International Journal of Fuzzy Systems, Vol. 15, No. 4, December 2013
423
Type-2 Fuzzy Logic Controller Using SRUKF-Based State Estimations for
Biped Walking Robots
Liyang Wang, Zhi Liu, Yun Zhang, C. L. Philip Chen, and Xin Chen
Abstract1
Walking motions of biped robots have features of
humanoid and intelligent. Generally, when we human
are walking, we may control the walking motions using qualitative instructions, such as linguistic “slow”
or “fast,” “forward” or “backward,” “big step” or
“small step”. This motivates us to design a fuzzy controller for biped robots. Furthermore, to improve the
robustness of the biped walking system, a framework
of biped gait control based on a predictable type-2
fuzzy logic controller (T2FLC) is proposed to ensure
the dynamic balance of the biped under the condition
of complex process noises and measurement noises.
Different with existing controllers for biped robots, a
state estimator based on square root unscented Kalman filter (SRUKF) is incorporated in the proposed
control strategy. Using the estimated biped states as
inputs, the proposed T2FLC can predictably adjust
the posture of the trunk timely and properly to ensure the dynamic balance of the whole legged system.
Simulation results are presented to verify the effectiveness of the proposed method.
Keywords: Biped robot, square root unscented Kalman
filter, state estimation, type-2 fuzzy logic system.
1. Introduction
Biped robots always suffer from complex process
noises and measurement noises. As a result, to ensure the
dynamic balance of biped robots under the noises beCorresponding Author: Liyang Wang is with the Department of
Automation, Guangdong University of Technology, Guangzhou
Guangdong 510006, China. Liyang Wang is also with the Department
of Electronic Engineering, Shunde Polytechnic, Foshan Guangdong
528300, China.
E-mail: [email protected]
Zhi Liu and Yun Zhang are with the Department of Automation,
Guangdong University of Technology, Guangzhou Guangdong
510006, China.
C. L. Philip Chen is with the Faculty of Science and Technology,
University of Macau, Macau, China.
Xin Chen is with the Department of Mechatronics Engineering,
Guangdong University of Technology, Guangzhou Guangdong
510006, China.
Manuscript received 27 June 2012; revised 9 Nov. 2012; accepted 19
April 2013.
comes an important and interesting problem.
Walking motions of biped robots have features of
humanoid and intelligent. Generally, when we human
are walking, we do not compute the accurate value of
our step length, the angle of our trunk rotation or the
speed of our joints. However, we may control the walking motions using a qualitative instruction, such as linguistic “slow” or “fast,” “forward” or “backward,” “big
step” or “small step”. This is just what the fuzzy controllers [1-7] are good at. Fuzzy logic was first proposed
by Zadeh, and it is has been successfully applied to
many fields like uncertainty modeling and control. As an
extended version of the standard type-1 fuzzy system
(T1FS), type-2 fuzzy system (T2FS) [8-9] is proposed by
Mendel and his colleagues. The membership value of a
type-2 fuzzy set in T2FSs is a type-1 fuzzy number. That
is, type-2 fuzzy sets allow researchers to model and
minimize the effects of uncertainties in rule-based systems. Due to the advantages mentioned above, type-2
fuzzy sets and fuzzy logic systems have been used in
many different fields, such as system modeling and control [10-12], motion control and robotics [13-16], wireless sensor network [17], computer network [18,19] and
data processing [20-23]. Furthermore, to cope with a
specific sort of stochastic uncertainty, Liu and Li have
presented a probabilistic fuzzy logic system (PFLS) [24]
to handle the fuzziness and randomness together, and
further integrating with the neural computation to enhance its performance in stochastic and time-varying
environment [25]. The T2FS employs two fuzzy membership functions (a primary membership function and a
secondary membership function) to improve the ability
of handling the uncertainty of linguistic expression, and
it has been proved that the T2FSs is a more promising
method than the T1FSs for handling uncertainties such
as noisy data [26-27]. For biped robots, in general, both
the biped model and the measurement data in the control
systems could be drawn into various noises and uncertainties. To deal with these complex noises and uncertainties more effectively, the type-2 fuzzy controller is a
better choice instead of the type-1 fuzzy controller.
However, the existing fuzzy controllers pay little attention to the state estimation of the controlled object,
which could lead to degradations of the controller. To
enhance the stability and robustness of dynamic biped
walking, the biped states should be estimated precisely
© 2013 TFSA
International Journal of Fuzzy Systems, Vol. 15, No. 4, December 2013
424
and timely.
Nonlinear filters, such as extended Kalman filter
(EKF) [28], particle filter (PF) [29], unscented Kalman
filter (UKF) [30] and SRUKF [31-32], are known as
powerful tools for nonlinear state estimations. SRUKF is
an improved version of the UKF. The most computationally expensive operation in the UKF is to calculate
the new set of sigma points at each time update. This
requires the computation of a matrix square-root of the
state covariance matrix, and an integral part of the UKF
requires computation of full covariance matrix be recursively updated. In the SRUKF implementation,
square-root of the state covariance matrix will be propagated directly, avoiding the requirement of re-factorizing
at each time step. Therefore, the SRUKF provides the
same results as the UKF to within machine accuracy and
it can be reposed with less complexity for nonlinear state
estimations. From the above, the SRUKF is a kind of
potential estimator for the biped states.
In this work, a SRUKF-based predictable T2FLC is
proposed to ensure the dynamic balance of biped robots
under complex process noises and measurement noises.
First, a method of SRUKF-based biped state estimation
is proposed, which provides an optimal estimation of the
biped states. Secondly, a T2FLC is built, which realizes
a dynamical compensation for the biped gait using trunk
rotations. Two following modules are implemented to
realize the proposed SRUKF-based predictable T2FLC.
To move the trunk timely and properly, the state of the
biped robot for the coming sampling time of the system
is predicted by using the SRUKF-based biped state estimator with low computational and the high preciseness.
Furthermore, the T2FLC is proposed to compensate the
biped gait according to the dynamic kinematics relationships between the legs and the trunk. The essential of
the T2FLC is moving the ZMP far away from the edge
of the support area to ensure the dynamic balance of the
biped walking robots.
The organization of this paper is as follows. In Section
II, criteria for the stability of biped motions are presented, and the proposed SRUKF-based biped state estimation is presented in Section III. The T2FLC with
estimated biped states is proposed in Section IV. In Section V, simulations are conducted and the conclusions
are presented in Section VI.
2. Background of Biped Robots
A. Criteria for the stability of biped motions
The general definition of motion stability was first
proposed by Lyapunov, which mainly describes the stability of equilibrium points [33-34]. However, biped robots have inherent characteristics of unilateral constraint,
hybrid and variable topology of the drive mode. As a
result, it is difficult to analyze the stability of biped robots using Lyapunov stability theory. Some researches
[35-36] proposed that stable gaits of biped robots show
themselves in the form of limit cycles in the phase space,
and limit cycles are fixed points of Poincaré mappings.
So the study on stable gaits of biped robots can be simplified as the study on the stability of the fixed points of
Poincaré mappings. However, it is difficult to get the
analytic solution of the Poincaré mappings because of
the complexity of the biped dynamic. The fixed points of
Poincaré mappings can only be found using numerical
method, which determines that this kind of method can
only be applied to passive robots, biped robots without
feet and jumping robots based on simple models.
On the other hand, the concept of ZMP has been applied to many famous biped robots successfully, such as
ASIMO of Honda [37]. ZMP [38] is a point on the
ground at which the net moment of the inertial forces
and the gravity forces has no component along the horizontal axes. At a given time instant, dynamic balance of
legged systems is ensured if the ZMP is inside the support area. ZMP is the most popular criterion for the stability of the biped locomotion up to now. Therefore,
ZMP theory is used as the criterion of dynamic biped
balance in this work.
B. Ensuring the dynamic balance of the biped using
ZMP theory
According to the ZMP theory, the dynamic balance of
the biped can be ensured if the ZMP is inside the support
area, as shown in Figure 1. In addition, the minimum
distance between the ZMP and the boundary of the support area is called ZMP stability margin. To avoid the
falling down of the biped, the biped motions should
guarantee the ZMP criterion, and the ZMP stability margin should be positive.
Furthermore, the ZMP can be computed using the
following equations [39]:
xzmp 

