Adaptive Attitude Control of Redundant Time-Varying Complex Model of Human Body in The Nursing Activity

Dong, H., Luo, Z., and Nagano, A.
Paper:
Adaptive Attitude Control of Redundant Time-Varying Complex
Model of Human Body in The Nursing Activity
Haiwei Dong, Zhiwei Luo, and Akinori Nagano
Department of Computational Science, Graduate School of System Informatics, Kobe University
1-1 Rokkodai-cho, Nada-ku, Kobe 657-8501, Japan
E-mail: haiwei@stu.kobe-u.ac.jp
[Received December 14, 2009; accepted April 13, 2010]
With the development of human society, there are
more and more elderly people need to be taken care of.
However, there is not enough labor force to take the
nursing jobs. Nowadays robots play more and more
important roles in our daily life, especially in nursing
activities. In this paper, we illustrate a new attitude
control approach to lift human regardless of the indi-
vidual differences, such as height, weight, and so on.
In detail, considering our daily experience that only
very few joints are critical for accomplishing the lifting
up task, we treats the human body as a redundant sys-
tem. We use robust adaptive control to eliminate the
effects from the “uninterested joints” and identify the
human parameters in real time. In addition, the con-
vergence analysis, including tracking time and track
error, is also given. The approach is simulated by lift-
ing a human skeleton with two robot arms, which ver-
ifies the efficiency and effectiveness of our strategy.
Keywords: human body, model reduction, adaptive atti-
tude control, human parameters identification
1. Introduction
As more and more countries step into the aging soci-
ety, much attention is drawn to the aging of population
than ever before. The fact is that increasing number of
elderly people need to be looked after; however, there is
not enough labor force to take the nursing jobs. How to
deal with the aging problem has drawn attention of all
the countries in the world. Apart from many social so-
lutions, as the advanced robotics technology is becoming
mature, we believe that the mentioned aging problem can
be partially solved by using robots in the nursing activi-
ties, which inspires our research [1].
Specifically, this paper focuses on how to carry up the
human body to a desired position and posture, which can
be utilized in the nursing activities in hospitals. However,
few efforts have been made in the literatures to change
the attitude of human body [2, 3]. From a theoretical
viewpoint, the most relevant approach is whole arms co-
operation control. However, many preceding researches
consider simple objects with small Degrees of Freedom
(DOF) whereas the human body is much more complex
with considerable DOF [4–6]. From a practical stand-
point, the most successful work on lifting human is the de-
velopment of a robot named RI-MAN which was selected
as one of the best inventions by the TIME Magazine in
2006 [7–9]. RI-MAN has many perceptive functions such
as sight, smell, hearing, touch, etc. Our research is based
on some of the dynamics simulations of RI-MAN.
It has been thought that the human body has about 206
bones and numerous joints connecting adjacent bones.
Based on the physiological structure of human joints, the
joints can be mainly divided into five types as hinge (1-
DOF), pivot (1 DOF), saddle (2 DOF), gliding (2 DOF),
ball socket (3 DOF). While in dynamic equations, each
DOF is expressed as one differential equation. Hence, it is
predicted that the overall set of equations of human body
dynamics is very complex to handle. Hence, the difficulty
arises to compute such a big dynamic model.
Actually, the human body can be considered as a free-
floating multi-link rigid object with passive moments.
The objective is to change the attitude of the mentioned
object by external forces. Hence, two difficulties come
out: the first one is about free-floating multi-link rigid
object. The previous studies on free-floating object are
mainly in controlling spacecraft. In the literatures, the
spacecrafts were accurately modeled. Based on the pre-
cise models, various methods, e.g., generalized Jacobian
methods, were proposed to complete the attitude control
task [10, 11]. However, in our case, the model of human
body cannot be modeled accurately. That is not only be-
cause there are some human parameters which can not be
measured, but also because human bodies have individ-
ual differences. The second one is about external forces.
As the human body is such a complex model with very
high dimension, application of external forces on the hu-
man body is also very complicated. The calculation needs
considerable time and real-time performance is impossi-
ble. In addition, the process of lifting human must be ab-
solutely safe. If we cannot make sure that the computation
is done in real-time, the safety cannot be guaranteed.
In consideration of the two difficulties above, the ba-
sic idea for solution comes from our daily experience.
When human lift a person we do not care about the an-
gle of ankle, the position of hands and so on. What we
do have to care about are the position of the head, the ver-
418 Journal of Robotics and Mechatronics Vol.22 No.4, 2010

