FRAMEWORK OF AN ESTIMATION ALGORITHM OF TIME
VARYING MULTIJOINT HUMAN ARM VISCOELASTICITY
Mingcong Deng, Ni Bu and Akira Yanou
Graduate School of Natural Science and Technology
Okayama University 3-1-1 Tsushima-Naka, Okayama 700-8530, Japan
Keywords:
Multijoint human arm viscoelasticity, Non-Gaussian system, Robust estimation.
Abstract:
The paper concerns a framework of an estimation of multijoint human arm viscoelasticity in a small sufficient
time period. The uncertainties have to be considered in estimating the viscoelasticity of the multijoint human
arm. In general, the uncertainties existing in the structure of the human arm and the motor command from
the central nervous system are subject to the non-Gaussian noises. A generalized Gaussian ratio function is
brought in to deal with the non-Gaussian noises. The momotonicity of the generalized Gaussian ratio function
is studied based on the approximation formula of Gamma functions, then a robust condition is proposed for
the computation of even moments using shape parameters. That is, we can guarantee the accuracy of the
simulation results and experimental results by the robust condition. The effectiveness of the proposed method
is confirmed by the experimental results.
1 INTRODUCTION
One of the key characteristics that human beings sur-
pass the animals is the human arm. According to the
central nervous system (CNS), human can do many
things by making full use of human arm. Simply,
when human beings want to do something by arm,
(e.g. taking something in front of him) the arms move
forward following the guide of the CNS, adjust the di-
rection and gradually reduce the distance between the
hand and the object, finally the object can be taken.
Actually, the human arm is derived by the multijoint
muscle generated torque, which is assumed to be a
function of angular position, velocity and motor com-
mand of CNS (Gomi and Kawato, 1996; Gomi and
Kawato, 1997). The change of the torque is caused by
multijoint arm viscoelasticity which consists of joint
stiffness. Joint stiffness is regulated by muscle inher-
ent spring-like properties, neural feedbacks, and vis-
cosity. In the fields of industrial robots and medical
service, the study of the human arm viscoelasticity
plays an important part. For example, if the knowl-
edge of how the arm moves according to the CNS is
known, some artificial limbs can be designed to help
the disabled. So in order to get some corresponding
knowledge about the moving human arms, the esti-
mation of the human arm viscoelasticity is discussed
in this paper.
The estimation of the viscoelasticity of the hu-
man arm has been studied by many researchers
(Deng, Inoue, Gomi and Hirashima, 2006; Gomi and
Kawato, 1996; Gomi and Kawato, 1997; Deng, Saijo,
Gomi and Inoue, 2006; Kim, Kang, Kim and Park,
2009). A high-performance manipulandum was de-
veloped to measure human arm stiffness based on
the equilibrium-point control hypothesis (Gomi and
Kawato, 1996). The authors discussed the manipu-
landum in details in (Gomi and Kawato, 1997): by
using the manipulandum and a new estimation al-
gorithm, human multi-joint arm stiffness parameters
during multi-joint point-to-point movements on a hor-
izontal plane were measured. Later, online estima-
tion algorithm of the human arm viscoelasticity was
proposed in (Deng et al, 2006), (Deng, Inoue and
Zhu) and (Iseki, Deng, Inoue and Bu, 2009). An in-
tegrated procedure to study on real time estimation
of time varying multijoint human arm viscoelastic-
ity was proposed in (Deng et al.) concerning the
uncertainty factor consisting of time-varying motor
command from central nervous system, measurement
noises and modeling error of the rigid body dynam-
ics. However, there exist some problems, for ex-
ample, in (Gomi and Kawato, 1996) and (Gomi and
Kawato, 1997), the stiffness, viscosity, and inertia pa-
258
Deng M., Bu N. and Yanou A. (2010).
FRAMEWORK OF AN ESTIMATION ALGORITHM OF TIME VARYING MULTIJOINT HUMAN ARM VISCOELASTICITY.
In Proceedings of the Third International Conference on Bio-inspired Systems and Signal Processing, pages 258-263
DOI: 10.5220/0002727202580263
Copyright
c
 SciTePress

