IEEE Robotics & Automation Magazine - September 2013 - 76

CJs Comparison Criteria
In the present implementation, the helical (HEL) joint connects
the proximal phalanx to the palm whereas the spiral (SPIR)
joint connects the medial and distal phalanges. The proposed
large-displacement CJs are compared by defining a set of
compliance matrices, composed by entries of the nonuniform
physical dimensions, which describe how the CJ reacts to
external wrenches. Naturally, being a differential operator,
each matrix measures only local configuration-dependent
CJ properties. Hence, the comparison is based on a set of
normalized, dimensionless local matrices [C ]k, evaluated by
means of FEM within the CJ workspace. (Having defined
I * = l/EC i y M y, C i y M y being the CJ principal compliance,
displacements are normalized by the overall joint length L
[Figure 4(a)], forces by EI */L2 , and moments by EI */L .)
Given the normalized dimensionless loads vector 3 f acting
on a reference point, the normalized dimensionless joint
displacement 3 u is expressed by

(S1)

where the translational and rotational components along the
reference axis of both the load and the displacement vector are
specified. The general structure of the compliance matrix is
C 12
C 22
C 32
C 42
C 52
C 62

C 13
C 23
C 33
C 43
C 53
C 63

C 14
C 24
C 34
C 44
C 54
C 64

C 15
C 25
C 35
C 45
C 55
C 65

V
C 16W
C 26W
C 36WW
.
C 46W
W
C 56
W
C 66W
X

(S2)

-22.5° 0° +22.5°
-45°
+45°

I Rk = C Rk F ,

*

IEEE ROBOTICS & AUTOMATION MAGAZINE

F

.

(S4)

N

IR =

/ I Rk

k =1

N

N

,

g

IT =

/ I Tk

k =1

N

.

(S5)

A smaller GPI indicates a better global CJ behavior.
Note that small secondary rotations at joint level can be
dramatically amplified at the end of serial articulated chains,
hence the evaluation of g I R is usually more significant. The
GPIs obtained from the analysis are g I T = 9.88 $ 10 -3 and
g
I R = 1.00 $ 10 -2 concerning the SPIR CJ, and g I T = 1.59 $ 10 -2
and g I R = 1.60 $ 10 -2 concerning the HEL CJ.

0.16
0.14
0.12
SPIR
HEL

0.1
0.08
0.06
-50

0
Principal Angular Task
(a)

0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35

50

SPIR
HEL

0
Principal Angular Task
(b)

50

(b)

Figure S1. The FEM models and workspace discretization:
(a) SPIR joint and (b) HEL joint.

76

I Tk = C Rk

A smaller LPI indicates a better local CJ behavior. Starting
from this local evaluation of the joint behavior, global
performance indexes (GPIs), which summarize the overall
joint performance over the whole workspace are defined and
evaluated,

-50

(a)

(S3)

The two local performance indexes (LPIs) that characterize
the joint performance in each configuration, specifically a
rotational LPI, IR , and a translation LPI, IT, are evaluated by
using the weighted Frobenius norm [10] (see Figure S2):

g

In ideal revolute CJs, pure rotation along the principal
direction (the y axis) is expected. Hence, its normalized
compliance matrix is characterized by a finite unit entry in C55,
all the other entries being null. The term C55 is reported in (S2)
and arises from a rotation around the principal axis under the
action of the load m y . On the contrary, a real CJ will present
finite values of normalized compliance coefficients along the
other directions.
To evaluate the compliance matrices in the different joint
configurations and assuming that SPIR and HEL joints are
capable of 90° rotation ! 45 °, the joint workspace is divided
into N = 5 equally spaced intervals, as shown in Figure S1,
where 0° represents the joint undeformed configuration. Then,
a fraction of the maximum principal load m y is applied to
reach the kth (where k = 1, ..., N ) configuration, and a small

-22.5° 0° +22.5°
-45°
+45°

T

Rotational LPI

R
SC 11
SC 21
SC
31
C =S
SC 41
SC 51
SS
C 61
T

C k = 6C TR C TT@k , C Rk = C i, j, C Tk = C l, j
i = 1, ..., 3, l = 4, ..., 6, j = 1, ..., 6.

Translational LPI

3u = C $3f
3 u = [ 3 ux 3 uy 3 u z 3 ix 3 iy 3 iz ] T
3 f = [ 3 fx 3 fy 3 fz 3 m x 3 m y 3 m z ] T,

variation of the load vector 3 f is applied while maintaining
the previous principal load. As a result, the related joint
displacement 3 u and the related compliance matrix can
be evaluated. Each matrix is then split into two submatrices
C R and C T containing, respectively, the coefficients relative
to angular and linear displacements along the reference
directions:

*

september 2013

Figure S2. LPI trend for joints SPIR and HEL [Figure 4(a) and
(b)] (a) Trend of LPI IR for SPIR and HEL. (b)]. Trend of LPI IT
for SPIR and HEL.



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