1. 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.

2. 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 T
pass rob
d
q G q Hq q Hq q Hq
dt
τ 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 T
pass rob pass robq G q G Cq q Hqτ τ τ τ+ − = + − − + (4)

3. 560 H. Dong and Z. Luo
After simplification, the result is
( 2 ) 0T
q 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
2
2
ij ij
ji ji
if i j
H 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 s
i i i i i im i i i i i imi i i
s s s s s s s s s
im 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
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ +⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦
1
s
i
s
im
τ
τ
⎡ ⎤
⎢ ⎥
⎢ ⎥= ⎢ ⎥
⎢ ⎥
⎥ ⎢ ⎥⎣ ⎦
(9)
According to the relation of equation (7), the new system satisfies
( ), ,
, ,
0
2
2
iu iv iu iv
iv iu iv iu
if iu iv
H 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 [ ]
T
H C GP P P P= , where

4. Adaptive Control Scheme with Parameter Adaptation 561
11 12 1 1 2
11 12 1 1 2 1 2,
Ts s s s s s
H n n n nn
T Ts s s s s s s s s
C 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 s
H n n n nn
T Ts s s s s s s s s
C 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
1
1 1
( )
2 2
T T
sV 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 T
s s s s s s r s
T T
s 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 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
⎛ ⎞⎟⎜ = − + − − +⎟⎜ ⎟⎜⎝ ⎠
= − − − + −
(16)
According to equation (11), 2s sH 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
⎛ ⎞⎟⎜ = − − −⎟⎜ ⎟⎜⎝ ⎠
(17)

5. 562 H. Dong and Z. Luo
Therefore, 1( )V t can be simplified as
' '
1 , ,
1 1 ˆ( ) ( )
2 2
T T T T
s 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 T
s s r s s r s
T T T T T T
s 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 , , 1
ˆ( ) 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
1
1 , , 1 , , 1
ˆT 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⎢ ⎥⎣ ⎦
(22)
Then
1( ) sgn( ) 0V t k s= − ⋅ ≤ (23)
Hence, the state sq converges to ,s dq and meanwhile the estimate parameter ˆP
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 r
e λ− −
where λ and r are positive number. By integrating equation (24), we get

6. 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 T
s s
t t HT T
t r t r s s
s C
G
T T
s s
q q
P
q q
e r dr e dr P
P
q q
λ λ
τ− − − −
⎡ ⎤
⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦⎢ ⎥
⎢ ⎥⎣ ⎦
∫ ∫ 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 T
sq 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
T
s
t T
t r s
T
s
t
T T
s s
T T
t r t rs s
T T
s s
q
q
e dr
q
q q
q qd
e 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 e
P
P P
∂ − ⋅ −∂
= −Ξ = −Ξ
∂ ∂
(30)
where Ξ is a diagonal matrix gain with positive number. Hence,
( )ˆ ˆ ˆ2 ( ) 2 2 ( ) ( )T T T
P W WP WP W e W T t T t= − Ξ − = − Ξ = − Ξ − (31)

7. 564 H. Dong and Z. Luo
If we consider the parameter change much slower with respect to the parameter
identification, we have
ˆ 2 T
P P P W WP= − = − Ξ (32)
Choose a Lyapunov candidate
2
1
( )
4
T
V t P P= (33)
then the derivative of 2 ( )V t is
( ) ( ) ( )2
1 1
( ) 2 0
2 2
TT T T
V 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 2
1 1
( ) ( ) ( ) 2
2 4
T T
sV 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 T
s 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

8. 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[ ]T
sq 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]T
sq = and the desired state values are
[ ]
[ ]
[ ]
,
,
,
sin( ) 1 1 1 1 1
cos( ) 1 1 1 1 1
sin( ) 1 1 1 1 1
T
s d
T
s d
T
s 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
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦

9. 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)
to
rq
u
e
(N
m
)
τs,1
τs,2
τs,3
τs,4
τs,5
(b)
Hs
hat
(0)
1 2 3 4 5
1
2
3
4
5
Hs
hat
(7)
1 2 3 4 5
1
2
3
4
5
Hs
hat
(14)
1 2 3 4 5
1
2
3
4
5
Hs
hat
(20)
1 2 3 4 5
1
2
3
4
5
Cs
hat
(0)
1 2 3 4 5
1
2
3
4
5
Cs
hat
(7)
1 2 3 4 5
1
2
3
4
5
Cs
hat
(14)
1 2 3 4 5
1
2
3
4
5
Cs
hat
(20)
1 2 3 4 5
1
2
3
4
5
-8
-6
-4
-2
0
2
4
x 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.

10. 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.
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