Similar to a space flying net, the capture field of the space
netted pocket system is large and it can be applied to capture space
non-cooperative targets flexibly. To maintain the stability of the space
netted pocket system, eight inflatable rods are used as the supporting
structure of the net surface. In this paper, a space netted pocket system is
designed and modeled. Based on ANCF (absolute nodal coordinate formulation),
a dynamic model of the complex space rope net system is established, and
then an accurate model of closing rope considering the variable length is
derived by introducing mass flow element. A double closed-loop sliding
control method is designed to maintain the stable attitude of the service
spacecraft. An extended observer is applied to estimate and compensate for
the disturbances due to the uncertainty of the contact and flexibility in
the system. Finally, the dynamic model and control method is verified
through the simulation of the virtual prototype. Results show that the
service spacecraft can maintain the attitude stability during target
captured process and can track the desired angle during attitude maneuver.
The flexible deformation and collision cause great disturbance to the
service spacecraft, and the extended observer can improve the control
accuracy from 10-3 to 10-4.
Introduction
In recent years, with the increase of human space activities, threats to
space activities from space debris are
worsening (Barmin et al., 2014; Chen, 2011).
The contact capture method is a simple and feasible way to capture space debris. There are mainly two types of contact capture: rigid capture and
flexible capture (Nishida and Kawamoto,
2011; Shan et al., 2016). Like space manipulator and capture claw, the
rigid capture methods will inevitably cause collision between the capture
target and operation platform. Therefore, the rigid capture methods are more
complicated and require high control accuracy. The space net as a typical
flexible acquisition method is more lightweight and simple
(Zhang et al., 2017). The capture domain of the space net is
larger, which can reduce the control accuracy requirements and reduce the
cost. Therefore, the space net is an active space debris removal method with
broad application prospects.
The space fly net launches a flexible rope net to cover the space debris in
a large envelope and drags space debris to the atmosphere burned or grave
orbit. In recent years, a lot of theoretical and experimental researches
has been carried out around the space fly net. In terms of large-scale
experiments, RemoveDEBRIS project has now completed the first space
rope net capture test in
orbit (Forshaw et
al., 2016, 2017) and e.Deorbit project has carried out parabolic flight
test (Biesbroek et al., 2017). At the same time, many
theories about space net dynamics modeling, space net deployment control,
space net capture performance analysis and space net structure optimization
design are deeply
researched (Bonnal
et al., 2013; Ming et al., 2017; Xu et al., 2019). However, it is difficult
for the fly net system to maintain the net shape fully deployed for a long
time. And the capture success rate will be reduced due to the space fly net
hardly maintaining the net shape for a long time. Some scholars have
proposed the motorized fly net to control the net shape
(Huang et al., 2015; Meng et
al., 2017), but multiple motorized devices around the net will greatly
increase the complexity and cost of the system. The space rope netted pocket
capture mechanism described in this paper can inherit the advantages of
lightness and simplicity. Furthermore, the inflatable rods in the system can
maintain the net shape for a long time, which can improve the success rate
of capture. The rope net surface of the netted pocket capture mechanism in
Fig. 1 is supported in an umbrella-like shape by
eight flexible rods. First, the service spacecraft approaches the captured
target until the target enters the net. Second, the closing devices at the
end of inflatable rods are tightened to complete the closing. Finally, with
attitude maneuver of the service spacecraft, the space netted pocket drags
the captured target to change orbit.
Space netted pocket capture mechanism.
Due to the strong nonlinearity dynamics of the net and the large uncertainty
of the target, the precise control of the rope net spacecraft is more
difficult. The traditional PD (proportional differential) controller can hardly meet the accuracy
requirements of the mission. For the stability control of tethered
satellites, Huang et al. considered the effect of the flexibility of rope,
and designed an adaptive control method to achieve attitude stability
control of combinations with unknown parameters
(Huang et al., 2016); Wei et al. (2019) designed
a spacecraft attitude stability ADRC (active disturbance rejection control) controller which can estimate and
compensate uncertainties of system parameters in real
time (Wei et al., 2019).
However, the current research generally only focuses on the attitude control
during capture or maneuver, and the process of capture and maneuver is not
continuous. Moreover, the dynamic model and simulation prototype for
controller verification is imperfect, which cannot simulate the nonlinear
properties and contact collision of the rope accurately. So, it is untrue to
verify controllers using the imperfect dynamic model for the space netted
pocket system.
