IEEE Robotics & Automation Magazine - December 2018 - 88

(k)
Ji

ti(k)

(k)

ai

(k)

bi

(k)

Li

(k)

mi

Arm 1

(k)

Ii

ar (k)
i

(k)
ab
01

Base

∑I : Inertial Coordinate

b0

m0 I0
a0

∑G
∑B

xB
zB

yB

zG

xT
yG

ar
0

g

ar

yT

∑G : Space Robot Coordinate
∑B : Base Coordinate
∑T : Target Coordinate
: Center of Mass

Target

t

xI

Reaction Wheels

∑T
zT

ar

twi

ar (k)
e

tp(k)

xG

∑I

zI
yI
Arm 2

Figure 2. A model of a dual-arm space robot.

overlap with each other. The contact points on the target
surfaces are located in the x T - y T plane. Therefore, the
motion of the space robot arms will be restricted within the
x I - y I plane in the inertial space, and the spacecraft orientation will be regulated only in the x I - y I plane.
On the basis of these assumptions, define a vector of con1
1
2
2
trol variables X = [c, r ex
, r ey
, r ex
, r ey
, i e1, i e2] T ! R 7, where c
k
k
is the spacecraft attitude; r ex, r ey (k = 1, 2) denote the endeffector position of arm k along the x and y directions; and
k
i e represents end-effector rotation angles of arm k. The
dynamic model of the dual-arm space robot in task space can
be represented as [13]
AXp + BXo = x,
7#7

(1)

where A ! R
represents the inertia matrix and
BXo ! R 7 # 1 denotes the nonlinear terms. x ! R 7 is a vector
of control torques x = 6x w, x 11, x 12, x 13, x 21, x 22, x 23@T , where x w
and x ki (i = 1, 2, 3, k = 1, 2), respectively, denote reactionwheel torque and the ith joint torque of arm k.
Define an error state vector e as e = [e T1 , e T2 ] T with
e 1 = X - X d, e 2 = Xo - Xo d, where X d is the vector of desired
motion. Rearrange the dynamic equation (1) in state-space
form as
88

*

IEEE ROBOTICS & AUTOMATION MAGAZINE

*

december 2018

eo 1 = e 2,
eo 2 = - A - 1 B ^e 2 + Xo d h - Xp d + A - 1 x.

(2)

To generalize the control application to scenarios in which
the space robot parameters cannot be exactly known, e.g., a
space robot manipulating a target with unknown parameters,
system uncertainties are considered (i.e., matrix A and B are
not exactly known). Assume At = A + DA, where At denotes
the nominal value of A and is known, and DA represents the
uncertainty. To decouple the multiple input torques, construct
a virtual torque x * = At -1 x according to a diagonalization
method [14]. The substitution of x = At x * and At = A + DA
into (2) produces
eo 2 = - A - 1 B ^e 2 + Xo d h - Xp d + A - 1 ^ A + DAh x *
= f (e, t) + x * + h (e, t),
w here f (e, t) = - A -1 B ^e 2 + Xo d h - Xp d
A -1 D Ax * .

and

(3)

h (e, t) =

Smoothed Quasicontinuous Second-Order
Sliding Mode Controller
In [15], the authors compared different types of sliding
mode controllers for spacecraft-attitude-tracking maneuvers,



IEEE Robotics & Automation Magazine - December 2018

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - December 2018

Contents
IEEE Robotics & Automation Magazine - December 2018 - Cover1
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