- Email: [email protected]
Deployment dynamics of tethered-net for space debris removal
Author’s Accepted Manuscript Deployment Dynamics of Tethered-Net for Space Debris Removal Minghe Shan, Jian Guo, Eberhard Gill
www.elsevier.com/locate/actaastro
PII: DOI: Reference:
S0094-5765(16)30996-1 http://dx.doi.org/10.1016/j.actaastro.2017.01.001 AA6149
To appear in: Acta Astronautica Received date: 3 October 2016 Revised date: 29 December 2016 Accepted date: 1 January 2017 Cite this article as: Minghe Shan, Jian Guo and Eberhard Gill, Deployment Dynamics of Tethered-Net for Space Debris Removal, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2017.01.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Deployment Dynamics of Tethered-Net for Space Debris Removal Minghe Shana , Jian Guoa,∗, Eberhard Gilla a Faculty
of Aerospace Engineering, Delft University of Technology, Delft, 2629HS, The Netherlands
Abstract A tethered-net is a promising method for space debris capturing. However, its deployment dynamics is complex because of the flexibility, and its dependency of the deployment parameters is insufficiently understood. To investigate the deployment dynamics of tethered-net, four critical deployment parameters, namely maximum net area, deployment time, travelling distance and effective period are identified in this paper, and the influence of initial deployment conditions on these four parameters is investigated. Besides, a comprehensive study on a model for the tethered-net based on absolute nodal coordinates formulation (ANCF) is provided. Simulations show that the results based on the ANCF modeling method present a good agreement with that based on the conventional mass-spring modeling method. Moreover, ANCF model is capable of describing the flexibility between two nodes on the net. However, it is more computationally expensive. Keywords: Space debris, Tethered-Net, Mass-spring model, ANCF model, Deployment dynamics
1. Introduction
5
10
Operational space missions, vital for the main services, in low-Earth orbits (LEO), are more and more endangered by millions of space debris. In order to mitigate this situation, many space debris capturing and removal methods have been investigated [1], such as the robotic arm removal method [2], tethered space robot [3], [4], [5], harpoon [6], manoeuvrable tether-net space robot system [7], etc. Among these methods, the tethered-net capturing method, also termed net capturing method, is regarded as one of the most promising capturing methods due to its multiple advantages: it allows a large distance between chaser satellite and target, such that close rendezvous and docking is not mandatory; it is compatible with various dimensions and shapes of space debris objects; and, finally, the net is flexible, lightweight and cost efficient. Figure. 1 shows the ∗ Corresponding
author. Tel: +31-15-2785990; Email: [email protected]
Preprint submitted to Acta Astronautica
January 3, 2017
conceptual diagram of net capturing of a dysfunctional satellite in the initial deployment phase.
Dysfunctional Satellite
Chaser Satellite (a)
(b)
(c)
(d)
(e)
Figure 1: Conceptual diagram of initial deployment phase of the net capturing 15
20
25
Many institutes and universities have studied the net capturing method. ESA has sponsored the Robotic Geostationary Orbit Restorer (ROGER) whose objective is to transport a target into a graveyard orbit using a net [8]. The net capture mechanism consists of four flying weights in each corner of a net. The flying weight is called ”bullet”, shot by a spring system, named net gun. These four bullets help expand the large net thus wrapping a target up. Based on this concept, the feasibility of net capturing method has been analyzed. In the e.Deorbit project of ESA, simulations with different parameters such as relative position, relative rotation, tether length and tether stiffness have been performed to investigate the dynamics of a net [9]. REsearch and Development for the Capture and Removal of Orbital Clutter (REDCROC) proposed by University of Colorado is comprised of inflatable boom structures and nets configured aside [10]. The dimension of the space debris object which can be handled by REDCROC is 30 cm in diameter, and the whole system can be scaled when choosing 2
30
35
40
45
50
55
60
a larger target [10]. In the project Debris Collecting Net (D-CoNe) [11], the net is modeled as a mass-spring system. A mass-spring model is the most commonly used modeling method for tethered-net. In a mass-spring model, a tether is usually assumed to be formed by several small pieces. Since a net is comprised of many small square meshes, the interaction knot is simplified as a mass point, and the tethers between these knots are regarded as spring-damping elements. Many simulations using mass-spring model for a space tethered-net have been carried out. Botta et al. have analyzed the effect of bending stiffness of cables on the deployment dynamics and capturing motion using a net [12]. Benvenuto et al. focus on the overall dynamics of the net and interface between the net and the free-tumbling object during the disposal pull [13]. Parabolic flight experiments performed by GMV and ESA have validated the simulation results of net deployment and capturing [14]. However, little attention has been paid so far on the dynamic characteristics of the net during its initial deployment phase, and there is little discussion on the influence by the initial deployment conditions. In this paper, criteria to evaluate a tethered-net deployment, e.g., maximum deployed area, deployment time, travelling distance and effective period, have been identified, defined and investigated. The effect of the initial parameters on these criteria is studied by extensive numerical simulations. The absolute nodal coordinate formulation (ANCF) has been initially proposed by Shabana and has been utilized in solving large displacement and deformation problems [15]. In ANCF, absolute positions and the gradients of the positions act as the element nodal coordinates to describe the configuration of a flexible system. Liu et al have provided a modeling method for tethered-net based on ANCF [16]. However, the model is not verified sufficiently, and the dynamic characters of the net are not further investigated. Therefore, a model for a net based on ANCF is established, further discussed and compared with the conventional mass-spring model in this paper. To summarize, the contributions of this paper are twofold: • The dynamic characteristics of a tethered-net during its deployment phase is investigated, and the dependency on the initial deployment conditions is analyzed; • A comprehensive study on modeling of tethered-net based on ANCF is provided, and the simulation results are compared with the conventional mass-spring model.
65
70
The paper is organized into six parts. The modeling methods of a space tethered-net using a mass-spring model in Section 2, and a model based on ANCF are described in detail in Section 3. Section 4 presents the simulations based on these two modeling methods and some important parameters describing the dynamic characteristics of a tethered-net. In Section 5, the dynamic characteristics of the net is investigated using the mass-spring model. Comparison with ANCF model is also carried out in this section. Section 6 concludes the paper.
3
2. Mass-Spring Model Modeling of a net is an indispensable step to investigate the dynamic characteristics of the net capturing system. A mass-spring model for a net is introduced in this section. A mass-spring model, also named as lumped parameters model in other references, is a simplified model by discretising a flexible cable as a series of mass points and the cable between them as massless spring-damper elements. As noted earlier, a tethered-net for space debris removal is weaved as many combined square meshes by hundreds of thin, flexible and well-knit cables, and the interaction knots of the net are modeled as mass points, the cables between them as massless spring-damper elements. Assuming the net material is homogeneous isotropic linear material, the axial stiffness of the cable k is defined as EA k= (1) l0 where E is the elastic Young’s modulus of cable material, A the cross section of the cable and l0 the initial unstretched mesh length. Net
Bullet Mass Point
Spring Damping Element
Figure 2: Mesh modeling
The mass lumped on each node is not always the same. It depends on the number of adjacent cables with which is connected. The mass lumped on each node is modeled as the sum of half masses of the cables connected to the node. Thus, the point mass on four edges of the net is different from the inside ones. The mass at i-th node can be expressed as ⎧ i at four corners, ⎨ mb 3/2ρAl0 i at four edges but corners, mi = (2) ⎩ 2ρAl0 i is inside the net. 75
where mb is the bullet mass and ρ is the material density of the cable. According to the nature of cable material, cable elements are not able to withstand compression and the tension force will be generated only when the cables are elongated. The linear Kelvin-Voigt model is the most efficient and 4
commonly used method to characterize the tension force generated in a cable. Based on the linear Kelvin-Voigt model, the vector tension force between node i and j can be expressed as rij rij > l0 , (−k(rij − l0 ) − cr˙ij )ˆ Tij = (3) 0 rij ≤ l0 ,
80
where rij is the absolute distance between i-th and j-th node, c is the damping coefficient of the cable material which needs to be determined experimentally, r˙ij is the relative velocity between i-th and j-th node, and rˆij is the unit direction vector along i-th and j-th node expressed in Local Vertical-Local Horizontal (LVLH) reference frame. Orbit H-Bar V-Bar
Dysfunctional Satellite
P Chaser Satellite R-Bar
Z
Y X
Earth
Figure 3: Target approaching along V-Bar
The chaser satellite is approaching along track with the target before casting the net. The absolute position in the Earth Centered Inertial (ECI) reference frame of mass points is expressed as Ri = RP + Rri , where R is the rotation matrix that transforms the position from LVLH to ECI. With such a model, the resultant force on each node can be defined as the sum of the tension forces from the adjacent cables connected to the specified node plus the perturbations. Thus, based on Newton’s second law, the dynamic equations of motion for the entire net is discretized as N
mi
M
i i dR˙ i = RTij + Fis + Gi dt s=1 j=1
(4)
where Ri is the absolute position vector of the i-th node in ECI, Ni is the number of adjacent cables connected to the i-th node, Tij is the force on the i-th node generated by the j-th cable connected to it, Mi is the number of external forces on the i-th node, and Fis is the sum of external forces, e.g., 5
aerodynamic drag, solar radiation pressure and other perturbations on the i-th node. The micro gravitational force Gi on the i-th node is given by: Gi = −μMe mi
