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Acta Astronautica
journal homepage: www.elsevier.com/locate/actaastro
Coordinated stabilization for space robot after capturing a noncooperative
target with large inertia
Bo Zhang
a,⁎
, Bin Liang
a,b
, Ziwei Wang
b
, Yilin Mi
c
, Yiman Zhang
b
, Zhang Chen
c
a
Department of Automation, Tsinghua University, Beijing, China
b
School of Astronautics, Harbin Institute of Technology, Harbin, China
c
Graduate School at Shenzhen, Tsinghua University, Shenzhen, China
ARTICLE INFO
Keywords:
Space robot
Large noncooperative target
Momentum reduction
Coordinated stabilization
Aadaptive sliding Mode control
ABSTRACT
A noncooperative target with large inertia grasped by space robot may contain a large unkonwn initial angular
momentum, which will cause the compound system unstable. Unloading the unkonwn angular momentum of
the compound system is a necessary and diffcult task. In the paper, a coordinated stabilization scenario is
introduced to reduce the angular momentum, which has two stages, Momentum Reduction and Momentum
Redistribution. For the Momentum Reduction, a modified adaptive sliding mode control algorithm is proposed
and used to reduce the unknown angular momentum of target, which uses a new signum function and time-
delay estimation to assure fast convergence and achieve good performance with small chattering effect. Finally,
a plane dual-arm space robot is simulated, the numerical simulations show that the proposed control algorithm
is able to stabilize a noncooperative target with large inertia successfully, while the attitude disturbance of base
is small. The control algorithm also has a good robust performance.
1. Introduction
Space robot is one of the most effective methods for on-orbit
servicing. Taking into account the economy and complexity of the
future mission, a small space robot is a more appropriate choice to
perform different mission, which has become the frontiers of technol-
ogy competition in the field of robot. Especially in the geostationary
orbit, the project of SUMO/FREND [1] and Phoenix Program [2] are
all typical orbital application of small space robot. The significant
characteristics of these projects are that the space robot has more than
one arm and the inertial parameters of space robot is less than the
target spacecraft.
Most of on-orbit servicing can be divided into four main phases in
order to describe the interactive relationship between space robot and
target, pre-grasping, contact, post-grasping and compound stabiliza-
tion [3,4]. In the pre-grasping phase, space manipulator is controlled
and planned to approach and follow the motion of target gradually. In
the contact phase, the space robot is free floating, the end of space
manipulator comes into contact with the target until the gripper is
closed. In the post-grasping phase, the end of space manipulator
captures target firmly, the space robot damps out the motion of the
target. In the compound stabilization phase, the space robot stabilizes
the tumbling motion of the space robot and target. In the paper, the
third phase and the fourth phase are categorized together as one phase,
which is a stabilization task to deal with angular momentum of the
compound system. Generally speaking, a noncooperative target may
contain a large initial angular momentum in the compound system,
which will cause the system unstable [5–7]. Unloading the angular
momentum of the compound system is a necessary task, which has
been attracting more and more attentions.
Many researchers have studied the issue of compound stabilization.
From the view of target, all studies are divided into two categories, the
one target has small inertial, the other target has large inertial. For
small inertial target, Cyril [8] establishes the dynamics models of
compound system after capture and finds that if the postimpact
dynamics are uncontrolled, the response can be unexpected, then
proposes a feedback linearization control scheme to keep the system
stable. Yoshikawa [9] discusses an angular momentum control of a
tumbling spacecraft by applying repetitive impulses from a space robot
arm, the simulation results shows that the rotational motion of the
target spacecraft is well damped. Aghili [10,11] focuses on controlling
the space manipulator in the postcapture phase to bring a tumbling
satellite to rest. In the literature [12], the manipulator damps out the
target's angular and linear momentums as quickly as possible subject to
the constraint that the magnitude of the exerted force and torque
remain below their prespecified values in the postgrasping phase. Xu
http://dx.doi.org/10.1016/j.actaastro.2017.01.041
Received 25 October 2016; Accepted 28 January 2017
⁎
Corresponding author.
E-mail address: zhboyan@hit.edu.cn (B. Zhang).
Acta Astronautica 134 (2017) 75–84
Available online 02 February 2017
0094-5765/ © 2017 IAA. Published by Elsevier Ltd. All rights reserved.
MARK
[13] proposes a method for berthing a target and reorientating the base
using manipulator motion only after the capture. Dimitrov [14]
presents a new strategy for capturing a free floating satellite initially
having angular momentum, and proposes two control laws for angular
momentum management during the post-impact phase. Wang [15]
addresses a novel sliding mode control scheme in the postcapture
phase, which is used for achieving attitude stabilization by coordination
of the tethered space manipulator. For large inertial target, Rekleitis
[16] developes a planning and control methodology for manipulating
passive objects by cooperating orbital free-flying robots, both on-off
base thrusters and manipulator continuous forces are used in handling
on-orbit passive objects and eliminating the effects of on-offcontrol.
