Parameter Identification and Vibration Control in Modular ... - CiteSeerX

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The controller functionality was validated with a model of the vehicle dynamics, similar to [5], using a Simulink environment before tests in the real vehicle were tried (see Section VI). The test of the controller software was done in a two-step procedure. For a first test of the implemented C code modules, they were substituted for the original Simulink modules in the simulation environment so that their output could be directly compared with that of the original simulation modules. As a second test, the C code modules were run in their final embedded environment together with vehicle and motor control on the target MCU hardware in an off-line mode: The responses of the controller to artificially excited inputs were recorded and were again compared with the simulation results. After that preparation, the commissioning of the controller in the vehicle was performed without problems in a very short time because functional faults and coding bugs were already eliminated.

[4] G. Brandenburg, “Einfluß und Kompensation von Lose und Coulombscher Reibung bei einem drehzahl- und lagegeregelten, elastischen Zweimassensystem,” Automatisierungstechnik at (37), H. 1 and H. 3, pp. 23–31, 1989. [5] H.-P. Willumeit, Modelle und Modellierungsverfahren in der Fahrzeugdynamik Teubner, Stuttgart, Leipzig, Germany, 1998. [6] H.-P. Beck, “Aktive Schwingungsdämpfung in Triebachsen mittels Zustandsregelung,” Elektrische Bahnen, pp. 393–401, Dec. 2000. [7] F. Schütte, “Automatisierte Reglerinbetriebnahme für Elektrische Antriebe mit schwingungsf higer mechanik,” Ph.D. dissertation, Univ. Paderborn, Paderborn, Germany, 2002. [8] S. Richard, P. Chevrel, and B. Maillard, “Active control of future vehicles drivelines,” in Proc. 38th IEEE Conf. Decision and Control, vol. 4, Phoenix, AZ, 1999, pp. 3752–3757. [9] S. N. Vukosavic´ and M. R. Stojic´, “Suppression of Torsional oscillations in a high-performance speed servo drive,” IEEE Trans. Ind. Electron., vol. 45, pp. 108–117, Feb. 1998. [10] K. Sugiura and Y. Hori, “Vibration suppression in 2- and 3-mass system based on the feedback of imperfect derivative of the estimated Torsional torque,” IEEE Trans. Ind. Electron., vol. 43, pp. 56–64, Feb. 1996.

VI. VEHICLE TEST RESULTS Fig. 7 shows measurements without the oscillation control. The measurements are taken on a test track with moderate speed. The driver accelerates unevenly which excites the drive train oscillations as it can particularly be seen from the motor speed !1 and the wheel torque Tload . The oscillations are poorly damped. After each excitation it takes a few seconds for fading away. It is evident that these oscillations cause high mechanical stress and are felt as shuddering by the driver. The large difference between wheel torque Tload and motor torque Tmot in the last few seconds of the plot is due to the mechanical braking. Now, Fig. 8 shows a situation with the oscillation control turned on. Because the accelerator pedal was manually operated by the driver, the situation is not exactly the same as in Fig. 7, but fairly similar. The oscillations are just about gone. The torque and the motor speed are very smooth. Only when the torque demand crosses zero so that the gear backlash is traversed, a small flicker can be seen in the motor speed. The figures show very impressively the effectiveness of the proposed control. Obviously, the control can also cope well with the backlash nonlinearity, which is an advantage compared with other linear controllers.

Parameter Identification and Vibration Control in Modular Manipulators Yangmin Li, Yugang Liu, Xiaoping Liu, and Zhaoyang Peng Abstract—The joint parameters of modular manipulators are prerequisite data for effective dynamic control. A method for identifying these parameters using fuzzy logic was devised to study modular redundant robots. Experimental modal analysis and finite element modeling were exploited to model the dynamics. The joint parameters of a nine degrees-of-freedom (9-DOF) modular robot have been identified. In addition, active vibration control based on a neural network and a genetic algorithm were investigated. Ideal control simulation results for a reduced dynamic model of the 9-DOF modular robot were then derived. Index Terms—Genetic algorithm, modular robot, neural network, parameter identification, vibration control.

