Dynamics Analysis of Space Netted Pocket System Capturing Non-Cooperative Target

Abstract

With the increasing amount of space debris, it is necessary to develop space debris active cleaning technology. The space netted pocket system is a flexible mechanism system which can be used to capture a non-cooperative target flexibly due to the advantages of having a stable structure, strong adaptability and a large capture range. In this paper, the dynamics of the space netted pocket system capturing a non-cooperative target are investigated. The dynamics model of the space netted pocket system is established based on the absolute nodal coordinate formulation (ANCF). The mathematic model of the closing rope is modeled using an ANCF flexible cable dynamics model. Then, the contact collision force model is presented to describe the collision characteristics during the space netted pocket system capturing the target. Finally, the dynamic characteristics of the space netted pocket system capturing the non-cooperative target are analyzed and discussed for different capturing strategies. The simulation results indicate that the target is captured successfully and the target contacts and collides with the rope net in the capturing process, which is subjected to contact and collision forces.

1. Introduction

The Earth’s orbital environment is a limited resource [1]. With the increase in human space activities, the number of satellites has risen sharply. Space debris resulting from spacecraft disintegration or failure has become an increasing threat to space activities [2,3]. To deal with the threat of space debris to space activities, many debris-capturing methods have been proposed. Generally, contact capturing methods include net capturing [4], harpoon capturing [5], gripper capturing [6] and link space manipulators [7]. The net capturing methods are lightweight, low cost, and more flexible for capturing missions than other capture methods [8]. Furthermore, because of the flexible capturing range and the larger permission for the shape and size of space targets, net capturing technologies are a more promising solution for space debris and have been extensively studied.

The space net system can throw a flexible net to wrap around space debris and then drag it to the expected orbit. It usually deploys to a predetermined shape and captures the space debris. Accordingly, the whole capturing task includes three phases, namely the launch phase, deployment phase and capture phase of the space net. Several dynamics and simulations studies have been carried out on these phases [9,10]. However, the space fly net cannot maintain its configuration for a long time in the deployment phase [11,12], which reduces the success rate of capture [13]. There is a proposed solution to maintain the configuration of the net by maneuvering satellites at the corner of the space net using the maneuverable tethered space net robot (TSNR) [14,15]. However, the drawbacks of the maneuverable TSNR are the high production cost and difficulty of control. Moreover, the space inflatable net (SIN) can also help to maintain the configuration of the net [16]. The SIN consists of inflatable rods and rope nets, which are supported by inflatable rods to unfold and form a large capture net. The stiffness of the inflatable rods can maintain the net configuration to improves the capture success rate. The SIN inherits the advantages of the space fly net, and it also has a more stable configuration and controllable attitude [17].

Due to the high nonlinear dynamics of the space inflatable net capture system, it is important and difficult to derive a precise dynamic model. Generally, the methods successfully applied to flexible rope dynamic modeling are mainly divided into two kinds: mass-spring models [18,19] and absolute nodal coordinate formulation (ANCF) models [20,21]. Zhang et al. [22] proposed the centralized mass method to simplify the modeling of complex flexible net structures, and the dynamics mechanism of the space net was studied. In [23], the mass-spring model was used to establish a dynamic model of a square fly net, including the bending stiffness of the rope. Shan et al. [24,25] performed a parabolic flight experiment to validate the simulation results of the mass-spring model and the ANCF model. After that, to analyze the influence of the flexibility modeling on the net simulation, the dynamics models of a space net based on the ANCF and the mass-spring models was presented, respectively. Zhang et al. [26] studied a capture device which can throw a sticky rope which winds around and captures the target based on the ANCF model. Wang et al. [27] presented a dynamics simulation method of capturing a space debris cloud, in which the gradient-deficient beam element of the ANCF is employed to discretize threads which are woven into the net. The normal contact force between the net and the debris cloud and among debris particles is computed by using the penalty method. Liang et al. [28] designed a new net projectile mechanical structure to capture the target and simulated the projection process to make references on the projectile strategy of the space net. Meanwhile, the deployment dynamics for the inflatable rods mainly include a nonlinear finite element model [29], a finite volume inflation model [30] and an energy model [31]. Contact and collision are important features for the space netted pocket system in capturing the target. The nonlinear spring-damper method has been widely used to derive the dynamics for contact and collision force analysis of flexible multibody systems [32,33,34]. Zhao et al. [35] analyzed the deployment dynamics based on the contact behavior, and three preferable releasing conditions were given. Huang et al. [36] utilized a fully implicit method to analyze the nonlinear dynamics of a slender rod net, involving interactions. Thereafter, Si et al. [37] derived the model of thread-ring sliding joints based on the mass-spring-damper method, which can be used for simulating the net closure process. Gołębiowski et al. [38] developed a method based on the Cosserat rod model, which can be used in space nets to simulate full contact dynamics. Botta et al. [39] studied the influence of the stiffness of the rope on the capture motion of the net and presented a contact dynamic model. In [40], the contact model is derived from the penalty-based method and the impulse-based method.

