Research Journal of Applied Sciences, Engineering and Technology 4(13): 2009-2016, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: March 19, 2012 Accepted: April 06, 2012 Published: July 01, 2012
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2
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Abstract: In this study, sliding mode observer for fault detection and isolation is applied to a Robotic
Wheelchair. In addition, a general review has been done on dynamic model of robotic wheelchair along with detailed study of the mathematical model of robotic wheelchair system. Then, sliding mode observer is investigated with detailed comment. In order to isolate and estimate the possible actuator faults a bank of
Sliding Mode Observer (SMO) is designed. Also a simple canonical form for sliding mode observer is presented. A conclusion of the work will end the report giving future recommendation and comments.
Key words: Canonical form for sliding mode observer, fault detection and isolation, robotic wheelchair, sliding mode observer this field.
INTRODUCTION
Robotic wheelchair helps disabled people to perform their daily tasks. Commonly, for someone who uses a wheelchair, the route that wheelchair moves along it may include some unevenness. Even in the case that the route includes a little unevenness, an individual with disabilities will require assistance to bypass unevenness. In
(Takahashi and Tsubouchi, 2005; Takahashi et al ., 2003;
Takahashi et al ., 2000) some attempt have been made in
Primarily, the recommended system purpose is to help the wheelchair gainer to be capable of climbing to a height of 10 cm without the assistance of another person.
The recommended system composed of two phases of step rising. First, frontal wheel goes up. In the second stage, after climbing the wheelchair, an inverse pendulum control is used. For climbing of the frontal wheel, the location of the back wheel axle is changed by a little force. Once an individual elevates the wheelchair, a rapid increase can be predicted. The force needed to elevate the frontal wheel 10 cm is inversely commensurate with the location of the back wheel axle. Therefore, a mechanical sliding of back wheel axle to modify situation for approximately 50 mm is configured once an elevating mechanism is performed. After completion the section that belongs to the wheelchair’s frontal wheel elevating, the system’s conditions are similar to the effect of inverted pendulum (Takahashi et al ., 2000).
Control problem of uncertain systems that have been exposed to an external perturbation has been an enabled field of study during the last decade. The majority of systems that we facing them in real terms are exposed to different uncertainties like nonlinearities, actuator faults, variations in parameters and etc. In majority of proposed control strategies is assumed that all state variables are existing; this assumption is not always correct in real terms, therefore the state vector should be appraised for using in the control rules. The basic objective of a fault identification plan is to produce a warning when faults happen. Among research activities performed in this field can cited to Kalman filter (Kalman, 1976), adaptive observers (Gevers and Bastin, 1986), high gain observers
(Hammouri and Othman, 1992), sliding mode observers
(SMO) (Utkin, 1992; Walcott and Zak, 1986; Edwards and Spurgeon, 1994), etc. (Thein and Misawa, 1995) to perform comparing.
A certain situation amongst observer based methods is taken by sliding mode observers. Largely, the sliding mode observer takes advantage from discontinuous control operation to move the observer fault direction toward a certain hyper-plane in the fault space and then the direction is preserved to slide on this so long as the origin of the state space is attained. Basically, Observer generated the signals which are used to discover data associated with the fault. Specifically, remaining generation statements, using as linear observers, have been extensively applied. In this method, difference
Corresponding Author: Mohammad Hassan Khooban, Department of Electrical and Robotic Engineering, Garmsar Branch Islamic
Azad University of Iran, Garmsar, Iran
2009

