Research Journal of Applied Sciences, Engineering and Technology 4(20): 3875-3884, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: December 18, 2011 Accepted: April 20, 2012 Published: October 15, 2012
Abstract: A trajectory tracking approach is proposed for end-point trajectory control of elastic manipulator based on model predictive control. In practice measuring lag is inevitable for flexible manipulator. To solve the problem, a simplified linear predictive model is derived from the manipulator’s dynamics and special prediction strategy based on this model is designed to predict the robot’s future behavior with delayed feedback.
Then the control law aimed to minimize a quadratic performance index of predicted tracking error is proposed.
Since the conventional predictive control involves large amount of calculation, which is intolerable for high sampling speed, proper simplifications are made to achieve high computation efficiency through obtaining the analytical expression of the control signal. Theoretical analysis proves the boundness and stability convergence of the end-point tracking error. Finally the proposed controller is applied to a one-link manipulator in simulation section, where good control effect demonstrates the advantages of the proposed controller.
Keywords: Flexible manipulator, high computation efficiency, measuring lag, predictive control
INTRODUCTION
WIDE application of robots and increasing demands for high-speed and light robot arms generate lots of interest in the study of flexible manipulators. Different from traditional rigid robots, the flexible manipulator is more difficult to control due to its elastic deformation and dynamic coupling between joints. Moreover, this issue is further complicated since there is no analytical solution to manipulator’s dynamics equation.
Considerable researches concentrate on tracking of a desired end-effector trajectory (Benosman and Le Vey,
2004). Because of flexible link’s non-collocated endeffector output, the system dynamics is featured with nonminimum phase characteristic, which makes the end-point difficult to be controlled along the desired trajectory. To address the problem, various types of approaches are put forward. Wang and Vidyasagar (1990) comes up with the output redefinition method to obtain a minimum phase system through modifying the output objective; singular perturbation approach is used by Siciliano and Book
(1988) and Sun et al.
(2005) for approximate tracking control; Eduardo (Bayo, 1987; Kwon and Book, 1994) develop stable inversion method to cancel the nonminimum phase features but the unmodeled dynamics exert significant effects on its accuracy.
Predictive control has been gaining increasing popularity in industry applications in these years.
Traditional predictive control algorithm is featured with receding optimization which needs to be solved repeatedly every sampling time on-line. Its common applications are in slow processes such as refinery and chemical industries (Duchaine et al ., 2007), where a standard numerical solver which usually takes several seconds every time is normally adopted for the online optimization, whose computation efficiency is tolerable since the process is slow. However, in nonlinear robotic systems, where a high servo rate is necessary for controlling accuracy, predictive control’s computation efficiency becomes an increasingly critical burden.
Some researchers make attempts to apply predictive control to servo control problems, including manipulator end-point trajectory tracking control. Lu (1994) proposes a nonlinear predictive controller to track desired trajectory through minimizing predicted tracking errors. In Lu
(1994) the controller is successfully applied in missile autopilot design and relevant theoretical analysis on the controller’s tracking performance proves its robustness.
Yim (1996) for the first time applies this approach to flexible manipulators. He derives a predictive control law through minimizing a quadric function of end-point trajectory tracking errors and input joint torques as the performance index. However, his control scheme is openloop and suffers from potential risks of being inaccurate and unstable. Shuai et al . (2010) modifies Yim’s controller and proposed a closed-loop control scheme where feedback is incorporated for better performance in the presence of model inaccuracies.
Since it is a complicated process to measure the states of the manipulator and the measurement device would
Corresponding Author: Xiaoya Wei, Department of Control Science and Engineering, Zhejiang Univ, Hangzhou 310027, P.R.
China
3875