Consisting of inflatable rods and flexible nets, the space netted pocket system is a flexible mechanical system which can be used to capture a non-cooperative target flexibly due to the advantages of having a stable structure, strong adaptability and a large capture range [17,40]. Therefore, the space netted pocket system presents a wide application prospect in space debris active cleaning technology. The objective of this paper to study the dynamics characteristics of the space netted pocket mechanism capturing non-cooperative targets. A mathematic modeling and simulation method is proposed for the space netted pocket mechanism. Different capturing strategies are presented to analyze the capturing process. First, the dynamics model of the space netted pocket system is established based on ANCF. Then, the contact collision force model is presented to describe the contact collision characteristics during the space netted pocket system capturing the target. Finally, the dynamics characteristics of the space netted pocket mechanism capturing the target are analyzed and discussed for different capturing strategies.

This paper is organized as follows. In Section 2, the structure of the space netted pocket system is presented. In Section 3, the dynamics modeling of space netted pocket is established. In Section 4, the dynamic simulation for different cases are presented, and the simulation results are analyzed and discussed. Finally, Section 5 concludes the presented work.

2. Structure of the Space Netted Pocket System

The structure of the netted pocket system is presented in Figure 1. There are eight flexible support inflatable rods for the netted pocket system, which are connected by a rope net [17]. The inflatable rods are evenly distributed in a regular octagon, and the diameter of each rod is 0.1 m. The upper square rope net and the lower triangular rope net maintain their shape by relying on the adjacent inflatable rods. The maximum diameter of the net pocket is 15.12 m, and the maximum depth is 12 m.

The diameter of the opened net is adjusted by controlling the length of the closing rope, which is pulled by the rope retractor installed at the end of each inflatable rod.

3. Dynamics Modeling of Space Netted Pocket

The space netted pocket system is a flexible mechanism system, and contact and collision are important characteristics during the space netted pocket system capturing the target. In this section, the dynamics model of the space netted pocket system is established based on ANCF, and the contact collision characteristics are presented by the nonlinear contact force models.

3.1. Dynamics Model of the Closing Ropes

In this subsection, the dynamics model of the closing ropes is established based on the ANCF flexible cable dynamics model [17]. As shown in Figure 2, the variable length flexible cable element model is presented, where $v_1$ is the mass flow velocity at the element boundary node.

The generalized coordinates of the flexible cable element ${}^j\mathit{q}$ can be obtained based on ANCF and it is expressed as:

$$\begin{array}{cc}\hfill {}^j\mathit{q}& =\left[{}^j{\mathit{q}}_1{}^j{\mathit{q}}_2\right]\hfill \\ & ={\left[\begin{array}{cccc}{}^j{\mathit{r}}^{\mathrm{T}}\left(0\right)& {}^j{\mathit{r}}_l^{\mathrm{T}}\left(0\right)& {}^j{\mathit{r}}^{\mathrm{T}}\left(L\right)& {}^j{\mathit{r}}_l^{\mathrm{T}}\left(L\right)\end{array}\right]}^{\mathrm{T}}\hfill \end{array}$$

(1)

where ${}^j{\mathit{r}}^T\left(0\right)$ and ${}^j{\mathit{r}}^T\left(\mathrm{L}\right)$ are the position vectors of nodes at both ends of the element, and ${}^j{\mathit{r}}^T\left(0\right)$ and ${}^j{\mathit{r}}^T\left(\mathrm{L}\right)$ are the gradient vectors of nodes at both ends of the element.

