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An analysis of the flexibility modeling of a net for space debris removal
Journal Pre-proofs An analysis of the flexibility modeling of a net for space debris removal Minghe Shan, Jian Guo, Eberhard Gill PII: DOI: Reference:
S0273-1177(19)30791-4 https://doi.org/10.1016/j.asr.2019.10.041 JASR 14524
To appear in:
Advances in Space Research
Received Date: Accepted Date:
31 July 2019 31 October 2019
Please cite this article as: Shan, M., Guo, J., Gill, E., An analysis of the flexibility modeling of a net for space debris removal, Advances in Space Research (2019), doi: https://doi.org/10.1016/j.asr.2019.10.041
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An analysis of the exibility modeling of a net for space debris removal Minghe Shana,∗, Jian Guob , Eberhard Gillb a Department b Faculty
of Aerospace Engineering, University of Maryland, College Park, 20740, The United States
of Aerospace Engineering, Delft University of Technology, Delft, 2629 HS, The Netherlands
Abstract
Operational spacecraft are facing a risk of collision with space debris objects. The net capturing method has been proposed to mitigate this risk on spacecraft. The mass-spring model is usually applied for net modeling by discretizing a cable into one or several mass-spring-damper elements in simulation. The absolute nodal coordinates formulation (ANCF) has also been applied to model the net, and this model is able to describe the exibility of a net using less elements. However, the in uence on the net behavior in simulation by the exibility modeling of a net is not well understood and barely discussed. In this paper, exibility models of a net are established based on the mass-spring model and the ANCF model,respectively. The in uence on the net behavior by the exibility modeling is, for the rst time, analyzed via simulations. Two case studies of capturing a ball and a cube shaped targets are performed. It is found that the exibility modeling has little in uence on the net dynamics in simulation. Finally, the characteristics and bene ts of the ANCF model are described and analyzed. A drawback of the ANCF model was found to be its inferior computational performance. Keywords:
space debris, net capturing, ANCF, exibility modeling
∗ Corresponding
author. Email: [email protected]
Preprint submitted to Advances in Space Research
November 6, 2019
1. Introduction
According to the Space Debris O ce (SDO) of the European Space Agency (ESA), space debris is de ned as "all non-functional, human-made objects, including fragments and elements thereof, in Earth orbit or re-entering into Earth's atmosphere". The Earth orbit is in a serious predicament caused by millions pieces of space debris. According to the United States Space Surveillance Network (US-SSN), more than 19 000 objects larger than 10 cm in diameter have been catalogued by radar and/or other techniques as of May 2019 (NASA Orbital Debris Program O ce, May 2019). Figure 1 shows the increasing of space debris objects in Earth orbit o cially cataloged by the US-SSN. 20000
https://orbitaldebris.jsc.nasa.gov
1
Number of Objects
15000
10000
2 1 Total Objects 2 Fragmentation Debris
5000
3
4 Mission-related Debris
4
5 Rocket Bodies 2017
1997
1987
1977
1967
1957
2007
5
0
3 Spacecraft
Year
Figure 1: Number of objects in Earth orbit by object type (NASA Orbital Debris Program Of ce, May 2019)
As of Sep., 2019, about 2400 payloads on orbit are active. Comparing it to the total number of objects on orbit, which is about 19 000, it indicates that around 12% of objects are operating in an environment where 88% of the objects are space debris. The collision of Cosmos 2251 and Iridium 33 in 2009 has highlighted the threat by space debris objects, since it signaled a trend that the future space environment will be dominated by fragmentation debris generated via similar collisions, instead of explosions (Liou, 2011). Even more serious, space debris is a threat to the lives of astronauts. In March 2009, a ve inches 2
space debris object passed particularly close to the International Space Station (ISS). Fortunately, the alarm was cleared 10 minutes later (Website, 2019). To mitigate the risk of collision and stabilize the space environment, active debris removal (ADR) is of great relevance. Net capturing method has been proposed to mitigate the collision risk on satellites (Bischof et al., 2004). Extensive researches on net capturing method have been performed, including the sensitivity analysis (Shan et al., 2017a), simulators design (Shan et al., 2017a; Botta et al., 2016b; Benvenuto et al., 2015; Goª¦biowski et al., 2016) and parabolic ight experiments (Goª¦biowski et al., 2016; Medina et al., 2017; Shan et al., 2017b) etc. Concretely, Shan et al. have analysed the input parameters, such as the bullet mass, the shooting velocity and the shooting angle's in uence on the output parameters, such as the net opening area, the deployment time and the travelling distance during the net deployment phase (Shan et al., 2017a). Botta et al. have built a net model in a commercial software, Vortex, and investigated the capturing dynamics (Botta et al., 2016a). Benvenuto et al. have developed a simulator based on a mass-spring model, compared the planar and conical net deployment process, and carried out an analysis on the interface between a net and a target during de-orbiting (Benvenuto et al., 2015, 2016). Zhang and Huang have proposed a novel space robot system called the maneuverable tethered space net robot with four microsatellites installed at four corners of the net (Zhang and Huang, 2016; Zhang et al., 2017; Huang et al., 2015). Si et al. have proposed a line-line self-collision detection algorithm to accurately simulate the capture process (Si et al., 2019). Goª¦biowski et al. have developed a simulator based on the Cossret rod theory and validated the model with a parabolic ight experiment (Goª¦biowski et al., 2016). Another parabolic ight experiment has been performed by shooting a net on a xed Envisat mock-up in the cabin, by which the mass-spring model has been veri ed and validated (Medina et al., 2017). The net modelling in most of the existing researches is based on the massspring (MS) model. However, the in uence on the net behavior in simulation by the exibility modeling of a net is not well understood and barely discussed. 3
Obviously, the exibility between two nodes can be described by discretising the cable into more segments. However, this dramatically increases the degrees of freedom and the complexity of the system equations. In addition, the mass-spring model has a limitation in describing the contact between a net and a target because that the ctitious penetration of the massless spring into the target cannot be avoided. The Aboslute Nodal Coordinates Formulation (ANCF), which is initially proposed by Shabana, can be utilized in solving large displacement and large deformation problems (Shabana, 2013). In ANCF, absolute positions and the gradients of the positions are acting as the element nodal coordinates to describe the con guration of a exible system. The nodal coordinates of the elements are de ned 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 (named 'global slopes') instead of in nitesimal or nite 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 (Shabana, 2013). In a net system for space debris removal, an ANCF cable element can be applied to build the net. Only the deformation along the longitudinal direction is necessary to be considered based on the nature of a cable. 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 ignores the deformations along the lateral and transverse directions which accurately describes the nature of a cable (Gerstmayr and Shabana, 2006). For the contact problem between cable and other rigid object based on ANCF, Takehara et al. have developed a numerical model that can describe the behaviour of the contact between a wire rope and a pulley (Takehara et al., 2016). Wang et al. have developed a new approach to simulate the frictional contact dynamics of cable elements based on penalty method (Wang et al., 2014). Yu et al. have proposed a method to represent contact zones of ANCF elements by a set of basic geometries, like sphere, brick cylinder and triangular patch (Yu et al., 2010). Khude et al. have proposed a 4
method to model the frictional contact between exible cable elements and rigid body by combining the Discrete Element Method (DEM) with ANCF (Khude et al., 2011). This paper rst presents the modeling of exibility of a net based on both the mass-spring model and the ANCF model. In the mass-spring model, the exibility of a cable is described by discretising the cable into several segments. In the ANCF model, a cable element is discretised into multiple contacting spheres in contact detection process. The segments between contacting spheres are not detected, thus penetration between the segment with the target might happen. Therefore, the more contacting spheres are discretised in one cable element, the more accurate of the model is, and undoubtedly more computational time will be spent. The nonlinear damping model is applied to calculate the contact force on each contacting sphere once the contact on them are detected. Frictional force on contacting spheres are derived based on the Hollars (Brown and McPhee, 2016). The in uence on the net behavior by the exibility modeling is, for the rst time, analyzed via simulations. Two case studies of capturing a ball- and a cube-shaped targets are performed. Moreover, the characteristics and bene ts of ANCF are described and analysed. A drawback of the ANCF was found to be its inferior computational performance.
