Capture dynamics and control of tethered space net robot for space debris capturing in unideal capture case

https://doi.org/10.1016/j.jfranklin.2020.04.037Get rights and content

Highlights

  • The dynamic model and contact dynamic model of tethered space net robot are derived.

  • The contact dynamics of tethered space net robot in an unideal capture case are analyzed.

  • An integral version of Super-Twisting Adaptive Sliding Mode Control scheme is designed for tethered space net robot to complete the net closing action.

  • Natural and controlled net capture cases are simulated. Compared with the nonlinear integral sliding mode control with equivalent control technique and the original Super-Twisting algorithm, the designed control method shows a better performance to close the net in this unideal case.

Abstract

Tethered Space Net Robot (TSNR) is proposed as a hopeful solution for space debris capturing. An unideal case of net capture is studied in this paper, in which the net does not contact the target in the center of the net. For this case, the net cannot naturally close to wrap around the target completely after collision due to the asymmetric net configuration. Therefore, an integral adaptive Super-Twisting sliding mode control (IASTSMC) is designed to close the net. The control algorithm consists of a nonlinear integral sliding surface and adaptive Super-Twisting sliding mode control (ASTSMC), which not only has the property of “saturate the large error, magnify the small error”, but also provides a positive chattering alleviation effect. Compared with the nonlinear integral sliding mode control (ISMC) with equivalent control technique and the original Super-Twisting algorithm (STA), the proposed control method shows a better performance to close the net in this unideal case.

Introduction

Space debris capturing has received more and more attention since space debris jeopardizes operation of spacecraft [1]. To deal with this risk, many capturing methods have been proposed, which can be summarized as a stiff connection method and flexible connection method [2]. Between these two methods, flexible connection method, where the end capture device and the platform satellite are connected by a tether, is thought to be better for capturing the debris due to its less mass and cost [3]. One of the flexible connection methods with a flexible net as the end effector, Tethered Space Net (TSN), has been focused by many institutes and universities [4], [5], [6] for its several advantages: lower docking precision requirement [7], long distance capture [8], and possible to capture uncooperative target [9]. It consists of a chaser satellite, a connection tether, a flexible net and four flying weights located on the corner of the net. However, it is uncontrolled after launching. To overcome the drawback, Tethered Space Net Robot (TSNR) is proposed [10] (Fig. 1), in which the flying weights are substituted by Maneuvering Units (MUs). The MUs are treated as micro-satellites, which means that the TSNR can be controlled after launching from the platform.

During the deployment phase, maintaining the net in a desired configuration can improve the possibility of successfully capturing the target. And the controller design of maintaining net configuration for TSNR during deployment has already been investigated in [10,11]. During post-contact phase, based on the criterion of successful net capture defined in [12], whether the four MUs converge to one position and the net closes to minimum area has a significant influence on the success of whole capture mission. Therefore, a net closing control is needed when the net cannot completely close around the target. The net closing control of an ideal net capture case with symmetric net configuration has already been investigated in [12], in which the net captures the target in the center of the net, as shown in Fig. 2. However, nearly no studies has discussed the cases which are deviated from the ideal condition of net state. In this paper, an unideal net capture case is studied. As shown in Fig. 3, the case considered in the paper is that the net does not contact the target in the center of the net. Due to the offset, the net cannot maintain the symmetric configuration after collision. Moreover, the net opening area may never reduce to zero because of the different distances of each MU to the target. Therefore, an effective control algorithm for net closing action in this unideal capture case is necessary and challenging.

Sliding Mode Control (SMC) is chosen as the net closing control algorithm in this paper for its good performance even in the existence of uncertainties [13]. However, the chattering effect of traditional SMC cannot be ignored, which may excite high-frequency dynamics and lead to instability. To overcome this drawback, the second-order Sliding Mode Control scheme has been proposed. One of the second-order SMC algorithms, adaptive super-twisting sliding mode control (ASTSMC) [14], is considered in the paper. ASTSMC is the second-order SMC which can be directly applied in the controlled system when the relative degree of the system equals to one [15]. Therefore, it can provide a positive chattering alleviation effect. Additionally, due to the offset, the desired position of MU may be far away from the real position of MU, which will lead to a very large control force in the beginning when the controller is switched on. Therefore, inspired by [16], a nonlinear integral sliding surface is introduced, which has a property of “saturate the large error, magnify the small error”. Therefore, the proposed control algorithm can be named as integral adaptive super-twisting sliding mode control (IASTSMC), which maintains chattering alleviation properties of the original ASTSMC approach, but also makes the control force not very large. These features are highly beneficial in net closing control of TSNR.