The position vector ${}^j\mathit{r}\left(l\right)$ of the arbitrary node on the central axis of the element can be written as:

$${}^j\mathit{r}\left(l\right)=S{\left(l\right)}^j\mathit{q}$$

(2)

$$\begin{array}{cc}\hfill S\left(l\right)& =\left[\begin{array}{cccc}{S}_{1}I& S_{2}I& S_{3}I& S_{4}I\end{array}\right]\hfill \\ & \left\{\begin{array}{c}{S}_1=\frac{1}{4}{(\xi -1)}^2\left(\xi +2\right)\\ S_2=\frac{L\left(t\right)}{8}{(\xi -1)}^2\left(\xi +1\right)\\ S_3=\frac{1}{4}{(\xi +1)}^2\left(-\xi +2\right)\\ S_4=\frac{L\left(t\right)}{8}{(\xi +1)}^2\left(\xi -1\right)\end{array}\right.\hfill \end{array}$$

(3)

where $S\left(l\right)$ is the shape function; I is the unite matrix. $\xi$ can be written as following:

$$\xi \left(l,t\right)=\frac{2l\left(t\right)-L\left(t\right)}{L\left(t\right)}$$

(4)

where $L\left(t\right)$ is the length of the rope element and $l\left(t\right)$ is the position of the material mass particle. Both $L\left(t\right)$ and $l\left(t\right)$ are time variables. Therefore, the variable length element shape function $S\left(l,t\right)$ is a function of time, and the velocity and acceleration of any point can be expressed as:

$${}^j\dot{\mathit{r}}\left(l,t\right)=\mathit{S}{(l)}^j\dot{\mathit{q}}\left(t\right)+\dot{\mathit{S}}{(l)}^j\mathit{q}\left(t\right)$$

(5)

$${}^j\ddot{\mathit{r}}\left(l,t\right)=\mathit{S}{(l)}^j\ddot{\mathit{q}}\left(t\right)+2\dot{\mathit{S}}{(l)}^j\dot{\mathit{q}}\left(t\right)+\ddot{\mathit{S}}{(l)}^j\mathit{q}\left(t\right)$$

(6)

$$\dot{\mathit{S}}=\frac{dS}{d\xi }\dot{\xi },\ddot{\mathit{S}}=\frac{d^{2}S}{d{\xi }^2}{\dot{\xi }}^2+\frac{dS}{d\xi }\ddot{\xi }$$

(7)

With the assumption of the mass flow at the boundary, the Lagrange method cannot be applied in the case of varying element length. D’Alembert’s principle [41] states that the sum of virtual work done by inertial force and acting force on arbitrary virtual displacement is 0, by which the dynamics equations Equation (8) is obtained.

$${{\displaystyle \sum }}_i\left({\mathit{F}}_i-m{\ddot{\mathit{r}}}_i\right)\cdot \delta {\mathit{r}}_i=0$$

(8)

where $\delta {\mathit{r}}_i$ is the material mass $i$ in any moment to meet the constraints of the virtual displacement. The force on the rope element consist of the elastic force ${\mathit{F}}_{\mathrm{E}}$ and the external force ${\mathit{F}}_{\mathrm{f}}$, then the dynamics equation of the cable element can be written as:

$$\underset{0\left(t\right)}{\overset{L\left(t\right)}{{\displaystyle \int }}}{\delta }^j{\mathit{r}}^{\mathrm{T}}\left({\mathit{F}}_{\mathrm{f}}+{\mathit{F}}_{\mathrm{E}}-{\rho }^j\ddot{\mathit{r}}\right)\mathrm{d}l=0$$

(9)

Due to the material mass of the cable element being variable, the upper and lower limits of the above integral changes with time. Considering the axial and bending deformation of the cable, neglecting the rotational motion of the cable, the virtual work of each part can be expressed as:

$$\begin{array}{cc}\hfill \underset{0\left(t\right)}{\overset{L\left(t\right)}{{\displaystyle \int }}}{\delta }^j{\mathit{r}}^{\mathrm{T}}\left({\mathit{F}}_{\mathrm{f}}\right)\mathrm{d}l& ={\delta }^j{\mathit{q}}^{\mathrm{T}}\underset{0\left(t\right)}{\overset{L\left(t\right)}{{\displaystyle \int }}}{\mathbf{S}}^{\mathrm{T}}\left(l\right)\mathbf{f}\left(l,t\right)\mathrm{d}l\hfill \\ & ={\delta }^j{\mathit{q}}^{\mathrm{T}}\frac{\mathrm{d}l}{\mathrm{d}\xi }\underset{-1}{\overset{1}{{\displaystyle \int }}}{\mathbf{S}}^{\mathrm{T}}\left(\xi \right)\mathit{f}\left(\xi ,t\right)\mathrm{d}\xi \hfill \end{array}$$

