IEEE Robotics & Automation Magazine - June 2012 - 28

5

Maintaining Manipulability
and Realizing Desired Impedances
The structural redundancy of the robot can be utilized to
maintain a proper manipulability. In this respect, a singularity avoidance for the arms has been designed to prevent
rank deficiency of the Jacobian matrix J C (q) of the Cartesian impedance. A spatially bounded potential function is
set up to keep the kinematic manipulability measure

0
-5
-10
tstart

8

10

12

14
(a)

16

18

20

tend

tstart

8

10

12

14
(b)

16

18

20

tend

5
0
-5

10

mkin (q) ¼

Hardware (ICR Control On)
Simulation (ICR Control On)
Simulation (ICR Control Off)

5
0
tstart

8

10

12 14 16
Time (s)
(c)

18

20

tend

Figure 12. Parts (a)-(c) show the steering velocities of wheels
1-3 for activated and deactivated ICR repulsion in simulations
and experiments. The peaks directly correspond to the arising
singularities indicated in Figure 11.

parameterization of the impedance, i.e., the spatial stiffness. Correspondingly, Vimp (q, X odo , t) can be expressed by
Vimp (q, X odo , t) ¼ Vs (H r (q, X odo ), H dr (t), Kr )
þ Vs (H l (q, X odo ), H dl (t), Kl )

(6)

x (m)
0

0.2

0.4

0.6

0.8

Yb
0

Xb
Platform
Center

þ Vs (H b (X odo ), H db (t), Kb ),

B

-0.6

0.25
0.2
0.15
0.10
0.05
0
-0.05
0

Reference x
Current x
Current y
Current z

0.5 1

-0.8

Figure 13. The simulated ICR in the body frame (blue/solid:
deactivated leg motion, red/dashed: activated leg motion, and
black/chain dotted: possible wheel location).

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IEEE ROBOTICS & AUTOMATION MAGAZINE

1.5 2 2.5 3
Time (s)
(a)

3.5 4

4.5 5

0.1
0.05
0

-0.05

Wheel 2

28

(8)

where subscript t stands for torso and b for mobile base.
The parameterizations of the impedances to realize the
postures are specified by Kt and Kb .

Base Command
νxd (m/s)

y (m)

A

(7)

Vtb, imp (q, t, X odo ) ¼ Vs (H t (q), H dt (t), Kt )

-0.2

-0.4

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
det(J C (q)J C (q)T )

on a sufficiently high level. More details on the algorithm
by Ott can be found in [32].
In terms of providing unrestricted stereo vision, specific
postures of the torso support the neck joint actuators in
positioning the camera system in the head. Likewise, a separate trajectory for the mobile base has the ability to handle
obstacles in the workspace of the robot, which are known
in advance or given by a higher level planning instance
online. A potential function can be set up including these
two issues:

Location (m)

Steering Velocity
Wheel 1 (rad/s)
Steering Velocity
Wheel 2 (rad/s)
Steering Velocity
Wheel 3 (rad/s)

since both right and left TCPs are considered (subscripts r
and l).

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JUNE 2012

-0.1
0

0.5

1

1.5

2 2.5 3
Time (s)
(b)

3.5

4

4.5

5

Figure 14. Parts (a) and (b) show the step response for a
translational TCP motion in the x direction with a stiffness of
Kt ¼ 500 N/m.
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - June 2012