Gasbarri [17] introduces two control strategies used in the pre-
grasping and post-grasping phases, the one controller is used to control
torques applied to the joints of the robotic arms, the other controller is
used to control torques applied to the base. Suresh [18] addresses
stabilization and control issues in autonomous capture and manipula-
tion of noncooperative space objects and presents robust dissipativity-
based control laws. Bandyopadhyay [19] proposes an attitude control
strategy and a new nonlinear tracking controller for a spacecraft
carrying a large object such as an asteroid or a boulder. The proposed
nonlinear tracking control law guarantees global exponential conver-
gence of tracking errors.
Unloading the angular momentum of the compound system is
diffcult due to the state and parameters of target can’t be known
exactly, so its model uncertainties not only contains kinematic terms
but also involves dynamic parameters. To solve the problem of
kinematics and dynamic uncertainties, a feedback control law with
uncertain parameters is designed to track the desired position [20].In
literature [21,22], a limited torque input controller is used to achieve
the stabilization of the setpoint control error in cartesian space. A
Jacobian adaptive controller is proposed for robot and some experi-
ments are conducted to illustrate the performance of this controller
[23]. In literature [24], according to the actuator model, an adaptive
controler is used to track the trajectory. In literature [25], the
parameterization problem in dynamic structure and adaptive control
of a space robot system with an attitude-controlled base is introduced.
A passivity based adaptive Jacobian controller is proposed for free-
floating space manipulators [26], which consists of a transposed
Jacobian feedback and a dynamic compensation term, and the para-
meter adaptation laws are derived by Lyapunov-like stability analysis
tools. Theoretically, all the mentioned methods can achieve the control
objects under both kinematic and dynamic uncertainties. These
methods are useful for controlling a robot with small kinematics and
dynamic uncertainties or without uncertainties. When a large unco-
operative target with large unknown initial angular momentum is
grasped by space robot, the compound system is more nonlinear and
stronger coupling among base, manipulators and target. Adaptive
sliding mode control is an efficient and effective tool to resolve the
control problems of nonlinear robot systems [27,28]. Sliding mode
control is mainly used to improve the robustness in terms of system
construction uncertainties and unmodeled dynamics. If the state
variables arrive at the sliding mode, they will slide along the designed
linear mode no matter what the detailed system is. Furthermore,
adaptive scheme is introduced with respect to system parameter
uncertainties. Particularly, adaptive scheme alternatives the conver-
gence velocity, which matters before the state variables arrive at the
sliding mode. Thus, the controller, combined sliding mode and
adaptive scheme, covers both advantages synthetically with the state
variables arriving to the desired values.
In the paper, a coordinated stabilization scenario is introduced to
unload the angular momentum of the compound system in Section 2.
which has two stages, momentum reduction and momentum redis-
tribution. For the momentum reduction, an adaptive sliding mode
control is proposed in Section 4, the controller is designed to output the
control torques applied to the joints of the robotic arms and base,
which is used to stablize the large inertial target carried the unknown
momentum. Before the Section 4, the fundamental knowledge about
compound system is described in the Section 3. To confirm the validity
and robustness of the proposed control law, the numerical simulations
are demonstrated in Section 5
a
.
2. Coordinated stabilization scenario
When a space robot has grasped a noncooperative target with large
inertia, the momentum of the compound system may be large and
unknown. As we all know, one of the main characteristics of a operation
in orbit is the momentum conservation if there are no external forces.
If the compound system is considered, it might undergo momentum
change but the conservation law will hold. So the base rotational
motion may be not expected because the small space robot's ability to
absorb momentum is limited, which should be avoided as far as
possible. In this paper a coordinated stabilization scenario is proposed
to solve the problem of unloading the compound system's large and
unknown momentum while minimizing the disturbances to the base.
The coordinated stabilization scenario is a strategy about angular
momentum management, which has two stages, the first stage is the
momentum reduction, the second stage is the momentum redistribu-
tion. To clearly illustrate the two stages for the compound system, there
are three variables to be defined as follows:
L
t
- angular momentum in the target.
L
b
- angular momentum in the base.
L
m
- angular momentum in the manipulators, which has the maximum
amount of angular momentum Lm
max.
Lw- angular momentum in the attitude stabilization devices such as
reaction wheels, which also has the maximum amount of angular
momentum Lw
ma
x
.
Ls- angular momentum in space robot, and LL L L=+ +
sbmw
.
It's clear that the maximum amount of angular momentum
absorbed by space robot is limited. In other words, if
LL L L≤+ +
sbm
max
w
max is satisfied, the space robot is able to maintain
a stable state without external force and torque. The angular momen-
tum of space robot can be distributed to different manipulators or
reaction wheels. After contacting between space robot and target, the
compound stabilization phase is formed. If LLL L L+≤ + +
stbm
max
w
max
is satisfied, the compound system can be stable by space robot using
angular momentum management without external force and torque.
Especially, when angular momentum of reaction wheels is saturated,
multi-arm coordinated motion planning and control algorithm can
absorb extra momentum of compound system, in other words, the
extra momentum can be transferred to some manipulators. The
coordinated planning and control between the base and manipulators
is very useful for keeping the base stable. This method has been
successfully demonstrated in ETS-VII. However, if a noncooperative
large inertia target is grasped by a space robot and
LLL L L+> + +
stbm
max
w
max may be satisfied, it's a challenging issue
for maintaining the compound system stability. To achieve the
compound stabilization, the coordinated stabilization strategy is pro-
posed as follows.