To solve these problems, this paper designs an extended observer to compensate
the uncertainty and disturbance of the model based on the principle of
active disturbance rejection control (ADRC). Combined with the robust
sliding mode control method, the attitude controller of the service
spacecraft is constructed. At the same time, a complex dynamic model of the
net pocket system is established based on the dynamic theories such as
variable flexible cable, ANCF (absolute nodal coordinate formulation) and collision theory. The closed-loop
simulation of dynamics and control can be completed through the virtual
simulation prototype.
Dynamics modeling of space netted pocketNetted pocket system composition and size design
As shown in Fig. 2, the netted pocket system
consists of eight flexible support inflatable rods, which are connected by
rope net. The inflatable rods are evenly distributed in a positive octagonal
shape, with a single rod diameter of 0.1 m. The rope-retracting mechanism at
the end of each rod pulls the rope to close the net. The size and braided
shape of the single net piece between support rods are designed as shown in
Fig. 3. The upper part of the square rope net is
14×12 (row × column) configuration and the lower part of the
triangular rope net is 10×12 (row × column) configuration.
The maximum diameter of the net pocket is 15.12 m and the maximum depth is
12 m.
Space netted pocket system composition.
Space netted pocket system size.
Dynamic model of the space netted pocketDynamic model of the closing ropes based on variable flexible ropes
A rope retractor is installed at the end of each inflatable rod, and the
diameter of the opened net is adjusted by controlling the length of the
closing rope. Due to the rope retractor, the system structure is complex,
and it is difficult to simulate the ropes inside and outside the retractor
at the same time. In this case, the simulation takes a long time, and the
rope inside the retractor has little impact on the rope net capture system.
Therefore, in order to simplify the dynamic model of the rope net capture
device and improve the simulation speed, this paper establishes the dynamic
model of closing ropes based on the ANCF flexible cable dynamics model.
Using v1 as the mass flow velocity at the element boundary node,
the variable length flexible cable element model of the closing rope is
established in Fig. 4.
Model of closing rope.
Based on ANCF, the generalized coordinates of flexible cable element jq can
be expressed as
jq=jq1jq2=jrT(0)jrlT(0)jrT(L)jrlT(L)T,
where jrT(0) and jrT(L) are the position vector of nodes at
both ends of the element; and jrlT(0) and jrlT(L) are the
gradient vector nodes at both ends of the element. And the position vector
jr(l) of any node on the central axis of the element can be
written as
2jr(l)=Sljq3Sl=S1IS2IS3IS4IS1=14ξ-12ξ+2S2=Lt8ξ-12ξ+1S3=14ξ+12-ξ+2S4=Lt8ξ+12ξ-1,
where S(l) is the shape function; ξ can be written as follows:
ξl,t=2lt-LtLt.
The length of the rope element L(t) and the position of the particle
l(t) are time variables, so the variable length element shape function
Sl,t is a function of time. Then the velocity and
acceleration of any point are expressed as
5jr˙l,t=Sljq˙t+S˙ljqt6jr¨l,t=Sljq¨t+2S˙ljq˙t+S¨ljqt7S˙=dSdξξ˙,S¨=d2Sdξ2ξ˙2+dSdξξ¨.
In the case of varying element length, the Lagrange method is no longer
applicable with the assumption of the mass flow at the boundary. The
dynamical equations are derived using D'Alembert's principle, which states
that the sum of virtual work done by inertial force and acting force on any
virtual displacement is 0.
∑iFi-mr¨i⋅δri=0.δri is the particle i in any moment to meet the constraints of
the virtual displacement. The force on the rope element can be decomposed
into elastic force FE and external force Ff, then the dynamics
equation of the cable element can be written as
∫0tLtδjrTFf+FE-ρjr¨dl=0.
