S. Jeschke, H. Liu, and D. Schilberg (Eds.): ICIRA 2011, Part II, LNAI 7102, pp. 558–568, 2011. © Springer-Verlag Berlin Heidelberg 2011

Adaptive Control Scheme with Parameter Adaptation - From Human Motor Control

to Humanoid Robot Locomotion Control

Haiwei Dong1 and Zhiwei Luo2

1 Japan Society for the Promotion of Science and Kobe University 1-8 Chiyoda-ku, 1028472 Tokyo, Japan

2 Department of Computational Science, School of System Informatics 1-1, Rokkodai-cho, Nada-ku, 6578501 Kobe, Japan

{haiwei,luo}@gold.kobe-u.ac.jp

Abstract. As the origin intention of humanoid robot is showing the possibility of the biped walking and explaining the principle, there are many common issues between human motor control and humanoid robot locomotion. This paper considers two major common issues of the two researches. First is modeling. Both in human dynamics simplification and humanoid dynamics modeling, we actively or passively choose parts of the variable states because of dynamics simplification and unmodeled dynamics. In these cases, it is questionable that the dynamics represented by the partial variables states still corresponds to a physical system. In this paper, we discuss this problem and prove that the partial dynamics satisfies the conditions of a physical system, which is the basis of control scheme design. Second is control. To tolerate all the errors or perturbations, we design a control scheme which is composed of variable state control and parameter adaptation. The former can tolerate modeling error; the latter can identify the dynamic system in real time. Finally, we apply the proposed control scheme into a humanoid robot control case, which shows the effectiveness of the proposed control scheme.

Keywords: Unmodeled dynamics, model simplification, adaptive control.

1 Introduction

From the viewpoint of biomechanics, human body can be seen as a multi-rigid-object with numerous joints. Adding muscles and tensors, human body can move as desired by the neural system. The whole system is called neural-skeleton-muscle system [1,2]. One of the features of this system is that there are much more muscles than required to generate movement. Hence, the human body is an over redundant system [3]. One of the important roles of neural system is to control these muscles, which is called human motor control [4]. While in the research area of robotics, there is one research filed focusing on locomotion control of the humanoid robot [5,6]. As the origin intention of humanoid robot is showing the possibility of the biped walking and explaining the principle, there are many common issues between the two research areas.

Adaptive Control Scheme with Parameter Adaptation 559

In this paper, we address the common problems of human motor control and humanoid robot locomotion and give solution.

Specifically, this paper considers two issues in common. First is modeling. As we all know, human body has about 206 bones and numerous joints connecting adjacent bones. Based on the classification criteria of human joints, the joints can be mainly divided into hinge (1 DOF), pivot (2 DOF), saddle (2 DOF), gliding (2 DOF) and ball socket (3 DOF). In dynamic equations, each DOF is expressed as one differential equation. Hence, the overall human dynamics is too large to handle. It is necessary to pick up some state variables which are crucial [7]. The same case is with humanoid robot. To model a humanoid robot, we can not model all the dynamics. The unmodeled dynamics is inevitable. These unmodeled dynamics shows the dynamics corresponding with the omitted state variables. Thus, both in human dynamics simplification and humanoid dynamics modeling, we actively or passively choose parts of the variable states. In these cases, it is questionable that these dynamics represented by the partial variables states corresponds to a physical system. In Part II, we discuss this problem and prove that the partial dynamics satisfies conditions of a physical system, which is the basis of control scheme design.

The second issue is control. For simplified human dynamics, the variable states which are not picked up as crucial variable state also influence the total human dynamics in the form of disturbance. On the other hand, the humanoid model does not only have unmodeled dynamics, but also have many perturbations and modeling errors because of measurement error. To accomplish human motor control or humanoid robot locomotion control, we have to make sure that the designed control scheme is able to tolerate the mentioned disturbances, perturbations and modeling error. In Part III, we design a control scheme consisting of variable state control and parameter adaptation. The former can tolerate modeling error; the latter can identify the dynamic system in real time. The proposed control scheme is verified by applying it into a humanoid robot control case.