m
m j ( 
z j  g ) x j   j 1 m j 
xj zj
m
j 1
 j 1 m j (z j  g )
m
 I 
 m (z  g )
m

j 1
jy
jy
m
j 1
j
j
(1)
 j 1 m j (z j  g ) y j   j 1 m j y j z j
m
y zmp 
m

m
j 1
m j ( 
z j  g)
 j 1 I jx  jx
m


m
j 1
m j ( 
z j  g)
(2)
where ( xzmp , yzmp , 0) is the coordinate of the ZMP, and
( x j , y j , z j ) is the coordinate of the mass center of link j
on an absolute Cartesian coordinate system. mj is the
mass of link j , I jx and I jy are the inertial components,
 jx and  jy are the absolute angular acceleration
components around x axis and y axis at the center of
L. Wang et al.: Type-2 Fuzzy Logic Controller Using SRUKF-Based State Estimations for Biped Walking Robots
gravity of link j , g is the gravitational acceleration.
ZMP stability margins can be calculated as follows:
(3)
d xzmp  min[ x forward  xzmp , xzmp  xbackward ]
d yzmp  min[ yleft  y zmp , y zmp  yright ]
walking
direction
supporting leg

ZMP
support area

ZMP
swing leg
double support phase
where
1
0



  0

0  M 1  C      M 1[  G ]




T
2n


is the
  1 ,, n ,1 ,, n   R
(8)
joint variable
vector, and the definitions of the other symbols are the
same with Eq. (5). And Eq. (8) can be rewritten as
   ()  B() (t )
(9)

