txt/2201.02942.txt

See discussions, st ats, and author pr ofiles f or this public ation at : https://www .researchgate.ne t/public ation/357188680
Fast Solver for J2-Pertu rbed Lambert Problem Using Deep Neu ral Network
Article    in  Journal of Guidanc e, Contr ol, and Dynamics  · Dec ember 2021
DOI: 10.2514/1.G006091
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Bin Y ang
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  Fast solver for J2-perturbed Lambert problem using deep 
neural network 
Bin Yang1 and Shuang Li2* 
Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China 
Jinglang Feng3 and Massimiliano Vasile4 
University of Strathclyde, Glasgow, Scotland G1 1XJ, United Kingdom 
This paper presents a novel and fast solver for the J2-perturbe d Lambert problem. The 
solver consists of an intelligent initial guess generator combi ned with a differential 
correction procedure. The intelligent initial guess generator i s a deep neural network that is 
trained to correct the initial velocity vector coming from the solution of the unperturbed 
Lambert problem. The differential correction module takes the i nitial guess and uses a 
forward shooting procedure to further update the initial veloci ty and exactly meet the 
terminal conditions. Eight sample forms are analyzed and compar e d  t o  f i n d  t h e  o p t i m u m  
form to train the neural network on the J2-perturbed Lambert pr oblem. The accuracy and 
performance of this novel approach will be demonstrated on a re presentative test case: the 
solution of a multi-revolution J2-perturbed Lambert problem in the Jupiter system. We will 
compare the performance of the proposed approach against a clas sical standard shooting 
                                                           
1 Ph.D. candidate, Advanced Space Technology Laboratory, No. 29 Yudao Str., Nanjing 211106, China. 
2 Professor, Advanced Space Technology Laboratory, Email: lishua [email protected], Corresponding Author. 
3 Assistant Professor, Department of Mechanical and Aerospace En gineering, University of Strathclyde, 75 
Montrose Street, Glasgow, UK. 
4 Professor, Department of Mechanical and Aerospace Engineering,  University of Strathclyde, 75 Montrose Street, 
Glasgow, UK. 2 
 method and a homotopy-based perturbed Lambert algorithm. It wil l  be  s ho w n  t h a t, f o r  a  
comparable level of accuracy, th e proposed method is significan tly faster than the other two. 
I. Introduction 
The effect of orbital perturbations, such as those coming from a non-spherical, inhomogeneous gravity field, 
leads a spacecraft to depart from the trajectory prescribed by t h e  s o l u t i o n  o f  t h e  L a m b e r t  p r o b l e m  i n  a  s i m p l e  
two-body model [1], [2]. Since the perturbation due to the J2 z onal harmonics has the most significant effect around 
all planets in the solar system, a body of research exists that  addressed the problem of solving the perturbed Lambert 
problem accounting for the J2 effect [3], [4]. This body of res earch can be classified into two categories: indirect 
methods and shooting methods [5]. Indirect methods transform th e perturbed Lambert problem into the solution of a 
system of parametric nonlinear algebraic equations. For instanc e, Engles and Junkins [1] proposed an indirect 
method that uses the Kustaanheimo-Stiefel (KS) transformation t o derive a system of two nonlinear algebraic 
equations. Der [6] presented a superior Lambert algorithm by us ing the modified iterative method of Laguerre that 
h a s  g o o d  c o m p u t a t i o n a l  p e r f o r m a n c e  i f  g i v e n  a  g o o d  i n i t i a l  g u e s s. Armellin et al. [7] proposed two algorithms, 
based on Differential Algebra, for the multi-revolution perturb ed Lambert problems (MRPLP). One uses homotopy 
over the value of the perturbati on and the solution of the unpe rturbed, or Keplerian, Lambert problem as initial guess. 
The other uses a high-order Taylor polynomial expansion to map the dependency of the terminal position on the 
initial velocity, and solves a system of three nonlinear equati ons. A refinement step is th en added to obtain a solution 
with the required accuracy. A common problem of indirect method s is the need for a good initial guess to solve the 
system of nonlinear algebraic equations. A bad initial guess in creases the time to solve the algebraic system or can 
lead to a failure of the solution procedure, especially when th e transfer time is long.  3 
 Shooting methods transcribe the perturbed Lambert problem into the search for the initial velocity vector that 
provides the desired terminal conditions at a given time. Kraig e et al. [8] investigated the efficiency of different 
shooting approaches and found that a straightforward differenti al correction algorithm combined with the 
Rectangular Encke’s motion predictor is more efficient than the  analytical KS approach. Junkins and Schaub [9] 
transformed the problem into a two-point boundary value problem  and applied Newton iteration method to solve it. 
The main problem with shooting methods is that, with the increa se of the transfer time, the terminal conditions 
become more sensitive to the variations of the initial velocity  and the derivatives of the final states with respect to 
the initial velocity are more affected by the propagation of nu merical errors. In order to mitigate this problem, Arora 
et al. [10] proposed to compute the derivatives of the initial and final velocity vectors with respect to the initial and 
final position vectors, and the time of flight, with the state transition matrix. Woollands et al. [11] applied the KS 
transformation and the modified Chebyshev–Picard iteration to o btain the perturbed solution starting from the 
solution of the Keplerian Lambert problem, which is to solve th e initial velocity vector corresponding to the transfer 
between two given points with a given time of free flight in a two-body gravitational field [12]. For the 
multi-revolution perturbed Lambert problem with long flight tim e, Woollands et al. [13] also utilized the modified 
Chebyshev-Picard iteration and the method of particular solutio ns based on the local-linearity, to improve the 
computational efficiency, but its solution relies on the soluti on of the Keplerian Lambert problem as the initial 
guesses. Alhulayil et al. [14] proposed a high-order perturbati on expansion method that accelerates convergence, 
compared to conventional first-order Newton’s methods, but requ ires a good initial guess to guarantee convergence. 
Yang et al. [15] developed a targeting technique using homotopy  to reduce the sensitivity of the terminal position 
errors on the variation of the initial velocity. However, often  techniques that improve robustness of convergence by 
reducing the sensitivity of the terminal conditions on the init ial velocity vector, incur in a higher computational cost.  4 
 The major problem of both classes of methods can be identified in the need for a judicious initial guess, often 
better than the simple solution of the Keplerian Lambert proble m. To this end, this paper proposes a novel method 
combining the generation of a first guess with machine learning  and a shooting method based on finite-differences. 
We propose to train a deep neural network (DNN) to generate ini tial guesses for the solution of the J2-perturbed 
Lambert problem and which has been a growing interest in the ap plication of machine learning (ML) to space 
trajectory design [16], [17]. In Ref. [18] one can find a recen t survey of the application of ML to spacecraft guidance 
dynamics and control. Deep neural network is a technology in th e field of ML, which has at least one hidden layer 
and can be trained using a back-propagation algorithm [18]. Sán chez-Sánchez and Izzo [19] used DNNs to achieve 
online real-time optimal control for precise landing. Li et al.  [16] used DNN to estimate the parameters of low-thrust 
and multi-impulse trajectories in multi-target missions. Zhu an d Luo [20] proposed a rapid assessment approach of 
low-thrust transfer trajectory using a classification multilaye r perception and a regression multilayer perception. 
Song and Gong [21] utilized a DNN to approximate the flight tim e of the transfer trajectory with solar sail. Cheng et 
al. [22] adopted the multi-scale deep neural network to achieve  real-time on-board trajectory optimization with 
guaranteed convergence for optimal transfers. However, to the b est of our knowledge ML has not yet been applied 
to improve the solution of the perturbed Lambert problem.  
The DNN-based solver proposed in this paper was applied to the design of trajectories in the Jovian system. The 
strong perturbation induced by the J2 harmonics of the gravity field of Jupiter creates significant differences 
between the J2-perturbed and Keplerian Lambert solutions, even for a small number of revolutions. Hence Jupiter 
was chosen to put the proposed DNN-based solver to the test. Th e performance of the combination of the DNN first 
guess generation and shooting will be compared against two solv ers: one implementing the homotopy method of 
Yang et al. [15], the other implementing a direct application o f Newton method starting from a first guess generated 5 
 with the solution of the Keplerian Lambert problem. The homotop y method in Ref. [15] was chosen for its 
simplicity of implementation and robustness also in the case of  long transfer times. 
The rest of this paper is organized as follows. In Sec. II, the  J2-perturbed Lambert problem and the shooting 
method are presented. Sec. III investigates eight sample forms and their learning features for the DNN. With 
comparative analysis of the different sample forms and standard ization technologies, the optimal sample form for 
the J2-perturbed Lambert problem is found. The algorithm using the deep neural network and the finite 
difference-based shooting method is proposed and implemented to  solve the J2-perturbed Lambert problem in Sec. 
IV. Considering Jupiter’s J2 perturbation, Sec. V compares the numerical simulation results of the proposed 
algorithm, the traditional shooting method and the method with homotopy technique. Finally, the conclusions are 
made in Sec. VI. 
II. J2-perturbed Lambert Problem 
This section presents the dynamical model we used to study the J2-perturbed Lambert problem and the shooting 
method we implemented to solve it. 
A. Dynamical modeling with J2 perturbation 
The J2 non-spherical term of the gravity field of planets and m oons in the solar system induces a significant 
variation of the orbital parameters of an object orbiting those  celestial bodies. Thus, the accurate solution of the 
Lambert problem [12] needs to account for the J2 perturbation, especially in the case of a multi-revolution transfer. 
The dynamic equations of an object subject to the effect of J2 can be written, in Cartesian coordinates, in the 
following form: 6 
  2 2
2 32
2 2
2 32
2 2
2 32311 52
311 52
313 52x
y
z
x
y
zxv
yv
zv
xR zvJr rr
yR zvJr rr
zR zvJr rr