The remainder of this paper is organized as follows. In Section 2, the dynamic models of TSNR are derived. The unideal case of net capture is demonstrated and analyzed in Section 3. IASTSMC algorithm is proposed to close the net in Section 4. Finally, the proposed control scheme is verified by numerical simulations in Section 5.

Section snippets

Dynamic model

The mass-spring model is applied to model the net, which approximates the net tether as a series of mass points and massless spring-damper elements. The maneuvering units are treated as four corner mass points on net. The geometrical and physical configuration of TSNR is demonstrated in Fig. 4. The motion equations for TSNR have been derived by Zhao et al. [12] in detail. Therefore, the motion equation for each net node is given here briefly based on Newton's second lawmijd2Rijdt2=FGij+i=±1,j=±

Case study

The previous studies about net capture are focused on the ideal case that the target was located at the center of the net [4,12]. In that ideal case, the net was symmetric with respect to the target during the whole capture mission. Considering the truth that the measurement error of sensor and the existence of disturbances in space environment, this ideal capture case is hard to achieve. Therefore, a net capture case, in which the initial conditions are deviated from the ideal assumption of

Problem statement

Based on the dynamic equations of TSNR derived in Section 2, the equations can be rewrite as{x¨Aa=f(xAa,xAb,xBa,x˙Aa,x˙Ab,x˙Ba)+1mAauAa_xy¨Aa=f(yAa,yAb,yBa,y˙Aa,y˙Ab,y˙Ba)+1mAauAa_yz¨Aa=f(zAa,zAb,zBa,z˙Aa,z˙Ab,z˙Ba)+1mAauAa_z{x¨Ff=f(xFf,xFe,xFg,xEf,xGf,x˙Ff,x˙Fe,x˙Fg,x˙Ef,x˙Gf)y¨Ff=f(yFf,yFe,yFg,yEf,yGf,y˙Ff,y˙Fe,y˙Fg,y˙Ef,y˙Gf)z¨Ff=f(zFf,zFe,zFg,zEf,zGf,z˙Ff,z˙Fe,z˙Fg,z˙Ef,z˙Gf)where (4) are the equations for MU Aa, and other MUs have the same function form like Aa. Eq. (5) are the equations

Simulation environment

Based on the analysis of natural capture case in Section 3, a net closing control is necessary for the unideal net capture case by TSNR. Therefore, numerical simulations have been performed in this section to verify the effectivity of the proposed control scheme in Section 4. The parameters for TSNR and the target are the same as Table 1. As the aforementioned capture scenario in Section 3, the net first freely flies to the target and the control scheme is not involved in MUs. Based on the

Conclusion

An unideal case of Tethered Space Net Robot (TSNR) for space debris capturing was focused in this paper, in which the net did not contact the target in the center of the net. After deriving the dynamic models of TSNR, the capture sequence of TSNR and the target was simulated and analyzed. Notably, the net cannot naturally close to zero due to the asymmetric net configuration. Thus, a net closing control scheme was proposed to close the net. Inspired by nonlinear integral sliding mode surface,

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant Nos. 91848205, 61725303, 61803313, in part by the Fundamental Research Funds for the Central Universities under Grant No. 3102019HTQD003, and in part by the Young Talent Fund of University Association for Science and Technology in Shaanxi, China under Grant No. 20190102, and in part by the Natural Science Basic Research Plan in Shaanxi Province of China (No. 2019JQ-345, 2019JQ-411, 2019JM-406). And

References (23)

  • P. Huang et al.

    A Review of Space Tether in New Applications

    Nonlinear Dynamics

    (2018)
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