-1-

Abstract— As the human society steps into the aging society,

robots play more and more important roles in our daily life,

especially in nursing activities. As we all know, the human body

has numerous joints and each joint has at least one degree of

freedom. Hence, changing the attitude of human body is very

complex. In this paper, we consider the human body as a

redundant system and only interested parts of the joint states are

controlled. The new strategy proposed in this paper is able to lift

up the human body to the pre-defined position and posture while

regardless of individual differences. In addition, the convergence

analysis, including tracking time analysis and tracking error

analysis, is also given. The approach is simulated by lifting a

skeleton human model with two robot arms, which verifies the

efficiency and effectiveness of the approach.

I. INTRODUCTION

S more and more countries step into the aging society,

nursing service for elderly people becomes much more

important than ever before. Our research focuses on the

social problem and we propose to use robots to carry elderly

people in nursing care. Until now, there have been many

researches on rigid object manipulation; however, the objects

they dealt with were simple rigid objects. They did not

consider manipulating complex multi-link object like human

body [1-4]. Moreover, dealing with floating object, e.g. lifting

up human body, has also not been fully considered yet.

RI-MAN which was selected as the best invention of TIME

Magazine 2006 achieved a great success in lifting up human

body [5-7]. Some of the simulations in this paper are based on

the research of RI-MAN.

Actually, in our research the human body can be considered

as a free-floating multi-link rigid object with passive moments.

Our objective is to change the shape of the above mentioned

object by external forces. Hence, two difficulties come out:

the first one is about free-floating multi-link rigid object.

Previous studies on free-floating object are mainly in

controlling spacecraft. In these cases, the spacecraft can be

accurately modeled. Based on the precise model, various

methods, e.g. generalized Jacobian methods, can complete the

attitude control task [17]. However, in our case, the model of

Manuscript received January 29, 2010. This work was supported in part

by the Japan Society for the Promotion of Science under Grant No.

21240019.

Haiwei Dong, Zhiwei Luo and Akinori Nagano are with the Department

of Computer Science and Systems Engineering, Kobe University, Kobe,

657-8501, Japan (e-mail: haiwei@stu.kobe-u.ac.jp; luo@gold.kobe-u.ac.jp;

aknr-ngn@phoenix.kobe-u.ac.jp).

human body can not be modeled accurately. That is not only

because there are some human parameters which can not be

measured, but also because human bodies have individual

differences. The second one is about external forces. As we all

know, the human body consists of 206 bones and much more

joints. Each joint has one, two or three degrees of freedom

(DOF). And each degree of freedom is described by a

differential equation. Hence, the model of human body is

really large. The calculation on such a big model is also a

tough work. Moreover, considering passive moments, the

model becomes much more complex. As the human body is

such a complex model with very high dimension, application

of external forces on the human body is also very complicated.

The calculation needs much time and real-time performance is

almost impossible. Furthermore, the process of lifting up

human body must be absolutely safe. If we can not make sure

that the computation is real-time, the safety can not be

guaranteed correspondingly.

In consideration of the two difficulties mentioned above,

the basic idea for solution comes from our daily experience.

When we lift up a human body, we do not care about the angle

of ankle, the position of hands and so on. What we do have to

care about are the position of the head, the vertical deflection

of upper limb and the angle of hip. Here we call them

“interested states”. From the view point of system theorem, we

treat the human body as a large redundant system whose

dimension can be reduced by diverting the effects of other

uninterested joints to the ones of interested joints. The newly

constructed small model of human body has very few DOF

while having huge uncertainties, unfortunately. In order to

eliminate these effects of uncertainties, robust adaptive

controller is designed. Moreover, because we assume no

priori knowledge of the human body is given in advance, the

human model estimator is built to identify the parameters of

human body. Thus, the whole attitude control approach with

human model estimation overcomes the individual differences,

such as height, weight, and so on when lifting up human body.

This paper is organized as follows. The second section

demonstrates the fundamental idea of our approach. The third

section illustrates the detailed derivation of the controller and

the estimator. The fourth section analyzes the convergence of

our strategy, including 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.

Reduced Model Adaptive Force Control for Carrying Human Beings

with Uncertain Body Dynamics in Nursing Care

Haiwei Dong, Zhiwei Luo and Akinori Nagano

A

2010 IEEE/ASME International Conference onAdvanced Intelligent MechatronicsMontréal, Canada, July 6-9, 2010

978-1-4244-8030-2/10/$26.00 ©2010 IEEE 193

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The sixth section concludes the whole paper.