2 Dynamic Reduction in Modeling

Consider the general form of a dynamic system

( ) ( , ) ( ) pass robH q q C q q q G q τ τ+ + = + (1)

where robτ is an active torque which is the power to drive the system and passτ is a

passive torque which can not be controlled. From previous research [8], in Hamiltonian form we can write conservation of energy in the form

1 1

( ) ( )2 2

T T T Tpass rob

dq G q Hq q Hq q Hq

dtτ τ+ − = = + (2)

From equation (1), we have

pass robHq G Cqτ τ= + − − (3)

Taking equation (3) into equation (2), we obtain

1

( ) ( )2

T T Tpass rob pass robq G q G Cq q Hqτ τ τ τ+ − = + − − + (4)

560 H. Dong and Z. Luo

After simplification, the result is

( 2 ) 0Tq H C q− = (5)

i.e. the matrix of 2H C− is a skew-symmetric matrix. Specifically, for any mechanical system in the form

11 12 1 11 12 11 1 1 1

1 2 1 2

n n

n n nn n n n nn n n n

H H H C C Cq q G

H H H q C C C q G

τ

τ

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ + =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

(6)

it holds that 2H C− is a skew-symmetric matrix. Hence, the following relation satisfies

( )0

22ij ij

ji ji

if i jH C

H C otherwise

⎧ =⎪⎪⎪− = ⎨⎪− −⎪⎪⎩ (7)

During modeling process, without loss of generality, we choose

,1 ,2 ,[ ]s i i i mq q q q= as new state vector which we are interested in. Thus, we

generate a new system with dynamic reduction

( ) ( ) ( ) ( )s s s s s sH t q C t q G t tτ+ + = (8)

Following the same system simplification procedures, the dynamics equation of the new system is

1, 1 1, 2 1, 1, 1 1, 2 1,1 1 1

, 1 , 2 , , 1 , 2 ,

s s s s s ss s si i i i i im i i i i i imi i i

s s s s s s s s sim i im i im im im im i im i im im im im

H H H C C Cq q G

H H H q C C C q G

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

1si

sim

τ

τ

⎡ ⎤⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥

⎥ ⎢ ⎥⎣ ⎦

(9)

According to the relation of equation (7), the new system satisfies

( ), ,, ,

02

2iu iv iu iviv iu iv iu

if iu ivH C

H C otherwise

⎧ =⎪⎪⎪− = ⎨⎪− −⎪⎪⎩ (10)

Therefore,

2 is a skew symmetric matrixs sH C− (11)

which means the new system after dynamic reduction satisfies conditions of a physical system. Based on it, we design an adaptive control scheme as follows.

3 Control Scheme Design

From now on, we consider the system after dynamic reduction (equation (8)). For the convince of derivation, we define some parameter variables as follows. The actual

parameter vector is [ ]TH C GP P P P= , where

Adaptive Control Scheme with Parameter Adaptation 561

11 12 1 1 2

11 12 1 1 2 1 2,

Ts s s s s sH n n n nn

T Ts s s s s s s s sC n n n nn G n

P H H H H H H

P C C C C C C P G G G

⎡ ⎤= ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

estimate parameter vector is ˆ ˆ ˆ ˆ T

H C GP P P P⎡ ⎤= ⎢ ⎥⎣ ⎦ , 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 ˆ ˆ ˆ ˆˆ ˆ,

Ts s s s s sH n n n nn

T Ts s s s s s s s sC n n n nn G n

P H H H H H H

P C C C C C C P G G G

⎡ ⎤= ⎢ ⎥⎣ ⎦

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

and estimate error vector is ˆP P P= − .

3.1 Basic Control Scheme

Define a Lyapunov function candidate

11 1

( )2 2

T TsV t s H s P P= + Γ (12)

where Γ is a symmetric positive definite matrix. A tracking error vector s is defined as

, ,( ) ' ( )s s s s d s s ds q q q q q q= +Λ = − +Λ − (13)

where Λ is a symmetric positive definite matrix. ,s dq is the desired value of sq . In

addition, a velocity-reference vector is defined as

,s r sq q s= − (14)

Then the first part of 1( )V t can be written as

'

,

,

1 1 1( )

2 2 2

1( )

2

T T T T Ts s s s s s r s

T Ts s s s r s

s H s s H s s H s s H q q s H s

s H q H q s H s

⎛ ⎞⎟⎜ = + = − +⎟⎜ ⎟⎜⎝ ⎠

= − +

(15)

From equation (8), s s s s s sH q C q Gτ= − − , then

( )

'

, ,

, ,

1 1( )

2 2

1( ) ( 2 )

2

T T Ts s s s r s s s r s

T Ts 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

⎛ ⎞⎟⎜ = − + − − +⎟⎜ ⎟⎜⎝ ⎠

= − − − + −

(16)

According to equation (11), 2s sH C− is a skew-symmetric matrix. Hence,

'

, ,1

( )2

T Ts s s s r s s r ss H s s H q C q Gτ

⎛ ⎞⎟⎜ = − − −⎟⎜ ⎟⎜⎝ ⎠ (17)

