Recently, the tethered satellite system (TSS) has been used in Earth observations, space interferometry and other space missions, due to potential merits of TSS. The tethered TSAR (tomographic synthetic aperture radar) system is a group of tethered SAR satellites that can be rapidly deployed and provide a stable baseline for 3-dimensional topographic mapping and moving target detection. Successful deployment is critical for TSAR tethered system. Several control methods, including length, length rate, tension, and thrust-aided control, have been proposed over the years. Among them, adjusting tension is a viable yet challenging approach due to tether's strong nonlinearity and underactuated traits. Current tether deployment schemes focus on two-body TSS, with little emphasis on multi-TSSs. In a research article recently published in *Space: Science & Technology*, the team led by Zhongjie Meng from Northwestern Polytechnical University develops a new deployment strategy for a 3-body chain-type tethered satellite system in a low-eccentric elliptical orbit.

First, authors establishe the motion model of a 3-body chain-type TSS in a low-eccentric elliptical orbit. Two assumptions are made: (a) the tethers are massless; (b) only the planar motion is considered. The proposed model consists of 3 point masses (*m*_{1}, *m*_{2}, and *m*_{3}) and 2 massless tethers (*L*_{1} and *L*_{2}), as illustrated in Fig. 1. The orbit of *m*_{1} is defined by its orbital geocentric distance * r* and true anomaly

*; the position of*

*α*

*m*_{2}relative to

*m*_{1}is determined by tether

*L*_{1}and in-plane libration angle

*θ*_{1}; the position of

*m*_{3}relative to

*m*_{2}is determined by

*L*_{2}and

*θ*_{2}. The dynamic model of 3-body TSS is derived using Lagrangian formulation, and the motion equations are expressed in the Euler–Lagrange form as

**(**

**M****)**

**q****+**

**q̈****(**

**C****,**

**q****)**

**q̇****+**

**q̇****(**

**G****) =**

**q****with generalized coordinates**

**Q****= (**

*q**,*

*r**,*

*α*

*θ*_{1},

*θ*_{2},

*L*_{1},

*L*_{2})

^{T}. Since the TSS model in is a typical underactuated systems, the generalized coordinates are decomposed into 2parts, i.e., the actuated configuration vectors (

*q**= (*

_{a}

*L*_{1},

*L*_{2})

^{T}) and the unactuated configuration vectors (

*q**= (*

_{ua}*,*

*r**,*

*α*

*θ*_{1},

*θ*_{2})

^{T}).

Then, authors introduce a novel deployment scheme for the 3-body chain-type TSS. Sequential deployment strategy, ejecting satellites one by one, is employed to avoid collisions, this method utilizes the deployment techniques for a 2-body system directly; Poincaré’s recurrence theorem, Poisson stability, and the Lie algebra rank condition (LARC) are used to analyze the controllability of underactuated TSS system. A combination of exponential and uniform deployment law yields a simple and efficient deployment scheme, providing the requisite reference trajectory for satellite deployment. During the deployment process, positive tension must be guaranteed due to the characteristic tether, and to avoid tether rupture, tension could not exceed the given boundaries. So, deployment process can be simplified to a underactuated control with constrained control inputs. To address this limitation, a hierarchical sliding mode controller (HSMC) was designed for accurate trajectory tracking. The controller framework is shown in Fig. 2. In the controller, an auxiliary system is introduced to mitigate the input saturation caused by tether tension constraint. A 3-layer sliding surface for the whole TSS is constructed. A disturbance observer (DO) was introduced to estimate second derivative signal ** q̈**. The uncertainty of sliding surface and its time derivative for orbit motion (

*,*

*r**) are estimated by a sliding mode-based robust differentiator.*

*α*Finally, authors present the numerical simulation and draw the conclusion. To verify the effectiveness of the proposed deployment scheme (marked as Scheme 3), 2 alternative deployment schemes were used for comparison. In Scheme 1, the system is regarded as 2 independent 2-body, in which the tether length *L*_{2} remains constant, and only tension *T*_{1} is adjustable. In Scheme 2, the system is regarded as two 2-body, but the coupling between adjacent tethers is neglected. That is to say, tether *L*_{1} only affects angle *θ*_{1} and *L*_{2} only affects *θ*_{2}. In Schemes 1 and 2, the deployment controller in the literature (Murugathasan L, Zhu ZH. Deployment control of tethered space systems with explicit velocity constraint and invariance principle. Acta Astronaut. 2019;157:390–396.) is adopted. The results show that the tether deployment error and libration angle converge to zero asymptotically in 3 h (a little more than one orbital period) under Scheme 3, and the deployment error under Schemes 1 and 2 is quite larger than that under the proposed Scheme 3. A comparison is made between Schemes 2 and 3 based on the integration of tracking error and tether tension, and the normalized results are illustrated in Fig. 3. Compared to Scheme 2, the proposed HSMC explicitly takes the 3-body TSS couple into account, resulting in faster and more accurate tether deployment with a smaller in-plane angle, which further shows that a fairly better deployment process is achieved under the proposed scheme, and confirm the effectiveness of the proposed deployment scheme.

Reference

Article Title: Deployment of Three-Body Chain-Type Tethered Satellites in Low-Eccentricity Orbits Using Only Tether

Journal:* Space: Science & Technology*

Authors: Cheng Jia, Zhongjie Meng*, and Bingheng Wang

Corresponding Author Affiliation: School of Astronautics, Northwestern Polytechnical University, Xi'an, China.

Link: https://spj.science.org/doi/10.34133/space.0070

#### Journal

Space Science & Technology

#### Article Title

Deployment of Three-Body Chain-Type Tethered Satellites in Low-Eccentricity Orbits Using Only Tether

#### Article Publication Date

18-Sep-2023