II. FUNDAMENTAL IDEA

The problem we focus on is how to lift up a human body

from bed, and then move him (or her) to another place. As

pointed before, the problem we want to solve is how to change

the shape of a multi-link object with passive moments by

external forces. Considering the characters of human body,

the points of force application are the back and the lap of the

body. The effect of attitude change by forces is equivalent to

applying torques to the joints of the body as

1 1 2 2

T T

robJ F J Fτ = + (1)

where 1F and

2F are the forces acted by the arms of robot.

robτ is the torque equivalent to the effects by

1F and

2F .

1J

and 2J are Jacobian matrices of the human body model. Thus,

the general dynamic equations of human body can be written

as

( ) ( , ) ( )pass rob

H q q C q q q G q τ τ+ + = +&& & & (2)

where

1nq × Generalized states of human body which include

the position of head and the angles of all the

joints.

( )n n

H q × Inertia matrix. This is a symmetric and positive

semi-definite matrix which contains information

regarding the instantaneous mass distribution of

the human body model.

( , )n n

C q q ×& Centripetal and coriolis torques. The terms of

( , )C q q& contain products of angular speeds.

When the degrees of freedom are rotational, the

terms of ( , )C q q& represent the moments of

centrifugal forces.

1( )

nG q × Gravitational torques. Because ( )G q changes

with the posture configuration of the human body

model, the terms are functions of the generalized

states.

1pass nτ × Passive joint torques. It contains the torques and

moments arising from muscular activations and

passive elastic structures surrounding the human

joints.

1rob nτ × The torques acted by the robot arms, which is

controllable.

It is noted that the subscript means the dimension of matrix

(or vector). Besides, as the passive torques pass

τ is actuated

from the internal organs and tissues, pass

τ can not be

controllable.

By defining [ ]1 1, ,

mq q q= L a state vector composed of the

states of human body which we are interested in, and

[ ]2 1, ,

m nq q q+= L a state vector consisting of other states, we

can get [ ]1 2q q q= . Thus the dynamics of human body can

be rewritten as

11 12 1 11 12 1 1

21 22 2 21 22 2 2

,1 ,1

,2 ,2

pass rob

pass rob

H H q C C q G

H H q C C q G

τ ττ τ

+ +

= +

&& &

&& &

(3)

where the dimensions of sub-block matrices of 11

H , 12

H ,

21H ,

22H are m m× , ( )m n m× − , ( )n m m− × ,

( ) ( )n m n m− × − , respectively. And the dimensions of

sub-block matrices of 11

C , 12

C , 21

C , 22

C are m m× ,

( )m n m× − , ( )n m m− × , ( ) ( )n m n m− × − , respectively.

The dimensions of vectors 1

G , ,1pass

τ , ,1rob

τ are 1m× , and

2G ,

,2passτ ,

,2robτ are ( ) 1n m− × .

Extracting the parts of the dynamics of human body which we

are interested in, we get

1 1

11 12 11 12 1 ,1 ,1

2 2

pass rob

q qH H C C G

q qτ τ

+ + = +

&& &

&& & (4)

Considering that the dynamic model is time-varying, after

arranging equation (4), we obtain

( )11 1 11 1

1 12 2 12 2 ,1 ,1

( ) ( )

( ) ( ) ( ) ( ) ( )pass rob

H t q C t q

G t H t q C t q t tτ τ

+

+ + + − =

&& &

&& & (5)

By defining the inertia matrix, centripetal matrix, gravitational

matrix and torque vector of the small system as

11

11

1 12 2 12 2 ,1

,1

( ) ( )

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )

s

s

s pass

s rob

H t H t

C t C t

G t G t H t q C t q t

t t

τ

τ τ

=

=

= + + −

=

&& & (6)

we obtain the general mechanical form of the reduced model

of human body

( ) ( ) ( ) ( )s s s s s s

H t q C t q G t tτ+ + =&& & (7)

where the subscript s denotes the reduced system. Actually,

our basic idea is to consider the influences from uninterested

human joints (in this case from state 2q ) as perturbations. Our

idea is to change the attitude of reduced human body model

adaptively by estimating the parameters of s

H , s

C and s

G in

real time. The detailed estimation meanings are

� Estimating s

H and s

C --- make the system adaptively

adjust itself to various people with different weights.

