Dynamic modeling and control for flexible bodies with mobile control moment gyroscopes

doi:10.1007/s10483-026-3431-9

Abstract

Abstract:

This paper investigates the dynamic modeling and vibration suppression of a flexible cantilever body with mobile control moment gyros (CMGs). A nonlinear dynamic model capturing the rigid-flexible coupling between the structure and the mobile CMG is systematically derived by Kane’s method. Vibration suppression is realized via the gyroscopic coupling caused by the direction change of the CMG’s angular momentum. To address the practical challenge of unmeasurable states, a neural network (NN)-based cascaded observer with a super-twisting differentiator is designed for real-time state estimation. A radial basis function (RBF) NN is used to approximate the nonlinear terms and disturbances in the system. Based on the observed states and the NN, a sliding mode controller based on the observer’s estimations is synthesized to achieve simultaneous trajectory tracking of the mobile CMG and active vibration suppression of the flexible structure. Numerical simulations demonstrate the effectiveness of the proposed method. Comparisons between the fixed CMG and mobile CMG are carried out to illustrate the advantages of the CMG acting as a mobile actuator.

Key words: mobile actuator, gyroscopic coupling, rigid-flexible, vibration suppression, position tracking

2010 MSC Number:

O326

Zhou YANG, Xinyuan LI, Dongping JIN, Ti CHEN. Dynamic modeling and control for flexible bodies with mobile control moment gyroscopes. Applied Mathematics and Mechanics (English Edition), 2026, 47(9): 1943-1968.

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URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3431-9

https://www.amm.shu.edu.cn/EN/Y2026/V47/I9/1943

Figures/Tables 11

d,	input dimensions of the radial basis function (RBF);
D_b,	damping coefficient matrix of the flexible body;
E,	Young’s modulus;
E_a,	modal mass coefficients of the system;
E_b,	modal mass coefficients of the flexible body;
f_d,	unknown external bounded disturbances;
F_C,	Coriolis force vector;
F_in,	internal force vector of the structural vibration;
F_Ai,	generalized active force;
F_Ii,	generalized inertia force;
F_Ei,	generalized elastic force;
G_i,j,	transformation matrix from $⁢ Γ i$ to $⁢ Γ j;$
$J g i a,$	moment of inertia of the ith control moment gyroscope (CMG);
J_gi,	moment of inertia of the ith CMG’s gimbal;
J_ri,	moment of inertia of the ith CMG’s rotor;
k,	mode order;
l,	output dimensions of the RBF;
m_gi,	mass of the gimbal;
m_ri,	mass of the rotor;
$P g i a,$	modal momentum coefficients of the CMGs;
Q_gi,	mass center of the ith CMG;
s,	number of RBF basis functions;
v_gi,	translational velocity of the ith CMG;
α_i,	angular velocity of the gimbal;
β_i,	angular velocity of the rotor;
λ_b,	stiffness coefficient matrix of the flexible body;
υ,	Poisson’s ratio;
ρ_m,gi,	position of the mass microelement of gimbal in $⁢ Γ g ⁢ i;$
ρ_gi,	position of $Q g ⁢ i$ in $⁢ Γ b;$
ρ_m,ri,	position of the rotor’s mass microelement in $⁢ Γ r ⁢ i;$
η_b,	modal coordinate of $B;$
$B,$	flexible body;
Γ_b,	body reference frame of $B;$
Γ_I,	inertial reference frame;
Γ_gi,	gimbal-fixed frame of the ith CMG;
Γ_ri,	rotor-fixed frame of the ith CMG;
θ_m,b,	rotational modal matrices;
$Λ m,b,$	translational modal matrices;
C_gi,1,	coupling coefficients corresponding to $v ˙ g ⁢ i;$
C_gi,2,	coupling coefficients corresponding to $⁢ η ¨ b;$
C_gi,3,	coupling coefficients corresponding to $⁢ η ˙ b .$

Fig. 1

Fig. 2

Table 1

Parameter values for the beam with mobile CMGs"

Parameter	Value	Unit
Density of the flexible body $B$	2 700	kg/m³
Length of the flexible body $B$	10	m
External diameter of the flexible body $B$	0.04	m
Inside diameter of the flexible body $B$	0.038	m
Young’s modulus, E	70	GPa
Poisson’s ratio, ν	0.3	–
Mass of the gimbal, $m g ⁢ i$	0.5	kg
Gimbal’s moment of inertia, $⁢ J g ⁢ i$	$10 − 2 × diag ⁢ (1.3, 2.5, 2.5)$	$kg ⋅ m 2$
Mass of the rotor, $m r ⁢ i$	0.5	kg
Rotor’s moment of inertia, $⁢ J r ⁢ i$	$10 − 2 × diag ⁢ (2.5, 1.3, 2.5)$	$kg ⋅ m 2$
Structural damping ratio	0.001	–
Rotor’s angular velocity, $β i$	100	rad/s
Rotor’s angular momentum, $H i = J r ⁢ i y ⁢ β i$	1.3	$N ⋅ m ⋅ s$
Rated gimbal-torque capacity	4	$N ⋅ m$
Rated translational-force capacity	3	N

Table 1

Table 2

Parameters for the observer and controller"

Parameter	Value
Diagonal gain matrices for the multivariable STD, $⁢ χ 1, ⁢ χ 2$	$20 ⁢ ⁢ I 6, 5 ⁢ ⁢ I 2$
Boundary layer for the element-wise operation in the STD, δ	0.001
Observer feedback gain matrix, $⁢ K v$	$10 ⁢ ⁢ I 4, ⁢ I 2$
Learning-rate matrix for the NN-based observer, $⁢ Γ$	$⁢ I 20$
Diagonal parameter matrix for the sliding surface, $⁢ C s$	$20 ⁢ ⁢ I 4, 0.5 ⁢ ⁢ I 2$
Sliding mode gain matrix, $⁢ K s$	$25 ⁢ ⁢ I 4, 0.5 ⁢ ⁢ I 2$
Robust switching gain, η	2
Boundary layer for the robust switching term, $Δ 1$	0.001
Input dimensions of RBF, d	12
Number of RBF basis functions, s	20
Output dimensions of RBF, l	6
Widths of the ith receptive field, $σ i$	0.5
Center vectors of RBF receptive fields, $⁢ μ i$	$(− 5 + 10 ⁢ (i − 1) / 19) ⁢ ⁢ I 12, 1$

Table 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Table 3

Commanded CMG input extrema"

Configuration	$T g, 1, max cmd / (N ⋅ m)$	$T g, 2, max cmd / (N ⋅ m)$	$F v, 1, max cmd / N$	$F v, 2, max cmd / N$
Mobile CMGs	3.728 7	3.752 6	2.509 1	2.462 3
Fixed CMGs	3.730 4	3.752 6	–	–

Table 3

Table 4

RMS tip displacement and component-wise absolute-power metric over the full simulation"

Configuration	RMS tip displacement/m	$E rotation$ /J	$E translation$ /J	$E total$ /J
Fixed CMGs	$4.3 × 10 − 4$	6.06	0	6.06
Mobile CMGs	$2.9 × 10 − 4$	4.34	11.95	16.29

Table 4

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