2. Modeling of Flexibility
2.1. Based on the mass-spring model
The mass-spring model is usually applied for net modeling by discretizing cables into mass-spring-damper elements as shown in Fig. 2. Four mass points at four corners of the net are called 'bullets', which is shot by a net gun and helps deploy the net. Based on the nature of a cable, it is not able to withstand compression. Therefore, a special spring between two mass points which can only be tensioned is de ned. References (Shan et al., 2017a) and (Shan et al., 2017b) have introduced the derivation of the model of the net.
5
Bullet 1
Bullet 3
One cable element, no additional mass points, one segment
{
Mass Point
Spring Damping Element Bullet 4
Bullet 2
Figure 2: mass-spring model
However, the model in these references does not consider the exibility of the cable, neither of the in uence on the net behavior by the exibility. In other words, there is only one spring-damper element between two mass points, and the shape of this element is always being straight. Obviously, the exibility of a cable can be described by discretising the cable into several segments as shown in Fig. 3. Additional mass points should be added between two mass points in order to describe the exibility. However, the number of the added mass points will dramatically increase the degrees of freedom and the complexity of the system. Moreover, since the stiness of the cable segment ki can be expressed as
ki =
EA ∆l
(1)
where E is the elastic Young's modulus of the cable, A is the area of the cross section of the cable, and ∆l is the length of the cable segment. The more mass points are added, the shorter the cable segment is, thus the larger the stiness is. To compute system equations with larger stiness, the smaller time step in the simulation is required which will also reduce the computing e ciency. On
6
the contrary, adding more mass points can better describe the contact between a net and a target. This is because only the contact between the mass points and the target is detected in the simulation, thus the ctitious penetration of the spring-damping elements into the target's body always happens. Adding more mass points can prevent this phenomena to some extend and make the simulation results more reliable and accurate. Bullet 1
Bullet 3
{
One cable element, two additional mass points, three segments
Bullet 4
Bullet 2
Figure 3: Flexibility modeling based on the mass-spring model.
2.2. Based on the ANCF model
The ANCF model has also been applied to model the net, and this model is able to describe the exibility of the net with less elements. The ANCF method is initially proposed by Shabana, and it is specialized to be utilized in solving large displacement and large deformation problems (Shabana, 2013). In ANCF, absolute positions and the gradients of the positions are acting as the element nodal coordinates to describe the con guration of a exible system. For the net which is weaved by cables (as shown in Fig. 4), one cable element contains two 7
nodes at its ends. The arbitrary position on the deformed cable element can be expressed using the shape function S and coordinates of these two nodes as, (2)
r = Se = [S1 I, S2 I, S3 I, S4 I][e1 , e2 ]T
where 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
(3)
where ri is the global displacement and rix is the global slope of the element. The shape function Si is de ned 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 ) (4) where ξ = x/l0 , and x is the coordinate of an arbitrary point on the element.
x 0 Original configuration
Cable
elem ent
rix Node i
Node j rjx
r ri
rj
Z X
Y o
Bullet
Deformed configuration
Figure 4: Flexibility modeling based on the ANCF model.
Using the principle of virtual power and introducing Lagrange multipliers (De Jalon and Bayo, 2012), one can derive the system equations of motion for
8
the tethered-net as
Mb
0 Φb
0 Me Φe
ΦT b
q¨b
Qb
= , ΦT e ¨ Q e e 0 λ Q
(5)
where Mb and Qb are the mass matrix of the bullets and the external forces acting on the bullets, 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, λ represents the Lagrange multipliers; Φe is the Jacobian matrix of the constraint equations associated with the absolute nodal coordinates; Φb represents the constraints coupled by the bullet masses and the cable elements; Q is quadratic velocity vectors derived by dierentiating the constraint equations twice with respect to time. The detailed derivation of the system equations of motion is given in (Shan et al., 2017a). In the ANCF model, the penetration between the cable elements and the target's body can be avoided by distributing ctitious contacting spheres on the cable element. Contacting spheres on one ANCF cable element can dier based on the requirements of accuracy and computational performance. In other words, the contacting spheres can be several spherical points (Fig. 5 (a), (b)) distributed along the centreline of a cable element, or even a chain of spheres (Fig. 5 (c)). The segment between contacting spheres are not detected in a contact detection algorithm. Thus, penetration between the segment with the target might happen in that region. With increasing number of spheres, the computational time to evaluate the contact detection and the contact force will increase. On the other hand, the accuracy of the net motion can be enhanced and the penetration can be avoided.
3. Contact Dynamics of a Flexible Net with a Target
Since a net is assumed to be multiple mass points connected with springdamper elements in a certain pattern, net contacting with a target can be con9
(a)
Contacting point
Cable element
(b)
(c)
Figure 5: Contacting spheres on one cable element.
sidered as multiple mass points contacting with the surfaces of the target as shown in Fig. 6. The mass points have to move outside the space of the target and they are not allowed to penetrate the target. According to this description, net capturing can be considered as an inequality constraint problem. The
Figure 6: Net contacting with a target (left) and simpli ed contact scenario (right).
penalty-based method, which is usually used in the optimization problem, is a very attractive method to solve the inequality constraint problem. In the optimization problem, an inequality constraint problem can be formulated as Minimize f (x),
subjext to c(x) ≥ 0
(6)
where f (x) is the objective function, c(x) is the constraint equation. It is possible to convert this inequality constraint problem into a suitable unconstraint
10
problem as Minimize f (x) + µmin{0, c(x)},
subjext to x ∈ Rn
(7)
The term "µmin{0, c(x)}" is called the penalty function. If c(x) ≥ 0, then min{0, c(x)} = 0 and no penalty occurs. If by any chance, c(x) < 0, then min{0, c(x)} = c(x). The penalty should be added into the objective function to prevent this from happening. Taking the point-surface contact problem as an example, the point is constrained to be always moving above the surface, which is c(x) ≥ 0. If the constraint equation holds, which means the point is not contacting with the surface, thus min{0, c(x)} = 0. On the contrary, if it is detected that c(x) < 0, which means the point is penetrating the surface. The penalty force, here the contact force, µc(x) will be added to f (x) to push the point away from the surface and prevent the penetration from continuing. According to this de nition, the penalty function, "µmin{0, c(x)}", can be considered as a function of the penetration depth. In fact, a contact between the elastic bodies will lead to a deformation on colliding bodies, and the contact force can be expressed as a function of the contact deformation. In the simulation of a contact scenario, the contact deformation can be parameterized as a function of the penetration depth between the colliding bodies. In the penalty-based method, the response force depends on the penetration depth: the larger the penetration, the higher the penalty. 3.1. Normal Contact Force
According to the review of the contact dynamics modeling by (Gilardi and Sharf, 2002), two main categorises of contact dynamics models can be distinguished: discrete models and continuous models. Discrete models assume that the impact process is instantaneous. Therefore they are not applicable to the penalty-based method. Commonly used continuous contact dynamics models are: spring-dashpot model, Hertz's model and non-linear damping model. Those models account for the normal force based on the deformation. The direction of the force is along the normal line of the contact. Table 1 lists the expressions 11
of the normal contact forces in those models. In the expressions, k represents the contact stiness, d represents the damping coe cient, δ represents the deformation or the penetration in the simulation, and n represents a coe cient depending on the materials and geometries of the contacting bodies. Table 1: Comparison of Three Contact Dynamic Models
Contact Models
Expressions
Spring-dashpot model
fn = kδ + dδ˙
Hertz's model
fn = kδ n
Nonlinear damping model
fn = kδ n + dδ˙ n δ
0
0
0
(a)
(b)
(c)
Figure 7: Continuous Contact dynamics models: (a) Spring-dashpot model. (b) Hertz's model. (c) Non-linear damping model.
