IEEE Robotics & Automation Magazine - June 2021 - 15

environments [5]. To resolve this issue, parallel or distributed
actuation has been introduced in industrial settings [6], [7],
but these methods inevitably use additional motors, which
makes the actuation system bulky and unfit for lightweight
and compact applications.
Therefore, there needs to be an alternative option to add
elasticity to the system instead of solely relying on series
elasticity. Parallel compliance (PC) is one viable option to
complement the necessary stiffness. PC with series actuation
has been implemented by several researchers [8], [9].
However, these works consider only the performance of
noncompliant robotic end-effectors under position control
and neither investigate nor mathematically formulate the
effects of PC on the control stability. Some researchers have
used PC to save energy consumption during periodic
motions [10], but periodic motions are not common in
dexterous manipulation. Additionally, our aim is to use
coupled or nonlinear PC to achieve stability with minimal
stiffness increment.
This article addresses the effects of PC in tendon-driven
robotic hands on stiffness control stability, inspired by the
profound effects of joint compliance on human hand performance
[11]. In terms of control, the incorporation of PC is
an effective yet simple solution toward stability as it introduces
no time delay to the system and needs only a feed-forward
term in the control input. PC can simply be installed in
tendon-driven systems by adding extension springs along
with pulleys at the joint level without interfering with the
existing tendons.
This article demonstrates, for the first time, the optimization
process and stabilization effects of coupled and nonlinear
PC and proposes the possibility of minimizing the size,
weight, and cost of the overall actuation system with a stiffness-efficient
design. With this outcome, it is possible to stably
render a wide range of stiffness values without using
power-demanding motors or higher series stiffness, which
can be convenient in various problems such as dexterous
manipulation in unknown environments.
Stiffness Control Stability
To maintain the passivity of the overall system and
mechanically ensure that the system returns to its equilibrium
state, the controller stiffness must not exceed the robot's
passive stiffness [3], [4]. This limits the range of stiffness
values that a robotic system can stably realize, thereby limiting
the type of tasks that the system is able to carry out. As
demonstrated in [12], adding PC to the system is a good
option to shift the upper bound of the stable range. In this
research, we demonstrate that optimal passive stiffness is
achievable by NPC and CTR between joints. This enables
the system to perform tasks that require lower stiffness and
also reduces actuator effort.
For tendon-driven robotic fingers, the passivity-based
stability criteria can be formulated as follows [12]:
,
JK JK K,,
T
pc
xd
x
--1## passive
(1)
where K ,x passive
defined in
KJ () (2)
x,,
TT
sc
pc
where R and Ksc
passive=+ --j
CCT
ance matrices, respectively, and K ,j CCT
RK RK KJ 1
,
denote the routing and series complirefers
to the external
force-based stiffness term [13]. As seen from (1), the
addition of the PC term Kpc
influences both the lower
and upper boundaries of the passivity range. Therefore, as
long as the desired stiffness is known throughout the task,
the optimal form of PC can be determined through convex
optimization.
The Optimization of PC
The overall stiffness of the end effector and also the
torque requirements all depend heavily on Kpc
as suggested
by (2). Therefore, it is crucial to formulate an optimization
problem to identify the amount of PC that
strikes a balance between the necessary stiffness and stability.
Note that this method can be extended to other
physical configurations with different tendon routing
strategies or link lengths. From the optimization, we seek
to determine Kpc
for a predefined task environment and
synthesize the linear and nonlinear PCs that will physically
realize this Kpc
.
In [12], the possibility of using linear PCs to adjust
the stable range of stiffness was investigated. However, it
is rare to find characteristics that are purely linear and
decoupled from one another, as dealt with in [12]. For
instance, it is easy to see human fingers exhibit highly
coupled and nonlinear PCs depending on their configuration.
In this study, notable improvements have been
made: we introduce NPC along with CTR, which stabilizes
the system with minimal change to the overall stiffness.
Through simulation and experiments using the
NuFingers illustrated in Figure 1, this article demonstrates
the validity and effectiveness of the proposed
method.
Optimization
To satisfy the stability criteria throughout n points in a given
workspace while minimizing the resultant overall stiffness,
the fully populated Kpc
and K ,xd
denote the system's passive and
represents the PC matrix. K ,x passive
desired Cartesian stiffness matrices, respectively; J is the Jacobian
matrix; and Kpc
is
is optimized by a multiobject search
algorithm based on interior-point methods [14]. One can
relate the matrix inequalities found in (1) to the size comparison
of the hyperellipsoids represented by the matrices, regardless
of the number of dimensions.
To avoid unnecessary stiffness, however, the cost function
was defined as the Frobenius norm of Kpc
since it is desirable
to minimize the area of the stiffness ellipse. By trimming
out the excess stiffness, we can also minimize the chance of
damaging the unknown environment as well as reduce the
motor effort.
JUNE 2021 * IEEE ROBOTICS & AUTOMATION MAGAZINE *
15
IEEE Robotics & Automation Magazine - June 2021

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