Adaptive Attitude Control of Time-Varying Model of Human Body
tical deflection of upper limb and the angle of hip. Here
we call them “interested states.” From the viewpoint of
system theorem, we treat the human body as a large re-
dundant system whose dimension is reduced by diverting
the effects of other “uninterested joints” to the ones of
“interested joints.” The newly constructed reduced human
model has very few DOF while having unmodeled uncer-
tainty, unfortunately. In order to eliminate the unmod-
eled uncertainty, human attitude controller is designed.
Furthermore, the human parameter estimator is also de-
signed.Thus, the whole attitude control approach for lift-
ing up human body is capable to overcome the individual
differences, such as height, weight, and so on.
This paper is organized as follows. The second sec-
tion models the lifting problem and reduces the human
model. The third section illustrates the detailed derivation
of the controller and the estimator. The fourth section
analyzes the convergence of the proposed approach, in-
cluding tracking error and tracking time. The fifth section
takes a normal human body for example in simulation to
test the effectiveness of the proposed approach. The sixth
section concludes the whole paper.
2. Adaptive Attitude Control of Human Body
2.1. Human Model Simplification
If we consider human body as a rigid multi-link object,
each bone of the human body corresponds to a link and
each joint of human body corresponds to a human joint
connecting adjacent links. Moreover, the joints of human
body has passive torques corresponding to the constric-
tion forces and moments developed by ligaments, joint
capsules and other soft tissues. Hence, we write the dy-
namics of human body model in the form
H(q) ¨q+C(q, ˙q) ˙q+G(q) = τpass, . . . . . (1)
where qn×1 is generalized states of human body, which
include the position of head and the angles of all the
joints. H(q)n×n is inertia matrix. It is a symmetric and
positive semi-definite matrix, which contains informa-
tion with regard to the instantaneous mass distribution
of the human body. C(q, ˙q)n×n is centripetal and cori-
olis torques. The terms of C(q, ˙q) contain products of
angular speeds. When the degrees of freedom are rota-
tional, the terms of C(q, ˙q) represent the moments of cen-
trifugal forces. G(q)n×1 is gravitational torques. Because
G(q) changes as the posture configuration of the human
body model, the terms of it are functions of the general-
ized states. τpass n×1 is passive joint torques. It contains
the torques and moments arising from muscular activa-
tions and passive elastic structures surrounding the human
joints. τrob n×1 is the torques exerted by the robot arms
which is controllable.
The task assigned to robot is to lift up the human body,
and control the position and posture of the body to the
desired states. In this process, the human body is exerted
by external forces. Hence, the dynamics of human model
Fig. 1. RI-MAN lifts up human skeleton. The robot RI-
MAN was designed for taking care of elderly in the nursing
activity. Hence, he has many sensors for perception, includ-
ing sight, smell, hearing, and touch.
can be modified into the following form
H(q) ¨q+C(q, ˙q) ˙q+G(q) = τpass +τrob, . . . (2)
where τrob n×1 is the torques exerted by the robot arms,
which is controllable.
Actually, the robot considered in this paper is RI-MAN
which has two manipulation arms (Fig. 1). In this case,
the location of force application is the back and the knee
of human body, which is shown as F1 and F2. The forces
F1 and F2 have relation with τrob as
τrob = JT
1 F1 +JT
2 F2, . . . . . . . . . . . (3)
where J1 and J2 are Jacobian matrix of the human model.
After introducing the external forces in the human dy-
namics, the case we have to deal with becomes attitude
control of a free-floating multi-link rigid object. The ba-
sic idea of our approach is to reduce the human model into
a small one with less DOF, which includes the following
three steps:
(a) Choose “interested states.” These states include the
fundamental performance indexes of the lifting task. In
other words, based on these states, we can easily check
whether the task is completed or not. Let us define
¯q1 = [q1,...,qm]T
m×1 . . . . . . . . . . (4)
as the “interested states” and
¯q2 = [qm+1,...,qn]T
(n−m)×1 . . . . . . . . (5)
as the “uninterested states” consisting of the other states.
We can easily obtain that q = [ ¯q1 ¯q2]T
.
(b) Arrange the dynamic equation set. Based on the divi-
sion of “interested states” in (a), we change the positions
of elements of H, C, G, τpass and τrob extract the dynam-
ics of the “interested states”. Then we divide the dynamic
equation set into two parts where H, C, G, τpass and τrob
are written into the form of block matrix (or block vector).
Journal of Robotics and Mechatronics Vol.22 No.4, 2010 419

The expanded form of new dynamics equation set is
¯H11 ¯H12
¯H21 ¯H22
¨¯q1
¨¯q2
+
¯C11
¯C12
¯C21 ¯C22
˙¯q1
˙¯q2
+
¯G1
¯G2
=
¯τpass,1
¯τpass,2
+
¯τrob,1
¯τrob,2
, . . (6)
where the dimensions of sub block matrices of ¯H11, ¯H12,
¯H21, ¯H22 are m× m, m× (n−m), (n−m)× m, (n−m)×
(n−m), respectively. And the dimensions of sub block
matrices of ¯C11, ¯C12, ¯C21, ¯C22 are m × m, m × (n−m),
(n−m)×m, (n−m)×(n−m), respectively. The dimen-
sions of vectors ¯G1, ¯τpass,1, ¯τrob,1 are m × 1, and ¯G2,
¯τpass,2, ¯τrob,2 are (n−m)×1.
(c) Construct the reduced human dynamics. First of all,
we define the generalized states of the reduced human
model as
qs = ¯q1. . . . . . . . . . . . . . . (7)
Extracting the parts of the human dynamics which we are
interested in, we get
¯H11 ¯H12
¨¯q1
¨¯q2
+ ¯C11 ¯C12
˙¯q1
˙¯q2
+ ¯G1
= ¯τpass,1 + ¯τrob,1. . . . (8)
Considering that the dynamic model is time-varying, after
arranging Eq. (8), we obtain
¯H11(t) ¨¯q1 + ¯C11(t) ˙¯q1
+ ¯G1(t)+ ¯H12(t) ¨¯q2 + ¯C12(t) ˙¯q2 − ¯τpass,1(t)
= ¯τrob,1(t). . . . . . . . . . . . (9)
By defining the inertia matrix, centripetal matrix, gravi-
tational matrix and torque vector of the reduced human
model as
Hs(t) = ¯H11(t)
Cs(t) = ¯C11(t)
Gs(t) = ¯G1(t)+ ¯H12(t) ¨¯q2 + ¯C12(t) ˙¯q2 − ¯τpass,1(t)
τs(t) = ¯τrob,1(t)
(10)
we obtain the general mechanical form of the reduced hu-
man model
Hs(t) ¨qs +Cs(t) ˙qs +Gs(t) = τs(t) . . . . . (11)
where the subscript s denotes the small system. Actu-
ally, our basic idea is to consider the influences from the
“uninterested human joints” (in this case from state ¯q2) as
perturbations and then change the attitude of human adap-
tively by estimating the human parameters of Hs, Cs and
Gs in real time. The detailed estimation meanings are
• Estimating Hs and Cs – make the system adap-
tively adjusts itself to various people with different
weights;
• Estimating Gs – eliminate the perturbations from the
“uninterested joints.”
Considering the basic idea above, the approach to be pro-
posed in this paper should be able to identify and control
the dynamics of the reduced human model at the same
time. Assuming that the human model is totally unknown
in advance, for the safety in the nursing activity, the iden-
tification process needs to be performed in real-time. On
the other hand, the weights and heights etc. of the human
bodies are different between individuals. Hence, the strat-
egy also has to be able to tolerate these individual differ-
ences.
2.2. Human Attitude Control and Parameter Iden-
tification
First of all, we assume that we do not have any pri-
ori knowledge before lifting human, i.e., the initial value
of Hs, Cs, Gs are set as zero matrices (or zero vectors).
The benefit of such assumption is that the proposed trajec-
tory is much more robust and can be adaptive to various
people with different heights and weights. Whereas, the
shortcoming is also obvious, i.e., generating unmodeled
dynamics. To overcome the above disadvantage, we use
robust controller to change the human attitude. Moreover,
online human parameter identification is also done so as
to estimate the human body in real time.
For the convenience of mathematical derivation, we de-
fine the actual human parameter vector
P = PT
H PT
C PT
G
T
, . . . . . . . . . . (12)
where
PH = [Hs,11 Hs,12 ... Hs,1n ... Hs,n1 Hs,n2 Hs,nn]T
PC = [Cs,11 Cs,12 ... Cs,1n ... Cs,n1 Cs,n2 Cs,nn]T
PG = [Gs,1 Gs,2 ... Gs,n]T
. (13)
and estimated human parameter vector as
ˆP = ˆPT
H
ˆPT
C
ˆPT
G
T
, . . . . . . . . . . (14)
where
ˆPH = ˆHs,11 ˆHs,12 ... ˆHs,1n ... ˆHs,n1 ˆHs,n2 ˆHs,nn
T
ˆPC = ˆCs,11 ˆCs,12 ... ˆCs,1n ... ˆCs,n1 ˆCs,n2 ˆCs,nn
T
ˆPG = ˆGs,1
ˆGs,2 ... ˆGs,n
T
(15)
then the estimation error matrix can be defined as
˜P = ˆP−P. . . . . . . . . . . . . . (16)
In fact, not any combination of H, C and G corresponds to
a physical system. Therefore, the first step is to prove that
the reduced human model represents a physical system. It
is easy to prove that by verifying that ˙Hs −2Cs is a skew-
symmetric matrix, i.e., the reduced human model satisfies
conservation of energy (the detailed derivation is in the
proof of Theorem 1).
We proposed a theorem for changing the “interested
states” of the large complex human body as in Theorem 1.
Theorem 1 is composed of a human attitude control law
and a human parameter update law. In fact, the two pro-
cesses of control and identification run at the same time.
In the proof of Theorem 1, the global stability is shown
by proving that the derivative of Lyapunov function can-
didate is less than zero.