Ri |Ri |
3
(5)
where μ is the universal gravitational constant, Me the mass of the Earth. 3. ANCF Model
85
90
95
100
105
110
115
The mass-spring model is the most commonly used modeling method for a tethered-net. However, to fully simulate the flexibility of a cable, a large number of mass points and massless spring-damper elements are required, which will lead the multibody system super large degrees of freedom. Moreover, little attention has been paid on the verification and validation of this mass-spring model for a net. In this aspect, the ANCF model for a net is established to be used for such an application and compared with the mass-spring model. ANCF originates from the finite element method. In the classical finite element method, infinitesimal or finite rotations are used as nodal coordinates. Assumptions of small deformations and rotations are however made, which causes that the model can not accurately describe the dynamic characteristics of a flexible system. In ANCF, which is initially proposed by Shabana and utilized in solving large displacement and deformation problems, absolute positions and the gradients of the positions act as the element nodal coordinates to describe the configuration of a flexible system [15]. In ANCF, the nodal coordinates of the elements are defined in a global inertial coordinate frame. As a result, coordinate transformations are not necessary when deriving the dynamic equations of motion. Moreover, the gradients of the absolute positions, hereinafter named as the global slopes, instead of infinitesimal or finite rotations are used to describe the orientation of the elements. This important feature leads to a constant mass matrix in ANCF and consequently no centrifugal and Coriolis forces are involved in the derivation of the dynamic equations. On the contrary, the expressions of the elastic forces in ANCF are non-linear and coordinate or time dependent. Several simplified models of elastic forces for beam element have been derived by Berzeri and Shabana [17]. In a fully parameterized beam element, stiff terms of the shear deformation remain in the equations of motion and consequently make the element inefficient. Moreover, the beam element takes all the three directions deformation into consideration. While in a cable element, only the deformation along the longitudinal direction is necessary to be considered. Gerstmayr and Shabana have developed a low order cable element, in which it has half degrees of freedom of a fully parameterized beam element, and it can not describe the deformations along the lateral and transverse directions which more accurately describes the nature of the cable [18]. Since four bullets are attached to the four corners of the net to help it extend, a tethered-net system becomes a rigid-flexible coupled system. In this section, the dynamic equations of motion for the rigid flexible coupled multibody system of a tethered-net are derived. 6
For the net which is weaved by cables, one cable element contains two nodes and the arbitrary position in a cable element can be expressed as r = Se = [S1 I, S2 I, S3 I, S4 I][e1 , e2 ]T
(6)
where S is the global shape function, I is a 3×3 identity matrix, ei is the absolute nodal coordinates at x = 0 and x = l0 . ei = [ri , rix ]; rix =
∂ri ∂x
(7)
where ri is the global displacement and rix is the global slope of the element. The shape function Si is defined as S1 = 1 − 3ξ 2 + 2ξ 3 , S2 = l0 (ξ − 2ξ 2 + ξ 3 ), S3 = 3ξ 2 − 2ξ 3 , S4 = l0 (−ξ 2 + ξ 3 ) (8) 120
where ξ = x/l0 , and x is the coordinate of arbitrary point on the element. In a cable element, y and z dimensions are not taken into account due to the small magnitude in these two directions and this is the main difference between a cable element and a beam element. The constant mass matrix Mi can be derived by evaluating the kinetic energy expression. The kinetic energy of an element can be expressed as 1 1 Ti = ρr˙ T r˙ i dV = e˙ Ti Mi e˙ i (9) 2 V i 2 where ρ is the density of the element material, r˙ i is the velocity of the i-th element. The constant mass matrix Mi is consequently written as ρS T SdV. (10) Mi = V
The elastic forces of a cable element can be derived from the elastic energy. The energy of a deformed cable element contains two parts: the strain energy due to longitudinal deformation and the strain energy due to bending. In this paper, only the elastic forces due to longitudinal deformation are taken into account because of the flexibility in bending of the cable material. The strain energy due to longitudinal deformation is 1 l0 Ul = EAε2l dx (11) 2 0 where εl can be expressed based on Cauchy-Green longitudinal strain as εl =
1 T (r r − 1). 2
(12)
Using the expression of the strain energy, one obtains the vector of elastic forces Qk =
∂U T ∂e 7
= Ke
(13)
125
and K is the nonlinear stiffness matrix of the element. In order to minimize the number of mathematical operations in the computer implementation, Gaussian quadrature was used for the integration of the non-rational strain energy expressions to approximate the elastic forces. In the tethered-net, each cable element is constrained with other cable elements under a specific configuration, which will be introduced in Section 4. Moreover, the cable elements on the four corners of the net are also connected with the rigid bodies (bullet mass) under constrain conditions. This makes the system a coupled rigid-flexible multibody system. Using the principle of virtual power and introducing the Lagrange multipliers, one can derive the system equations of motion for the tethered-net as ⎡ ⎤⎡ ⎤ ⎡ ⎤ Mb 0 0 ΦT q¨b Qb qb ⎢ ⎥⎢ ⎢ ⎥ M e ΦT ΦT ¨ ⎥ ⎢ 0 e eq ⎥ ⎢ e ⎥ = ⎢ Qe ⎥ (14) ⎢ ⎥ Φe 0 0 ⎦ ⎣ λ1 ⎦ ⎣ Q1 ⎦ ⎣ 0 λ2 Q2 Φqb Φeq 0 0 where Mb and Qb are the mass matrix and the external forces, e.g., microgravity, aerodynamic drag and other perturbations of the rigid bodies, namely the bullet masses, respectively; Me and Qe are the constant ANCF mass matrix and the generalized forces associated with absolute nodal coordinates e, respectively. In this case, Qe is the external forces plus the elastic forces. Here, λ1 and λ2 represent the Lagrange multipliers; Φe is the Jacobin matrix of the constraint equations (Eq. 15) associated with the absolute nodal coordinates; Φqb and Φeq describe the constraints coupled by the bullet masses and the cable elements; Q1 and Q2 are quadratic velocity vectors derived by differentiating the constraint equations (Eq. 15) twice with respect to time. According to the folding scheme introduced in Section 4.2, the elements are connected end by end, and the bullets are connected at four corners of the net. The constraint equations of connected absolute nodal coordinates and constraint equations of bullets can be expressed as: eis − eje = 0, qb − Sec = 0 (15)
130
in which eis is the absolute nodal coordinates of the start of element i, and eje is the end of element j where element i and j are connected; ec is the absolute nodal coordinates at four corners of the net. 4. Simulations
135
4.1. Physical Properties of a Net The physical properties of a net, like strength and stiffness, are key features for a tethered-net system. They will directly impact in the effect of capturing and removal, e.g., less strong net material might be cut by sharp parts on a space debris object and, consequently, cause a failure of the mission. Since the net material is required to be lightweight, strong and flexible in bending, Honeywell
8
Table 1: Physical properties of Kevlar
Parameter Young’s Modulus E, GPa Elongation at break −, % Density ρ, g/cm3 Tensile Strength σ, GPa
140
145
150
155
160
165
Honeywell Spectra 79 3.6 0.97 2.6
Kevlar 130 2.8 1.44 3.6
Zylon 180 3.5 1.54 5.8
Carbon fiber 230 1.5 1.76 3.5
Spectra, Kevlar, Zylon and Carbon fiber are possible material candidates. According to numerical simulations of net casting and disposal strategies, a Kevlar net with the cable diameter of 1 mm is able to carry out all the required operations with target space debris object up to 1000 kg without failures [19]. The different stiffness of material has an influence on the flexibility of the net. In order to study how the different stiffness affect the dynamic characteristics of a tethered-net, these four materials are applied and compared. Table 1 shows the physical properties of these four materials. Based on previous studies on space webs, a quadrangular mesh is suggested since it revealed to be the optimum both on mass and stiffness [20]. Therefore, a square is chosen as the mesh shape in this study. 4.2. Folding Scheme To be easily folded in a canister is a fundamental characteristic for a net and also for a successful deployment. Many folding schemes, such as Miura-Ori folding scheme [21] and Hub-wrapping folding scheme [22] for various solar sails have been proposed. A two-step folding sequence has also been provided by Gardsback and Tibert [23]. However, this two-step folding sequence applies for a spin-deployed space web. For a tethered-net, consideration is only taken in the folding of cables due to the nonexistence of a membrane in the configuration. To clarify the folding scheme of a tethered-net, Fig. 4 displays the folding sequence with four units. Each cable is folded at its center point, and four bullets are successively coming close, which turns a folded net to be a bundle of cables under a specific configuration. This folding scheme also simplifies the modeling because the numbering of each cable and each node is following some specific formulations, either arithmetic sequence or geometric sequence. 4.3. Output and Input Parameters The focus of most previous research about tethered-net is on the capturing and control part, while little attention has been paid on the dynamic characteristics of the net deployment phase. In order to describe the dynamic characteristics of a tethered-net, four criteria to evaluate a net are defined with the help of Ref. [24] and Fig. 5 showing the net area changing with time during the net deployment: 9
Cable Element Bullet
Figure 4: Folding sequence
100 Maximum Area 90 80 Effective Period
70
Area [m 2 ]
60 50 40 30 20
Deployment Time
10 0 0
0.5
Time t [s]
1
1.5
Figure 5: Three parameters are shown during the net deployment
10
170
• Maximum Area The maximum area of a net is defined as the largest area the net is able to reach during the deployment. The deployment area is defined as the square area composed by four bullets. • Deployment Time Deployment time is the period starting from the time of shooting to when the net reaches its maximum area.
175
• Travelling Distance Travelling distance is the distance the mass center of a net travels during the deployment time. • Effective Period A period in which the area of a net is beyond 80% of its designed maximum area (80 m2 in this paper).
