Dynamic analysis of a rigid-flexible inflatable space structure coupled with control moment gyroscopes

The vibration generated by the inflatable structure after deployment has a great impact on the performance of the payloads. In this paper, the influence of the control moment gyroscopes (CMGs) on the dynamic responses and characteristics of an inflatable space structure is studied, based on the flexible multibody dynamics in a combination of the absolute nodal coordinate formulation (ANCF) and the natural coordinate formulation (NCF). Firstly, the ANCF and NCF are used to accurately describe the large deformations and large overall motions of flexible inflatable tubes and rigid satellites, respectively. Then, instead of modeling gyroscopic flexible bodies, this paper pioneers a rigid body dynamic model of the CMG in detail by using the NCF modeling scheme, which can be attached to and coupled with any flexible bodies without any assumptions. Then, the orbital dynamic equations of the inflatable space structure coupled with distributed CMGs are obtained by considering the effects of Coriolis force, centrifugal force, and gravity gradient through coordinate transformation. The dynamic characteristics of the inflatable space structure are also analyzed by deriving the eigenvalue problem of a flexible multibody system. Finally, the accuracy of the CMG dynamic model is verified via a classic heavy top example. Several numerical examples are presented to study the influence of the magnitudes and directions of the rotor angular momentum of the CMGs on the dynamic responses and characteristics of the inflatable space structure.

equipment, the inflatable space structure has to suppress the structural deformation and vibration effectively. Therefore, when using inflatable space structures to build very large aperture observation systems, it is necessary to focus on the in-orbit dynamic responses of inflatable space structures and study the corresponding vibration suppression methods.
Due to the rapid development of inflatable technology, many researchers have carried out studies on the theoretical methods of inflatable structures [5]. For example, Elsabbagh [6] established a nonlinear finite element model of an axisymmetric inflatable structure by using beam elements and studied the relationship between the wrinkling load and the inflation pressure. Gimadiev [7] used the finite difference method to solve the inflatable spatial problem of a shell with reinforcement rings. Wei et al. [8] presented a deployable sail with four triangular membranes and established a finite element model, which was verified to be useful for membrane structure analysis through experimental tests. Glaser et al. [9] proposed a scheme for controlling volume and arbitrary Lagrange-Eulerian inflation method by the means of a real-time quasi-static inflation experiment combining photogrammetry and finite element analysis. Zhao et al. [10] proposed a new computational method for the finite element model of inflatable membrane structures based on geometrical shape measurement and verified the finite element model through experiments. Li et al. [11] used the extended position-based dynamics method and control volume method to establish the simulation system of the inflatable folded membrane structure and simulated the deployment process. Han et al. [12] proposed a new space inflatable deployment system and used the spacecraft's flexible compartment to carry out an experiment and simulate the whole expansion process. Nobuhisa et al. [13] investigated the deployment behavior of inflatable booms from the experiment and used commercial software to simulate the folding and expansion process. Li et al. [14] proposed a novel static joint model to optimize the design and modeling of a joint for inflatable robotic arms for long-range inspection. San et al. [15] developed a numerical model to study the shape's error of initially curved inflatable antennas, to investigate errors and analyzed errors in the elastic modulus of the membrane, pressure variations, and boundary deviations. Ledkov et al. [16] simulated a tethered satellite system motion with an inflatable spherical balloon during a spacecraft orbit injection by using Lagrange equations of the second kind. Renata et al. [17] analyzed the nonlinear static and dynamic breathing motions of a hyperelastic spherical membrane subjected to internal pressure. Marco et al. [18] integrated the nonlinear finite element analysis of Nastran-SOL400 and the multibody function of Adams in a unique solver and processed an inflatable manned space module through this solver. It can be concluded that the inflatable structures are mostly analyzed by traditional nonlinear finite element methods. However, the inflatable space structures may undergo both large overall motions and large deformations when spinning in space. An accurate rigid-flexible multibody system dynamic model is therefore established for the inflatable space structure by using the absolute nodal coordinate formulation (ANCF) and the natural coordinate formulation (NCF).