Stage 1: Momentum Reduction.
After a large inertia target is grasped firmly by a space robot, its
residual angular momentum may be large. In other words, the
condition LLL L L+> + +
stbm
max
w
max is satisfied. If there are no
external forces, the space robot can not absorb or store the residual
angular momentum absolutely. So the external forces are needed. In
the stage of momentum reduction, there are external forces acting on
the base centroid, which are come from the thrusters, control moment
gyroscope or magnetorquers and used to reduce the angular momen-
B. Zhang et al. Acta Astronautica 134 (2017) 75–84
76
tum of compound system by the coordinated movement of the base and
manipulators, while avoiding collision between the space robot and
target. After the stage of momentum reduction, the residual angular
momentum of compound system is reduced to be small, then the
coordinated stabilization can enter the next stage.
Stage 2: Momentum Redistribution.
When the residual angular momentum is small, or the condition
LLL L L+≤ + +
stbm
max
w
max is satisfied, by applying internal torques in
the joints or reaction wheel, the space robot can absorb or store the
residual angular momentum absolutely, which means that the com-
pound stabilization can be easily achieved in comparison to the
momentum reduction.
3. Problem formulation
Before designing the adaptive sliding mode control law used for the
stage of momentum reduction, the dynamic model of space robot and
its properties are described in this section. It is known that space robot
is a spacecraft system that equips one or more robotic arms. A dual-
arm space robot is shown in Fig. 1(a), which is made up of base,
arm a−and arm b−. The arm a−is a
n−
aDOF serial link manipulator,
the arm b−is a
n−
b
DOF serial link manipulator. The arm a−is a
mission arm used for grasping a target, the arm b−is an auxiliary arm
used for balancing the whole system.
It is a basic problem to derive the kinematic and dynamic equation
of space robot system. In the paper, we use Roberson-Wittenburg
method to derive the rigid dynamics of multi-body system and get the
following set of differential equation to describe the dual-arm space
robot.
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
HH H
HHH
HH H
v
ω
c
c
c
F
τ
τ
Θ
˙
˙
¨
+=
H
vbωbθ
bω
T
ωωθ
bθ
T
ωθ
T
θ
b
b
v
ω
θ
b
b
l
l
(1)
The inertia matrix
H
is uniformly positive definite,
v
˙band
ω
˙bare the
linear and angular acceleration of base,
Θ
¨
is a vector about all
acceleration of joints.
cv
,
c
ωand
cθ
are nonlinear terms.
F
b
and
τ
b
l
are
external forces and torques acting on the base centroid.
τ
l
are torques
acting on joints.
When the external force acting on the base centroid F
0
=
b, and
HI
w=
vis a postive matrix(wis the mass of the space robot), so the Eq.
(1) can be rewritten as,
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
⎡
⎣
⎢⎤
⎦
⎥
⎡
⎣
⎢⎤
⎦
⎥⎡
⎣
⎢⎤
⎦
⎥
⏟
⏟
⏟
HH
HH
ωc
c
τ
τ
Θ
˙
¨+=
Hqcτ
ωωθ
ωθ
T
θ
bω
θ
b
¨
l
l
l
l
ll
ll
l
l
(2)
where,
H
H HHH H H HHH H H HHH=− , = − ,=−
ωωbω
T
vbωωθ ωθ bω
T
vbθ θ θ bθ
T
vb
θ
−1 −1 −1
ll l
c c HHcc c HHc=− ,=−
ωω bω
T
vvθ θ bθ
T
vv
−1 −1
ll
Eq. (2) reveals the dynamic variation of the compound system
under the torque of the base centroid and each joint. When the tip of
mission arm has grasped the target firmly, as shown in Fig. 1(b), the
target can be considered as a part of the last link in mission arm, which
may change the inertia parameters of space robot. Then the Eq. (2)
should be written as
H
qc
τ
¨+=
∼
͠
(3)
There are uncertainties in
H
͠
and
c
∼
due to unknown target grasped by
the mission arm. We define the following bound for the uncertainties,
there exist positive definite matrices
H
͠
max and
H
͠
min such that
H
HH≤≤
͠ ͠ ͠
min max
In other words, when the parameters of control method and the
max spin speed of faulty spacecraft are determined, the range of
H
͠
is
bounded and satisfied. The Eq. (3) can be rewritten as,
qH
τ
Γ
¨=+−1 (4)
where HHτHcΓ=( − ) +
∼
͠ ͠
−1 −1 −1 and
H
diag h h h=(,,…,)
n12
is a con-
stant matrix to be determined for guaranteeing the stability.
In the stage of momentum reduction, the goal of control is to
decrease the spinning speed of the target while minimize the distur-
bances to the base attitude by coordinated movement of the base and
manipulators.
