2.1 Equivalent model of the inflatable rod

The analysis of conventional inflatable structures is mostly based on the finite element model, but this cannot be easily and efficiently employed for a rigid-flexible-control coupling model of the spacecraft with a large flexible-tether-net system. Therefore, this paper conducts an equivalent modeling analysis based on the ideal-pressure charging theory for inflatable structures.

The bending failure process of the inflatable rod is divided into two parts: the linear load-bearing phase and the buckling failure phase. In the linear load-bearing stage, the deformation of the inflatable beam is small; no wrinkles are formed on the pipe wall; and the inflatable beam exhibits overall buckling. In the buckling failure stage, the inflated beam undergoes large deflection deformation, and the inflatable beam will produce wrinkles in some parts; that is, local buckling and the wrinkle area will no longer participate in the bearing. When the pleated area extends over the entire circumference of the inflatable beam section, the inflatable beam loses its load-carrying capacity (Comer and Levy, 1963).

In this paper, the case is where the inflatable rod is completely inflated and the deformation is small, so the inflatable beam is in the linear load-bearing stage, and the bending stiffness is approximately constant under the allowable air pressure. At this time, the inflatable beam can be equivalent to the Euler–Bernoulli beam.

The bending stiffness of the equivalent beam could be modeled and depends on the section and material properties of the inflated beam. Considering that the bending stiffness of the inflatable beam E_iI_i and the equivalent beam E_eI_e are equal, the elastic modulus E_e of the equivalent beam is

where D and d represent the outer diameter and inner diameter of the inflatable beam, respectively, and E_i is the elastic modulus of the inflatable-beam material.

It can be found that the factors affecting the bending behavior of the inflatable beam in the linear load-bearing stage are mainly the material itself and the shape of the section, and the overall buckling is independent of the inflation pressure.

2.2 Dynamic modeling of the tether net using the ANCF

In this section, the dynamic model of the tether has been established based on the ANCF (Gerstmayr and Shabana, 2006), which can describe the large flexibility and large deformation of the space tether net. The configuration and node numbering of the tether net are shown in Fig. 3.

Figure 3Configuration of SITNCS.

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Figure 4Nodes and element numbering of SITNCS.

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The numbering rule of one side surface is shown in Fig. 4 and Table 1; the nodes are ${}^k{N}_i^j(k=\mathrm{1},\mathrm{\dots },\mathrm{4}$, $i=\mathrm{1},\mathrm{\dots },p$, $j=\mathrm{1},\mathrm{\dots },(p-(i-\mathrm{1}\left)\right)\cdot \mathrm{2}+\mathrm{1})$, where k is the surface index, i is the row index of kth surface, j is the row index of kth surface and p is the row number (e.g., p=10 in Fig. 4).

The rods consists of the supporting rods and warping rods, where the supporting rods are ${}^{k}b{\mathrm{1}}^m(m=\mathrm{1},\mathrm{\dots },p-\mathrm{1})$ and the warping rods are ^kb2ⁿ(m=1) and ^kb3ⁿ(m=5), $n=\mathrm{1},\mathrm{\dots },(p-(m-\mathrm{1}\left)\right)\cdot \mathrm{2}$. The horizontal tethers are ${}^{k}c{\mathrm{1}}_i^n(i\ne \mathrm{1},i\ne \mathrm{5})$, where the vertical tethers are ${}^{k}c{\mathrm{2}}_i^j(i\ne \mathrm{1})$.

Table 1Connection between nodes and elements.

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A flexible cable element considering the axial and bending deformation is obtained on the basis of the theoretical hypothesis of a Euler–Bernoulli beam. Figure 5 shows the undeformed and deformed configurations of the three-dimensional cable element using two nodes.

Figure 5Absolute nodal coordinate formulation model of the cable element.

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Set the length of the element as L and the generalized coordinates of the cable element to be

where ^jr and ^jr_x represent the position vector and gradient vector at the end point, respectively.

The position vector of the cable element at a point on the axis of the cable element can be expressed in generalized coordinates as

where S(x) is the shape function of the three-dimensional flexible ANCF cable element.