         
        
          



  (1) 
where 
, R ,and J2 represent the gravitational constant, mean equator radius and oblateness of the celestial body, 
respectively. ( x, y, z, vx, vy, vz) is the Cartesian coordinates of the state of the spacecraft, and 22 2rx y z   i s  
the distance from the spacecraft to the center of the celestial  body. 
B. Shooting Method for the J2-perturbed Lambert Problem 
The classical Lambert problem (or Keplerian Lambert problem in the following) considers only an unperturbed 
two-body dynamics [12]. However, perturbations can induce a sig nificant deviation of the actual trajectory from the 
solution of the Keplerian Lambert problem. One way to take pert urbations into account is to propagate the dynamics 
in Eqs. (1) and use a standard shooting method for the solution  of two-point boundary value problems.  
Fig. 1 depicts the problem introduced by orbit perturbations. T he solution of the Keplerian Lambert problem, 
dashed line, provides an initial velocity v0. Because of the dynamics in Eq.(1), the velocity v0 corresponds to a 
difference f0 f f0rr r  between the desired terminal position fr and the propagated one f0r, when the dynamics 
is integrated forward in time, for a period tof, from the initial conditions [ r0, v0]. In order to eliminate this error, one 
can use a shooting method to calculate a velocity v that corrects v0. Fig. 1 shows an example with two subsequent 
varied velocity vectors vi and the corresponding terminal conditions. 7 
 
0rfr
0r
ir0viv0v
iv
nvf0r
fir
 
Fig. 1 Illustration of the shooti ng method based on Newton’s it eration algorithm for the J2-perturbed 
Lambert problem  
As mentioned in the introduction, shooting methods have been ex tensively applied to solve the perturbed 
Lambert problem. Different algorithms have been proposed in the  literature to improve both computational 
efficiency and convergence, e.g. the Picard iteration [11] and the Newton’s iteration [23]. In this section, the 
standard shooting method based on Newton’s algorithm is present ed [23]. Given the terminal position rfi = [xi, yi, zi]T 
and the initial velocity vi = [vxi, vyi, vzi]T at the i-th iteration, the shooting method requires the Jacobian matrix : 
 =iii
xiy iz i
iii
i
xiy iz i
iii
xiy iz ix xx
vvv
y yy
vvv
zzz
vvv    
  
 
  
   H , (2) 
to compute the correction term: 
 1
f iivHr r , (3) 
where J-1 is the inverse of the Jacobian matrix Hi, and rf is the desired terminal position, as shown in Fig. 1. The 
corrected initial velocity then becomes 1ii i vvv . 8 
 Here the partial derivatives in the Jacobian matrix are approxi mated with forward differences. Finite differences 
are computed by introducing a variation 610v in the three components of the initial velocity and computing t he 
corresponding variation of the three components of the terminal  conditions ixr, iyr, and izr. Consequently, the 
Jacobian matrix can be written as follows. 
 =iy ix iz
ivvv
 
 
 r rrH   (4) 
Because of the need to compute the Jacobian matrix in Eq. (2), finite-difference-based shooting methods need to 
perform at least three integrations for each iteration. Further more, if the accuracy of the calculation of the Jacobian 
matrix in Eq.(2) is limited, this algorithm could fail to conve rge to the specified accuracy or diverge, which is a 
common situation if the time of flight is long (e.g., tens of r evolutions). Homotopy techniques are an effective way 
to improve the convergence of standard shooting methods for MRP LP but still require an initial guess to initiate the 
homotopy process and can require the solution of multiple two-p oint boundary value problems over a number of 
iterations. Here a DNN is employed to globally map the change i n the initial velocity to the variation of the terminal 
position for a variety of initial state vectors and transfer ti mes. This mapping allows one to generate a first guess for 
the initial velocity change ivby simply passing the required initial state, transfer time and  terminal condition as 
input to the DNN.  
In the following, we will present how we trained the DNN to gen erate good first guesses to initiate a standard 
shooting method. We will show that an appropriately trained DNN  can generate initial guesses that provide 
improved convergence of the shooting method even for multi-revo lution trajectories. It will be shown that the use of 
this initial guess improves the robustness of convergence of a standard shooting method and makes it significantly 
faster than the homotopy method in [15]. 9 
 III. Sample Learning Feature Analysis 
DNN consists of multiple layers of neurons with a specific arch itecture, which is an analytical mapping from 
inputs to outputs once its parameters are given. The typical st ructure of DNN and its neuron computation is 
illustrated in Fig. 2. The output of each neuron is generated f rom the input vector x, the weights of each component 
w, the offset value b, and the activation function y=f(x). The inputs are provided according to the specific problem or  
the outputs of the neurons of the previous layer. The weight an d offset values are obtained through the sample 
training. The activation function is fixed once the network is built. The training process includes two steps: the 
forward propagation of the input from the input layer to the ou tput layer; and then the back propagation of the output 
error from the output layer to the input layer. During this pro cess, the weight and the offset between adjacent layers 
are adjusted or trained to reduce the error of the outputs.  
ii sbw x
  yf s
 