Theorem 1
Consider a time-varying system
Hs(t) ¨qs +Cs(t) ˙qs +Gs(t) = τs(t) . . . . . (17)
without any pre-knowledge about Hs, Cs and Gs. The vec-
tor qs,d means the desired states. Define a sliding term s
as
s = ˙˜qs +Λ ˜qs = (qs −qs,d) +Λ(qs −qs,d) . . (18)
where Λ is a positive diagonal matrix. From the concep-
tual view of velocity, we define the reference velocity ˙qs,r
and reference acceleration ¨qs,r as
˙qs,r = ˙qs −s
¨qs,r = ¨qs − ˙s
. . . . . . . . . . . . . (19)
If we choose the human attitude control law
τs = ˆHs(t) ¨qs,r + ˆCs(t) ˙qs,r + ˆGs(t)−k ·sgn(s) . (20)
and human parameter update law
˙ˆP = −Γ−1
s1 ¨qT
s,r ... sn ¨qT
s,r s1 ˙qT
s,r ... sn ˙qT
s,r s1 ... sn (21)
under the assumption of
k ·sT
sgn(s) > ˜PT
Γ ˙P . . . . . . . . (22)
where k and Γ are positive diagonal matrices, sgn(·) is a
signal function, then the whole system tracks the desired
trajectory and the parameter matrices Hs, Cs and Gs con-
verge to actual values globally.
Proof:
Define a Lyapunov function candidate
V(t) =
1
2
sT
Hss+
1
4
˜PT
(2Γ+I) ˜P . . . . . (23)
where Γ is a positive diagonal matrix. The first part of
V(t) can be written as
1
2
sT
Hss = sT
(Hs ¨qs −Hs ¨qs,r)+
1
2
sT ˙Hss. (24)
From Eq. (17), Hs ¨qs = τs −Cs ˙qs −Gs, then
1
2 sT Hss
= sT (τs −Cs(s+ ˙qs,r)−Gs −Hs ¨qs,r)+ 1
2 sT ˙Hss
= sT (τs −Hs ¨qs,r −Cs ˙qs,r −Gs)+ 1
2 sT ( ˙Hs −2Cs)s
(25)
According to the previous research on mechanical system,
the system in the form of Eq. (2) satisfies
˙qT ˙H −2C ˙q = 0 . . . . . . . . . . . (26)
i.e., ˙H − 2C is a skew-symmetric matrix. Hence, the fol-
lowing relation satisfies
˙Hi j −2Ci j =
0 if i = j
− ˙Hji −2Cji otherwise
. (27)
Without loss of generality, we choose ¯q1 as the new state
vector which we are “interested in” (Eq. (4)) and follow
the same system reduction procedures in Eqs. (4)-(11).
According to the relation in Eq. (27), the reduced system
satisfies
˙¯H11(iu,iv) −2 ¯C11(iu,iv)
=
0 if iu = iv
− ˙¯H11(iv,iu) −2 ¯C11(iv,iu) otherwise
(28)
where ¯H11 and ¯C11 are defined in Eq. (6). Hence, ˙¯H11 −
2 ¯C11 is a skew-symmetric matrix. Based on the definitions
of Hs and Cs in Eq. (10), ˙Hs − 2Cs is a skew-symmetric
matrix, hence
1
2
sT
Hss = sT
(τs −Hs ¨qs,r −Cs ˙qs,r −Gs) . (29)
Therefore, ˙V(t) can be simplified as
˙V(t) =
1
2
sT
Hss +
1
2
˜PT
Γ ˜P
= sT
(τs −Hs ¨qs,r −Cs ˙qs,r −Gs)+( ˙ˆP− ˙P)T
Γ ˜P (30)
Applying the human attitude control law Eq. (20)
τs= ˆHs(t) ¨qs,r + ˆCs(t) ˙qs,r + ˆGs(t)−ksgn(s)
=