(10)

$$\begin{array}{l}\underset{0\left(t\right)}{\overset{L\left(t\right)}{{\displaystyle \int }}}{\delta }^j{\mathit{r}}^{\mathrm{T}}\left({\mathit{F}}_{\mathrm{E}}\right)\mathrm{d}l=-{\delta }^j{\mathit{q}}^{\mathrm{T}}\frac{\mathrm{d}l}{\mathrm{d}\xi }\underset{-1}{\overset{1}{{\displaystyle \int }}}\left({\left(\frac{\partial {\epsilon }_0}{{\partial }^j\mathit{q}}\right)}^{\mathrm{T}}\cdot \right.\\ \left.EA\left({}^j{\epsilon }_0+\stackrel{\cdot }{c^j{\epsilon }_0}\right)+{\left(\frac{{\partial }^j\kappa }{{\partial }^j\mathit{q}}\right)}^{\mathrm{T}}\cdot EJ\left({}^j\kappa +\stackrel{\cdot }{c^j\kappa }\right)\right)\mathrm{d}\xi \end{array}$$

(11)

$$\underset{0\left(t\right)}{\overset{L\left(t\right)}{{\displaystyle \int }}}{\left({\delta }^j\mathit{r}\right)}^{\mathrm{T}}\left(-{\rho }^{j}A\ddot{\mathit{r}}\right)\mathrm{d}x=-{\delta }^j{\mathit{q}}^{\mathrm{T}}\frac{\mathrm{d}l}{\mathrm{d}\xi }\underset{-1}{\overset{1}{{\displaystyle \int }}}\rho A{\mathbf{S}}^{\mathrm{T}}{}^j\ddot{\mathit{r}}\mathrm{d}\xi$$

(12)

where ${}^j{\epsilon }_0^j\kappa$ is the axial strain and curvature, $E$ is elasticity modulus, $A$ is the cross-sectional area and $J$ is the moment of inertia of the flexible cable element. $c$ is the damping coefficient and $\frac{\mathrm{d}l}{\mathrm{d}\xi }=\frac{L\left(t\right)}{2}$.

Combining Equations (10)–(12), the dynamics equation of the variable flexible cable element can be obtained as:

$${}^j\mathbf{M}{}^j\ddot{\mathit{q}}+{}^j\mathbf{M}{}_v^j\dot{\mathit{q}}+{}^j\mathbf{M}{}_q^j\mathit{q}+{}^j\mathbf{Q}=0$$

(13)

where

$$\begin{array}{cc}\hfill {}^{j}M& =\frac{L(t)}{2}{\displaystyle \underset{-1}{\overset{1}{\int }}\rho S^{\mathrm{T}}S\mathrm{d}{\xi }^j{M}_v}=\frac{L(t)}{2}{\displaystyle \underset{-1}{\overset{1}{\int }}\rho S^{\mathrm{T}}\dot{S}\mathrm{d}{\xi }^j{M}_q}\hfill \\ & =\frac{l(t)}{2}{\displaystyle \underset{-1}{\overset{1}{\int }}\rho S^{\mathrm{T}}\ddot{\mathbf{S}}\mathrm{d}\xi }\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill {}^j\mathbf{Q}& ={}^j\mathbf{Q}{}_e+{}^j{\mathbf{Q}}_f^j\mathbf{Q}{}_e\hfill \\ & =\frac{L(t)}{2}{\displaystyle \underset{-1}{\overset{1}{\int }}\left({\left(\frac{{\partial }^j{\epsilon }_0}{{\partial }^j\mathit{q}}\right)}^{\mathrm{T}}EA\left({}^j{\epsilon }_0+c^j{\dot{\epsilon }}_0\right)\right.\left.+{\left(\frac{{\partial }^j\kappa }{\partial \mathit{q}}\right)}^{\mathrm{T}}EJ\left({}^j\kappa +c^j\dot{\kappa }\right)\right)}\mathrm{d}\xi \hfill \end{array}$$

(15)

$${}^j{\mathbf{Q}}_f=-\frac{L(t)}{2}{\displaystyle \underset{-1}{\overset{1}{\int }}{\mathbf{S}}^{\mathrm{T}}}f\mathrm{d}\xi$$

(16)

Consequently, the equation for a variable length cable can be obtained by combining the dynamic equations of all the elements, which is written as:

$$\mathbf{M}\ddot{\mathit{q}}+\mathbf{Q}=0$$

(17)

where $\mathit{q}$ represents the generalized coordinates of the system, consisting of the generalized coordinates of each node. $\mathbf{M}$ and $\mathbf{Q}$ are the relative generalized mass matrix and generalized forces of the cable, respectively.

