Vibroimpact Dynamics of a Tethered Satellite System

This paper presents the vibroimpact dynamics of an in-plane tethered subsatellite caused by sudden braking during deployment or retrieval. The full dynamics of the subsatellite are composed of its free-flight and the instantaneous impacts. At the moment of impact, the reflective angle of the subsatellite is envisioned to be equal to its incident angle such that the impact law is obtained. Then, the stability of the periodic vibroimpacts is analyzed using the composite Poincaré map. Further, the vibroimpact responses that do not exceed a specified region are numerically determined via the cell mapping method.


Introduction
The vibroimpact phenomena of tethered satellites have been observed in experiments since the first on-orbit launch.In the H-9M-69 and S-520-2 missions, for instance, the deployment process was braked suddenly when the tether was deployed to only 38 m and 65 m, and then a series of impacts occurred [1].In the TSS-1 mission, impact-like phenomena were observed when the space tether was jammed unexpectedly by a screw installed at an inappropriate position [2].Moreover, a similar impact behavior was witnessed in a multiapplication survivable tether (MAST) experiment [3].Furthermore, numerical simulations have indicated that vibroimpacts occurred at the end of the free deployment of a flexible tethered satellite system, as shown in Figure 1 [4].Matunaga et al. [5] pointed out that the impact phenomena in a tethered satellite system are caused by a sudden tensioning force in the tether.To avoid the impact due to the slack of the tether, Wen et al. [6,7] designed a nonlinear control law with a positive tension constraint for the deployment.Markeyev [8] analyzed the stability of the vibroimpact oscillation of an ideal tethered particle under a uniform gravitational field in terms of a strict analytical solution of periodic motion.
The aim of this paper is to reveal the impact dynamics of a tethered subsatellite system under tether constraint.
The paper is organized as follows.First, the velocities of the subsatellite before and after impact are obtained.Then, a set of composite Poincaré maps is established.Next, the periodic vibroimpacts and their stability via the composite Poincaré map are analyzed.The cell mapping method is used to determine the parameter regions where the impact occurs.Finally, case studies are numerically examined.

Impact Problem of a Tethered Subsatellite
An in-plane tethered satellite system orbiting Earth under consideration consists of a mother satellite , a subsatellite , and a flexible tether, as shown in Figure 2. The satellites connected by tether are regarded as particles.The tether can be deployed/retrieved by a deployment/retrieval device mounted on two satellites.The mass of the mother satellite is sufficiently larger than that of the subsatellite, so the motion of the mother satellite is immune to the subsatellite motion.Furthermore, it is assumed that the mother satellite operates in a circular orbit with an angular velocity Ω around the Earth.To describe the motion of the system, an orbital reference frame - with coordinate origin  in the mother satellite is established, wherein the -axis points are in the opposite direction of the motion of the mother satellite, and the -axis is on the orbital plane and perpendicular to the axis.
Provided that the deployment or retrieval process is stopped suddenly at time   with the tether length (  ) = , there follows a subsatellite impact under the tether constraint (, ) = √ 2 +  2 −  = 0, as illustrated in Figure 2.
The impact is divided into the perfect elastic and inelastic impacts, that is, impact without/with damping due to the elasticity of the tether.As shown in Figure 2, the incident and reflective angles before and after impact are equal to each other; that is,  in =  re .By letting  1 = ,  2 = ẋ ,  3 = , and  4 = ẏ , the impact law of the subsatellite can be obtained as follows: where the superscripts "−" and "+" represent the moment before and after impact, respectively, and  is the factor of restitution in impact.The pitch angle of the tether at the impact moment   is Equation ( 1) can be rewritten in the following form: where cos   = − − 3 (  )/ and sin   =  − 1 (  )/.After one impact, the subsatellite could freely move without constraint until the beginning of the next impact.For such a free-flight phase, by the straightforward application of Hill's equations, the motion equations of the subsatellite are as follows: where the dot represents the derivative with respect to time .Equation (4) can be expressed in the following state equation: where By taking x + (  ) as the initial states, the analytical solutions of motion of the subsatellite can be obtained as Impact process of a tethered subsatellite system.

