IEEE Robotics & Automation Magazine - June 2021 - 18

addition to NPC, our proposed setup also uses a CTR strategy
for PC.
After careful examination, we have found that a configuration-dependent
change in optimal stiffness occurs primarily
Single Finger
Object Manipulation
in the first joint. Hence, we chose to synthesize the NPC associated
with the first joint and additionally introduce CTR on
top of the entire mechanism, as seen in Figure 5. In the next
section, we derive the stiffness of the extension springs attached
to the antagonistic pairs based on the structure of the mechanism
using the optimization results from (3).
The Computation of Extension Spring Stiffness
In the proposed structure, we utilized two different sets of
torques to impose the required stiffness
K :pc nonlinear
torques applied at the first joint and the additional coupled
torques on top. Details of the physical structure of the mechanism
are shown in Figure 5. The resulting torques from this
configuration can be expressed as
(a)
(b)
Figure 4. The proposed structure (NPC and CTR) is validated
through a set of experiments in Cartesian space. (a) Scenario
1: step input to a single finger. (b) Scenarios 2 and 3: object
manipulation with two fingers.
x==
;
x
x
xnl
1
2
where ,,kc
E =
xnl
+- +-+
-
2
kr qq qq
2
cc 1102 20
cc 1102 20
((
((
)(
))
and rc
kr qq )(qq ))
2
2
G ,
(4)
represent torque from the NPC, the
extension spring stiffness, and the moment arm for the coupled
PC tendons, respectively. Each joint's current and equilibrium
positions are represented as ,, ,qqq12 10
and q .20
Table 1. The physical parameters for the different
structures.
Joints
1
2
1
2
1
Structure
[12]
[12] and NPC
Proposed
Stiffness ( /Nm)
1,548.04
534.57
749.5426
398.3
749.5426
1 and 2 NPC and CTR 238.1724
Moment Arm
(mm)
18.75
18.75
26.7-49.5
18.75
22-25.2
18.75
Based on the definition of the joint stiffness, the stiffness
matrix is the partial derivative of joint torques:
K ==>
pc
2
2x
q
dq
d
xnl + kr2
cc
1
() 2 cc
2kr2
2
2
kr
kr
dd ()q nl
2
2
cc
cc
H .
(5)
This is the overall stiffness matrix for the mechanism. Notice
that only the nonlinear element
and dd ()q nl
optimization results. The radius of the moment arms rc
designed to be the same as the actuated joint radius
1 x varies throughout
the workspace. Therefore, we carried out the optimization
process shown in (3) with a constraint that Kpc
of (5). Next, kc
takes the form
1 x were computed from the
was
Grounds for CTR
CTR Pulleys
CTR-Related Tendons
Actuated Tendons
NPC-Related Tendons
Grounds for NPC
Joint 1
Joint 2
NPC Pulley
(a)
Grounds for CTR
(b)
Figure 5. Tendon routing for the proposed structure. (a) In the proposed setup, there are two notable changes from [12]. The top layer
uses CTR that accounts for the off-diagonal terms in
K .pc In addition, NPC in the bottom layer provides optimized stiffness that depends
on the finger configuration. (b) The actual set up of the mechanism is depicted. Note that all of the tendons are connected to the linear
extension springs.
18 * IEEE ROBOTICS & AUTOMATION MAGAZINE * JUNE 2021
Actuators
IEEE Robotics & Automation Magazine - June 2021

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