IEEE Robotics & Automation Magazine - June 2021 - 21

the actuation system. For example, it is
easy to verify that nominal torque,
size, weight, and cost of the motor all
increase proportionally for commercialized
dc motors [18]. In turn, the
battery size can also be reduced, further
lowering the weight.
In addition, the desired trajectories
and the actuation mechanism remained
the same across the experiments.
Therefore, the striking difference
depicted in Figure 9 is entirely due to
the difference in stiffness for the different
physical structures depicted in Figure
6. Based on these figures and Table 1,
we can safely conclude that the previous
approach [12] would result in
higher torque demands due to a higher
stiffness, further requiring heavier
actuation mechanisms.
1
0.8
0.6
0.4
0.2
-0.5
0102030
Time (s)
(a)
40 50
Torque From PC
0.5
Joint 1
Joint 2
0.4
0.3
0.2
0.1
0102030
Time (s)
(b)
40 50
Energy Added to PC
[12] and NPC
Proposed Approach
Modeling Limitations
The nonlinear effects of stiction and
damping on stability underscore the
shortcomings with classical passivitybased
stability analysis. The formulation of the stability criteria
does not take into account dynamic parameters such as
damping and inertia [12]. This assumption makes the criteria
a conservative measure of stability as more damping or
inertia will help the system become stable. Indeed, we included
no damping in the simulations.
However, in real-life applications, we cannot ignore inertia
Figure 9. (a) The maximum restoring torques at the first joint and (b) the potential energy
in PC elements are shown to have about 26% and 50% reduction, respectively, in the
proposed NPC and CTR setup (dark lines) compared to [12] and NPC (light lines). Potential
energy was calculated according to the principle of virtual work. These plots demonstrate
that the proposed setup can be much more efficient in terms of stiffness and torque
requirements-and possibly also energy consumption-while preserving the overall passivity.
the desired Cartesian stiffness stays constant. The interaction
force affects the K ,j CCT
term in (2). Therefore, it will be interesting
to look for a new design that allows the stiffness to vary
along with the interaction force.
and damping as they contribute to the stability of the overall
system. We have to take into account that the controller stiffness
may be greater than the theoretical passive stiffness of
the system, but the system can remain stable due to damping
and inertial effects. Thus, to better reflect the stability criteria
derived in (1), it is desirable to maintain the experimental systems
to be as frictionless and massless as possible.
The Effects of Conservative Congruence
Transformation Term
Throughout the optimization process, it is interesting to see
that as the K ,j CCT
Conclusions and Future Work
We have introduced the systematic optimization process of
PC and demonstrated that the stability of the system could
be achieved from efficient and optimal design choices that
lead to overall passivity. Using the NPC and CTR strategy,
we have successfully reduced the overall stiffness of the PC
components as compared to the previous approach [12].
The reduction of passive stiffness is not only beneficial in
terms of power consumption but is also safer around
unknown environments.
Optimal Parallel Stiffness
term enlarges, the amount of coupling
between the first and second joint that exists in K ,j CCT
increases. This effect is illustrated in Figure 10, suggesting that
a single component of the optimal stiffness matrix is affected
by the displacement of all of the joints. This effect is not as
exaggerated in our experiments as depicted in Figure 10
because the degree of this coupling effect depends on the ratio
between the interaction force and the desired stiffness.
Thus, it is important to know 1) the range of interaction
forces required for a particular task before it is executed and
2) whether the generated optimal stiffness covers the whole
range. This plot also implies that, if the interaction force
changes, the optimal stiffness changes along with it, even if
20
10
20
-20
Q1
-40
-60
-40
Q2
100 N
50 N
0 N
-80
20
18
16
14
12
10
8
6
4
2
Figure 10. The amount of coupling and nonlinearity becomes
larger as the interaction force increases with respect to the desired
stiffness. The top, middle, and bottom layers each indicate the (1,1)
element of the optimal PC matrix across the workspace where
the interaction force is assumed to be
0 N , 50 N , and 100 N ,
respectively. The desired stiffness is set to 150 Nm / . for all. q1
represent the joint positions.
q2
JUNE 2021 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
21
and
Torque (Nm)
Stiffness (Nm/rad)
Potential Energy (J)

IEEE Robotics & Automation Magazine - June 2021

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