4. Adaptive sliding mode control design
4.1. Sliding mode control design
In the paper, the tracking error is defined as
e
q
q
=−
d
, then the
sliding variable can be defined as
s
eKet()=˙+·
λ(5)
where
s
tdiagstst st()= ( (), (),…, ())
n
T
12 and Kdiag λ λ λ=(,,…,
)
λn12 .itis
noted that the K
λin Eq. (5) is a design parameter to be determined for
guaranteeing the stability. According to the sliding variable
s
t(
)
, we can
design the following control,
τ
HHqKeβstttΓ=− ()+ (
¨+˙()+ ()
)
∼
dλ(6)
where
β
diag β β β=(,,…,
)
n12 ,
Γ
is an estimate of
Γ
in Eq. (4), and it can
be obtained from one sample-delayed measurement of
Γ
. In other
words, we have
qHτttLtL tLΓΓ()= ( − )= ¨(− )− (−
)
−1 (7)
where Lis the sampling time period. Substituting Eq. (7) into Eq. (7),
we have
τ
Hq τ H q K e βstL tL t t=− ¨(− )+ (− )+ (
¨+˙()+ ()
)
∼
dλ(8)
Supposed
τ
τ=
∼
, and substituting the control input Eq. (8) into Eq. (8),
the closed loop system is
e
Ke Ke ttΓΓ0
¨+˙++()−()=
dp (9)
where KK
β
=+
dλ and KβK=
pλ
.
If we can estimate tΓ(
)
exactly, or
0
ttΓΓ()− ()= , the tracking error
in Eq. (9) goes to zero by choosing proper K
λand β. If the sampling
time Lis sufficiently small, the estimation in Eq. (7) implies that tΓ(
)
can be as close to tΓ(
)
as possible. In other words, the closed loop
system is stable. However, when a space robot has grasped an unknown
target, tΓ(
)
cannot estimate exactly even for small sampling period L.
We propose a new version of adaptive sliding mode control and add it
to the control in Eq. (8).
Fig. 1. A Dual-arm Space Robot and Target.
B. Zhang et al. Acta Astronautica 134 (2017) 75–84
77
4.2. Adaptive sliding mode control design
In the section, an adaptive sliding mode control is proposed and
used to control a space robot that has grasped an large inertia tumbling
target. The control algorithm can be shown as follows:
τ
Hq τ H q K e s H K stL tL t βt tsgnt=− ¨(− )+ (− )+ (
¨+˙()+ ())+ ( ()· (())
)
dλ
l
(10)
where
Ktdiagktkt kt()= ( (), (),…, ())
n12
ll l
l
and the signum function
ssgn t sgn s t sgn s t sgn s t( ( )) = [ ( ( )), ( ( )), …, ( ( ))
]
n12 defined by
⎧
⎨
⎩
sgn s t st δ
st δ
(())= 1if0≤()≤
−1 if ( ) >
i
i
i(11)
Where the δis a small postive number. The proposed adaptive sliding
mode control algorithm employs a adaptive law as follows:
⎪
⎪
⎧
⎨
⎩
k
tϕα st θt kt
ϕα st k t
˙()= ·{ ·| ( )|} · ( ) if ( ) > 0
··|()| if()=0
i
iii
θt
i
iii i
−1 ( )
−1
ll
l
(12)
where ϕ
i
and α
i
are positive gains,
θt(
)
is defined as
ssgn t ε(∥ ( )∥ −
)
∞
with a positive parameter ε. The adaptation speed of switching gains
k
t(
)
i
l
is seriously affected by ε.
As seen in Eq. (12), for
k
t()>0
i
l
there are two different cases in
terms of the signum function
ssgn t ε(∥ ( )∥ −
)
∞
. When st
ε∥
()∥ ≥
∞, the
gain
k
t(
)
i
l
increases until st
ε∥
()∥ <
∞. As the gains
k
t(
)
i
l
are increasing,
the sliding variable
s
t(
)
goes to the vicinity of the sliding manifold. That
means st
ε∥
()∥ <
∞, then the gain
k
t(
)
i
l
decreases while the sliding
variable
s
t(
)
stays in the vicinity of the sliding manifold.
The above description shows that the proposed adaptive law can be
said to provide better tracking performance and chattering reduction
simultaneously due to its high switching gains and fast adaptation
speed. Before showing the stability of the proposed adaptive law Eq.
(12), we introduce two Lemmas that will be helpful in the proof of the
main results.
Lemma 1. If the matrix
H
in Eq. (10) is chosen to satisfy the
following condition:
IHH
∥
−∥<
1
͠
−1
2
for all
t>0
,then qqtL t
∥
¨(− )−
¨()∥ →0
2as
L
→0 and the errors
between Γt(
)
iand
Γt(
)
i
are bounded by constant
Γ
*
i
for all
in=1,2,…,
,i.e.,
Γt Γt Γ
|
()− ()|≤
*
ii i
proof:.The proof is given in the literature[15].□.
Lemma 2. For a space robot controlled by Eq. (10),the switching
gain
k
t(
)
i
l
is upper bounded by a positive constant
k*
i
l
as follows:
k
tk()<
*
ii
ll
for all
t≥0
.
proof:.The proof is given in Appendix A.□.