VII. CONCLUSION

I. INTRODUCTION

The proposed active damping control scheme can very effectively reduce the drive train oscillations. In comparison with other linear control structures, the controller copes with nonlinear backlash effects that are included in the torque observer. Mechanical stress is diminished and riding comfort is increased. The control algorithm is implemented in the existing motor control unit without any additional hardware cost. The algorithmic effort is moderate. Damping of a mechanical system by control electronics is a good example of mechatronic design, where a well designed system turns out a better performance than it was initially intended by the design of the components.

The development of a robot requires that it be able to adopt as many configurations as possible using limited links, so as to allow the construction of new types of robot without redesign and remanufacturing. The modular robot concept can be traced back to the 1970s [1]. Along with the development of modular robots, the current trend is toward the design of self-organizing, self-reconfigurable, self-assembling, and self-repairable modular robots [2], [3]. A snakelike Polybot was developed several years ago, which has had wide application in pipe exploration and biomedical engineering [4]. A hormone-inspired control strategy was proposed to control the CONRO self-reconfigurable robots that form different types of robots to suit various tasks [5]. Dynamics and control are becoming the next key issues after design. A robot’s joint parameters (stiffness and

REFERENCES [1] A. Lohner, “Aktive Dämpfung von niederfrequenten Torsionsschwingungen in momentengeregelten Antrieben,” VDI Berichte Nr. 1533, pp. 307–322, 2000. [2] M. Menne, “Drehschwingungen im Antriebsstrang von Elektrostraßenfahrzeugen,” Ph.D. dissertation, Aachen University of Technology, Wissenschaftsverlag, Mainz, Germany, 2001. [3] M. Menne and O. Bitsche, “Comparison of drivetrain-oscillation damping-algorithms for electrical vehicles,” EVS 18, Berlin, Germany, 2001.

Manuscript received October 2, 2002; revised July 14, 2003. This work was supported by the Research Committee of the University of Macau, Macau, P.R.C., under Grants RG025/00-01S/LYM/FST and RG037/01-Macao S.A.R. 02S/LYM/FST. Y. Li and Y. Liu are with the Faculty of Science and Technology, University of Macau, Macao S.A.R. (e-mail: [email protected]). X. Liu and Z. Peng are with the Automation School, Beijing University of Posts and Telecommunications, Beijing 100876, China. Digital Object Identifier 10.1109/TMECH.2004.839035

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damping) are essential for establishing an accurate dynamic model for a modular robot. Identifying the physical parameters of a modular robot is much more complicated than for a general industrial robot with few joints [6]. Yet parameter identification is a prerequisite step in accurate dynamic modeling and effective control of modular robots [7], [8]. Researchers pay more attention to control issues with robots moving over a large range. Few studies have focused on vibration control in robots moving over a small range. A modular robot is composed of many modules connected via joints, and the dynamic characteristics of the elastic joints have an important influence on the behavior of the whole robot. This is because there are more joints in a modular robot than in a general robot. The modules are rigid, while the joint connections are elastic and weak. Robot vibration will not only induce errors in the robot’s operational accuracy and repeatability, but also speed up the wear of its parts. If resonance occurs, the robot can not be controlled to perform motions over a large range unless the vibration is controlled. Therefore, more research work on robot vibration control is required [9]. This paper addresses two issues for modular manipulators: how to identify joint parameters using the finite element modeling and the experimental modal analysis technique, and how to realize vibration control based on the identified parameters. A new active vibration control method is presented combining genetic algorithm (GA) and back propagation (BP) neural networks (NN) with suboptimal control. Controllers are simulated using a simplified nine degrees-of-freedom (9-DOF) modular robot and the reduction of degrees of freedom method.

701

Fig. 1.

A 9-DOF modular robot diagram.

fx_ g and fxg are generalized acceleration, velocity and displacement vectors, respectively, in [n 2 1]. ff g is the generalized force vector in [n 2 1]. n is the number of generalized coordinates. Equation (1) then yields the eigenvalues and eigenvectors of the robot.