¨qT
s,r 0 ··· 0 ˙qT
s,r 0 ··· 0 1 0 ··· 0
0 ¨qT
s,r ··· 0 0 ˙qT
s,r ··· 0 0 1 ··· 0
...
...
...
...
...
...
...
...
...
0 0 ··· ¨qT
s,r 0 0 ··· ˙qT
s,r 0 0 ··· 1





m×2m2
×


ˆPH
ˆPC
ˆPG


2m2×1
−k




sgn(s1)
sgn(s2)
...
sgn(sm)




m×1
. . . . (31)
into ˙V(t), which leads to
˙V(t) = sT ˆHs(t) ¨qs,r + ˆCs(t) ˙qs,r + ˆGs(t)−Hs(t) ¨qs,r
−Cs(t) ˙qs,r −Gs(t))−k ·sT
sgn(s)+ ˜PT
Γ(˙ˆP− ˙P)
= sT ˜Hs(t) ¨qs,r + ˜Cs(t) ˙qs,r + ˜Gs(t)
−k ·sT
sgn(s)+ ˜PT
Γ(˙ˆP− ˙P)
= [s1 ¨qT
s,r ... sn ¨qT
s,r] ˜PH +[s1 ˙qT
s,r ... sn ˙qT
s,r] ˜PC
+[s1 ... sn] ˜PG −sT
sgn(s)+ ˜PT
Γ(˙ˆP− ˙P) (32)
where
¨qs,r = ¨qs
r,1 ¨qs
r,2 ... ¨qs
r,m
T
m×1
˙qs,r = ˙qs
r,1 ˙qs
r,2 ... ˙qs
r,m
T
m×1
We obtain
˙V(t) = ˜PT
1×2m2
× s1 ¨qT
s,r ... sn ¨qT
s,r s1 ˙qT
s,r ... sn ˙qT
s,r s1 ... sn
T
2m2×1
−k ·sT
sgn(s)+ ˜PT
Γ(˙ˆP− ˙P). . . . . . (33)
Taking the human parameter update law of Eq. (21),
finally we obtain
˙V(t) = − ˜PT
Γ ˙P−k ·sT
sgn(s). . . . . . . (34)

Fig. 2. Scheme of the proposed attitude control. The input signals are the desired trajectories of the “interested states” of the human
model. The output is the actual motions of the human model. The proposed approach both controls the position and posture of the
human model and identifies the human parameters online at the same time.
According to the assumption of Eq. (22), ˙V(t) < 0.
Hence, the tracking error and parameter estimation error
converge to zero asymptotically.
It is noted that the signals required in Eqs. (20) and (21)
are s, ˙qs,r, ¨qs,r. According to the definitions of s, ˙qs,r, ¨qs,r
in Eqs. (18) and (19) , the basic signals required are qs,
˙qs, ¨qs. Actually, the “interested states” qs, ˙qs represent
the attitude position and acceleration of the human body.
Hence, they are obvious by binocular vision technology.
However, ¨qs is hard to measure. To avoid the accelera-
tion term, we use filtering technology. Specifically, let
w(t) be the impulse response of a stable, proper filter. For
example, for the first-order filter ε/(p + ε) where ε > 0,
p = d/dt, the impulse response is e−εt . Then using partial
integration, ¨qs can be integrated as
t
0
w(t −r) ¨qrdr = w(t −r) ˙qs|t
0 −
t
0
dw
dr
˙qsdr
= w(0) ˙qs −w(t) ˙qs(0)
−
t
0
[w(t −r) ˙qs − ˙w(t −r) ˙qs]dr (35)
which means ¨qs = f( ˙qs,w), i.e., the acceleration signal
can be obtained from velocity signal.
To illustrate the strategy more clearly, the scheme il-
lustration is shown in Fig. 2. The input of the proposed
approach is the desired attitude of human body and the
output is the actual attitude. There are four blocks in the
scheme figure as signal transform unit, attitude controller,
model parameter estimator and human body model. De-
tailed explanations are as follows.
• Signal transform unit. This unit has two input sig-
nals, including desired attitude and actual attitude.
Actually, we can get many kinds of error signals,
such as position error signal, angular velocity er-
ror signal and angular acceleration signal and so on.
Meanwhile, the combination of the above error sig-
nals can also be obtained. Specifically, in the pro-
posed approach, we create the sliding signal s, ve-
locity reference error signal ˙qs,r and acceleration ref-
erence error signal ¨qs,r.
• Attitude controller. The controller is a robust con-
troller which can tolerate the uncertainty of reduced
human model. In fact, the controller has its own es-
timated human model in the form of
ˆH(q) ¨q+ ˆC(q, ˙q) ˙q+ ˆG(q) = τpass +τrob. (36)
Each time, the controller gives orders to robot arms
to lift human body up according to its own estimated
human model.
• Model parameter estimator. Although there is an in-
ternal estimated human model in the controller, the
uncertainty of human model estimation has a strong
impact on the control performance. In other words,
if we can get a more accurate approximation model
of human body, the performance is better. Based on
the above consideration, model parameter estimator
uses system identification technology to refine the
estimated human model. In the proposed approach,
the calculation of the derivative of ˆP revises the esti-
mated human model.
• Human body model. The control orders are trans-
ferred to control the manipulators of the robot for
exerting external forces.
3. Convergence Analysis
First of all, we assume that the position and velocity
cannot “jump,” so that any desired trajectory feasible from
time t = 0 necessarily starts with the same position and
velocity as those of plant. Let ˜qs = qs − qs,d be the track-
ing error in the variable qs, i.e.,
˜qs =
qs −qs,d
˙qs − ˙qs,d
. . . . . . . . . . . (37)
Furthermore, define a time-varying surface in the state-
space Rn by the scalar equation s(qs;t) = 0, where
s = ˙˜qs +λ ˜qs. . . . . . . . . . . . . (38)