Theorem 1. For a space robot controlled by Eq. (10),the sliding
variables enter the vicinity of the sliding manifold,st
ε∥
()∥ <
∞,within
afinite time t>
0
ε,and then they are guaranteed to be uniformly
ultimately bounded for
tt≥
ε
as follows:
∑
stεk
∥
()∥ < +
i
n
m2
=1
2
where k
m
is the maximum value of Γkt
∑(*−())
i
nα
ϕii
=1
2
i
i
l
.
proof:.We choose a Lyapunov function as
∑
ss
V
ttt α
φΓkt()= 1
2()()+ 1
2(*−())
T
i
n
i
i
ii
=1
2
l
(13)
Taking the derivative of the Lyapunov function with respect to the time
t, we have
∑
ss
V
ttt α
φΓktkt
˙()= ()
˙()− ( *−())
˙(
)
T
i
n
i
i
iii
=1
ll
(14)
Substituting Eqs. (4) and (5) into Eq. (14) yields
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
∑
∑∑∑
∑∑
∑
ssKsVt t t t β t tsgn t α
φΓktkt
st Γt Γt β s t st kt
α
φΓktkt st α
φkt Γ kt
βst
ΓΓ
˙()= ()( ()− ()− ()− () (()))− ( *−())
˙(
)
≤ () ()− () − ()− () ()
−(
*−())
˙()≤ () − ˙()( *−())
−()
T
i
n
i
i
iii
i
n
ii i
i
n
i
n
ii
i
n
i
i
iii
i
n
i
i
i
ii
i
n
=1
=1 =1
2
=1
=1 =1
=1
2
i
i
i
ll
l
ll l l
l
(15)
For Eq. (15), when
st
ε
() ≥
∞
,
sθt sgn t ε()= ( () − )=
1
∞
Fig. 2. A planar 6DOF dual-arm space robot.
Table 1
Simulation Parameters for Dual-arm Space Robot.
M
as
s
Length (m) Inertia (
k
gm·
2
)
(kg) a
∥∥
i
kb
∥∥
i
kI
k
i
other
base 40.0 0.0 0.3 44.0 0.0
arm-a Link1 2.49 0.2 0.2 0.21 0.0
arm-a Link2 2.49 0.2 0.2 0.21 0.0
arm-a Link3 2.49 0.2 0.2 0.21 0.0
arm-b Link1 2.49 0.2 0.2 0.21 0.0
arm-b Link2 2.49 0.2 0.2 0.21 0.0
arm-b Link3 2.49 0.2 0.2 0.21 0.0
Table 2
The simulation parameters for the large target.
M
ass(kg) Length (m) Inertia (
k
gm·
2
)Angular velocity (rad/s)
m
t
b
∥∥
t
I
t
ω
t
target 200.0 0.3 220.0 −0.175234
Table 3
Initial state for the dual-arm space robot.
Angle(rad
)
Angular velocity (rad/s)
base 0.1752 0.000566
arm-a [π144
180 ,−π−102
180 ,
−
π−34
180 ][−0.19894,0.40194,−0.3788]
arm-b [−π−137
180 ,
π92
180
,
π43
180
][0.00934,−0.0108,0.0107]
B. Zhang et al. Acta Astronautica 134 (2017) 75–84
78
Fig. 3. The Tracking Errors of the Base in Space Robot and the Torque Acting on the Base Centroid.
Fig. 4. The Motion State of the Large Target.
Fig. 5. The momentum of compound system.
B. Zhang et al. Acta Astronautica 134 (2017) 75–84
79
then
k
tφαst θt φ
αst
˙()= ·( · ()) · ()= · ()
iiii
θt i
i
i
−1 ()
l
(16)
Substituting Eq. (16) into Eq. (15) yields
∑
V
tβstβε
˙()≤− ()≤− ·
i
n
=1
2
2
i
(17)
Due to Vt V0 ≤ ( ) ≤ (0) <
∞
and
V
t
˙()≤0
, sliding variable
s
t(
)
is able to
enter the vicinity of the sliding manifold in a finite time t>
0
ε. That
means the condition
st
ε
() <
∞
is satisfied. When the sliding variable
s
t(
)
enters the region
st
ε
() <
∞
, it may move in and out since
V
t
˙(
)
is
not guaranteed to be nonpositive in this vicinity of the sliding manifold.
If sliding variable
s
t(
)
leaves the region
st
ε
() <
∞
,
V
t
˙(
)
becomes
negative again according to Eq. (17), which steers it back toward the
sliding manifold.