� Estimating s

G --- eliminate the perturbations from other

uninterested joints.

Considering the basic idea above, the approach to be

proposed in our paper should be able to identify and control

the dynamics of the reduced human body model at the same

time. Assuming that the human model is totally unknown in

advance, for the safety in the nursing activity, the

identification 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 strategy

also has to be able to tolerate these individual differences.

194

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III. STRATEGY OF ATTITUDE CONTROL

First of all, we define some human parameter matrices for

the convenience of derivation. Define actual parameter matrix

as

T

T T T

H C GP P P P = (8)

where

11 12 1 1 2

11 12 1 1 2

1 2

Ts s s s s s

H n n n nn

Ts s s s s s

C n n n nn

Ts s s

G n

P H H H H H H

P C C C C C C

P G G G

=

=

=

L L

L L

L

and define estimation parameter vector as

ˆ ˆ ˆ ˆT

T T T

H C GP P P P = (9)

where

11 12 1 1 2

11 12 1 1 2

1 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆˆ

ˆ ˆ ˆˆ

Ts s s s s s

H n n n nn

Ts s s s s s

C n n n nn

Ts s s

G n

P H H H H H H

P C C C C C C

P G G G

=

=

=

L L

L L

L

then the estimation error matrix can be defined as

ˆP P P= −% (10)

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 small system represents a physical system. It is

easy to prove that by proving 2s s

H C−& is a skew-symmetric

matrix, i.e. the reduced system 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 control law and a parameter update law to

realize the human model attitude control and human model

parameter identification, respectively. In fact, the two

processes 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 candidate is

less than zero.

Theorem 1

Consider a time-varying system with m-order

( ) ( ) ( ) ( )s s s s s s

H t q C t q G t tτ+ + =&& & (11)

without any pre-knowledge about s

H , s

C and s

G . The

vector ,s d

q means the desired states. Define a new vector s

as

, ,

( ) ' ( )s s s s d s s d

s q q q q q q= + Λ = − + Λ −&% % (12)

where Λ is positive diagonal matrix. From the conceptual

view of velocity, we define the reference velocity ,s r

q& as

,s r s

q q s= −& & (13)

If we choose the control law

, ,

ˆ ˆˆ ( ) ( ) ( ) sgn( )s s s r s s r s

H t q C t q G t k sτ = + + − ⋅&& & (14)

and parameter update law

1

1 , , 1 , , 1ˆ T T T T

s r n s r s r n s r nP s q s q s q s q s s− = −Γ

&&& && & &L L L (15)

under the assumption of

sgn( )T T

k s s P P⋅ > Γ% & (16)

where k and Γ are positive diagonal matrixes, sgn( )⋅ is

signal function, then the whole system tracks the desired

trajectory and the parameter matrices s

H , s

C and s

G

converge to actual values globally.

Proof:

Define a Lyapunov function candidate

( )1 1( ) 2

2 4

T T

sV t s H s P I P= + Γ +% % (17)

then the first part of ( )V t can be written as

'

,

1 1( )

2 2

T T T

s s s s s r ss H s s H q H q s H s

= − +

&&& && (18)

From equation (7), s s s s s s

H q C q Gτ= − −&& & , then

( )

'

, ,

, ,

1 1( )

2 2

1( ) ( 2 )

2

T T T

s s s s r s s s r s

T T

s s s r s s r s s s

s H s s C s q G H q s H s

s H q C q G s H C s

τ

τ

= − + − − +

= − − − + −

&& &&

&&& &

(19)

According to the previous research on mechanical system, the

system in the form of equation (2) satisfies

( )2 0Tq H C q− =&& & (20)

i.e. 2H C−& is a skew-symmetric matrix. Hence, the following

relation satisfies

( )0

22

ij ij

ji ji

if i jH C

H C otherwise

=− = − −

&&

(21)

Without loss of generality, we choose 1q& as the new state

vector in which we are interested (equation (3)) and follow the

same system simplification procedures in equation (3)-(7).