Res. J. Appl. Sci. Eng. Technol., 4(20): 3875-3884, 2012 consume some time to finish a procedure of measurement, the measurement feedback, especially those related to flexible vibration, commonly suffers from certain lag. In both Yim and Shuai’s control scheme (Shuai et al ., 2010) this feedback delay is not incorporated. Because of the flexible manipulator’s high nonlinearity and nonminimum phase characteristics, neglecting feedback lag may result in the system’s instability. Due to the influence of various factors, the length of the lag is usually stochastic, in which case the conventional Smith predictor cannot be used to modify Shuai’s controller because it can only compensate for constant time delay. In the proposed controller a special prediction strategy is specifically designed to compensate for the measuring lag and ensure the system’s robustness. In practice, the controller and the measurement device are incorporated in the whole control system. Therefore the time when the measurement device starts a procedure of measurement and the time when the controller receive the measurement results are available to the controller. That is to say, in implementation the controller can tell how long a certain piece of measurement feedback is delayed, although the lengths of different feedback lag’s lasting time may be different.
Moreover, since both Yim’s and Shuai’s controller adopt a standard numerical solver for the on-line optimization, whose computation efficiency is relatively low, they take a single-step prediction horizon to minimize calculation so that their algorithms’ efficiency can be guaranteed for real-time control, because a longer prediction horizon means more calculation is needed.
However, with a short prediction horizon they cannot take all major responses of the system into account and thus cannot guarantee satisfying control performance. To fix this drawback, this study makes proper simplifications to derive a computationally efficient control law with a prediction horizon of any length, where an analytical expression for the optimal control can be applied that does not involve a normal numerical procedure, as opposed to
Yim and Shuai.
With special computationally efficient prediction strategy and optimization method, this study presents a predictive controller with prediction horizon of any length through minimizing quadratic function of the predicted tracking errors. A special prediction strategy is designed to predict the model’s future responses with delayed measurement feedback and ensure the system’s robustness, in which case the stochastic measurement lag can be compensated for. Proper simplifications are also made to achieve high computational efficiency for the online optimization and state feedback is incorporated to compensate for the model’s inaccuracy. The analysis on the boundness and convergence of tracking errors is conducted to show the controller’s robustness even with inaccurate prediction model. A manipulator with rotatingprismatic joint, which is proposed and studied by prismatic joints
Kalyoncu (2008), is controlled with the propsed controller as an example. The simulation results show that the end point could well follow the desired trajectory even in the presence of stochaistic measuring lag.
In this study, we presents a predictive control scheme to control a flexible manipulator’s end-effector along the desired trajectory with measuring lag on its feedback channel. An approximate linear model is reduced from the complex nonlinear dynamics equation. Then a prediction strategy based on the approximate model is designed to predict the robot future behavior with delayed measurement feedback with a recursive equation and the general form of the equation is also obtained. In addition, proper simplifications are made and relevant techniques are proposed to achieve high computation efficiency for the online optimization needed in the predictive control scheme. Theoretical analysis proves the boundness and stability convergence of the end-point tracking. At last, a simulation comparison demonstrates the effectiveness and robustness of the control scheme in this study.
PROBLEM FORMULATION
A m-link flexible manipulator with rotating-prismatic joints (Kalyoncu, 2008) is shown in Fig. 1 where the elastic link can both rotate and slide with the joints. The initial length of the links is denoted as l = [l
1
Torque T = [T and force F[F joints. The manipulators is specified by joint angles
2
=
[
2
1
2
= [
8
1
2
…
2 m
8
2
] T
…
8 m
]
1
and m variations in flexible link’s length
8
T
1
T
2
F
2
… T m
… F m
] T
] T
l
2
… l m
] T
rotates the arms about Z axis
pushes them to slide in the
. With assumed-mode method the flexible displacement of the i-th link
0 i
can be written in the
.
following form:
 i
 s i  j

1
 ij
( )
 ij
( )
 T
.
i
(1)
3876

where, t is time, r represents the distance between the specific point and the joint,
M
= [
N i1
N i2
...
N isi
] T and
) i
=
[
* i1
* i2
…
* isi
] T respectively denote the assumed mode basis function and corresponding modal coordinates. It is assumed that s i
is the number of assumed modes that are sufficient to obtain a satisfying approximation of the i-th link’s deformation. Define the total number of assume s
 m  i

1 s
With the kinetic energy and potential energy of the system and apply the Lagrange’s equation calculated, the dynamics equations for the flexible manipulator can be expressed as follows (Kalyoncu, 2008):
M q q
 h q q
  where, generalized coordinate q = [
8 T 2 T )
1
T …
) T m
] T and the joint input
J
= [F T T T ] T . M(q) is the positive definite symmetric inertia matrix of the manipulator and can be expressed as h q q

(  , )