Fig. 2 The diagram of the DNN s tructure and neuron computation 
The ability of a DNN to return a good initial guess depends hig hly on the representation and quality of samples 
used to train the network. High-quality samples cannot only imp rove the output accuracy of the network, but also 10 
 reduce the training cost. Therefore, in the following, we prese nt the procedure used to generate samples with the 
appropriate features. 
A. Definition of Sample Form and Features  
In this work two groups of sample forms have been considered: o ne has the initial velocity v0 solving the 
J2-perturbed Lambert problem as output, the other has the veloc ity correction 0v to an initial guess of v0 a s  
output.  
For the first group of sample forms, the input to the neural ne twork includes the known initial and terminal 
positions 0f,rr and the time of flight tof. The output is only the initial velocity v0 as the terminal velocity can be 
obtained through orbital propagation once the initial velocity is solved. This type of sample form is defined as 
    0f 0,, ,vSt o frr v   (5) 
where the subscript 0 and f denotes the start and end of the tr ansfer trajectory, respectively. Thus, when trained with 
sample form in Eq. (5), the DNN is used to build a functional r elationship between 0f,,tof rr and 0v. 
The second group of sample forms was further divided in two sub groups. One that uses the initial state of the 
spacecraft 0r, the time of flight tofand the terminal error fras input and the other that uses the initial state 0r, 
the time of flight tof, the terminal position error frand the initial velocity vector from the Keplerian solution 
dvas inputs. These two sample forms are defined as follows:  
   
  d1 0 f 0
d2 0 d f 0,, ,
,, , ,v
vSt o f
St o f 
 rr v
rv r v  (6) 
In Eq. (6) the output 0v is always the initial velocity correction 00 dvv v , in which 0v is the initial 
velocity that solves the J2-perturbed Lambert problem. Thus, wh en trained with sample forms Sdv1 and Sdv2, the DNN 11 
 realizes a mapping between 0v a n d  0f,,tof rr or  0d f,, ,tof rv r respectively. The difference between Sdv1 
and Sdv2 is whether the input includes the initial velocity vd t h a t  i s  n e c e s s a r y  f o r  s o l v i n g  t h e  J a c o b i a n  m a t r i x .  
Therefore, it is theoretically easier to obtain the desired map ping with the input including the initial velocity, i.e. Sdv2. 
However, this increases the dimensionality of the sample and mi ght increase the difficulty of training. 
For each group of sample forms there are three main ways of par ameterizing the state of the spacecraft: Cartesian 
coordinates, spherical coordinates and the mean orbital element s. Cartesian coordinates provide a general and 
straightforward way to describe the motion of a spacecraft but state variables change significantly over time even for 
circular orbits with no orbital perturbations. Spherical coordi nates can provide a more contained and simpler 
variation of the state variables but are singular at the poles.  Double averaged mean orbital elements present no 
variation of semimajor axis, eccentric and inclination due to J 2 and a constant variation of argument of the perigee 
and right ascension of the ascending node [24]. Which parameter ization to choose for the training of the DNN will 
be established in the remainder of this section. The structures  of Eqs. (5) and (6) expressed in terms of these three 
coordinate systems are as follows: 
  
  
  

Car 0 0 0 f f f 0 0 0
S p h 0 0 0 fff 0 0 0
OEm 0 f 0 0 0
d1 C a r 0 0 0 f f f 0 0 0d1 S p h 0 0 0 f f f,,,,,, , ,
,,, ,,, , ,
,, ,,
,,, , , , , , ,
,,, , , ,vx y z
vv v
vv v
vx y zvSx y z x y z t o f v v v
Srrt o f v
So e o e t o f v
Sx y z x y z t o f v v v
Srr t o f v   

  


 


      
   ，
，
，
， 

  
  00 0
d2 C a r 0 0 0 d d d f f f 0 0 0
d2S p h 0 0 0 d d d f f f 0 0 0
d2 O E m d f f f 0 0 0,,
,,, , , , , , , , , ,
,,, ,,, , , , , ,
,, , , , ,vv
vx y z x y z
v vv
vv vSx y z v v v x y z t o f v v v
Sr vr t o f v
So e r t o f v
     
  


         
     
     ，
，  (7) 
where the subscript Car, Sph and OEm  denote the Cartesian coordinate, the spherical coordinate and mean orbital 
elements, respectively. And x, y, and z are the Cartesian coordinates of the position vector. And r, , and  a r e  12 
 the distance, azimuth, and elevation angle of position vector i n the spherical coordinate system. 
,, , , ,Toe a e i w M  represents the mean orbital elements. 
B. Performance Analysis of Different Sample Forms 
In this section the performance of the eight sample forms defin ed in Eq.(7) is assessed in order to identify the 
best one to train the DNN. We always generate a value for the i nitial conditions starting from an initial set of orbital 
elements. Values of the orbital parameters for each sample are randomly generated with the rand  function in 
MATLAB using a uniform distribution over the intervals defined in Table 1. Note that semimajor axis and 
eccentricity are derived from the radii of the perijove and apo jove. Considering the strong radiation environment of 
Jupiter and the distribution of Galilean moons, we want to limi t the radius of the pericentre rp of the initial orbit of 
each sample to be in the interval [5 RJ, 30RJ], where RJ = 71492 km is the Jovian mean radius. The value of the 
inclination is set to range in the interval [0, 1] radians. The  time of flight does not exceed one orbital period T, which 
is approximately calculated using the following formula 
 3
J=2aT  (8) 
where a is the semi-major axis, a = ( ra + rp ) / 2. 
Table 1 Parameters’ ranges of the sample 
Parameters Range 
Apojove radius ra (×RJ) [ rp, 30] 
Perijove radius rp (×RJ) [5, 30] 
inclination (rad) [0, 1] 
RAAN (rad) [0, 2) 
Argument of perigee (rad) [0, 2) 
Mean anomaly (rad) [0, 2) 
tof (T) (0, 1) 13 
 The following procedure is proposed to efficiently generate a l arge number of samples without solving the 
J2-perturbed Lambert problem: 
Step 1: The initial state [ r0, v0] and time of flight tof are randomly generated. 
Step 2: The terminal state [ rf, vf] is obtained by propagating the initial state [ r0, v0] under the J2 perturbation 
dynamics model, for the propagation period tof. 
Step 3: The Keplerian solution vd is solved from the classical Lambert problem with the initial and terminal 
position r0, rf and flight time tof. 
Step 4: The end state [ rfd, vfd] is obtained by propagating the initial Keplerian state [ r0, vd] under the J2 
perturbation dynamics model, and for the propagation period tof. 
Step 5: The initial velocity correction 0v and the end state error fr are computed with00 dvv v  and 
ff f drr r . 
Using these five steps, we generated 100000 samples and then gr ouped them in the eight sample forms given in 
Eq.(7). Before training, a preliminary learning feature analysi s is performed on the distribution of sample data and 
the correlation between the inputs and the output. Specifically , the mean, standard deviation, and magnitude 
difference coefficients are used to describe the distribution o f the data, and the Pearson correlation coefficient is 
chosen to evaluate the correlation of the data. Their mathemati cal definitions are given as follows 
 