rameters of the human arm are estimated by applying
small perturbations which means a apparatus of a very
high cost. This may hinder many researchers from
the studying of human arm. So in order to reduce
the cost of the experiment, we consider to estimate
the viscoelasticity of the human arm in a small suffi-
cient time period without perturbation. Moreover, the
online estimation is based on the former experimen-
tal data, and there may be some difference or some
inaccurate information from the actual value. So in
this study, the monotonicity of the generalized Gaus-
sian function is proved to be monotonically decreas-
ing and a robust condition of the computation of the
even moment is given, that is: when the varying of the
shape parameter is bounded, the variance of the even
moment is guaranteed to be bounded. Then we can
guarantee the accuracy of the simulation results and
experimental results by the robust condition.
The outline of the paper is given as follows: the
human arm model and the estimation filter of vis-
coelasticity for human arm model are introduced in
Section 2; Section 3 includes the main results, the
monotonicity of the even moments and the robust
condition; The experimental results are shown in Sec-
tion 4, and the last part is the conclusion.
2 HUMAN ARM DYNAMIC
MODEL AND ESTIMATING
FILTER FOR
VISCOELASTICITY
Two-link rigid human arm dynamics on the horizontal
plane can be described as the following equation:
Ψ(¨q, ˙q,q) = τ
in
(˙q,q,u) (1)
Here, Ψ(·) denotes a two-link arm dynamics, and q, ˙q
and ¨q are angular position, velocity and acceleration
vector, respectively. τ
in
can be regarded as a function
of angular position, velocity, and motor command, u
descending from the supraspinal central nervous sys-
tem, where
q = (θ
1
(t),θ
2
(t))
T
τ
in
= (τ
s
,τ
e
)
T
(2)
θ
1
(t) is shoulder angle and θ
2
(t) is elbow angle
shown in Fig. 2, where τ
s
= τ
1
, τ
e
= τ
2
. Taking the
derivative of (1):
∂Ψ
∂¨q
d¨q
dt
+
∂Ψ
∂˙q
d˙q
dt
+
∂Ψ
∂q
dq
dt
=
∂τ
in
∂˙q
d˙q
dt
+
∂τ
in
∂q
dq
dt
+
∂τ
in
∂u
du
dt
(3)
If the arm is assumed to be rigid body serial link
system, such that:
Ψ(¨q, ˙q,q) = I(q)¨q+ H(˙q,q) (4)
where, D and R present muscle viscosity and stiffness
matrix, and
−
∂τ
in
∂˙q
= D =

D
ss
D
se
D
es
D
ee

−
∂τ
in
∂q
= R =

R
ss
R
se
R
es
R
ee

(5)
The subscript ss of D and R represent the shoulder
single-joint effect on each coefficient. Similarly, se
and es denote cross-joint effects, and ee denotes the
elbow single-joint effect. Then according to (3) (4)
and (5), the following equation can be established:
I(q)
d
¨
q
dt
+
∂H(
˙
q,q)
∂
˙
q
¨q+ [
∂I(q)
¨
q
∂q
+
∂H(
˙
q,q)
∂q
]˙q
= −D¨q−R˙q+
∂τ
in
∂u
du
dt
(6)
Here the corresponding parameters of I and H are
given as follows:
I =

I
11
I
12
I
21
I
22

(7)
I
11
= m
1
l
2
g1
+ m
2
(l
2
1
+ l
2
g2
) +
˜
I
1
+
˜
I
2
+ 2m
2
l
1
l
g2
cosθ
2
= Z
1
+ 2Z
2
cosθ
2
I
12
= I
21
= m
2
l
2
g2
+
˜
I
2
+ m
2
l
1
l
g2
cosθ
2
= Z
3
+ Z
2
cosθ
2
I
22
= m
2
l
2
g2
+
˜
I
2
= Z
3
(8)
H =
"
−m
2
l
1
l
g2
sinθ
2
(
˙
θ
2
2
+ 2
˙
θ
1
˙
θ
2
)
m
2
l
1
l
g2
˙
θ
1
2
sinθ
2
#
=
"
−Z
2
sinθ
2
(
˙
θ
2
2
+ 2
˙
θ
1
˙
θ
2
)
Z
2
˙
θ
1
2
sinθ
2
#
(9)
from the above equations (6)-(9), we can get the fol-
lowing relationship:
τ
in
= −D˙q−Rq+
Z
∂τ
in
∂u
du (10)
Since the sampling time ∆t → 0, then du → 0, so the
value of
R
∂τ
in
∂u
du → 0. By using a band-pass filter for
(10), the high frequency and low frequency measure-
ment noise can be removed. The filtered torque τ
f
in
,
position θ
f
1
(t) and θ
f
2
(t), vecocities
˙
θ
f
1
(t) and
˙
θ
f
2
(t)
satisfy the following relationship:
τ
f
in
= XU + ∆+ ζ
1
(11)
FRAMEWORK OF AN ESTIMATION ALGORITHM OF TIME VARYING MULTIJOINT HUMAN ARM
VISCOELASTICITY
259

where X is the regression vector, U is the time-
varying parameter vector to be estimated, and
X =