II. EXPERIMENTAL AND MODELING METHODS

III. IDENTIFICATION METHOD

A. Experimental Setup and Results A 9-DOF modular redundant robot is composed of ten modules including wrist, finger, and general rotation modules, as shown in Fig. 1. The joint parameters for such modular robots have previously been specified in general using fuzzy logic theory and a genetic algorithm. The experimental setup, methods, and results in details can be found in [7]. B. Dynamic Modeling Although the external shape of a 9-DOF robot’s modules is simple, their internal parts are complicated. The representation of each module can be simplified as a thin shell with a centralized mass inside. Then dynamic model for each module can then easily be obtained using the finite element method (FEM). A dynamic model of the entire robot can then be assembled from the models of all the modules using the substructure synthesis method. The joints of the robot are quite complicated. In modeling the robot, they can be substituted by elements between the finite link points in the joints with equivalent stiffness and damping in the X , Y , and Z directions. To link the results of any experimental modal analysis with FEM results, the measured points in the experiment should correspond to the nodes of the FEM model. The general equation for the whole robot can be expressed as

fg

fg

[M 0 ] x  + [Cj ] x_ + ([K0 ] + [Kj ])

fxg = ff g

A. Parameter Identification Model The method of identifying the joint parameters presented in [7] only considered the first mode. However, the other modes may also affect the identification results. An identification method including all the modes obtained from the experiment can be expected to produce results which will be more reasonable and accurate. Consider the ith mode. Its identification model can be obtained from method of [7] by replacing S with si . For each module, adjust the sequence of DOF for the modal equations, and extract the last row from them. This produces the following (9) (1) (9) T s2i diag ([ Mj(1) Mj(2) . . . Mj 9j 10 ]) [ j 1 j(2) . . . j 9 ]i 1j 2 2j 3 2 (1) (2) (9) (1) (2) T + diag ([ Kj 1 Kj 2 . . . Kj 9 ]) [ j 1 j 2 . . . j(9) 9 ]i (1) (9) + si diag ([ cj 1 cj(2) . . . cj 9 ]) 2 T 1 [ j(1)1 0 j(2)1 j(2)2 0 j(3)2 . . . j(9)9 0 j(10) 9 ]i (1) (9) + diag ([ kj 1 kj(2) . . . kj 9 ]) 2 (1) (2) (2) (3) T (2) 1 [ j1 0 j1 j2 0 j2 . . . j(9)9 0 j(10) 9 ]i = 0:

If Ai and Bi are defined, as in the equation at the bottom of the page, then (2) can be written as

(1)

where [M0 ] and [K0 ] are the mass and stiffness matrices representing by each module in [n 2 n], and [Cj ] and [Kj ] are the joint damping and stiffness matrices representing by each joint surface in [n 2 n]. fx g,

Ai + Bi fKCj gi = 0 where

fKCj gi =

(1) (2) (2) (9) (9) si cj(1) 1 + kj 1 ; si cj 2 + kj 2 ; . . . ; si cj 9 + kj 9

(9) T (9) (1) . . . j 9 ]i j(2) . . . Mj 9j 10 ]) [ j 1 Mj(2) Ai = si2 diag ([ Mj(1) 2 1j 2 2j 3 (9) T (1) (9) (1) . . . j 9 ]i j(2) . . . Kj 9 ]) [ j 1 Kj(2) + diag ([ Kj 1 2 2 (10) (9) (3) (2) (2) (1) Bi = diag [ j 1 0 j 1 j 2 0 j 2 . . . j 9 0 j 9 ]i

(3) T i

.

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Furthermore, fKCj gi can be derived from (3). If Bi01 (0Ai ) is deT fined as Bi01 (0Ai ) = ai1 ; ai2 ; . . . ; ai9 , then (3) becomes

fKCj gi = Bi0 (0Ai ) = ai ; ai ; . . . ; ai 1

1

2

T

9

:

(4)

The identified parameters can be varied using different order measured joint modal parameters. The larger the relative vibrational displacement, also the higher the signal-to-noise ratio (SNR) in the experiment, the higher the precision of the identified joint parameters. Weighting factors improve the precision, so the formula for joint n in the ith mode becomes (n+1) (jnn) 0 jn

wni =

t

Fig. 2. i

(n) (n+1) jn 0 jn

k=1

Sketch of the sixth joint.