Figure 7 shows contact force history of those models. The spring-dashpot model assumes that a spring-dashpot element is set between the colliding bodies. However, this spring can only be compressed, i.e., it only pushes bodies away from each other rather than pull them together. The linear and explicit normal force fn depends on the penetration depth δ and the penetration rate
δ˙ . However, due to the damping term, the spring-dashpot model involves a discontinuity at the beginning of the contact as shown in Fig. 7 (a). Similarly, the contact force holds the colliding bodies together by the end of a contact, which is not realistic. The Hertz's model assumes the contacting bodies interacting 12
via a non-linear spring. Such a model has overcome the defect of the drift at the zero penetration. The normal force is continuous at the beginning and the end of a contact (Fig. 7 (b)). However, it does not take the damping into consideration, such that the energy dissipation is not accounted for. The non-linear damping model combines the advantages of the two aforementioned models. It is taking the damping into account and remaining the continuity at zero deformation (Fig. 7 (c)). Therefore, the non-linear damping model is selected as the contact model in the penalty-based method throughout this paper. While we have selected the contact model for the net capturing, it is necessary to calculate the coe cients, such as the stiness k and the damping coe cient d, in the model. As previously mentioned, the net is modeled as a series of connected mass points. Because of the lightness of the net and the thinness of the cable, the value of the radii of the mass points are assumed to be the radius of the cable, which are negligible compared to the surface of a target. Therefore, the contact of a mass point with the surface of a target can be treated as a sphere-plane contact. According to the Hertzian contact theory (Johnson and Johnson, 1987), for a sphere-plane contact, the contact stiness can be obtained by
√ 4 r k= , 3π(h1 + h2 )
with
hi =
1 − νi2 , πEi
(i = 1, 2),
(8)
(9)
where Ei and νi are Young's modulus and Poisson's ratio of the material of the contacting bodies, respectively, and r is the radius of the contacting mass point. Here, i = 1 indicates the parameters of the chaser material and i = 2 indicates the parameters of the target material. For the damping coe cient d, Hunt et al. (Hunt and Crossley, 1975) have established
d=
3αk , 2
(10)
where α is an experimental parameter that varies in the range of 0.08-0.32 s/m according to (Benvenuto et al., 2016). 13
3.2. Friction
There are several models to describe the friction of two contacting bodies. The Coulomb friction model is the most commonly used model for friction. It has a linear relation with the normal force by a frictional coe cient. However, this model neglects the static friction, which describes the friction at or near zero relative velocity. It also neglects the Stribeck eect, which describes the friction force reduction with increasing relative velocity of contacting surfaces. Andersson et al., Specker et al. and Brown et al. have developed separate friction models to describe the static friction and the Stribeck eect (Andersson et al., 2007; Specker et al., 2014; Brown and McPhee, 2016). Among the existing models, Hollars friction model is the most commonly used one due to its advantage of the physical meaning at the transition velocity vt0 (Brown and McPhee, 2016). The friction force reaches its maximum at vt0 and decreases with the increasing of the relative velocity. Figure 8 shows the comparison of the Coulomb friction model and Hollars friction model, from which it is seen that the Coulomb model is a simpli ed version of the Hollars model. The expression of the friction force in Hollars model is,
ft = [min{
vt 2(µs − µk ) , 1}(µk + )]fn vt0 1 + ( vvt0t )2
(11)
where vt is the relative velocity of colliding bodies, vt0 is the transition velocity,
fn is the magnitude of normal force, and µs , µk are the coe cients of static and kinetic friction, respectively.
Stribeck effect
Static friction
0
0 (a)
(b)
Figure 8: Friction models: (a) Coulomb friction model. (b) Hollars friction model.
14
The frictional force vector generated from the contact is always opposite to the relative velocity along the tangential direction of the contact surface. Assume a mass point pi is in contact with an object on point O and the normal vector of point O on the object is n (see Fig. 9). Then the relative velocity vr of the points pi and O can be expressed as
vr = vpi − vO .
(12)
fn n
pi ft
vt
O vNr vr
Figure 9: Normal and frictional force on contacting point.
The component of the relative velocity along the normal vN r is
vN r = (nT vr )n.
(13)
Therefore, the relative velocity vector along the tangential direction vt is obtained as
vt = vr − vN r .
(14)
The friction force ft can therefore be obtained by Eq. 11,
ft = −ft nt ,
(15)
with nt = vt /vt . The net contact dynamics have been introduced in this section. The contact model introduced above can be used in both the mass-spring model and the ANCF model. The only thing needs to pay attention is, for the contacting spheres distributed on cable elements in the ANCF model, the contact force fc 15
needs to be converted to the generalized force Qc to t the system equations based on the ANCF model. Using the principle of virtual work, the generalized contact force distributed on cable element is
Qc = S(ξi )fc .
(16)
with S being the shape function matrix of the cable element, ξi the element coordinates of the ith contacting sphere on the same cable element. Finally, implement fc and Qc into Qb and Qe , respectively in Eq. 5 to solve the system equations therefore to achieve the motion of the net with contacts.
4. Numerical Simulations and Discussions
The mass-spring model is the most commonly used modeling method for a tethered-net. However, to fully describe the exibility 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. In the classical nite element method, in nitesimal or nite 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 exible system with large displacement and large deformation. Gerstmayr and Shabana have developed a low order cable element using ANCF, in which it has half degrees of freedom of a fully parameterized beam element, and it does not consider the deformations along the lateral and transverse directions which more accurately describes the nature of a cable (Gerstmayr and Shabana, 2006). Even though the exibility modeling can be described based on the above mentioned models, the in uence on the net dynamics in simulation by the exibility modeling of a net is not well understood so far. For example, to express the exibility of a cable using the mass-spring model, the number of mass points added on one cable element can be 2, 3 or even more (Fig. 3). The way the net dynamics in uenced by the number of those added mass points is unclear. In 16
this section, the in uence on the net dynamics in simulation by the exibility modeling is investigated via simulations of capturing a ball and a cube shaped targets using the mass-spring model and the ANCF model, respectively. The con gurations of the net and the targets along with other simulation parameters are summarized in Table. 2. To investigate the in uence on the net dynamics in simulation by the exibility modeling, here we compared three groups of simulation for both ball and cube capturing, which are simulations based on the mass-spring model without added middle mass points, the massspring model with three middles mass points on one cable element, and the ANCF model with three virtual contacting spheres on one cable element. The process of capturing a space debris object using a net includes three steps, namely, net deployment, target capturing and de-orbiting. During the net deployment, four bullets are shot by a net gun to help expand the net. The simulations are performed in the reference frame shown in Fig. 10. The reference frame is centred at the centre of mass of the net container. The axis is de ned as: x direction is along the traveling direction; y axis is orthogonal to x and pointing to the left side of the target, z axis follows the right-hand rule and orthogonal to x and y axis. The target is captured by a 6 by 6 mesh squared net. The net is folded in a certain pattern introduced in (Shan et al., 2017a) and it is fully deployed around 0.7 s, then contact with the target. Four bullets move towards each other and pass by each other which makes the net embracing and wrapping the target. The mass and the moment of inertia of the target in this study is assumed to be in nite. In other words, the target is immovable during the net capturing. However, the contact force acting on the target is so small due to the lightness and low shooting velocity of the net that the target can hardly move even with a smaller mass during capturing. Figure 10 - 15 show the screenshots of those simulations. Figure 10 and 13 show the simulation of the ball and the cube capturing based on the mass-spring without any added middle mass points on one cable element. Figure 11 and 14 show the simulation of the ball or the cube capturing based on the mass-spring with three middle mass points on one cable element. Figure 12 and 15 show the simulation 17
Table 2: Simulation Parameters
Parameter Net
Value
Net size A, [m2 ] Mesh square −, [−]
6×6
Mesh length l0 , [m]
0.08
Bullet mass mb , [kg]
0.05×4
Shooting velocity v , [m/s]
1.5
Shooting angle θ, [◦ ]
25
Distance to the target dt , [m] Young's modulus E , [Pa]
Targets
0.48×0.48
1 4.456e8
Poisson's ratio v , [-]
0.3
Cube length a, [m]
0.12
Ball radius r, [m]
0.08
of the ball or the cube capturing based on the ANCF model with three virtual contacting spheres on one cable element. It can be clearly seen in the above gures that the penetration occurs when not adding any middle mass points, while adding three middle points on one cable element can eectively avoid the penetration, so as the ANCF model with three virtual contacting spheres on one cable element. With these results, it would be interesting to compare those simulation results and analyze the in uence of the exibility modeling quantitatively. We compare the trajectory of one of the bullets due to the symmetric con guration of the net. In this case, we choose the bullet one, which is located at the left up corner of the net. Figure 16 and 18 show the bullet displacements along three axes, from which we notice that the bullet displacements under three cases (without middle points using the mass-spring model, with three middle points using the massspring model and with three contacting spheres using the ANCF model) are actually quite close to each other. The maximum dierence is 6 cm, and the relative dierence comparing to the traveling distance of the net is only 4%, which indicates that the exibility modeling has trivial in uence on the net
18
z y
x
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 10: Ball capturing based on the mass-spring model. (without middle points)
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 11: Ball capturing based on the mass-spring model. (with 3 middle points)
19
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 12: Ball capturing based on the ANCF model.