θ
1
θ
2
˙
θ
1
˙
θ
2
0 0 0 0
0 0 0 0 θ
1
θ
2
˙
θ
1
˙
θ
2

U =

R
ss
R
se
D
ss
D
se
R
es
R
ee
D
es
D
ee

T
(12)
where ∆ = [∆
1
,∆
2
]
T
consists of the structural un-
certainties which are assumed to be Gaussian. ζ
1
=
[
¯
ζ
11
,
¯
ζ
22
]
T
is the non-Gaussian measurement error
matrix of filtered measurement noise.
First, the above model needs to be converted into
its discrete time state-space form as follows.
U(t + 1) = U(t) + ζ
2
, t = 1,2,···
τ
f
in
(t + 1) = X(t + 1)U(t + 1) + ∆(t+ 1) + ζ
1
(t + 1)
where, ζ
2
is white noise. The shape parameters of
the probability density function (pdf) are known to
control the shape of the distribution. For the gener-
alized Gaussian uncertainty factor ∆
i
(t) +
¯
ζ
ii
(t)(i =
1,2) with zero mean, variance σ
2
i
and shape parame-
ter γ
i
is given by:
p
i
(x
i
;σ
i
,γ
i
) =
α
i
(γ
i
)γ
i
2σ
i
Γ(1/γ
i
)
e
−[α
i
(γ
i
)|x
i
/σ
i
|]
γ
i
x
i
∈ R, i = 1,2 (13)
α
i
(γ
i
) =
s
Γ(3/γ
i
)
Γ(1/γ
i
)
(14)
where Γ(·) is the Gamma function. In this paper,
the generalized Gaussian ratio function is given as
follows (Niehsen, 1999; Niehsen, 2002; Sharifi and
Leon-Garcia, 1995).
φ
(2m)
i
(γ
i
) =
Γ(
2m+1
γ
i
)Γ
m−1
(1/γ
i
)
Γ
m
(3/γ
i
)
,m = 1, 2, ···(15)
where
σ
2
i
= σ
2
∆
i
+ σ
2
¯
ζ
ii
(16)
E(τ
2m
i
) is a function of σ
2
∆
i
, γ
∆
i
, σ
2
¯
ζ
ii
and γ
¯
ζ
ii
. Variables
σ
2
∆
i
, γ
∆
i
, σ
2
¯
ζ
ii
and γ
¯
ζ
ii
are variance of ∆
i
, shape param-
eter of ∆
i
, variance of
¯
ζ
ii
and shape parameter of
¯
ζ
ii
,
respectively. The odd moments vanish because of the
symmetrical pdf.
3 MAIN RESULTS
Before the robust condition is given, some mathemat-
ical preliminaries are given as follows:
First, the form of the Stirling’s formula is given as
follows:
n! ≈
√
2πn

n
e

n
(17)
The Stirling’s formula can be applied to estimate the
Gamma function Γ(z), if Re(z) > 0. The correspond-
ing approximation formula is in the following form:
Γ(z) =
r
2π
z

z
e

z

1+ O

1
z


(18)
where O denotes the Big O notation. In 2007, an esti-
mation form was proposed by Gergo Nemes (Stirling
approximation)which has the same computational ac-
curacy with formers’, but it is simpler for calculator.
Γ(z) ≈
r
2π
z