( ) where is the vector of Coriolis and centripetal forces and G(q) is the generalized gravitational force vector.
The following properties of the model are pointed out by Spong et al.
(1989)
Res. J. Appl. Sci. Eng. Technol., 4(20): 3875-3884, 2012


T 0
1
 s

T
(2)
Property 1: Since M(q) is uniformly positive definite, there exists positive constants m and m so that for all q(t) 0ú 2m+s , m

|| ( )||
 m where |.|| represents the
Euclidian norm.
Property 2: The matrix is linear on bounded on q(t), therefore there exists k
1
 ( ) and
0ú
+
such that
C q q
 k q .
Property 3: The generalized gravitational force vector
G(q) satisfies |G(q)||
# k
2
for some k
2
0ú
+
.
In practice the flexible manipulator’s end-effector is usually expected to track a given reference trajectory and the following part of this study aims to deal with this problem. To achieve this goal, the model predictive control algorithm is specifically adapted and applied on the flexible manipulator so that its endpoint can track the desired trajectory. The robustness of the controller is also guaranteed.
It should be noted that since it is a complicated process to measure the states of the manipulator, a measuring lag is usually inevitable. In practice, the time when the measurement system starts to measure and the time when the real measured generalized coordinates obtained are available. In other words, the controller can tell the exact length of each measuring lag, which can be viewed as feedback time delay. In the design of the controller this measuring lag should also be taken into account.
Predictive control law: In general model predictive control scheme (Duchaine et al ., 2007), in order to give the correct control signal to be applied to the system, it is usually required to minimize a quadratic cost function over the prediction horizon as performance index. The cost function is composed of two components, namely, a quadratic function of deterministic and stochastic components of the process and a quadratic function of the constraints. The function to be optimized can be written as Rawlings (2000):
J

H n
 P

1
( y
 r ) T Q y
 r )

H  c m

1
  m m
(3) where k is the current time step; H horizon; H c p
is the prediction
is the prediction horizon; Q and
8
are the weighting factors; r k+n
is the reference input; y output of the system; u k+m is the constraint function, usually
Q m
= u m
-u m
!
1
.
k+n
is the
is the control input signal;
Q m
The conventional model predictive control scheme can be illustrated in Fig. 2 (Kouvaritakis and Cannon,
2001).
The conventional Model Predictive Control (MPC) scheme usually uses a standard numerical solver to deal with the optimization problem, namely to minimize the performance index (3). However, because of the flexible
Reference r k r
0

