Deployment of three-body chain-type tethered satellites in low-eccentricity orbits using only tether

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 (m₁, m₂, and m₃) and 2 massless tethers (L₁ and L₂), as illustrated in Fig. 1. The orbit of m₁ is defined by its orbital geocentric distance r and true anomaly α; the position of m₂ relative to m₁ is determined by tether L₁ and in-plane libration angle θ₁; the position of m₃ relative to m₂ is determined by L₂ and θ₂. 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) = Q with generalized coordinates q = (r, α, θ₁, θ₂, L₁, L₂)^T, where the nonconservative force including atmospheric drag and solar radiation pressure is also considered. 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 = (L₁, L₂)^T) and the unactuated configuration vectors (q_ua = (r, α, θ₁, θ₂)^T).

Fig. 1. Model of 3-body chain-type tethered satellite system.

Then, authors introduce a novel hierarchical sliding mode control (HSMC) deployment law for the 3-body chain-type TSS. For deployment strategy, the satellites are ejected one by one, in order to avoid collisions and apply deployment techniques used for a 2-body system directly to the single element. Here, a simple and efficient deployment scheme, combining exponential and uniform deployment law, is selected. Poincaré’s recurrence theorem, Poisson stability, and the Lie algebra rank condition (LARC) are used to analyze the controllability of underactuated TSS system. When the motion model of TSS is converted to the state space form ẋ = f(x) + g(x)u, it can be demonstrated that f(x) is weakly positively Poisson stable based on Poincaré’s recurrence theorem and the system meets LARC via complex Lie brackets. Therefore, the underactuated TSS is controllable. 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. To address this limitation, a controller 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,α) are estimated by a sliding mode-based robust differentiator.

Fig. 2. Schematic of the deployment control framework.

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 L₂ remains constant, and only tension T₁ 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 L₁ only affects angle θ₁ and L₂ only affects θ₂. In Schemes 1 and 2, the deployment controller in the literature 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.

Fig. 3. Error and tension integral in Schemes 2 and 3.