Nonlinear dynamic modeling of a tether-net system for space debris capture

Abstract

In this paper, a flexible tether-net system is applied to capture the space debris and a numerical framework is established to explore its nonlinear dynamic behaviors, which comprises four principal phases: folding, spreading, contacting, and closing. Based on the discretization of the whole structure into multiple nodes and connected edges, elastic force vectors and associated Jacobian matrix are derived analytically to solve a series of equations of motion. With a fully implicit method applied to analyze the nonlinear dynamics of a slender rod network, the involved mechanical responses are investigated numerically accounting for the interactions. Contact between the deformable net and a rigid body is handled implicitly through a cost-effective modified mass algorithm while the catenary theory is utilized to guide the folding process (from planar configuration to origami-like pattern). The dragging and spreading actions for the folded hexagon net could be realized by shooting six corner mass toward a specific direction; next, the six corners would be controlled to move along a prescribed path producing a closing gesture, when touch between the flying net and the target body is detected, so that for the space debris could be captured and removed successfully. We think the established discrete model could provide a novel insight in the design of active debris removal (ADR) techniques and promote further development of the model-based control of tether tugging systems.

Appendices

Nonlinear bending energy

In this appendix, we discuss the difference between the original bending curvature derived in Eq. (7) and the modified bending curvature formulated in Eq. (8). The discrete bending curvature in Eq. (8) would be singular and may experience numerical issue once the turning angle, $\phi$, between two consecutive edges is relatively large, which is the case during the folding process, referring to Fig. 8a2. Here, we demonstrate that the adapted discrete curvature given in Eq. (7) could not only get rid of this numerical discontinuity, but also still be accurate enough.

First of all, the variation of normalized curvatures, $\{ {\hat{\kappa }}, \kappa \}$, on the turning angle, $\phi$, between two tangential directions are detailed in Fig. 13a2. Quantitative agreement has been found when the turning angle is small, e.g., $\phi < 40^{\circ }$; however, ${\hat{\kappa }}$ would grow much faster than $\kappa$ when $\phi > 60^{\circ }$. This error could be eliminated with a dense mesh, because the turning angle $\phi$ could decrease as the increasing of discrete nodal number with a specified continuous curve. Notice that it would not be physically correct to use discrete curvature, ${\hat{\kappa }}$, to formulate a 1D curve with $C^{1}$ continuity, referring to Fig. 7a2. Also, ${\hat{\kappa }} = 2 \tan (\phi /2)$ and would be infinite when $\phi$ is close to $180^{\circ }$, which is the case when a net is folded completely.

The accuracy of the bending curvature proposed in Eq. (7) is rendered by using a nonlinear cantilever beam, as shown in Fig. 14a. For a clamped-free beam with length L, density $\rho$, cross-section area A, Young’s modulus E, moment of inertia $I=\pi r_0^{4}/4$, and gravity g, its tip displacement in geometrically linear phase can be obtained classical Euler–Bernoulli beam theory,

$$\begin{aligned} \frac{\delta }{L} = \frac{1}{8} \frac{\rho A g L^3}{EI}. \end{aligned}$$

(33)

Fig. 15

Dynamics of a nonlinear cantilever beam under gravity. a1 First-order Euler method without damping. b1 First-order Euler method with damping. a2 Second-order Newmark-beta method without damping. b2 Second-order Newmark-beta method with damping

In this numerical setup, we choose $L=1$m, $\rho =1000\mathrm {kg}/\mathrm {m}^3$, $E=1$ GPa, $r_0=1$ mm, and g is varied for different loading conditions. The structure is discretized into $N=50$ nodes, and the first two nodes, $\{{\mathbf {x}}_{0},{\mathbf {x}}_{1} \}$, are manually fixed to achieve a clamped-like boundary condition. In Fig. 14b, we plot the relations between the beam tip deflection and its own weight from two different bending models and linear analytical solution. Both numerical results match well with analytical solution provided in Eq. (33) when the normalized weight is moderate; also, the variation between the curvature formulated in Eqs. (7) and in (8) are negligible, and the overlapped markers indicate the accuracy of our simplified bending formulation for folded tether-net system, even in a geometrically nonlinear range.

