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Fast and accurate analyses of spacecraft dynamics using implicit time integration techniques

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  • Control Theory and Applications
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Abstract

This paper deals with the implementation techniques of an implicit integrator to achieve fast and accurate analyses of spacecraft dynamics. For this purpose, the pseudospectral method is adopted to directly integrate the second-order system of equations for both the spacecraft dynamics and corresponding state transition matrix. Various implementation techniques are proposed to enhance the numerical efficiency and integration accuracy, which include a moving horizon approach, the decoupled integration of the second-order dynamics for the state transition matrix, and the grid adaptation method. The numerical features of the proposed techniques are investigated through their applications to a spacecraft’s motion around highly eccentric elliptic orbits, and the resultant numerical errors and computing times are compared with those from the Runge-Kutta method to show the relative efficiency and accuracy of the presented methods. In addition, an optimal two-impulse orbit transfer from the Earth to the Moon is analyzed by implementing the proposed methods using a multiple-shooting framework. The results show that the proposed techniques are extremely effective for dynamical problems requiring intensive and accurate time integrations, and can provide much better accuracy and efficiency than the explicit Runge-Kutta integrator.

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Abbreviations

A :

Jacobian matrix of the force vector

a m,n :

coefficient of the generalized spline interpolation coefficient at the mth time interval

d EM :

distance from the Earth to the Moon

D :

element of the differentiation matrix or parameter to define the orbit direction(= ±1)

\(\bar D\) :

differentiation matrix

e, e :

error or eccentricity, error vector

f, g :

Lagrange coefficients

\(\tilde f\) :

force function vector for the nonlinear algebraic equations

F :

function vector defining the nonlinear algebraic equations

h :

time step size

h :

function vector for nonlinear algebraic equations

I :

element of the integration matrix

\(\bar I\) :

integration matrix

J :

identity matrix

N :

number of nodes for the pseudospectral integrator

N MS :

number of multiple shooting nodes

N node :

number of nodes per time segement

N seg :

number of time segements

N x :

number of the states, x

p :

order of system dynamics

Q :

Jacobian matrix

r :

position vector

R SOI :

sphere of influence of the Moon

Δ v :

delta velocity vector due to the impulsive thrust

w :

quadrature weight

x, y, z :

position of the satellite in the synodic coordinate system

x :

system state vector

δx :

variation in the system state vector

\(\tilde x\) :

unknown variable vector in the nonlinear algebraic equations

y :

system state vector

α:

coefficient of the spline interpolation function or angular position of departure point

β:

factor for the number of statesx or angular position of arrival point

δ:

maximum grid adaptation ratio

ε:

prescribed error tolerance

\(\tilde \phi\) :

generalized spline interpolation function

ϕ :

sub-matrix in the state transition matrix

σ:

mass ratio parameter in the Earth-Moon system

μ :

mass parameter for the Earth and the Moon

τ:

nondimensional time for the pseudospectral method

τ m :

nondimensional time at the mth time interval

Φ:

state transition matrix

Ω:

angular velocity of the Earth-Moon system

0:

initial condition and initial time

C :

computed solution

E :

exact solution, the Earth’s parameter, or relative error

f :

final states or final time

N :

nominal trajectory

M :

the Moon’s parameters

RK :

Runge-Kutta integrator

PS :

pseudospectral integrator

r :

radial position

x :

gradient with respect to the state variable

\(\dot x\) :

gradient with respect to the state derivative

v :

speed

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Authors and Affiliations

  1. Department of Aerospace Information Systems Engineering, Konkuk University, 1 Hwanyang-dong, Gwanjin-gu, Seoul, 135-703, Korea

    Chang-Joo Kim, Do Hyeon Lee, Sung Wook Hur & Sangkyung Sung

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  1. Chang-Joo Kim
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  3. Sung Wook Hur
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Corresponding author

Correspondence to Sangkyung Sung.

Additional information

Recommended by Associate Editor Wen-Hua Chen under the direction of Editor Fuchun Sun. This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIP) (No. 2014M1A3A3A03034589).

Chang-Joo Kim is a Professor of the Department of Aerospace Engineering at Konkuk University, Korea. He received the Ph.D. degree in Aeronautical Engineering from Seoul National University in 1991. His research interests include nonlinear optimal control, helicopter flight mechanics, and helicopter system design.

Dohyeon Lee received his B.S. and M.S. degrees in Aerospace Engineering from University of Ulsan, Ulsan, Korea, in 2011 and 2013, respectively. Presently, he is a Ph.D. student with Department of Aerospace Information Engineering, Konkuk University. His research interests include aircraft flight dynamics, optimal control, and trajectory generation.

Sung Wook Hur received his B.S. and M.S. degrees in Aerospace Information Engineering from Konkuk University, Seoul, Korea, in 2013 and 2015, respectively. Currently, he is a Ph.D. student with Department of Aerospace Information Engineering, Konkuk University. His research interests include nonlinear optimal control, spacecraft trajectory generation, and flight dynamics.

Sangkyung Sung received his B.S. and Ph.D. degrees in Electrical Engineering from Seoul National University, Seoul, Korea, in 1996 and 2003, respectively. Currently, he is an Associate Professor of the Department of Aerospace Information Engineering, Konkuk University. His research interests include inertial sensors, integrated navigation, and application to mechatronics and unmanned systems.

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Kim, CJ., Lee, D.H., Hur, S.W. et al. Fast and accurate analyses of spacecraft dynamics using implicit time integration techniques. Int. J. Control Autom. Syst. 14, 524–539 (2016). https://doi.org/10.1007/s12555-014-0486-5

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  • DOI: https://doi.org/10.1007/s12555-014-0486-5

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