where
1
0

 0 
     M 1  G 
1

M

C
0




 ( )  
 0 
B (  )   1 
M 
supporting leg
swing
leg
support area
Submitting   [ x1 , x2 ]T and   [ x1 , x2 ]T to Eq. (7), we
get
(4)
where d xzmp and d yzmp are the ZMP stability margins.
xzmp and yzmp are the positions of the ZMP, which can
be obtained using Eq.(1) and Eq.(2). x forward and xbackward
are the positions of the forward edge and backward edge
of the support area, yleft and yright are the positions of
the left edge and the right edge of the support area.
425
single support phase
Figure1. Support areas for biped robots.
3. Estimating Biped States Using a SRUKF
To estimate biped states using a SRUKF, a discrete-time model for biped state estimation (including
the state equation and the measure equation) is developed first. After that, a SRUKF filter algorithm for the
biped state estimation is deduced in detail.
A. System modeling for biped state estimation
To design a SRUKF for biped robots, the dynamic of
the robot should be described using state equation firstly.
A discrete-time model for predicting joint states of biped
robots is built based on Lagrange dynamics. The dynamic of biped robots in single support phase can be
written as
(5)
M ( )  C ( , )  G ( )  
T
n
where   [1 ,,n ]  R is the joint variable vector,
M (q )  R nn is the inertia matrix, and C (q, q )  R nn
represents Coriolis and centripetal forces, G(q)  R n is
the gravity term.  is the control input torque.
Let x1   and x2   , the dynamic of biped robots
can be written as follows:
 x1  x2

1
1
 x2  M ( x1 )   M ( x1 )[C ( x1 , x2 ) x2  G ( x1 )]
(6)
i.e.,
1
0
 x1  0
  x1  

 x   0  M 1 ( x )  C ( x , x )   x    M 1 ( x )  [  G ( x )]
 2 

1
1
2  2
1
1 
(7)
(10)
(11)
A zero-holder between the controller and the plant is
considered, and the process noise is considered. Referring to [40], the sampled-data representation of Eq. (9)
can be expressed as
(k  1)  A(k )  G  (k )  (k )  W  (k ), (k )  +w(k ) (12)
where (k  1) and (k ) are the biped states at time
k  1 and time k respectively.  (k ) is the biped control
torque at time k . A , G  (k )  and W  (k ), ( k )  can
be referenced in [40] for detail. w(k ) is the process
noise. And Eq. (12) can be rewritten as
(13)
(k  1)  f ((k ), (k ))  w(k )
where
f ((k ), (k ))  A(k )  G  ( k )  ( k )  W  (k ), (k )  (14)
Eq. (13) is the needed discrete time state equation of biped robots.
Taking the joint angles and the angle velocities as the
variables to be estimated and measured, the measurement equation of biped robots is defined as follows:
(15)
y (k )  (k )  v(k )
where y(k ) is the measurement matrix of the joint angle
and the angle velocity. v(k ) is the measurement noise.
is the joint
(k )  [1 ( k ), , n (k ), 1 (k ), , n ( k )]T  R 2 n
variable vector, and the joint angle n (k ) and the angle
velocity n (k ) are physical quantities at time k .
A discrete-time model for biped state estimation is
developed as follows:
(k  1)  f ((k ), (k ))  w(k )

 y ( k )  ( k )  v ( k )
(k )  [1 ( k ), , n (k ), 1 (k ), , n ( k )]T  R 2 n
(16)
where
is the
state vector of the SRUKF-based models for biped state
estimations.  n (k ) is the joint angle and n (k ) is the
angle velocity (They are physical quantities at time k ).
w(k ) and v(k ) are the process and measurement noise
respectively.  (k ) is the control input torque. f () is
defined in the Eq.(14). y(k ) is the measurement matrix
International Journal of Fuzzy Systems, Vol. 15, No. 4, December 2013
426
of the joint angle and the angle velocity. Without loss of
generality, w(k ) and v(k ) are both white, zero-mean,
uncorrelated, and have known covariance matrices Q(k )
and R(k ) , respectively.
B. SRUKF algorithm for biped state estimation
Here we denote the joint angle estimate of (k  1) as

ˆ ( k  1) when all of the measurements before (but not

including) time k  1 are available for our joint angle
estimate. The “-” superscript denotes that the estimate is
a priori.
Step 1: Initialization
The estimation of the initial state (0) is denoted as
ˆ (0) , E () is the expectation. Then we have

ˆ (0)  E[(0)]
(17)

ˆ (0))((0)  
ˆ (0))T ]
S  (0)  E[((0)  
Step 2: Calculate Sigma Points
The scaled 2L  1 sigma points set {i (0); i  0, , 2 L} ,
where the 0 inside the parentheses denotes time step
k  0 , is generated by
ˆ (0) , i  0
(18)
i (0)  
ˆ
i (0)  (0)  L   Si (0) , i  1,, L
ˆ (0)  L   S (0) , i  L  1, , 2 L
i (0)  
i
m
c
Let i and i denote the weights associated with the
i th sigma point. These weights can be calculated by
0m   /( L   )
0c   /( L   )  (1   2   )
im  ic  1/(2 L  2 ) , i  1, , 2 L
where    2 ( L   )  L ,  controls dispersing extent of
sampling points in a small positive value range 103 ~1.
 is a secondary scaling factor,   2 is optimal under
the condition of the Gaussian noise.  is a tertiary
scaling factor and is usually set equal to 0. The superscripts m and c denote mean and covariance, respectively.
Step 3: Time update
ˆ  (k  1)   2 L  m  (k  1 k )

i
i 0 i


ˆ  (k  1))]
S  (k  1)  qr [ 1c (1:2 L (k  1 k )  
S ( k  1)  cholupdate S


(k  1),  0 (k  1 k ), 0c 
yˆ  ( k  1)   i  0 im yi ( k  1 k )
2L
Step 4: Measurement update
S y  ( k  1)  qr{[ 1c ( y1:2 L ( k  1 k )  yˆ  ( k  1))]}
S y ( k  1)  cholupdate S y  ( k  1), y0 ( k  1 k ), 0c 
P y (k  1) 