Actuators have many applications in the aerospace engineering for vibration suppression and attitude control. For example, Hasan et al. [19] designed an efficient and cost-effective finite-time fault-tolerant attitude controller by considering the vibrations of flexible appendages. Wu et al. [20] used a fixed-time extended state observer to simultaneously compensate for the negative effects of unknown time-varying actuator failures and external disturbances. Few studies, however, focus on the dynamics of inflatable space structures coupled with actuators. In this paper, the coupled dynamics of an inflatable space structure with distributed control moment gyroscopes (CMGs) is studied and analyzed. As important actuators for large space structures, CMGs have the advantages of large output torque and low energy loss compared with other actuators. In recent years, with the miniaturization of CMGs, they have been successfully applied to large space structures and miniature agile satellites for vibration suppression and attitude control [21]. By rotating the rotor at a high speed through the motor to obtain a certain angular momentum and changing the angular momentum of the rotor through the control component, a CMG can output control torques in the space structures, to suppress the vibrations and maneuver the attitude [22].
Due to the increasingly high requirements of maneuverability in space structures, CMGs are required to realize its vibration suppression and attitude control. For example, Zhang et al. [23] constructed a whole satellite dynamic model including the CMGs and the vibration isolation platform with the Newton-Euler method and derived the analytical CMGs disturbance model. Luo et al. [24,25] constructed a dynamic model to analyze the dynamic characteristics of a CMG by energy method and performed experiments to validate the availability of the dynamic model. By using Kane's equation, Hu et al. [26] derived the dynamic equation of a truss and a plate equipped with CMGs and studied the modal characteristics of a beam structure and a plate structure. Xu et al. [27] used the Lagrange equation to develop the dynamic model of the CMG-isolator coupling system with the exciting force of the dynamic unbalance torque, the CMG's gyro torque, and the gimbal motor torque. Xiu et al. [28] studied static and modal characteristics of a framework structure of a CMG system under a large load by ANSYS finite element method software. Frederick [29] derived the equations of motion from the first principles for a very general case of a spacecraft with n-CMG and described each contribution of the dynamic equation and its effect on CMG performance. Papakonstantinou et al. [30] dealt with the problem of singularity avoidance for a 4-CMG pyramid cluster by using the attitude control of a satellite using machine learning techniques. Indeitsev et al. [31] presented the results of modeling the nonlinear dynamics of vibrating wheel gyroscopes. Wang et al. [32] employed the differential transformation method to analyze and control the dynamic behavior of a gyroscope system. Jan et al. [33] studied the dynamics behavior of the micromechanical gyroscope designed for measuring one component of the angular speed. Amer et al. [34][35][36] studied the good effects of different forces and moments on the body's motion by methods such as the Krylov-Bogoliubov-Mitropolski asymptotic method. When CMGs act on a flexible structure, most of the existing studies are to establish a model of the gyroscopic flexible body. However, for different flexible structures, different models need to be established, which is undoubtedly inconvenient. Therefore, this paper proposes a rigid body model of a CMG based on the NCF, which can be directly used for arbitrary flexible structures and greatly simplifies the problem.
The rest of the paper is organized as follows. Section 2 establishes a dynamic model of the inflatable space structure via the ANCF and the NCF and proposes a rigid body dynamic model of a CMG based on the NCF. Section 3 derives the orbital dynamic equations of the gyroscope-rigid-flexible coupling dynamic model of the inflatable space structure through coordinate transformation and presents the eigenvalue problem of the flexible multibody system. Section 4 gives several numerical examples to validate the dynamic model of the CMG and study the dynamic responses and characteristics of the inflatable space structure coupled with distributed CMGs. Section 5 concludes the study.