Time marching scheme

In this appendix, we discuss the first-order Euler method and the second-order Newmark-beta method for the time integration. It is known that the first-order Euler method would experience artificial damping when the time step size h is relatively large, and this issue can be overcome if the symplectic Newmark-beta method is used [41].

The Euler method is:

$$\begin{aligned}&{\mathbf {E}} \equiv {{\mathbb {M}}} \left[ \Delta {{\mathbf {q}}}(t_{k+1}) - h {\dot{{{\mathbf {q}}}}(t_{k})} \right] \nonumber \\&\qquad - h^2 \left[ {{\mathbf {F}}}^{\text {int}}(t_{k+1}) + {\mathbf {F}}^d(t_{k+1}) + {\mathbf {F}}^g(t_{k+1}) \right] = {\mathbf {0}} \end{aligned}$$

(34a)

$$\begin{aligned}&{{\mathbf {q}}}(t_{k+1}) = {{\mathbf {q}}}(t_{k}) + \Delta {{\mathbf {q}}}(t_{k+1}) \end{aligned}$$

(34b)

$$\begin{aligned}&\dot{{\mathbf {q}}}(t_{k+1}) = \frac{1}{h} \Delta {{\mathbf {q}}}(t_{k+1}). \end{aligned}$$

(34c)

The Newmark-beta method is:

$$\begin{aligned}&{\mathbf {E}} \equiv {{\mathbb {M}}} \left[ \Delta {{\mathbf {q}}}(t_{k+1}) - h {\dot{{{\mathbf {q}}}}(t_{k})} \right] \nonumber \\&\qquad - h^2 \beta ^2 \left[ {{\mathbf {F}}}^{\text {int}}(t_{k+1}) + {\mathbf {F}}^d(t_{k+1}) + {\mathbf {F}}^g(t_{k+1}) \right] \nonumber \\&\qquad - h^2 \beta (1-\beta ) \left[ {{\mathbf {F}}}^{\text {int}}(t_{k}) + {\mathbf {F}}^d(t_{k}) + {\mathbf {F}}^g(t_{k}) \right] = {\mathbf {0}} \end{aligned}$$

(35a)

$$\begin{aligned}&{{\mathbf {q}}}(t_{k+1}) = {{\mathbf {q}}}(t_{k}) + \Delta {{\mathbf {q}}}(t_{k+1}) \end{aligned}$$

(35b)

$$\begin{aligned}&\dot{{\mathbf {q}}}(t_{k+1}) = \frac{1}{h \beta } \Delta {\mathbf {q}}(t_{k+1}) - \frac{1-\beta }{\beta } \dot{{\mathbf {q}}}(t_{k}). \end{aligned}$$

(35c)

Here, the nonlinear vibration of a cantilever beam under gravity is considered. In this numerical setup, we choose $L=0.1$ m, $\rho =1000\mathrm {kg}/\mathrm {m}^3$, $E=10$ MPa, $r_0=1$ mm, and $g=10.0\mathrm {m}/\mathrm {s}^2$. Time step size h is varied to show the accuracy of dynamic simulation. In Fig. 15, we plot the beam tip deflection as a function of time for both Euler method and Newmark-beta method. When the environmental damping is ignored, artificial damping is observed when the time-step size h in Euler method is relatively large. However, the numerical error can be overcome if the symplectic Newmark-beta method is employed (here $\beta = 0.5$). On the other side, if the environmental damping is taken into account, the artificial damping from Euler integration would be trivial and the solution is acceptable. For the flexible system, the environmental damping and material damping are usually relatively large, such that it would not experience a long time vibratory motion, and, therefore, the naive Euler method is largely used. If the energy conservation is concerned, the time marching scheme can be easily switched based on our previous work [41].

Fig. 16

The ratio between computational time and wall-clock time as a function of total nodal number, N, for different time step size, $h \in \{0.001, 0.01 \}$s

Computational time

Here, referring to Fig. 16, we show the reliance of the computational time on the number of nodes for the dynamic vibration case. The time-step size in this figure is set to be $h \in \{0.001, 0.01\}$ s. The numerical framework is implemented within C++ environment by Eigen [50]. We can see that real time simulation can be achieved when $N \le 1000$ if $h=0.01$ s. It should be noticed that the computational time would almost double if the contact is considered. PARDISO package is utilized when solving a sparse linear system formulated in Eq. (18) [51,52,53]. The simulations are performed on a single thread of Intel Core i$7-6600$U Processor @3.4GHz. In the future, parallel computing could be implemented to speed up the solver.