Considering only the axial and bending deformation of the cable, the
expression of the virtual work of each part can be written as
10∫0tLtδjrTFfdl=δjqT∫0tLtSTlfl,tdl=δjqTdldξ∫-11STξfξ,tdξ11∫0tLtδjrTFEdl=-δjqTdldξ∫-11(∂ε0∂jqT⋅EAjε0+cjε˙0+∂jκ∂jqT⋅EJjκ+cjκ˙)dξ12∫0tLtδjrT-ρjAr¨dx=-δjqTdldξ∫-11ρASTjr¨dξ,
where jε0jκ are axial strain and curvature, and E is
elasticity modulus. A is the cross-sectional area and J is the moment
of inertia of the flexible cable element. c is the damping coefficient and
dldξ=Lt2.
Combining Eqs. (10), (11) and (12), the dynamic equation of the variable
flexible cable element can be written as
jMjq¨+jMvjq˙+jMqjq+jQ=0,
where
14jM=Lt2∫-11ρSTSdξjMv=Lt2∫-11ρSTS˙dξjMq=lt2∫-11ρSTS¨dξ15jQ=jQe+jQfjQe=Lt2∫-11(∂jε0∂jqTEAjε0+cjε˙0+∂jκ∂qTEJjκ+cjκ˙)dξ16jQf=-Lt2∫-11STfdξ.
Considering a long cable with N-1 flexible elements, only the lengths of the
flexible elements at the ends can be varied, while the lengths of the other
elements remain constant. The equation for a variable length cable can be
obtained by combining the dynamic equations of all the elements.
Mq¨+Q=0,
where q represents the generalized coordinates of the system, consisting
of the generalized coordinates of each node. M and Q represent the
relative generalized mass matrix and generalized forces of the cable.
Dynamic model of space rope netted system based on ANCF
In Gerstmayr and Shabana (2006), the ANCF
equations for the dynamics of the space rope netted system can be expressed
as
Mq¨+CqTλ=Qk+QεC=0,
where M is the mass matrix, q is the generalized coordinates,
Qε generalized force vector, C is constraint
equation, Qk=∂U∂qT and U is
strain energy. The details in Eq. (17) can also be obtained by taking
v1=0 from the variable length flexible cable model.
Contact collision force model
According to the Hertz contact model, the contact force at the collision
point consists of the normal collision force Fn and the tangential
friction force Ft.
F=Fn+Ft.
Using the nonlinear spring damping model, the normal collision force is
calculated as
Fn=knδnn-gnvn,
where n is the normal unit vector and v is the normal relative velocity.
The expressions of kn and gn are as follows:
kn=43Ereff⋅δngn=-256⋅β⋅Sn⋅meff,
where reff is the radius of curvature of the contact point,
m1 and m2 are the mass of the contact pair, E is the elastic modulus
of the material, and other unknown quantities are as follows:
β=lnreff/lnreff2+π2Sn=2Ereff⋅δnmeff=m1m2(m1+m2).
Coulomb friction is used in the tangential direction as follows:
Ft=μ(vt)Fn,
where μ is the coefficient of friction, which is determined by the
tangential relative velocity vt.
In the process of catching and towing target, the cable element collides
with the target. Based on the principle of virtual work, the contact force
can be transformed to a generalized nodal force.
Attitude dynamics of the service spacecraft
The attitude Euler angle of the service spacecraft can be selected as
θ=[γϕφ]T, then the spacecraft attitude dynamic equation can be
written as
24Icω˙=-ΩIcω-dq+Mcθ˙=R(θ)ω25R(θ)=1tanϕsinγtanϕcosγ0cosγ-sinγ0sinγcosϕcosγcosϕ,
where Mc is the control torque of the service spacecraft. Ic∈R3×3 is the rotational inertia of the service spacecraft and
ω is the angular velocity of the service spacecraft. dq is the
system uncertainty terms such as the interference generated by the mutual
collision and the large deformation of flexible netted pocket. Ω is
the skew–symmetric matrix of the form as follows:
Ω=0-ωz-ωyωz0-ωx-ωyωx0.
Design of spacecraft attitude double closed-loop sliding mode controller
In this paper, the spacecraft attitude controller uses an extended observer
to estimate dq, and a two-loop sliding mode control method to design the
attitude control torque Mc, which can ensure the service spacecraft
attitude θ tracking the desired attitude angle θd. The
outer-loop sliding mode control law calculates the attitude angular
velocity ωd and transmits ωd to the inner loop; the
inner-loop sliding mode control law tracks the attitude angle ωd.
The entire two-loop control system is shown in
Fig. 5.