2 Dynamic modeling

Physical description
As shown in Fig. 1, the rigid-flexible inflatable space structure of concern consists of one main satellite, three sub-satellites, three flexible inflatable tubes, and distributed control moment gyroscopes. The main satellite has a shape of hexagonal prism with an edge length of l 1 and a height of h. The sub-satellite has a cubic shape with an edge length of l 2 . The masses of the main satellite and sub-satellite are m 1 and m 2 , respectively. The flexible inflatable tube has a length of l 3 and crosssection radius of R. Young's modulus, Poisson's ratio, and mass density of the flexible inflatable tube are E, v, and q, respectively. The masses of the satellites are uniformly distributed, and three identical CMGs are mounted on the centroid of the three sub-satellites, i.e., points A, B, and C in Fig. 1, respectively.

Dynamic model of a rigid-flexible inflatable space structure
In the rigid-flexible inflatable space structure, the satellites are regarded as rigid bodies which can be described via the NCF. As shown in Fig. 2, the vector of generalized coordinates of a rigid body can be expressed as [37] where r i and r j are the position vectors of points i and j, respectively. u and v are unit vectors which are perpendicular to each other and to vector (r j -r i ). Thereby, the global position vector of an arbitrary point P in the rigid body is where I is a third-order identity matrix and c 1 , c 2 , and c 3 can be determined by Herein, r is the position vector of point P described in the body-fixed coordinate frame i À ngf of the rigid body, as shown in Fig. 2.
The mass matrix of the rigid body described via NCF is [38] where m is the mass of the rigid body, that is, m 1 for the main satellite and m 2 for the sub-satellite. L is the distance between points i and j. x G , y G , and z G are the local coordinates of the center of mass of the rigid body. I x , I y , and I z are related to the moments of inertia of the rigid body, and I xy , I yz , and I xz are the corresponding products of inertia. Due to the flexibility of the inflatable tubes, the tubes may undergo large deformations during the spinning of the inflatable space structure. Hence, the ANCF is employed to establish the dynamic model of the flexible inflatable tubes. As shown in Fig. 3, the vector of generalized coordinates of the gradientdeficient shell element of ANCF reads with Fig. 1 A schematic diagram of the rigid-flexible inflatable space structure where r A is the global position vector of the node A and r A;x , r A;y are the corresponding slope vectors at node A. The global position vector of an arbitrary point P in the shell element can be given by where S denotes the shape function matrix of the shell element [39]. The mass matrix and elastic force vector of the shell element can be expressed as where U e and U j are the strain energy arising from inplane deformation and out-of-plane deformation of the shell element, respectively [40]. Considering the gas in the inflatable tubes, the generalized forces of the gas pressure and corresponding Jacobian matrix can be derived as where P is the gas pressure in the inflatable tubes and n is the unit normal vector of the inflatable tubes.

Dynamic model of a control moment gyroscope
Previously, most of the processing of CMG is to establish the dynamic model of distributed gyro flexible body structure, which assumes that there is a continuous distribution of angular momentum in the element of the flexible body. The problem is that different dynamic equations need to be derived when the CMG acts on different structures. This paper proposes a new method for establishing a rigid body dynamic model of CMG. In this method, the mounting base of the CMG is regarded as a rigid body, and the gyro force generated by rotor rotation is acted on the base. Based on this, the rigid body dynamic model of CMG can be applied to any rigid-flexible coupling space structure. As shown in Fig. 4, the CMG can be simplified as an angular momentum exchange device, which consists of a mounting base, a gimbal, and a rotor, all located at the centroid C of the CMG. In the base-fixed frame C À x b y b z b and the gimbal-fixed frame C À x g y g z g of the CMG, the x b axis coincides with the x g axis. The x b and y g axes are the rotation axes of the gimbal and the rotor, respectively.