562 H. Dong and Z. Luo

Therefore, 1( )V t can be simplified as

' '

1 , ,1 1 ˆ( ) ( )2 2

T T T Ts s s s r s s r sV t s H s P P s H q C q G P Pτ

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜= + Γ = − − − + Γ⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠ (18)

Taking the control law as

, ,ˆ ˆˆ ( ) ( ) ( ) sgn( )s s s r s s r sH t q C t q G t k sτ = + + − ⋅ (19)

where k is a symmetric positive matrix and sgn( )⋅ is a signal function. Applying

the control law into 1( )V t , which leads to

( )1 , ,

1 , , 1 , ,

ˆ( ) ( ) ( ) ( ) sgn( )

ˆ[ ] [ ] sgn( )

T Ts s r s s r s

T T T T T Ts r n s r H s r n s r C G

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

s q s q P s q s q P s P k s P P

= + + − ⋅ + Γ

= + + − ⋅ + Γ(20)

where

, ,1 ,2 , , ,1 ,2 ,,T Ts s s s s s

s r r r r n s r r r r nq q q q q q q q⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Therefore,

1 1 , , 1 , , 1ˆ( ) sgn( )T T T T T

s r n s r s r n s r nV t s q s q s q s q s s P k s P P⎡ ⎤= − ⋅ + Γ⎢ ⎥⎣ ⎦ (21)

Taking the parameter adaptation law

11 , , 1 , , 1

ˆT 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⎢ ⎥⎣ ⎦ (22)

Then

1( ) sgn( ) 0V t k s=− ⋅ ≤ (23)

Hence, the state sq converges to ,s dq and meanwhile the estimate parameter P̂

converges to the actual parameter P .

3.2 Additional Parameter Adaptation

In practical application, when the modeled system has large dynamic reduction, it is of great importance to have quick convergence speed of parameter. Here we add additional adaptation law into basic control scheme. Considering the normal usage of system identification, we rewrite the dynamics of the modeled system as

( ) ( ) ( ) ( )s s s s st H t q C t q G tτ = + + (24)

In practice, sq is hard to measure. To avoid the joint acceleration in equation (24),

we use filtering technique. Specifically, multiply both sides of equation (24) with ( )t re λ− − where λ and r are positive number. By integrating equation (24), we get

Adaptive Control Scheme with Parameter Adaptation 563

( ) ( )

0 0

0 0 0 0 1 0 0

0 0 0 0 0 1 0( )

0 0 0 0 0 0 1

T Ts s

t t HT Tt r t r s s

s C

GT Ts s

q qP

q qe r dr e dr P

Pq q

λ λτ− − − −

⎡ ⎤⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

∫ ∫ i

(25)

where ,1 ,2 , ,1 ,2 ,,T T

s s s s n s s s s nq q q q q q q q⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ .

By using partial integration, the term consisting of Tsq on the right side can be

rewritten as

( )

0

( ) ( )

0

0 0

0 0

0 0

0 0 0 0

0 0 0 0

0 0 0 0

Ts

t Tt r s

Ts

tT T

s s

T Tt r t rs s

T Ts s

q

qe dr

q

q q

q qde e

dr

q q

λ

λ λ

− −

− − − −

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

⎛ ⎞⎡ ⎤ ⎡ ⎤⎟⎜⎢ ⎥ ⎢ ⎥⎟⎜ ⎟⎜⎢ ⎥ ⎢ ⎥⎟⎜ ⎟⎜⎢ ⎥ ⎢ ⎥⎟⎜ ⎟= −⎢ ⎥ ⎢ ⎥⎟⎜ ⎟⎜⎢ ⎥ ⎢ ⎥⎟⎜ ⎟⎜⎢ ⎥ ⎢ ⎥⎟⎜⎜⎢ ⎥ ⎢ ⎥⎜⎝ ⎠⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∫

0

t

dr

⎟⎟⎟

∫

(26)

Therefore, equation (25) can be written in the form

( ) ( , , )s sT t W t q q P= (27)

In fact, from this view, we consider ( )T t as “output” of the system; ( , , )s sW t q q as

signal matrix; P as real parameters, respectively. We can predict the value of the output ( )T t based on the parameter estimate. The prediction model is

ˆ ˆT W P= ⋅ (28)

Then the prediction error e is defined as

ˆ ˆ( ) ( ) ( )e t T t T t W P W P W P= − = ⋅ − ⋅ = ⋅ (29)

The basic idea to update the unknown parameters is that the parameters should be updated so that the prediction error is reduced.

