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UDC 521.3:629.7:681.5 DOI: 10.18698/1812-3368-2022-6-77-89
ON ONE APPROACH TO THE SOLUTION OF THE LAMBERT PROBLEM USING
THE DECOMPOSITIONAL METHOD OF MODAL CONTROL
N.E. Zubov1' 2 [email protected]
V.N. Ryabchenko2 A.V. Proletarsky2 A.A. Volochkova1
1 PJSC S.P. Korolev Rocket and Space Corporation Energia, Korolev, Moscow Region, Russian Federation
2 Bauman Moscow State Technical University, Moscow, Russian Federation
Abstract
A new approach to the solution of the Lambert's problem in spaceflight mechanics is proposed for elliptical orbits. The system of four transcendental algebraic equations is solved using the method of modal synthesis which is based on multilevel decomposition of discrete dynamic system and applied to solve the problem of identification of parameters of discrete system by a state observer. The solution algorithm is as follows: conditional and identification discrete models (systems) are built for the specified system of equations; initial values of estimates are given; initial conditions in the equations of residuals are formed. Using the method of modal synthesis, the problem of search for control of the auxiliary system is solved, as a result of which the matrix of state observer feedback coefficients is calculated. This matrix is used to predict the state vector and to obtain refined estimates — parameters of the planar orbit. A numerical example of the Lambert's problem solution using the proposed algorithm is given. In essence, an approach to the solution of nonlinear algebraic systems of the fourth order, which can be extended to systems of any observable order, is proposed. The peculiarity of the proposed algorithm is that the convergence of the iterative process of finding a solution can have a different "adjustable" speed using the control law
Keywords
Lambert's problem, elliptic orbits, discrete modelling, state observer, modal synthesis
Received 21.02.2022 Accepted 27.06.2022 © Author(s), 2022
Introduction. The Lambert's problem is one of the important problems of spaceflight mechanics. Despite numerous and exhaustive investigations in this direction, the Lambert's problem still remains very attractive for researchers [1-9].
We restrict ourselves to the case of elliptical orbits and consider it as a problem of determining elements of a Kepler orbit within one revolution by two positions of the spacecraft (SC). In the considered case, the Lambert's problem is formulated as follows: from the two positions of the SC in the orbit in the form of measurements of the moduli of the radius vectors (r0, r1), of time At, elapsed during the flight from one position to another, the difference of the true anomalies A& is required to determine the semi-major axis a, eccentricity e and eccentric anomalies E0, E1. Accordingly, for elliptic orbits there is a system of four transcendental algebraic equations [10, 11]:
which must be resolved with respect to the four orbital parameters (a, e, E0, E1).
In (1) p, — gravitational parameter, for the Earth p, = 398600.4418 km3/s2. Depending on the variants of the two measurements on the orbit, the possible variants of the classification of trajectories of motion of the solution of the Lambert's problem within one turn are presented in figure [11]. When considering system (1), it should be noted that due to the peculiarities of the arctangent associated with the domain of its values, the application of additional measures is required. For single rotation determination of elliptical orbits, one of the measures is the condition of limited time elapsed between the two positions of the object for which the measurement parameters were determined. In this case, the algorithm can only use the variant shown in figure a. It is also necessary to satisfy the rank criterion, the essence of which will be described below. These two circumstances allow us to exclude non-existence and ambiguity of the system solution (1).
In general, all methods for solving this problem are reduced to obtaining and solving a single Lambert equation or a system of two Kepler equations based on the energy integral and the area integral for unperturbed motion.
r0 = a (1 - e cos E0), r1 = a (1 - e cos E1),
(1)
a3/2
At = - E0 - e (sin E1 - sin E0 )),
Classification of elliptic trajectories of flight in the Lambert's problem:
а — sector 5 does not contains F1 and F2 focuses; b — sector 5 contains Fi and F2 focuses; c — sector 5 does not contain F2 focus and contains F1 focus; d — sector 5 does not contain F1 focus and contains F2 focus; e — sector 5 does not contain F1 focus, F2 focus is on the border
of this sector
In this paper, we consider an approach based on solving the above system of four equations using the method of modal synthesis. The method is based on a multilevel decomposition of a discrete dynamical system and is applied to solve the problem of identifying the parameters of a discrete system using a state observer. The essence of the solution search is as follows.