3.2. Dynamics Model of the Rope Netted System Based on ANCF

According to [20], the ANCF equations for the dynamics of the space rope netted system can be expressed as

$$\left\{\begin{array}{l}\mathbf{M}\ddot{\mathit{q}}+{\mathit{C}}_q^{\mathrm{T}}\mathsf{\lambda }={\mathit{Q}}_{\mathit{k}}+{\mathit{Q}}_{\mathit{\epsilon }}\\ \mathit{C}=\mathbf{0}\end{array}\right.$$

(18)

where $\mathbf{M}$ is the mass matrix, $\mathit{q}$ represents the generalized coordinates, ${\mathit{Q}}_{\epsilon }$ is the generalized force vector, $\mathit{C}$ is the constraint equation, ${\mathit{Q}}_k={\left(\frac{\partial U}{\partial \mathit{q}}\right)}^{\mathrm{T}}$ and $U$ is strain energy. The details in Equation (17) can also be obtained by taking $v_1=0$ from the variable length flexible cable model.

3.3. Contact Collision Force Model

Contact and collision are important features for the space netted pocket system grabbing the target. In this work, the contact and collision characteristics are described by the contact force models, which include the normal collision force model and the tangential friction force model.

Therefore, the contact and collision force at the collision point consists of the normal contact collision force ${\mathit{F}}_n$ and the tangential friction force ${\mathit{F}}_t$. Then, the contact and collision forces can be expressed as:

$$\mathit{F}={\mathit{F}}_n+{\mathit{F}}_t$$

(19)

For the normal collision force, a nonlinear spring damping model including the elastic force and damping force is used, which is expressed as [42]:

$${\mathit{F}}_n=K_n{\delta }_n^{{}^{1.5}}\mathit{n}-D_n{\mathit{v}}_n$$

(20)

where ${\delta }_n$ is the penetration depth, $\mathit{n}$ is the normal unit vector and ${\mathit{v}}_n$ is the normal relative velocity. The expressions of $k_n$ and $g_n$ are expressed as follows:

$$\begin{array}{l}{K}_n=\frac{4}{3}{E}_{\mathrm{eff}}\sqrt{r_{\mathrm{eff}}}\\ D_n=\alpha \sqrt{K_n{m}_{\mathrm{eff}}}{\delta }^{\frac{1}{4}}\end{array}$$

(21)

where $r_{\mathrm{eff}}$ is the radius of curvature of the collision point, and it is expressed as $r_{\mathrm{eff}}={(r_1^{-1}+r_2^{-1})}^{-1}$. $r_1$ and $r_2$ are the radii of the two collision bodies. $\frac{1}{E_{\mathrm{eff}}}=\frac{1-{\nu }_1^2}{E_1}+\frac{1-{\nu }_2^2}{E_2}$, in which ${\nu }_1$ and ${\nu }_2$ are Poisson’ s ration of the collision bodies. $E_1$ and $E_2$ are the elasticity modulus of the collision. $\alpha$ and $m_{\mathrm{eff}}$ are expressed as the following:

$$\begin{array}{cc}\hfill \alpha =& \sqrt{\frac{5{\ln }^2{c}_r}{{\ln }^2{c}_r+{\pi }^2}}\hfill \\ \hfill m_{\mathrm{eff}}=& \frac{m_1{m}_2}{m_1+m_2}\hfill \end{array}$$

(22)

where $c_r$ is the Newton’s coefficient of restitution, and $m_1$ and $m_2$ are the mass of the collision bodies.

The tangential friction is modeled using the Coulomb friction law to describe the contact and collision in tangential direction, which is presented as following:

$${\mathit{F}}_t=\mu (\left|v_t\right|){\mathit{F}}_n$$

(23)

where $\mu$ is the coefficient of friction, which is determined by the tangential relative velocity $v_t$.

The cable element contacts and collides with the target in the process of capturing and towing the target. The contact force can be transformed to a generalized nodal force based on the principle of virtual work.