According to Konish theorem and neglecting the inertia tensor of the gimbal, one can obtain the kinetic energy of the CMG as follows where m is the mass of the CMG and _ r C is the speed of centroid C in O-XYZ. J is the inertia matrix of the rotor in C À x g y g z g , which can be expressed as x g is the angular speed of the rotor in C À x g y g z g , which can be given by where A g is the coordinate transformation matrix from C À x b y b z b to C À x g y g z g , x b is the angular velocity of the mounting base in C À x b y b z b . _ a and _ b are the angular speeds of the gimbal and the rotor, respectively, x b and y g are unit vectors in corresponding directions. Since the mounting base is viewed as a rigid body, the angular velocity x b in O-XYZ can be obtained by solving the following linear equations wherex b is the skew symmetric matrix of vector x b . Then, the angular velocity x b can be expressed as where L is the length between point i and point j and e C is the vector of generalized coordinates of the CMG, in the same form as the generalized coordinates of the rigid body. Hence, the entries of x b read where G i (i = 1, 2, 3) is a constant matrix and can be expressed as where I is a third-order identity matrix and 0 is a thirdorder zero matrix.
According to the Lagrange equation, the dynamic equation of a CMG described via NCF is d dt where M C and F C are the mass matrix and vector of gyroscopic forces of the CMG, respectively. a ij and _ a ij are the i-th row and j-th column element in A g and _ A g , respectively. x i and y i are the i-th element in x b and y g , respectively.
3 Dynamic analysis

Orbital dynamics
The dynamic equation of the inflatable space structure in the inertial frame O-XYZ can be obtained as where M is a constant matrix which can be assembled from M R , M S , and M C . q a is the vector of generalized coordinates assembled from e R , e S , and e C . U is the kinematic constraints, k is the Lagrange multiplier, and U T ;q a k represents the vector of constraint forces with U ;q a ¼ oU=oq a . F is the vector of elastic forces and the gas pressure forces. F C and F g are the vector of gyroscopic forces of the CMG and the universal gravitation, respectively.
It is assumed that the inflatable structure runs in a geostationary earth orbit whose orbital plane coincides with the Earth's equatorial plane. The inertial coordinate frame O-XYZ and the orbital coordinate frame As shown in Fig. 5, the relationship between the absolute position vector r a of the an arbitrary point P in the inertial coordinate frame and the relative position vector r r in the orbital coordinate frame can be expressed as where R 0 is the absolute position vector of O o described in O-XYZ and A 0 is the rotation matrix between the frames of O-XYZ and O o -X o Y o Z o , respectively, expressed as where x e ¼ 0 0 x e ½ T is the angular velocity of the orbital revolution of the structure with x e ¼ ffiffiffiffiffiffiffiffiffiffi l=R 3 0 p . l and R 0 are the geocentric gravitational constant and the radius of geostationary orbit, respectively. h ¼ x e t is the deflection angle of the orbital frame relative to the inertial frame.
Similarly, the relationship between the absolute and relative slope vectors of point P is By combining Eqs. (20) and (22), one has where ;q a k.Ĉ _ q r andKq r þF L represents the vectors of Coriolis and centrifugal forces, respectively. The detailed derivations of matricesM,Ĉ,K andF L can be referred to Ref. [41]. Due to the extremely small angular velocity x e of the orbital coordinate frame compared with the angular velocity Fig. 5 The inflatable structure in inertial coordinate frame and orbital coordinate frame x g of the rotor of the CMG, the influence of the orbital dynamics on the gyroscopic forces F C is neglected.
According to the principle of virtual work, the generalized gravitation of a rigid body and a shell element of ANCF described in O o -X o Y o Z o can be, respectively, given bŷ where q Rr and q Sr are the generalized coordinates of the rigid body and the shell element of ANCF in O o -X o Y o Z o andF g can be assembled fromF Rg andF Sg . According to Eq. (24), it is obvious that the inflatable space structure is a high-dimensional and strong nonlinear dynamic system, for which analytical solutions cannot be directly obtained via conventional analytical methods. Therefore, numerical computational methods are required for nonlinear vibration analysis [42]. In this study, the generalized a method [43] is used to solve the dynamic Eq. (24) of the system.

Modal analysis
with / ¼ x r t being the deflection angle of the main satellite-fixed frame relative to the orbital frame.