Algorithm of the problem solution. Let us construct a conditional dynamic discrete system (model) of the following type:
xD (t + 1) = ADxf (t), 7(t) = CpxD (t), (2)
where xD = (r0 r1 Ad At a e E0 E1) isastate vector, t = 0,1, 2,..., N is a discrete time determined on the basis of the introduction of a uniform grid of N intervals with a step on the segment of the search time of integration constants h = At / N. In fact, N determines the number of iterations to be performed to solve the problem with a given accuracy. The corresponding matrices in equations (2) have the form AD = I8x 8, Cp = (I4 x4 04 x4 ), where In x n is the unit matrix of size n x m.
From (2) it follows that for the subvector xD = x = (r0 r1 AS At) the iteration xD (x +1) = xD (x) is valid.
T
For the other subvector we denote x i — x ^ = (a e E0 E1) and present the system (1) in vector form x(x) = G(x T):
C a (1 - ecosE0) ^
a (1 - ecosE1)
G(xT)=
f
2arctg
1 + e
1 - e
Л
tg у ~ 2arctg
f
1 + e 1 - e
Л
t Eo
tgT
,3/2
Vi
(£1 - E0 - e (sin E1 - sin E0 ))
For system (2) let us construct a full rank observer of the state, which in general form is defined by the equation [8]:
xD(t + 1) = ( AD - LpCp ) (x) + Lpy(x). (3)
Here Lp is the matrix of state observer feedback coefficients [8]. The estimate is written as
X(x) = G(X T ). (4)
We linearize the function G(x T ) in the vicinity of X T, using the Taylor series expansion:
G(x i ) = G(x% ) + —-x i,
OXx
where
X ^ - X ^ X ^ ; ôG(XT )/ôxi is the Jacobian matrix. By calculating the discrepancy vector of the X (X = x - X), we obtain
x(x + 1) = xT(x).
dxz
(5)
Combining x and xT into a single vector and using (5) taking into account (3) we obtain the equations of the discrete model of discrepancies:
Xp(x + 1) = (Ap -LpCp)Xp(x),
(6)
ad -
Ap -
r dG( xx, t ) >
I4 X 4
0
dx 1 14 x4
If the condition of full observability is fulfilled
CP CpAl>
rank CP (Ad ) = 4 + 4 = 8, (7)
Cp (AD )'
Cp (AD )4
the choice of the coefficient matrix Lp with known matrices Ap and Cp it is always possible to provide any given placement on the complex plane Cstab (in the considered example, the area inside a circle of unit radius) of a characteristic polynomial root (poles) [12] det (XIn AD -LpCp )) orequivalent to the set of eigenvalues
eig (AD - LpCp ) = e C: det (XIn+m ~ (aD - LpCp )) = o)
state observer and thereby solve the equations system (1). In this case it is necessary to consider an auxiliary discrete model of the form
v(x +1) = DTv(x) + BTt|(t), t|(t) = -LTpv(x), (8)
where v is a vector having the dimensionality of the extended vector xp; ^
is the control vector; DT = (a^ f; CT = (Cp f.
The search for the Lp matrix belongs to the classical problem of modal control and in the case under consideration is in fact the goal in determining a, e, E0, E1, since their identification is the solution of the problem posed earlier.
If the condition of complete observability is fulfilled, the matrices necessary for the solution of the control problem exist. To solve the observer synthesis problem, any of the modal control methods can be applied. Here, as in [12], we use the decomposition method [12-15].
Let us introduce a multilevel decomposition of the discrete system (8) represented by a pair of matrices (DT, BT ). We have zero (initial) level
(9)
An = DT, Bo = С
-nT
the first level
Ai - Bq AoBQ , Bi - BQ AoBo.
(10)
Here Bq~ is the annulator (divisor of zero) of the matrix B0, i.e., Bq-B0 = 0; Bq~~ is a two-semi-inverse (2-semi-inverse) matrix for BQ" [12], i.e., a matrix satisfying the regularity conditions
БХдХ—DJ__D! dI^^I- _ D!-
0 B0 B0 - B0 , B0 B0 B0 - B0 •
'0
(11)
Then, according to [8], the required matrix L = L0 e №2 x 4 is calculated by the recursive formulas
and provides an exact given placement of the poles. This is indeed true since all elements of the set of eigenvalues eig (A - LB) coincide with the eigenvalues of the given stable matrices Ok (k = 0,1), i.e., eigenvalues lying inside the unit circle.