180
185
190
These four criteria determine the scope of a tethered-net space debris removal mission. The maximum area a net deploys determines the maximum volume of a space debris object the net is capable of capturing; the distance a chaser satellite should keep with the target in close rendezvous phase depends on the net’s travelling distance. In addition, the deployment time determines how efficient the capturing is and the effective period describes how long the net can stay in an effective configuration. From this respect, the maximum area of a net, travelling distance, deployment time and effective period are the four most critical parameters which characterize the net capturing method. However, these four critical parameters are dependent on several initial shooting parameters, such as shooting velocity, shooting angle, bullet mass and material of a net. The shooting angle θ is defined as the angle between the direction of the shooting velocity and the travelling direction of the net, which is shown in Fig. 6. The shooting angle for every bullet is the same due to the symmetric configuration of the net. To investigate how initial parameters affect
Travelling Direction
Bullet Figure 6: Definition of the shooting angle
195
the dynamic characteristics of a net, a set of simulations have been executed. Some input parameters for the simulations are given in Table 2. 4.4. Net Deployment Process In this study, two modeling methods, mass-spring model and ANCF model, were used to simulate the net deployment process. Firstly, the bullets are shot in the same velocity by the shooting mechanism along its shooting direction 11
Table 2: Input Parameters for Simulations
Parameter Cable diameter d, [mm] Mesh length l0 , [m] Bullet mass mb , [kg] Net mass mn , [kg] Shooting velocity v, [m/s] Shooting angle θ, [◦ ] Net size A, [m2 ]
200
205
Value 1 1 1×4 0.2 10 55 10×10
as an angle θ which is defined as shooting angle above. The successive mass points and the rest part of the net will follow the bullet to deploy. The velocity of the bullet is decreased when the cables are elongated and tensioned. The velocities of four bullets are consequently decreased to zero when the net reaches its maximum area, after which the net starts to shrink, i.e., the bullets start to travel back towards the center of the net. The motion after shrinkage is not considered because a space debris object is supposed to be captured before the net shrinks to a certain configuration. The net deployment simulations by mass-spring model and ANCF model are shown in Section 5. 5. Results and Discussion
210
215
220
225
Since the maximum area, deployment time, travelling distance and effective period are the four most critical parameters for the net capturing method, different initial parameters, such as initial bullet velocity, shooting angle and bullet mass affect these parameters in a different way. Thus it is necessary to investigate the dependency of initial bullet velocity, shooting angle and bullet mass on these parameters, respectively. On the other side, ANCF model for space tethered-net has been proposed but not further investigated. Therefore, we discussed the modeling method based on ANCF for tethered-net and applied it to compare with the mass-spring model and improve the flexibility description. To study how the input parameters influence the dynamic characteristics of a tethered-net and to further investigate the ANCF modeling method for a tethered-net, a series of simulations based on these two modeling methods introduced in this section were performed, compared, and analyzed. 5.1. Investigation on Dynamic Characteristics of Tethered-Net In all the simulations, initial parameters are applied as given in Table 1 and Table 2. A univariate analysis was applied to study the influence on the four critical parameters by different initial conditions. Firstly, the initial bullet velocity was set as changing from 10 to 28 m/s, while bullet mass and shooting angle are fixed at: mb = 1kg, θ = 55◦ . Simulations with four materials were 12
230
235
carried out using the mass-spring model, and the result is shown in Fig. 7, from which we notice the maximum area is slightly increasing, and the net with the less stiff material is able to extend to a lager area. While the deployment time is decreasing w.r.t. the increase of initial bullet velocity, i.e., the higher the initial bullet velocity is, the larger the net can be deployed and the more efficient the capture is. On the other hand, while the initial bullet velocity is increasing, the effective period for capturing is reduced, and the travelling distance is slightly varying. The reason for this slight variation is due to the fact that the travelling distance depends both on initial bullet velocity and deployment time that is changing in the other way around. 1.2 Deployment Time [s]
Maximum Area [m2]
100 95 90
Honeywell Spectro Kevlar Zylon Carbon Fiber
85 80
0.8 0.6 0.4 0.2 0.0
10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
5.5 5.4 5.3 5.2 5.1 5.0 4.9 4.8 4.7 4.6 4.5
26
28
10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
0.35 Honeywell Spectro Kevlar Zylon Carbon Fiber
0.30 Effective Period [s]
Travelling Distance [m]
Honeywell Spectro Kevlar Zylon Carbon Fiber
1.0
Honeywell Spectro Kevlar Zylon
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
0.20 0.15 0.10 0.05
Carbon Fiber 10
0.25
26
0.00 28
10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
Figure 7: Dynamic parameters changing with different initial bullet velocities
240
245
In the second case, with v = 10 m/s, mb = 1 kg, and shooting angle θ changes from 35◦ to 70◦ . It is shown in Fig. 8 that larger shooting angle also helps contribute to a larger net area and makes a more efficient capture. Also, travelling distance is significantly decreased w.r.t. increasing shooting angles. It is because the velocity component in the travelling direction is decreased w.r.t. the increasing shooting angels. In the third case, v = 10m/s, θ = 55◦ , and bullet mass changes from 0.4 to 2.2 kg. From Fig. 9, it is concluded that larger bullet mass dramatically helps extend the net but barely affect the deployment time. Moreover, travelling distance and effective period share a similar trend w.r.t. bullet mass. Following this analysis, it is concluded that the mass of the bullet and the shooting angles are key parameters influencing the net deploying area. The shooting angle influences the travelling distance dramatically. The effective period is primarily influenced by shooting velocity. To clarify the sensitivity of output parameters on different input parameters, Table 3 is made to show the 13
1.6 1.4
100
Deployment Time [s]
Maximum Area [m2]
105
95 90 Honeywell Spectro Kevlar Zylon Carbon Fiber
85 80
1.0 0.8
Honeywell Spectro Kevlar Zylon Carbon Fiber
0.6 0.4 0.2
75
0.0 35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
35
12
40
45
0.6 Honeywell Spectro Kevlar Zylon Carbon Fiber
10 8 6
Effective Period [s]
Travelling Distance [m]
1.2
4 2 0
50 55 60 65 Shooting Angles [deg]
70
75
Honeywell Spectro Kevlar Zylon Carbon Fiber
0.5 0.4 0.3 0.2 0.1 0
35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
Figure 8: Dynamic parameters changing with different shooting angles
105
95 90 Honeywell Spectro Kevlar Zylon Carbon Fiber
85 80 75
Deployment Time [s]
Maximum Area [m2]
100
70 0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
2.2
5.3
0.35
5.2
0.30
5.1
0.25
5.0 4.9 Honeywell Spectro Kevlar Zylon Carbon Fiber
4.8 4.7 4.6
Honeywell Spectro Kevlar Zylon Carbon Fiber
0.4
Effective Period [s]
Travelling Distance [m]
0.4
1.02 1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
2.2
0.20 Honeywell Spectro Kevlar Zylon Carbon Fiber
0.15 0.10 0.05 0.00
4.5 0.4
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
0.4
2.2
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
Figure 9: Dynamic parameters changing with different bullet masses
14
1.8
2.0
2.2
Table 3: Output patameters’ dependency on initial input parameters. A ’-’ denotes an inverse dependency
Initial bullet velocity (180% 10) Shooting angles (85.71% 35) Bullet mass (450% 0.4)
Maximum Area
Deployment Travelling Time Distance
Effective Period
1.28%
-35.61%
0.31%
-35.01%
8.30%
-37.13%
-71.58%
-30.82%
3.93%
-1.75%
0.74%
11.34%
dependency quantitatively. The dependency on input parameters Di j can be described as δOi Di j = (16) δIj where δOi and δIj are the relative deviation of the i-th output parameter and j-th input parameter, respectively. They can be expressed as δOi =
250
255
260
265
ΔOi ΔIj , δIj = O1 I1
(17)
where ΔOi and ΔIj are the absolute deviation of the i-th output parameter and j-th input parameter, respectively. From Eq. 17, it is known that output parameters dependency also relies on the starting value of the input parameters. Table 3 displays the dependency in percentage (take Kevlar as an example), in which the numbers in the brackets show the relative deviation and starting value of each input parameter. 5.2. Comparison and Analysis of Mass-Spring Model and ANCF Model As stated above, a mass-spring model has been the most commonly used method to model a space tethered-net so far. Another modeling method based on ANCF was developed and applied as well in this paper to compare with the mass-spring model. A set of simulations have been performed with the same initial conditions. In the simulations of ANCF model, the folding scheme of the net is used as shown in Fig. 4. Kevlar is chosen as the net material in this comparison. Fig. 10 and Fig. 11 show the net deployment simulation by mass-spring model and ANCF model, respectively. As shown in Fig. 12 to Fig. 14, we also investigated the dependency on the four critical parameters by the initial deployment conditions using both massspring model and ANCF model. It is seen that the results obtained from the mass-spring model and ANCF model are very close to each other, which indicates that ANCF modeling method is also suitable to model a net. Moreover, from Fig. 10 and Fig. 11, we notice that the net configurations from these two 15
t = 0.1 s
t = 0.2 s
t = 0.4 s
t = 0.6 s
t = 0.8 s
t=1s
Figure 10: Net deployment simulation by Mass-spring model
16
t = 0.1 s
t = 0.2 s
t = 0.6 s
t = 0.4 s
t = 0.8 s
t=1s
Figure 11: Net deployment simulation by ANCF model
17
Table 4: Comparison of single step computational time of ANCF and Mass-spring model
Mass-spring Model ANCF Model
100 98 96 94 92 90 88 86 84 82 80
Single Step Computational Time 7.883ms 0.114s
Deployment Time [s]
1.2
MS_Kevlar ANCF_Kevlar
MS_Kevlar ANCF_Kevlar
1.0 0.8 0.6 0.4 0.2 0.0
10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
10
5.7
0.35
5.5
0.30 Effective Period [s]
280
Maximum Area [m2]
275
Degrees of Freedom 363 5280
models at a certain moment have a good agreement with each other. However, with the mass-spring model, the massless spring-damper elements always keep being straight in any configuration. It requires more nodes to achieve the description of the same flexibility as ANCF model. In another word, in ANCF model, it is more capable to describe the flexible configuration of the net with fewer nodes than the mass-spring model. The cables between two nodes are not approximated as straight lines, and the flexibility of the cable is properly described, which makes ANCF more powerful to reflect the nature of the net. However, our conclusion is not that ANCF model is better than the conventional mass-spring model in any case. Actually ANCF model is much more computationally expensive compared to the conventional mass-spring model because of the higher degrees of freedom and the evaluations of the elastic forces. Table 4 shows the comparison of the computational time in a single step of both modeling methods.