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 13: Cube capturing based on the mass-spring model. (without middle points)
20
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 14: Cube capturing based on the mass-spring model. (with 3 middle points)
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 15: Cube capturing based on the ANCF model.
21
dynamics during capturing in simulation. It is also noticed that in Fig. 11 there is a dierence at the starting point of the displacement in x direction. This is because the initial con gurations of ANCF model and MS model are set dierently (Shan et al., 2017a). Figure 17 and 19 show the bullet velocity along three axes. There are bigger diverse at the velocity level, especially along the traveling direction.
Displacement [m]
1.5
1
x (w.o.m) y (w.o.m) z (w.o.m) x (w.m) y (w.m) z (w.m) x (ANCF) y (ANCF) z (ANCF)
0.5
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time [s] Figure 16: Bullet displacement comparison of ball capturing (w.o.m represents 'without mass points'; w.m represents 'with mass points').
Velocity [m/s]
2 1 x (w.o.m) y (w.o.m) z (w.o.m) x (w.m) y (w.m) z (w.m) x (ANCF) y (ANCF) z (ANCF)
0 -1 -2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time [s] Figure 17: Bullet velocity comparison of ball capturing.
Even though adding more mass points in the mass-spring model or adding more contacting spheres on cable element in the ANCF model will better de22
Displacement [m]
1.5
1
x (w.o.m) y (w.o.m) z (w.o.m) x (w.m) y (w.m) z (w.m) x (ANCF) y (ANCF) z (ANCF)
0.5
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time [s] Figure 18: Bullet displacement comparison of cube capturing.
Velocity [m/s]
2 1 x (w.o.m) y (w.o.m) z (w.o.m) x (w.m) y (w.m) z (w.m) x (ANCF) y (ANCF) z (ANCF)
0 -1 -2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time [s] Figure 19: Bullet velocity comparison of cube capturing.
23
scribe the exibility of the net, it will also increase the degree of freedoms of the system or the contact detection eort thus increasing the cost of computing. The simulator is developed in Matlab throughout this paper. All the simulations are performed by a processor of Intel Xeon CPU E3-1245 with 16 Gigabytes (GB) installed memory. The system equations established in this thesis are either dierential algebraic equations (DAEs) or the ordinary dierential equations (ODEs). Some DAEs are converted to ODEs then solved by the 4th order of Runge-Kutta method (RK4) that is a single-step and explicit integration algorithm which indicates no use of the data calculated in the previous steps. However, the step-size of RK4 or even all the explicit algorithms should not be too small or too large. This is because a small step-size might increase the steps of the integration, the round-o errors etc., while a large step-size might aect the stability of the equation and the accuracy. 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 3 shows the comparison of the computational time in a single step of both modeling methods. It is found that the mass-spring model with three added middle mass points consumes 5 times more than the mass-spring model without adding mass points at a single step computational time, and the ANCF model consumes 7.5 times more than the conventional mass-spring model. Based on the comparison of all the three cases, we noticed that the exibility modeling has little in uence on the net dynamics in simulation, and the computational time is, however, increased when considering the exibility. Therefore, a mass-spring model without any added mass points can be suggested when the accuracy requirement is not high.
5. Conclusion
This paper presents two ways of modeling the exibility of a net for space debris removal. One is based on mass-spring model and the other is based on ANCF model. The contact dynamics between a net and a target based on both
24
Table 3: Comparison of single step computational time of ANCF and Mass-spring model
Single Step Compu-
# Nodes
# DoF
Step Size
MS (w.o.m)
49
147
2e-5 s
4.14 ms
MS (w.3.m)
301
903
2e-5 s
23.13 ms
ANCF
49
2016
2e-6 s
31.20 ms
tational Time
models are provided. The nonlinear damping model is applied to calculate the contact force on each contacting sphere once the contact on them are detected. Static friction and Stribeck eect are considered in the Hollars friction model. Numerical simulations of a ball and a cube shaped targets are performed to investigate the in uence on the net dynamics by the exibility modeling. It is found that the exibility modeling has little in uence on the net dynamics in simulation. Adding more mass-points or contacting spheres is able to accurately describe the nature of a net and avoid the penetration with the target, but the computational time is, however, increased when considering the exibility. The mass-spring model with three added middle mass points consumes 5 times more than the mass-spring model without adding mass points at a single step computational time. The ANCF model consumes 7.5 times more than the conventional mass-spring model at a single step computational time. Therefore, a mass-spring model without any added mass points is encouraged to use when the accuracy requirement is not high; a mass spring with three middle points or even more points and the ANCF model can be used when researchers are interested in the exibility of a net and/or trajectories of points along a cable. Andersson, S., Söderberg, A., Björklund, S., 2007. Friction models for sliding dry, boundary and mixed lubricated contacts. Tribology international 40, 580587.
25
Benvenuto, R., Lavagna, M., Salvi, S., 2016. Multibody dynamics driving gnc and system design in tethered nets for active debris removal. Advances in Space Research 58, 4563. Benvenuto, R., Salvi, S., Lavagna, M., 2015. Dynamics analysis and gnc design of exible systems for space debris active removal. Acta Astronautica 110, 247265. Bischof, B., Kerstein, L., Starke, J., Guenther, H., Foth, W., et al., 2004. ROGER- robotic geostationary orbit restorer. Science and Technology Series 109, 183193. Botta, E.M., Sharf, I., Misra, A.K., 2016a. Contact dynamics modeling and simulation of tether nets for space-debris capture. Journal of Guidance, Control, and Dynamics , 114. Botta, E.M., Sharf, I., Misra, A.K., Teichmann, M., 2016b. On the simulation of tether-nets for space debris capture with vortex dynamics. Acta Astronautica 123, 91102. Brown, P., McPhee, J., 2016. A continuous velocity-based friction model for dynamics and control with physically meaningful parameters. Journal of Computational and Nonlinear Dynamics 11, 054502. De Jalon, J., Bayo, E., 2012. Kinematic and dynamic simulation of multibody systems: the real-time challenge. Springer Science & Business Media. Gerstmayr, J., Shabana, A.A., 2006. Analysis of thin beams and cables using the absolute nodal co-ordinate formulation. Nonlinear Dynamics 45, 109130. Gilardi, G., Sharf, I., 2002. Literature survey of contact dynamics modelling. Mechanism and Machine Theory 37, 12131239. Goª¦biowski, W., Michalczyk, R., Dyrek, M., Battista, U., Wormnes, K., 2016. Validated simulator for space debris removal with nets and other exible tethers applications. Acta Astronautica 129, 229240. 26
Huang, P., Wang, D., Meng, Z., Zhang, F., Guo, J., 2015. Adaptive postcapture backstepping control for tumbling tethered space robottarget combination. Journal of Guidance, Control, and Dynamics 39, 150156. Hunt, K., Crossley, F., 1975. Coe cient of restitution interpreted as damping in vibroimpact. Journal of applied mechanics 42, 440445. Johnson, K.L., Johnson, K.L., 1987. Contact mechanics. Cambridge university press. Khude, N., Melanz, D., Stanciulescu, I., Negrut, D., 2011. A parallel gpu implementation of the absolute nodal coordinate formulation with a frictional/contact model for the simulation of large exible body systems, in: Proceedings of the 8th International Conference on Multibody Systems, Nonlinear Dynamics and Control. Liou, J.C., 2011. An active debris removal parametric study for leo environment remediation. Advances in Space Research 47, 18651876. Medina, A., Cercós, L., Stefanescu, R.M., Benvenuto, R., Pesce, V., Marcon, M., Lavagna, M., González, I., López, N.R., Wormnes, K., 2017. Validation results of satellite mock-up capturing experiment using nets. Acta Astronautica 134, 314332. NASA Orbital Debris Program O ce, N., May 2019. Orbital debris quaterly news. National Aeronautics and Space Administration 23, 114. Shabana, A.A., 2013. Dynamics of multibody systems. Cambridge university press. Shan, M., Guo, J., Gill, E., 2017a. Deployment dynamics of tethered-net for space debris removal. Acta Astronautica 132, 293302. Shan, M., Guo, J., Gill, E., Goª¦biowski, W., 2017b. Validation of space net deployment modeling methods using parabolic ight experiment. Journal of Guidance, Control, and Dynamics , 19. 27
Si, J., Pang, Z., Du, Z., Cheng, C., 2019. Dynamics modeling and simulation of self-collision of tether-net for space debris removal. Advances in Space Research . Specker, T., Buchholz, M., Dietmayer, K., 2014. A new approach of dynamic friction modelling for simulation and observation. IFAC Proceedings Volumes 47, 45234528. Takehara, S., Kawarada, M., Hase, K., 2016. Dynamic contact between a wire rope and a pulley using absolute nodal coordinate formulation. Machines 4, 4. Wang, Q., Tian, Q., Hu, H., 2014. Dynamic simulation of frictional contacts of thin beams during large overall motions via absolute nodal coordinate formulation. Nonlinear Dynamics 77, 14111425. Website, 2019. Space debris. Wayana Sofware. Web. http://www.geosats.