Considering the above, we write down the algorithm for solving the problem of finding the parameters a, e, E0, E1.
1. Using the system of equations (1) models are constructed: conditional (2) and identification (9).
2. Initial values of xx estimates are set. It should be noted that an approximate method of calculation proposed in [1, 6] can be used to select the initial values of XT, which will significantly reduce the number of iterations to find the exact solution.
3. Based on the given initial conditions according to (5), estimates of the state vector x are determined, thus, initial conditions for the difference discrete equation of residuals (6) are formed.
4. Using the method of modal control [12], defined by (9)-(13), we solve the problem of search for control of auxiliary system (8), which results in the transpose matrix Lp of observer's feedback coefficients.
5. Using Lp values on the basis of (3), we find new estimates of the vector XT and then, according to (4), new estimates of the predicted state vector X.
The matrix Ap included in the equation of residuals of the form (6):
Li = B+Ai -Ф1Б+,
L0 = B0A0 - Ф0Bo , Bo = LiB^ + B,
'0
(12) (13)
' 1 0 0 0 b11 b12 b13 b14 ^
0 10 0 b21 b22 b23 b24
0 0 10 b31 Ьз2 ЬЗЗ ЬЗ4
0 0 0 1 b41 b42 b43 b44
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
V
0 0 0 0 0 0 0 1
J
where
b11 = 1 - e cos E0; b12 = -a cos E0; b13 = ae sin E0; b14 = 0; b21 = 1 ~ e cos E1; b22 = -a cos E1; b23 = 0; b24 = ae sin E0; b31 = 0; b32 = sin E1 / (-(e +1)/ ((e - 1)1/2(e-1) (e cos E1 -1 )-- sin E0 ))/((-(e + 1)/(e- 1))1/2(e-1) (ecos E0 )-1)); b33 =-((2(- (e + 1)/(e - 1))1/2(e-1))/ (2e cos E0 - 2)); b34 =(2(- (e + 1)/(e - 1))1/2(e -1))/(2e cos E1 - 2 ); b41 = ( 3a1/2 ( E1 - E0 + e ( sin E0 - sin E1) ) ) / ( 2^1/2 ); b42 = a3/2 (sin E0 - sin E1) / ^,1/2; b43 = a3/2 (e cos E0 -1)/ ^1/2; b44 = - (a3/2 (e cos E1 -1)) / n1/2.
Main result. Let us apply the above approach (expressions (8)-(13)) to the problem of identification of elliptical orbit parameters. According to (8) and on the basis of (9), we have
AT =
( 1 0 0 0 0 0 0 0 > f 1 0 0 0 >
0 1 0 0 0 0 0 0 0 1 0 0
0 0 1 0 0 0 0 0 0 0 1 0
0 0 0 1 0 0 0 0 , CT = 0 0 0 1
ЪЦ Ъ21 Ъз1 Ъ41 1 0 0 0 0 0 0 0
Ьц Ъ22 Ъ32 Ъ42 0 1 0 0 0 0 0 0
Ъ13 Ъ23 Ъзз Ъ43 0 0 1 0 0 0 0 0
v Ъ14 Ъ24 Ъз4 Ъ44 0 0 0 1, v 0 0 0 0 V
According to (9), the zero level of decomposition for the considered subsystems is A0 = AT, B0 = CT. To find the first level of decomposition let us calculate the matrices-annulators:
(д, Г
( 0000100 0 ^
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
, (B0 )±_=(B0 )
±T
Then according to (10), we obtain
(1 0 0 0 ^ 0 10 0
A =
0 0 10 0 0 0 1
в =
( b11 b12 b13 b14
b21 b22 b23 b24
Ьз1
b32
ЬЗЗ Ьз4
b41
b42
b43 b44
Л
In order to use expressions (12), (13), let us define a generalized inverse
(1000000 0 ^ 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
and select (assign) the 01,00 matrices for the corresponding subsystems in the following simple form:
Ф1 =
fXn 0 0 0 ^
0 ^12 0 0
0 0 Ä43 0
0 0 0 ^14
Ф0 =
(Ь 01 0 0 0
0
^ 02 0 0
0 0
^ 03 0
stab ,
0 ^
0
0
^04
rT -L1 -
Assume that all the elements are located inside the Cstab =C ^ Then ac cording to (13), we obtain
f (Xn -1 )lnl D -(^12 -1 )Z12 / D (^13 -1 )l13 / D -(A,14 -1 )WD) ~(Xn -1 )l21 /D (A,12 - 1 )l22 /D -(A.13 - 1)l23/D (A,14-1)l24/D (Xn -1 )l31 /D -(A.12 - 1)l32/D (^13 - 1)l33/D -(A,H -1 >/34/D ~(Xn - 1 )l41 / D (A.12 - 1)l42/D -(A.13 - 1 )l43 / D (^14 - 1)l44/D
where
ln = ^22^33^44 - ^22^34^43 - ^23^32^44 + ^23^34^42 + ^24^32^43 - ^24^33^42!