According to the relation in equation (21), the small system

satisfies

( ) ( )( ) ( )( )11 , 11 ,

11 , 11 ,

0

22

iu iv iu iv

iv iu iv iu

if iu iv

H CH C otherwise

=− =

− −

&& (22)

where 11

H and 11

C are defined in equation (3). Hence,

11 112H C−& is a skew-symmetric matrix. Based on the

definitions of s

H and s

C in (6), 2s s

H C−& is a

skew-symmetric matrix, hence

'

, ,

1( )

2

T T

s s s s r s s r ss H s s H q C q Gτ = − − −

&& & (23)

Therefore, ( )V t& can be simplified as

' '

, ,

1 1( )

2 2

ˆ( ) ( )

T T

s

T T

s s s r s s r s

V t s H s P P

s H q C q G P P Pτ

= + Γ

= − − − + − Γ

& % %

& & %&& &

(24)

195

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Applying the control law (14)

2

2

, ,

, ,

, ,

2 1, , 2

1

2

ˆ ˆˆ ( ) ( ) ( ) sgn( )

0 0 0 0 1 0 0ˆ

0 0 0 0 0 1 0ˆ

ˆ0 0 0 0 0 0 1

sgn( )

sgn( )

sgn( )

s s s r s s r s

T T

s r s r

HT T

s r s r

C

GT T ms r s r m m

m

H t q C t q G t k s

q qP

q qP

Pq q

s

sk

s

τ

××

= + + −

= ⋅

−

&& &

&& &L L L

&& &L L L

M O M M O M M O M

&& &L L L

M

1m×

(25)

into ( )V t& , which leads to

( )

( )

, , , ,

, ,

1 , , 1 , ,

( )

ˆ ˆˆ ( ) ( ) ( ) ( ) ( ) ( )

ˆsgn( ) ( )

ˆ( ) ( ) ( ) sgn( ) ( )

[ ] [ ] [

T

s s r s s r s s s r s s r s

T T

T T T

s s r s s r s

T T T T

s r n s r H s r n s r C

V t

s H t q C t q G t H t q C t q G t

k s s P P P

s H t q C t q G t k s s P P P

s q s q P s q s q P

= + + − − −

− ⋅ + Γ −

= + + − ⋅ + Γ −

= + +

&

&& & && &

&% &

&% %% % &&& &

% %&& && & &L L1

]

ˆsgn( ) ( )

n G

T T

s s P

k s s P P P− ⋅ + Γ −

%L

&% &

(26)

where

, ,1 ,2 , 1

, ,1 ,2 ,1

Ts s s

s r r r r m m

Ts s s

s r r r r mm

q q q q

q q q q

×

×

=

=

&& && && &&L

& & & &L

We can obtain,

2 21 , , 1 , , 11 2 2 1

( )

ˆsgn( ) ( )

TT T T T T

s r n s r s r n s r nm m

T T

V t

P s q s q s q s q s s

k s s P P P

× × =

− ⋅ + Γ −

&

% && && & &L L L

&% &

(27)

Taking the parameter adaptation law of equation (15), finally

we obtain

( ) sgn( )T TV t P P k s s= − Γ − ⋅& % & (28)

According to the assumption of (16), ( ) 0V t <& . Hence, the

tracking error and parameter estimation error converges to

zero asymptotically.

■

The assumption (16) means that the control signal should

be, in some ways, large than the uncertainty of parameters. In

practice, the condition (16) is satisfied by giving a relative

large k . The scheme illustration is shown in Figure 1.It is

noted that we use the combined error s , reference errors ,s r

q&

and ,s r

q&& to estimate the parameters s

H , s

C and s

G , while the

output signals required to measure from the physical system

are sq and

sq& . Such an adaptation scheme not only maintains

the global stability, but also leads to fast parameter

convergence and small tracking errors.

IV. CONVERGENCE ANALYSIS

First of all, assume that ,(0) (0)

s d sq q= which means there

is no “initial jump” of position and velocity between the

desired state and actual state. Let sq% be the tracking error in

the variable sq , i.e.,

,s s s d

q q q= −% (29)

Let us define a time-varying surface in the state-space nR by

the scalar equation ( ; ) 0s

s q t = , where

s s

s q qλ= +&% % (30)

Given initial assumption ,(0) (0)

s d sq q= , the problem of

tracking ,s s d

q q≡ is equivalent to the one of remaining sq% on

the surface ( )S t for all 0t > ; indeed 0s ≡ represents a

linear differential equation whose unique solution is 0sq ≡% .