com/spacedebris.html. Accessed: 2019-07-27. Yu, L., Zhao, Z., Tang, J., Ren, G., 2010. Integration of absolute nodal elements into multibody system. Nonlinear Dynamics 62, 931943. Zhang, F., Huang, P., 2016. Releasing dynamics and stability control of maneuverable tethered space net. IEEE/ASME Transactions on Mechatronics 22, 983993. Zhang, F., Huang, P., Meng, Z., Zhang, Y., Liu, Z., 2017. Dynamics analysis and controller design for maneuverable tethered space net robot. Journal of Guidance, Control, and Dynamics 40, 28282843.
28
S0273-1177(19)30791-4 https://doi.org/10.1016/j.asr.2019.10.041 JASR 14524
To appear in:
Advances in Space Research
Received Date: Accepted Date:
31 July 2019 31 October 2019
Please cite this article as: Shan, M., Guo, J., Gill, E., An analysis of the flexibility modeling of a net for space debris removal, Advances in Space Research (2019), doi: https://doi.org/10.1016/j.asr.2019.10.041
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© 2019 Published by Elsevier Ltd on behalf of COSPAR.
An analysis of the exibility modeling of a net for space debris removal Minghe Shana,∗, Jian Guob , Eberhard Gillb a Department b Faculty
of Aerospace Engineering, University of Maryland, College Park, 20740, The United States
of Aerospace Engineering, Delft University of Technology, Delft, 2629 HS, The Netherlands
Abstract
Operational spacecraft are facing a risk of collision with space debris objects. The net capturing method has been proposed to mitigate this risk on spacecraft. The mass-spring model is usually applied for net modeling by discretizing a cable into one or several mass-spring-damper elements in simulation. The absolute nodal coordinates formulation (ANCF) has also been applied to model the net, and this model is able to describe the exibility of a net using less elements. However, the in uence on the net behavior in simulation by the exibility modeling of a net is not well understood and barely discussed. In this paper, exibility models of a net are established based on the mass-spring model and the ANCF model,respectively. The in uence on the net behavior by the exibility modeling is, for the rst time, analyzed via simulations. Two case studies of capturing a ball and a cube shaped targets are performed. It is found that the exibility modeling has little in uence on the net dynamics in simulation. Finally, the characteristics and bene ts of the ANCF model are described and analyzed. A drawback of the ANCF model was found to be its inferior computational performance. Keywords:
space debris, net capturing, ANCF, exibility modeling
∗ Corresponding
author. Email: [email protected]
Preprint submitted to Advances in Space Research
November 6, 2019
1. Introduction
According to the Space Debris O ce (SDO) of the European Space Agency (ESA), space debris is de ned as "all non-functional, human-made objects, including fragments and elements thereof, in Earth orbit or re-entering into Earth's atmosphere". The Earth orbit is in a serious predicament caused by millions pieces of space debris. According to the United States Space Surveillance Network (US-SSN), more than 19 000 objects larger than 10 cm in diameter have been catalogued by radar and/or other techniques as of May 2019 (NASA Orbital Debris Program O ce, May 2019). Figure 1 shows the increasing of space debris objects in Earth orbit o cially cataloged by the US-SSN. 20000
https://orbitaldebris.jsc.nasa.gov
1
Number of Objects
15000
10000
2 1 Total Objects 2 Fragmentation Debris
5000
3
4 Mission-related Debris
4
5 Rocket Bodies 2017
1997
1987
1977
1967
1957
2007
5
0
3 Spacecraft
Year
Figure 1: Number of objects in Earth orbit by object type (NASA Orbital Debris Program Of ce, May 2019)
As of Sep., 2019, about 2400 payloads on orbit are active. Comparing it to the total number of objects on orbit, which is about 19 000, it indicates that around 12% of objects are operating in an environment where 88% of the objects are space debris. The collision of Cosmos 2251 and Iridium 33 in 2009 has highlighted the threat by space debris objects, since it signaled a trend that the future space environment will be dominated by fragmentation debris generated via similar collisions, instead of explosions (Liou, 2011). Even more serious, space debris is a threat to the lives of astronauts. In March 2009, a ve inches 2
space debris object passed particularly close to the International Space Station (ISS). Fortunately, the alarm was cleared 10 minutes later (Website, 2019). To mitigate the risk of collision and stabilize the space environment, active debris removal (ADR) is of great relevance. Net capturing method has been proposed to mitigate the collision risk on satellites (Bischof et al., 2004). Extensive researches on net capturing method have been performed, including the sensitivity analysis (Shan et al., 2017a), simulators design (Shan et al., 2017a; Botta et al., 2016b; Benvenuto et al., 2015; Goª¦biowski et al., 2016) and parabolic ight experiments (Goª¦biowski et al., 2016; Medina et al., 2017; Shan et al., 2017b) etc. Concretely, Shan et al. have analysed the input parameters, such as the bullet mass, the shooting velocity and the shooting angle's in uence on the output parameters, such as the net opening area, the deployment time and the travelling distance during the net deployment phase (Shan et al., 2017a). Botta et al. have built a net model in a commercial software, Vortex, and investigated the capturing dynamics (Botta et al., 2016a). Benvenuto et al. have developed a simulator based on a mass-spring model, compared the planar and conical net deployment process, and carried out an analysis on the interface between a net and a target during de-orbiting (Benvenuto et al., 2015, 2016). Zhang and Huang have proposed a novel space robot system called the maneuverable tethered space net robot with four microsatellites installed at four corners of the net (Zhang and Huang, 2016; Zhang et al., 2017; Huang et al., 2015). Si et al. have proposed a line-line self-collision detection algorithm to accurately simulate the capture process (Si et al., 2019). Goª¦biowski et al. have developed a simulator based on the Cossret rod theory and validated the model with a parabolic ight experiment (Goª¦biowski et al., 2016). Another parabolic ight experiment has been performed by shooting a net on a xed Envisat mock-up in the cabin, by which the mass-spring model has been veri ed and validated (Medina et al., 2017). The net modelling in most of the existing researches is based on the massspring (MS) model. However, the in uence on the net behavior in simulation by the exibility modeling of a net is not well understood and barely discussed. 3
Obviously, the exibility between two nodes can be described by discretising the cable into more segments. However, this dramatically increases the degrees of freedom and the complexity of the system equations. In addition, the mass-spring model has a limitation in describing the contact between a net and a target because that the ctitious penetration of the massless spring into the target cannot be avoided. The Aboslute Nodal Coordinates Formulation (ANCF), which is initially proposed by Shabana, can be utilized in solving large displacement and large deformation problems (Shabana, 2013). In ANCF, absolute positions and the gradients of the positions are acting as the element nodal coordinates to describe the con guration of a exible system. The nodal coordinates of the elements are de ned 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 (named 'global slopes') instead of in nitesimal or nite 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 (Shabana, 2013). In a net system for space debris removal, an ANCF cable element can be applied to build the net. Only the deformation along the longitudinal direction is necessary to be considered based on the nature of a cable. 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 ignores the deformations along the lateral and transverse directions which accurately describes the nature of a cable (Gerstmayr and Shabana, 2006). For the contact problem between cable and other rigid object based on ANCF, Takehara et al. have developed a numerical model that can describe the behaviour of the contact between a wire rope and a pulley (Takehara et al., 2016). Wang et al. have developed a new approach to simulate the frictional contact dynamics of cable elements based on penalty method (Wang et al., 2014). Yu et al. have proposed a method to represent contact zones of ANCF elements by a set of basic geometries, like sphere, brick cylinder and triangular patch (Yu et al., 2010). Khude et al. have proposed a 4
method to model the frictional contact between exible cable elements and rigid body by combining the Discrete Element Method (DEM) with ANCF (Khude et al., 2011). This paper rst presents the modeling of exibility of a net based on both the mass-spring model and the ANCF model. In the mass-spring model, the exibility of a cable is described by discretising the cable into several segments. In the ANCF model, a cable element is discretised into multiple contacting spheres in contact detection process. The segments between contacting spheres are not detected, thus penetration between the segment with the target might happen. Therefore, the more contacting spheres are discretised in one cable element, the more accurate of the model is, and undoubtedly more computational time will be spent. The nonlinear damping model is applied to calculate the contact force on each contacting sphere once the contact on them are detected. Frictional force on contacting spheres are derived based on the Hollars (Brown and McPhee, 2016). The in uence on the net behavior by the exibility modeling is, for the rst time, analyzed via simulations. Two case studies of capturing a ball- and a cube-shaped targets are performed. Moreover, the characteristics and bene ts of ANCF are described and analysed. A drawback of the ANCF was found to be its inferior computational performance.