Based on Eq. (23), the system state equation can be written as
θ¨=-Ic-1ΩIcω+RθIc-1Mc+d̃=RθIc-1dq+R˙θω.
Referring to Freidovich and Khalil (2008), the
spacecraft attitude extended observer is designed as
y˙1=y^2+α1δθ-y^1y˙2=y^3+RθIc-1u-RθIc-1ΩIcω+α2δ2θ-y^1y˙3=α3δ3θ-y^1,
where y^1 is the state estimation of θ, y^2 is the
state estimation of θ˙ and y^3 is the state estimation
of d̃. δ is taken as follows:
1δ=100t3100.
Design of outer loop sliding mode control
The integral sliding mode (Liu
and Wang, 2011) is adopted to realize the sliding mode surface design.
Taking eθ=θd-y^1, the sliding mode surface of the
outer loop can be written as follows:
sw=eθ+Kw∫0teθdt,
where Kw>0 is 3×3 diagonal matrix.
The derivation of Eq. (29) can be obtained as follows:
s˙w=e˙θ+Kweθ=θ˙d-θ˙+Kweθ=θ˙d-R(θ)ωd+R(θ)ωe+Kweθ,
where ωe=ωd-y2 and outer loop control law is
ωd=R-1(θ)(θ˙d+Kweθ)+R-1(θ)ρwsw,
where ρw>0. Taking Lyapunov function Vw=12swTsw
(Pham et al., 2019), it can be proved that
V˙w=-ρwsw2+swTR(θ)ωe,
when the inner loop converges quickly and ωe is sufficiently
small. There is V˙w≤ and the error eθ is
asymptotically stable.
Design of inner loop sliding mode control
The sliding surface of the inner loop is designed as
sn=ωe+Kn∫0tωedt,
where Kn>0 is 3×3 diagonal matrix. The derivation of Eq. (33)
can be obtained as follows:
s˙n=ω˙e+Knωe=ω˙d+Ic-1(ΩIcω+dq-Mc)+Knωe.
The control law of the inner loop is designed as
Mc=RθIc-1ΩIcω+IcR-1θ(ω˙d+Knωe-y3+ρnsat(sn)+λsn),
where ρn>maxdi, λ=λλ>0. To further eliminate the “chattering”, the
saturation function sat(s) is used as the switching function.
Taking Lyapunov function, Vn=12snTIcsn, it can be
proved (Slotine and Sastry, 1983) that
V˙n=-ρn∑sn-snTd-λsn2≤-λsn2≤-Vn.
The Eq. (36) proves that Vn is exponential convergence.
Simulation analysis of capture and towing process
This paper utilizes the multi-body dynamic simulation software (MBDyn)
developed by the author's laboratory to perform dynamic simulation. The
dynamic parameters of the service spacecraft and captured target are in
Table 1.
The dynamic parameters of the service spacecraft and captured target.
The material parameters of the inflatable deployment rod and rope net are in
Table 2.
The material parameters of the inflatable deployment rod and rope net.
Parameter nameNetsInflatable rodsDiameter (m)0.0040.01Density (kg m-3)1430164Poisson ratio0.30.3Modulus of elasticity (GPa)120.75
Capturing and attitude maneuver process.
Service spacecraft attitude maneuvering process simulation
The process of the service spacecraft capture target and attitude maneuver
is designed as follows: 0–20 s is the process of net retraction during which
the service spacecraft remains stationary; 20–60 s is the process of service
spacecraft attitude maneuvering 90∘ around x axis; and 60–100 s is
the process of service spacecraft attitude maintenance. In order to ensure
the smooth and continuous motion process, the rope retrieval length in the
0–20 s and the spacecraft attitude maneuvering in 20–60 s are planned using
the 5 times interpolation, and then the angular velocity and angular
acceleration are 0 at 20 and 60 s can be determined.
Set the parameters of the sliding mode controller kw=1, kn=10, ρn=0.01, the parameters in the saturation function Δ=0.5 , and
the parameters of the expansion observer R=100. The dynamics and control
closed-loop simulation of the attitude maneuvering process is performed, and
the calculation results are in Fig. 6.
Service spacecraft attitude motion state.
Attitude extended observer results.