112 = ^21^33^44 " ^21^34^43 - ^3^44 + ^23^34^41 + ^3^43 - ^24^33^41;
113 = ^21^32^44 " ^21^34^42 " ^22^44 + ^22^34^41 + ^24^31^42 " ^24^32^41;
114 = ^21^32^43 - ^21^33^42 - ^22^31^43 + ^22^33^41 + ^23^31^42 - ^23^32^41;
121 = ^12^33^44 " ^12^34^43 " ^13^32^44 + ^13^34^42 + ^4^32^43 " ^14^33^42;
122 = bUb33b44 - bUb34b43 - ^3^44 + ^13^34^41 + ^3^43 - ^14^33^41;
123 = bnb32b44 " bnb34b42 " b12b31b44 + b12b34b41 + b14b31b42 " b14b32b41;
124 = bnb32b43 " bUb33b42 " b^^ + b12b33b41 + b13b31b42 " bUb32b41;
¡31 = ЪпЪ2зЪ44 - ЪпЪ24Ъ43 ~ b^22b44 + ^3^42 +
¡32 = ЪцЪ23Ъ44 ■
l33 = ЪцЪ22Ъ44 -¡34 = ЪцЪ22Ъ43 ■ ¡41 = Ъ12Ъ23Ъ34 " ¡42 = ЪцЪ23Ъ34 ¡43 = ЪцЪ22Ъ34 ■ ¡44 = ЪцЪ22Ъ33
■ЪПЪ24Ъ43 ■ЪцЪ24Ъ42 ■ Ь11Ъ23Ъ42 Ъ12Ъ24Ъ33 -ЪцЪ24Ъ33 ЪцЪ24Ъ32 -ЪцЪ23Ъ32
"b13b21b44 -Ъ12Ъ21Ъ44
-Ъ\2Ъ2\Ъ43
"b13b22b34 -Ъ13Ъ21Ъ34
"b12b21b34
-Ъ12Ъ21Ъ33
-Ъ13Ъ24Ъ41 - Ъ12Ъ24Ъ41
"b12b23b41
Ъ13Ъ24Ъ32 -Ъ13Ъ24Ъ31 "b12b24b31
-Ъ12Ъ23Ъ31
b14b22b43 Ъ14Ъ21Ъ43 ■ Ъ14Ъ21Ъ42 ■
b13b21b42 -
Ъ14Ъ22Ъ33 ■ Ъ14Ъ21Ъ33 -b14b21b32 -
Ъ13Ъ21Ъ32 -
"b14b23b42; -bx4b23b41; -b14b22b41; b13b22b41; -b14b23b32; bnb23b31; b14b22b31; b13b22b31;
D = bnb22b33b44 - bnb22b34b43 - ЬцЬ23Ь32Ъ44 + ЪцЪ23Ъ34Ъ42 + ЬцЬ24Ь32Ъ43 "
- ЬцЬ24Ь33Ъ42 - Ъ12Ъ21Ъ33Ъ44 + b^21b34b43 + Ъ12Ъ23Ъ31Ъ44 - Ъ12Ъ23Ъ34Ъ41 "
- Ъ12Ъ24Ъ31Ъ43 + Ъ12Ъ24Ъ33Ъ41 + bBb2^32b44 - bBb2^34b42 - Ъ13Ъ22Ъ31Ъ44 + + Ъ13Ъ22Ъ34Ъ41 + bBb24b31b42 - Ъ13Ъ24Ъ32Ъ41 - bUb21b32b43 + b14b2feb42 +
+ Ъ14Ъ22Ъ31Ъ43 - Ъ14Ъ22Ъ33Ъ41 - Ъ14Ъ23Ъ31Ъ42 + bMb23b32b41;
DZ - bZbZ ■ D — b11b22
bz bZ Ь12Ъ21"
Using (13), for each subsystem we can calculate the transpose LTp matrix of observer's feedbacks. Taking into account its cumbersomeness, we allocate a block concerning the estimation of orbit parameters resulting in
Lx =
'-(A.01 " 1)(^11 " 1H1 (^02 " 1 )(XU - 1 )/21 -(X03 " 1 )(XU - 1 )/31 (^04 " 1)(XU - 1 )/41 ^ = (^01 - 1)(^12 " 1 )l12 "(^02 " 1 )(X12 " 1 )/22 (^03 " 1) (^12 " 1)^32 "(^04 " 1 )(^12 " 1)*42
" D -(X01 - 1 )(X13 - 1)l13 (^02 - 1 )(X13 - 1 )*23 -(^03 - 1)(X13 - 1 )l33 04 - 1 )(X13 - 1 )l43
v (^01 - 1 )(^14 - 1) l14 -(^02 - 1 )(^14 - 1) l24 (^03 - 1) (^14 _ 1) l34 -(^04 _ 1) (^14 _ 1) l44
According to (5), the equations of estimates will be written as follows:
r a л
e
aE, АД
= lT.
f Г0 ^
П At
accordingly, the values of estimates for the next calculation step with regard to the previous one will be equal to
( a e
AJEQ AjEj
л
'n+1
( a e
AJEQ AjEj
л
n
( Г0 ^ h
AS A t