Travelling Distance [m]
270
Number of Nodes 121 121
5.3 5.1 4.9 MS_Kevlar ANCF_Kevlar
4.7
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
MS_Kevlar ANCF_Kevlar
0.25 0.20 0.15 0.10 0.05
4.5
0.00 10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
Figure 12: Comparison of ANCF and Mass-spring model with different initial bullet velocities
The difference between these two methods is shown quantitatively in Table
18
Deployment Time [s]
Maximum Area [m2]
100 98 96 94 92 90 88 86 84 82 80
MS_Kevlar ANCF_Kevlar 35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
MS_Kevlar ANCF_Kevlar 35
12
40
45
50 55 60 65 Shooting Angles [deg]
0.45 MS_Kevlar ANCF_Kevlar
10
Effective Period [s]
Travelling Distance [m]
1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
8 6 4 2 0
70
75
MS_Kevlar ANCF_Kevlar
0.40 0.35 0.30 0.25 0.20 0.15
35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
Figure 13: Comparison of ANCF and Mass-spring model with different shooting angles
1.09 Deployment Time [s]
Maximum Area [m2]
100 95 90 85 MS_Kevlar 80
ANCF_Kevlar
75
0.99 0.94 0.89 0.84
0.4
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
5.5 5.4 5.3 5.2 5.1 5.0 4.9 4.8 4.7 4.6 4.5
1.8
2.0
2.2
0.4
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
2.2
0.35 Effective Period [s]
Travelling Distance [m]
MS_Kevlar ANCF_Kevlar
1.04
MS_Kevlar ANCF_Kevlar
0.30 0.25 0.20 0.15 0.10
MS_Kevlar ANCF_Kevlar
0.05 0.00
0.4
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
2.2
0.4
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
2.2
Figure 14: Comparison of ANCF and Mass-spring model with different bullet masses
19
Table 5: Maximum difference between ANCF and Mass-spring model
Initial bullet velocity Shooting angles Bullet mass
Maximum Area 4.29% 1.71% 3.90%
Deployment Time 4.68% 1.11% 5.95%
Travelling Distance 7.22% 6.72% 6.27%
Effective Period 10.32% 15.01% 32.61%
5 where the maximum difference was obtained using: Δ=
285
max|xM − xA | . xM
(18)
Here xM is the data from the mass-spring model and xA is the data from ANCF model. From the table, it is seen that the maximum differences between these two modeling methods are below 10% except for the effective period due to its sensitivity itself. 6. Conclusion
290
295
300
305
310
In this paper, the dynamic characteristics of a tethered-net in its deployment phase has been investigated. Four critical parameters describing the deployment dynamic characteristics of tethered-net, namely maximum area, deployment time, travelling distance and effective period have been identified and defined. These parameters have been investigated by simulations under different initial input parameters, namely initial bullet velocity, shooting angles and bullet mass. The physical properties of the net have been addressed, and a new folding pattern is provided for simulations. It is concluded that a higher bullet mass, shooting angles and more flexible net material contribute to a larger net area. Net material has little influence on deployment time and travelling distance. However, a higher bullet mass consumes more energy and involves a more complex system, such as the shooting mechanism. Moreover, a higher initial bullet velocity and a larger shooting angle lead to a shorter effective period which leads the capturing a higher risk to fail and less reliable. The dependency of the output parameters on different initial input parameters has been provided quantitatively. Besides, a comprehensive study on modeling of a tethered-net based on ANCF has been provided. Rigid-flexible coupled system of the tetherednet has been established. Simulations based on ANCF have been performed and compared with the conventional mass-spring model. The results from both methods show a good agreement on the changing of four critical parameters, which indicates the ANCF model is very suitable to describe the dynamics of a tethered-net. The maximum differences between two methods, which are below 10% except for the sensitive effective period, have been calculated showing the agreement between these two modeling methods. Furthermore, ANCF model is 20
315
more capable of describing the flexibility of the net with fewer nodes than the conventional mass-spring model. However, it is more computationally expensive. Based on the comparison of single-step computational time, Mass spring model is nearly 15 times faster than ANCF model. References
320
[1] M. Shan, J. Guo, E. Gill, Review and comparison of active space debris capturing and removal methods, Progress in Aerospace Sciences 80 (2016) 18–32. [2] D. Reintsema, J. Thaeter, A. Rathke, W. Naumann, P. Rank, J. Sommer, Deos–the german robotics approach to secure and de-orbit malfunctioned satellites from low earth orbits, in: Proceedings of the i-SAIRAS, Sapporo, Japan, (2010), pp. 244–251.
325
330
[3] P. Huang, D. Wang, Z. Meng, F. Zhang, J. Guo, Adaptive postcapture backstepping control for tumbling tethered space robot–target combination, Journal of Guidance, Control, and Dynamics 39 (1) (2016) 150–156. [4] P. Huang, D. Wang, Z. Meng, F. Zhang, Z. Liu, Impact dynamic modeling and adaptive target capturing control for tethered space robots with uncertainties, IEEE/ASME Transactions on Mechatronics 21 (5) (2016) 2260–2271. [5] P. Huang, M. Wang, Z. Meng, F. Zhang, Z. Liu, Attitude takeover control for post-capture of target spacecraft using space robot, Aerospace Science and Technology 51 (2016) 171–180.
335
340
[6] J. Reed, J. Busquets, C. White, Grappling system for capturing heavy space debris, in: 2nd European Workshop on Active Debris Removal,Paris, France, (2012). [7] P. Huang, Z. Hu, F. Zhang, Dynamic modelling and coordinated controller designing for the manoeuvrable tether-net space robot system, Multibody System Dynamics 36 (2) (2016) 115–141. [8] B. Bischof, L. Kerstein, J. Starke, H. Guenther, W. Foth, et al., Rogerrobotic geostationary orbit restorer, Science and Technology Series 109 (2004) 183–193.
345
350
[9] C. Billot, S. Ferraris, R. Rembala, F. Cacciatore, A. Tomassini, R. Biesbroek, e.deorbit: Feasibility study for an active debris removal, in: 3rd European Workshop on Space Debris Modeling and Remediation, Paris, France, (2014). [10] N. Zinner, A. Williamson, K. Brenner, J. B. Curran, A. Isaak, M. Knoch, A. Leppek, J. Lestishen, Junk hunter: Autonomous rendezvous, capture, and de-orbit of orbital debris, in: Proceedings of AIAA SPACE 2011 Conference & Exposition, Long Beach, CA, USA, (2001). 21
[11] M. Lavagna, R. Armellin, A. Bombelli, R. Benvenuto, Debris removal mechanism based on tethered nets, in: Proceedings of the i-SAIRAS, Turin, Italy, (2012). 355
360
[12] E. M. Botta, I. Sharf, A. K. Misra, M. Teichmann, On the simulation of tether-nets for space debris capture with vortex dynamics, Acta Astronautica 123 (2016) 91–102. [13] R. Benvenuto, M. Lavagna, S. Salvi, Multibody dynamics driving gnc and system design in tethered nets for active debris removal, Advances in Space Research 58 (1) (2016) 45–63. [14] A. Lorenzo, R. Stefanescu, R. Benvenuto, M. Marcon, M. Lavagna, Validation results of satellite mock-up capturing experiment using nets, in: 66th International Astronautical Congress, Jerusalem, Israel, (2015).
365
[15] A. A. Shabana, Flexible multibody dynamics: review of past and recent developments, Multibody system dynamics 1 (2) (1997) 189–222. [16] L. Liu, J. Shan, Y. Ren, Z. Zhou, Deployment dynamics of throw-net for active debris removal, in: 65th International Astronautical Congress, Toronto, Canada, (2014).
370
[17] M. Berzeri, A. Shabana, Development of simple models for the elastic forces in the absolute nodal co-ordinate formulation, Journal of Sound and Vibration 235 (4) (2000) 539–565. [18] J. Gerstmayr, A. A. Shabana, Analysis of thin beams and cables using the absolute nodal co-ordinate formulation, Nonlinear Dynamics 45 (1-2) (2006) 109–130.
375
[19] R. Benvenuto, R. Carta, Active debris removal system based on tetherednets: experimental results, in: Proceedings of the 9th PEGASUS-AIAA Student Conf., Milano, Italy, (2013). [20] G. Tibert, M. Gardsback, Space webs final report, in: ESA/ACT, AdrianaID: 05, Noordwijk, The Netherlands, (2006).
380
[21] K. Miura, Method of packaging and deployment of large membranes in space, The Institute of Space and Astronautical Science report 618 (1985) 1–9. [22] H. Furuya, Y. Inoue, T. Masuoka, Deployment characteristics of rotationally skew fold membrane for spinning solar sail, AIAA 2045 (2005) 1–10.