1
2
1=
1
max
log
min 0n
j
j
n
j
jX
Xn
XXn
X
X




   (9) 14 
 where X and  are the mean and standard deviation of the data, respectively.  And n is the total number of data. 
 denotes the magnitude difference coefficients that assesses th e internal diversity of the data.  
The statistical characteristics of the variables in the sample are given in Table 2. For the variables described in 
Cartesian coordinate, the mean values are close to 0 but the st andard deviations are generally large. Furthermore, 
their magnitude difference coefficients are all more than 5, wh ich indicate a large difference in the absolute values 
of the variables. For the variables described in spherical coor dinate, the most of their standard deviations are less 
than these described in the Cartesian coordinate. In addition, the magnitude difference coefficients of the magnitude 
of the position and velocity vectors are less than 1. The varia bles with smaller standard deviation have better 
performance in the training process. Therefore, the samples wit h the variables represented in spherical coordinate 
are easier to learn than those described in Cartesian coordinat es.  
Table 2 The statistical distributi ons of the variables in the s amples  
Parameters 
of sample Mean Standard deviations Magnitude difference 
coefficients 
r0-Car [-0.014; 0.087; 0.001] [11.424; 11.424; 1.145] [5.125; 5.949; 7.077] 
r0-Sph [14.954; 0.007229; 0.000193] [6.221; 1.815; 0.070] [0.777; 4.9 26; 6.607] 
rf-Car [0.001; -0.031; -0.005] [12.438; 12.469; 1.246] [5.094; 4.371;  6.442] 
rf-Sph [16.503; -0.005966; -0.000386] [6.275; 1.813; 0.070] [0.777; 4 .454; 6.321] 
v0-Car [-0.088351; -0.032116; 
-0.006130] [8.916; 8.887; 0.895] [5.450; 5.185; 6.338] 
v0-Sph [12.082577; -0.002034; 
-0.000529] [3.647; 1.821; 0.071] [0.773; 5.257; 5.851] 
oe0 [15.771; 0.257895; 0.087045; 
3.148016;  3.137227; 3.138286] [5.225; 0.177; 0.050; 
1.813;1.812;1.816] [0.774; 5.273; 4.145; 
4.653; 5.422; 4.634] 
oef [15.771; 0.257850; 0.087045; 
3.147935; 3.137600; 3.151625] [5.225; 0.177; 0.050; 
1.813; 1.812; 1.528] [0.774; 4.721; 4.145; 
5.917; 5.228; 5.163] 
vd-Car [-0.087568; -0.031006; 
-0.006340] [8.915; 8.886; 0.895] [5.654; 5.578; 6.673] 
vd-Sph [12.081729; -0.001599; 
-0.000538] [3.647; 1.821; 0.071] [0.774; 5.651; 6.658] 
f-Carr [-3.162; -11.384; -0.075] [1369.838; 1395.080; 
187.322] [10.495; 10.828; 11.371]15 
 f-Sphr  [1154.249; -0.004; -0.001] [1589.222; 1.817; 0.135] [10.283; 4. 831; 8.576] 
oed [15.769; 0.258264; 0.087096; 
3.147709; 3.136805; 3.139263] [5.226179; 0.177; 0.051; 
1.813; 1.812; 1.814] [9.556; 4.800; 5.049; 
5.240; 5.927; 4.639] 
tof 4.023 3.220 5.481 
0-Carv  [-0.000782; -0.001109; 0.000210] [0.326; 0.283; 0.063] [9.917; 1 0.432; 10.471]
0-Sphv  [0.013321; 0.003913; 0.002884] [0.436; 1.818; 0.503] [8.948; 4. 672; 5.924] 
It is also known that the learning process is easier if the cor relation between the input and output of the sample is 
stronger. Here the Pearson correlation coefficient is used to d escribe this correlation and is defined as follows  
 
1n
jj
j
XYXX Y Y
R

 ( 1 0 )  
where  n is the total number of sample data. Y and Yrepresent the mean and standard deviation of the data Y. 
X a n d  Xdenote the mean and standard deviation of the data X. 
The matrix of the Pearson correlation coefficients of the propo sed sample’s inputs and outputs are given in Table 
3. The elements of Pearson correlation coefficients matrix are the correlation coefficient between the corresponding 
input and output variables. The signs of the elements indicate positive and negative correlations, respectively. The 
absolute values of elements represent the strength of correlati on. The greater the absolute value is, the stronger the 
correlation is.  
Table 3 The matrix of the Pearson correlation coefficients of t he input and output for different sample forms 
Sample 
Forms Pearson correlation coefficients matrix 
Sv-Car 0.003 0.004 0.000 0.002 0.002
0.002 0.003 0.001 0.001 0.003
0.005 0.000 0.002 0.001 0.002 0.000 0.002 

  

-0.764 0.126
0.764 -0.122  
Sv-Sph 0.005 0.001 0.003 0.000
0.002 0.000 0.004 0.000 0.001 0.003
0.001 0.001 0.002 0.003 0.002 0.001 0.003

  

-0.898 -0.459 -0.344
-0.116  
Sv-OEm 0.002 0.002 0.002 0.000 0.002 0.003 0.002 0.001
0.002 0.000 0.000 0.002 0.003 0.007 0.002 0.000 0.000 0.002 0.003 0.001 0.0 03
0.002 0.001 0.007 0.004 0.002 0.001 0.002 0.001 0.007 0.004 0.000 
 -0.587 0.291 -0.587 0.291 -0.344
0.008 0.003

16 
 Sdv1-Car 0.006 0.002 0.002
0.004 0.000 0.002
0.003 0.006 0.003 0.004 0.004
 


-0.011 -0.049 -0.053 -0.046
0.010 0.037 -0.041 -0.013
0.011 -0.090 
Sdv1-Sph 0.003 0.005 0.004 0.002
0.002 0.000 0.002 0.000 0.001
0.001 0.002 0.001 0.001 0.004



-0.025 0.081 0.010
0.377 0.254
0.512 0.045 
Sdv2-Car 0.006 0.002 0.000 0.002
0.004 0.000 0.002 0.002
0.003 0.006 0.003 0.004 0.004 0.004
  


-0.011 -0.017 -0.014 -0.049 -0.053 -0.046
0.010 0.011 -0.012 0.037 -0.041 -0.013
0.010 -0.040 0.011 -0.090 
Sdv2-Sph - . 0.003 -0.005 . 0.004 -0.005 . 0.004 -0.002 .
-0.002 . 0.000 0.005 . -0.001 0.002 . 0.000 -0.001
-0.001 -0.002 . 0.003 0.004 . 0.001 -0.001 . 0.004      
      
            0 025 0 032 0 081 0 010
0 377 0 259 0 254
05 1 2 02 9 7 00 4 5
Sdv2-OEm  0.001 0.004 0.001 0.004 0.002
0.002 -0.003 0.002 0.004 0.002 0.001 0.002 0.000 0.001
0.000 0.001 0.002 0.003 0.002 0.001 0.001 0.004 