2. Modeling of Flexibility
2.1. Based on the mass-spring model
The mass-spring model is usually applied for net modeling by discretizing cables into mass-spring-damper elements as shown in Fig. 2. Four mass points at four corners of the net are called 'bullets', which is shot by a net gun and helps deploy the net. Based on the nature of a cable, it is not able to withstand compression. Therefore, a special spring between two mass points which can only be tensioned is de ned. References (Shan et al., 2017a) and (Shan et al., 2017b) have introduced the derivation of the model of the net.
5
Bullet 1
Bullet 3
One cable element, no additional mass points, one segment
{
Mass Point
Spring Damping Element Bullet 4
Bullet 2
Figure 2: mass-spring model
However, the model in these references does not consider the exibility of the cable, neither of the in uence on the net behavior by the exibility. In other words, there is only one spring-damper element between two mass points, and the shape of this element is always being straight. Obviously, the exibility of a cable can be described by discretising the cable into several segments as shown in Fig. 3. Additional mass points should be added between two mass points in order to describe the exibility. However, the number of the added mass points will dramatically increase the degrees of freedom and the complexity of the system. Moreover, since the stiness of the cable segment ki can be expressed as
ki =
EA ∆l
(1)
where E is the elastic Young's modulus of the cable, A is the area of the cross section of the cable, and ∆l is the length of the cable segment. The more mass points are added, the shorter the cable segment is, thus the larger the stiness is. To compute system equations with larger stiness, the smaller time step in the simulation is required which will also reduce the computing e ciency. On
6
the contrary, adding more mass points can better describe the contact between a net and a target. This is because only the contact between the mass points and the target is detected in the simulation, thus the ctitious penetration of the spring-damping elements into the target's body always happens. Adding more mass points can prevent this phenomena to some extend and make the simulation results more reliable and accurate. Bullet 1
Bullet 3
{
One cable element, two additional mass points, three segments
Bullet 4
Bullet 2
Figure 3: Flexibility modeling based on the mass-spring model.
2.2. Based on the ANCF model
The ANCF model has also been applied to model the net, and this model is able to describe the exibility of the net with less elements. The ANCF method is initially proposed by Shabana, and it is specialized to be utilized in solving large displacement and large deformation problems (Shabana, 2013). In ANCF, absolute positions and the gradients of the positions are acting as the element nodal coordinates to describe the con guration of a exible system. For the net which is weaved by cables (as shown in Fig. 4), one cable element contains two 7
nodes at its ends. The arbitrary position on the deformed cable element can be expressed using the shape function S and coordinates of these two nodes as, (2)
r = Se = [S1 I, S2 I, S3 I, S4 I][e1 , e2 ]T
where 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
(3)
where ri is the global displacement and rix is the global slope of the element. The shape function Si is de ned 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 ) (4) where ξ = x/l0 , and x is the coordinate of an arbitrary point on the element.
x 0 Original configuration
Cable
elem ent
rix Node i
Node j rjx
r ri
rj
Z X
Y o
Bullet
Deformed configuration
Figure 4: Flexibility modeling based on the ANCF model.
Using the principle of virtual power and introducing Lagrange multipliers (De Jalon and Bayo, 2012), one can derive the system equations of motion for
8
the tethered-net as
Mb
0 Φb
0 Me Φe
ΦT b
q¨b
Qb
= , ΦT e ¨ Q e e 0 λ Q
(5)
where Mb and Qb are the mass matrix of the bullets and the external forces acting on the bullets, 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, λ represents the Lagrange multipliers; Φe is the Jacobian matrix of the constraint equations associated with the absolute nodal coordinates; Φb represents the constraints coupled by the bullet masses and the cable elements; Q is quadratic velocity vectors derived by dierentiating the constraint equations twice with respect to time. The detailed derivation of the system equations of motion is given in (Shan et al., 2017a). In the ANCF model, the penetration between the cable elements and the target's body can be avoided by distributing ctitious contacting spheres on the cable element. Contacting spheres on one ANCF cable element can dier based on the requirements of accuracy and computational performance. In other words, the contacting spheres can be several spherical points (Fig. 5 (a), (b)) distributed along the centreline of a cable element, or even a chain of spheres (Fig. 5 (c)). The segment between contacting spheres are not detected in a contact detection algorithm. Thus, penetration between the segment with the target might happen in that region. With increasing number of spheres, the computational time to evaluate the contact detection and the contact force will increase. On the other hand, the accuracy of the net motion can be enhanced and the penetration can be avoided.
3. Contact Dynamics of a Flexible Net with a Target
Since a net is assumed to be multiple mass points connected with springdamper elements in a certain pattern, net contacting with a target can be con9
(a)
Contacting point
Cable element
(b)
(c)
Figure 5: Contacting spheres on one cable element.
sidered as multiple mass points contacting with the surfaces of the target as shown in Fig. 6. The mass points have to move outside the space of the target and they are not allowed to penetrate the target. According to this description, net capturing can be considered as an inequality constraint problem. The
Figure 6: Net contacting with a target (left) and simpli ed contact scenario (right).