1
e

z+
1
12z−
1
10z

z
=
r
2π
z

1
e
120z
3
+ 9z
120z
2
−1

z
(19)
Next, let’s prove that the generalized Gaussian ra-
tio function decreases with the increasing of the shape
parameter γ.
Based on the former results about generalized
Gaussian ratio function, the following equality is es-
tablished:
φ
(2m)
(γ) =
Γ(
2m+1
γ
)Γ
m−1
(
1
γ
)
Γ
m
(
3
γ
)
(20)
when m = 2, we get the equation (21):
φ
(4)
(γ) =
Γ(
5
γ
)Γ(
1
γ
)
Γ
2
(
3
γ
)
(21)
From the equation (21), we can find that the following
equation is satisfied if z=
1
γ
and named the equivalent
of φ
(4)
(γ) to be Φ(z):
Φ(z) =
Γ(5z)Γ(z)
Γ(3z)
=
3
√
5
(
5
9
)
5z
(
1
9
)
z
[
1000z
2
+3
120×25z
2
−1
]
5z
[
40z
2
+3
120z
2
−1
]
z
[
120z
2
+1
120×9z
2
−1
]
6z
(22)
For simplicity, we consider the natural logarithm of
Φ(z):
f(z) = ln(Φ(z)) = ln3−
1
2
ln5+ (5ln
5
9
+ ln
1
9
)z
+ [5ln(1000z
2
+ 3) + ln(40z
2
+ 3)
+ 6ln(120 ×9z
2
−1)]z−[5ln(120×25z
2
−1)
+ ln(120z
2
−1) + 6ln(120z
2
+ 1)]z (23)
BIOSIGNALS 2010 - International Conference on Bio-inspired Systems and Signal Processing
260

and take the derivative of the above function f(z):
f
′
(z)=5ln
5
9
+ ln
1
9
+ 5ln
1000z
2
+ 3
120×25z
2
−1
+ln
40z
2
+ 3
120z
2
−1
+ 6ln
120×9z
2
−1
120z
2
+ 1
−
30
1000z
2
+ 3
−
6
40z
2
+ 3
+
12
120×9z
2
−1
−
10
120×25z
2
−1
−
2
120z
2
−1
+
12
120z
2
+ 1
We can assume that
F(z) = 5ln
1000z
2
+ 3
120×25z
2
−1
+ ln
40z
2
+ 3
120z
2
−1
+ 6ln
120×9z
2
−1
120z
2
+ 1
H(z) = −
30
1000z
2
+ 3
−
6
40z
2
+ 3
+
12
120×9z
2
−1
−
10
120×25z
2
−1
−
2
120z
2
−1
+
12
120z
2
+ 1
According to the monotonicity of the functions F(z)
and H(z), we can find that when z ∈ [
1
8
,
2
3
], H(z) ∈
[−0.84, −0.1558] and F(z) ∈ [ln792, ln1280], then
the estimation of the f
′
(z) can be obtained:
f
′
(z) ≥5ln5−12ln3+ ln792−0.84
≥ln
5
5
×792
3
12
−0.84
≥ln4.657 −0.84
≥0.16 ≥0
Therefore, the function f(z) is monotonically increas-
ing in z ∈ [
1
8
,
2
3
], so the even moment φ
(4)
(γ) is mono-
tonically decreasing with the shape parameter γ in
γ ∈ [
3
2
,8], so the similar results can be obtained for
the case of m = 3.
Then, the robust condition can be obtained as fol-
lows:
Using the result in (Deng et al, 2006), the follow-
ing relationship is established:
E(τ
6
i
)(γ) = σ
6
∆
i
φ
(6)
i
(γ
∆
i
) + 15σ
4
∆
i
φ
(4)
i
(γ
∆
i
)σ
2
¯
ζ
ii
+ 15σ
2
∆
i
φ
(4)
i
(γ
¯
ζ
ii
)σ
4
¯
ζ
ii
+ σ
6
¯
ζ
ii
φ
(6)
i
(γ
¯
ζ
ii
)
where φ
(4)
(γ
∆
i
) = 3 and φ
(6)
(γ
∆
i
) = 15. So the rela-
tionship can be simplified to be:
E(τ
6
i
)(γ
i
)=15σ
6
∆
i
+ 45σ
4
∆
i
σ
2
¯
ζ
ii
+ 15σ
2
∆
i
φ
(4)
i
(γ
¯
ζ
ii
)σ
4
¯
ζ
ii
+σ
6
¯
ζ
ii
φ
(6)
i
(γ
¯
ζ
ii
) (24)
Therefore, the similar results can be obtained for
the case of γ
¯
ζ
ii
= 1.5, where φ
(4)
(1.5) = 3.76 and
φ
(6)
(1.5) = 26.7.
E(τ
6
i
)(1.5)=15σ
6
∆
i
+ 45σ
4
∆
i
σ
2
¯
ζ
ii
+ 56.4σ
2
∆
i
σ
4
¯
ζ
ii
(1.5)
+26.7σ
6
¯
ζ
ii
(1.5) (25)
Then from the equations (24) and (25), the differ-