Given initial assumption qs,d(0) = qs(0), the problem of
tracking qs ≡ qs,d is equivalent to that of remaining ˜qs on
the surface S(t) for all t > 0; indeed s ≡ 0 represents a lin-
ear differential equation whose unique solution is ˜qs ≡ 0,
given initial condition qs,d(0) = qs(0). Thus, the problem
of tracking the n-dimensional vector qs,d can be reduced
to that of keeping the scalar quantity s at zero. More pre-
cisely, the problem of tracking the n-dimensional vector
qs,d can in effect be replaced by a 1st-order stabilization
problem in s. Actually, the stabilization process in s can
be divided into two phases. The fist phase is to make s ap-
proach and finally hit the manifold B(t) which is defined
as
B(t) = {qs|s(qs;t) ≤ φ} . . . . . . . . . (39)
where φ > 0 denotes the boundary layer thickness. While
the second phase is to make s converge to zero asymptot-
ically in the boundary layer (Fig. 3).
3.1. Tracking Time Analysis
In the proof of Theorem 1, we define the Lyapunov
function as
V(t) =
1
2
sT
Hss+
1
4
˜PT
(2Γ+I) ˜P = V1(t)+V2(t) (40)
After taking the control law Eq. (20) and parameter update
law Eq. (21), finally the derivative of V(t) can be written
as
˙V(t) = −k ·sT
sgn(s)− ˜PT
Γ˙P = ˙V1(t)+ ˙V2(t) . (41)
Extracting parts of the elements in V1(t) as
n
∑
k=1
s2
kHs,kk and
differentiating the ordinary element, we obtain
1
2
s2
kHs,kk ≤ −kksksgn(sk) = −kk |sk|. . (42)
Eq. (42) states that the “distance” to the surface, as mea-
sured by s2
k, decrease along all system trajectories. Thus,
it constrains trajectories to point towards the manifold
B(t), as illustrated in Fig. 3. In detail, let treach
k be the re-
quired time of the k-th generalized coordinate qs
k to reach
the surface sk = 0. Integrating the left side of Eq. (42)
between t = 0 and t = treach
k leads to
treach
k
0
1
2
d
dt
s2
kHs,kkdt =
1
2
Hs
kks2
k
t=treach
k
t=0
= −
1
2
Hs,kksk(t = 0)2
. (43)
while the integration of the right side between t = 0 and
t = treach
k can be written as
−
treach
k
0
kk |sk|dt ≤ −
treach
k
0
kk |sk(t = 0)|dt
= −kk |sk(t = 0)|treach
k . (44)
Applying the inequality relation in Eq. (42), we get the ac-
quired time for any generalized coordinate qs
k to get sk = 0
treach
k ≤
Hs,kk
2kk
|sk(t = 0)|. . . . . . . . . (45)
Fig. 3. Two phases for transferring interested state to the de-
sired state. The first phase is to drive the initial state qs,d,k(0)
to reach the boundary layer; the second phase is to make the
state qs,k(t) converge to the desired state qs,d,k(tf ) asymptot-
ically.
Fig. 4. Position error of the “interested states” in the second
phase. The error between the desired state and actual state
of the human joint decreases to zero exponentially.
Furthermore, manifold definition of B(t) implies that once
on the surface, s(t) = 0 , i.e.,
˙˜qs +λ ˜qs = 0. . . . . . . . . . . . . (46)
The solution to the Eq. (46) is
˜qs = e−λt
. . . . . . . . . . . . . . (47)
Plugging the definition in Eq. (37), we can obtain qs −
qs,d = e−λt. Hence, the tracking error qs − qs,d tends to
zero with a time constant λ exponentially (Fig. 4). In
other words, the proposed approach is exponentially sta-
ble.
3.2. Static Tracking Error Analysis
At the second phase, bounds on s can be directly trans-
lated into bounds on the tracking error vector ˜qs, and
therefore the scalar s represents a true measure of tracking