From Eq. (13), the Lyapunov function is bounded as
∑
sstVt t α
φΓkt
1
2() ≤ ()≤ 1
2() + 1
2(*−())
i
n
i
i
ii2
22
2
=1
2
l
(18)
It's noted that Γkt
∑(*−())
i
nα
φii
1
2=1
2
i
i
l
is bounded because the
Γ
*
i
is
constant and
k
t(
)
i
l
is bounded in terms of Lemma 2. Then we have
∑
V
tεk()< 1
2+1
2
i
n
m
=1
2
(19)
Putting Eqs. (18) and (19) together yields
∑
stεk
1
2() < 1
2+1
2
i
n
m2
2
=1
2
(20)
which means that
∑
stεk() < +
i
n
m2
=1
2
(21)
Eq. (21) implies that the sliding variable
s
t(
)
is uniformly ultimately
bounded for
tt≥
ε
. Although the sliding variable
s
t(
)
moves in and out
of the small vicinity of the sliding manifold due to the attractivity from
Eq. (17), it is guaranteed to be upper-bounded by Eq. (21).□
5. Numerical simulations
5.1. Verification of the adaptive sliding mode control
To verify the effectiveness of the adaptive sliding mode control
algorithm for unloading the compound system's momentum using a
small space robot, a planar 6DOF dual-arm space robot is depicted in
Fig. 2. The space robot's initial parameters are shown in Table 1. All
symbols in Fig. 2 are defined as follows: m
ij
is the mass of the rigid
body,
I
i
j
is the inertia of the rigid body, m
0
and
I
0is the mass and
inertia of base, m
t
and
I
t
is the mass and inertia of target, ai
jis the
vector from the ith joint to the ith rigid body centroid for
arm j−
,
b
i
j
is
the vector from the ith(−1) rigid body centroid to the ith() joint for
arm j−
.
b
t
is the vector from the tip of mission arm to the target
centroid, bb=
ab
00
. For the planar 6DOF dual-arm space robot, the
inertia
I
i
k
is a one-dimensional matrix I
ik
. The simulation stepsize is
0.005 s.
After the contact between space robot and target, the motion state
of compound system is constrained strictly, its initial motion state for
simulation is shown in Tables 2 and 3. From the Tables 1 and 2,itis
clear that the inertia of target is larger than the space robot, and
because of the initial angular velocity, the compound system's mo-
mentum is not zero as shown in Tables 2 and 3.
Fig. 6. The Tracking Errors of the arm a−in Space Robot and the Torque Acting on Each Joint.
B. Zhang et al. Acta Astronautica 134 (2017) 75–84
80
The matrix
H
diag= (1.05, 0.02, 0.02, 0.02, 0.36, 0.36, 0.36
)
, so that
Lemma 1 is satisfied. Actually, the term
IHH
∥
−∥
͠
−1
2
is computed to
be between 0.9835 and 0.9868. The matrix
Kdiag= (3.2 × 10 , 1.5 × 10 , 0.9 × 10 , 0.3 × 10 , 8.0 × 10 , 4.8
× 10 , 1.6 × 10 )
λ
−3 −3 −3 −3 −3
−3 −3
and
β
diag= (0.5 × 10 , 0.06 × 10 , 0.04 × 10 , 0.01 × 10 , 0.06
× 10 , 0.04 × 10 , 0.01 × 10 )
−3 −3 −3 −3
−3 −3 −3
.
When we apply the proposed signum function (11) and adaptive law
(12) to the simulation, δ=0.00
1
,
ϕ=1.00
4
1
,
α
=0.00
5
1
,
ε
=0.1
2
1. When
k
t()=0.0
i
l
is satisfied, ϕϕϕ===0.00
1
234 ,
α
αα===0.00
5
234
,
ϕϕϕ===0.00
5
56 7 ,
α
αα===0.00
1
567 . While
k
t()>0.0
iis satisfied,
ϕϕϕ= = = 360
234 ,
α
αα= = = 30.
5
234 ,ϕϕϕ= = = 300.0
56 7 ,
α
αα= = = 30.
5
567 . The positive parameter
ε
=0.0
8
2∼7 is setted.
Using the proposed adaptive sliding mode control algorithm, the
simulation results of momentum reduction are shown from Figs. 3–8.
The tracking errors of the base and the torque acting on the base
centroid are displayed in Fig. 3. Under the action of the torque(Fig.
3(d)), the base attitude is gradually stable, the tracking errors of
angular, angular velocity and angular acceleration are shown as
Fig. 3(a), Fig. 3(b) and Fig. 3(c) respectively. From the Fig. 3, the
attitude of base is converged and near rad
2
.73 × 10−3 finally.
The noncoorperative target is also gradually stable as shown in
Fig. 4 and its error of angular velocity is near zero as shown in Fig. 4(a).
From the Fig. 5, the angular momentum of the compound system is
reduced. In other words, the unknown momentum of noncoorperative
target is unloaded by space robot, while the base's attitude disturbance
is small according to the Fig. 3(a). The tracking errors of the arm a−,
arm b−and its torques acting on each joint are depicted as Figs. 6 and
7respectively. In the compound stabilization phase, the coordinated
motion between the base and manipulators is useful to unloading the
entire momentum. For the target, there are external forces acting on
the target. The forces are come from the joints in manipulators and
thrusters or CMG in base. Because of the configuration of the
manipulator arm a−, the curve changes of torque of each joint and
the base are almost the same, however its magnitudes are different,
which are shown in Fig. 6(c), (d) and (e).
The sliding variables can be seen in Fig. 8. When the sliding
variables leave the vicinity of sliding manifold, they are strongly
influenced by the switching gains. From the Fig. 8, the condition
st
ε
()<
iis satisfied for all sliding variables from the time
t
s
=16
.