2L
i 0
ˆ  (k  1)][ y (k  1 k )  yˆ  (k  1)]
ic [i (k  1 k )  
i
K (k  1)  P y (k  1) [ S y  (k  1)]T S y  (k  1)
ˆ ( k  1)  
ˆ  ( k  1)  K ( k  1)[ y ( k  1)  yˆ  ( k  1)]

U  K (k  1) S y  (k  1)
S  ( k  1)  cholupdate S  ( k  1), U , 0c 
where qr () is a QR decomposition (also called a QR
factorization) of a matrix, which is a decomposition of a
matrix into a product QR of an orthogonal matrix Q
and an upper triangular matrix R . cholupdate 
 means
Cholesky first-order update. For example, if R  chol ( A)
is the original Cholesky factorization of A , then
cholupdate{R, X ,  } returns the Cholesky factor of
A   XX T .
Step 5: The estimated centroid positions of the supporting hip and the swing ankle
By submitting the estimated states into the calculating
equations of the joint positions, we can get the estimated
centroid positions of the supporting hip and the swing
ankle respectively.
Up to now, the optimal estimations of the biped states
are obtained using the five steps mentioned above, and
the estimated states will be used as the inputs of the
T2FLC discussed in the next section.
4. T2FLC Using Estimated Biped States
Trunk rotation is a usual way for effective dynamic
human walking [41-43]. When the bipeds try to maintain
the dynamic balance by trunk rotations, the controller
should make the decision accurately and timely, even
there exists unavoidable noise from the biped system and
the sensors.
The T2FLC with estimated biped states is shown in
Figure 2. To make the decision accurately and timely,
the inputs of the proposed T2FLC are preprocessed by a
SRUKF. Using the estimated ankle trajectory and hip
trajectory as inputs, and the corresponding trunk trajectory as outputs, the essential of the T2FLC is moving the
ZMP far away from the boundary of the support area, for
ensuring the dynamic balance of the biped walking robots.
This section designs trunk trajectory corresponding to
the estimated biped states using a T2FLC. The antecedent part of the T2FLC uses interval type-2 fuzzy sets,
and the consequent part is of the Mamdani-type. The
i th rule in the system has the following form:
L. Wang et al.: Type-2 Fuzzy Logic Controller Using SRUKF-Based State Estimations for Biped Walking Robots
427
Outer-Loop, Type-2 fuzzy controller
with SRUKF-based biped state
estimation
 (k ),(k )
 (k )
Internal-Loop, Controller of Joint Motions
SRUKF-based
biped state estimation
Gait Planning
Gait Planning
yˆ hip (k  1)
yˆ ankle (k  1)
Type-2 fuzzy controller
－


e

＋
－
 e

＋ 
ytrunk (k  1)
Inverse
Kinematics
( , ,)
Articulation
Controller

Biped
Robot
Figure 2. T2FLC using estimated biped states as inputs.
Rule i : IF yˆ hip is Ai ,1 AND yˆ ankle is Ai ,2
(19)
THEN ytrunk is G i , i  1, , M
where yˆ hip and yˆ ankle denote the estimated centroid positions of the supporting hip and the swing ankle respectively, ytrunk is the centroid positions of the trunk. Ai ,1
and Ai ,2 are interval type-2 fuzzy sets, G i is the output
interval type-2 fuzzy set of the i th rule, and M is the
number of rules. Based on the rule base in (19), the
computation of the T2FLC involves the fuzzifier, inference engine, type reducer, and defuzzifier, which will be
described in detail next.
A. Fuzzification
The fuzzifier maps crisp input values to fuzzy sets.
For the i th fuzzy sets Ai ,1 and Ai ,2 in the input variables, a Gaussian primary membership function is used,
which has a fixed standard deviation and an uncertain
mean that takes on values in [m, m] .
For example, the membership degree of the centroid
positions of the supporting hip is
h 2

 1  yˆ hip  mi  
 
i ,1
 2   i
 
h
h
h
h
 N (mi ,  i ; yˆ hip ), mi  [mi , mih ]