Fig. 5. Relation between ˜qs,k, ˙˜qs,k and sk (1 ≤ k ≤ n) in
Laplace space. From the knowledge of Laplace transform, it
is possible to get the upper bound of joint angle by integra-
tion.
performance. Indeed, by definition Eq. (38), the tacking
error ˜qs is obtained from s through a first-order lowpass
filters (Fig. 5), where p = d/dt is the Laplace operator.
From the definition of s in Eq. (38), for the first term of
s, we obtain
s1 = ˙˜qs,1 +λ1 ˜qs,1 . . . . . . . . . . . . (48)
i.e.,
˜qs,1 = (p+λ1)s1 . . . . . . . . . . . . (49)
where p = d/dt. According to the signal processing
knowledge, we have
L(e−λ1t
) =
∞
0
e−λ1t
e−pt
dt =
1
p+λ
. . . . (50)
where L(·) is Laplace operator. By applying the convolu-
tion theorem, we obtain
˜qs,1 =
t
0
e−λ1(t−T)
s1(T)dT. . . . . . . (51)
According to the assumption ˜qs(0) = 0 and |s(t)| < φ,
|sk(t)| < φk (1 ≤ k ≤ n) the upper bound of ˜qs,1(t) can
be obtained
| ˜qs,1(t)| ≤ φ1
t
0
e−λ1(t−T)
dT =
φ1
λ1
e−λ1T−λ1t
T=t
T=0
=
φ1
λ1
1−e−λ1t
≤
φ1
λ1
. . . . . (52)
The derivation is the same of ˜qs,k, where 1 ≤ k ≤ n. In all,
we obtain
˜qs,k(t) ≤
φk
λk
, (1 ≤ k ≤ n). . . . . . . (53)
For the purpose of making use of the above derivation, we
rewrite the lowpass filter unit as
p
p+λ
= 1−
λ
p+λ
. . . . . . . . . . . (54)
Then the upper bounds of derivatives of ˜qs,1 can be ob-
tained as
˙˜qs,1(t) ≤ |s1(t)| 1−
t
0
λ1eλ1T−λ1t
dT
= |s1(t)| 1+
1
λ1
λ1eλ1T−λ1t
T=t
T=0
= φ1 1+
λ1
λ1
−e−λ1t
≤ 2φ1. . (55)
The same derivation to the other generalized joints, we
obtain
˙˜qs,k(t) ≤ 2φk, 1 ≤ k ≤ n. . . . . . . . (56)
4. Simulation and Analysis
In the simulation, at first we used AUTOLEV to con-
struct the human model and then exported the model as a
MATLAB code. After that, we added the proposed strat-
egy codes, including human attitude control and human
parameter update, into the MATLAB code. By running
the code, we obtained all the information about the po-
sitions, velocities and accelerations of the human model.
The animation was done based on these data with VOR-
TEX where the skeleton model was constructed by con-
necting the bones composed of polygon points. It is noted
that in the simulation, we assume the robot realize perfect
force control.
4.1. Human Model Configuration
In the simulation, we take a normal human body into
account which is composed of 16 parts, including head,
chest, mid-trunk, lower-trunk, upper arms (left and right),
lower arms (left and right), hands (left and right), upper
legs (left and right), lower legs (left and right) and feet
(left and right). Each two parts (or two links) are con-
nected by one joint. According to the physiological struc-
ture of the human body, the joints vary from 1 DOF to
3 DOF. In all, the human body model we considered has
35 DOF with 1.7142 m height and 72.81 kg weight. The
detailed parameters of the human body model are shown
in Table 1 [12, 13].
As we all know that the human joint can not rotate
from 0◦ to 360◦. For example, the neck can only rotate in
the interval [−π/2 π/2] when we turn around our head.
Moreover, there are passive joint moments corresponding
to the constriction forces and moments developed by lig-
aments, joint capsules and other soft tissues around the
joints. Based on the previous researches by Anderson et
al. and Yamaguchi, we used the passive moment τpass in
the simulation as
τpass = α+eβ+(q−q+)
+α−eβ−(q−q−)
. . . . (57)
where q+ and q− are the threshold angle beyond which
the passive moment takes effect. The passive moment is
small in the interval q− ≤ q ≤ q+ and it becomes large
very quickly if q > q+ or q < q− as shown in Fig. 6.
From the viewpoint of geometrics, α+ and α− denote

Table 1. Anthropological parameter values. The human body model used in the simulation is a normal body with 1.7142 m height
and 72.81 kg weight which totally has 35 DOF. Specifically, the anthropological items include length, mass center position and
inertia coefficients
Segment Length (m) Mass (kg) MCS pos (m) I11 (kg.m2) I22 (kg.m2) I33 (kg.m2)
Head 0.2429 5.07 0.1215 0.027 0.020 0.030
Chest 0.2421 11.65 0.1226 0.174 0.148 0.070
Mid-Trunk 0.2155 11.92 0.0970 0.129 0.121 0.081
Lower-Trunk 0.1457 8.15 0.0891 0.065 0.060 0.053
Upper Arm 0.2817 1.98 0.1626 0.013 0.004 0.011
Lower Arm 0.2689 1.18 0.1230 0.007 0.001 0.006
Hand 0.0862 0.45 0.0681 0.001 0.001 0.001
Upper Leg 0.3960 10.34 0.1622 0.175 0.036 0.175
Lower Leg 0.4300 3.16 0.1890 0.037 0.006 0.035
Foot (antero posterior) 0.1788 0.90 0.0652 0.001 0.004 0.004
Foot (vertical) 0.0420 — 0.0210 — — —
Hip Width 0.0835
−0.5 0 0.5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x 10
5
q (rad)
Passivemoment(kg.m2
)
q+
q−
−0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2
−6
−4
−2
0
2
4
6
q (rad)
Passivemoment(kg.m
2
)
(a) (b)
Fig. 6. Passive moment term. The parameters are set as q+ = 0.175, q− = −0.175, α+ = 0.085, α− = 0.085, β+ = 30.72,
β− = −30.72. It is easy to see that the value of passive moment in the interval of [q−,q+] is very small while increasing rapidly
outside that interval.
how sensitive the passive moment is and β+ and β− de-
note what magnitude level the passive moment is. In
Fig. 6, it is shown that α+ and α− decide the steepness of
the curve and β+ and β− determine the vertical extension
of the curve. In the simulation, the passive moments were
implemented in the joints of chest-midtrunk, midtrunk-
lowertrunk, lowertrunk-upperleg, upperleg-lowerleg and
lowerleg-foot where the parameters of passive moment
are shown in Table 2 [14].
For the other joints, there were no passive moment
models available in the literature. However, we have to
make sure the angles of these joints are in the reasonable
range. Hence, we add passive angle joint moment
τang cont = α+eβ+(q−q+)
+α−eβ−(q−q−)
−γ ˙q . (58)
into the human model where the passive term −γ ˙q acts as
a damping component.
4.2. Simulation Results
The simulation was implemented by coordination of
three software packages, including AUTOLEV, MAT-
LAB and VORTEX. The detailed cooperation relations
are explained as follows. AUTOLEV is used to construct
the dynamic model of human body for further computa-
tion [15]. As MATLAB is very powerful in computing,
we choose it to do the main computation as a solution tool
for ordinary differential equations; although VORTEX is
able to do physical simulation, the programming grammar
is a bit complex. Hence, we just use its stereoscopic pre-
sentation function to make animations. More specifically,
the simulation process is explained as the following three
phases:
• Construct the dynamics of human body model. We
imported the human body parameters, including set-
tings of link, joint and passive moment, into AU-
TOLEV. For efficient computation, AUTOLEV auto-