( )'ˆ ˆ( ) ( )( )ˆ

ˆ ˆ

T Wp Wp Wp Wpe eP

P P

∂ − ⋅ −∂=−Ξ =−Ξ

∂ ∂ (30)

where Ξ is a diagonal matrix gain with positive number. Hence,

( )ˆ ˆ ˆ2 ( ) 2 2 ( ) ( )T T TP W WP WP W e W T t T t=− Ξ − =− Ξ =− Ξ − (31)

564 H. Dong and Z. Luo

If we consider the parameter change much slower with respect to the parameter identification, we have

ˆ 2 TP P P W WP= − =− Ξ (32)

Choose a Lyapunov candidate

21

( )4

TV t P P= (33)

then the derivative of 2 ( )V t is

( ) ( ) ( )21 1

( ) 2 02 2

TT T TV t P P P W WP WP WP= = − Ξ =−Ξ ≤ (34)

which means the estimate error of parameters converge to zero. In all, considering the Lyapunov function candidate in equation (12), we choose the Lyapunov candidate as

( )1 21 1

( ) ( ) ( ) 22 4

T TsV t V t V t s H s P I P= + = + Γ+ (35)

Thus, in all, the control law and parameter adaptation law are chosen as

( ), ,

1 , , 1 , , 1

ˆ ˆˆ ( ) ( ) ( ) sgn( )

ˆ ˆ2 ( ) ( )

s s s r s s r s

T T T T T Ts r n s r s r n s r n

H t q C t q G t k s

P s q s q s q s q s s W T t T t

τ = + + − ⋅

⎡ ⎤=− Γ− Ξ −⎢ ⎥⎣ ⎦

(36)

From equation (23) and equation (34), it is easy to prove ( ) 0V t ≤ , which indicates

the tracking error as well as parameter estimate error converge to zero. In the total control scheme, we use two kinds of errors to adjust the estimate parameters. One is tracking error s and the other is prediction error e , both of which contain the parameter information. Such an adaptation scheme leads to fast parameter convergence and finally smaller tracking error.

4 Humanoid Robot Control Application

We apply the proposed control scheme into postural control of a humanoid robot (Fig. 1). The robot is composed of torso, upper legs, lower legs and feet. For this

Fig. 1. Humanoid robot model

1θ

2θ

3θ

4θ

5θ

1d

2d

3d

4d

5d1l

2l

3l/ 2w

x

y

z

Adaptive Control Scheme with Parameter Adaptation 565

humanoid robot model, all the body parts are modeled as cylinder and the material is aluminum. As foot is considered as a cube whose thickness is infinite small, the mass of the foot is set as zero. The parameter settings of the robot are shown in Table 1. It is noted that as the material is aluminum, the mass and moment of inertia are small.

Table 1. Parameter of the humanoid robot

link mass mi

(kg) moment of inertia Ii (kg m)

length li (m)

location of center of mass di (m)

Width of robot w (m)

1, 5 0.0211 1.787×10-5 0.1 0.05 2, 4 0.0211 1.787×10-5 0.1 0.05 3 0.0211 1.787×10-5 0.1 0.05 0.1

In this application, we use a software package AUTOLEV to model the humanoid

robot and output it in MATLAB code. Considering the unmodeled dynamics and consequence from modeling error, we picked up parts of the variable states

1 2 3 4 5[ ]Tsq θ θ θ θ θ= . The dynamic equation is as follows.

( ) ( ) ( ) ( )s s s s s sH t q C t q G t tτ+ + = (37)

Then we apply the control scheme into the dynamic equation (37). Specifically, the

initial state is (0) [0 0 0 0 0]Tsq = and the desired state values are

[ ]

[ ]

[ ]

,

,

,

sin( ) 1 1 1 1 1

cos( ) 1 1 1 1 1

sin( ) 1 1 1 1 1

Ts d

Ts d

Ts d

q t

q t

q t

=

=

=−

(38)

The parameters in the control scheme are shown in Table 2. To show the adaptivity of the proposed control scheme, we just initial the estimation of dynamics equation parameters at the beginning of simulation as

ˆ ˆˆ (0), (0), (0)s s sH H C C G G= = = (39)

As ( )sH t , ( )sC t , ( )sG t are time-variant, the ˆ ( )sH t , ˆ ( )sC t , ˆ ( )sG t adapt values

to the actual ones by the parameter adaptation law.