To obtain the equivalent static equilibrium configuration, let € q b and _ q b be zeros, By solving Eq. (28) with the Newton-Raphson iteration method, the generalized coordinate q 0 of the equivalent static equilibrium position can be obtained. Assuming a small perturbation in the equilibrium position, the linearized equation of Eq. (26) can be expressed as [44] Md€ yþCd _ yþKdy ¼0 ð29Þ Substituting the general solution dyðtÞ ¼e rt V in Eq. (29) will lead to the eigenvalue problem as Since the inflatable space structure has rigid body modes, in order to ensure the invertibility of the Fig. 6 The orbital coordinate frame and main satellite-fixed coordinate frame stiffness matrix, the equation can be solved by using the frequency shift technique [45].

Heavy top
This classic example is used to validate the dynamic model of the CMG. As shown in Fig. 7, a heavy top is fixed to the ground by a spherical joint [46]. The mass and inertial matrix of the heavy top are set as m = 15 kg, J = diag[0.234, 0.469, 0.234] kgÁm 2 . The initial position of the centroid in the inertial frame is [0, 1, 0] T m. The gravitational acceleration is 9.81 m/ s 2 . The gimbal and rotor of the CMG are introduced into the heavy top model as gyroscopic factors. The structural parameters of the rotor are the same as those of the top. The spin angular speed of the gimbal with respect to the base is set as 0 rad/s, and the spin angular speed of the rotor with respect to the gimbal is set as 150 rad/s. Figure 8 presents the comparison of Z-displacement of centroid between the CMG model described by NCF and the heavy top model [46]. It can be easily seen that the curve of the CMG model described by NCF coincides with the curve of the heavy top model perfectly. From Fig. 9, it can be obviously seen that within 2 s, since the CMG is only affected by gravity, the sum of the gravitational potential energy and kinetic energy of the CMG remains unchanged. Figure 10 shows the motion trajectory of the centroid of the CMG within 10 s. It can be found that the trajectory of the centroid of the model conforms to the motion law of the rotating rigid body under the action of gyroscopic moment.    Poisson's ratio v, and the mass density of tube are set as 3.14 9 10 9 Pa, 0.34, and 1250 kg/m 3 , respectively. The pressure in the inflatable tube is 1 psi. And the spin angular speed of the inflatable space structure is p=8 rad/s. The inertia of the rotor is set as J ¼ diag[0:1; 0:04; 0:04 kg m 2 . In Fig. 11, X b1 , Y b1, and Z b1 represent directions of the out-of-plane bending deformation, in-plane bending deformation, and in-plane stretching deformation of the sub-satellite 1, respectively. The inflatable space structure operates in the geosynchronous orbit. The gravitational constant is G = 6.67 9 10 -11 NÁm 2 /kg 2 , the mass of the earth is M = 5.965 9 10 24 kg, and the rotation period T of Earth is 23 h, 56 min, and 4 s. Hence, the radius of the orbit can be obtained by In this subsection, several numerical cases are presented to study the influence of the magnitudes and directions of the angular momentum provided by CMGs on the dynamic response of sub-satellites. Table 1 gives ten numerical cases for comparison. The calculation results are presented as follows. Figure 12 presents the deformations and phase trajectories of the sub-satellite 1 in three directions for case 0. In the X b direction, it is clear that when the CMG does not work, the out-of-plane bending vibration of the sub-satellite 1 is quasi-periodic. In the Y b and Z b directions, the in-plane bending and stretching vibrations of the sub-satellite 1 all remain quasiperiodic. It can also be seen that in the Y b and Z b directions, the magnitudes of the in-plane vibrations are significantly greater than that of the out-of-plane vibration. Figure 13 presents the deformations and phase trajectories of the sub-satellite 1 in three directions for cases 1-3. It can be observed from the comparison between Figs. 13 and 12 that the amplitude of out-ofplane bending vibration decreases slightly with an increase of the rotor angular momentum of the CMGs in the X b direction. Besides, the out-of-plane bending vibrations of the sub-satellite 1 are also quasi-periodic but more complicated. The vibrations of the subsatellite 1 in the Y b and Z b directions remain quasiperiodic and are consistent with those for case 0, which indicates that the rotor angular momentum of the CMGs in the X b direction has no effects on the vibrations of the sub-satellites in Y b and Z b directions. Figure 14 presents the deformations and phase trajectories of the sub-satellite 1 in three directions for cases 4-6. From the comparison between Figs. 14 and 12, it can be found that