The kinetic energy of the flexible element can be written as follows:

where ρ and A are the density and cross-sectional area of the cable element, respectively, and ${}^j\mathit{M}={\int }_{\mathrm{0}}^L\mathit{\rho }\left(A{\mathbf{S}}^T\mathbf{S}\right)\mathrm{d}x$ is the constant mass matrix of the ANCF cable element.

The elastic energy of the flexible cable element is

where E is the modulus of elasticity, J_κ is the moment of inertia of the flexible cable section, ${}^j{\mathit{\epsilon }}_{\mathrm{0}}=\sqrt{{}^j\mathit{r}{{}_x^T}^j{\mathit{r}}_x}-\mathrm{1}$ is the axial strain and ${}^j\mathit{\kappa }=\left|{}^j{\mathit{r}}_x{\times }^j{\mathit{r}}_{xx}\right|/{\left|{}^j{\mathit{r}}_x\right|}^{\mathrm{3}}$ is the curvature.

The total kinetic energy and strain energy of the system can be written as follows:

Since the dynamics of the element are described by generalized coordinates varying with time, the dynamic equations of the rigid body and flexible body are as follows:

where Q_e is the generalized force vector, λ is the Lagrange multiplier, C is the constraint equation, and q and λ are both unknown quantities.

It is derived from the expressions of kinetic energy and strain energy.

The dynamic equation of the flexible cable system can be written as follows:

3.1 Spacecraft dynamics

The space-inflated unwinding tether net capture system consists of large-scale flexible inflatable beams and a tether net. The relationship between beam nodes makes the dynamic model very complex and computationally inefficient, which also cannot be directly applied to the design of the control system. The attitude dynamics model considering the rigid-flexible coupling used for controller design (Di Gennaro, 2003) would be expressed as

where J is the moment of the inertia matrix, u is the control torque, d is the external disturbance, δ is the coupling matrix between the spacecraft and the flexible mechanism, η is the flexible modal coordinates, C is the damping matrix, and K is the stiffness matrix. While J₀ is the nominal inertia and ΔJ is the unknown inertia, the equation could be derived as

This yields

where ${\mathit{d}}^{\prime }={\mathit{J}}_{\mathrm{0}}^{-\mathrm{1}}\left(\mathit{d}-\stackrel{\mathrm{̃}}{\mathit{\omega }}\mathrm{\Delta }\mathit{J}\mathit{\omega }-\mathrm{\Delta }\mathit{J}\dot{\mathit{\omega }}-\stackrel{\mathrm{̃}}{\mathit{\omega }}{\mathit{\delta }}^T\dot{\mathit{\eta }}-\mathit{\delta }\ddot{\mathit{\eta }}\right)$.

The attitude dynamic equation of satellite is given by

where θ=[γ  ψ  ϕ]^T is the attitude angle of the satellite, γ is the roll angle, ψ is the pitch angle and ϕ is the yaw angle. R is given by

so the satellite dynamic equation is

3.2 Design of the proposed ADRC control

From the system dynamics in Eq. (15), it is shown that the system is a typical cascade system. Considering that the angular velocity ω and attitude angle θ of spacecraft are measurable, a double-loop controller based on active disturbance rejection control for the spacecraft attitude is proposed.

In Fig. 6 the transient procedure is first arranged by the tracking differentiator and then the external-loop feedback controller output virtual angular speed ω^∗. Internal-loop feedback is designed by the ADRC. System uncertainties and disturbances are estimated by the extended state observer and compensated for during each sampling period, with the result of achieving a good tracking effect for the angular velocity ω^∗. The controller consists of an arrangement of the transient procedure, angle feedback law of the outer loop and disturbance compensation.

3.2.2 Angle feedback law of the outer loop

The feedback control law of the outer loop is designed for the angular error of the spacecraft, resulting in the virtual control volume ω^∗ in the inner loop. The outer-loop angle feedback control law is

$$\begin{array}{}\text{(19)}& {\mathit{\omega }}^{*}=R^{-\mathrm{1}}\left(\mathit{\theta }\right)k_{\mathrm{1}}\mathrm{fal}({\mathit{\theta }}_d-\mathit{\theta },\mathit{\alpha },\mathit{\delta }),\end{array}$$