4. Dynamic Simulation and Results

In this section, the dynamics simulation of the netted pocket system capturing the target is implemented. Three different capturing strategies with different closing times of the netted pocket system are investigated. The dynamics results for different cases are presented and discussed.

4.1. Parameters of the Service Spacecraft and Captured Target

The parameters of the service spacecraft and captured target used in the simulation are presented in

Table 1. Parameters of the spacecraft and target.

	Mass (kg)	Moment of Inertia (kg m2)
		Ixx	Iyy	Izz
Spacecraft	1210	4282	12,736	14,498
Target	200	1210	1210	1210

and the parameters of the netted pocket system are presented in

Table 2. Parameters of the netted pocket system.

	Diameter (m)	Density (kg/m3)	Poisson Ratio	Modulus of Elasticity
Nets	0.004	1430	0.3	12
Inflatable rods	0.01	164	0.3	0.75

. The process of the service spacecraft capturing the target is designed as follows: at the beginning, the closing distance L between the ends of adjacent inflation rods is 5.78 m in the initial state; then, the netted pocked system is closed up and the final distance L between the ends of adjacent inflation rods is shortened to 2.1 m. After that, when the target is captured, the service spacecraft tows the target and the service spacecraft and the target motion together.

4.2. Simulation Results and Discussion

In this subsection, the dynamics simulation results for three different capturing strategies with different closing times of the netted pocket system are investigated. Here, the closing time is 10 s, 7.5 s and 5 s for each case, respectively. Then, after capturing the target, the service spacecraft starts to tow the target at t = 10 s. In the simulation, the initial positions of the spacecraft and the target are (0, −1 m, 0) and (0, 7.5 m, 0), respectively.

For the first case, the closing time is 10 s. At the initial state (t = 0 s), the distance L between the ends of adjacent inflation rods is 5.78 m. Then, L shortens from 5.78 m to 2.1 m. At 10 s, L decreases to the minimum value of 2.1 m, and the capturing phase ends. After the capturing phase, the service spacecraft tows the target. The capturing rule of the distance L is shown in Figure 3, in which the distance L and velocity of adjacent inflation are presented. The dynamics simulation of capturing process is performed and the simulation results are presented in Figure 4.

The simulation results for the first case are presented in Figure 5 and Figure 6. Figure 5 presents the collision forces on the captured target. Figure 6 presents the positions of the target and the service spacecraft, respectively. Figure 5 shows that the collision force in the Y direction is larger than the collision forces in X and Z directions. The maximum collision force occurs before the end of the closing phase. After the closing phase is completed, the service spacecraft tows the target. In the towing phase, the collision force on the target is decreased obviously. Both the collision forces in the X and Z directions appear after the end of the closing phase.

Comparing with the collision forces in Figure 5 and the positions of the target and spacecraft in Figure 6, it can be found that the target collides with the rope net in the closing phase, and is subjected to a negative collision force in the Y direction, resulting in a decrease in the distance between the target and the service spacecraft. Subsequently, the target collides with the rope net near the service spacecraft, resulting a positive collision force in the Y direction, which leads to the distance between the target and the spacecraft increasing. The target contacts and collides with the rope net at the closing point again, resulting a negative collision force in the Y direction, and the closing capture process is completed. After the closing process is completed, the service spacecraft tows the target. From Figure 6, it clearly shows the service spacecraft and target motion together in the Y direction in the towing phase, indicating that the target is captured successfully. In the towing phase, the target and the spacecraft have continuous contact and there will be smaller collision forces generated in the Y direction. Moreover, the positions of the center of mass of the target and the service spacecraft still stay before the contact and collision between the target and the rope net in the capturing phase. After that, due to the contact and collision, as shown in Figure 5, the positions of the center of mass of the target and the service spacecraft present slight fluctuations in the Y direction before the end of the closing, as shown in Figure 6b,d. Furthermore, in the towing phase, the positions of the center of mass in the X and Z directions are not changed obviously, which is represented by the small magnitude for the spacecraft and target. The order of magnitude for the spacecraft is 10⁻⁵ m and the order of magnitude for the target is 10⁻³ m. This is due to the contact and collision between the spacecraft and the captured target, as shown in Figure 5.