Res. J. Appl. Sci. Eng. Technol., 4(13): 2009-2016, 2012 among output of the system and output of the observer is calculated by a scaling matrix to create known as remaining. The remaining will equal zero if fault does not occur in the process, however will reply specifically once a special fault happens (Frank, 1990; Magni and Mouyon,
1994).
Utkin designed an ordinary observer, just with discontinuous part getting fed back along a suitable gain
(Edwards et al ., 1998). Walcott and Zak (Walcott and
Zak, 1988) designed an observer where the output fault getting fed back linearly and applied a Lyapunov method to demonstrate the sustainability. Edwards and Spurgeon
(Edwards et al ., 2000; Edwards et al ., 1994) suggested a canonical form for designing of sliding mode observer depending on special circumstances associated with the output and input distribution matrices and as well as the invariable zeros of the system. Their procedure expressed in (Edwards et al ., 2000; Edwards et al ., 1994) used both linear and nonlinear output error injection. A procedure for calculation the gain as related with the linear output error injection part is provided. The solution is obvious, however does not utilize all grades of freedom. Tan and
Edwards (Edwards et al ., 2000) offered another canonical form based on an adequate status according to linear matrix inequality (LMI). The study attempts to utilize the freedom in (Edwards et al ., 2000) for sectional pole attribution. However, they did not determine the most appropriate place for eigenvalues in the favorable district.
Their procedure is partly complicated, although their procedure is obvious (Xiang et al ., 2005).
In this study, we consider simple linear models of the robotic wheelchair system with actuator fault. And, we propose the design of the simultaneous multiplicative faults detection for robotic wheelchair using a sliding mode observer. The effectiveness of the proposed method is verified by simulations.
Problem Statement:
Wheelchair mechanism : In this part, the suggested hardware mechanism will be studied. The equilibrating mechanism is created by controlling the rear wheel spin.
Rear-wheel drive system is composed of DC motor, gearbox, chain and housing. The gear ratio is 772 which results the reduction ratio of 1/772.
In order to achieve the needed measurements two sensors are placed. An optical encoder is embedded in order to identify the spin of the rear-wheel. A gyro sensor is placed to measure deviation change rate of the wheelchair body. The control system is composed of personal computer, counter board, Analog-to-Digital (Ato-D) converter and Digital-to-Analog (D-to-A) converter.
Figure 1 demonstrates an overall shape of the hardware system of the wheelchair.
The mechanism is considered as follows:
C
Gyro sensor identifies the deviation rate of the wheelchair
C
Deviation rate data is transferred into the PC via
Analog-to-Digital (A-to-D) converter
C Deviation rate is subtracted from a bias signal which is measured formerly
C
Deviation rate is integrated for giving the deviation angle of the wheelchair
C
The error is calculated by subtracting the favorable amount from the measured deviation
C The error signal is transferred into the controller
C
A control input is generated by Digital-to-Analog (Dto-A) converter to provide the needed current to set up the DC motor… and so
The sampling period applied in the main work was 17 ms.
Wheelchair Dynamics: In this part, the general dynamics of wheelchair is studied. Figure 2 demonstrates a scheme of the dynamic model of wheelchair.
The model in Fig. 2 can be attributed to Fig. 3 of the inverted pendulum on a cart. As Mb in Fig. 2 being the center-of-mass mass of the wheelchair body is projected
Fig. 1: Hardware system of the Wheelchair
2010
Fig. 2: Wheelchair Dynamics

Res. J. Appl. Sci. Eng. Technol., 4(13): 2009-2016, 2012
Table 1: Wheelchair parameters and variables
N
Inclination angle of the wheelchair body [rad]
2
Rotation angle of the rear wheels [rad] v
R
Input voltage of the DC motor [V]
DC motor resistance [ohm]
Mb Total mass of the wheelchair body and person onboard [kg]
Mw Mass of the rear wheels [kg]
Jb Total moment of inertia of the wheelchair body and person
[kg.m
2 ]
Jw Moment of inertia of the rear wheels [kg.m2]
L Length between wheelchair shaft & gravity center of r wheelchair body [m]
Radius of the rear wheels [m]
Kcf Damping constant between floor and rear wheels [N.m/(rad/s)]
Kcs Damping constant of the wheel shaft [N.m/(rad/s)]
Kt Torque constant of the DC motor [N.m/A]
Fig. 3: Schematic of an Inverse Pendulum on a Cart as the point mass of the pendulum. And
N is the deviation angle of the pendulum. The uniform movement of ‘x’ in
Fig. 3 is here in Fig. 2 performed by turning of the rear wheel here is shown by
2
. Also r and Mw indicate radius of the rear wheel and mass of rear wheels, respectively. It is supposed that the behavior of the two rear wheels is similar to one wheel without considering dynamics of each wheel. Thus the ?
only ?
impact of the wheelchair on an inverted pendulum system is that the uniform movement is now obtaining through the motor controlled rear wheel spin.
The obtained results have been used to form the full motion’s equations of the wheelchair nonlinear dynamic system. The two differential equations expressing the motion is:

M L
2 
J b

J K
2 g cos
 
J K g
2

  
K cs

 


M gL b sin
  
K K t v
R
(1)

M L b
2 
J b

J K
2 m g
M rL b

J K
2 m g


K cs

 


M gL b
  
K K g t v
R

M rL

J K g
2

  
 
M b

M w
 r
2 
J w

J K
2 g

  
K cs
  

K cs

K
CS

  
K K t
R
(3)
(4)
State-Space Model: In this part, the whole state-space model of the system will be introduced. From the motion’s linearized equations expressed in Eqs (3) and (4) of prior part, the full State-Space Model of the Robotic
Wheelchair when using the Inverted Pendulum Control is
 ( )

( )

( )
( )

