Dynamics and control for capture mode of drag-free satellite considering nonlinear electrostatic effect

doi:10.1007/s10483-025-3297-8

Abstract

Abstract:

A drag-free satellite is an important platform for space-borne gravitational wave (GW) observation. To achieve the high-precision control of a drag-free satellite in practical engineering, an accurate dynamic model is essential. This paper presents a nonlinear model of the electrostatic effect between a satellite and a test mass (TM), and designs a model predictive controller based on the drag-free satellite model with the nonlinear electrostatic effect. To determine the analytical form of the electrostatic effect, a comprehensive theoretical analysis is performed for gravitational reference sensors (GRSs). An electrostatic force and a torque are simulated with the displacement as a varying parameter through a commercial software. Then, the results are fitted to derive the nonlinear expressions of the electrostatic effect. The model predictive controllers based on the models with the nonlinear and linear electrostatic effects are designed in the capture mode. Finally, the control results are given to show the advantages of the nonlinear electrostatic effect.

Key words: drag-free satellite, nonlinear electrostatic effect, capture mode, model predictive control (MPC)

2010 MSC Number:

O313.7

Ti CHEN, Songyuan HE, Yankai WANG, Zhengtao WEI, Yingjie CHEN, J. TAYEBI. Dynamics and control for capture mode of drag-free satellite considering nonlinear electrostatic effect. Applied Mathematics and Mechanics (English Edition), 2025, 46(9): 1631-1648.

TrendMD

Figures/Tables 21

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Table 1

Area and distance parameters of the electrode plates"

Axis	Area of the electrode plate	Distance relative to the TM surface
$x$	$1.45 cm × 3.60 cm = 5.220 cm 2$	4.0 mm
$y$	$0.71 cm × 3.82 cm = 2.712 cm 2$	2.9 mm
$z$	$0.65 cm × 3.70 cm = 2.405 cm 2$	3.5 mm

Table 1

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Table 2

Fitting absolute residuals of cubic polynomial"

Force or torque	$x$	$y$	$z$	$α$	$β$	$γ$
$F x$	$1.462 7 × 10 − 12$	$8.692 0 × 10 − 13$	$7.689 4 × 10 − 13$	$6.332 6 × 10 − 13$	$9.003 4 × 10 − 13$	$7.062 5 × 10 − 13$
$F y$	$1.397 0 × 10 − 12$	$1.053 3 × 10 − 11$	$9.338 8 × 10 − 13$	$1.075 7 × 10 − 12$	$1.065 1 × 10 − 12$	$1.104 5 × 10 − 12$
$F z$	$9.792 7 × 10 − 13$	$1.597 1 × 10 − 12$	$2.131 6 × 10 − 12$	$1.330 7 × 10 − 12$	$9.356 8 × 10 − 13$	$1.089 6 × 10 − 12$
$M x$	$2.447 9 × 10 − 14$	$2.004 7 × 10 − 14$	$1.977 8 × 10 − 14$	$3.495 0 × 10 − 13$	$2.082 1 × 10 − 14$	$2.694 9 × 10 − 14$
$M y$	$2.618 4 × 10 − 14$	$2.338 4 × 10 − 14$	$2.214 0 × 10 − 14$	$2.566 1 × 10 − 14$	$6.357 8 × 10 − 14$	$2.009 0 × 10 − 14$
$M z$	$2.196 1 × 10 − 14$	$1.859 4 × 10 − 14$	$1.793 8 × 10 − 14$	$1.727 7 × 10 − 14$	$2.231 7 × 10 − 14$	$1.198 6 × 10 − 13$

Table 2

Table 3

Fitting absolute residuals of linear polynomial"

Force or torque	$x$	$y$	$z$	$α$	$β$	$γ$
$F x$	$1.264 5 × 10 − 11$	$4.579 7 × 10 − 12$	$1.664 5 × 10 − 12$	$6.482 5 × 10 − 13$	$9.097 2 × 10 − 13$	$7.136 4 × 10 − 13$
$F y$	$2.335 8 × 10 − 12$	$6.268 9 × 10 − 11$	$2.051 6 × 10 − 12$	$1.081 9 × 10 − 12$	$1.062 7 × 10 − 12$	$1.129 5 × 10 − 12$
$F z$	$1.368 1 × 10 − 12$	$3.266 9 × 10 − 12$	$2.032 3 × 10 − 11$	$1.440 6 × 10 − 12$	$9.762 8 × 10 − 13$	$1.135 9 × 10 − 12$
$M x$	$2.437 5 × 10 − 14$	$2.293 9 × 10 − 14$	$3.988 9 × 10 − 14$	$1.730 5 × 10 − 12$	$2.278 1 × 10 − 14$	$2.719 6 × 10 − 14$
$M y$	$4.731 6 × 10 − 14$	$2.699 7 × 10 − 14$	$2.279 9 × 10 − 14$	$2.662 5 × 10 − 14$	$5.124 6 × 10 − 13$	$2.214 8 × 10 − 14$
$M z$	$2.702 1 × 10 − 14$	$5.497 1 × 10 − 14$	$1.852 0 × 10 − 14$	$1.875 7 × 10 − 14$	$2.299 9 × 10 − 14$	$7.432 4 × 10 − 13$

Table 3

Fig. 11

Table 4

Initial state of the TM"

Parameter	Value	Parameter	Value
$r M j$	$(111) T mm$	$r ˙ M j$	$(0.2 0.2 0.2) T mm/s$
$θ M j$	$(0.1 0.1 0.1) T rad$	$θ ˙ M j$	$(0.05 0.05 0.05) T rad/s$

Table 4

Table 5

Parameters of the drag-free satellite"

Parameter	Value
$m S$	1 500 kg
$m M$	1.93 kg
$J S$	$diag (800, 800, 1 000) kg ⋅ m 2$
$J M$	$diag (7.053 × 10 − 4, 7.053 × 10 − 4, 7.053 × 10 − 4) kg ⋅ m 2$

Table 5

Table 6

Parameters of the PD controller"

Parameter	Value	Parameter	Value
$K P1, K P2$	4	$K D3$	20
$K D1, K D2$	8	$K P4, K P5$	5
$K P3$	10	$K D4, K D5$	10

Table 6

Fig. 12

Fig. 13

Fig. 14

Fig. 15

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