penalty-based method, which is usually used in the optimization problem, is a very attractive method to solve the inequality constraint problem. In the optimization problem, an inequality constraint problem can be formulated as Minimize f (x),
subjext to c(x) ≥ 0
(6)
where f (x) is the objective function, c(x) is the constraint equation. It is possible to convert this inequality constraint problem into a suitable unconstraint
10
problem as Minimize f (x) + µmin{0, c(x)},
subjext to x ∈ Rn
(7)
The term "µmin{0, c(x)}" is called the penalty function. If c(x) ≥ 0, then min{0, c(x)} = 0 and no penalty occurs. If by any chance, c(x) < 0, then min{0, c(x)} = c(x). The penalty should be added into the objective function to prevent this from happening. Taking the point-surface contact problem as an example, the point is constrained to be always moving above the surface, which is c(x) ≥ 0. If the constraint equation holds, which means the point is not contacting with the surface, thus min{0, c(x)} = 0. On the contrary, if it is detected that c(x) < 0, which means the point is penetrating the surface. The penalty force, here the contact force, µc(x) will be added to f (x) to push the point away from the surface and prevent the penetration from continuing. According to this de nition, the penalty function, "µmin{0, c(x)}", can be considered as a function of the penetration depth. In fact, a contact between the elastic bodies will lead to a deformation on colliding bodies, and the contact force can be expressed as a function of the contact deformation. In the simulation of a contact scenario, the contact deformation can be parameterized as a function of the penetration depth between the colliding bodies. In the penalty-based method, the response force depends on the penetration depth: the larger the penetration, the higher the penalty. 3.1. Normal Contact Force
According to the review of the contact dynamics modeling by (Gilardi and Sharf, 2002), two main categorises of contact dynamics models can be distinguished: discrete models and continuous models. Discrete models assume that the impact process is instantaneous. Therefore they are not applicable to the penalty-based method. Commonly used continuous contact dynamics models are: spring-dashpot model, Hertz's model and non-linear damping model. Those models account for the normal force based on the deformation. The direction of the force is along the normal line of the contact. Table 1 lists the expressions 11
of the normal contact forces in those models. In the expressions, k represents the contact stiness, d represents the damping coe cient, δ represents the deformation or the penetration in the simulation, and n represents a coe cient depending on the materials and geometries of the contacting bodies. Table 1: Comparison of Three Contact Dynamic Models
Contact Models
Expressions
Spring-dashpot model
fn = kδ + dδ˙
Hertz's model
fn = kδ n
Nonlinear damping model
fn = kδ n + dδ˙ n δ
0
0
0
(a)
(b)
(c)
Figure 7: Continuous Contact dynamics models: (a) Spring-dashpot model. (b) Hertz's model. (c) Non-linear damping model.
Figure 7 shows contact force history of those models. The spring-dashpot model assumes that a spring-dashpot element is set between the colliding bodies. However, this spring can only be compressed, i.e., it only pushes bodies away from each other rather than pull them together. The linear and explicit normal force fn depends on the penetration depth δ and the penetration rate
δ˙ . However, due to the damping term, the spring-dashpot model involves a discontinuity at the beginning of the contact as shown in Fig. 7 (a). Similarly, the contact force holds the colliding bodies together by the end of a contact, which is not realistic. The Hertz's model assumes the contacting bodies interacting 12
via a non-linear spring. Such a model has overcome the defect of the drift at the zero penetration. The normal force is continuous at the beginning and the end of a contact (Fig. 7 (b)). However, it does not take the damping into consideration, such that the energy dissipation is not accounted for. The non-linear damping model combines the advantages of the two aforementioned models. It is taking the damping into account and remaining the continuity at zero deformation (Fig. 7 (c)). Therefore, the non-linear damping model is selected as the contact model in the penalty-based method throughout this paper. While we have selected the contact model for the net capturing, it is necessary to calculate the coe cients, such as the stiness k and the damping coe cient d, in the model. As previously mentioned, the net is modeled as a series of connected mass points. Because of the lightness of the net and the thinness of the cable, the value of the radii of the mass points are assumed to be the radius of the cable, which are negligible compared to the surface of a target. Therefore, the contact of a mass point with the surface of a target can be treated as a sphere-plane contact. According to the Hertzian contact theory (Johnson and Johnson, 1987), for a sphere-plane contact, the contact stiness can be obtained by
√ 4 r k= , 3π(h1 + h2 )
with
hi =
1 − νi2 , πEi
(i = 1, 2),
(8)
(9)
where Ei and νi are Young's modulus and Poisson's ratio of the material of the contacting bodies, respectively, and r is the radius of the contacting mass point. Here, i = 1 indicates the parameters of the chaser material and i = 2 indicates the parameters of the target material. For the damping coe cient d, Hunt et al. (Hunt and Crossley, 1975) have established
d=
3αk , 2
(10)
where α is an experimental parameter that varies in the range of 0.08-0.32 s/m according to (Benvenuto et al., 2016). 13
3.2. Friction
There are several models to describe the friction of two contacting bodies. The Coulomb friction model is the most commonly used model for friction. It has a linear relation with the normal force by a frictional coe cient. However, this model neglects the static friction, which describes the friction at or near zero relative velocity. It also neglects the Stribeck eect, which describes the friction force reduction with increasing relative velocity of contacting surfaces. Andersson et al., Specker et al. and Brown et al. have developed separate friction models to describe the static friction and the Stribeck eect (Andersson et al., 2007; Specker et al., 2014; Brown and McPhee, 2016). Among the existing models, Hollars friction model is the most commonly used one due to its advantage of the physical meaning at the transition velocity vt0 (Brown and McPhee, 2016). The friction force reaches its maximum at vt0 and decreases with the increasing of the relative velocity. Figure 8 shows the comparison of the Coulomb friction model and Hollars friction model, from which it is seen that the Coulomb model is a simpli ed version of the Hollars model. The expression of the friction force in Hollars model is,
ft = [min{
vt 2(µs − µk ) , 1}(µk + )]fn vt0 1 + ( vvt0t )2
(11)
where vt is the relative velocity of colliding bodies, vt0 is the transition velocity,
fn is the magnitude of normal force, and µs , µk are the coe cients of static and kinetic friction, respectively.
Stribeck effect
Static friction
0
0 (a)
(b)
Figure 8: Friction models: (a) Coulomb friction model. (b) Hollars friction model.
14
The frictional force vector generated from the contact is always opposite to the relative velocity along the tangential direction of the contact surface. Assume a mass point pi is in contact with an object on point O and the normal vector of point O on the object is n (see Fig. 9). Then the relative velocity vr of the points pi and O can be expressed as
vr = vpi − vO .
(12)
fn n
pi ft
vt
O vNr vr
Figure 9: Normal and frictional force on contacting point.
The component of the relative velocity along the normal vN r is
vN r = (nT vr )n.
(13)
Therefore, the relative velocity vector along the tangential direction vt is obtained as
vt = vr − vN r .
(14)
The friction force ft can therefore be obtained by Eq. 11,
ft = −ft nt ,
(15)
with nt = vt /vt . The net contact dynamics have been introduced in this section. The contact model introduced above can be used in both the mass-spring model and the ANCF model. The only thing needs to pay attention is, for the contacting spheres distributed on cable elements in the ANCF model, the contact force fc 15
needs to be converted to the generalized force Qc to t the system equations based on the ANCF model. Using the principle of virtual work, the generalized contact force distributed on cable element is
Qc = S(ξi )fc .
(16)
with S being the shape function matrix of the cable element, ξi the element coordinates of the ith contacting sphere on the same cable element. Finally, implement fc and Qc into Qb and Qe , respectively in Eq. 5 to solve the system equations therefore to achieve the motion of the net with contacts.
