IEEE Robotics & Automation Magazine - June 2021 - 20

that NuFingers would track the desired trajectory as well, but
at the expense of a very high stiffness.
On a side note, object manipulation in an elliptical path
was more robust against a higher controller stiffness compared
to a step response. We were able to reach above the robot's passive
stiffness in the elliptical path. This is because the frequency
of the control input is much lower
^h1
+10
/s
compared to
the step input, where oscillations are usually at a higher frequency
and
stiction play a more important role in stabilizing the system,
and thus it is easier to reach stability [16], [17].
Discussion
We presented a stiffness-efficient method of stabilization
through passivity and demonstrated its positive effects via a
simulation and experiments. In the simulation, we assumed
that there is no damping to better reflect the derived stability
criteria. As expected, the proposed mechanism with NPC and
CTR successfully stabilizes the system with a minimal addition
of stiffness. Using NuFingers, we carried out experiments, further
validating the efficacy of the proposed structure. Through
three different scenarios, we verified the stabilization effects in
an otherwise unstable system. After careful analysis, we make
the following interesting observations.
Torque Requirements
The amount of torque required to overcome the passive stiffness
element is profoundly different depending on the
No PC
0.2
0.1
0.1
0.05
-0.05
0.01
0.05
-0.05
01 2
Time (s)
(a)
3
0.5 1 1.5 2
-0.05
-0.02
0.02
02 4
Time (s)
(b)
6
24 6
-0.05
0.02
-0.02
0.02
05 10
Time (s)
(c)
Figure 8. Experiment results from the NuFingers setup are shown. In all cases, the desired Cartesian stiffness is set high: -70 Nm / in
the case of a single manipulator and 140 Nm / in the case of object manipulation. (a) Without any PC, the system shows an oscillatory
behavior or loses the grip on the object instantly. (b) With [12] and NPC, the system becomes stable at the expense of higher stiffness.
(c) With the proposed setup (NPC and CTR), the system maintains the overall passivity and stability even with significantly smaller
overall stiffness. The steady-state error is due to unmodeled friction, which is shown to be more profound under low velocities.
Regardless of the friction under low velocities in elliptical paths, the system without optimized PC cannot render high stiffness.
20 * IEEE ROBOTICS & AUTOMATION MAGAZINE * JUNE 2021
0 24 68 10
0.05
024 6 0102030
0.05
0.1
05 10
0.05
15
^h+10 /.s At lower frequencies and velocities, friction
structure. Due to the absence of CTR and the use of the stiffer
PC requirement for [12] and NPC, the size of the pulley is
considerably larger than the proposed structure, as illustrated
in Figure 7. Detailed moment arms can be found in Table 1.
Therefore, the amount of torque and stiffness at the first joint
differs in the two structures despite the equal stiffness of the
extension springs.
As mentioned, the torques originating from PC are fairly
simple to calculate as they depend only on the configuration of
the robot. Therefore, when controlling a robot with PC, one
can simply add a feedforward term that would compensate for
the restoring torques generated by the PC components. The
restoring torques can be computed by integrating the stiffness
curves from the corresponding equilibrium position, which
are all known parameters in this work, as found in (4).
Figure 9 demonstrates that the maximum torque required
to compensate for the restoring torques from PC has been
reduced by up to 26% at joint 1 in Scenario 1. As a consequence,
the potential energy added to the PC elements also
decreased by up to 50%. This suggests that the incorporation
of NPC and CTR can reduce the torque requirements greatly
while maintaining a similar level of stability.
From the potential energy plots, we can further observe
the possibility of energy expenditure reduction. This can be
of great importance in the fields of wearable platforms, such
as prostheses or exoskeletons, where power consumption is
crucial to battery life. It is also known that a reduction in the
required motor torques affects the size, weight, and cost of
[12] and NPC
Proposed Approach
xd
yd
y
x
Object Manipulation
(Ellipse)
Object Manipulation
(Step)
Position (m)
Single
Manipulator

IEEE Robotics & Automation Magazine - June 2021

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - June 2021