385
[23] M. G¨ardsback, G. Tibert, Deployment control of spinning space webs, Journal of guidance, control, and dynamics 32 (1) (2009) 40–50. [24] Y. Yu, H. Baoyin, J. Li, Modeling and simulation of projecting deployment dynamics of space webs, Journal of Astronautics 5 (2010) 1–8. 22
www.elsevier.com/locate/actaastro
PII: DOI: Reference:
S0094-5765(16)30996-1 http://dx.doi.org/10.1016/j.actaastro.2017.01.001 AA6149
To appear in: Acta Astronautica Received date: 3 October 2016 Revised date: 29 December 2016 Accepted date: 1 January 2017 Cite this article as: Minghe Shan, Jian Guo and Eberhard Gill, Deployment Dynamics of Tethered-Net for Space Debris Removal, Acta Astronautica, http://dx.doi.org/10.1016/j.actaastro.2017.01.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Deployment Dynamics of Tethered-Net for Space Debris Removal Minghe Shana , Jian Guoa,∗, Eberhard Gilla a Faculty
of Aerospace Engineering, Delft University of Technology, Delft, 2629HS, The Netherlands
Abstract A tethered-net is a promising method for space debris capturing. However, its deployment dynamics is complex because of the flexibility, and its dependency of the deployment parameters is insufficiently understood. To investigate the deployment dynamics of tethered-net, four critical deployment parameters, namely maximum net area, deployment time, travelling distance and effective period are identified in this paper, and the influence of initial deployment conditions on these four parameters is investigated. Besides, a comprehensive study on a model for the tethered-net based on absolute nodal coordinates formulation (ANCF) is provided. Simulations show that the results based on the ANCF modeling method present a good agreement with that based on the conventional mass-spring modeling method. Moreover, ANCF model is capable of describing the flexibility between two nodes on the net. However, it is more computationally expensive. Keywords: Space debris, Tethered-Net, Mass-spring model, ANCF model, Deployment dynamics
1. Introduction
5
10
Operational space missions, vital for the main services, in low-Earth orbits (LEO), are more and more endangered by millions of space debris. In order to mitigate this situation, many space debris capturing and removal methods have been investigated [1], such as the robotic arm removal method [2], tethered space robot [3], [4], [5], harpoon [6], manoeuvrable tether-net space robot system [7], etc. Among these methods, the tethered-net capturing method, also termed net capturing method, is regarded as one of the most promising capturing methods due to its multiple advantages: it allows a large distance between chaser satellite and target, such that close rendezvous and docking is not mandatory; it is compatible with various dimensions and shapes of space debris objects; and, finally, the net is flexible, lightweight and cost efficient. Figure. 1 shows the ∗ Corresponding
author. Tel: +31-15-2785990; Email: [email protected]
Preprint submitted to Acta Astronautica
January 3, 2017
conceptual diagram of net capturing of a dysfunctional satellite in the initial deployment phase.
Dysfunctional Satellite
Chaser Satellite (a)
(b)
(c)
(d)
(e)
Figure 1: Conceptual diagram of initial deployment phase of the net capturing 15
20
25
Many institutes and universities have studied the net capturing method. ESA has sponsored the Robotic Geostationary Orbit Restorer (ROGER) whose objective is to transport a target into a graveyard orbit using a net [8]. The net capture mechanism consists of four flying weights in each corner of a net. The flying weight is called ”bullet”, shot by a spring system, named net gun. These four bullets help expand the large net thus wrapping a target up. Based on this concept, the feasibility of net capturing method has been analyzed. In the e.Deorbit project of ESA, simulations with different parameters such as relative position, relative rotation, tether length and tether stiffness have been performed to investigate the dynamics of a net [9]. REsearch and Development for the Capture and Removal of Orbital Clutter (REDCROC) proposed by University of Colorado is comprised of inflatable boom structures and nets configured aside [10]. The dimension of the space debris object which can be handled by REDCROC is 30 cm in diameter, and the whole system can be scaled when choosing 2
30
35
40
45
50
55
60
a larger target [10]. In the project Debris Collecting Net (D-CoNe) [11], the net is modeled as a mass-spring system. A mass-spring model is the most commonly used modeling method for tethered-net. In a mass-spring model, a tether is usually assumed to be formed by several small pieces. Since a net is comprised of many small square meshes, the interaction knot is simplified as a mass point, and the tethers between these knots are regarded as spring-damping elements. Many simulations using mass-spring model for a space tethered-net have been carried out. Botta et al. have analyzed the effect of bending stiffness of cables on the deployment dynamics and capturing motion using a net [12]. Benvenuto et al. focus on the overall dynamics of the net and interface between the net and the free-tumbling object during the disposal pull [13]. Parabolic flight experiments performed by GMV and ESA have validated the simulation results of net deployment and capturing [14]. However, little attention has been paid so far on the dynamic characteristics of the net during its initial deployment phase, and there is little discussion on the influence by the initial deployment conditions. In this paper, criteria to evaluate a tethered-net deployment, e.g., maximum deployed area, deployment time, travelling distance and effective period, have been identified, defined and investigated. The effect of the initial parameters on these criteria is studied by extensive numerical simulations. The absolute nodal coordinate formulation (ANCF) has been initially proposed by Shabana and has been utilized in solving large displacement and deformation problems [15]. In ANCF, absolute positions and the gradients of the positions act as the element nodal coordinates to describe the configuration of a flexible system. Liu et al have provided a modeling method for tethered-net based on ANCF [16]. However, the model is not verified sufficiently, and the dynamic characters of the net are not further investigated. Therefore, a model for a net based on ANCF is established, further discussed and compared with the conventional mass-spring model in this paper. To summarize, the contributions of this paper are twofold: • The dynamic characteristics of a tethered-net during its deployment phase is investigated, and the dependency on the initial deployment conditions is analyzed; • A comprehensive study on modeling of tethered-net based on ANCF is provided, and the simulation results are compared with the conventional mass-spring model.
65
70
The paper is organized into six parts. The modeling methods of a space tethered-net using a mass-spring model in Section 2, and a model based on ANCF are described in detail in Section 3. Section 4 presents the simulations based on these two modeling methods and some important parameters describing the dynamic characteristics of a tethered-net. In Section 5, the dynamic characteristics of the net is investigated using the mass-spring model. Comparison with ANCF model is also carried out in this section. Section 6 concludes the paper.
3
2. Mass-Spring Model Modeling of a net is an indispensable step to investigate the dynamic characteristics of the net capturing system. A mass-spring model for a net is introduced in this section. A mass-spring model, also named as lumped parameters model in other references, is a simplified model by discretising a flexible cable as a series of mass points and the cable between them as massless spring-damper elements. As noted earlier, a tethered-net for space debris removal is weaved as many combined square meshes by hundreds of thin, flexible and well-knit cables, and the interaction knots of the net are modeled as mass points, the cables between them as massless spring-damper elements. Assuming the net material is homogeneous isotropic linear material, the axial stiffness of the cable k is defined as EA k= (1) l0 where E is the elastic Young’s modulus of cable material, A the cross section of the cable and l0 the initial unstretched mesh length. Net
Bullet Mass Point
Spring Damping Element
Figure 2: Mesh modeling
The mass lumped on each node is not always the same. It depends on the number of adjacent cables with which is connected. The mass lumped on each node is modeled as the sum of half masses of the cables connected to the node. Thus, the point mass on four edges of the net is different from the inside ones. The mass at i-th node can be expressed as ⎧ i at four corners, ⎨ mb 3/2ρAl0 i at four edges but corners, mi = (2) ⎩ 2ρAl0 i is inside the net. 75
where mb is the bullet mass and ρ is the material density of the cable. According to the nature of cable material, cable elements are not able to withstand compression and the tension force will be generated only when the cables are elongated. The linear Kelvin-Voigt model is the most efficient and 4
commonly used method to characterize the tension force generated in a cable. Based on the linear Kelvin-Voigt model, the vector tension force between node i and j can be expressed as rij rij > l0 , (−k(rij − l0 ) − cr˙ij )ˆ Tij = (3) 0 rij ≤ l0 ,
80
where rij is the absolute distance between i-th and j-th node, c is the damping coefficient of the cable material which needs to be determined experimentally, r˙ij is the relative velocity between i-th and j-th node, and rˆij is the unit direction vector along i-th and j-th node expressed in Local Vertical-Local Horizontal (LVLH) reference frame. Orbit H-Bar V-Bar
Dysfunctional Satellite
P Chaser Satellite R-Bar
Z
Y X
Earth
Figure 3: Target approaching along V-Bar
The chaser satellite is approaching along track with the target before casting the net. The absolute position in the Earth Centered Inertial (ECI) reference frame of mass points is expressed as Ri = RP + Rri , where R is the rotation matrix that transforms the position from LVLH to ECI. With such a model, the resultant force on each node can be defined as the sum of the tension forces from the adjacent cables connected to the specified node plus the perturbations. Thus, based on Newton’s second law, the dynamic equations of motion for the entire net is discretized as N
mi
M
i i dR˙ i = RTij + Fis + Gi dt s=1 j=1
(4)
where Ri is the absolute position vector of the i-th node in ECI, Ni is the number of adjacent cables connected to the i-th node, Tij is the force on the i-th node generated by the j-th cable connected to it, Mi is the number of external forces on the i-th node, and Fis is the sum of external forces, e.g., 5