Stability of Periodic Vibroimpacts
The Poincaré map will be used to analyze the stability of the periodic vibroimpacts.The Poincaré section is defined as The full dynamics of the subsatellite are determined by the following two maps: where P  and P  represent the maps for the free-flight and impact phases of the subsatellite, respectively.The composite Poincaré map is Clearly, the fixed point x of the composite map corresponds to a vibroimpact oscillation of period .Taking the Taylor expansion of P in the vicinity of the fixed point gives the following: where  = ∑  =1   , and   is the free-flight time of the subsatellite after the th impact.Based on the perfect elastic impact, J(x, ) is the Jacobian of (10) calculated at x using the chain rule; namely, ＇Ｌ c q = 0? . . ., , . . ., N z Thus, the stability of the periodic vibroimpacts can be determined by the roots of the Jacobian J(x, ),   ( = 1, 2, 3, 4); that is,

Algorithm of Parameter Domain
To obtain an insight into the vibrodynamics of a tethered system, the parameter domain state not exceeding a specified region Π is calculated using the cell mapping method [9].The specified region Π is also a system state in an expected pitch angle or radial impact velocity range of the subsatellite.
According to the polar coordinate system shown in Figure 2, The study on the parameter domain is divided into two parts.First, the three-dimensional state space is divided into   =      θ   l cells as   ( = 1, 2, . . .,   ) with    ,   θ , and   l elements in , θ , and l.The sink cell is defined as    +1 =   +1.The minute flow chart of the cell mapping algorithm is given in Figure 3, where the group numbers Gr(  ) are assigned for cell sequences {  }, and the state information Da(  ) is recorded, so that the maximum value for each cell mapping can be obtained based on Gr(  ) and Da(  ).After completion of the calculation, the states of all cells can be analyzed.The corresponding cell will be not in the domain of attraction (i.e., the parameter domain) as long as the maximum of any mapping cell of the cell exceeds the specified region Π.Hence, the domain of attraction is dependent upon the attractor, all cells of which are maintained in the specified region Π.
To extend the application to all system states (, θ , , l), a similar flow process of the cell mapping method is conducted.The ranges [ min ,  max ], [ θ min , θ max ], [ min ,  max ], and [ lmin , lmax ] are uniformly divided into   ,  θ ,   , and  l elements.Then, the state space of the system is divided into   =    θ    l cells as   ( = 1, 2, . . .,   ).Similarly, the sink cell is defined as    +1 =   + 1.For each cell,   ( = 1, 2, . . .,   ), the corresponding cell   = (  ) can be computed by (7), as shown in the dashed box of Figure 3. Finally, the parameter domain for the cells of all system states is acquired via Gr(  ) and Da(  ).

Case Studies
5.1.Nonlinear Dynamics.Suppose that the tethered system moves in a circular orbit at an altitude of 300 km with an orbital angular velocity Ω = 1.1591 × 10 −3 rad/s.The total tether length  = 10 km, and  = 1.
The initial velocity is set as [ + 20 ,  + 40 ]  = [−10 m/s, 10 m/ s]  .The bifurcation responses of the pitch angle in Figure 4(a) show that the pitch angle for the impact moment increases with an increase of  + 10 .The pitch angle will particularly exist throughout [−, ] once the abscissa  + 10 > 7113 m. Figure 4(b) shows the bifurcation for  + 40 = 2m/s, in which the critical value of the impact point's coordinate is (8627 m, −5057 m).To elaborate the relationship between the coordinate position and the sudden change, the mechanical energy  of the subsatellite in the orbital frame is introduced [10]  where   = 3.986 × 10 14 m 3 /s 2 is the gravitational parameter of the Earth,   = 6371 km is the mean radius of the Earth, and   = 50 kg is the mass of the subsatellite.The critical values  0 of the initial energy are −2.987554706× 10 9 J ± 10 J.The phenomenon that the pitch angle is throughout [−, ] will appear when the initial energy of the system is larger than  0 .Figure 5 gives the bifurcation responses of the pitch angle when the initial positions ( + 10 ,  + 30 ) are (0, −).For the above-determined system parameters, the periodic, quasi-periodic, and irregular motions will appear as the initial energy changes.For  + 10 = 0,  + 20 = −2.12496m/s,  + 30 = −, and  + 40 = 10m/s, for example, a periodic motion is exhibited in Figure 6. Figure 6(a) plots the motion trajectory of the subsatellite in the orbital frame -.As shown in Figure 6(a), the periodic motion is constituted by one free-flight and one impact.The subsatellite flies with only the Coriolis force acting along the solid curve and comes back to the initial impact position (0, −) for 561 s.The stability of the periodic motion is studied via the Poincaré map in (12).The eigenvalues of the Jacobian can be calculated as  1 = 0.5360,  2,3 = 0.5202 ± 0.9136, and  4 = −1.6880,with | 1 | = 0.5360, | 2,3 | = 1.0513, and | 4 | = 1.6880.This means that the periodic motion with one impact is unstable.With the initial state of [0 m, −2.2 m/ s, −, 9.98376 m/s]  , the impact positions are located on a segment of C(x), and the phase trajectory is a closed curve, as shown in Figure 7.
A periodic motion with four impacts is depicted in Figure 8. Figure 8(a) presents the flight trajectory of the subsatellite.As seen in Figure 8(a), starting with the initial point 0, the motion of the subsatellite without the tether constraint will impact at points 1, 2, and 3 and return to the origin.Figure 8(b) shows the coordinate position of the impacts.The change of state  2 with each impact is displayed in Figure 8(c).The phase trajectory on (x) is shown in Figure 8(d), where the four converging points correspond to the states of four impacts.Similarly, the roots of the Jacobian of the periodic motion can be obtained as  1 = 0.0284,  2 = −0.0981, 3 = 10.8714, and  4 = −32.9549.Clearly, | 3,4 | > 1 means that the periodic motion is unstable.
A quasi-periodic motion is shown in Figure 9.With the initial state [0 m, −5 m/s, −, 10 m/s]  , Figure 9(a) depicts the coordinate positions of the impacts.The impacts are all situated on a segment of (x), and Figure 9(b) shows the phase points on (x).As observed from Figure 9(b), the phase points constitute a closed curve, which concludes that this is a quasi-periodic motion.
Irregular motion will occur if the initial mechanical energy exceeds the critical value  0 .With the assumption of the initial state [0 m, −20 m/s, −, 10 m/s]  , an irregular motion is presented in Figure 10.The coordinate positions of the impacts are distributed densely on the entire (x), as shown in Figure 10(a).Figure 10(b) sketches the distribution of the phase points on (x).Evidently, an oscillation with such a wide range caused by the irregular motion might cause serious disaster to other spacecraft in the neighborhood of the tethered system.In terms of the above irregular motion, a numerical case for the impacts with damping is given in Figure 11.The recovery factor in the impact was set as  = 0.999; that is, there exists energy dissipation of 0.1% for each impact.The change of  1 coordinate in each impact is presented in Figure 11(a).The corresponding phase trajectory on (x) is drawn in Figure 11(b).It can be observed from Figure 11(b) that the impact points eventually converge to origin (0, 0).