where $k_{\mathrm{1}}=\mathrm{diag}\mathit{\{}{k}_{\mathrm{11}},k_{\mathrm{12}},k_{\mathrm{13}}\mathit{\}}$ is a gain matrix for adjusting the speed of tracking the desired value of attitude angle. $\mathrm{fal}(e,\mathit{\alpha },\mathit{\delta })=\left[\begin{array}{c}\mathrm{fal}(e_{\mathrm{11}},\mathit{\alpha },\mathit{\delta })\\ \mathrm{fal}(e_{\mathrm{12}},\mathit{\alpha },\mathit{\delta })\\ \mathrm{fal}(e_{\mathrm{13}},\mathit{\alpha },\mathit{\delta })\end{array}\right]$ The function fal is a nonlinear function, and its form is as follows:

$$\begin{array}{}\text{(20)}& \mathrm{fal}(e,\mathit{\alpha },\mathit{\delta })=\left\{\begin{array}{ll}e{\mathit{\delta }}^{\mathit{\alpha }-\mathrm{1}},& \left|e\right|\le \mathit{\delta }\\ {\left|e\right|}^{\mathit{\alpha }}\mathrm{sgn}e,& \left|e\right|>\mathit{\delta }\end{array},\right.\end{array}$$

where e is the state error $\mathrm{0}<\mathit{\alpha }<\mathrm{1}$, 0<δ.

Remark 2.

Because R(θ) could be calculated, the virtual control variable ω^∗ could be compensated for by this ascertained function. Then Eq. (13) could transform into $\dot{\mathit{\theta }}=k_{\mathrm{1}}\mathrm{fal}({\mathit{\theta }}_d-\mathit{\theta },\mathit{\alpha },\mathit{\delta })$. The nonlinear feedback control law is adopted so that θ will track θ_d.

Figure 7Attitude-tracking curve.

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Figure 8Attitude error in the steady-state process.

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Figure 9Angular-velocity estimation.

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Figure 10Estimated disturbance.

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3.2.3 Extended state observer design

By using the input and output data of the spacecraft, the ESO could estimate the angular velocity ω and the internal and external disturbances of the system in real time. By expanding the first-order system of Eq. (12) into a two-order system, we can get

$$\begin{array}{}\text{(21)}& \left\{\begin{array}{l}\dot{\mathit{\omega }}=-{{\mathit{J}}_{\mathrm{0}}}^{-\mathrm{1}}\stackrel{\mathrm{̃}}{\mathit{\omega }}{\mathit{J}}_{\mathrm{0}}\mathit{\omega }+{{\mathit{J}}_{\mathrm{0}}}^{-\mathrm{1}}\mathit{u}+{\mathit{d}}^{\prime }\\ {\dot{\mathit{d}}}^{\prime }=\frac{\mathrm{d}{\mathit{d}}^{\prime }}{\mathrm{d}t}\end{array}.\right.\end{array}$$

A discrete two-order expansion observer is designed as follows:

$$\begin{array}{}\text{(22)}& \left\{\begin{array}{l}{\mathit{\xi }}_{\mathrm{1}}=z_{\mathrm{1}}-\mathit{\omega },\mathrm{fe}=\mathrm{fal}({\mathit{\xi }}_{\mathrm{1}},\mathit{\alpha },\mathit{\delta })\\ z_{\mathrm{1}}=z_{\mathrm{1}}+h(z_{\mathrm{2}}-{\mathit{\beta }}_{\mathrm{1}}{\mathit{\xi }}_{\mathrm{1}}-{J_{\mathrm{0}}}^{-\mathrm{1}}\mathrm{\Omega }{J}_{\mathrm{0}}\mathit{\omega }+{J_{\mathrm{0}}}^{-\mathrm{1}}u)\\ z_{\mathrm{2}}=z_{\mathrm{2}}+h(-{\mathit{\beta }}_{\mathrm{2}}\mathrm{fe})\end{array}.\right.\end{array}$$

Remark 3.

The observer states z₁ and z₂ converge to the state variables ω and d^′, respectively. β₁ and β₂ are observer gains. Because of the limited estimation capability of the ESO, the known part $-J_{\mathrm{0}}^{-\mathrm{1}}\mathrm{\Omega }{J}_{\mathrm{0}}\mathit{\omega }$ of the nominal model is used in the design process, which could reduce the burden of the ESO and improve its performance and accuracy.