Fig. 7. The tracking errors of the
arm b−
in space robot and the torque acting on each joint.
Fig. 8. The sliding variables
s
t(
)
.
Table 4
The simulation parameters for different large inertia target.
M
ass(kg) Length (m) Inertia (
k
gm·
2
)Angular velocity (rad/s)
m
t
b
∥∥
t
I
t
ω
t
Case A 180.0 0.3 198.0 −0.17523
4
Case B 160.0 0.3 176.0 −0.17523
4
Case C 120.0 0.3 132.0 −0.17523
4
Case D 80.0 0.3 88.0 − 0.17523
4
Case E 20.0 0.3 22.0 −0.17523
4
B. Zhang et al. Acta Astronautica 134 (2017) 75–84
81
Fig. 9. The validation result for different large inertia target.
B. Zhang et al. Acta Astronautica 134 (2017) 75–84
82
5.2. Verification of the different large inertia target
In Section 5.1, simulation is conducted to demonstrate the feasi-
bility of the proposed adaptive sliding mode control algorithm, which is
able to reduce the momentum of compound system successfully. In this
section, a validation for different large inertia target is carried out to
further verify the feasibility of this coordinated control scheme. Two
different cases are studied as shown in Table 4 and other parameters
involved keep the same values with those in Section 5.1. The validation
results of momentum reduction and sliding variables are depicted as
Fig. 9.
From the Fig. 9, we can see that the proposed adaptive sliding mode
control algorithm is able to adapt to different large inertia target, which
has a good robustness and can be used to reduce the momentum of
different compound system.
The numberical simulations show that momentum reduction is able
to achieved by an adaptive sliding mode control algorithm. Because of
the proposed control algorithm has a good robust performance, it can
be used to reduce different large initial angular momentum while the
base's attitude disturbance is small. However, it's not considered that
the torques acting on the base and joints are limited, so the calculated
torques by control law are large.
6. Conclusions
In the paper, a coordinated stabilization scenario used for space
robot after capturing a noncooperative large inertia target is intro-
duced. For the stage of momentum reduction, an adaptive sliding mode
control algorithm is proposed. From the numerical simulations, the
proposed control algorithm is able to reduce the momentum of
compound system effectively and has a robust performance, the
attitude of a noncooperative large inertia target can be near zero by
the coordinated stabilization, while minimizing the disturbance of base.
So a small space robot can stabilize a noncooperative large inertia
target successfully.
Taking into account of the limited torques acting on the base and
joints, the proposed adaptive sliding mode control algorithm should be
modified in order to adapt the actual application.
Acknowledgement
This research is supported by the National Natural Science
Foundation of China (Grant no. 61673239) and Science and technology
project of Shenzhen (Grant no. JCYJ20160428182227081).
Appendix A. A
We consider a sufficiently large number,
V*
. We assume that the Lyapunov function (13) satisfies the condition
V
tV()≤
*
. Since the Lyapunov
function (13) has two terms, at least one of them should be sufficiently large. When the ttss()(
)
Tis sufficiently large, in other words,
β
st
∑(
)
i
n
=1
2
iis
sufficiently large, the derivative of the Lyapunov function (14) is negative according to (17). If the second term of the Lyapunov function (13) is
sufficiently large, in other words, when the condition
⎪
⎪
⎪
⎪
⎧
⎨
⎩
⎫
⎬
⎭
∑α
φΓktmax ( *−())<0
i
n
i
i
ii
=1
l
is satisfied, then we have
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
∑∑∑∑
V
tst
α
φkt Γ kt β s t st α
st Γktβst
˙()≤ () − ˙()( *−())− ()≤ ()+ () (*−())− ()<0
i
n
i
i
i
ii
i
n
i
n
i
i
i
i
i
n
=1 =1
2
=1
2
=1
2
ii ii
ll l
In both cases where one of two terms in Eq. (13) is significantly large,
V
t
˙()≤0
holds. In other words, if
V
t(
)
has the value
V*
, the derivative of V(t)is
negative, which means that V(t) can not exceed
V*
and hence we have
V
tV()≤
*
. Finally, it follows that V(t) is globally upper bound. Due to the
Γ
*
i
is
constant,
k
t(
)
i
l
is also upper bounded as follows:
k
tk()<
*
ii
ll
for all
t≥0
.
References
[1] Debus, Thomas, S. Dougherty. Overview and Performance of the Front-End
Robotics Enabling Near-Term Demonstration (FREND) Robotic Arm, AIAA
Infotech@Aerospace Conference, 2009.
[2] DARPA, Broad Agency Announcement Phoenix Technologies Tactical Technology
Office (TTO) DARPA- BAA-12-02. 2011.
[3] W. Xu, J. Peng, B. Liang, Z. Mu, Hybrid modeling and analysis method for dynamic
coupling of space robots, IEEE Trans. Aerosp. Electron. Syst. 52 (2016) 85–98.
[4] W. Xu, B. Liang, Y. Xu, Survey of modeling, planning, and ground verification of
space robotic systems, Acta Astronaut. 68 (2011) 1629–1649.