the amplitudes of the out-ofplane bending, in-plane bending, and in-plane stretching vibrations all increase significantly. When the rotor angular momentum points to the Y b direction, the out-of-plane bending vibrations of sub-satellite 1 in the X b direction remains quasi-periodic, and the   Fig. 12 Dynamic responses of sub-satellite 1 for case 0 amplitude of the vibration decreases with an increase in the rotor angular momentum of the CMGs. The inplane bending and in-plane stretching vibrations of the sub-satellite 1 are still quasi-periodic when the rotor angular momentum points to the Y b direction. It can also be seen that the vibrations of the sub-satellite 1 in Y b direction and Z b direction are almost the same for cases 4-6. Figure 15 presents the deformations and phase trajectories of the sub-satellite 1 in three directions for cases [7][8][9]. From the comparison between Figs. 15 and 12, it can be found that the amplitudes of out-of-plane bending, in-plane bending, and in-plane stretching vibrations all increase significantly. When the rotor angular momentum of the CMGs points to the Z b direction, the amplitude of the out-of-plane vibration of the sub-satellite 1 decreases with an increase in the rotor angular momentum of the CMGs, while the amplitudes of the in-plane bending and stretching vibrations are essentially the same for the three cases. Similar to the results for cases 4-6, the vibrations of the sub-satellite 1 in all the three directions for cases 7-9 are all quasi-periodic.

Modal analysis of the inflatable space structure with distributed CMGs
In subsection 4.2, the dynamic responses of the inflatable space structure in the orbital frame are calculated and analyzed. This subsection will analyze the dynamic characteristics of the inflatable space structure, that is, the influence of the rotor angular Fig. 16 The first six frequencies of the inflatable space structure for cases 0, 1, 2, and 3 momentum of the CMGs and the rotating angular speed x r of the system on the eigenmodes of the inflatable space structure. The parameters of the inflatable space structure and the CMGs are the same as those in subsection 4.2. Figure 16 presents the first six bending modal frequencies for cases 0, 1, 2, and 3. It can be observed from Fig. 16a that when the CMGs are out of service, the curves for the in-plane and out-of-plane bending modes of the same order are almost coincide with other. Besides, all of the first six bending frequencies increase with an increase in the rotating angular speed of the inflatable space structure. By comparing Fig. 16a and b, it can be found that when the rotor angular momentum of the CMGs points to X b , the frequencies of the in-plane bending modes are almost unchanged, while the frequencies of the first-order and third-order out-of-plane bending modes increase and decrease, respectively, and the second-order out-ofplane bending mode disappears. From Fig. 16b, c, and d, it can be observed that the magnitudes of the rotor angular momentum of the CMGs in the X b direction have little influence on the bending modal frequencies.
The corresponding mode shapes of the first six bending modes are presented in Fig. 17. Figure 18 presents the first six bending modal frequencies for cases 0, 4, 5, and 6. By comparing Fig. 18a and b, it can be found that when the rotor angular momentum of the CMGs points to Y b , the frequencies of the out-of-plane bending modes are almost unchanged, while the frequencies of the firstorder and third-order in-plane bending modes increase and decrease, respectively, and the second-order inplane bending mode disappears. From Fig. 18b-d, it Fig. 17 The first six bending mode shapes of the inflatable space structure can be observed that the magnitudes of the rotor angular momentum of the CMGs in the Y b direction have little influence on the bending modal frequencies. By comparing Fig. 16b and Fig. 18b, it can be found that when the rotor angular momentum of the CMGs points to different directions, the frequencies of the inplane and out-of-plane bending modes are totally switched. Figure 19 presents the first six bending modal frequencies for cases 0, 7, 8, and 9. By comparing Fig. 19a and b, it can be found that when the rotor angular momentum of the CMGs points to Z b , the frequencies of both first-order in-plane and out-ofplane bending modes increase, and the frequencies of both third-order in-plane and out-of-plane bending modes decrease. Similarly, both second-order in-plane and out-of-plane bending mode disappear for this case. From Fig. 19b-d, it can be found that the magnitudes of the rotor angular momentum of the CMGs in the Z b direction have little effects on the bending modal frequencies.
