IEEE Robotics & Automation Magazine - June 2021 - 19

^18.75 mmh for the sake of manufacturing simplicity. The
optimization results from various approaches for a given
point in the workspace are compared in Figure 6. As expected,
the approach used in [12] produces the stiffest PC, and a
more compliant result can be acquired through NPC. The
stiffness gets reduced further with the CTR strategy, which is
the proposed method.
From the optimized Kpc
, we computed kc
is known. Then, dd ()q nl
using (5) since rc
1 x can be computed point-wise and
curve-fitted, leading us to nlx from the integration. We can use
x nl
to synthesize a nonlinear pulley profile that imposes the
desired joint stiffness [15]. This pulley will act as a nonlinear
torsional spring. Note that the degree of nonlinearity depends
heavily on multiple variables in (3). Similarly, an additional
nonlinear pulley profile was generated to see the performance
of [12] and NPC. One of the advantages of utilizing PC to stabilize
the system is that it can be modeled as an additional feedforward
actuator effort without any changes in the control loop.
We utilize the extended methodology [15] by using the torque
profile nlx to generate the pulleys depicted in Figure 7.
Table 1 describes the necessary stiffness of the extension
springs based on the optimization results for 70 Nm / . We can
see that NPC by itself significantly reduces the stiffness of the
required springs compared to the previous approach. In the
proposed structure, where NPC and CTR are employed
together, the stiffness of the extension springs is further minimized.
Notice that the maximum moment arm of the proposed
structure is about 50% smaller than [12] and NPC,
making it preferable in compact settings.
Experiment Results
For the experiments, we installed the synthesized NPC
along with the linear extension springs based on the given
structures. However, due to the size constraints of the extension
springs, we could not find proper springs that satisfy
Table 1 and fit our mechanism for the approach taken in
[12]. Therefore, for the experiments, we compared only the
previous approach combined with NPC and the proposed
approach. We carefully examined the results to draw a reasonable
conclusion about the performance difference
between the proposed and past [12] approaches. The step
response and elliptical trajectory tracking results using
NuFingers are expressed in Figure 8. As the results suggest,
the system is unable to render high controller stiffness without
any PC and quickly diverges. Consistent with the simulation
results, stability is recovered with the addition of PC.
Both of the structures enable the system to quickly converge
to the equilibrium configurations.
From the experiments, we verified that the desired trajectories
could be stably tracked using the proposed structure
(NPC and CTR), which utilizes a much more compliant
PC component as compared to the other two structures.
The same trajectories were tracked using the previous
approach combined with NPC (i.e., [12] and NPC). The
performance difference was not noticeable in terms of the
Cartesian position error, but the difference was significant
in terms of the stiffness of the PC components, as seen in
Figure 6 and Table 1.
In unstructured settings, it is safer to keep a low passive
stiffness whenever applicable (i.e., a lower PC stiffness) due to
possible collisions, disturbances, or noise. This is especially
true in dexterous manipulation with fragile objects. Note that
sudden impacts or high frequency disturbances cannot be
handled by controller stiffness due to the control delay and
bandwidth of the system. Therefore, a high-frequency
response reduces to the system's passive dynamic response.
With a higher passive stiffness (i.e., a higher PC stiffness), the
system not only fails to passively reject high-frequency disturbances
but is also unable to render the desired stiffness,
becoming more prone to damaging the environment or the
robot itself because of its high passive stiffness. Therefore, utilizing
a more compliant PC component to render the same
desired stiffness is the safe, practical, and stiffness-efficient
solution. With the previous approach [12] alone, we suspect
Optimization Results Using Various Setups
150
100
50
-50
-100
-150
Kdesired
Kpassive
Kproposed
K[12] and NPC
K[12]
-150 -100 -50 050 100 150
Stiffness (N/m)
Figure 6. Kpc is optimized based on the criteria found in (3) for
70 Nm / . As expected, the previous work [12] results
in the stiffest Kpc
different setups as stated in the subscripts. The desired Cartesian
stiffness is
. The NPC alleviates the high stiffness, and CTR
further reduces the stiffness. The proposed structure incorporates
NPC and CTR and thus produces the most stiffness-efficient results.
74 mm
(a)
48 mm
(b)
Figure 7. The nonlinear pulley profiles are generated based on
the optimal stiffness to ensure stability for each configuration in
the workspace. The desired stiffness is set to
70 Nm / . (a) The
nonlinear pulley profile combined with the previous approach
(i.e., [12] and NPC). (b) The nonlinear pulley profile for the
proposed method (NPC and CTR).
JUNE 2021 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
19
Stiffness (N/m)
IEEE Robotics & Automation Magazine - June 2021

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