 d

  e k x n
Standard
Optimization solver

1
Predictive
Model
Plant x
0
Fig. 2: General model predictive control scheme
3877

Res. J. Appl. Sci. Eng. Technol., 4(20): 3875-3884, 2012
Reference r k r
0

 d


 x n e ˆ k
Sp eci al
Opt imization
Sol ver

1 M anip ul at or
Pred ictiv e
Mo del x
0
Co mpen sato r
M easu remen t
Dev ice
Fig. 3: Predictive control scheme for flexible manipulator manipulator’s severe nonlinearity and non-minimum phase characteristics, the inevitable feedback lag cannot be neglected in the control of flexible manipulator, otherwise the control accuracy cannot be guaranteed and the system can even go unstable. In MPC there is no component designed to compensate for such feedback lag.
Besides, it is usually involved lots of calculation to solve a general optimization problem. In this case, it would take a relatively long time to minimize (3) and thus the sampling time is difficult to be set small enough for realtime control to guarantee the accuracy. Therefore, certain improvements on the MPC scheme should be made so as to reduce the needed calculation to a tolerable size.
In this study, a compensator is designed to offset the measuring lag caused by the measurement device based on the MPC scheme. In practical implementation, the time when the measurement device starts a procedure of measurement and when the controller receives the result of the measurement can be respectively recorded. With these records, the specific length of the feedback lag, which is viewed to equal to t
2
-t
1
, can be calculated and used at every sampling period, although the lag is stochastic and can be different at different times.
Moreover, by taking advantage of the flexible manipulator’s characteristics, some exquisite simplifications are made in the derivation of the control law, which makes it possible to find the exact analytical expression of the optimal control signal. In this case the standard optimization solver is replaced by a special one so that the computing time is dramatically reduced.
Therefore the sampling period can be set small enough for accurate control performance. The measurement lag compensator and high computation efficiency are the two distinctions of the proposed predictive control technique in this study. The block diagram of the control scheme is illustrated by Fig. 3.
With Fig. 2 and 3 compared, a compensator is placed on the feedback channel to offset the feedback time lag caused by the measurement device. Moreover, the proposed control scheme uses a special optimization solver, instead of a standard one, to calculate the optimal control signal with high computational efficiency. In the proposed control scheme, the predictive model predicts
3878 the manipulator’s future behavior through prediction strategy at each sampling time based on the delayed measurement feedback. The compensator functions to compensate for the feedback lag caused by the measurement device to strengthen the algorithm’s accuracy. Then the optimization solver gives the optimal control signal with high efficiency. The detailed control law is formulated in the following sections.
Prediction strategy: As pointed out in previous sections, measuring lag is inevitable on the feedback channel. In order to compensate for the lag and estimate the manipulator real-time state based on the delayed feedback, two simplifications and an assumption are made as follows:
Simplification 1: Within the duration of the measurement lag, the nonlinear components M(q) and have no obvious variations and thus can be viewed as constant.
Simplification 2: The term remains constant during the measurement lag.
Assumption 1: edure of measurement t
1
and when the controller receives the result of the measurement t
2
are obtainable. That is, the length of the feedback delay t
2
-t
1 is available for the controller.
Define n = [t
2
-t
1
/T s
], where, T s
denotes the sampling period of the control system and [.] means the Gaussian
Function. In this case, the length of measuring lag is n sampling period.
For convenience of the prediction strategy’ derivation, two simplifications are made as follows:
Simplification 3: Within prediction horizon, the terms
M(q) and are constant.
Simplification 4: The term q can be considered as constant within the prediction horizon (Duchaine et al .,
2007).
Equation (2) can be converted into the following form:

Res. J. Appl. Sci. Eng. Technol., 4(20): 3875-3884, 2012
 
M

1
( q
 n
)
 u

( 
 n
, q
 n
)

(4) controller gets the measurement information as mentioned in previous sections. where, q 
 n
and q
!
n
are the measurement feedback the controller receives at the sampling instant, whose length of delay is n sampling periods.
With S1, S2, S3 and S4, one can estimate the manipulator past states or predict its behavior within prediction horizon with the following recursive equations:
Remark 3: When the feedback lag is neglected, or there is no feedback lag, the prediction strategy can be written as:
  q  k

1

M

1
(
0
) s
( k

( 
0
,
0
))

( s
) (8)
 k
 q k

1

M

1
( q
 n
) s
( k

(
 n
, q
 n
))

( s
) (5) q k
 q k

1

T q  k

1

1
2
2
T M

1
( q
0
)( u k

( 
0
,
0
))

( s
2
) (9) q k
 q k

1 s
 k

1

1
2
T M

1
( q
 n
)
( u k
 h q
 n
, q
 n
)

( s
2
))
(6) where, q  k
and q k
represents the predictive value of and q after k sampling times and T s
denotes the sampling time.
Combining (8) and (9), the general form of the predictive generalized coordinate q can be expressed as:
When !
n<k # 0 k
and q k
denotes the estimated value of the manipulator states k sampling periods before the current sampling instant; when k>0 k
and q k
denotes the predicted value of the manipulator states sampling periods after the current sampling instant. Combining (5) and (6), the general form of the predictive generalized coordinate can be expressed as: q n
 q
0
 
0

(
0
)

1 2
T s


 i n 

1

 n
 
2

1
2 n h q q
0
)

D t
 i u i
(10) where, D(t) denotes the errors caused by the linearization in (8)-(9) and the S1-S4.
q k
 q
 n

( k

) 
 n
(
 n
)

1
T
5
2


 k  i n 1

 k n
1
2
 i u i

1
2
( k
 n h q
 n
, q
 n
)