h
h
 N (mi ,  i ; yˆ hip ),
 A ( yˆ hip )  
i ,1
 N (m h ,  h ; yˆ ),
i
i
hip

mih  mih
2
mih  mih

2
yˆ hip 
yˆ hip
Similarly, the membership degree of the centroid positions of the supporting ankle can be expressed as
(23)
 A ( yˆ ankle )  N (mia ,  ia ; yˆ ankle ), mia  [mia , mia ]
i ,2
And the membership degree of the centroid positions
of the trunk can be expressed as
(24)
G ( ytrunk )  N (mitr ,  itr ; ytrunk ), mitr  [mitr , mitr ]
i
B. Inference
The inference engine operation performs the fuzzy
meet operation by using an algebraic product operation.
The result of the input and antecedent operations Fi is
an interval type-1 set, i.e., Fi  [ f i , f i ] , where
fi   A  A
i ,1
i ,2
, fi   A  A
i ,1
(25)
i ,2
The i th rule fired output consequent set
G ( ytrunk )  
i
i
ytrunk
[ fi G ( ytrunk ), fi G ( ytrunk )]
i
i
1/ ytrunk
i
(20)
i
membership grades of G ( ytrunk ) . The output fuzzy set
i
G ( ytrunk ) is
Here, the membership degree  A ( yˆ hip ) is an interval set
G ( ytrunk ) 
and is denoted by  A ( yˆ hip )  [  A ( yˆ hip ),  A ( yˆ hip )] . The

mathematical functions of the lower and upper MFs,
 A ( yˆ hip ) and  A ( yˆ hip ) , are described as follows:
where ‘  ’ operation is the maximum operation.
i ,1
i ,1
i ,1
(26)
i
where G ( ytrunk ) and G ( ytrunk ) are the lower and upper
 A ( yˆ hip )  exp  

i ,1
(22)
i ,1
i ,1
 N (mih ,  ih ; yˆ hip ) , yˆ hip  mih

 A ( yˆ hip )  1
, mih  yˆ hip  mih
i ,1

h
h
h
 N (mi ,  i ; yˆ hip ) , yˆ hip  mi
(21)
ytrunk [ f1 G ( ytrunk )] [ f M G ( ytrunk )], [ f1 G ( ytrunk )  f M G ( ytrunk )]


1
M
1
M
1/ ytrunk
(27)
C. Type reduction
For T2FLCs, the final crisp output is the center of the
type-reduced set. There exist many kinds of
type-reduction, such as centroid, center-of-sets, height,
and modified height [44]. In this paper, center-of-sets
type-reduction is used as follows:
International Journal of Fuzzy Systems, Vol. 15, No. 4, December 2013
428
 fy
 f
M
l
r
[ ytrunk
, ytrunk
]


y1trunk

M
ytrunk
   1/
f1
fM
i 1
i
i
trunk
M
i 1
(28)
i
where l and r represent the left and right limits, rei
spectively. ytrunk
is the centroid of the type-2 interval

consequent set Gi .
l
r
and ytrunk
can be computed as folThe outputs ytrunk
lows:
l
trunk
y


r
ytrunk


L
i
i
(Qf )i ytrunk
  i  L 1 (Qf )i ytrunk
M
i 1

R
L
i 1
(Qf )i   i  L 1 (Qf )i
M
i
i
(Qf )i y trunk
  i  R 1 (Qf )i y trunk
(29)
M
i 1

R
i 1
(Qf )i   i  R 1 (Qf )i
M
(30)
where L and R denote the left and right crossover
,
and
points,
respectively.
ytrunk  Qytrunk
1
M
ytrunk  ( ytrunk ,, ytrunk ) denotes the original rule-ordered
1
2
M
. Q is an
consequent values and ytrunk
 ytrunk
   ytrunk
M  M permutation matrix. These two points vary with
different inputs. Karnik-Mendel iterative procedure can
be used here to find these two points [45].
1）Simulation conditions
Firstly, the biped system is as follows.
The details of the humanoid can be referenced in Table 1. Simulation model of the biped system is as follows:
(k  1)  f ((k ), (k ))  w(k )

 y ( k )  ( k )  v ( k )
(32)
where (k )  [1 (k ), , n (k ),1 (k ), ,n (k )]T  R14 is the state
vector, and the joint angle  n (k ) and the angle velocity
n ( k ) are physical quantities at time k . The process
noise w(k ) ~ (0,103 ) , and the measurement noise
v(k ) ~ (0,103 ) . f () is defined in the Eq. (14). y (k ) is the
measure matrix of the joint angle and the angle velocity.
Let k  1, , 41 , and the sampling interval T  0.025s .
The control input torque  (k )  { (1), (2), , (41)} is designed using the method of computer torque:
 (k )  Mˆ [ (k )][d (k )  K v e(k )  K p e(k )]
 Cˆ [ (k ), (k )]  Gˆ [ (k )], k  1, , 41
(33)
denotes the estimation parameter.

d
e(k )   (k )   (k ) , e(k )  (k )   d (k ) ,  d (k ) ,  d ( k ) and
d (k ) are the desired reference trajectories for the robot.
where
D. Defuzzification
The defuzzifier computes the system output variable
ytrunk by performing the defuzzification operation on the
l
r
interval set [ ytrunk
, ytrunk
] from the type reduction. Based
on the centroid defuzzification operation, the centroid of
l
r
l
and
the interval set [ ytrunk
, ytrunk
] is the average of ytrunk
r
ytrunk . Hence, the defuzzified output is
ytrunk 
l
r
ytrunk
 ytrunk
2
(31)
and Kv are the proportional and differential gains.
It is noted that discrete-time models are discussed in this
work. The variables such as e(k ) , (k ) ,  d (k ) and
d (k ) are all physical quantities. They are not continuous-time variables.
Secondly, initial state values of the biped system are
as follows.
Kp
Up to now, with type-2 fuzzy inputs of the centroid
positions for the supporting hip and the swing ankle, the
corresponding position for the trunk is deduced using a
T2FLC. The trunk trajectory provided by the T2FLC is
then used as the reference trajectory for control, which
ensures the dynamic balance of biped walking robots.
Thirdly, the three parameters of the SRUKF estimator
are set to be   1,   2 and   0 because the additive
noise is Gaussian [46].
5. Simulations Research
Table 1. Parameters of the biped system.
ˆ (0)  [0.0873, 0.1183, 0.7210, 0, 0.3550, 0.5989, 0, 0 ]