Remark 4.

The resulting observer estimation error system for errors ${\mathit{\xi }}_{\mathrm{1}}=z_{\mathrm{1}}-\mathit{\omega }$ and ${\mathit{\xi }}_{\mathrm{2}}=z_{\mathrm{2}}-d^{\prime }$ takes the following form:

$$\begin{array}{}\text{(23)}& \left\{\begin{array}{l}{\dot{\mathit{\xi }}}_{\mathrm{1}}={\mathit{\xi }}_{\mathrm{2}}-{\mathit{\beta }}_{\mathrm{1}}{\mathit{\xi }}_{\mathrm{1}}\\ {\dot{\mathit{\xi }}}_{\mathrm{2}}=-{\mathit{\beta }}_{\mathrm{2}}\mathrm{fe}-{\dot{d}}^{\prime }\end{array}.\right.\end{array}$$

If ${\mathit{\beta }}_{\mathrm{2}}\gg {\dot{d}}^{\prime }$, the errors ξ₁ and ξ₂ converge to the zeros. The ESO would achieve good performance, which means z₁→ω and $z_{\mathrm{2}}\to d^{\prime }$.

Figure 11Control torque without a filter.

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Figure 12Control torque with a filter.

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Figure 13Tension of booms.

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Figure 14Torque of booms.

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In practical systems, the signals measured by the sensors are unavoidably noisy, which will bring unpredictable disturbances to the controller. The filters need to be designed to extract or restore the original signals from noisy signals. The filter is established by the tracking differentiator.

$$\begin{array}{}\text{(24)}& \left\{\begin{array}{l}{v}_{\mathrm{1}}=v_{\mathrm{1}}+h{v}_{\mathrm{2}}\\ v_{\mathrm{2}}=v_{\mathrm{2}}+h\mathrm{fhan}(v_{\mathrm{1}}-\mathit{\omega },v_{\mathrm{2}},r,h_{\mathrm{0}})\\ {\mathit{\omega }}_{\mathrm{0}}=v_{\mathrm{1}}+k_{\mathrm{0}}h{v}_{\mathrm{2}}\end{array}\right.\end{array}$$

Remark 5.

The TD2 is used for noise filtering because of its simplicity and ease of use. The original values would not be delayed in predicting the differential signal v₂ and prediction step k₀.

So the ESO when the system output is polluted by the noise is designed as follows:

$$\begin{array}{}\text{(25)}& \left\{\begin{array}{l}\mathrm{fh}=\mathrm{fhan}(v_{\mathrm{1}}-\mathit{\omega },v_{\mathrm{2}},r,h_{\mathrm{0}})\\ v_{\mathrm{1}}=v_{\mathrm{1}}+h{v}_{\mathrm{2}}\\ v_{\mathrm{2}}=v_{\mathrm{2}}+h\mathrm{fh}\\ {\mathit{\omega }}_{\mathrm{0}}=v_{\mathrm{1}}+k_{\mathrm{0}}h{v}_{\mathrm{2}}\\ {\mathit{\xi }}_{\mathrm{1}}=z_{\mathrm{1}}-{\mathit{\omega }}_{\mathrm{0}},\mathrm{fe}=\mathrm{fal}({\mathit{\xi }}_{\mathrm{1}},\mathit{\alpha },\mathit{\delta })\\ z_{\mathrm{1}}=z_{\mathrm{1}}+h(z_{\mathrm{2}}-{\mathit{\beta }}_{\mathrm{1}}{\mathit{\xi }}_{\mathrm{1}}-{J_{\mathrm{0}}}^{-\mathrm{1}}\mathrm{\Omega }{J}_{\mathrm{0}}{\mathit{\omega }}_{\mathrm{0}}+{J_{\mathrm{0}}}^{-\mathrm{1}}u)\\ z_{\mathrm{2}}=z_{\mathrm{2}}+h(-{\mathit{\beta }}_{\mathrm{2}}\mathrm{fe})\end{array}.\right.\end{array}$$

Figure 15Procedure for attitude maneuvering.