The estimates of the attitude motion state and disturbance by the expansion
observer are shown in Fig. 8. Comparing
Figs. 8 and 7, it
can be seen that the estimates of the observer are consistent with the
motion process, which illustrates the validity of the observations.
Error eθ in each direction.
As shown in Fig. 9, compared with ordinary
sliding mode control, the error of extended observer controller is reduced
by 1 order of magnitude. Therefore, the control accuracy of the
double-loop sliding mode attitude controller based on the extended observer
is significantly higher than the ordinary sliding mode control. It indicates
that the extended observer compensates the disturbance effectively, and the
service spacecraft attitude controller designed in this paper can meet the
stability requirements of the attitude maneuvering process.
Service spacecraft control torque.
As shown in Fig. 10, the net is a centrosymmetric
structure and the initial state of target is located in the center, so the
control torque Mc in each direction is small during 0 to 20 s. Due to
the collision increases at 20 and 60 s, the control torque Mcz,Mcy produces buffeting. And during 20 to 60 s, the service
spacecraft attitude maneuvers around the x axis, so the control torque
Mcx is much higher than Mcy,Mcz. The
flexible structure of the system generates certain perturbations. So, in the
attitude holding process after 60 s, the control torque curve still fluctuates
until the system stabilizes.
Service spacecraft attitude dynamic response.
It can be seen from the collision force curve in
Fig. 11a that the collision force between the
capture target and the rope net is mainly concentrated in the y direction.
During 0 to 15 s, tensile force of closing rope is mainly caused by the
bending of the inflatable rod, and the value is small as shown in
Fig. 11b. During 15 to 20 s, the collision
force is large, so the tensile force of closing rope is also with large
step. As shown in Fig. 11b, the tension force
of the closing rope tends to be stable and non-zero after 20 s, indicating
that the rope net is tightly wrapped in the target and there is no relative
movement between them. It can also be seen from the collision force curve
that the close wrapping of the rope net around the target significantly
reduces the collision in the maneuvering process. Thus, the interference is
further reduced, and the control torque of spacecraft is facilitated as shown
in Fig. 10.
Conclusion
In this paper, the dynamics and stability control of the space net pocket
capture and towing process are studied, and the following conclusions are
obtained.
The dynamic model of the space netted pocket capture system is carried
out. Based on the ANCF method, the dynamic model of the space rope system
can respond to the large deformation properties of the catching mechanism.
Based on the dynamic model of the closing rope established from the ANCF
flexible cable theory, the simulation of the non-cooperative target capture
process is realized, and the dynamic response of the closing rope recovery
can be analyzed.
A sliding mode controller based on the extended observer is designed with
reference to the service spacecraft dynamic equations. Only the attitude of
the spacecraft is needed to complete the maneuver control, as the extended
observer can observe the velocity angular velocity of the spacecraft.
Moreover, based on spacecraft attitude extended observer, the sliding mode
controller can ensure the high accuracy and stable attitude control of the
service spacecraft.
Closed-loop dynamics and control simulations are performed for the
capture and towing process of the service spacecraft. The simulation results
can reflect the large deformation of the net pocket and the contact
collision between the net pocket and the target during the capture and
towing process. The closed-loop simulation results verify that the control
accuracy is improved by 1 order of magnitude under the interference
compensation of the expansion observer, which is beneficial for realizing
the stable attitude control of service spacecrafts with large deformation
structure.
Code availability
The simulation software is jointly developed with a third party. The other party is for commercial purposes and does not want the software code to be disclosed.
Data availability
In this paper, the simulation conditions and relevant parameter data are listed in Tables 1 and 2. And the effective data results are shown by curves in Figs. 7–11.
Author contributions
CT was mainly responsible for the calculation and data analysis of the simulation examples in this paper.
ZH was mainly responsible for the modeling of simulation examples in this paper.
CW was mainly responsible for the development of the dynamics software in this paper.
YZ was mainly responsible for the research of dynamics and control theory in this paper.
Competing interests
The contact author has declared that none of the authors has any competing interests.
Disclaimer
Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Financial support
This research has been supported by the Heilongjiang Postdoctoral Fund (grant no. LBH-Z21141) and the National Natural Science Foundation of China (grant no. 12102316).
Review statement
This paper was edited by Daniel Condurache and reviewed by two anonymous referees.
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