1 7
S  (0)  diag{ 10 3 , , 10 3 }1414
Inertia moment
A. Reference trajectories for the biped joints
Mass( Kg )
Length( m )
Link
（ Kg  m 2 ）
A typical gait is planned for the biped robot to walk
on the horizontal ground. The whole walking period of
Trunk
0.175
1.060
0.01623
the biped robot is considered to be composed of a single
Thigh
0.130
0.075(*2)
0.00063(*2)
support phase and an instantaneous double support phase.
The supporting foot is assumed to remain in full contact
Shank
0.130
0.075(*2)
0.00037(*2)
with the ground during the single support phase, and
0.020
0.060(*2)
0.00001(*2)
Foot
both the feet revolve around the point contacting with
the ground during the double support phase. The walk
cycle is Tc , and let Tc  1s .
2）Simulation results of the biped state estimations
Although the reference trajectories are planned as preB. Implementation of SRUKF-based biped state estima- viously mentioned, the biped could deviate from the
tion
planned gait because of the process noise and measure-
L. Wang et al.: Type-2 Fuzzy Logic Controller Using SRUKF-Based State Estimations for Biped Walking Robots
ment noise. To reduce the influence of the noises,
SRUKF-based biped state estimation is implemented to
provide optimal estimated inputs to the T2FLC.
In this work, Matlab7.0 is used to model the biped
system and the estimators. The estimated trajectories of
the joint angles are shown in Figure 3. As we can see,
the SRUKF-based estimated results are close to the true
trajectories of the joint angles. One other thing should be
noted is that the true trajectories here in the simulations
are determined in advance while true trajectories in the
practical engineering are unknown because of various
noises. That is why biped state estimations are needed.
429
of the particles is 200 here) have been shown to present
better performance than the others in term of MSE.
However, the execution time for the PF is much longer
than that of the others, which is disadvantageous for the
real-time control of the robot. On the other hand, the
execution time of the PF can be reduced if the number of
the particles is decreased, while this will leads to significant decreasing of the MSE. As a result, the performance of the SRUKF is the best among the four typical nonlinear filters if both the MSE and the execution
time are considered.
0.75
(a)
0.08
(b)
0.7
UKF
0.07
0.65
joint angle (rad)
joint angle (rad)
0.06
0.05
0.04
0.03
EKF
0.6
0.55
0
0.5
0.2
0.4
0.6
0.8
0.02
0.45
EKF
0.01
0.4
0
-0.01
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
time step
0.6
0.5
0.8
0.9
1
Execution Time
(s)
MSE
0.00055
0.094
UKF
0.59175 0.00040
0.053
0.059
SRUKF
0.00032
0.053
0.15
0.1
joint angle (rad)
joint angle (rad)
0.7
(d)
0.2
0.45
0.4
0.35
0.3
0
-0.05
-0.1
0.2
-0.15
0.15
-0.2
0.1
-0.25
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 4. Comparison of EKF/PF/UKF/SRUKF (mean of 100
runs).
0.05
0.25
0
0.1
0.2
0.3
0.4
time step
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
time step
0.9
0.8
(e)
0.8
(f)
0.7
0.6
joint angle (rad)
0.7
joint angle (rad)
0.6
0.25
(c)
0.55
0.6
0.5
0.4
0.3
0.5
0.4
0.3
0.2
0.2
0.1
0.5
time step
PF
0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0
0.1
0.2
0.3
time step
0.4
0.5
time step
Figure 3. The true trajectories and the SRUKF-estimated joint
angles: true (dotted black line with circles), SRUKF-estimated
(blue solid line) (a)supporting ankle (b)supporting knee (c)
supporting hip (d)swing hip (e) swing knee (f) swing ankle.
3）Comparison of different filters
In order to evaluate the performance of the estimator,
the mean of the squared error (MSE) of the SRUKF estimator is calculated using the following formula, and
the result is compared with those of the EKF, the PF, and
the UKF:
MSE 
1
Nr
Nr
N
 [(k )  ˆ (k )]
2
r 1 k 1
r
where k  1, , N ( N  41) is the index of the samples,
ˆ (k )
r  1, , N r ( N r  100) is the index of different runs. 
r
is computed using the r th realization of the recursive
filters. In Figure 4, the SRUKF and the PF (the number
C. Implementation of the proposed T2FLC
The proposed T2FLC has the estimated positions of
the supporting hip and the swing ankle as inputs, and the
centroid positions of the trunk as outputs. The i th rule
in the system has the following form:
Rule i : IF yˆ hip is Ai ,1 AND yˆ ankle is Ai ,2
THEN ytrunk is G i , i  1,,35
where yˆ hip and yˆ ankle denote the estimated centroid positions of the supporting hip and the swing ankle respectively, ytrunk is the centroid position of the trunk. Ai , j ,
j  1, 2 is an interval type-2 fuzzy set, G i is the output
interval type-2 fuzzy set of the i th rule, and the number
of rules is 35.
As shown in Figure 2, in this work, ZMP stability of