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3.2.4 Disturbance rejection and compensation

The ESO can estimate the total disturbance value z₂ in real time, and then the active disturbance rejection function can be achieved by compensation in the control law. The inner-loop angular velocity feedback control law is

$$\begin{array}{}\text{(26)}& \left\{\begin{array}{l}{e}_{\mathrm{1}}={\mathit{\omega }}^{*}-\mathit{\omega }\\ u=J_{\mathrm{0}}\left(k_{\mathrm{2}}\mathrm{fal}\right(e_{\mathrm{1}},\mathit{\alpha },\mathit{\delta })-z_{\mathrm{2}})+\mathrm{\Omega }{J}_{\mathrm{0}}\mathit{\omega }\end{array},\right.\end{array}$$

where $k_{\mathrm{2}}=\mathrm{diag}\mathit{\{}{k}_{\mathrm{21}},k_{\mathrm{22}},k_{\mathrm{23}}\mathit{\}}$ is the gain matrix.

Remark 6.

Bringing Eq. (24) into Eq. (20), we get $\dot{\mathit{\omega }}=k_{\mathrm{2}}\mathrm{fal}(e_{\mathrm{1}},\mathit{\alpha },\mathit{\delta })+d^{\prime }-z_{\mathrm{2}}$; when the estimated error is small enough, $\dot{\mathit{\omega }}=k_{\mathrm{2}}\mathrm{fal}(e_{\mathrm{1}},\mathit{\alpha },\mathit{\delta })$. It can be proved that ω can track ω^∗.

Considering the above design procedure, the ADRC law for the spacecraft is obtained as follows:

$$\begin{array}{}\text{(27)}& \left\{\begin{array}{l}{\mathit{\omega }}^{*}=R^{-\mathrm{1}}\left(\mathit{\theta }\right)k_{\mathrm{1}}\mathrm{fal}({\mathit{\theta }}_d-\mathit{\theta },\mathit{\alpha },\mathit{\delta })\\ \mathrm{fh}=\mathrm{fhan}(v_{\mathrm{1}}-\mathit{\omega },v_{\mathrm{2}},r,h_{\mathrm{0}})\\ v_{\mathrm{1}}=v_{\mathrm{1}}+h{v}_{\mathrm{2}}\\ v_{\mathrm{2}}=v_{\mathrm{2}}+h\mathrm{fh}\\ {\mathit{\omega }}_{\mathrm{0}}=v_{\mathrm{1}}+k_{\mathrm{0}}h{v}_{\mathrm{2}}\\ {\mathit{\xi }}_{\mathrm{1}}=z_{\mathrm{1}}-{\mathit{\omega }}_{\mathrm{0}},\mathrm{fe}=\mathrm{fal}({\mathit{\xi }}_{\mathrm{1}},\mathit{\alpha },\mathit{\delta })\\ z_{\mathrm{1}}=z_{\mathrm{1}}+h(z_{\mathrm{2}}-{\mathit{\beta }}_{\mathrm{1}}{\mathit{\xi }}_{\mathrm{1}}-{J_{\mathrm{0}}}^{-\mathrm{1}}\mathrm{\Omega }{J}_{\mathrm{0}}{\mathit{\omega }}_{\mathrm{0}}+{J_{\mathrm{0}}}^{-\mathrm{1}}u)\\ z_{\mathrm{2}}=z_{\mathrm{2}}+h(-{\mathit{\beta }}_{\mathrm{2}}\mathrm{fe})\\ e_{\mathrm{1}}={\mathit{\omega }}^{*}-{\mathit{\omega }}_{\mathrm{0}}u=J_{\mathrm{0}}\left(k_{\mathrm{2}}\mathrm{fal}\right(e_{\mathrm{1}},\mathit{\alpha },\mathit{\delta })-z_{\mathrm{2}})+\mathrm{\Omega }{J}_{\mathrm{0}}{\mathit{\omega }}_{\mathrm{0}}\end{array}.\right.\end{array}$$

Figure 16Attitude-tracking curve.

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Figure 17Angular-velocity estimation.

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Figure 18Estimated disturbance.

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Figure 19Control torque.

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4.1 Simulation parameters

The service spacecraft is assumed to be a single rigid body which is represented by a cuboid with dimensions of 2.5 m × 2.5 m × 4 m. For the capture mechanism, the length of its short side is l_d=0.4 m; the length of its long side is l_u=4 m; and its height is h=4 m. The constraint between the inflatable rods and tether net is a spherical joint. The constraint between the inflatable booms and the satellite is fixed.