D t
(7)
Control law derivation:
Definition 1: In convenience for the control law’s deduction, define the pseudo angle a i
of the i-th link as follows: where, D(t) denotes the errors caused by the linearization in (5)-(6) and S1-S4. The model uncertainties would also introduce certain error between (7) and the real model and this error is also incorporated in D(t). Although this term will be disregarded in the derivation of control law, theoretical analysis in the following sections would prove the controller’s robustness even with this error term discarded.
 i
  i

  i
 j
 s i

1
 ij
( , )
 ij
( ) / (
 i
 l i
)
  i
T
.
i
/ (
 i
 l i
)
(11)
The practical implications of (11) is illustrated in
Fig. 1 With definition of the pseudo angle, system’s pseudo generalized coordinate x is defined as x = [
8 T " T ] T , where,
"
= [
"
1
"
2
…
" m
] T . With (11) x can be written in the following compact form:
Remark 1: S1 implies that M(q) and substituted with M(q
-n
) and h q 
 n
, q
 n
)
can be
since they would not experience significant variations, where q
 n
and q
-n
are the measurement feedback the controller receives at the sampling instant, whose length of delay is n sampling periods. This is reasonable with appropriate choice of sampling time and prediction horizon in high servo rates.
S2 is also plausible since there shouldn be aggressive variation in controller output. Moreover, the analysis on dynamic performance shows that the controller is robust even in the presence of errors caused by these two simplifications.
Remark 2: S3 and S4 are in fact similar to S1 and S2. A1 is grounded since the control system can practically identify when a measurement is conducted and when the x

C q

[ I where,
  


0 0
0 E




].
 z
T

T

T
(12)
(13)
E





  T ,
1 l i

0


T
1
/ l i
 
0
T m

/ l m





The predictive tracking error
~ n
can be denoted as:
~ n
 x n
 r n
(14)
3879

Res. J. Appl. Sci. Eng. Technol., 4(20): 3875-3884, 2012 in which r n
refers to reference desired trajectory and the predictive pseudo generalized coordinate x n
= C.q
n
. Since in predictive control the model’s inaccuracy and unmodeled dynamics may significantly influence the controller’s performance, therefore feedback should be incorporated to correct the model for the uncertainties.
Definition 2: Let the initial tracking error d = r define corrected predictive tracking error d = r
0
0
!
x
0
and
!
x in the following form:
 n
 x n
 d (15)
Definition 3: Note that
 n
 x n
 d
 x n
 r n
 d
 x n

( r n
 d ) (16)
Define the corrected reference trajectory r  n
as:
 n
 r n
 d (17)
The corrected tracking error can be denoted as follows: e  n
 x n
 r  n
(18)
For deduction of the predictive control law, consider the following cost function to be optimized as performance index on the prediction horizon:
J
 i
H p 
 

1 e Pe i
(19) where, H p
refers to the prediction horizon and P is the weighting matrix to obtain satisfying performance.
Note that J is a positive definite quadric form of u, the minimum of J could be reached as long as the derivatives of J with respect to u equal to zero. Therefore the optimization problem of J is reduced to solving the following linear equation with unique solution:

J
 u i

0 , i

, ,..., H p
(20)
Remark 4: Since (20) is a linear equation, its solution is exact and unique. However, (6) indicates that the manipulator’s predictive behavior has to be determined over a number of time periods depending on prediction horizon every sampling instant. In addition, solving (20) involves calculating the inversion of the linear equation’s coefficient matrix whose sizes increase with the prediction horizon and the inertia matrix M(q) has to be inverted as a whole. The process involves a lot of computation while one has to choose small sampling time to guarantee the satisfying performance. Therefore the implementation of this control strategy greatly relies on the performance of the control computer and it is necessary to propose some techniques for high computation efficiency for online optimization.
ANALYTICAL SOLUTION FOR
OPTIMIZATION
Lemma 1: There exists analytical expression for
J
1 optimization of the performance index (19):
in
J

Hp 
  i

1 e Pe i
.
Proof: According to (20), to minimize the cost function is equivalent to solve the following linear equations: k
H
 p

1

1
(
0
)
T
S
2


M q T k
M 0 ( q 0 )

T s 


 j k


1

 k
 
2
 
2



 j u j
[ (
0
 kT q 
0


1
2 k h q q
0
))



  r  k
]

0
(21)
The inversion of inertia matrix is partitioned as:
M q
0
)

1
 

 m m
11 12 m
21 m
22



(22)
Combining (7), (11), (12) and (22), (21) can be written in the following form:





1
2 



 a
 a
H p
, 1




1


H p










 b
1 b

H p




 a
1 ,

Hp a
H
P
, H p





( m
11
  m
21
) T s
2 where, a
( H p i

 ( H p j )
2 p i ) j
 i i j H p
(23) bi

Hp k


1

 k
 
2

 
1
2
T s
2

 1

2 i



 1
6
H p
( H p

H p

1 )

1
6 i i

)( i

1 )]

1
4
H
2 p
( H p

1 )
2 
1
4
(

1 ) }

(
  m h
2
 m h
  m h
2
)

1
2
( H p i )
2

( z
0




1
2

 
0
)
 i




1
2
H
T s
{
1
6
H p
( H p p
( H p

1 )


H p

1
1
2 i i

1 )


( z
0
)


1
6 i i

)( i

1 )
 
0
)
(24)
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obtained as:

1

2 ( m
11
  m
21
)
Res. J. Appl. Sci. Eng. Technol., 4(20): 3875-3884, 2012
Note that a
1, i
= a
2,i
(i = 2, 3, …, H c
), without calculating the inversion of coefficient matrix,
J
1
can be b
1
 b
2
T s
2
( a
 a )
(25)
Since the main interest is focused on
J
1
and its analytical solution can be expressed as (25), no additional calculation is necessary for minimization of the cost function. With these simplifications and deductions, a standard on-line numerical solver for quadratic programming is no longer needed and high computational efficiency can be guaranteed.
Remark 5: Note that (m
= [I
'
] and ||C|| is bounded; by virtue of P1, ||M|| is also bounded, therefore||(M
11
11
+
' m
+
' m
21
)
21
-1
)
!
1 = (C. M)
!
1 , where C
|| is bounded. In addition, a
1,1
!
a
2,1
is a positive integer under any circumstances.
Therefore the optimal control signal
J
1
can be guaranteed to be finite and thus the control law is practical.
Remark 6: Note that M(q) is a positive definite symmetric partitioned matrix, relevant theorems about the inversion of partitioned matrix can be applied to further reduce the calculation involved in getting M(q)’s inversion.
Summary of the algorithm:
Algorithm 1: The basic procedure of the algorithm proposed in previous sections can be summarized as follows:
Step 1 (Initialization): Calculate
' with (13), which can be done offline.
Step 2 (Feedback correction): With the length of the feedback lag t
2
-t
1
, estimate the model’s current state with
(7) based on the delayed feedback q
-n
and calculate the corrected reference trajectory r
 k
with (17).
Step 3 (Receding optimization): Calculate the partitioned inverse matrices m
11
, m
12
, m
21
and m
22
with
(22), the coefficients a
1,1
, a
2, 1
, b
1
and b insert
'
, a
1, 1
, a
2, 1
, b
1 control signal
J
1
.
and b
2
2
with (24) and
into (25) to obtain the optimal
Step 4 (Controller output): Hold the controller output
J
1 for one sampling period and then get the next feedback measurement result.
Step 5: Go to Step 2 until the process is done.
Remark 7: In this algorithm the amount of necessary calculation involved is proper since no normal numerical optimization is needed. In this case calculation is no longer a burden for the algorithm and the sampling time can be set small enough to ensure the stability and accuracy, which will be analyzed in the following section.
Dynamics performance: In convenience for analysis of stability the following assumption is made:
Assumption 2: The desired reference trajectory is assumed to be available as bounded functions of time in terms of the generalized coordinate. That is, there exist three positive constants r
0
, r
1
and r
2
so that the following inequalities hold: r   r
0
, r   r and r   r
2
(26)
Note that  ( )

( )

( )
 x t
0
 r t
0
, combining
(4) and (12) the dynamics equation in terms of tracking error 
can be expressed as follows:
 ( )

CM

1 ( )(
 h q q
 r t

(  ( )
 r t
0
(27)
Inserting (25) into (27), using (4) and (14) gives the error dynamics:
T e s
 
[ H p
( H p

1 )

1 ] T e s
 
( 2 H p

1 ) e 

T s
2


1
6
H p
( H p

)( H p

1 )

1


( r   
CM h
0 0
)

D t
(28) where, M
0
= M(q
0
), h
0
= h ( 
0
,
0
) and D(t) is the perturbed term.
Theorem 1: As for the flexible manipulator model (2), use (7), (13), (17), (22), (24) and (25), which makes up the control law discussed in previous sections, to control the manipulator. If
 