Table 2. Passive moment parameters of human joints. The values of α+, α−, β+, β−, q+ and q− are based on the research of
human lower extremity model.
Joint Joint Joint Axes α+ β+ α− β− q+ q−
Type
Chest-MidTrunk 3 DOF
1 0 0 −0.250 20.360 0.250 −20.360 0.085 −0.085
0 1 0 −0.250 20.360 0.250 −20.360 0.085 −0.085
0 0 1 −0.350 30.720 0.250 −20.360 0.085 −0.085
MidTrunk-LowerTrunk 3 DOF
1 0 0 −0.250 20.360 0.250 −20.360 0.085 −0.085
0 1 0 −0.250 20.360 0.250 −20.360 0.085 −0.085
0 0 1 −0.350 30.720 0.250 −20.360 0.085 −0.175
LowerTrunk-UpperLeg 3 DOF
1 0 0 −0.030 14.940 0.030 −14.940 0.500 −0.500
0 1 0 −0.030 14.940 0.030 −14.940 0.920 −0.920
0 0 1 −0.244 5.050 1.510 −21.880 1.810 −0.470
UpperLeg-LowerLeg 1 DOF 0 0 1 −6.090 33.940 11.030 −11.330 0.130 −2.400
LowerLeg-Foot 1 DOF 0 0 1 −2.030 38.110 0.180 −42.120 0.520 −0.740
Fig. 7. Flow diagram of simulation program. The sim-
ulation uses three software packages including AUTOLEV,
MATLAB and VORTEX. After adding control module and
identification module, the MATLAB source code generated
by AUTOLEV is executed. Such computation results (in-
cluding how the human body model moves) are used for
making animations by VORTEX.
matically generated 6773 intermediate variables and
constructed 35 dynamic equations. After compiling,
the final dynamic equations were exported as exe-
cutable source code of MATLAB (or C) for further
computation.
• Compute the motion of human body. The MATLAB
source code generated by AUTOLEV does not con-
tain any attitude control strategy. Hence, we have
to add the codes of human attitude control and hu-
man parameter update into the basic MATLAB code.
More specifically, one execution cycle can be ex-
plained as follows (Fig. 7). The function ReadUser-
Input() initiates all the coefficients and parameters
for simulation, such as the mass, inertia coefficient,
and initial posture of human body model. Then Ope-
nOutputFilesAndWriteHeadings() makes text files
(from human.1 to human.45) for storing final motion
results. The new mdlDerivatives() with adaptive con-
trol function calculates and returns the derivatives
of the continuous “interested states” of the human
model. After that, the ordinary differential equations
representing the dynamics of human body under con-
trol and identification is solved by solver Ode45()
and the final results are outputted to VORTEX for
animation.
• Animate the whole dynamics. At first, the poly-
gons of human body bones were built by the soft-
ware PRO/ENGINEER. Then the polygons of bones
were imported into VORTEX and presented in stereo
display. After that, the bones were connected accord-
ing to the physiological annexation. Finally, the mo-
tion data resolved by MATLAB was used to drive
the joints. With the movement of skeleton model,
the animation was presented.
In the simulation, we chose the position of head (denoted
as Ph,x, Ph,y, Ph,z), the angle drift off the horizontal line
of lower-trunk (denoted as θ1,x, θ1,y, θ1,z), the angle of
lower-trunk and upper-leg (denoted as θ2,x, θ2,y, θ2,z) to
constitute the “interested states” (Fig. 8), i.e.,
qs = Ph,x,Ph,y,Ph,z,θ1,x,θ1,y,θ1,z,θ2,x,θ2,y,θ2,z
T
. (59)
The initial velocity and acceleration are set as qs(0) =
˙qs(0) = 0 and the desired states is set as
qs,d =
[0.2,0.8,0.01,0.01,0.01,−0.7854,0.01,0.01,1.5708]T
˙qs,d = ¨qs,d = [0, 0, 0, 0, 0, 0, 0, 0, 0]T
.
. . . . . . . . . . . . . . . . . . (60)
Applying the control law in Eq. (20) and adaptation law
in Eq. (21), we obtain
τs = Fh,x, Fh,y, Fh,z, τ1,x, τ1,y, τ1,z, τ2,x, τ2,y, τ2,z
T
(61)

Fig. 8. Choice of the “interested states” in lifting up hu-
man body. When lifting up human body, not all the motions
are essential to measure and control. Based on the proposed
approach, the “interested states” are chosen as follows: po-
sition of the head, angle drift off the horizontal line of lower-
trunk and angle of lower-trunk and upper-leg.
Then we choose
F1 =


Fh,x
Fh,y
Fh,z


F2 ≈







(τ1,zl1,z sinθ1,z +τ2,zl2,z sinθ2,z)
+(τ1,yl1,y cosθ1,y +τ2,yl2,y cosθ2,y)
(τ1,zl1,z cosθ1,z +τ2,zl2,z cosθ2,z)
+(τ1,xl1,x sinθ1,x +τ2,xl2,x sinθ2,x)
(τ1,xl1,x cosθ1,x +τ2,xl2,x cosθ2,x)
+(τ1,yl1,y sinθ1,y +τ2,yl2,y sinθ2,y)







