Similar to a space flying net, the capture field of the space
netted pocket system is large and it can be applied to capture space
non-cooperative targets flexibly. To maintain the stability of the space
netted pocket system, eight inflatable rods are used as the supporting
structure of the net surface. In this paper, a space netted pocket system is
designed and modeled. Based on ANCF (absolute nodal coordinate formulation),
a dynamic model of the complex space rope net system is established, and
then an accurate model of closing rope considering the variable length is
derived by introducing mass flow element. A double closed-loop sliding
control method is designed to maintain the stable attitude of the service
spacecraft. An extended observer is applied to estimate and compensate for
the disturbances due to the uncertainty of the contact and flexibility in
the system. Finally, the dynamic model and control method is verified
through the simulation of the virtual prototype. Results show that the
service spacecraft can maintain the attitude stability during target
captured process and can track the desired angle during attitude maneuver.
The flexible deformation and collision cause great disturbance to the
service spacecraft, and the extended observer can improve the control
accuracy from 10

In recent years, with the increase of human space activities, threats to space activities from space debris are worsening (Barmin et al., 2014; Chen, 2011). The contact capture method is a simple and feasible way to capture space debris. There are mainly two types of contact capture: rigid capture and flexible capture (Nishida and Kawamoto, 2011; Shan et al., 2016). Like space manipulator and capture claw, the rigid capture methods will inevitably cause collision between the capture target and operation platform. Therefore, the rigid capture methods are more complicated and require high control accuracy. The space net as a typical flexible acquisition method is more lightweight and simple (Zhang et al., 2017). The capture domain of the space net is larger, which can reduce the control accuracy requirements and reduce the cost. Therefore, the space net is an active space debris removal method with broad application prospects.

The space fly net launches a flexible rope net to cover the space debris in a large envelope and drags space debris to the atmosphere burned or grave orbit. In recent years, a lot of theoretical and experimental researches has been carried out around the space fly net. In terms of large-scale experiments, RemoveDEBRIS project has now completed the first space rope net capture test in orbit (Forshaw et al., 2016, 2017) and e.Deorbit project has carried out parabolic flight test (Biesbroek et al., 2017). At the same time, many theories about space net dynamics modeling, space net deployment control, space net capture performance analysis and space net structure optimization design are deeply researched (Bonnal et al., 2013; Ming et al., 2017; Xu et al., 2019). However, it is difficult for the fly net system to maintain the net shape fully deployed for a long time. And the capture success rate will be reduced due to the space fly net hardly maintaining the net shape for a long time. Some scholars have proposed the motorized fly net to control the net shape (Huang et al., 2015; Meng et al., 2017), but multiple motorized devices around the net will greatly increase the complexity and cost of the system. The space rope netted pocket capture mechanism described in this paper can inherit the advantages of lightness and simplicity. Furthermore, the inflatable rods in the system can maintain the net shape for a long time, which can improve the success rate of capture. The rope net surface of the netted pocket capture mechanism in Fig. 1 is supported in an umbrella-like shape by eight flexible rods. First, the service spacecraft approaches the captured target until the target enters the net. Second, the closing devices at the end of inflatable rods are tightened to complete the closing. Finally, with attitude maneuver of the service spacecraft, the space netted pocket drags the captured target to change orbit.

Space netted pocket capture mechanism.

Due to the strong nonlinearity dynamics of the net and the large uncertainty of the target, the precise control of the rope net spacecraft is more difficult. The traditional PD (proportional differential) controller can hardly meet the accuracy requirements of the mission. For the stability control of tethered satellites, Huang et al. considered the effect of the flexibility of rope, and designed an adaptive control method to achieve attitude stability control of combinations with unknown parameters (Huang et al., 2016); Wei et al. (2019) designed a spacecraft attitude stability ADRC (active disturbance rejection control) controller which can estimate and compensate uncertainties of system parameters in real time (Wei et al., 2019).

However, the current research generally only focuses on the attitude control during capture or maneuver, and the process of capture and maneuver is not continuous. Moreover, the dynamic model and simulation prototype for controller verification is imperfect, which cannot simulate the nonlinear properties and contact collision of the rope accurately. So, it is untrue to verify controllers using the imperfect dynamic model for the space netted pocket system.

To solve these problems, this paper designs an extended observer to compensate the uncertainty and disturbance of the model based on the principle of active disturbance rejection control (ADRC). Combined with the robust sliding mode control method, the attitude controller of the service spacecraft is constructed. At the same time, a complex dynamic model of the net pocket system is established based on the dynamic theories such as variable flexible cable, ANCF (absolute nodal coordinate formulation) and collision theory. The closed-loop simulation of dynamics and control can be completed through the virtual simulation prototype.

As shown in Fig. 2, the netted pocket system
consists of eight flexible support inflatable rods, which are connected by
rope net. The inflatable rods are evenly distributed in a positive octagonal
shape, with a single rod diameter of 0.1 m. The rope-retracting mechanism at
the end of each rod pulls the rope to close the net. The size and braided
shape of the single net piece between support rods are designed as shown in
Fig. 3. The upper part of the square rope net is

Space netted pocket system composition.

Space netted pocket system size.