 
 
(28) is bounded. Particularly, if the prediction horizon H p
= 1 and the perturbed term D(t) = 0, the tracking error goes to zero as t 64 whatever the initial condition is.
Proof: Consider the following Lyapunov function candidate:
V e
 e e

( 2 H p

1 ) e e (29)
Function (29) is positive definite since H p
0
N
+
. Its time derivative is then given as:

( )
  s
2



1
6
H p
( H p

H p

1 )

1 ( r  
CM h
0
)


( )}
 s
[ p
( H p

1 )

1 ]
 
(30)
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positive real k p
so that for all the following inequality holds true: that ( )
  p s
2  ( ) . The time derivative of V(e) is then bounded as:

( )

T s
2

Thus, if
 
, there also exists k p
 

ú
+
such

1
H p
( H


6 p
( H p p

H p

1 )

1

 k p
  p

2
) e 
2 setting a small enough sampling time T s t.
|
By virtue of P1-P3, A2 and noting (26) there exists a r  
CM h
Since
T s
2





 


1
6
0
H p p

}
( k e
H p


T H s
C
H
C p
(
Hp
H
Res. J. Appl. Sci. Eng. Technol., 4(20): 3875-3884, 2012 p

1 )

1


 
)]
In particular, with D(t) = 0 and H p


 
In this case the origin is the globally asymptotically stable equilibrium point and the tracking error zero with t
64
.
e 
goes to
Remark 8: It should be noted that the perturbed term D(t) also incorporates the errors introduced by the model uncertainties. Therefore T1 proves the proposed controller robustness even if there exists uncertainties in the model.
SIMULATIONS k p becomes the equilibrium point of the (28) and the time derivative of the Lyapunov function is:
(31)
(32)
(33) dynamics are determined by the sampling time T s
and the prediction horizon H p
. It should be noted that through
(33) can always be satisfied and the error bound becomes small for large
= 1, the origin
To demonstrate the performance of the proposed control scheme, a one-link flexible manipulator, as a special case of the manipulator modeled in previous sections, is investigated here as an example. Two assumed modes are supposed to be sufficient to specify the flexible link’s vibration. Simulation parameters are taken from
Kalyoncu’s (Kalyoncu, 2008) work and listed in Table 1.
Since the controller proposed in this study makes improvements on Shuai’s control scheme (Shuai et al.
,
Table 1: Simulation parameters for the one-link flexible manipulator and the contorller
Parameter
Link density
Cross sectional area
Moment of inertia
Joint’s rotational inertia
Sampling time
Modulus of elasticity
Payload
Initial length of link
Gravity acceleration
Value
2729.5 kg/m
3
0.001471 m
2
1.14197×10
!
8 m 4
0.5 kg.m
2
0.001s
6.626×10 10 N/m 2
0.2 kg
1m
9.81m/s
2
Table 2: Probability distribution of stochastic measuring lag
Measuring lag 1T
5
Probability 0.1
2T
0.1
5
3T
0.2
5
4T
0.3
5
5T
0.2
5
6T
5
0.1
0.7
0.6
Reference
Proposed controller
Shuai’s controller
0.5
0.4
0.3
0.2
0.1
0
0 0.2
0.4
0.6 0.8
1.0 1.2
1.4
1.6 1.8
2.0
Time (sec)
Fig. 4: Actual trajectory of pseudo angle
1.04
1.02
1.00
0.08
0.06
0.04
0.02
Reference
Proposed controller
Shuai’s controller
0
0 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (sec)
Fig. 5: Actual trajectory of length's variation
Reference
Shuai’s controller
25
20
15
10
5
0
-5
-10
-15
0 0.2
0.4
0.6 0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (sec)
Fig. 6: Actual output of the force F
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Res. J. Appl. Sci. Eng. Technol., 4(20): 3875-3884, 2012
Reference
Shuai’s controller
30
20
10
0
-10
-20
-30
0 0.2
0.4
0.6 0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (sec)
Fig. 7: Actual output of the torque T
Reference
Shuai’s controller
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
0
0 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (sec)
Fig. 8: Elastic deformation of the end-point
Table 3: Performance index of different controllers
Performance
Shuai’s
Ours
Performance
Shuai’s
Ours
Maximum tracking error for
>
1
0.0360(rad)
0.0091(rad)
Maximum tracking error for u
1
0.0092m
0.0030m
Standard deviation of tracking error for
0.03103(rad)
0.03103(rad)
Standard deviation
0.0022m
0.0011m
> of tracking error for u
1
1