Parameter Domain.
The parameter domain is analyzed through the cell mapping method.First, the state space of impacts on  in the polar coordinate system is discussed.The study areas of the states are  ∈ [−/2, /2], θ ∈ [−10 −3 rad/s, 10 −3 rad/s], and l ∈ [0, 18 m/s], and the state variables are uniformly divided into 120, 120, and 90 elements, respectively.Thus, the state space of the system on  has 1296000 three-dimensional cells.Then, the corresponding parameter domain (i.e., domain of attraction) can be calculated provided that the specified region (i.e., attractor) is given.
For the specified region Π = {(, θ , , l) |  ∈ [−/ 6, /6],  = }, the effect of the radial impact velocity is as shown in Figure 12.As seen from Figure 12, the section area  − θ of the parameter domain decreases as l increases.Therefore, the expansion of the range of pitch angle with the initial kinetic energy increase is in accordance with the above statement on the initial system energy.For l = 6 m/s, a comparison of the parameter domain with different specified regions is presented in Figure 13, where the specified regions are  The influence of the tangential velocity θ on the parameter domain is shown in Figure 14. Figure 14 shows that the maximal section area of the parameter domain appears when θ = −5 × 10 −4 rad/s because the flight of the subsatellite after impact is towards the positive direction of θ under the Coriolis force.The effect of pitch angle on the parameter domain is provided in Figure 15.Considering the deployment/retrieval process of the tether before its first impact, the tether length variable  is introduced.The parameter domain on all the system states (i.e., , θ , , and l) is proposed using the above results for the domain on .The study areas of the pitch angle and pitch angular velocity are  ∈ [−/2, /2] and θ ∈ [−10 −3 rad/s, 10 −3 rad/s], respectively, and both are divided into 120 elements.Moreover, the ranges of the radial position and radial velocity are  ∈ [/20, ] and l ∈ [−18m/s, 18 m/s], respectively, both of which are divided into 30 elements.Accordingly, the state space of the subsatellite is divided into 12960000 four-dimensional cells.The parameter domain determined by the specified region Π = {(, θ , , l) |  ∈ [−/6, /6],  = } is depicted in Figure 16.For all of the state cells with an initial radial position  0 > 8 km and an initial radial velocity l0 = 0.6 m/s, the parameter domain corresponds to the blue zone in Figures 16(a) and 16(b).These figures show that the contrasted red zone decreases as  0 decreases or l0 increases.
In addition, it is important to study the specified region based on the radial impact velocity.The break of the space

Figure 3 :
Figure 3: Flow chart of the cell mapping algorithm.
Figure 5 shows that the amplitudes of pitch angle   decrease suddenly if the initial velocities  + 20 > −17.44 m/s or  + 20 > −20.00 m/s.Both the absolute values of the critical velocities can be calculated as approximately √ ( + 20 ) 2 + ( + 40 ) 2 = 20.10 m/s, and the critical energy is in accord with only  0 .Consequently, it can be concluded that the initial energy  0 plays an important role in the impact dynamics of the tethered satellite system.

Figure 6 :
Figure 6: Periodic motion with one impact.