In order to investigate how the frequencies of the bending modes are switched or disappeared when the rotor angular momentum of the CMGs points to different directions, the first six frequencies of the inflatable space structure with an increase in the rotor angular momentum of the CMGs are presented in Fig. 20. The rotating speed of the inflatable space structure is chosen as 1 rad/s. From Fig. 20a-c, it can be found that when the rotor angular momentum is 0 at different directions, the first six frequencies of the inflatable space structure are all the same for the initial Fig. 18 The first six frequencies of the inflatable space structure for cases 0, 4, 5, and 6 case. However, with the increase of the rotor angular momentum in X b direction, as shown in Fig. 20a, the frequencies of the first-order and third-order out-ofplane bending modes gradually increase and decrease, respectively, while the frequency of the second-order out-of-plane bending mode decreases to 0 rapidly and disappears. When the rotor angular momentum points to Y b , as shown in Fig. 20b, the changes of the frequencies of the bending modes are right opposite to those in Fig. 20a when the rotor angular momentum points to X b . In Fig. 20c, with the increase in the rotor angular momentum in Z b direction, the frequencies of the bending modes of the same order are coincide with each other all the time. Similarly, the frequencies of the first-order and third-order bending modes slightly increase and decrease, respectively, while the frequencies of the second-order bending mode decrease to 0 rapidly. The zero frequency of a vibration mode occurs probably due to the instability problem of the structure, which will be studied in detail in future work.
The effects of the CMGs on the frequencies and mode switching of the inflatable space structure are quite complex. To summarize, the in-plane directional CMGs, that is, the rotor angular momentum in X b direction, influence the out-of-plane bending modes significantly, while the out-of-plane directional CMGs affect the in-plane bending modes of the inflatable space structure critically. The axial directional CMGs in Z b direction, however, have important and equal effects on both in-plane and out-of-plane bending modes, in which the in-plane and out-of- Fig. 19 The first six frequencies of the inflatable space structure for cases 0, 7, 8, and 9 plane bending vibrations of the inflatable space structure are coupled in a complex manner.

Conclusions
The paper presents an accurate dynamic model for an inflatable space structure coupled with distributed control moment gyroscopes (CMGs) by jointly using the absolute nodal coordinate formulation (ANCF), the natural coordinate formulation (NCF), and orbital dynamics. The paper also proposes a rigid body dynamic model of the control moment gyroscope (CMG) and verifies its accuracy via a classic heavy top example, which shows that the CMG model can be directly attached to any flexible bodies for dynamic analysis without modeling the gyroscopic flexible body. Based on the established dynamic model, the paper studies the influence of the magnitudes and directions of the rotor angular momentum of the CMGs on the dynamic responses and characteristics of the inflatable space structure through ten cases. The results show that the X b -directional rotor angular momentum of the CMGs can inhibit the out-of-plane bending vibrations of the inflatable structure to a certain extent. When the rotor angular momentum of the CMGs points to Y b or Z b directions, the vibrations of the inflatable space structure become more intense. This phenomenon indicates that the in-plane and outof-plane vibrations of the inflatable structure are coupled with each other due to the CMGs, which makes the further vibration control more complicated. In terms of dynamic characteristics, the distributed CMGs with different directional rotor angular momentums have significant and complicated effects on the frequencies of the inflatable space structure. The frequency veering and crossing facilitate the optimal design of the inflatable space structure so as to control the frequencies of the structure to avoid resonances and internal resonances.