Further, the dynamics simulation for the other two cases with different closing times of the netted pocket system are presented, respectively. For the second case, the closing time is 7.5 s. At the initial state (t = 0 s), the distance L between the ends of adjacent inflation rods is 5.78 m. Then L shortens from 5.78 m to 2.1 m. At 7.5 s, distance L decreases to the minimum value of 2.1 m, and the capturing phase ends. After the capturing phase, the service spacecraft starts to tow the target at t = 10 s. For the third case, the closing time is 5 s. At the initial state (t = 0 s), the distance L between the ends of adjacent inflation rods is 5.78 m. Then L shortens from 5.78 m to 2.1 m. At 5.0 s, distance L decreases to the minimum value of 2.1 m, and the closing phase ends. After the capturing phase, the service spacecraft starts to tows the target at t = 10 s.

The dynamics simulation is performed and the capturing processes of both cases are presented in Figure 7 and Figure 8, respectively.

The simulation results for cases 2 and 3 are presented in Figure 9, Figure 10 and Figure 11. Figure 9 presents the contact collision forces on the captured target. Figure 10 and Figure 11 present the positions of the target and the service spacecraft, respectively.

Comparing the three cases for different closing times of the netted pocket system, Figure 5 and Figure 9 show that the collision force in the Y direction is larger than the collision forces in the X and Z directions. When the closing time is decreased, the collision forces in the Y direction are larger and the first collision is earlier. However, both the collision forces in the X and Z directions appear after the closing, and their amplitude is not significantly affected by the closing time. Figure 6, Figure 10 and Figure 11 show similar phenomena, which indicates that the target is captured successfully for different capture strategies. After the closing process is completed, the service spacecraft tows the target, which clearly shows the service spacecraft and target motion together in the Y direction in the towing phase. The positions of the center of mass of the target and the service spacecraft still stay before the contact and collide between the target and the rope net of the capturing phase of each case. Then, due to the contact and collision, as shown in Figure 9, the positions of the center of mass of the target and the service spacecraft present slight fluctuations in the Y direction just before the end of the closing for each case, as shown in Figure 10b,d as well as Figure 11b,d. However, in the towing phase, there will be smaller collision forces generated, but the positions of the center of mass in the X and Z directions are not changed obviously, which is represented by the small magnitude for the spacecraft and target. This is also due to the contact and collision between the spacecraft and the captured target, as shown in Figure 9.

5. Conclusions

In this paper, the dynamics modeling and simulation of the space netted pocket mechanism capturing the target are presented. The dynamics model of the space netted pocket system is established based on the absolute nodal coordinate formulation (ANCF). The mathematic model of the closing rope is modeled based on the ANCF flexible cable dynamics model. Furthermore, the contact collision force model is presented by using the nonlinear spring-damping model to describe the contact collision characteristics during the space netted pocket system capturing the target. Three different capturing strategies with different closing times of the netted pocket system are implemented and investigated. The simulation results indicate that:

(1)

The target contacts and collides with the rope net in the capturing process, which is subjected to contact and collision forces. The collision force in the Y direction is larger than the collision forces in the X and Z directions when capturing the target. The maximum collision force occurs before the end of the closing phase. Furthermore, there is continuous contact and collision between the service spacecraft and target in the towing phase. However, in the towing phase, the collision force on the target is decreased obviously.

(2)

When the closing time is decreased, the collision forces in the Y direction are larger and the first collision appears earlier. However, the collision forces in the X and Z directions appear after the closing phase, and their amplitude is not significantly affected by the closing time.

(3)

Different cases show similar phenomena, which indicates that the target is captured successfully for different capture strategies. The positions of the center of mass of the target and the service spacecraft still stay before the contact and collision between target and the rope net of the capturing phase of each case. After the closing phase is completed, the service spacecraft tows the target and the service spacecraft and target move together in the Y direction. Furthermore, in the towing phase, the positions of the center of mass in the X and Z directions are not changed obviously, which is represented by the small magnitude for the spacecraft and target. This is due to the slight contact and collision between the spacecraft and the captured target.

Dynamics modeling and simulation for the space netted pocket mechanism capturing the target play a crucial role in predicting the characteristics and success rate of the capturing task. This work proposed a computational method for the space netted pocket mechanism capturing the target, which is the basis of the design and performance evaluation of the space netted pocket mechanism in real engineering applications. In this work, we studied different capturing strategies with different closing times of the netted pocket system. More capturing cases related to real engineering requirements will be researched in the future, such as the different inertia of targets, different initial state of the target and so on, which will improve the engineering application.