Thus, the problem of tracking the n-dimensional vector ,s d

q

can be reduced to the one of keeping the scalar quantity s at

zero. More precisely, the problem of tracking the

n-dimensional vector ,s d

q 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 approach and finally reach the

manifold ( )B t which is defined as

Signal Transform UnitAttitude Controller Human Body Model

Human Body Model Estimator

Input: Desired Attitude

Output: Actual Attitude

s

s

q

q&s s s s s sH q C q G τ+ + =&& &( ), ,

ˆ ˆˆ sgns s s r s s r sH q C q G k sτ = + + −&& &sτ

11 , , 1 , , 1

ˆ T T T Ts r n s r s r n s r nP s q s q s q s q s s

− = −Γ &

&& && & &L L L

,s dq

,s dq&

,s dq&&

Λ

Λ

Λ

+

+

s+

−

sq%

sq

sq&%

sq&

+

−

+

+

−

−,s rq&

,s rq&&

Signal Transform UnitAttitude Controller Human Body Model

Human Body Model Estimator

Input: Desired Attitude

Output: Actual Attitude

s

s

q

q&s s s s s sH q C q G τ+ + =&& &( ), ,

ˆ ˆˆ sgns s s r s s r sH q C q G k sτ = + + −&& &sτ

11 , , 1 , , 1

ˆ T T T Ts r n s r s r n s r nP s q s q s q s q s s

− = −Γ &

&& && & &L L L

,s dq

,s dq&

,s dq&&

Λ

Λ

Λ

+

+

s+

−

sq%

sq

sq&%

sq&

+

−

+

+

−

−,s rq&

,s rq&&

Fig.1 Scheme block diagram. The input signals are the desired trajectories of generalized human joints in the reduced

model. The output is the actual motion trajectories of the human body model. Our strategy controls the position and posture

of human body model and identifies the human body model online at the same time.

196

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{ }( ) | ( ; )s s

B t q s q t φ= ≤ (31)

where 0φ > denotes the boundary layer thickness. The

second phase is to make s converge to the desired state

asymptotically. In the following, tracking time and static

tracking error are analyzed.

A. Tracking Time Analysis

In the proof of Theorem 1, we define the Lyapunov function

as

( ) 1 2

1 1( ) 2 ( ) ( )

2 4

T T

sV t s H s P I P V t V t= + Γ + = +% % (32)

After taking the control law (14) and parameter adaptation law

(15), we can rewrite the derivative of ( )V t as

1 2

( ) sgn( ) ( ) ( )T TV t k s s P P V t V t= − ⋅ − Γ = +& % & & & (33)

Extracting parts of the elements in 1( )V t as

2

,

1

n

k s kk

k

s H=∑ and

differentiating the ordinary element, we obtain

'

2

,

1sgn( )

2k s kk k k k k ks H k s s k s

≤ − = −

(34)

Equation (34) states that the “distance” to the surface, as

measured by 2s , decrease along all system trajectories. Thus,

it constrains trajectories to point towards the manifold ( )B t .

In detail, let reach

kt be the required time of the k th generalized

coordinate s

kq to hit the surface 0

ks = . Integrating the left

side of (34) between 0t = and reach

kt t= leads to

2 2

, ,0

1 1( 0)

2 2

reachkt

k s kk s kk k

ds H dt H s t

dt= − =∫ (35)

while the integration of the right side between 0t = and reach

kt t= can be written as

0 0( 0)

( 0)

reach reachk kt t

k k k k

reach

k k k

k s dt k s t dt

k s t t

− ≤ − =

= − =

∫ ∫ (36)

Applying the inequality relation in (34), we get the acquired

time for any generalized coordinate s

kq to get 0

ks =

,

( 0)2

s kkreach

k k

k

Ht s t

k≤ = (37)

Furthermore, manifold definition of ( )B t implies that once

on the surface, ( ) 0s t = , i.e.,

0s sq qλ+ =&% % (38)

The solution to the equation (38) is

t

sq e λ−=% (39)

which means the tracking error tends exponentially to zero

with a time constant λ as shown in Fig.2.

B. Static Tracking Error Analysis

For the static error, we can analyze the error in the second

phase. Bounds on s can be directly translated into bounds on

the tracking error vector sq% , and thus the scalar s represents

a true measure of tracking performance. Indeed, by definition

(30), the tacking error sq% is obtained from s through a

first-order lowpass filters (Fig.3), where /p d dt= is the

Laplace operator.

Fig.2 Position error of one human joint in the second phase.

The error between the desired state and actual state of the

joints in human body decrease to zero exponentially.

Fig.3 Relation between ,s k

q% , ,s k

q&% and ks (1 k n≤ ≤ ) in the

Laplace field. It is possible to get the upper bound of joint

angle by integration from the knowledge of Laplace

transform.

