajectory Optimization 33 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 4.2. OPTIMAL CONTROL T J Cost function: 0 Hamiltonian: H P dt min w 2 P 1 pm 3 pF P h A cos cos Ar sin cos An sin w h m A hp h 1 cos F e x p ex 1 sin F e y p ey Ar sin F p ex cos F p ey 1 An e y p ex e x p ey ~ cos F p ix sin F p iy p F 2 cos Optimal control: sin Ar A A Ar 2 2 r A A 2 2 r cos sin A2 Ar2 A A A An 2 2 r 2 n A2 Ar2 An2 1, s 0 0, s 0 s или 1 12 1 pm h A2 Ar2 An2 w m 12 P 2 Ph 2 H 1 pm 3 pF A Ar2 An2 w h m Optimal Hamiltonian: Averaged Hamiltonian does not depends on F, so after averaging dp F H 0 . So as orbit-to-orbit transfers are dt F considered, the final value F=F(T) is not fixed pF(T)=0 (transversality condition) 12 p F 0 it can be missed terms including pF H P 1 pm P A~2 A~r2 A~n2 , where ~ h A ,A ~ h A ,A ~ hA A r r n n w m V.G. Petukhov. Low Thrust Trajectory Optimization 34 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 4.3. EQUATIONS OF OPTIMAL MOTION (P-SYSTEM) ~ ~ ~ A A dx H P ~ 2 ~ 2 ~ 2 1 2 ~ A ~ ~ n A Ar An A An An n , dt p m p p p dm H P , dt pm m ~ ~ ~ dp H P ~ 2 ~ 2 ~ 2 1 2 ~ A ~ An ~ An A Ar An A An An , dt x m x x x dpm H P ~2 ~2 ~2 1 2 2 A Ar An , dt m m where x h, ex , e y , i x , i y , p ph , pex , pey , pix , piy T T - state and co-state vectors, ~ A h Ai i , i , r , n. p p ~ A 1 i Ai h ~ A h Ai i i x i x ~ Ai A ; i h e x ~ A h Ai ; i i x i x h ~ A A A h cos F h sin F Ai i ; i Ai i e x e y e y ~ A A h e x sin F e y cos F ; i Ai i , i , r , n. F F V.G. Petukhov. Low Thrust Trajectory Optimization ; 35 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS A A A A A A h; 1 cos F e x ; 1sin F e y ; 0; p h pex pey pix piy p F Ar A A A A A 0; r sin F ; r cos F ; r r r 0; p h pex pey pix piy p F An A A A A A 1 1 0; n e y ; n e x ; n ~ cos F ; n ~ sin F ; n . p h pex pey pix 2 piy 2 p F A A A p h ; cos 2 F 1 p ex cos F sin F p ey ; sin 2 F 1 p ey cos F sin F p ex ; h e x e y A A 0; i x i y A e y cos F e x sin F cos F 1sin F p ex e y cos F e x sin F sin F 1 cos F p ey ; F Ar A A 0; r cos F pex sin F pey cos F ; r sin F pex sin F pey cos F ; h e x e y Ar Ar A 0; r e x sin F e y cos F pex sin F pey cos F pex cos F pey sin F ; i x i y F An A A A A 0; n pey p F cos F ; n pex p F sin F ; n n 0; h e x e y i x i y An 1 i x cos F i y sin F e x pey e y pex ~ piy cos F pix sin F F 2 e y cos F e x sin F p F i x cos F i y sin F p F . V.G. Petukhov. Low Thrust Trajectory Optimization 36 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 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 and pm 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: h(T ) hk e x (T ) e xk e (T ) e y yk 0 f i x (T ) i xk i y (T ) i yk p (T ) m h(T ) hk e x (T ) e xk e (T ) e y yk f 0 i x (T ) i xk i y (T ) i yk H (T ) T ph p ex p 1 ey dz f b, where z (minimum time) or z=p (fixed time); pix d z piy T 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 37 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 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 semimajor 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 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. V.G. Petukhov. Low Thrust Trajectory Optimization 38 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 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 of pF 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 0.1-0.5 mm/s2. An optimal thrust steering examples are presented below. V.G. Petukhov. Low Thrust Trajectory Optimization 39 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 4.7. OPTIMAL ORBITAL EVOLUTION AND OPTIMAL THRUST STEERING (MINIMUM-TIME PROBLEM) Р аPerigee д и у с п еdistance ригея 80000 Distance, km Ра сстояние , км Orbital evolution for suboptimal apogee altitude of initial orbit (ha = 30000 km, i = 75°) Р аApogee д и у с а пdistance огея 70000 Semi-major axis Б оль ш ая п олу ос ь 60000 50000 40000 30000 20000 10000 0 0 50 100 150 В р е мdays я , сут Time, Inclination, На клоне ние , deg гра дусы 80 1. Average apogee, semi-major axis, and eccentricity have maximum during transfer. 2. Perigee distance increases monotonously. 