(5) TABLE I IDENTIFIED STIFFNESS FOR THE SIXTH JOINT

k

where t is the order of mode, wni represents the weighting factor for (n) (n+1) joint n in the ith mode, and jn and jn represent the mode shapes of joint n in module n and module n + 1, respectively. Thus

fKCj g =

t

t

w1i ai1 ;

i=1

t

w2i ai2 ; . . . ;

i=1

T

w9i ai9

:

(6)

i=1

Set the real part and the imaginary part on each side of (6) equal to each other, and all the required joint parameters can be identified. B. Identifying Joint Parameters Using Fuzzy Logic In view of the fact that both the modeling and the experiments involve errors of unknown magnitude, applying a fuzzy equation can help to obtain more reasonable results. Define t

f (fKCj g) = fKCj g 0

w1i ai1 ;

i=1

t

w2i ai2 ; . . . ;

i=1

t

T

w9i ai9

:

i=1

(7) Equation (7) can then be expressed in the following fuzzy form:

kf (fKCj g)k  0:

(8)

Its membership function is

(x) = where x =

0x 0  x   0; x> 1



(9)

kf (fKCj g)k,  is a constant.

C. Optimization of Joint Parameters The objective function is composed of multiple objects, as follows: min u =



t i=1

(f i

0 fie )

2

+

t i=1

ki 0 ie k

(10)

where fi and fie , i , and ie are the ith order natural frequencies and mode shapes, obtained from the analytical model and the experiment, respectively.  and are weighting coefficients. Select (x) =  , then the constraint condition described in (8) becomes

kf (fKCj g)k = (1 0 ):

(11)

Under this constraint condition, taking (10) as an objective function results in a group of optimization solutions fKCj g. The optimal solution can then be selected from them. IV. ANALYSIS OF THE IDENTIFICATION RESULTS Since the complex frequencies and complex mode shapes were derived by experiment, both the joint stiffness and damping coefficients can, in theory, be identified using the above method, only the joint stiffness using the above method has been discussed. The analytical model of the sixth joint is shown in Fig. 2. The other joints are similar to this one. The sixth joint has four pairs of nodes in the four corners. Each pair is connected by three springs along the X , Y and Z axes. Only the spring along the axis Z is shown in Fig. 2. The stiffness coefficients of these springs are equivalent to the physical stiffness of the joint. In Fig. 2, the number beside the node represents the node number in the FEM. The stiffness in the Z direction represents the tension stiffness, and that in the X and Y direction represents the shear stiffness. The number of equivalent joint nodes is different for different joints, since the joint shape is different. There are eight pairs of nodes in joint 1, six pairs in joint 2, and four pairs of nodes in the other joints. All together, there are 42 pairs of equivalent joint nodes in the analytical model. For each pair of equivalent joint nodes, the stiffness values along the three axes are different from each other. This gives a total of 126 parameters to be identified. The constraint range of the stiffness parameters is defined as [1:0e + 3 1:0e + 9] (KN=m). When  = 1:0e + 3, its membership is 0.6, and the constraint condition is defined by (11). The joint stiffness coefficients identified for the 6th joint are shown in Table I. The results for the other joints are omitted. The natural frequencies of the whole robot can be calculated by substituting the identified joint stiffness coefficients into a dynamic model of the robot. Several low-order natural frequencies are compared with their experimental results in Table II, which demonstrates that the dynamic characteristics of the analytical model are to some extent equivalent to those of the physical model

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TABLE II COMPARISON OF THE IDENTIFIED RESULTS WITH EXPERIMENT

Fig. 4.

Control system diagram.

0M 0 K

=

1

111 111

Fig. 3.

