IEEE Robotics & Automation Magazine - December 2015 - 44

Table 1. The Cartesian and joint stiffness at the time instants from the reported experiment.
joint
Plotted values from Figure 7 are shown in red and blue. Passive joint stiffness k passive
reaching
a mechanical limit are marked with a " ) ."
Point of Time

t1

K des ; N E
M

>

K passive ; N E
M

>-1695

K active ; N E
M

>

116 5
0
5 97
1H
0 1 4545

>

116 5
0
5 97
1H
0 1 4545

>

K res ; N E
M

>

96 1
0
1 95
0H
0
0 280

>

96 1
0
1 95
0H
0
0 441

>

K passive ; Nm E
rad

R 394 )V
S
W
S 123 W
S 113 W
SS
W
)W
T 161 X

t2

100
0
0
0 100
0H
0
0 300

100
0
0
0 100
0H
0
0 300
922 -1705 12
8110 65 H
- 12
65 489

Cartesian Position (m)

0.2

xi
xq
zi
zq

0

592 - 1
11092 25H
-1
25 324

116 5
0
5 97
1H
0 1 4545

R541V
S
W
S122W
S124W
SS
WW
T250X

and passive stiffness result in the serial stiffness K res, which is
close to the desired one.
The compliant behavior of the impedance controller for
deflections from the desired position can be seen in the measurements at the time instants t 2 and t 3 . At t 2, the TCP is
deflected in the z-direction. As the passive stiffness in this
direction (489 N/m) is close to the desired one (300 N/m),
most of the deflection can be observed between motor and
link side, as shown by the deviations at t 2 between z i and z q
in Figure 7. Therefore, the motor barely moves at t 2 . At t 3, the
situation is completely different. Here, the manual disturbance is applied in the x-direction. Looking at the respective
elements in Table 1 (blue), one can see that the active stiffness
(116 N/m) is now significantly smaller than the passive one

KActive

xd
qd

xi

i
q

KPassive
t1

t2

IEEE ROBOTICS & AUTOMATION MAGAZINE

*

DECEMBER 2015

xq

F

t3

Figure 7. The measurement plots of the TCP position x q and z q and
motor TCP position x i and z i . The TCP is deflected in the x- and
z- directions. For the case of a low passive stiffness and a high active
stiffness ( z- direction, time instant t 2 ), the passive stiffness element
greatly deflects. In the opposite case ( x- direction, time instant t 3 ),
the passive stiffness element show little deflection, motor, and linkside move simultaneously.

*

879

>592

102 5
0
5 96
0H
0
0 303

R395V
S
W
S207W
S172W
SS
WW
T151X

Time (s)

44

>

>-1705

should approach K des . However, looking at the diagonal
entries, the desired value is only satisfactorily reached in the
z-coordinate with 299 N/m. The x- and y-coordinates are
far away from the desired values. This is due to the lower
passive joint stiffness bounds (see Figure  2) that prohibit
reaching the low Cartesian stiffnesses. The robot configuration requires high joint torques in the first axis to support the
joint
robot and high joint stiffnesses result (see Table  1, k passive
entries marked with a ")" reach the mechanical minimum
values). To compensate for the high passive Cartesian stiffness entries in K passive, the active compliance optimization
(step 2) computes low stiffness values for the x- and y-components (diagonal elements: 116 N/m, 97 N/m) and a large
value for the z-component (4545 N/m). Finally, the active

-0.2

100
0
0
0 100
0H
0
0 300

>

922 -1695
1
8398 -8 H
1
- 8 299

joint

t3

(a)

(b)

F

Figure 8. The deflection behavior of two simple VS systems. (a) A onelink VS joint is sketched. A force F acting at the TCP deflects the link and
the motor coordinate. The active stiffness K active is valid between the
motor and the desired position, while the passive stiffness K passive is
valid between the motor and the link. (b) The deflection behavior of a
planar robot is depicted. The motor TCP position x i is shown between
the desired TCP position x d and the TCP x q .
Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - December 2015