70 60 50 40 30 20 10 0 0 50 100 150 ВTime, р е м яdays , сут 0,8 Eccentricity Эксце нтрисите т 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0 0 50 100 150 В р е м яdays , сут Time, V.G. Petukhov. Low Thrust Trajectory Optimization 40 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS Accelerationbraking Brakingacceleration Acceleration Optimal thrust steering for suboptimal apogee altitude of initial orbit (ha = 30000 km, i = 75°) Yaw, deg градусы Угол рысканья, 90 60 30 0 -30 -60 -90 0 20 40 60 80 100 120 140 160 120 140 160 120 140 160 Тангаж, deg Pitch, градусы Время, сутки Time, days 180 150 120 90 60 30 0 -30 -60 -90 -120 -150 -180 0 20 40 60 80 100 deg of attack, Angle атаки, Угол градусы Время, сутки Time, days 180 150 120 90 60 30 0 -30 -60 -90 -120 -150 -180 0 20 40 60 80 100 Время, Time, daysсут V.G. Petukhov. Low Thrust Trajectory Optimization 41 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS pitch тангаж angle of attack угол атаки path angle угол траекторный 180 150 120 90 60 30 0 -30 -60 -90 -120 -150 -180 90 Yaw, deg градусы Угол рысканья, Угол, градусы deg Angle, OPTIMAL THRUST STEERING 60 30 0 -30 -60 -90 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 Time, daysсут Время, тангаж pitch angle of attack угол атаки path angle угол траекторный deg Angle,градусы Угол, 60 30 0 -30 -60 -90 1 60 30 0 -30 -60 -90 80 80.2 80.4 80.6 80.8 81 81.2 81.4 81.6 81.8 82 80 Время, сут Time, days 180 150 120 90 60 30 0 -30 -60 -90 -120 -150 -180 141 141.2 141.4 141.6 pitch тангаж angle of attack угол атаки path angle угол траекторный 141.8 142 142.2 142.4 142.6 142.8 143 Время, сут Time, days 80.2 80.4 80.6 80.8 81 81.2 81.4 81.6 81.8 82 Время, Time, daysсут 90 Yaw, deg градусы Угол рысканья, deg Angle, Угол, градусы 0.8 90 Yaw, deg градусы Угол рысканья, 90 0.6 Время, сут Time, days 60 30 0 -30 -60 -90 141 141.2 141.4 141.6 141.8 142 142.2 142.4 142.6 142.8 143 Время, Time, daysсут V.G. Petukhov. Low Thrust Trajectory Optimization 42 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 43 1 4t=141 1 - е dс у т к и 180 150 120 90 60 30 0 -3 0 -6 0 -9 0 -1 2 0 -1 5 0 -1 8 0 8 0t=80 - е сd у т к и Угол атаки, градусы deg of attack, Angle deg Pitch, градусы Тангаж, OPTIMAL THRUST STEERING 2 -t=2 е сdу т к и 0 30 60 90 120 150 180 210 240 270 300 330 360 Yaw,, deg Рысканье градусы deg И с ти н н аTrue я а нanomaly, ом алия , г р а д ус ы 80 1t=141 4 1 - еdс у т к и 70 8t=80 0 - е dс у т к и 60 1 4t=141 1 - е dс у т к и 180 150 120 90 60 30 0 -3 0 -6 0 -9 0 -1 2 0 -1 5 0 -1 8 0 8 0t=80 - е сd у т к и 2 -t=2 е сdу т к и 0 30 60 90 40 30 20 10 0 -1 0 -2 0 -3 0 0 30 60 90 120 150 180 210 240 270 210 240 270 300 330 360 И с ти н н аTrue я а нanomaly, о м а л и deg я , г р а д ус ы 2t=2 - е dс у т к и 50 120 150 180 300 330 360 И с ти н н а яTrue а н оanomaly, м а л и я ,deg г р а д ус ы V.G. Petukhov. Low Thrust Trajectory Optimization Distance, km OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 150000 140000 130000 120000 110000 100000 90000 80000 70000 60000 50000 40000 30000 20000 10000 0 Orbital evolution and optimal thrust steering for optimal apogee altitude of initial orbit (ha = 140000 км, i = 65°) rp ra a 0 10 20 30 40 50 60 70 80 90 100 44 110 t, days Perigee & apogee distance and semi-major axis 1 180 0.9 150 0.8 120 90 yaw 60 0.6 Angle ° Eccentricity 0.7 pitch 0.5 0.4 30 0 -30 -60 0.3 -90 0.2 -120 -150 0.1 0 -180 0 10 20 30 40 50 60 70 80 90 100 110 0 10 20 30 40 t, days 50 60 t, days Eccentricity Eccentricity V.G. Petukhov. Low Thrust Trajectory Optimization 70 80 90 100 110 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 250000 Orbital evolution and optimal thrust steering for superoptimal apogee altitude of initial orbit (ha = 240000 км, i = 65°) rp ra 200000 a Distance, km 45 150000 100000 50000 0 0 10 20 30 40 50 60 70 80 90 100 110 120 t, days Perigee & apogee distance and semi-major axis Braking 1.00 0.90 0.80 0.60 Angle ° Eccentricity 0.70 0.50 0.40 0.30 0.20 0.10 0.00 0 10 20 30 40 50 60 70 80 90 100 110 120 180 150 120 90 60 30 0 -30 -60 -90 -120 -150 -180 Braking-acceleration pitch yaw 0 10 20 30 40 50 60 t, days t, days Eccentricity Eccentricity V.G. Petukhov. Low Thrust Trajectory Optimization 70 80 90 100 110 120 OPTIMAL MULTI-REVOLUTION TRANSFER BETWEEN NON-COPLANAR ELLIPTIC ORBITS 4.8. OPTIMIZATION OF TRANSFER FROM ELLIPTIC ORBIT TO GEO Initial inclination ° Initial perigee altitude 250 km, SC mass in the GEO 450 kg, thrust 0.166 N, specific impulse 1500 s Transfer duration, days Initial apogee altitude, thousands km i0=75° i0=51.3° i0=65° i0=0° Initial apogee altitude, thousands km V.G. Petukhov. Low Thrust Trajectory Optimization 47 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 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 V.G. Petukhov. Low Thrust Trajectory Optimization