 

-0.019 0.082 0.076 0.081 0.010
0.254
0.010 0.045 
First, it is seen that most elements of the matrix are less tha n 0.01, indicating the correlations between the inputs 
and the outputs are generally weak. Second, for the first three  sample forms of Table 3, the absolute values of all 
elements for some rows are less than 0.01. This means that some  components of the output variable are in 
weak-correlation with all input variables, and hence the mappin g from these output components to the input 
variables is very difficult to capture. Therefore, samples with  the initial velocity as output, i.e. Sv-Car, Sv-Sph, and 
Sv-OEm, are not deemed to be ideal for the training of the neural net work. Third, by comparing the matrix listed in 
rows 4 to 7 of Table 3, the absolute values of the elements for  the samples described in Cartesian coordinates are 
smaller than those for the samples described in spherical coord inates. Furthermore, for the samples in spherical 
coordinates, it is seen that the submatrix of each input variab le in the Pearson correlation coefficients matrix is a 
diagonally dominant matrix, where the elements with large absol ute values for each input variable are distributed in 
different rows and columns, and are independent. Therefore, the  samples described in the spherical coordinate have 
better learning features and performance due to the strong corr elations. Additionally, for Sdv2-Sph that includes the 
Keplerian solution vd as one of the inputs, the correlation with the initial velocit y correction 0v i s  [0.032 , 0.004, 17 
 -0.005; 0.005, 0.259 , -0.001; 0.003, 0.004, 0.297 ], which is diagonally dominant with large diagonal values, whi ch 
demonstrates that the Keplerian solution is an important input.  Finally, for the sample in the mean orbital elements 
in the last row of Table 3, the matrix only contains a few elem ents whose absolute values are greater than 0.01, and 
most of them are distributed in the first row. The mean variati ons of semimajor axis, eccentricity and inclination are 
not affected by the J2 perturbation but only by the variation o f the initial velocity. Therefore, only the first row in the 
matrix displays larger values. In addition, the elements in the  first six columns of the Pearson correlation matrix of 
Sdv2-OEm  are generally smaller than others in Table 3, because the outp uts of the sample is the initial velocity 
correction, which is calculated using the osculating orbital el ements that contain both the long and the short term 
effects of the J2 perturbation. Thus the correlation using the mean orbital elements is moderate. This would suggest 
that the sample Sdv2-Sph is the best option for the training of the DNN among the eight  tested sample forms. We will 
now quantify the training performance for each of the eight sam ple forms by comparing the training convergence of 
a given DNN. It has to be noted that the structure of the DNN p lays a role as well. For example, a high dimensional 
sample with more variables needs a larger size DNN with more la yers and neurons. However, we argue that, since 
the sample form selection mainly depends on the problem and the  dynamics, a better sample form will have better 
training performance than other sample forms given the same DNN  structure. For this reason, it is reasonable to 
compare sample forms even on DNN structures that are not optima l. The effect of the structure of DNN on the 
training performance will be discussed in section V. 
Some data pretreatment is necessary to facilitate the training process and improve the prediction accuracy. 
Standardization, normalization and logarithms are used to pre-p rocess data with large ranges or magnitude 
differences. Tests in this section were performed using a four- layer fully connected DNN with 50 neurons per 
hidden layer. The activation functions of the hidden layers and  the output layer are all Tanh. The Adaptive moment  estimatio n
algorith m
The con s
training p
output o f 
 
where n i
Here MS E
From 
significa n
has a lar gn (Adam) [25 ]
m works throu g
struction and 
process, the va
fthe sample f o
s the number o
E has no unit s
Fig. 3 one c a
ntly smaller t h
ger range of v] was employ e
gh the entire t
training of t h
ariations of t h
or different sa m
of samples, a n
s because data 
Fig. 3 The tr
an see that th e
han that with t h
values and the r
ed for the opt i
training datas e
he DNN are b
he mean squa r
mple forms ar e
MSE
ndˆiyand y i are 
has been nor m
ainin g conve r
e MSE of the n
he initial velo c
refore has a m
imization. Th e
et) was set to 
based on the 
re error (MS E
e given in Fig

11ˆn
i
iE y
n
the output pr e
malized befor e
rgence histor y
neural netwo r
city as the ou t
more scattere d
e maximum e p
10000 and t h
Python impl e
E) between th e
. 3. The math e
2
iy
edicted by the 
e training. 
y for differe n
rk with the in i
tput. This is b e
d distribution. 
poch (or num b
he initial lear n
ementation o f 
e output of t h
ematical expr e
DNN and th e
 
nt sample for m
itial velocity c
ecause the ini t
Also, the M S
ber times that 
ning rate was s
fTensorFlow. 
he neural net w
ession of the M
e true output r e
ms 
correction as t
tial velocity i n
SE of sample S
18the learning 
set to 0.001. 
During the 
work and the 
MSE is: 
(11)
espectively . 
the output is 
n the sample 
Sdv1-Sph is an 19 
 order of magnitude higher than that of sample Sdv2-Sph. Therefore, the accuracy of predicting the initial velocity 
correction is effectively improved by including the Keplerian v elocity in the input of the sample. The blue line in 
Fig. 3 has obvious fluctuations due to the weak correlations be tween the output and the input of Sv-OEm, as shown in 
Table 3. Finally, the training results of the samples in spheri cal coordinate are better than those in Cartesian 
coordinates, which is consistent with the conclusions drawn in previous sections. 
In summary, for the J2-perturbed Lambert problem, the samples d escribed in spherical coordinate appear to be 
more suitable for the training of a DNN. In fact, among all eig ht sample forms, the sample form Sdv2-Sph yielded the 
best learning converge, given the initial position, Keplerian v elocity, the terminal position error of the Keplerian 
solution and time of flight as inputs and the initial velocity correction as output. Therefore, in the remainder of this 
paper, the Sdv2-Sph sample form is selected for the training of the DNN.  
IV. Solution of the J2-perturbed La mbert Problem Using DNN 
The proposed solution algorithm (see the flow diagram in Fig. 4 ) is made of an Intelligent initial Guess 
Generator (IGG) and a Shooting Correction Module (SCM). The DNN  is used in the IGG to estimate the correction 
of the Keplerian solution and provide an initial guess to the s hooting module. The shooting method discussed in part 
B of Section II is employed in th e SCM to converge to the requi red accuracy.   As s h
vectors. T
terminal p
form an d
shooting 
rendezvo u
approxi m
The m
terminal s
one call t o
method mFig. 4
hown in Fig. 4
Then the init i
position error 
d the generat i
method in S e
us constraint .
mated with the 
method propo s
state, where  i 
o the DNN ar e
mainly depend4 The flow c h
4, first the K e
ial conditions 
rfd. With th i
ion method o
ection II is a
. The Jacobi a
difference qu o
sed here perfo r
is the numbe r
e necessary t o
s on the SC M
hart of the pr o
eplerian Lam b
[r0, vd] are p
is erro r, the i n
of the sample s
applied to co r
an matrix is 
otient to redu c
rms a total of 
r of iterations.
o obtain the in i
M. As it will be 
oposed J2-pe
bert problem 
propagated f o
nitial velocity 
s a r e  d e s c r i b e
rrect the initi a
calculated a c
ce the comput a
4i+2 numeric
 Additionally ,
itial velocity g
shown in the 
rturbed La m
is solved wit
orward in tim e
correction is 
ed in Sectio n
al velocity to
ccording to E
ational load. 
al propagatio n
, one solution 
guess. Theref o
next section, 
 