Assuming that (0) 0sq =% and ( )s t φ< , for the first

element of ,1( )

sq t% , we have

1 ( )

,1 10

( ) ( )t

t T

sq t e s T dT

λ− −= ∫% (40)

According to the above assumption, ( ) (1 )k ks t k nφ< ≤ ≤

the upper bound of ,1( )

sq t% can be obtained

( )

1 1 1

1

( ) 1,1 1

01 0

1 1

1 1

( )

1

T tt

t T T t

s

T

t

q t e dT e

e

λ λ λ

λ

φφ

λ

φ φλ λ

=

− − − −

=

−

≤ =

= − ≤

∫%

(41)

The derivation is the same of ,s k

q% , where 1 k n≤ ≤ . In all, we

obtain

, ( ) (1 )ks k

k

q t k nφλ

≤ ≤ ≤% (42)

We rewrite the lowpass filter unit as

1

1

p λ+1s ,1sq%

L1

p

p λ+ ,1sq&%

1

np λ+ns ,s nq%

n

p

p λ+ ,s nq&%

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

( )t s

,s kq%

197

-6-

1p

p p

λλ λ= −

+ + (43)

Then the upper bounds of derivatives of ,1s

q% can be obtained

as

( )1 1 11

,1 1 1 1 10

1

( ) ( ) 1 1 2t

T t t

sq t s t e dT e

λ λ λλλ φ φ

λ− −

≤ − = + − ≤

∫&% (44)

Applying the same derivation to the other generalized joints,

we obtain

, ( ) 2 1s k kq t k nφ≤ ≤ ≤&% (45)

V. SIMULATION

In the simulation, we use AUTOLEV to construct the model

of human body [8-10] and output the model as a MATLAB

code. After that, we insert our own strategy codes, including

parameter identification of the human model and attitude

control, into the MATLAB code. By running the code, we get

all the information about the positions, velocities and

accelerations of the human model. The animation is done

based on these data with VORTEX where the skeleton model

is constructed by connecting the bones composed of polygon

points. It is noted that in the simulation, we assume the robot

can realize perfect force control.

A. Configuration of Human Body Model

In the simulation, we take a normal human body model 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 connected by one joint. According

to the physiological structure of the human body, the joints

vary from one DOF to three DOF. In all, the human body

model we considered has 35 DOF with 1.7142m height and

72.81kg weight [10].

There are passive joint moments corresponding to the

constriction forces and moments developed by ligaments,

joint capsules and other soft tissues around the joints. Based

on the previous researches [11-15], we use the passive

moment pass

τ in the simulation as

( ) ( )q q q q

passe eβ βτ α α+ + − −− −

+ −= + (46)

where q+ and q− are the threshold angles beyond which the

passive moment takes effect. α+ and α− denote how

sensitive the passive moment is and β+ and β− denote 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. The passive

moment is small in the interval [ , ]q q− + and it becomes large

very quickly in the interval [ , ]q+ +∞ or [ , ]q−−∞ as shown in

Fig.4. In the simulation, the passive moments are implemented

in the joints of chest-midtrunk, midtrunk-lowertrunk,

lowertrunk-upperleg, upperleg-lowerleg and lowerleg-foot.

−0.5 0 0.5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10

5

q (rad)

Passiv

e m

om

ent (k

g.m

2)

q+

q−

Fig.4 Curve of passive moment. The parameters are set as

0.175q+ = , 0.175q− = − , 0.085α+ = , 0.085α− = ,

30.72β+ = , 30.72β− = − . 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.

As we all know that the human joint can not rotate from 0

degrees to 360 degrees. For example, when we turn around

our head, the neck can only rotate in the interval

[ / 2 / 2]π π− . For the other joints with no passive moment

models available in the literature, it is necessary to make sure

the angles of these joints are in the reasonable range. Hence,

similar with the form of passive moment (46), we take passive

angle joint constraint as

( ) ( )

_

q q q q

ang conte e qβ βτ α α γ+ + − −− −

+ −= + − & (47)

where the passive term qγ− & acts as a damping component.

B. Simulation Process

The simulation is implemented by three software packages,

including AUTOLEV, MATLAB and VORTEX. The

detailed usage is explained as follows. AUTOLEV is used to

construct the dynamic model of human body for further

computation. As MATLAB is very powerful in computing, we

choose it to do the main computation tool for solving ordinary

differential equations; Although VORTEX is able to do

physical simulation, the programming grammar is a bit

complex. Hence, we only use its stereoscopic presentation

function to make animations.