the biped is ensured by trunk rotations [41, 43] based on
fuzzy rules. About the ZMP of biped robot, there are
some facts derived as follows [43]: The desired ZMP is
greatly influenced by the position of the supporting hip,
and in a less degree by the position of the foot of the
swing leg. According to these facts, to ensure ZMP stability of the biped, the rule base of the proposed T2FLC
is designed as shown in Table 2. Here, two steps are involved in the design procedure of the fuzzy rules: First,
the interval type-2 fuzzy sets of the variables and the rule
base of the T2FLC are designed manually. Secondly, the
final interval type-2 fuzzy sets are obtained using the
method of trial-and-error.
International Journal of Fuzzy Systems, Vol. 15, No. 4, December 2013
430
Table 2. Rule base of the proposed T2FLC.
yˆ hip
0.9
UB
VB
B
M
S
VS
US
UB
EB
EB
UB
B
S
VS
US
B
EB
UB
VB
M
S
US
US
M
EB
UB
VB
M
VS
US
ES
S
UB
UB
B
M
VS
US
ES
US
UB
VB
B
S
US
ES
ES
degree of membership function
yˆ ankle
1
0.8
0.7
0.6
VS
VB
B
M
S
0.5
0.4
0.3
0.2
0.1
0
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
the estimated centroid positions of the swing ankle
The details about Table 2 will be described next.
1) To express the linguistic and numerical uncertainty,
interval type-2 fuzzy membership functions of yˆ hip are
designed as Figure 5. Gaussian primary membership
functions are used, which have a fixed standard deviation   0.01 and uncertain means that takes on values
in the following intervals:
[mih1 , mih1 ]  [0027, 0.029] , [mih2 , mih2 ]  [0.044, 0.046] ,
[mih3 , mih3 ]  [0.061, 0.063] ,
[mih4 , mih4 ]  [0.078, 0.080] , [mih5 , mih5 ]  [0.095, 0.097] ,
[mih6 , mih6 ]  [0.112, 0.114] , [mih7 , mih7 ]  [0.128, 0.130]
2) Considering the linguistic and numerical uncertainty of the system, interval type-2 fuzzy membership
functions of the yˆ ankle are designed as Figure 6. Also,
Gaussian primary membership functions are used here,
which have a fixed standard deviation   0.02 and uncertain means that takes on values in the following intervals:
[mia1 , mia1 ]  [0, 0.002] , [mia2 , mia2 ]  [0.048, 0.052] ,
[mia3 , mia3 ]  [0.098, 0.102] ,
a
a
[mi 4 , mi 4 ]  [0.148, 0.152] , [mia5 , mia5 ]  [0.198, 0.200]
1
D. Analysis and comparisons of the dynamic biped
balance
To analyze the performance index of the dynamic biped balance, mean of the ZMP stability margin (MZSM)
is calculated during one whole walking step using the
next formula:
MZSM 
0.8
VS
US
S
B
M
1
N
N
 min{ x
toe
k 1
 xzmp ( k ) , xzmp (k )  xheel }
(34)
UB
VB
1
0.6
0.5
0.4
0.3
0.2
0.1
0
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
the estimated centroid positions of the supporting hip
Figure 5. Interval type-2 fuzzy membership function of yˆ hip .
degree of membership function:(ytrunk)
degree of membership function
3) Interval type-2 fuzzy membership functions for the
positions of the trunk are designed as Figure 7. Still,
Gaussian primary membership functions are used, which
have a fixed standard deviation   0.01 and uncertain
means that takes on values in the following intervals:
[mitr1 , mitr1 ]  [0.0728, 0.0708] , [mitr2 , mitr2 ]  [0.0593, 0.0573] ,
[mitr3 , mitr3 ]  [0.0468, 0.0448] ,
[mitr4 , mitr4 ]  [0.0343, 0.0323] , [mitr5 , mitr5 ]  [0.0218, 0.0198] ,
[mitr6 , mitr6 ]  [0.0093, 0.0073] , [mitr7 , mitr7 ]  [0.0032, 0.0052] ,
[mitr8 , mitr8 ]  [0.0157, 0.0177] , [mitr9 , mitr9 ]  [0.0282, 0.0302]
where xzmp (k ) is the position of the ZMP on the k th
sampling point in a walk cycle, which can be obtained
using Eq.(1), and k  1, 2, , N ( N  41) . xtoe and xheel are
the positions of the toe and the heel of the biped robot.
0.9
0.7
Figure 6. Interval type-2 fuzzy membership function of yˆ ankle .
0.9
0.8 ES
US
VS
S
M
B
UB
VB
EB
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0.01
0.02
centroid positions of the trunk: ytrunk(m)
Figure 7. Interval type-2 fuzzy membership function of ytrunk .
L. Wang et al.: Type-2 Fuzzy Logic Controller Using SRUKF-Based State Estimations for Biped Walking Robots
431
Table 3. Comparisons of mean of the ZMP stability margin (MZSM).
Methods
The value of the
MZSM when there is
no noise
The value of the MZSM when there
exists process noise w(k ) ~ (0,103 ) and
measurement noise v(k ) ~ (0,103 )
PID controller (without SRUKF)
0.0238m
-0.0003m
T1FLC (without SRUKF)
0.0323m
0.0028m
T2FLC (without SRUKF)
0.0347m
0.0040m
T1FLC (with SRUKF)
0.0340m
0.0325m
T2FLC (with SRUKF)
0.0346m
0.0341m
Comparisons for MZSM using different methods are
shown in Table 3. When the simulation is implemented
without adding the process noises and the measurement
noises, the T2FLCs exhibits better performance slightly