mbert proble m
th the desired 
e u n d e r  t h e  e
calculated us i
n I I I .  T h e n  t h
make the te r
Eq.(4), where 
ns to obtain t h
of the Keple r
ore, the calcul a
the initial gu e
m solver 
initial and f i
ef f e c t  o f  J 2  t o
ing the traine d
he finite diff e
rminal positi o
the partial d
he Jacobian ma
rian Lambert p
ation time of t
ess provided b y
20inal position 
o obtain the 
d DNN. The 
erence-based 
on meet the 
derivative is 
atrix and the 
problem and 
the proposed 
y the IGG is 21 
 close enough to the final solution that the number of iteration s required to the SCM to converge to the required 
accuracy is significantly reduced. 
V. Case Study of Jupiter Scenario 
In this section, taking the Jovian system as an example, some n umerical simulations are performed to 
demonstrate the effectiveness and efficiency of the proposed J2 -perturbed Lambert solver. Firstly, different network 
structures and training parameters are tested to find the optim al ones for this application. Then, we simulate the 
typical use of the proposed solver with a Monte Carlo simulatio n whereby a series of transfer trajectories are 
computed starting from a random set of boundary conditions and transfer times. To be noted that although the tests 
in this section use the J2, μ and R, of Jupiter the proposed method can be generalized to other ce lestial bodies by 
training the corresponding DNNs with a different triplet of val ues J2, μ and R, but using the same sample form. 
A. DNN Structure Selection and Training 
With reference to the results in Section III, the samples used to train the DNN include the initial position, the 
initial velocity, coming from the solution of the Keplerian Lam bert problem, the terminal position error of the 
Keplerian solution, and the time of flight. The output is the i nitial velocity correction of the Keplerian solution and 
all vectors in a sample are expressed in spherical coordinates.  In order to generalize the applicability of this method, 
the ranges of the parameters of the sample given in Table 1 hav e been appropriately expanded. The range of orbital 
inclinations is [0, ] in radian. The range of times of flight is now in the open in terval (0, 10 T), where T is calculated 
using Eq. (8) from the initial state ( r0, v0). The ranges of other parameters are consistent with Table 1. In total, 
200000 training samples are obtained using the rapid sample gen eration algorithm given in part B of Section III.  Since 
results, i n
one woul
sample f o
We s t
learning w
and ReL U
The sphe[-0.5
, 0
of the th r
chosen a s
Also i
other trai n
in Table 4the structure 
n this section 
d need to loo p
orm remains r e
tart by defini n
while Sigmoi d
U will be use d
rical coordin a
0.5]. Becau s
ree compone n
s the activatio n
in this case t h
ning paramet e
4. and training p
we analyze d
p back and ch e
easonably go o
ng the activa t
d functions ar e
d. The output r
ates (magnitu d
se the range o
nts of the sph e
n function of t
Fig
he Adaptive m
ers are the sa m
parameters o f 
different DNN 
eck the optim a
od even once t h
tion functions
e less used bec
ranges of Tan h
de, azimuth, a
f elevation a n
erical coordin
the output lay e
g. 5 The t ypica
moment estim a
me as in Secti o
fthe neural ne
structures a n
ality of the sa m
he DNN stru c
. Tanh and R
cause the gra d
h and ReLU a
and elevation) 
ngle can be tr a
ates can all m
er.  
al activation 
ation is used a
on III. The tr a
twork also pl a
nd settings. N o
mple form, ho
cture is chang e
ReLU are the 
dient tends to 
are [-1, 1] and 
of the outpu t
ansformed fro m
meet the requi
functions for 
as optimizer. T
aining results 
ays a signific a
ote that once t
wever, in this 
ed.  
common acti v
vanish [26], t h
[0, ∞] respec t
t of the samp l
m [-0.5, 0.5
rements of R e
 
 DNN  
The maximu m
of DNNs wit h
ant impact on 
the structure i
 paper we ass u
vation functi o
hus in the fol l
tively, as sho w
le are [0, ∞], 
5] to [0, ]
eLU. Therefo
m epoch is 5 0
h different si z
22the training 
is optimized 
ume that the 
on s  f o r  d e e p  
lowing Tanh 
wn in Fig. 5. 
[0, 2] and 
], the ranges 
re, ReLU is 
0000 and the 
zes are listed 23 
 Table 4 Training results of DNNs with different sizes 
Hidden 
Layers Neurons per 
hidden layer activation 
function MSE Training 
time (s) 
2 20 ReLU 9.423e-05 762 
Tanh 3.286e-05 839 
50 ReLU 1.435e-05 951 
Tanh 1.226e-05 1084 
100 ReLU 9.423e-06 1210 
Tanh 9.163e-06 1425 
3 20 ReLU 2.423e-06 1198 
Tanh 2.154e-06 1267 
50 ReLU 1.315e-06 1347 
Tanh 1.258e-06 1523 
100 ReLU 5.631e-06 1746 
Tanh 1.226e-06 1935 
4 20 ReLU 9.423e-06 1648 
Tanh 3.286e-06 1864 
50 ReLU 7.522e-07 1977 
Tanh 4.816e-07 2186 
100 ReLU 6.395e-05 2361 
Tanh 2.861e-05 2643 
The neural network with the minimum MSE has 4 hidden layers, ea ch with 50 neurons. The activation function 
of its hidden layers is Tanh. Additionally, some conclusions ca n be made from Table 4. Firstly, the networks with 
ReLU as the activation function take less time for training. Se condly, the networks with Tanh as the activation 
function achieve smaller MSEs. Thirdly, the network with 4 hidd en layers and 100 neurons in each hidden layer has 
overfitted during the training process. 
The variation of MSE of the neural network with 4 hidden layers  and with 50 neurons for each layer is shown in 
Fig. 6. MSE finally converges to 4.816e-07, which transforms in to the mean absolute error (MAE) of the DNN’s 
output: [0.004241 km/s; 0.000232 rad; 0.000152 rad].   In or d
trained s a
of the tr a
terminal p
of the tra i
 
