IEEE Robotics & Automation Magazine - December 2018 - 90

and T is the sampling time.
Phase 3 (constrained phase):
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xi

= - fi - m i eo 1i - M i sat (v i),

where a 0 and b 0 represent dimensions of the spacecraft base,
a (i k) (i = 1, 2, 3) denotes length from joint i to the center of
mass of link I, and b (i k) denotes the length from the center of
mass of link i to joint i + 1, as illustrated in Figure 2.
Because the translation of the base is not controlled and a
singularity problem is not considered, the initial system
configuration is specified such that the contact points stay
within the space robot workspace. The parameters of initial
configuration given in Table 2 can satisfy the previously
explained condition. On the other hand, the desired motion
in the task space is constrained to the x I - y I plane, and the
space robot end effectors are supposed to reach the contact
points in a desired orientation.

(10)

where m i denotes the ith positive diagonal entry of K, and
sat ^ $ h represents the commonly used saturation function.
Simulation
The space robot has two identical arms mounted centralsymmetrically with respect to the centroid of the base. It is
assumed that the pose of the target and the position of the
desired contact points can be observed or estimated. Parameters of the space robot and the target are listed in Table 1,

Table 1. The specifications of the dual-arm space robot and the target.
Body

Base

L1

L2

L3

Reaction
Wheel

Target

Mass (kg)

100

10

10

10

11.5

40

Inertia (kg . m2)

(k)

(k)

(k)

Ix

30

0.016

0.016

0.016

0.1

15

Iy

30

1.6

1.6

1.6

0.1

15

Iz

30

1.6

1.6

1.6

0.1

15

(m)

1.25

1

1

1

-

0.5

b 0 or b i (m)

1.25

1

1

1

-

0.5

a 0 or a

(k)
i
(k)

Table 2. The initial conditions of the approaching operation.
Description
Position

Symbol

Value

a (1)
e

[- 4.1563, - 1.2500, 0] T m

a (2)
e

[- 4.1563, 1.2500, 0] T

r

End effector

r

Contact points

a
a

Attitude/Angles

Linear velocity
Angular velocity

FFSR

a

Base

a

Target

a

Joint

p

(1)

p

(2)

Unit

[- 1.9852, - 0.2198, 0]
[- 2.3273, 0.7198, 0]

rg

[0, 0, 0]

r0

T

[1.0938, 0, 0]

rt

U
U

(2)
M

[60, - 120, 60]

m

m

T

[- 60, 120, - 60]
T

m

m
T

[- 2.1563, 0.25, 0]

(1)
M

T

m
T

T

m
°

T

°
°

Base

US

[0, 0, 0]

Target

Ut

[0, 0, 20] T

°

FFSR

vg

0

m/s

Target

vt

0

m/s

Joint

Uo M

0

rad/s

FFSR

~g

0

rad/s

Base

~B

0

rad/s

Target

~t

0

(1)

1

1

1

2

2

rad/s
2

2

Note: U S = [0, 0, c] T , U M = [z 1, z 2, z 3] T , U M = [z 1, z 2, z 3] T , U t = [0, 0, i t], i t represents
the target orientation in the x I - y I plane. FFSR: free-flying space robot.

90

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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december 2018

Path Tracking
The desired trajectories are defined in
task space, as required for many space
missions. Specifically, the end effectors are expected to reach the corresponding contact points along two
straight lines. Because the target is
assumed stationary, the end effectors
are expected to arrive the target points
with zero velocities in a perpendicular
configuration with respect to the target surfaces. During this process, the
spacecraft base must be maintained at
its desired attitude c d = 0. We introduce a polynomial [17] to describe
the desired trajectory as
X d (t) = X d (t 0) + ^15t 4n - 6t 5n - 10t 3nh
# ^ X d (t 0) - X d (t f )h,
where X d (t 0) is determined by the initial configuration, the specific parameters for which are given in Table 2;
X d (t f ) depends on the previously
mentioned requirements; t n = t/t f ; t
represents the elapsed time; t f represents the total motion time; and
t f = 150 s.
The simulated course of the dualarm space robot's postural change
under AVSC when approaching the
target is illustrated in Figure 4. Intuitively, this demonstrates that the space
robot's end effectors reach the contact
points following the predetermined
straight lines and touch the target in a
desired perpendicular configuration
with respect to the contact surface.
Meanwhile, the spacecraft base maintains its desired orientation, even with
the disturbance caused by the motion
of the manipulators.
IEEE Robotics & Automation Magazine - December 2018

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