4. Numerical Simulations and Discussions
The mass-spring model is the most commonly used modeling method for a tethered-net. However, to fully describe the exibility 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. In the classical nite element method, in nitesimal or nite 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 exible system with large displacement and large deformation. Gerstmayr and Shabana have developed a low order cable element using ANCF, in which it has half degrees of freedom of a fully parameterized beam element, and it does not consider the deformations along the lateral and transverse directions which more accurately describes the nature of a cable (Gerstmayr and Shabana, 2006). Even though the exibility modeling can be described based on the above mentioned models, the in uence on the net dynamics in simulation by the exibility modeling of a net is not well understood so far. For example, to express the exibility of a cable using the mass-spring model, the number of mass points added on one cable element can be 2, 3 or even more (Fig. 3). The way the net dynamics in uenced by the number of those added mass points is unclear. In 16
this section, the in uence on the net dynamics in simulation by the exibility modeling is investigated via simulations of capturing a ball and a cube shaped targets using the mass-spring model and the ANCF model, respectively. The con gurations of the net and the targets along with other simulation parameters are summarized in Table. 2. To investigate the in uence on the net dynamics in simulation by the exibility modeling, here we compared three groups of simulation for both ball and cube capturing, which are simulations based on the mass-spring model without added middle mass points, the massspring model with three middles mass points on one cable element, and the ANCF model with three virtual contacting spheres on one cable element. The process of capturing a space debris object using a net includes three steps, namely, net deployment, target capturing and de-orbiting. During the net deployment, four bullets are shot by a net gun to help expand the net. The simulations are performed in the reference frame shown in Fig. 10. The reference frame is centred at the centre of mass of the net container. The axis is de ned as: x direction is along the traveling direction; y axis is orthogonal to x and pointing to the left side of the target, z axis follows the right-hand rule and orthogonal to x and y axis. The target is captured by a 6 by 6 mesh squared net. The net is folded in a certain pattern introduced in (Shan et al., 2017a) and it is fully deployed around 0.7 s, then contact with the target. Four bullets move towards each other and pass by each other which makes the net embracing and wrapping the target. The mass and the moment of inertia of the target in this study is assumed to be in nite. In other words, the target is immovable during the net capturing. However, the contact force acting on the target is so small due to the lightness and low shooting velocity of the net that the target can hardly move even with a smaller mass during capturing. Figure 10 - 15 show the screenshots of those simulations. Figure 10 and 13 show the simulation of the ball and the cube capturing based on the mass-spring without any added middle mass points on one cable element. Figure 11 and 14 show the simulation of the ball or the cube capturing based on the mass-spring with three middle mass points on one cable element. Figure 12 and 15 show the simulation 17
Table 2: Simulation Parameters
Parameter Net
Value
Net size A, [m2 ] Mesh square −, [−]
6×6
Mesh length l0 , [m]
0.08
Bullet mass mb , [kg]
0.05×4
Shooting velocity v , [m/s]
1.5
Shooting angle θ, [◦ ]
25
Distance to the target dt , [m] Young's modulus E , [Pa]
Targets
0.48×0.48
1 4.456e8
Poisson's ratio v , [-]
0.3
Cube length a, [m]
0.12
Ball radius r, [m]
0.08
of the ball or the cube capturing based on the ANCF model with three virtual contacting spheres on one cable element. It can be clearly seen in the above gures that the penetration occurs when not adding any middle mass points, while adding three middle points on one cable element can eectively avoid the penetration, so as the ANCF model with three virtual contacting spheres on one cable element. With these results, it would be interesting to compare those simulation results and analyze the in uence of the exibility modeling quantitatively. We compare the trajectory of one of the bullets due to the symmetric con guration of the net. In this case, we choose the bullet one, which is located at the left up corner of the net. Figure 16 and 18 show the bullet displacements along three axes, from which we notice that the bullet displacements under three cases (without middle points using the mass-spring model, with three middle points using the massspring model and with three contacting spheres using the ANCF model) are actually quite close to each other. The maximum dierence is 6 cm, and the relative dierence comparing to the traveling distance of the net is only 4%, which indicates that the exibility modeling has trivial in uence on the net
18
z y
x
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 10: Ball capturing based on the mass-spring model. (without middle points)
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 11: Ball capturing based on the mass-spring model. (with 3 middle points)
19
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 12: Ball capturing based on the ANCF model.
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 13: Cube capturing based on the mass-spring model. (without middle points)
20
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 14: Cube capturing based on the mass-spring model. (with 3 middle points)
t = 0.25 s
t = 0.7 s
t = 1.1 s
t = 1.4 s
Figure 15: Cube capturing based on the ANCF model.
21
dynamics during capturing in simulation. It is also noticed that in Fig. 11 there is a dierence at the starting point of the displacement in x direction. This is because the initial con gurations of ANCF model and MS model are set dierently (Shan et al., 2017a). Figure 17 and 19 show the bullet velocity along three axes. There are bigger diverse at the velocity level, especially along the traveling direction.
Displacement [m]
1.5
1
x (w.o.m) y (w.o.m) z (w.o.m) x (w.m) y (w.m) z (w.m) x (ANCF) y (ANCF) z (ANCF)
0.5
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time [s] Figure 16: Bullet displacement comparison of ball capturing (w.o.m represents 'without mass points'; w.m represents 'with mass points').
Velocity [m/s]
2 1 x (w.o.m) y (w.o.m) z (w.o.m) x (w.m) y (w.m) z (w.m) x (ANCF) y (ANCF) z (ANCF)
0 -1 -2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time [s] Figure 17: Bullet velocity comparison of ball capturing.
Even though adding more mass points in the mass-spring model or adding more contacting spheres on cable element in the ANCF model will better de22
Displacement [m]
1.5
1
x (w.o.m) y (w.o.m) z (w.o.m) x (w.m) y (w.m) z (w.m) x (ANCF) y (ANCF) z (ANCF)
0.5
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time [s] Figure 18: Bullet displacement comparison of cube capturing.
Velocity [m/s]
2 1 x (w.o.m) y (w.o.m) z (w.o.m) x (w.m) y (w.m) z (w.m) x (ANCF) y (ANCF) z (ANCF)
0 -1 -2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time [s] Figure 19: Bullet velocity comparison of cube capturing.
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scribe the exibility of the net, it will also increase the degree of freedoms of the system or the contact detection eort thus increasing the cost of computing. The simulator is developed in Matlab throughout this paper. All the simulations are performed by a processor of Intel Xeon CPU E3-1245 with 16 Gigabytes (GB) installed memory. The system equations established in this thesis are either dierential algebraic equations (DAEs) or the ordinary dierential equations (ODEs). Some DAEs are converted to ODEs then solved by the 4th order of Runge-Kutta method (RK4) that is a single-step and explicit integration algorithm which indicates no use of the data calculated in the previous steps. However, the step-size of RK4 or even all the explicit algorithms should not be too small or too large. This is because a small step-size might increase the steps of the integration, the round-o errors etc., while a large step-size might aect the stability of the equation and the accuracy. 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 3 shows the comparison of the computational time in a single step of both modeling methods. It is found that the mass-spring model with three added middle mass points consumes 5 times more than the mass-spring model without adding mass points at a single step computational time, and the ANCF model consumes 7.5 times more than the conventional mass-spring model. Based on the comparison of all the three cases, we noticed that the exibility modeling has little in uence on the net dynamics in simulation, and the computational time is, however, increased when considering the exibility. Therefore, a mass-spring model without any added mass points can be suggested when the accuracy requirement is not high.
5. Conclusion
This paper presents two ways of modeling the exibility of a net for space debris removal. One is based on mass-spring model and the other is based on ANCF model. The contact dynamics between a net and a target based on both
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Table 3: Comparison of single step computational time of ANCF and Mass-spring model
Single Step Compu-
# Nodes
# DoF
Step Size
MS (w.o.m)
49
147
2e-5 s
4.14 ms
MS (w.3.m)
301
903
2e-5 s
23.13 ms
ANCF
49
2016
2e-6 s
31.20 ms
tational Time
models are provided. The nonlinear damping model is applied to calculate the contact force on each contacting sphere once the contact on them are detected. Static friction and Stribeck eect are considered in the Hollars friction model. Numerical simulations of a ball and a cube shaped targets are performed to investigate the in uence on the net dynamics by the exibility modeling. It is found that the exibility modeling has little in uence on the net dynamics in simulation. Adding more mass-points or contacting spheres is able to accurately describe the nature of a net and avoid the penetration with the target, but the computational time is, however, increased when considering the exibility. The mass-spring model with three added middle mass points consumes 5 times more than the mass-spring model without adding mass points at a single step computational time. The ANCF model consumes 7.5 times more than the conventional mass-spring model at a single step computational time. Therefore, a mass-spring model without any added mass points is encouraged to use when the accuracy requirement is not high; a mass spring with three middle points or even more points and the ANCF model can be used when researchers are interested in the exibility of a net and/or trajectories of points along a cable. Andersson, S., Söderberg, A., Björklund, S., 2007. Friction models for sliding dry, boundary and mixed lubricated contacts. Tribology international 40, 580587.
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