Comparison between beam model and cable theory

In this appendix, we discuss the difference between the bending-dominated beam model and the stretching-dominated cable theory, which is essential for the folding process. Beam is a structure that experiencing bending, while cable is an object that undergoing pure stretching. For a 1D rodlike structure with length L, stretching stiffness EA, bending stiffness EI, and experiencing external load F (or $\rho A L g$ for gravity), the deformed mode is based on the ratio among EA, $EI/L^2$, and F, i.e., the beam model would be effective as long as $EA \gg F \sim EI/L^2$, and the cable model should be considered in the phase of $EA \sim F \gg EI/L^2$.

We here consider a 1D object with pin-pin boundary condition, referring to Fig. 17a. The numerical parameters are: structure length $L=1.0$m, cross-section radius $r_{0}=1$cm, Young’s modulus is $E=1$GPa, and number of vertex $N=100$. If the magnitude of midpoint load F is similar to the normalized bending stiffness, the deformed solution should be a curved line and can be predicted by the classical Euler-Bernoulli beam theory

$$\begin{aligned} \frac{\delta }{L} = \frac{FL^2}{48 EI}. \end{aligned}$$

(36)

On the other side, if the external force is quite large, the deformed pattern should be a zig-zag line and can be computed based on equilibrium condition,

$$\begin{aligned} \frac{F}{EA} = 2 \left[ \frac{\sqrt{\delta ^2 + (L/2)^2} - L/2}{L/2} \right] \frac{\delta }{\sqrt{\delta ^2 + (L/2)^2}}.\nonumber \\ \end{aligned}$$

(37)

Our rod-based simulation can predict both bending-dominated phase and stretching-dominated phase, referring to Fig. 17b.

Fig. 17

Deformed configuration of a rodlike object undergoing compressive load. Predictions are from both beam model and cable model. a compressive distance $\Delta L / L = 0.1$; b compressive distance $\Delta L / L = 0.21$; c compressive distance $\Delta L / L = 0.7$

Fig. 18

Configuration of a compressed rod. Predictions are obtained from both beam buckling analysis and cable analysis

For a 1D structure with compressive load, e.g., folding process, the predictions from beam model and cable model would be different: Iif its bending stiffness is large and is similar to the external gravity, its deformation can be obtained based on Euler buckling theory, known as Euler Elastica; however, if the bending stiffness is quite small and the stretching stiffness is similar to external gravity, it becomes a classical catenary model. The deformed patterns for bending-dominated modes and stretching-dominated modes are plotted in Fig. 18. Different modes are also related to the ratio among EA, $EI/L^2$, and $\rho ALg$.

Usually, the net is soft and its bending rigidity would be small compared with EA and $\rho ALg$ during the folding phase (which is on earth), such that the bending effect (e.g., discontinuous point) would be trivial to be included. However, when the gravity is missing (e.g., on-orbit phase), the bending would be more important and cannot be ignored any longer. Moreover, the folding size is also determined based on the ratio among EA, $EI/L^2$, and $\rho ALg$. For example, if the distance between two folding points is large (which means $EI/L^2$ is small), the bending energy can be ignored (e.g., dashed red line in Fig. 18c); however, as the decreasing of folding distance L, $EI/L^2$ would gradually become large and the deformed pattern would switch from stretching to bending (e.g., solid green line in Fig. 18c), and the discontinuous bending would have large energy, such that catenary-based folding process would fail. Overall, the effective folding distance for catenary theory is determined by the ratio among EA, $EI/L^2$, and $\rho ALg$.

Video

We provide several videos corresponding to Fig. 2 (Dynamics.mov), Fig. 6 (Contact.mov), Fig. 8 (Fold.mov), Fig. 10 (Shot.mov), and Fig. 12 (Close.mov) of the main manuscript as Supplementary Materials.