than the other two methods, and the proposed method
provides similar result as that of the T2FLC without
SRUKF. However, when process noise w(k ) ~ (0,103 )
and measurement noise v(k ) ~ (0,103 ) are added in the
simulated model, an interesting phenomenon occurs.
Influenced by the added noises, the PID controller can
not keep the MZSM to be positive, which means that the
ZMP stability criterion can not be guaranteed any more.
Intelligent controller like T1FLC and T2FLC without
SRUKF lead to a positive MZSM with small value,
which is dangerous for the biped because the ZMP is
very close to the edge of the foot. On the other hand, the
proposed method improves the MZSM remarkably.
Compared with the traditional T2FLC, the proposed
SRUKF-based T2FLC obtains better performance using
“estimation and compensation” strategy.
Science Foundation of China under Projects 60974047
and U1134004, by the Natural Science Foundation of
Guangdong Province under Grant S2012010008967, by
the Science Fund for Distinguished Young Scholars
(S20120011437), by the 2011 Zhujiang New Star, by the
FOK Ying Tung Education Foundation of China under
Grant 121061, by the Ministry of Education of New
Century Excellent Talent, by the 973 Program of China
under Grant 2011CB013104, and by the Doctoral Fund
of Ministry of Education of China under Grant
20124420130001.
References
[1]
[2]
6. Conclusions
An effective method for biped robot based on a predictable T2FLC is proposed to ensure the dynamic balance of the biped walking. Simulation results demonstrate the superiority of the proposed methods, and the
proposed method shows superior performance when
compared to the PID controller, the T1FLC and the
T2FLC without SRUKF. We believe that the proposed
framework will be very promising for dynamic balance
of biped robot systems suffering from various uncertainties. Future works include automatic optimization on the
predictable T2FLC.
Acknowledgment
This work was supported by the National Natural
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Liyang Wang received the B.S. degree
in communication engineering from
Tongji University, Shanghai, China, in
2002, the M.S. degree in electrical and
communication
engineering
from
Guangdong University of Technology,
Guangzhou, China, in 2007. During
2010-2011, she was a Research Associate in Intelligent Systems Research Institute, Sungkyunkwan University, Korea.
She is currently pursuing a Ph.D. degree in control theory
and control engineering from Guangdong University of Technology, Guangzhou, China. Her research interests include robotics and learning control.
Zhi Liu received the B.S. degree from
Huazhong University of Science and
Technology, Wuhan, China, in 1997, the
M.S. degree from Hunan University,
Changsha, China, in 2000, and the Ph. D
degree from Tsinghua University, Beijing, China, in 2004, all in electrical engineering.
He is currently a Professor in the Department of Automation, Guangdong University of Technology,
Guangzhou, China. His research interests include fuzzy logic
systems, neural networks, robotics, and robust control.
C. L. Philip Chen received the M.S.
degree from the University of Michigan,
Ann Arbor, in 1985, and the Ph.D. degree from Purdue University, West Lafayette, IN, in 1988.
He is currently a Dean and Chair Professor with the Faculty of Science and
Technology, University of Macau,
Macau, China. He is also a Professor
and the Chair of the Department of Electrical and Computer
Engineering, Associate Dean for Research and Graduate Studies of the College of Engineering, University of Texas at San
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International Journal of Fuzzy Systems, Vol. 15, No. 4, December 2013
Antonio, San Antonio.
Yun Zhang received the B.S. and M.S.
degrees in automatic engineering from
Hunan University, Changsha, China, in
1982 and 1986, respectively, and the
Ph.D. degree in automatic engineering
from the South China University of Science and Technology, Guangzhou, China,
in 1998.
He is currently a Professor with the
Department of Automation, Guangdong University of Technology, Guangzhou, China. His research interests include intelligent control systems, network systems, and signal processing.
Xin Chen received the B.S. degree from
Changsha Railway Institute, Hunan,
China, in 1982, the M.S. degree from
Harbin Institute of Technology, Heilongjiang, China, in 1988, the Ph.D. degree
from Huazhong University of Science
and Technology, Wuhan, China, in 1995,
all in mechanical engineering.
He is currently a Professor with the
Department of Mechatronics Engineering, Guangdong University of Technology, Guangzhou, China. His research interests include Mechatronics and robotics.