Based on the special requirement of the space environment, the polyimide material is used for the inflatable rods, and the aramid fiber material is chosen for the net. The equivalent parameters of the system with an internal pressure of 25 KPa are listed in Table 2.

Assuming the moment of inertia of the capture mechanism is unknown, the main parameters of the spacecraft are listed in Table 3, which are the moment of inertia of the spacecraft J, environmental disturbance torque d, initial attitude angle θ, initial angular velocity ω, expected attitude angle θ_d and expected angular velocity ω_d.

It is reasonable to assume that the signal of the sensor is disturbed by white noise with a peak value of 0.1 % as its the output. The sampling period is h=1 ms. The parameters of the PID controller are $K_p=\mathrm{diag}\mathit{\{}\mathrm{1152},\mathrm{1024},\mathrm{1260}\mathit{\}}$ and $K_d=\mathrm{diag}\mathit{\{}\mathrm{1440},\mathrm{1280},\mathrm{1600}\mathit{\}}$. The parameters of the ADRC are listed in Table 4.

Figure 20Procedure of attitude maneuvering.

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4.2 Attitude control of the spacecraft based on the ADRC

It is assumed that the moment of inertia of the satellite can be accurately obtained and that of the capture mechanism is unknown. t_td is the transition time arranged by the TD1. t_θ is defined as the transition time when the attitude error is stable and is between $-{\mathrm{10}}^{-\mathrm{4}}$ and 10⁻⁴ rad s⁻¹. The simulation time is 20 s.

Comparing the two control schemes, the simulation results of the attitude tracking and steady-state error of the spacecraft are shown in Figs. 7 and 8. The ADRC has excellent dynamic and steady-state performance with an error of less than 10⁻⁴ rad. The PID controller is greatly affected by the vibration of the flexible tether net; the dynamic tracking has an obvious delay; and the steady-state error is larger; all of this cannot meet the high-precision control requirements. Table 5 shows that the spacecraft can track the desired signal in the transition time arranged by the TD1. The ADRC has high attitude accuracy and robust stability, which are to meet the critical requirements.

The performance of the ESO is shown in Figs. 9 and 10. The ESO can get the estimated value of the attitude angular velocity of the spacecraft and the disturbance caused by the flexible vibration of the capture mechanism. The maximum amplitude of disturbance can reach 6 N m, which would bring non-negligible disturbance to the satellite platform control.

If the TD2 is not used for filtering, we can get the filter performance of the controller. Figure 11 is the original control torque, and Fig. 12 is the control torque after using the filter. It shows that the noise has a great influence on the ADRC. Through filtering, the high-frequency vibration of the torque control is obviously suppressed, which is beneficial to the use of the actuator.

The flexible vibration of the inflatable rods and the tether net is accompanied with the attitude maneuver of the satellite. The largest deformation occurs in the connection point between the inflatable rods and the satellite, resulting in considerable tension and torque. In Figs. 13 and 14, the maximum tension is about 110 N, and the maximum torque is about 450 N m. Therefore, according to the material strength loading limit, it can be asserted whether the stress exceeds the stress limit of the inflatable rod. Otherwise, the applicable way is to increase the transition time to reduce the acceleration.

The whole procedure is shown in Fig. 15. It can be seen that the flexible capture mechanism vibrates in terms of the satellite attitude maneuver, but its amplitude is relatively small, which can maintain the configuration and tend to be stable.

4.3 Robustness of a larger SITNCS

In order to adapt to the larger capture target, we can increase the size of the capture mechanism. For the controller, it means having more parameter uncertainties and disturbances. The assumption is that l_d=0.8 m, l_u=8 m and h=8 m. The simulation time is still 20 s.

Figures 16 and 17 show that the transition time is t_θ=[12.66  12.64  12.62]^T. When the size of the capture mechanism is increased, the control performance of the system is still excellent. In Figs. 18 and 19, the amplitude of the flexible vibration increases, while the vibration frequency decreases, which improves the estimation effect of the ESO. Therefore, the ADRC has good robustness and disturbance resistance. The simulation (Fig. 20) shows the dynamic changes of the system during the control process.