A rope retractor is installed at the end of each inflatable rod, and the
diameter of the opened net is adjusted by controlling the length of the
closing rope. Due to the rope retractor, the system structure is complex,
and it is difficult to simulate the ropes inside and outside the retractor
at the same time. In this case, the simulation takes a long time, and the
rope inside the retractor has little impact on the rope net capture system.
Therefore, in order to simplify the dynamic model of the rope net capture
device and improve the simulation speed, this paper establishes the dynamic
model of closing ropes based on the ANCF flexible cable dynamics model.
Using

Model of closing rope.

Based on ANCF, the generalized coordinates of flexible cable element

Combining Eqs. (10), (11) and (12), the dynamic equation of the variable
flexible cable element can be written as

In Gerstmayr and Shabana (2006), the ANCF
equations for the dynamics of the space rope netted system can be expressed
as

According to the Hertz contact model, the contact force at the collision
point consists of the normal collision force

In the process of catching and towing target, the cable element collides with the target. Based on the principle of virtual work, the contact force can be transformed to a generalized nodal force.

The attitude Euler angle of the service spacecraft can be selected as

In this paper, the spacecraft attitude controller uses an extended observer
to estimate

Spacecraft attitude double-loop sliding mode controller.

Based on Eq. (23), the system state equation can be written as

The integral sliding mode (Liu
and Wang, 2011) is adopted to realize the sliding mode surface design.
Taking

The derivation of Eq. (29) can be obtained as follows:

The sliding surface of the inner loop is designed as

Taking Lyapunov function,

This paper utilizes the multi-body dynamic simulation software (MBDyn) developed by the author's laboratory to perform dynamic simulation. The dynamic parameters of the service spacecraft and captured target are in Table 1.

The dynamic parameters of the service spacecraft and captured target.

The material parameters of the inflatable deployment rod and rope net are in Table 2.

The material parameters of the inflatable deployment rod and rope net.

Capturing and attitude maneuver process.

The process of the service spacecraft capture target and attitude maneuver
is designed as follows: 0–20 s is the process of net retraction during which
the service spacecraft remains stationary; 20–60 s is the process of service
spacecraft attitude maneuvering 90

Set the parameters of the sliding mode controller

Service spacecraft attitude motion state.

Attitude extended observer results.

The estimates of the attitude motion state and disturbance by the expansion observer are shown in Fig. 8. Comparing Figs. 8 and 7, it can be seen that the estimates of the observer are consistent with the motion process, which illustrates the validity of the observations.

Error

As shown in Fig. 9, compared with ordinary sliding mode control, the error of extended observer controller is reduced by 1 order of magnitude. Therefore, the control accuracy of the double-loop sliding mode attitude controller based on the extended observer is significantly higher than the ordinary sliding mode control. It indicates that the extended observer compensates the disturbance effectively, and the service spacecraft attitude controller designed in this paper can meet the stability requirements of the attitude maneuvering process.

Service spacecraft control torque.

As shown in Fig. 10, the net is a centrosymmetric
structure and the initial state of target is located in the center, so the
control torque

Service spacecraft attitude dynamic response.

It can be seen from the collision force curve in
Fig. 11a that the collision force between the
capture target and the rope net is mainly concentrated in the

In this paper, the dynamics and stability control of the space net pocket
capture and towing process are studied, and the following conclusions are
obtained.

The dynamic model of the space netted pocket capture system is carried out. Based on the ANCF method, the dynamic model of the space rope system can respond to the large deformation properties of the catching mechanism. Based on the dynamic model of the closing rope established from the ANCF flexible cable theory, the simulation of the non-cooperative target capture process is realized, and the dynamic response of the closing rope recovery can be analyzed.

A sliding mode controller based on the extended observer is designed with reference to the service spacecraft dynamic equations. Only the attitude of the spacecraft is needed to complete the maneuver control, as the extended observer can observe the velocity angular velocity of the spacecraft. Moreover, based on spacecraft attitude extended observer, the sliding mode controller can ensure the high accuracy and stable attitude control of the service spacecraft.

Closed-loop dynamics and control simulations are performed for the capture and towing process of the service spacecraft. The simulation results can reflect the large deformation of the net pocket and the contact collision between the net pocket and the target during the capture and towing process. The closed-loop simulation results verify that the control accuracy is improved by 1 order of magnitude under the interference compensation of the expansion observer, which is beneficial for realizing the stable attitude control of service spacecrafts with large deformation structure.

The simulation software is jointly developed with a third party. The other party is for commercial purposes and does not want the software code to be disclosed.

In this paper, the simulation conditions and relevant parameter data are listed in Tables 1 and 2. And the effective data results are shown by curves in Figs. 7–11.

CT was mainly responsible for the calculation and data analysis of the simulation examples in this paper. ZH was mainly responsible for the modeling of simulation examples in this paper. CW was mainly responsible for the development of the dynamics software in this paper. YZ was mainly responsible for the research of dynamics and control theory in this paper.

The contact author has declared that none of the authors has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the Heilongjiang Postdoctoral Fund (grant no. LBH-Z21141) and the National Natural Science Foundation of China (grant no. 12102316).

This paper was edited by Daniel Condurache and reviewed by two anonymous referees.