( )

   cos

K cs
  

J K g
2


 
M b

M w
 r
2 
J w

J K g
2


K cs

M cf

    2 
K K t
R
(2)
The parameters and variables applied in the equations and over the whole report are described in Table 1.
In this part, just the linearization process is performed on the dynamic model to be prepared for analysis and design:
Sin
N .
N
Cos
N .
1
N
2
.
0
Thus, by that, the ultimate linearized mathematical equations are: with the state vector of: x



 
  







 



 with,
A






 
0
0
A
1
A
4
0
0
0
0
0
0

A
2
A
5
0
0

A
3
A
6






,
As,
A
1

A
3

A
2
B
1

 a b
 a
2
2
, a a
 a b a b a b a b a b a b




 a a a a
2
2
2
2 a b a
2
2
A
2
 a b a b

 a b a
2
2
,
,
,
A
B
4
A
6
2

 a b
 a
2
2 a b
 a b
 a b
 a
2
2 a b
 a a a b
 a
2
2
B







0
0

B
2
B
1






2011

Res. J. Appl. Sci. Eng. Technol., 4(13): 2009-2016, 2012
Table 2: System Parameters Values
M w
L
K t
J w
K cf
K cf
M b
JJ b r
K g
J m with,
1
 2   2
,
 b
 2 m g
1
3

,
4

,
5

R
 
2   2
,
  cf
6.52 [kg]
0.29 [m]
0.0239 [N.m/A]
84.16 [kg]
29.3 [kg. m
2
]
0.305 [m]
772
7.0×10
!
6 [kg. m 2 ] and D are supposed to be full rank. Here, u(t) and y(t) are accessible.
Suppose the observer of the form:

( )
 
( )

( )
 l y
( )

G v

( )
 
( ) v



 
D
2
2
0
2
( )
( ) e y

0 otherwise
(7) where, G l
, G n
0 R n×p are proper gain matrices. The discontinuous vector v is presented by:
(8)
Note that the measured outputs are optical encoder output (
N !
2
) and the output of the gyro sensor
N
. Thus, these causes:
C
 


1

1 0 0
0 0 1 0



And system input u = v the DC motor input voltage. In this summary part, only amounts of system’s parameters will be introduced and some simulations to present a general scheme about existing conditions.
Amounts of system’s parameters that have been applied in the whole system are introduced in Table 2.
Thus, the numeric complete state-space model is: x   x















0
0
.
0

.
.
.
0 .

0
0





 u ,
0
0 y
 


1
0
1

1 0
0
1
0
0 0 1 0








 x
FDI using sliding mode observer: Notice the following dynamical system:
 ( )

( )

( )
 i
( ) (5)
  f
0 t (6) where, A
0
R n×n , B
0
R n×m , C
0
R p×n , D
0
R n×q with q
# p
# n,
A 0 R n is the state vector, u 0 R m is the input vector and y
0
R p is the output vector. Also f i
(t) and f
0
(t) show the actuator and sensor faults, respectively. The matrices C where, e y t
 y t
 y t and P
2

R is symmetric positive definite. The definitions of the matrices P
2
and D
2 and the scalar can
D
(t) are presented in the later part. The disc ontinuous term is designed to drive the D
2
trajectories of the observer in a way that the state prediction error vector is forced into and as a result stays on a surface in the error space.
Now consider the dynamical system offered in (5) and (6) and suppose that:
* rank (CD) = q
* Invariable zeros of (A, D, C) must lie in the open LHP
First, suppose the case that f
0
(t) = 0. It can be demonstrated that under these hypotheses, there exists a resemblance transformation as T (The result expressed in
[19] confirms the existence of a nonsingular transform matrix to have this structure and two ways to gain it is presented in the Appendix) such a way that the system will be appear in the following form:

1
( )

11 1
( )

12
( )

1
( )
 ( )

21 1
( )

22
( )

2
( )

2 i
( )
(9) where, x
1
0
R n
!
p and the matrix A
11
has stable eigenvalues
(Tan and Edwards, 2001). The above transformation is applied to gain the following form of observer (7):

1
( )
 
( )

12

( )

1
( )

12 y
( )

( )
 
( )
 
( )
2
( )

( A
22

) y
( )
 v
(10) where, M is a stable design matrix that must be specified by designer. Also, P
2
is a Lyapunov Matrix for M and the scalar
D
is selected in such that:
||f i
(t)|| <
D
(t) (11)
2012

Res. J. Appl. Sci. Eng. Technol., 4(13): 2009-2016, 2012
From (10) and (11):

1
( )

11 1
( ) y
( )

21 1
( )
 y
( )
 
2 i
( )
(12) and
  
  
1 t D
2
D D
2
D
T
2
2
2
( )
( )
  where
1
( )
 