aerodynamic drag, solar radiation pressure and other perturbations on the i-th node. The micro gravitational force Gi on the i-th node is given by: Gi = −μMe mi
Ri |Ri |
3
(5)
where μ is the universal gravitational constant, Me the mass of the Earth. 3. ANCF Model
85
90
95
100
105
110
115
The mass-spring model is the most commonly used modeling method for a tethered-net. However, to fully simulate the flexibility of a cable, a large number of mass points and massless spring-damper elements are required, which will lead the multibody system super large degrees of freedom. Moreover, little attention has been paid on the verification and validation of this mass-spring model for a net. In this aspect, the ANCF model for a net is established to be used for such an application and compared with the mass-spring model. ANCF originates from the finite element method. In the classical finite element method, infinitesimal or finite rotations are used as nodal coordinates. Assumptions of small deformations and rotations are however made, which causes that the model can not accurately describe the dynamic characteristics of a flexible system. In ANCF, which is initially proposed by Shabana and utilized in solving large displacement and deformation problems, absolute positions and the gradients of the positions act as the element nodal coordinates to describe the configuration of a flexible system [15]. In ANCF, the nodal coordinates of the elements are defined in a global inertial coordinate frame. As a result, coordinate transformations are not necessary when deriving the dynamic equations of motion. Moreover, the gradients of the absolute positions, hereinafter named as the global slopes, instead of infinitesimal or finite rotations are used to describe the orientation of the elements. This important feature leads to a constant mass matrix in ANCF and consequently no centrifugal and Coriolis forces are involved in the derivation of the dynamic equations. On the contrary, the expressions of the elastic forces in ANCF are non-linear and coordinate or time dependent. Several simplified models of elastic forces for beam element have been derived by Berzeri and Shabana [17]. In a fully parameterized beam element, stiff terms of the shear deformation remain in the equations of motion and consequently make the element inefficient. Moreover, the beam element takes all the three directions deformation into consideration. While in a cable element, only the deformation along the longitudinal direction is necessary to be considered. Gerstmayr and Shabana have developed a low order cable element, in which it has half degrees of freedom of a fully parameterized beam element, and it can not describe the deformations along the lateral and transverse directions which more accurately describes the nature of the cable [18]. Since four bullets are attached to the four corners of the net to help it extend, a tethered-net system becomes a rigid-flexible coupled system. In this section, the dynamic equations of motion for the rigid flexible coupled multibody system of a tethered-net are derived. 6
For the net which is weaved by cables, one cable element contains two nodes and the arbitrary position in a cable element can be expressed as r = Se = [S1 I, S2 I, S3 I, S4 I][e1 , e2 ]T
(6)
where S is the global shape function, I is a 3×3 identity matrix, ei is the absolute nodal coordinates at x = 0 and x = l0 . ei = [ri , rix ]; rix =
∂ri ∂x
(7)
where ri is the global displacement and rix is the global slope of the element. The shape function Si is defined as S1 = 1 − 3ξ 2 + 2ξ 3 , S2 = l0 (ξ − 2ξ 2 + ξ 3 ), S3 = 3ξ 2 − 2ξ 3 , S4 = l0 (−ξ 2 + ξ 3 ) (8) 120
where ξ = x/l0 , and x is the coordinate of arbitrary point on the element. In a cable element, y and z dimensions are not taken into account due to the small magnitude in these two directions and this is the main difference between a cable element and a beam element. The constant mass matrix Mi can be derived by evaluating the kinetic energy expression. The kinetic energy of an element can be expressed as 1 1 Ti = ρr˙ T r˙ i dV = e˙ Ti Mi e˙ i (9) 2 V i 2 where ρ is the density of the element material, r˙ i is the velocity of the i-th element. The constant mass matrix Mi is consequently written as ρS T SdV. (10) Mi = V
The elastic forces of a cable element can be derived from the elastic energy. The energy of a deformed cable element contains two parts: the strain energy due to longitudinal deformation and the strain energy due to bending. In this paper, only the elastic forces due to longitudinal deformation are taken into account because of the flexibility in bending of the cable material. The strain energy due to longitudinal deformation is 1 l0 Ul = EAε2l dx (11) 2 0 where εl can be expressed based on Cauchy-Green longitudinal strain as εl =
1 T (r r − 1). 2
(12)
Using the expression of the strain energy, one obtains the vector of elastic forces Qk =
∂U T ∂e 7
= Ke
(13)
125
and K is the nonlinear stiffness matrix of the element. In order to minimize the number of mathematical operations in the computer implementation, Gaussian quadrature was used for the integration of the non-rational strain energy expressions to approximate the elastic forces. In the tethered-net, each cable element is constrained with other cable elements under a specific configuration, which will be introduced in Section 4. Moreover, the cable elements on the four corners of the net are also connected with the rigid bodies (bullet mass) under constrain conditions. This makes the system a coupled rigid-flexible multibody system. Using the principle of virtual power and introducing the Lagrange multipliers, one can derive the system equations of motion for the tethered-net as ⎡ ⎤⎡ ⎤ ⎡ ⎤ Mb 0 0 ΦT q¨b Qb qb ⎢ ⎥⎢ ⎢ ⎥ M e ΦT ΦT ¨ ⎥ ⎢ 0 e eq ⎥ ⎢ e ⎥ = ⎢ Qe ⎥ (14) ⎢ ⎥ Φe 0 0 ⎦ ⎣ λ1 ⎦ ⎣ Q1 ⎦ ⎣ 0 λ2 Q2 Φqb Φeq 0 0 where Mb and Qb are the mass matrix and the external forces, e.g., microgravity, aerodynamic drag and other perturbations of the rigid bodies, namely the bullet masses, respectively; Me and Qe are the constant ANCF mass matrix and the generalized forces associated with absolute nodal coordinates e, respectively. In this case, Qe is the external forces plus the elastic forces. Here, λ1 and λ2 represent the Lagrange multipliers; Φe is the Jacobin matrix of the constraint equations (Eq. 15) associated with the absolute nodal coordinates; Φqb and Φeq describe the constraints coupled by the bullet masses and the cable elements; Q1 and Q2 are quadratic velocity vectors derived by differentiating the constraint equations (Eq. 15) twice with respect to time. According to the folding scheme introduced in Section 4.2, the elements are connected end by end, and the bullets are connected at four corners of the net. The constraint equations of connected absolute nodal coordinates and constraint equations of bullets can be expressed as: eis − eje = 0, qb − Sec = 0 (15)
130
in which eis is the absolute nodal coordinates of the start of element i, and eje is the end of element j where element i and j are connected; ec is the absolute nodal coordinates at four corners of the net. 4. Simulations
135
4.1. Physical Properties of a Net The physical properties of a net, like strength and stiffness, are key features for a tethered-net system. They will directly impact in the effect of capturing and removal, e.g., less strong net material might be cut by sharp parts on a space debris object and, consequently, cause a failure of the mission. Since the net material is required to be lightweight, strong and flexible in bending, Honeywell
8
Table 1: Physical properties of Kevlar
Parameter Young’s Modulus E, GPa Elongation at break −, % Density ρ, g/cm3 Tensile Strength σ, GPa
140
145
150
155
160
165
Honeywell Spectra 79 3.6 0.97 2.6
Kevlar 130 2.8 1.44 3.6
Zylon 180 3.5 1.54 5.8
Carbon fiber 230 1.5 1.76 3.5
Spectra, Kevlar, Zylon and Carbon fiber are possible material candidates. According to numerical simulations of net casting and disposal strategies, a Kevlar net with the cable diameter of 1 mm is able to carry out all the required operations with target space debris object up to 1000 kg without failures [19]. The different stiffness of material has an influence on the flexibility of the net. In order to study how the different stiffness affect the dynamic characteristics of a tethered-net, these four materials are applied and compared. Table 1 shows the physical properties of these four materials. Based on previous studies on space webs, a quadrangular mesh is suggested since it revealed to be the optimum both on mass and stiffness [20]. Therefore, a square is chosen as the mesh shape in this study. 4.2. Folding Scheme To be easily folded in a canister is a fundamental characteristic for a net and also for a successful deployment. Many folding schemes, such as Miura-Ori folding scheme [21] and Hub-wrapping folding scheme [22] for various solar sails have been proposed. A two-step folding sequence has also been provided by Gardsback and Tibert [23]. However, this two-step folding sequence applies for a spin-deployed space web. For a tethered-net, consideration is only taken in the folding of cables due to the nonexistence of a membrane in the configuration. To clarify the folding scheme of a tethered-net, Fig. 4 displays the folding sequence with four units. Each cable is folded at its center point, and four bullets are successively coming close, which turns a folded net to be a bundle of cables under a specific configuration. This folding scheme also simplifies the modeling because the numbering of each cable and each node is following some specific formulations, either arithmetic sequence or geometric sequence. 4.3. Output and Input Parameters The focus of most previous research about tethered-net is on the capturing and control part, while little attention has been paid on the dynamic characteristics of the net deployment phase. In order to describe the dynamic characteristics of a tethered-net, four criteria to evaluate a net are defined with the help of Ref. [24] and Fig. 5 showing the net area changing with time during the net deployment: 9
Cable Element Bullet
Figure 4: Folding sequence
100 Maximum Area 90 80 Effective Period
70
Area [m 2 ]
60 50 40 30 20
Deployment Time
10 0 0
0.5
Time t [s]
1
1.5
Figure 5: Three parameters are shown during the net deployment
10
170
• Maximum Area The maximum area of a net is defined as the largest area the net is able to reach during the deployment. The deployment area is defined as the square area composed by four bullets. • Deployment Time Deployment time is the period starting from the time of shooting to when the net reaches its maximum area.
175
• Travelling Distance Travelling distance is the distance the mass center of a net travels during the deployment time. • Effective Period A period in which the area of a net is beyond 80% of its designed maximum area (80 m2 in this paper).