It should be noted that setting Xij (i = 0,1; j = 1, 4) within a unit circle regulates the speed of convergence of the iterative process, the minimum of which is ensured at Xij = 0.
Numerical calculations. The effectiveness of the proposed method for determining the parameters of elliptical orbits (semi-major axis, eccentricity, eccentric anomalies) by information from on-board measuring instruments, which includes data on the radius vector module of the SC position at the starting point r0, modulus of the radius-vector position of the SC at the next measuring point rx, time between measuring points Ai, position angle A9 between radius-vectors r0 and r was evaluated by the test results shown in Table 1.
Table 1
Test results
r0, km r1, km At, sec Д&, grad a, km e E>, grad E1, grad
6921 7471 190 15 7.9100 • 104 0.9169 5.6298 9.0375
7471 6921 190 15 7.9100 • 104 0.9169 351.0538 354.4615
6921 6971 480 30 6.9434 • 103 0.0139 76.6436 106.6579
6971 6921 480 30 6.9434 • 103 0.0139 253.4333 283.4477
6919.8 6923.2 479 30.1 6.9192 • 103 0.0010 95.2723 125.3827
6923.2 6919.8 479 30.1 6.9192 • 103 0.0010 234.7085 264.8189
6922.8 6922.9 479.2 30.11 6.9231 • 103 0.00005017 18.6367 48.7454
6921 6971 263 17 7.4503 • 103 0.0725 11.6389 27.5225
6921 6971 93 6.5 9.3098 • 103 0.2614 11.0577 16.0809
6921 7371 385 30 2.1747 • 104 0.6818 0.8378 14.1841
6921 7371 504 36 1.1774 • 104 0.4122 1.1590 24.9074
7371 6921 504 36 1.1774 • 104 0.4122 335.1839 358.9323
To ensure the minimum number of search iterations, the components of matrices Oi,O0: ^n = ^12 = ^13 = ^14 = ^01 = ^02 = ^03 = ^04 = 0, were taken equal to zero during the tests. In addition, in the measurements used as input data for the calculation, the time At was limited to a value that provides a class of trajectories (Figure a) called elliptical trajectories of the first kind [16]. The permissible value At was calculated according to the method outlined in [11]. During the algorithm's operation, the observance of the rank condition (7) was monitored at each calculation cycle. If it was not performed, then the parameters of the measurement information were changed. Using the developed algorithm for eight iterations with an accuracy of 1012, the required parameters
given in Table 2 are found. Accordingly, the values a, e, E0, E1 by iterations for the first row of Table 1 are given in Table 2.