1
( )
 
1
( )
( )
  ( )

( )
(13)
It is demonstrated in (Edwards et al ., 2000) that the nonlinear error system in (12) is stable and a sliding motion takes place forcing e y
(t) = 0 in limited time. The dynamical system in (12) may so be considered as an observer for the system in (5) and (6). It follows that if:
G l

T

1 

 A
A
12
22

M



, G n

T

1 


0
Ip 


(10) then the observer presented in (12) can be written in the basis of the main coordinates in the form of (7).
It is demonstrated that, offered a sliding motion can be achieved, the state of the system can be restored and also, estimates of fj (t) and fi (t) can be calculated as: x
















0 0 1 0
.
0 0
0 .
0
.
0

.
.

1
0
0





 u , y




1

1 0 0
0 0 1 2








 x
(14)
( )
 
( A
22
 
1
A A A
12
)

1
2
P e t
2 y
( )
( )
 
(15) where,
*
is a small positive scalar and f
0
(t) is supposed to be a slowly changing fault (Edwards et al ., 2000).
Simulation Results: Consider the robotic wheelchair equations (3) and (4) where the specific amounts of the system parameters such as masses and gear rato, ect.
Where presented in Table 2.
This results in the triple system below:
Fig. 4: SMO with fault estimation
2013

Res. J. Appl. Sci. Eng. Technol., 4(13): 2009-2016, 2012
Fig.5:Simulation design in Matlab
1.0
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
0 5 10
(a)
15 20 25 30
0
-0.2
-0.4
-0.6
-0.8
-1.0
1.0
0.8
0.6
0.4
0.2
0 5 10 15
(a)
20 25 30
1.0
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1.0
0 5 10
(b)
15 20 25 30
Fig.6: The Fault Signal
With supposed the fault distributation matrix D = B.
Using an algorithm like that suggested in [1] it can be demonstrated that in the canonical form of the system will be appeared in the following form:
0
-0.2
-0.4
-0.6
-0.8
-1.0
1.0
0.8
0.6
0.4
0.2
0 5 10 15
(b)
20
Fig. 7: The reconstructed Fault Signals
A








0

0

.

.

.

.
0 1

0 4464 .
0







25
2014
30

D








0
0
0






Using the suggested procedure:
G l






 

50
50
.


.
51






G n








0
0

0
0






In this specific design the scalar function
D
= 75 and design of the observer is perfect. General scheme of the designed Sliding Mode Observers is shown in Fig. 4 and 5.
Simulation results for fault estimation and determination are shown in Fig. 6 and 7. You can see that the robust term was able to track the fault occurred, the reconstructed fault from the Sliding mode.
CONCLUSION
This study demonstrates how Sliding Mode
Observers can be used to detect, estimate and isolate of the faults in actuators. In addition, conditions for the existence of a sliding mode linear functional observer are given. Then is demonstrated once the existing circumstances are satisfactory, how to discover parameters of the observer. The proposed FDI (Fault
Detection and Isolation) design is easy to implement and can be applied to a reasonably wide class of systems. The numerical example for Robotic Wheelchair illustrates that this approach is effective and easy to implement.
Appendix : Let state description of the system (1), (2) with m = p is:
 (

( )

( ) (A.1) y t

( )
(A.2)
C
1

CT
1


0 I p

,
Res. J. Appl. Sci. Eng. Technol., 4(13): 2009-2016, 2012
C




0 0 1 0
0 0 0 1



T
1

1
 


I
C
0 


Then
B
1

T B

T
1

1 


B
01
B
02


  


B
01
CB


  


B
11
B
12



(A.3)
(A.4)
If CB = B
12
is a regular matrix (in opposite case the pseudoinvers os CD = B
12
is possible to use), then the second transform matrix T
2
G 1 can be defined as follows:
T
2

1
 


I o

B B
12

1
Ip



, T
2
 

 
I
0

B B
12
I p

1


 
These results in:
B

T B
1
 

 
I o where,

B B
12
I p

1


 



B
11 b
12


  


0
B
2



B
11

B
01
, B
2

B
12

CB
(A.5)
(A.6)
(A.7) and
C

C T

 


 
I o

B B
12

1
I p


 
(A.8)
Finally, with T
!
1 con
= T
2
!
1 T
1
!
1 it yields:
A = T -1 con
AT con
(A.9)
Thus, (A.6), (A.8) and (A.9) represent the system canonical model.
Note: The structure of T
1
!
1 is not unique and others can be obtained by permutations of the first n
!
p rows in the structure defined in
(A.3) (Please check example in (Edwards and Tan, 2006)).
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2016