2010) and Shuai’s control technique is the most recently reported predictive control technique on flexible manipulator, to demonstrate the proposed controller’s advantages, his control scheme and the controller designed in this study are applied to control the flexible link’s end-point to track the same desired trajectory for comparison. The situation where there is measurement delay on feedback channel is investigated to demonstrate the effectiveness of the proposed controller since the proposed controller is specially designed to consider the feedback delay. Assume there is stochastic measuring lag in feedback channel and the probability distribution of the measurement lag length is shown in Table 2. As for the proposed controller, the prediction horizon H p
= 5 and the weighting matrix P = diag[1,1,1,1]. Numerical simulation results are shown from Fig. 4 to 8 and certain performance indices for the two controllers are listed in
Table 3. The situation of serious lag is also investigated and the controllers’ performance is shown in Fig. 9 and
10.
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0.5
0.4
0.3
0.2
0.1
0
0
1
0.9
0.8
0.7
0.6
Shuai's controller
0.2
0.4
0.6
0.8
1
Time (sec)
1.2
1. 4 1.6
1.8
2
Fig. 9: Shuai’s actual trajectory with serious lag
Proposed controller
0.6
0.5
0.4
0.3
0.50
0.45
0.2
0.40
0.1
0.7
0.8 0.9 1.0
0
0 0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
Time (sec)
Fig. 10: Actual trajectory with serious lag
Figure 4 and 5 show the actual end-point motion trajectory under the control of different controllers. It can be seen that the motion trajectory controlled with the controller proposed in this study is almost identical to the reference trajectory with stochastic measuring delay.
However, because Shuai’s controller fails to take the lag into account, its performance is not satisfying and there is obvious tracking error in his controller’s output trajectory.
Besides, since the controller proposed in this study takes a relatively longer prediction horizon while Shuai’s controller is limited with a single-step one, its control signal output is smoother than Shuai’s because it takes more model dynamics into account. Figure 6 and 7 demonstrate this point. Figure 8 illustrates the vibration of end-point with different controllers.
The actual trajectory with Shuai’s controller even cannot maintain stable with serious measuring lag on feedback channel. Figure 9 and 10 illustrate the situation with a spell of serious measuring lag on feedback channel: in addition to the stochastic delay in Table 2, from 0.80 to
0.82 s, the length measuring lag is set to be 0.2 s, which is much longer than common stochastic lag, to test the controllers’ performance. Shuai’s controller goes unstable under this circumstance as shown in Fig. 9. However, as

Res. J. Appl. Sci. Eng. Technol., 4(20): 3875-3884, 2012 for the controller proposed in this study, although tracking error occurs temporarily, it disappears after the serious lag and the controller’s overall performance is satisfying.
Figure 9 and 10 demonstrate the proposed controller’s robustness since it can better deal with uncertainties in model than Shuai’s controller does.
From Fig. 4 to 8 it shows that Shuai’s controller is not as competitive as that proposed in this study with stochastic feedback measuring delay. Figure 9 demonstrates Shuai’s controller’ weak robustness because it goes unstable when the measuring lag is serious. Since the control scheme proposed in this study can both better track reference trajectory and compensate for model uncertainties, it is superior to Shuai’s controller.
CONCLUSION
This study presents a predictive control scheme to control a flexible manipulator’s end-effector along the desired trajectory with measuring lag on its feedback channel. An approximate linear model is reduced from the complex nonlinear dynamics equation. Then a prediction strategy based on the approximate model is designed to predict the robot future behavior with delayed measurement feedback with a recursive equation and the general form of the equation is also obtained. In addition, proper simplifications are made and relevant techniques are proposed to achieve high computation efficiency for the online optimization needed in the predictive control scheme. Theoretical analysis proves the boundness and stability convergence of the end-point tracking. At last, a simulation comparison demonstrates the effectiveness and robustness of the control scheme in this study.
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