Fig. 7
DNN. [Δ v
Kepleria n
the termi n
points in 
also redu c
After the der to verify t
amples were r
ained DNN. T
position rf are 
ined DNN ( vc,
7 and Fig. 8 s
v0dx; Δv0dy; Δv
n solutions, re s
nal position a
Fig. 7 and Fi g
ced significa n
correction b yFig. 6 M S
the predictio n
randomly reg e
The initial vel o
used as refer e
, rfc) are calcu
show the co m
v0dz] and [Δ rfdx
spectively. [ Δ
after the DN N
g. 8) is much c
ntly after the c
y the DNN, th
SE of the sele c
n accuracy of 
enerated with t
ocity v0, whi c
ence values. T
lated as follo w
0d
0cv
v
mparison betw e
x; Δrfdy; Δrfdz]
Δv0cx; Δv0cy; Δv
N’s correc tion
closer to 0 aft e
correction, wh
e initial velo c
cted DNN du r
the trained D
the algorithm 
ch is the exac t
The errors of t h
ws 
0df d
0cf c,
,
vv r
vvr
een the Kepl e
are the errors
v0cz] and [Δ rfcx
s, respectivel y
er the DNN’s c
ich is indicat e
city error is li m
ring the trai n
DNN, 1000 n e
in part B of S
t solution of t
he Keplerian s
ff d
ff c
rr
rr
erian solution s
of the initial v
x; Δrfcy; Δrfcz]
y. It can be s
correction. T h
ed by the len g
mited to 10 m
 
ning process. 
ew samples t h
Section III to 
the J2-pertur b
solutions ( vd, r
s and the app r
velocity and t h
are the errors 
een that the m
he standard de v
gth of the blue 
m/s, and the te r
hat are differ e
examine the p
bed Lambert p
rfd) and the ap p
roximation o f
he terminal p o
of the initial v
mean of thes e
viation of the s
bars in Fig. 7
rminal positio n
24ent fro m t h e  
performance 
proble m an d 
proximation 
(12)
f t h e  t r a i n e d  
osition of the 
velocity and 
e errors (red 
se errors has 
7 and Fig. 8. 
n error does  not exce e
initial va l
Fig. 7 T
Fig.
B. Perfo
In this s e
method ued 100 km. T
lue with respe c
The statistica l
. 8 The statis t
ormance Ana l
ection the pr o
using NewtonThis proves th a
ct to a simple 
l results of th e
tical results o
lysis for MR P
oposed DNN- b
’s iteration a l
at the applic a
Keplerian La m
e initial velo c
of the termin a
PLP  
based metho d
lgorithm (SN )
ation of the D
mbert solutio n
city errors of t
al position er r
correctio n
d is compare d
) a n d  t h e  h o m
DNN has sign i
n. 
the Kepleria n
rors of the K e
n  
d a g a i n s t  o t h e
motopic pertu r
ificantly imp r
 
n solution an d
eplerian solu t
er two metho d
rbed Lamber t
roved the acc u
d the DNN’s c
tion and the D
ds: a traditio n
t algorith m (H
25uracy of the 
correction 
DNN’s 
nal shooting 
HL) in [15]. 26 
 When applying the HL, the C++ version of Vinit6 algorithm in li terature [27] is employed to implement the HL 
method in Ref. [15] and to decrease the CPU computation time of  HL. The HL is running in Matlab and the MEX 
function calls the Vinit6 algorithm that is running in visual s tudio 2015 C++ compiler to analytically propagate the 
perturbed trajectory. The accuracy tolerance of Vinit6 algorith m is set at 1 10-12. The homotopy parameter is 
defined as the deviation in the terminal position and other det ails of implementation and settings are the same as 
these given in Ref. [15]. For the SN and the proposed method, t heir dynamical models only include the J2 
perturbation. For the Vinit6 algorithm, the dynamical model inc ludes the J2, J3 and partial J4 perturbations. 
However, the magnitudes of J3 and J4 of Jupiter are much smalle r than that of J2. Their perturbation effects are very 
weak compared with that of J2. Therefore, the slight difference  in the dynamical model has very limited impact on 
the number of iterations and running time of the HL since the V init6 algorithm has high computational efficiency. 
Therefore, the comparison among the three methods is still vali d.  
The performance of the three methods is compared over 11 datase ts one per number of full revolutions from 0 
to 10. Each dataset has 1000 samples, which are regenerated wit h the method described in Section III to validate the 
DNN. The maximum iterations and tolerances of the three methods  are listed in Table 5.  
Table 5 The maximum iterations an d tolerance of three methods 
Algorithm Tolerance (km) Maximum iterations
SN 0.001 2000 
HL 0.001 10000 
DNN-based method 0.001 2000 
If the algorithm converges to a solution that meets the specifi ed tolerance within the set number of iterations, it is 
recorded as a valid convergence, otherwise, as a failed converg ence. The result is displayed in Fig. 9 and Fig. 10, in 
terms of convergence ratio (number of converged solutions over number of samples) and average number of 
iterations to converge.  Fig.
Fig. 10 A
Acco r
the valid 
the num b
the num b
requires t
revolutio n
problem. mitigates . 9 The conv e
Avera ge num
rding to Fig. 9
convergence r
ber of iteratio n
ber of iteratio n
the least nu m
ns is due the 
For the sam e
this problem ergence ratio 
mber of iterat i
9, the HL and t
ratio of the S N
ns of HL appe a
ns of SN and t
mber of iterati
growing dif f
e r e a s o n  t h e  H
by providing a
of different a
ions of differ e
the proposed m
N decreases a
ars to increas e
the proposed D
o n s .  T h e  l a c k
ference betwe e
HL progressi v
a good initial 
algorithms fo r
ent al gorith m
method coul d
s the number 
e linearly in l o
DNN-based m
k of converg e
en the exact s
vely requires 
guess for eve r
r the Jupiter J
ms on the Jup i
converge to t
of revolution s
og-scale as th e
method remai n
ence of the S N
solution and t
more iteratio n
ry number of r
 
J2-perturbe d
 
iter J2-pertu r
the required a c
s increases. T h
e number of r e
n nearly const a
N w i t h  t h e  i n
the solution o
ns to conver g
revolutions.  
d Lambert pr
rbed Lambe r
ccuracy in all 
hen, accordin g
evolutions inc r
ant. The prop o
ncrease in th e
of the Kepler i
ge .  T h e  p r o p o
 
27roblem 
rt problem 
cases, while 
g to Fig. 10, 
reases while 
osed method 
e number of 
ian Lambert 
osed method  The a
only acc o
DNN-Ba s
CPU cal c
of the re v
the SCM
the incre a
time of t
computatDNN eff
e
time wit h
number o
proposed shooting matrix. 
T
HL takes 
Fig. 11 Aaverage CPU c
ounts the ti m
sed method a n
culation time o
volution incre a
. In general, t
ase in the nu m
the SN and t h
tional time of H
ectively redu c
h the number o
of revolution s
method are r
algorith m, fo
Their computa t
less time, the 
Avera ge CPU computationa l
me of the S C
nd HL are 0. 0
of the propos e
ases because t h
the computati o
mber of iterat i
he proposed m
HL appears t o
ces the numb e
of revolution s
s tested in th
respectively 0
or which eac h
tional time pe r
HL requires m
computatio nl time of the t
CM .  F o r  z e r o
051 seconds, 0
ed method is t
he accurate i n
onal time inc r
ions and the l
method appe a
o increase mo r
er of iteration s
s. The comput
is pape r. The
0.0082 s, 0.0 0
h iteration ne e
r iteration is h
much more it e
 