Simplified model of a 9-DOF robot.

V. STRUCTURAL DYNAMICS MODELING Since each module can be modeled using the FEM, a FEM model of the whole robot can be obtained by the substructure synthesis method. In order to simplify the structural dynamics of the robot, the number of DOF of each module can be reduced. The results of the experiment described above show that the flexibility of the robot comes mainly from the joints, and the vibration of the robot is mainly along x direction. The model can therefore focus on DOF in the corners and joints along the x direction, and neglect the other DOFs. The simplified 9-DOF modular robot is shown in Fig. 3. After reduction, the equation for the whole robot becomes

fg fg

[M ] x  + [K ] x = f

(12)

fg

~ ][T ], [K ] = [T ]T [K ~ ][T ], and f where [M ] = [T ]T [M = T [T ] [f~][T ] are the reduced mass matrix, stiffness matrix, and force ~,K ~ , and f~ are the original mass matrix, stiffness vector respectively. M

matrix and force vector. [T ] is the transformation matrix.

0

, B

fg

C. BP Neural Network Controller Design The active vibration controller is composed of a BP NN with a 4-node input layer, a 10-node hidden layer and a 4-node output layer. wih and who denote the connection weights between the input layer and the hidden layer as well as between the hidden layer and the output (i) (i) layer. h and o denote the thresholds of the hidden layer and the output layer, respectively. The outputs of the BP NN are applied to the joints of the model in the form of torques. The relationship between the input and output in each layer of the neural network is defined as c(i) = K

The BP NN algorithm can be applied to address vibration control for vertical gestures of the simplified model of the robot. Let x1 = fx(1) x(2) . . . x(n) gT and x2 = fx_ (1) x_ (2) . . . x_ (n) gT be the displacement vector and the velocity vector of the nodes in the X direction. Equation (12) can then be written as

vo(i) =

0

M

x_ 1 x_ 2

=

0

0K

I 0

x1 x2

+

0

I

ff g:

(j )

s_ = As + Bu

(14)

y = Cs

(15)

(j ) vh =

2y i

( ) NN ;

j 10 j =1

yh = a

(13)

Define s = fx1 x2 gT as the state vector. The state equation of the simplified robot model can then be derived

fg

fg

A control system block diagram is shown in Fig. 4. Here, y denotes the state value measured by the sensors, which includes both displacement and velocity. yNN denotes the output of the neural network. yd denotes the desired state value, which should be a zero vector in this research. The state values measured by the sensors are used as the inputs to the NN, while the weights of the NN algorithm are adjusted via the difference between the practical state and the desired response. c = fc1 c2 c3 c4 gT is the controlled moment vector. K is the coefficient of the output amplifier.

A. State Equation of the Simplified Model of the Robot

0

0

B. Control System Structure

VI. ACTIVE VIBRATION CONTROL METHOD

I

=

f g

fg

of the robot. This shows that the joint parameter identification method presented in this paper is reasonable and effective.

fg

I

0

, and C = M 01 diag([0 1 0]) are the coefficient matrices of the state equation and the output equation. The number “1” denotes the positions of the sensors in the control system. u = f = f e + f c is a vector of force, in which fe represents the external force vector. f c represents the control force vector.

where A

(i) yNN = a

(j ) (ij ) yh who

0 exp 0bvoi i 1 + exp 0bvo 1

( ) ( )

0 oi

( )

0 exp 0bvhj j 1 + exp 0bvh

( )

1

m4

( )

(m) (jm) yin wih

m=1 yi yj (m) yin = yn+i yn+j

0 hj

( )

(m = 1) (m = 2) (m = 3) (m = 4).

(16)

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(j )

(i)

vh and vo represent the input neurons of the hidden layer and the output layer, respectively. Two constants are selected as a = 1:716 and b = 2=3.