[5] P. Huang, D. Wang, Z. Meng, F. Zhang, et al., Impact dynamic modelling and
adaptive target capturing control for tethered space robots with uncertainties,
IEEE/ASME Trans. Mechatron. 21 (2016) 2260–2271.
[6] P. Huang, D. Wang, Z. Meng, F. Zhang, et al., Adaptive postcapture backstepping
control for tumbling tethered space robot-target combination, J. Guid. Control Dyn.
39 (2016) 150–156.
[7] P. Huang, F. Zhang, Z. Meng, Z. Liu, Adaptive control for space debris removal with
uncertain kinematics, dynamics and states, Acta Astronaut. 128 (2016) 416–430.
[8] X. Cyril, et al., Postcapture dynamics of a spacecraft-manipulator-payload system,
J. Guid. Control Dyn. 23.1 (2000) 95–100.
[9] S. Yoshikawa, K. Yamada, Impulsive control for angular momentum management
of tumbling spacecraft, Acta Astronaut. 60.10 (2007) 810–819.
[10] F. Aghili, Optimal control of a space manipulator for detumbling of a target
satellite, IEEE Int. Conf. Robot. Autom. (2009) 3019–3024.
[11] F. Aghili, Coordination control of a free-flying manipulator and its base attitude to
capture and detumble a noncooperative satellite, International Conference on
Intelligent Robots and Systems, 2009, pp. 2365–2372.
[12] F. Aghili, Pre- and post-grasping robot motion planning to capture and stabilize a
tumbling/drifting free-floater with uncertain dynamics, 2013 IEEE International
Conference on Robotics and Automation (ICRA), 2013, pp. 5461–5468.
[13] W. Xu, C. Li, B. Liang, et al., Target berthing and base reorientation of free-floating
space robotic system after capturing, Acta Astronaut. 69 (2009) 109–126.
[14] N. Dimitrov, K. Yoshida, Momentum distribution in a space manipulator for
facilitating the post-impact control, IEEE/RSJ International Conference on
Intelligent Robots and Systems, 2004, pp. 3345–3350.
[15] D. Wang, P. Huang, Z. Meng, Coordinated stabilization of tumbling targets using
tethered space manipulators, IEEE Trans. Aerosp. Electron. Syst. 51.3 (2015)
2420–2432.
[16] G. Rekleitis, E. Papadopoulos, On-orbit cooperating space robotic servicers
handling a passive object, IEEE Trans. Aerosp. Electron. Syst. 51.2 (2015)
802–814.
[17] P. Gasbarri, A. Pisculli, Dynamic/control interactions between flexible orbiting
space-robot during grasping, docking and post-docking manoeuvres, Acta
Astronaut. 110 (2015) 225–238.
[18] J. Suresh, G. Kelkar, Spacecraft Stabilization and Control for Capture of Non-
Cooperative Space Objects, NASA/TMC2014-218668. 2014
B. Zhang et al. Acta Astronautica 134 (2017) 75–84
83
[19] S. Bandyopadhyay, J. Chung, Y. Hadaegh, Nonlinear Attitude Control of Spacecraft
with a Large Captured Object, J. Guid. Control Dyn. (2016) 1–16.
[20] C.C. Cheah, S. Kawamura, S. Arimoto, Feedback control for robotic manipulator
with uncertain kinematics and dynamics, IEEE International Conference on
Robotics and Automation, Proceedings, 4, 1998, pp. 3607–3612.
[21] C.C. Cheah, M. Hirano, S. Kawamura, S. Arimoto, Approximate Jacobian control
for robots with uncertain kinematics and dynamics, IEEE Trans. Robot. Autom.
19.4 (2003) 692–702.
[22] W.E. Dixon, Adaptive regulation of amplitude limited robot manipulators with
uncertain kinematics and dynamics, IEEE Trans. Autom. Control 52.3 (2007)
488–493.
[23] C.C. Cheah, C. Liu, J.J.E. Slotine, Approximate Jacobian adaptive control for robot
manipulators, IEEE International Conference on Robotics and Automation,
Proceedings, 3, 2004, pp. 3075–3080.
[24] C.C. Cheah, C. Liu, J.J.E. Slotine, Adaptive Jacobian tracking control of robots with
uncertainties in kinematic, dynamic and actuator models, IEEE Trans. Autom.
Control 51.6 (2006) 1024–1029.
[25] Y. Xu, H.Y. Shum, T. Kanade, et al., Parameterization and adaptive control of space
robot systems, IEEE Trans. Aerosp. Electron. Syst. 30.2 (1994) 435–451.
[26] H. Wang, Y. Xie, Passivity based adaptive Jacobian tracking for free-floating space
manipulators without using spacecraft acceleration, Automatica 45.6 (2009)
1510–1517.
[27] M. Jin, et al., Practical nonsingular terminal sliding-mode control of robot
manipulators for high-accuracy tracking control, IEEE Trans. Ind. Electron. 56.9
(2009) 3593–3601.
[28] J. Baek, M. Jin, S. Han, A new adaptive sliding mode control scheme for application
to robot manipulators, IEEE Trans. Ind. Electron. (2016) 3628–3637.
B. Zhang et al. Acta Astronautica 134 (2017) 75–84
84