180
185
190
These four criteria determine the scope of a tethered-net space debris removal mission. The maximum area a net deploys determines the maximum volume of a space debris object the net is capable of capturing; the distance a chaser satellite should keep with the target in close rendezvous phase depends on the net’s travelling distance. In addition, the deployment time determines how efficient the capturing is and the effective period describes how long the net can stay in an effective configuration. From this respect, the maximum area of a net, travelling distance, deployment time and effective period are the four most critical parameters which characterize the net capturing method. However, these four critical parameters are dependent on several initial shooting parameters, such as shooting velocity, shooting angle, bullet mass and material of a net. The shooting angle θ is defined as the angle between the direction of the shooting velocity and the travelling direction of the net, which is shown in Fig. 6. The shooting angle for every bullet is the same due to the symmetric configuration of the net. To investigate how initial parameters affect
Travelling Direction
Bullet Figure 6: Definition of the shooting angle
195
the dynamic characteristics of a net, a set of simulations have been executed. Some input parameters for the simulations are given in Table 2. 4.4. Net Deployment Process In this study, two modeling methods, mass-spring model and ANCF model, were used to simulate the net deployment process. Firstly, the bullets are shot in the same velocity by the shooting mechanism along its shooting direction 11
Table 2: Input Parameters for Simulations
Parameter Cable diameter d, [mm] Mesh length l0 , [m] Bullet mass mb , [kg] Net mass mn , [kg] Shooting velocity v, [m/s] Shooting angle θ, [◦ ] Net size A, [m2 ]
200
205
Value 1 1 1×4 0.2 10 55 10×10
as an angle θ which is defined as shooting angle above. The successive mass points and the rest part of the net will follow the bullet to deploy. The velocity of the bullet is decreased when the cables are elongated and tensioned. The velocities of four bullets are consequently decreased to zero when the net reaches its maximum area, after which the net starts to shrink, i.e., the bullets start to travel back towards the center of the net. The motion after shrinkage is not considered because a space debris object is supposed to be captured before the net shrinks to a certain configuration. The net deployment simulations by mass-spring model and ANCF model are shown in Section 5. 5. Results and Discussion
210
215
220
225
Since the maximum area, deployment time, travelling distance and effective period are the four most critical parameters for the net capturing method, different initial parameters, such as initial bullet velocity, shooting angle and bullet mass affect these parameters in a different way. Thus it is necessary to investigate the dependency of initial bullet velocity, shooting angle and bullet mass on these parameters, respectively. On the other side, ANCF model for space tethered-net has been proposed but not further investigated. Therefore, we discussed the modeling method based on ANCF for tethered-net and applied it to compare with the mass-spring model and improve the flexibility description. To study how the input parameters influence the dynamic characteristics of a tethered-net and to further investigate the ANCF modeling method for a tethered-net, a series of simulations based on these two modeling methods introduced in this section were performed, compared, and analyzed. 5.1. Investigation on Dynamic Characteristics of Tethered-Net In all the simulations, initial parameters are applied as given in Table 1 and Table 2. A univariate analysis was applied to study the influence on the four critical parameters by different initial conditions. Firstly, the initial bullet velocity was set as changing from 10 to 28 m/s, while bullet mass and shooting angle are fixed at: mb = 1kg, θ = 55◦ . Simulations with four materials were 12
230
235
carried out using the mass-spring model, and the result is shown in Fig. 7, from which we notice the maximum area is slightly increasing, and the net with the less stiff material is able to extend to a lager area. While the deployment time is decreasing w.r.t. the increase of initial bullet velocity, i.e., the higher the initial bullet velocity is, the larger the net can be deployed and the more efficient the capture is. On the other hand, while the initial bullet velocity is increasing, the effective period for capturing is reduced, and the travelling distance is slightly varying. The reason for this slight variation is due to the fact that the travelling distance depends both on initial bullet velocity and deployment time that is changing in the other way around. 1.2 Deployment Time [s]
Maximum Area [m2]
100 95 90
Honeywell Spectro Kevlar Zylon Carbon Fiber
85 80
0.8 0.6 0.4 0.2 0.0
10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
5.5 5.4 5.3 5.2 5.1 5.0 4.9 4.8 4.7 4.6 4.5
26
28
10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
0.35 Honeywell Spectro Kevlar Zylon Carbon Fiber
0.30 Effective Period [s]
Travelling Distance [m]
Honeywell Spectro Kevlar Zylon Carbon Fiber
1.0
Honeywell Spectro Kevlar Zylon
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
0.20 0.15 0.10 0.05
Carbon Fiber 10
0.25
26
0.00 28
10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
Figure 7: Dynamic parameters changing with different initial bullet velocities
240
245
In the second case, with v = 10 m/s, mb = 1 kg, and shooting angle θ changes from 35◦ to 70◦ . It is shown in Fig. 8 that larger shooting angle also helps contribute to a larger net area and makes a more efficient capture. Also, travelling distance is significantly decreased w.r.t. increasing shooting angles. It is because the velocity component in the travelling direction is decreased w.r.t. the increasing shooting angels. In the third case, v = 10m/s, θ = 55◦ , and bullet mass changes from 0.4 to 2.2 kg. From Fig. 9, it is concluded that larger bullet mass dramatically helps extend the net but barely affect the deployment time. Moreover, travelling distance and effective period share a similar trend w.r.t. bullet mass. Following this analysis, it is concluded that the mass of the bullet and the shooting angles are key parameters influencing the net deploying area. The shooting angle influences the travelling distance dramatically. The effective period is primarily influenced by shooting velocity. To clarify the sensitivity of output parameters on different input parameters, Table 3 is made to show the 13
1.6 1.4
100
Deployment Time [s]
Maximum Area [m2]
105
95 90 Honeywell Spectro Kevlar Zylon Carbon Fiber
85 80
1.0 0.8
Honeywell Spectro Kevlar Zylon Carbon Fiber
0.6 0.4 0.2
75
0.0 35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
35
12
40
45
0.6 Honeywell Spectro Kevlar Zylon Carbon Fiber
10 8 6
Effective Period [s]
Travelling Distance [m]
1.2
4 2 0
50 55 60 65 Shooting Angles [deg]
70
75
Honeywell Spectro Kevlar Zylon Carbon Fiber
0.5 0.4 0.3 0.2 0.1 0
35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
Figure 8: Dynamic parameters changing with different shooting angles
105
95 90 Honeywell Spectro Kevlar Zylon Carbon Fiber
85 80 75
Deployment Time [s]
Maximum Area [m2]
100
70 0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
2.2
5.3
0.35
5.2
0.30
5.1
0.25
5.0 4.9 Honeywell Spectro Kevlar Zylon Carbon Fiber
4.8 4.7 4.6
Honeywell Spectro Kevlar Zylon Carbon Fiber
0.4
Effective Period [s]
Travelling Distance [m]
0.4
1.02 1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
2.2
0.20 Honeywell Spectro Kevlar Zylon Carbon Fiber
0.15 0.10 0.05 0.00
4.5 0.4
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
0.4
2.2
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
Figure 9: Dynamic parameters changing with different bullet masses
14
1.8
2.0
2.2
Table 3: Output patameters’ dependency on initial input parameters. A ’-’ denotes an inverse dependency
Initial bullet velocity (180% 10) Shooting angles (85.71% 35) Bullet mass (450% 0.4)
Maximum Area
Deployment Travelling Time Distance
Effective Period
1.28%
-35.61%
0.31%
-35.01%
8.30%
-37.13%
-71.58%
-30.82%
3.93%
-1.75%
0.74%
11.34%
dependency quantitatively. The dependency on input parameters Di j can be described as δOi Di j = (16) δIj where δOi and δIj are the relative deviation of the i-th output parameter and j-th input parameter, respectively. They can be expressed as δOi =
250
255
260
265
ΔOi ΔIj , δIj = O1 I1
(17)
where ΔOi and ΔIj are the absolute deviation of the i-th output parameter and j-th input parameter, respectively. From Eq. 17, it is known that output parameters dependency also relies on the starting value of the input parameters. Table 3 displays the dependency in percentage (take Kevlar as an example), in which the numbers in the brackets show the relative deviation and starting value of each input parameter. 5.2. Comparison and Analysis of Mass-Spring Model and ANCF Model As stated above, a mass-spring model has been the most commonly used method to model a space tethered-net so far. Another modeling method based on ANCF was developed and applied as well in this paper to compare with the mass-spring model. A set of simulations have been performed with the same initial conditions. In the simulations of ANCF model, the folding scheme of the net is used as shown in Fig. 4. Kevlar is chosen as the net material in this comparison. Fig. 10 and Fig. 11 show the net deployment simulation by mass-spring model and ANCF model, respectively. As shown in Fig. 12 to Fig. 14, we also investigated the dependency on the four critical parameters by the initial deployment conditions using both massspring model and ANCF model. It is seen that the results obtained from the mass-spring model and ANCF model are very close to each other, which indicates that ANCF modeling method is also suitable to model a net. Moreover, from Fig. 10 and Fig. 11, we notice that the net configurations from these two 15
t = 0.1 s
t = 0.2 s
t = 0.4 s
t = 0.6 s
t = 0.8 s
t=1s
Figure 10: Net deployment simulation by Mass-spring model
16
t = 0.1 s
t = 0.2 s
t = 0.6 s
t = 0.4 s
t = 0.8 s
t=1s
Figure 11: Net deployment simulation by ANCF model
17
Table 4: Comparison of single step computational time of ANCF and Mass-spring model
Mass-spring Model ANCF Model
100 98 96 94 92 90 88 86 84 82 80
Single Step Computational Time 7.883ms 0.114s
Deployment Time [s]
1.2
MS_Kevlar ANCF_Kevlar
MS_Kevlar ANCF_Kevlar
1.0 0.8 0.6 0.4 0.2 0.0
10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
10
5.7
0.35
5.5
0.30 Effective Period [s]
280
Maximum Area [m2]
275
Degrees of Freedom 363 5280
models at a certain moment have a good agreement with each other. However, with the mass-spring model, the massless spring-damper elements always keep being straight in any configuration. It requires more nodes to achieve the description of the same flexibility as ANCF model. In another word, in ANCF model, it is more capable to describe the flexible configuration of the net with fewer nodes than the mass-spring model. The cables between two nodes are not approximated as straight lines, and the flexibility of the cable is properly described, which makes ANCF more powerful to reflect the nature of the net. However, our conclusion is not that ANCF model is better than the conventional mass-spring model in any case. Actually ANCF model is much more computationally expensive compared to the conventional mass-spring model because of the higher degrees of freedom and the evaluations of the elastic forces. Table 4 shows the comparison of the computational time in a single step of both modeling methods.