nal time of di f
three method s
o-revolution c
0.027 second s
the shortest. T
nitial guess ob t
reases with th e
longer propa g
ar to increase 
re rapidly. Th e
s and provide s
ational time o
e average co m
018 s, and 0. 0
eds additional
higher than th a
erations than t h
fferent metho
s is given in F
case, the av e
s and 0.329 s e
This advantag e
tained using I
e increase in t
gation time. A
linearly with 
e figure show s
s, as a result, 
of the propose
mputational t i
0078 s. The p r
 t h r e e  i n t e g r a
at of the HL. H
he other two m
ds for the Ju p
Fig. 11, in w h
erage CPU c o
econds, respe c
e becomes m o
GG reduces t h
the number o f
As shown in F
the number 
s that the initi a
a slower incr e
d method is b
ime per itera t
ropose d m e t h
al operations 
However, tho u
methods, as s h
 
piter J2-pert u
hich the prop o
omputation t i
ctively. It is s
ore obvious as 
he number of 
f revolutions, 
Fig. 11, the c o
of revolution s
al guess obtai n
ease of the c o
below 0.5 sec o
tion of SN, H
ho d  a n d  S N  u
to calculate t
ugh the singl e
hown in Fig. 1
urbed Lamb e
28osed method 
ime of SN, 
seen that the 
the number 
iterations of 
due to both 
omputational 
s, while the 
ned with the 
omputational 
onds, for the 
HL ,  a n d  t h e  
se the same 
the Jacobian 
e-iteration of 
1. 
ert problem C. Mon t
In thi
conditio n
generate s
time of s a
methods, 
transfer t i
DNN is t
called o n
while the computatfinal res
u
increase i
or larger sample g
ete Carlo Ana l
s section we 
ns and transf e
samples and t
ample genera t
four sets of M
imes are per fo
trained only o
ne time per M
 solutions of t
tions are perf o
ults are given 
in the numbe r 
than 5000, t h
eneration and 
Fig. 12 Tota llysis 
simulate the 
er times and 
train DNN be f
tion, the traini n
Monte Carlo s i
formed. For e a
once, using 2 0
MC simulation 
the J2-perturb
ormed on the 
in Fig. 12. I
r of Lambert s
he proposed m
the training o f
l CPU time o f
repeated use 
computing m
fore using the 
ng of the DN N
imulations wi t
ach set, the n u
00000 sample s
to generate t h
ed Lambert p r
personal co mp
It can be see n
olutions to b e
method outper f
f the DNN. 
f different m e
of the DNN-
multiple J2-pe r
proposed me t
N and the SC M
th 1000, 5000 ,
umber revolu t
s and the par a
he first guess
roblem using 
mputer with In t
n t h a t  t h e  e f fi
e compute d. In
forms the oth e
ethods for th e
-based metho d
rturbed Lam b
thod, the total
M. To compa r
, 10000, and 1
tions are equ a
ameters settin g
. The trainin g
the proposed 
tel Core-i7 4. 2
ficiency of th e
n particular, w
er two metho d
e Jupiter J2- p
d b y  t a k i n g  a  
bert solutions .
l computation a
re the total C P
100000 sets o f
ally distribute d
g presented i n
g of DNN wa
method, HL a
2 GHz CPU a
e p r o p o s e d  m
when the num b
ds even when 
 
perturbed La
random set o
. Since it is 
al time shoul d
PU time of the 
f boundary co n
d between 0 a
n previous se c
s implemente
and SN run in 
and 128GB o f
method i m p r o v
ber of simulat i
including th e
mbert probl e
29of boundary 
essential to 
d include the 
above three 
nditions and 
and 10. The 
ction, and is 
d in Python 
Matlab. All 
f RAM. The 
ves with the 
ions is equal 
e cost of the 
em   In ad
have bee n
converge 
Fig. 11. F
0.024 se c
due to its 
Fig. 13 A
 
A fas t
solve the 
neural ne t
the nove l
J2-pertur b
the Kepl e
applied t oddition, two s t
n teste d with t
successfully a
For the zero r e
conds and 0.0 2
longer time o
Avera ge CP U
t and novel m
J2-perturbed 
tworks, whic h
l method is t o
bed Lambert p
erian solutio n
o t h e  J u p i t e r  tress cases, w
the proposed m
and their ave r
evolution cas e
29 seconds, r e
of flight for ea c
U computatio n
method using D
Lambert pro b
h has an excel l
o u s e  a  D N N
proble m. We 
n and provide 
J2-pertur bed 
here the angl e
method. For e a
rage CPU co m
e, the CPU co m
espectively. T h
ch revolution.
nal time of t w
VI.
DNN and th e
blem. DNN co
lent performa n
N to generate 
demonstrated 
good initial 
Lambert pro b
e between the 
ach revolutio n
mputational ti m
mputation ti m
he case of 36 0
 
wo stress cas e
Conclus i
e finite-differ e
mposed of se v
nce on appro x
a first guess 
that the DN N
values for t h
bl e m ,  t h e  e r r o
initial and te r
n, 100 MC tes t
me is given in 
me of the 180 d
0 deg costs a 
es for the Jup i
ion  
ence-based sh o
veral layers is 
ximating nonl i
o f  t h e  c o r r e c
N is capable o
he subsequent 
ors in the ini t
rminal positio
ts are perfor m
Fig. 13, whic h
degree and th e
bit more tim e
 
iter J2-pertu r
ooting algorit h
the extensio n
inear system. T
ction of the i n
f correcting t h
differential c
tial velocity a
ns is 180 deg 
med for each c a
h is similar to 
e 360 degree s
e than the cas e
rbed Lambe r
hm has been 
n of conventio n
The major co n
nitial velocit y
he initial velo c
correction me
and terminal p
30or 360 deg, 
ase. All tests 
the trend in 
scenarios are 
e of 180 deg 
rt problem 
proposed to 
nal artificial 
ntribution of 
y t o  s o l v e  a  
city error of 
thod. When 
position are 31 
 limited to 5m/s and 100 km, respectively. In addition, when com pared to a direct application of a shooting method 
using Newton’s iterations and to a homotopy perturbed Lambert a lgorithm, the proposed method displayed a 
computational time that appears to increase linearly with a slo pe of 0.047 with the number of revolutions. In the 
application scenario presented in this paper the computational time is less than 0.5 seconds even for ten revolutions. 
It was also shown that compared to a direct application of a sh ooting method it provides convergence to the required 
accuracy in all the cases analyzed in this paper. Thus, we can conclude that the proposed DNN-based generation of a 
first guess is a promising method to increase robustness and re duce computational cost of shooting methods for the 
solution of the J2-pertubed Lambert problem. 
The method proposed in this paper can be used to solve the J2-p erturbed Lambert problem for other celestial 
bodies, by training the corresponding DNN with the correspondin g J2 a n d   parameters. Thus a library of 
pre-trained DNN could be easily used to have a more general app lication to missions around any celestial body. On 
the other hand, adding these dynamical parameters as part of th e training set would allow a single more general 
DNN to be used with all celestial bodies. This latter option is  the object of the current investigation. 
Acknowledgments 
The work described in this paper was supported by the National Natural Science Foundation of China (Grant No. 
11672126), sponsored by Qing Lan Project, Science and Technolog y on Space Intelligent Control Laboratory (Grant 
No. 6142208200203 and HTKJ2020KL502019), the Funding for Outsta nding Doctoral Dissertation in NUAA 
(Grant No. BCXJ19-12), State Scholarship from China Scholarship  Council (Grant No. 201906830066). The authors 
fully appreciate their financial supports. 
 32 
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