(62)
where [l1,x, l1,y, l1,z]T
denotes the distance between the
head position and the buttock. [l2,x, l2,y, l2,z]T
denotes
the distance between the buttock and the application point
of F2. By applying F1 and F2 in the human model, attitude
control is achieved.
The animation of lifting up human body in our ap-
proach is shown in Fig. 9. At the beginning of the simu-
lation, we assume that we do not have any pre-knowledge
about the human body. Hence, the initial values of ˆHs, ˆCs
and ˆGs are set to zero matrices (or zero vectors). As the
estimation of the human parameters goes on, ˆHs, ˆCs and
ˆGs converge to their true values of Hs, Cs and Gs.
The energy, position and angle changes are shown in
Fig. 10. It is easy to see that it takes about 1 sec to change
the attitude of human body. There is a peak of kinematics
energy at the time of about 0.2 sec which means at that
time, the attitude changes very quickly (Fig. 10(a)). That
is because we assume no pre-knowledge of the human
body at the beginning of the simulation. Moreover, as the
identification of human body needs only about 0.2 sec, the
identification process is fast enough for safe nursing.
We set the desired position of the head as (0.2 m, 0.8 m,
0.01 m). Compared with other joints rotating in x or y di-
rection, the joints rotating in z direction turn significantly.
Thus, the angle changes of these joints affect the head po-
sition in x direction greater (Fig. 10(b)).
(a)
(c)
(e)
(b)
(d)
Fig. 9. Snapshots of the lifting process. The five snapshots
labeled from (a) to (e) are taken in the equivalent time in-
terval, which shows the whole process of lifting up human
body.
In the proof of Theorem 1, it was shown that ˙Hs −2Cs is
a skew-symmetric matrix which indicates that parts of the
states (or their linear combination) can be controlled as
a whole. In the simulation, we constructed a new state
which is the angle sum of head, chest, mid-trunk, and
lower-trunk. The angle drift off the horizontal line of the
new state changes to −0.7854 rad (i.e., −45◦) as shown in
Fig. 10(c). The angle between lower-trunk and upper-leg
changes to 1.5708 rad (i.e., 90◦) at the time about 1 sec.
(Fig. 10(d)).
5. Conclusion
In this paper, a new reduced model adaptive force con-
trol strategy for lifting up human body was proposed.
Compared with previous researches, there are two signif-
icant advantages in the proposed attitude control. First is
that it is not necessary to measure human body, like height
and weight, in advance because the proposed approach
can automatically identify the human parameters online.
Second is that the human attitude control law guarantees
the accuracy. Moreover, the robust controller which we
used also tolerates the unmodeled uncertainty of the re-
duced human model. The proposed approach was ana-
lyzed completely from the view point of algorithm con-
vergence. From the derivation of tracking time and track-
ing error, the approach is reliable. The simulation verified
the proposed approach by lifting up a normal human body

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
200
400
Kinematicenergy(J)
Energy of the simplified human model
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−500
0
500
Potentialenergy(J)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−500
0
500
Time (s)
Totalenergy
(a)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.2
0.4
Xdirection(m)
Position of the head
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
Ydirection(m)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.005
0.01
0.015
Zdirection(m)
Time (s)
(b)
0 0.5 1 1.5 2
−1
−0.5
0
0.5
1
Angle(rad)
Angle and volicity off the hirizontal line of lower−trunk
0 0.5 1 1.5 2
−6
−4
−2
0
2
Time (s)
Volicity(rad/s)
(c)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
0.5
1
1.5
2
Angle(rad)
Angle and velocity of the joint of lowertrunk−upperleg
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−0.5
0
0.5
1
1.5
2
2.5
time
Velocity(rad/s)
(d)
Fig. 10. Energy and angle change with time when the hu-
man body is lifted up. (a) The change of energy. (b) The
position of the head. (c) The angle and velocity drift off the
horizontal line of the lowertrunk. (d) The angle and velocity
of the joint of lowertrunk-upperleg.
(35 DOF) with passive moments. It is novel that the ap-
proach proposed in this paper is not only designed for the
speciﬁc case of lifting human, but also can be used much
more widely for physically interacting with human. Our
future research will concern on how to change the lying
posture of human body by robot.
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Name:
Haiwei Dong
Affiliation:
Department of Computational Science, Graduate
School of System Informatics, Kobe University
Address:
Brief Biographical History:
2005 Received B.E. from Nanjing University of Science and Technology
2008 Received M.E. from Shanghai Jiaotong University
2008- Working towards Ph.D. degree in Kobe University
Main Works:
• H. Dong et al., “Novel Information Matrix Sparsification Approach for
Practical Implementation of Simultanesous Localization and Mapping,”
Advanced Robotics, Vol.24, pp.819-838, 2010.
Name:
Zhiwei Luo
Affiliation:
Professor, Kobe University
Address:
1992- Assistant Professor, Toyohashi University of Technology
1994- Frontier Researcher, The Institute of Physical and Chemical
Research
1999- Associate Professor, Yamagata University
2001- Team Leader, The Institute of Physical and Chemical Research
2006- Professor, Kobe University
Main Works:
• Development of a human-interactive robot RI-MAN
• Immersion-type dynamic simulation of human-robot interaction
• Innovation of health engineering
Membership in Academic Societies:
• The Society of Instrument and Control Engineers (SICE)
• The Robotics Society of Japan (RSJ)
• Japanese Neural Network Society (JNNS)
• The Institute of Electrical and Electronics Engineers (IEEE)
Name:
Akinori Nagao
Affiliation:
Department of Computational Science, Graduate
School of System Informatics, Kobe University
Address:
1-1 Rokkodai, Nada, Kobe, Hyogo
1996 B.L.A., University of Tokyo,
1998 M.A., University of Tokyo,
2001 Ph.D., Arizona State University
2002-2003 Boston University & Harvard Medical School
2003-2006 RIKEN
2006-2007 University of Aberdeen
2007- Kobe University
Main Works:
• Nagano et al., “An analysis of directional changes in the center of
pressure trajectory during stance,” Gait and Posture.
• Nagano et al., “Neuromusculoskeletal computer modeling and simulation
of upright, straight-legged, bipedal locomotion of Australopithecus
afarensis (A.L. 288-1),” American J. of Physical Anthropology.
• Nagano and Komura, “Longer moment arm results in smaller joint
moment development, power and work outputs in fast motions,” J. of
Biomechanics.
Membership in Academic Societies:
• International Society of Biomechanics (ISB)
• American Society of Biomechanics (ASB)
• Japanese Society of Biomechanics (JSB)
• Japanese Society of Physical Education

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