As illustrated before, the total DOF of the human body is 35.

But in the case of lifting up human body, we are interested in

only three states: the position of the head and the lower-trunk

angle drift off the horizontal line and the angle between

lower-trunk and upper-leg (Fig.5). The animation of lifting up

human body in our approach is shown in Fig.6. At the

beginning of the simulation, we assume that we do not have

any pre-knowledge about the human body. Hence, the initial

values of ˆs

H , ˆs

C and ˆs

G are set to zero matrices (or zero

vectors). As the estimation of the human body goes on, ˆs

H , ˆs

C and ˆs

G converge to their true values of s

H , s

C and s

G .

198

-7-

Fig.5 Three interested states. During the attitude control of

human body, not all of the motions are necessary to be

considered. In the proposed approach, three interested states

are chosen as the position of the head, the lower-trunk angle

drift off the horizontal line and the angle between lower-trunk

and upper-leg.

The energy, position and angle graphs are shown in Fig.7. It

is easy to see that it takes about 1 second to accomplish the

attitude change of human body. There is a peak of kinematics

energy at the time of about 0.2 second, which means at that

time, the attitude changes very quickly (Fig.7 (a)). That is

because we assume no priori knowledge of the human body at

the beginning of the simulation. Moreover, the main

identification process completes within a short time (about 0.2

second), which guarantees the safety in nursing care.

We set the desired position of the head as (0.2m, 0.8m,

0.01m). Compared with other joints rotating in x or y direction,

the joints rotating in z direction turn significantly. For this

reason, the angle changes of these joints affect the head

position in x direction greatly (Fig.7 (b)).

In the proof of Theorem 1, it was shown that 2s s

H C−& is a

skew-symmetric matrix which indicates that parts of the states

(or their linear combination ) can be controlled as a new

physical system. In the simulation, we construct 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 degrees) as shown in

Fig.7 (c). And the angle between lower-trunk and upper-leg

changes to 1.5708 rad (i.e. 90 degrees) at the time of 1 second

(Fig.7 (d)). The above tracking results also indirectly prove

that the estimations of s

H , s

C , s

G converge to their actual

values.

VI. CONCLUSION

In this paper, a new reduced model adaptive force control

approach for carrying human beings was proposed. Compared

with previous approaches, there are two significant

advantages in our strategy. First is that it is not necessary to

measure the anthropological parameters of human body, like

height and weight, in advance because our strategy can

identify the anthropological parameters of human body online.

Second is that attitude control law guarantees the

manipulation accuracy. Moreover, the robust controller which

we used also can tolerate the uncertainty of human body

model. The proposed approach is analyzed completely from

the viewpoint of algorithm convergence. From the analysis

results of tracking time and tracking error, the approach is

reliable. The simulation verifies our approach by lifting up a

normal human body (35 DOF) with passive moments. It is

highlighted that the approach proposed in this paper is not

only designed for the case of lifting up human body but also

can be used much more widely for controlling various

dynamics of human body.

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(a) (b) (c)

(d) (e)

Fig.6 Animation of the attitude control of the human skeleton model. The five snapshots labeled from (a) to (e) are taken in

the equivalent time interval, which represent the whole process of lifting up human body.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

200

400

Kin

em

atic e

ne

rgy (

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

Po

ten

tia

l e

ne

rgy (

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)

To

tal e

ne

rgy

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

X d

ire

ctio

n (

m)

Position of the head

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

Y d

ire

ctio

n (

m)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.005

0.01

0.015

Z d

ire

ctio

n (

m)

Time (s)

(a) ( b)

0 0.5 1 1.5 2−1

−0.5

0

0.5

1

Angle

(ra

d)

Angle and volicity off the hirizontal line of lower−trunk

0 0.5 1 1.5 2−6

−4

−2

0

2

Time (s)

Volic

ity (

rad/s

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

Angle

(ra

d)

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

Velo

city (

rad/s

)

(c) (d)

Fig.7 Energy and angle change with time when the human body is lifted up. (a) The change of energy. (b) The position of the

head. (c) The lower-trunk angle and angular velocity drift off the horizontal line. (d) The angle and angular velocity of the

joint between lower-trunk and upper-leg.

200