ence between E(τ
6
i
)(1.5 + ∆γ) and E(τ
6
i
)(1.5) is as
follows, where γ = 1.5+ ∆γ:
∆E =E(τ
6
i
)(1.5+ ∆γ) −E(τ
6
i
)(1.5)
=45σ
4
∆
i
[σ
2
¯
ζ
ii
(1.5+ ∆γ) −σ
2
¯
ζ
ii
(1.5)] + 15σ
2
∆
i
[φ
(4)
i
(1.5+ ∆γ)σ
4
¯
ζ
ii
(1.5+ ∆γ) −3.76σ
4
¯
ζ
ii
(1.5)]
+[σ
6
¯
ζ
ii
(1.5+ ∆γ)φ
(6)
i
(1.5+ ∆γ) −26.7σ
6
¯
ζ
ii
(1.5)]
(26)
Since the even moments φ
(4)
i
(1.5+∆γ) and φ
(6)
i
(1.5+
∆γ) decrease with the increasing of ∆γ, and the vari-
ance can be estimated using the method in (Iseki et
al, 2009), so there exists a bounded M to make the
following relationship satisfied:
∆E ≤ M (27)
Therefore, for ∆γ ≤6.5, the difference of the evenmo-
ment ∆E ≤M, so the even moment E(τ
6
i
)(γ) is robust
for boundedness of the varying of the shape parameter
γ.
4 EXPERIMENTAL ISSUES
The experimental system is shown in Fig. 1. In the ex-
periment, the parameters are given as follows, where
the external force between the handle and horizontal
plane is omitted. The arm parameters of the objec-
tive are l
1
= 0.26m, l
2
= 0.30m. The cut-off frequen-
cies of the third-order band-pass filter to generate τ
f
in
,
θ
f
i
(t) and
˙
θ
f
i
(t) are 0.5Hz and 9.5Hz. For designing
the filter, we use the case of m = 3 in (15), then:
E(τ
6
i
)(γ)=σ
6
∆
i
φ
(6)
i
(γ
∆
i
) + 15σ
4
∆
i
φ
(4)
i
(γ
∆
i
)σ
2
¯
ζ
ii
+15σ
2
∆
i
φ
(4)
i
(γ
¯
ζ
ii
)σ
4
¯
ζ
ii
+ σ
6
¯
ζ
ii
φ
(6)
i
(γ
¯
ζ
ii
) (28)
where, φ
(4)
(γ
∆
i
) = 3 and φ
(6)
(γ
∆
i
) = 15. The esti-
mated stiffness and viscosity are shown in Figs. 3 and
4, where l
11
= 9.472, l
12
= 7.750, l
21
= l
22
= 1.034e
9
are selected. From Figs. 3 and 4, we can find that
the proposed method is available to estimate the vis-
coelasticity of the human arm.
FRAMEWORK OF AN ESTIMATION ALGORITHM OF TIME VARYING MULTIJOINT HUMAN ARM
VISCOELASTICITY
261

Figure 1: The experimental system.
Figure 2: Schema of the experimental system and objective.
5 CONCLUSIONS
This paper considered the estimation of the viscoelas-
ticity of human arm and studied the monotonicity of
the generalized Gaussian ratio function, then a robust
condition of the generalized ratio function is proposed
for the varying of the shape parameter. So the ac-
curacy of the experimental data is guaranteed by the
robust condition. The effectiveness of the proposed
method is confirmed by the experimental results.
ACKNOWLEDGEMENTS
The authors would like to thank professor emeritus
A. Inoue and student Ms. A. Nishimura at Okayama
University for their contribution to the work.
0 1 2 3 4 5 6 7 8
-5
0
5
D
ss
&D
se
[Nm/(rad/s)]
0 1 2 3 4 5 6 7 8
-5
0
5
time [sec]
D
es
&D
ee
[Nm/(rad/s)]
Figure 3: Estimated viscosity by experiment.
0 1 2 3 4 5 6 7 8
-100
-50
0
50
100
R
ss
&R
se
[Nm/rad]
0 1 2 3 4 5 6 7 8
-60
-40
-20
0
20
40
60
R
es
&R
ee
[Nm/rad]
Figure 4: Estimated stiffness by experiment.
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