Table 2. Parameter values for the control scheme

Λ k Γ Ξ 1

1

1

1

1

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0.08

0.08

0.01

0.01

0.01

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0.01

0.01

0.01

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0.01

0.01

0.01

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

566 H. Dong and Z. Luo

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

time (s)

magnitude (rad)

qs,1

qs,2

qs,3

qs,4

qs,5

(a)

0 2 4 6 8 10 12 14 16 18 20-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

time (s)

torque (Nm)

τs,1

τs,2

τs,3

τs,4

τs,5

(b)

Hs

hat (0)

1 2 3 4 51

2

3

4

5

Hs

hat (7)

1 2 3 4 51

2

3

4

5

Hs

hat (14)

1 2 3 4 51

2

3

4

5

Hs

hat (20)

1 2 3 4 51

2

3

4

5

Cs

hat (0)

1 2 3 4 51

2

3

4

5

Cs

hat (7)

1 2 3 4 51

2

3

4

5

Cs

hat (14)

1 2 3 4 51

2

3

4

5

Cs

hat (20)

1 2 3 4 51

2

3

4

5

-8

-6

-4

-2

0

2

4x 10

-3

(c)

Fig. 2. Simulation results. (a) sq tracks the desired sine signal. (b) Torques during the control process. (c) Snapshots of ˆ ( )sH t and ˆ ( )sC t in initial time (0s), 7s, 14s, 20s.

Adaptive Control Scheme with Parameter Adaptation 567

The simulation results are shown in Fig. 2. It is shown that the sq converges to

the desired states with time. Considering ˆ ( )sH t , ˆ ( )sC t , ˆ ( )sG t are completely

updated by the parameter adaptation law (equation (36)), the tracking performance is satisfactory (Fig. 2 (a)). There is no torque with extremely large value, which indicates that proposed control scheme has advantage in energy expenditure (Fig. 2

(b)). The snapshots of ˆ ( )sH t , ˆ ( )sC t , (shown in the form of contour plot) during the

whole dynamics are shown in Fig. 2 (c). One obvious fact is that ˆ ( )sH t , ˆ ( )sC t ,

change all the time verifying they are time-varying. Another important phenomena is

that the patterns of ˆ ( )sH t and ˆ ( )sC t do not coincide with the initial values. The

explanation is that these estimated values can also have the same state output although they are not equal to the real values. When the dynamics gets more variety, the estimated values converge to the real ones.

5 Conclusion

This paper considered the human motor control and humanoid robot locomotion control together as one topic. Human motor control has to deal with the redundancy of the human movement system while humanoid robot locomotion control is influenced by unmodeled dynamics and modeling error. The two issue can be seen as one together, i.e., actively or passively selection of state variables. After proving the simplified dynamics also corresponds with a physical system, we designed a control scheme. One feature of the proposed scheme is adaptivity, which is verified by the simulation. For a time-varying system (i.e. dynamic process of humanoid robot), the system can track the desired signal very well under the condition that the time-varying parameters are given only in the initial moment. The contribution of this paper is giving an explanation on dynamic reduction in modeling process by a mathematical proof and further more, designing an adaptive control scheme for the model with reduced dynamics.

Acknowledgement. This work was supported in part by Japan Society for the Promotion of Science.

References

1. Ivancevic, V.G., Ivancevic, T.T.: Natural Biodynamics. World Scientific Publishing Co. Pte. Ltd. (2005)

2. Crago, P.E.: Creating Neuromusculoskeletal Models. In: Biomechanics and Neural Control of Posture and Movement, pp. 119–133. Springer, Heidelberg (2000)

3. Manal, K., Gonzalez, R.V., Lloyd, D.G., Buchanan, T.S.: A Real-time EMG-driven Virtual Arm. Computers in Biology and Medicine 32, 25–36 (2002)

4. Winter, D.A.: Biomechanics and Motor Control of Human Movement. John Wiley & Sons, Inc. (1990)

568 H. Dong and Z. Luo

5. Tzafestas, S., Raibert, M., Tzafestas, C.: Robust Sliding-mode Control Applied to a 5-Link Biped Robot. Journal of Intelligent and Robotic System 15, 67–133 (1996)

6. Kogakkai, S., Kumamoto, M.: Revolution in Humanoid Robotics: Evolution of Motion Control. Tokyo Denki University Publisher (2006)

7. Dong, H., Luo, Z., Nagano, A.: Adaptive Attitude Control of Redundant Time-Varying Complex Model of Human Body in the Nursing Activity. Journal of Robotics and Mechatronics 22, 418–429 (2010)

8. Slotine, J.J.E., Li, W.: Applied Nonlinear Control. Prentice Hall (1991)