Table 2
Parameter values by search iterations
Parameter Iteration
0 1 2 3 4 5 6 7 8
Л, km 45000 63866 60507 75006 76781 79026 79102 79102 79102
е 0.8544 0.9150 0.8927 0.9171 0.9144 0.9169 0.9169 0.9169 0.9169
g 0.1391 0.1007 0.1141 0.0985 0.0999 0.0982 0.0982 0.0982 0.0982
Ei, grad 0.2193 0.1627 0.1827 0.1583 0.1603 0.1577 0.1577 0.1577 0.1577
If we compare the algorithm with the algorithm presented in [2, 8], we can note the following by the results of the simulation proposed here. The developed algorithm has significant advantages in calculation accuracy. Thus, for the initial data (the first line of Table 2) we have a = 100800 km, e = 0.9343, Eo = 4.918°,
E1 = 7.933°, which is significantly worse than the exact values obtained by the developed algorithm. A similar picture for other data (the third line of Table 2 — the calculated data for the algorithm from [2-8] is a = 7965.5 km, e = 0.1602, E0 = 9.17°, E1 = 62.82°). As for the solution of the Lambert's problem using other algorithms, they have computational advantages in comparison with the developed algorithm. It should be noted that there are other problems of identification (for example, when controlling the approach of SC in the orbit plane), based on the use of systems of algebraic equations of the fourth order, which are solved on board the SC. In this case, on the basis of (9)-(13) it is possible to build a single computing module and thus unify the on-board software. The latter circumstance, as well as the presence of an additional rank control in the algorithm defined by (7), which increases the reliability of the algorithm and determines the preference of the proposed method for solving the Lambert's problem on board a SC within a single turn with respect to other methods.
Conclusion. To solve the Lambert's problem on board the SC developed an algorithm that allows you to apply methods of modal control. It is shown that on the basis of four transcendental algebraic equations (1) it is necessary to construct a discrete model of orbit parameter estimation and using the decomposition method of modal synthesis [12-15] to solve the classical problem of state observer synthesis.
If we look a little wider, then we can assume that in the work proposed approach to the solution of nonlinear algebraic systems of the fourth order, which
can be extended to similar (1) systems of any observable order. The peculiarity of the proposed algorithm consists in the fact that the convergence of the iterative process of solution search can have different adjustable speed. Thus, using the considered decomposition method of modal synthesis it is possible to solve not only matrix equations as in [17, 18] but also non-linear algebraic systems of equations.
If we look a little wider, then we can assume that in the work proposed approach to the solution of nonlinear algebraic systems of the fourth order, which can be extended to similar (1) systems of any observable order. The peculiarity of the proposed algorithm consists in the fact that the convergence of the iterative process of solution search can have different adjustable speed. Thus, using the considered decomposition method of modal synthesis it is possible to solve not only matrix equations as in [17, 18] but also non-linear algebraic systems of equations.
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Zubov N.E. — Dr. Sc. (Eng.), Professor, post-graduate education, PJSC S.P. Korolev Rocket and Space Corporation Energia (Lenina ul. 4A, Korolev, Moscow Region, 141070 Russian Federation); Professor, Department of Automatic Control Systems, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation).
Ryabchenko V.N. — Dr. Sc. (Eng.), Assoc. Professor, Professor, Department of Automatic Control Systems, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation).
Proletarsky A.V. — Dr. Sc. (Eng.), Professor, Dean of the Faculty of Informatics, Ar-tifical Intelligence and Control Systems, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation).
Volochkova A.A. — Post-Graduate Student, PJSC S.P. Korolev Rocket and Space Corporation Energia (Lenina ul. 4A, Korolev, Moscow Region, 141070 Russian Federation).
Please cite this article as:
Zubov N.E., Ryabchenko V.N., Proletarsky A.V., et al. On one approach to the solution of the Lambert problem using the decompositional method of modal control. Herald of the Bauman Moscow State Technical University, Series Natural Sciences, 2022, no. 6 (105), pp. 77-89. DOI: https://doi.org/10.18698/1812-3368-2022-6-77-89