Travelling Distance [m]
270
Number of Nodes 121 121
5.3 5.1 4.9 MS_Kevlar ANCF_Kevlar
4.7
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
MS_Kevlar ANCF_Kevlar
0.25 0.20 0.15 0.10 0.05
4.5
0.00 10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
10
12
14
16 18 20 22 24 Initial Bullet Velocity [m/s]
26
28
Figure 12: Comparison of ANCF and Mass-spring model with different initial bullet velocities
The difference between these two methods is shown quantitatively in Table
18
Deployment Time [s]
Maximum Area [m2]
100 98 96 94 92 90 88 86 84 82 80
MS_Kevlar ANCF_Kevlar 35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
MS_Kevlar ANCF_Kevlar 35
12
40
45
50 55 60 65 Shooting Angles [deg]
0.45 MS_Kevlar ANCF_Kevlar
10
Effective Period [s]
Travelling Distance [m]
1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0
8 6 4 2 0
70
75
MS_Kevlar ANCF_Kevlar
0.40 0.35 0.30 0.25 0.20 0.15
35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
35
40
45
50 55 60 65 Shooting Angles [deg]
70
75
Figure 13: Comparison of ANCF and Mass-spring model with different shooting angles
1.09 Deployment Time [s]
Maximum Area [m2]
100 95 90 85 MS_Kevlar 80
ANCF_Kevlar
75
0.99 0.94 0.89 0.84
0.4
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
5.5 5.4 5.3 5.2 5.1 5.0 4.9 4.8 4.7 4.6 4.5
1.8
2.0
2.2
0.4
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
2.2
0.35 Effective Period [s]
Travelling Distance [m]
MS_Kevlar ANCF_Kevlar
1.04
MS_Kevlar ANCF_Kevlar
0.30 0.25 0.20 0.15 0.10
MS_Kevlar ANCF_Kevlar
0.05 0.00
0.4
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
2.2
0.4
0.6
0.8
1.0 1.2 1.4 1.6 Bullet Mass [kg]
1.8
2.0
2.2
Figure 14: Comparison of ANCF and Mass-spring model with different bullet masses
19
Table 5: Maximum difference between ANCF and Mass-spring model
Initial bullet velocity Shooting angles Bullet mass
Maximum Area 4.29% 1.71% 3.90%
Deployment Time 4.68% 1.11% 5.95%
Travelling Distance 7.22% 6.72% 6.27%
Effective Period 10.32% 15.01% 32.61%
5 where the maximum difference was obtained using: Δ=
285
max|xM − xA | . xM
(18)
Here xM is the data from the mass-spring model and xA is the data from ANCF model. From the table, it is seen that the maximum differences between these two modeling methods are below 10% except for the effective period due to its sensitivity itself. 6. Conclusion
290
295
300
305
310
In this paper, the dynamic characteristics of a tethered-net in its deployment phase has been investigated. Four critical parameters describing the deployment dynamic characteristics of tethered-net, namely maximum area, deployment time, travelling distance and effective period have been identified and defined. These parameters have been investigated by simulations under different initial input parameters, namely initial bullet velocity, shooting angles and bullet mass. The physical properties of the net have been addressed, and a new folding pattern is provided for simulations. It is concluded that a higher bullet mass, shooting angles and more flexible net material contribute to a larger net area. Net material has little influence on deployment time and travelling distance. However, a higher bullet mass consumes more energy and involves a more complex system, such as the shooting mechanism. Moreover, a higher initial bullet velocity and a larger shooting angle lead to a shorter effective period which leads the capturing a higher risk to fail and less reliable. The dependency of the output parameters on different initial input parameters has been provided quantitatively. Besides, a comprehensive study on modeling of a tethered-net based on ANCF has been provided. Rigid-flexible coupled system of the tetherednet has been established. Simulations based on ANCF have been performed and compared with the conventional mass-spring model. The results from both methods show a good agreement on the changing of four critical parameters, which indicates the ANCF model is very suitable to describe the dynamics of a tethered-net. The maximum differences between two methods, which are below 10% except for the sensitive effective period, have been calculated showing the agreement between these two modeling methods. Furthermore, ANCF model is 20
315
more capable of describing the flexibility of the net with fewer nodes than the conventional mass-spring model. However, it is more computationally expensive. Based on the comparison of single-step computational time, Mass spring model is nearly 15 times faster than ANCF model. References
320
[1] M. Shan, J. Guo, E. Gill, Review and comparison of active space debris capturing and removal methods, Progress in Aerospace Sciences 80 (2016) 18–32. [2] D. Reintsema, J. Thaeter, A. Rathke, W. Naumann, P. Rank, J. Sommer, Deos–the german robotics approach to secure and de-orbit malfunctioned satellites from low earth orbits, in: Proceedings of the i-SAIRAS, Sapporo, Japan, (2010), pp. 244–251.
325
330
[3] P. Huang, D. Wang, Z. Meng, F. Zhang, J. Guo, Adaptive postcapture backstepping control for tumbling tethered space robot–target combination, Journal of Guidance, Control, and Dynamics 39 (1) (2016) 150–156. [4] P. Huang, D. Wang, Z. Meng, F. Zhang, Z. Liu, Impact dynamic modeling and adaptive target capturing control for tethered space robots with uncertainties, IEEE/ASME Transactions on Mechatronics 21 (5) (2016) 2260–2271. [5] P. Huang, M. Wang, Z. Meng, F. Zhang, Z. Liu, Attitude takeover control for post-capture of target spacecraft using space robot, Aerospace Science and Technology 51 (2016) 171–180.
335
340
[6] J. Reed, J. Busquets, C. White, Grappling system for capturing heavy space debris, in: 2nd European Workshop on Active Debris Removal,Paris, France, (2012). [7] P. Huang, Z. Hu, F. Zhang, Dynamic modelling and coordinated controller designing for the manoeuvrable tether-net space robot system, Multibody System Dynamics 36 (2) (2016) 115–141. [8] B. Bischof, L. Kerstein, J. Starke, H. Guenther, W. Foth, et al., Rogerrobotic geostationary orbit restorer, Science and Technology Series 109 (2004) 183–193.
345
350
[9] C. Billot, S. Ferraris, R. Rembala, F. Cacciatore, A. Tomassini, R. Biesbroek, e.deorbit: Feasibility study for an active debris removal, in: 3rd European Workshop on Space Debris Modeling and Remediation, Paris, France, (2014). [10] N. Zinner, A. Williamson, K. Brenner, J. B. Curran, A. Isaak, M. Knoch, A. Leppek, J. Lestishen, Junk hunter: Autonomous rendezvous, capture, and de-orbit of orbital debris, in: Proceedings of AIAA SPACE 2011 Conference & Exposition, Long Beach, CA, USA, (2001). 21
[11] M. Lavagna, R. Armellin, A. Bombelli, R. Benvenuto, Debris removal mechanism based on tethered nets, in: Proceedings of the i-SAIRAS, Turin, Italy, (2012). 355
360
[12] E. M. Botta, I. Sharf, A. K. Misra, M. Teichmann, On the simulation of tether-nets for space debris capture with vortex dynamics, Acta Astronautica 123 (2016) 91–102. [13] R. Benvenuto, M. Lavagna, S. Salvi, Multibody dynamics driving gnc and system design in tethered nets for active debris removal, Advances in Space Research 58 (1) (2016) 45–63. [14] A. Lorenzo, R. Stefanescu, R. Benvenuto, M. Marcon, M. Lavagna, Validation results of satellite mock-up capturing experiment using nets, in: 66th International Astronautical Congress, Jerusalem, Israel, (2015).
365
[15] A. A. Shabana, Flexible multibody dynamics: review of past and recent developments, Multibody system dynamics 1 (2) (1997) 189–222. [16] L. Liu, J. Shan, Y. Ren, Z. Zhou, Deployment dynamics of throw-net for active debris removal, in: 65th International Astronautical Congress, Toronto, Canada, (2014).
370
[17] M. Berzeri, A. Shabana, Development of simple models for the elastic forces in the absolute nodal co-ordinate formulation, Journal of Sound and Vibration 235 (4) (2000) 539–565. [18] J. Gerstmayr, A. A. Shabana, Analysis of thin beams and cables using the absolute nodal co-ordinate formulation, Nonlinear Dynamics 45 (1-2) (2006) 109–130.
375
[19] R. Benvenuto, R. Carta, Active debris removal system based on tetherednets: experimental results, in: Proceedings of the 9th PEGASUS-AIAA Student Conf., Milano, Italy, (2013). [20] G. Tibert, M. Gardsback, Space webs final report, in: ESA/ACT, AdrianaID: 05, Noordwijk, The Netherlands, (2006).
380
[21] K. Miura, Method of packaging and deployment of large membranes in space, The Institute of Space and Astronautical Science report 618 (1985) 1–9. [22] H. Furuya, Y. Inoue, T. Masuoka, Deployment characteristics of rotationally skew fold membrane for spinning solar sail, AIAA 2045 (2005) 1–10.
385
[23] M. G¨ardsback, G. Tibert, Deployment control of spinning space webs, Journal of guidance, control, and dynamics 32 (1) (2009) 40–50. [24] Y. Yu, H. Baoyin, J. Li, Modeling and simulation of projecting deployment dynamics of space webs, Journal of Astronautics 5 (2010) 1–8. 22