(ij )

The state equation and the output equation of the discrete-time system corresponding to the (14) and (15) are s(k + 1) y (k )

= G 1 s (k ) + H 1 u (k ) = C 1 s (k )

5w 5

(17) (18)

5w

where G = eA1t , and H = 0 eA1t dt 1 B = A01 (eA1t 0 E )B . E is a unit matrix, t is the sampling time, and A, B , and C are constant coefficient matrices. For a robot with revolute joints, the forces can be converted to equivalent torques in order to apply the model. Substituting torques in stead of forces into (17) t

s(k + 1)

= G 1 s(k) + H 1 D 1 fo (k) + c (k)g

= y (k ) 0 y d = y (k ):

(k )

(20)

J

1 T = y (k ) 1 y (k ) + 2 k=0

0 1 T  c (k ) 1  c (k ): 2

x(N )

1 H 1 D 1 c(0) = G 1 x(0) + G N 02 1 H 1 D 1 c (1) + 1 1 1 + H 1 D 1 c (N 0 1) +G N 1

then @x(N ) @c (k )

= GN 010k 1 H 1 D (k = 0; 1; 2 . . . N 0 1):

= Gi010k 1 H 1 D (i = 1; 2; . . . ; N; k = 0; 1; . . . ; i 0 1): (22)

Therefore @y (i) @c (k )

= C 1 Gi010k 1 H 1 D:

(23)

Considering (18), (21) and (23) together yields

0 1) = y (N ) 1 C 1 H 1 D + c (N 0 1) 111 @J = y T (N ) 1 C 1 G N 0 1 H 1 D @c (0) + yT (N 0 1) 1 C 1 GN 0 1 H 1 D + 1 1 1 + yT (1) 1 C 1 H 1 D + c (0):

@J @c (N



i 4

T

(i)

J

=



0 y i 1 yhj

(i)

1

a

0 y i 1 (01)

b

1

a + yNN

2a

2

1 K 1 4ba 1 2

T

(i)

(

2

2

( )

( ) NN

(i)

(j )

( ) ho

1 K 1 4ba 1

( ) NN

a + yNN

1 a 0 y i w ij 1 1 a 0 yhj 1 yinm

i 4 i=1

a

a + yNN

a + yh

)

(i)

a + yNN

1 a 0 y i w ij 1 1 a 0 yhj 1 (01)

(j )

( ) ho

a + yh

( )

(i)

where T (i) = @J=@c (k) can be obtained from (24). The adjustment formula for the NN parameters can be obtained as follows: (ij )

who

(ij ) (k) = who (k 0 1) 0  5w

= o (k 0 1) 0  5 (i)

wih

(i)

J; o

(k )

J

(k) = wih (k 0 1) 0  5w

(j )

(jm)

= h(j ) (k 0 1) 0  5

J; h

(k )

J

(25)

D. Initialization of Weights and Thresholds in the Neural Network via Genetic Algorithms In this study, a floating-point array vector was taken as the decisionmaking variable, whose values were the weights and thresholds of the BP network. The group size was selected as 200, and the objective function was J

Similarly @x(i) @c (k )

=

1

1

where the learning rate is  = 1 2 1003 .

0

N

J

(i)

b

2a

( ) NN

(21)

= G 1 x(0) + H 1 D 1 c (0)

111

= T (i) 1 K 1

(jm)

Since x(1)

J

i=1

5

N 1 k=0

= T (i) 1 K 1

( )

In order to reduce the vibration validly and economically, combining with the suboptimal control method, the objective function for the NN controller can be defined as N

J

( ) NN

(19)

where s(k) and s(k + 1) are the state values in the k th and (k + 1)th sampling instants, respectively. c (k) and o (k) are the control torques and external forces acting on the robot after the k th sampling. D is a coefficient matrix. If the desired output is zero (without vibration), then the errors are

(jm)

The adjustable parameters of the NN controller are who , wih , (j ) (i) o , and h . The gradients of the output c with respect to these parameters are (j )

T

1

2

(24)

=

1 T y (N ) 1 y (N ) + 2

0 1

N 1 k=0

2

y

T

(k) 1 y(k) + cT

1 c (k )

: (26)

The fitness function was taken as the reciprocal of the objective function in order to keep it consistent with the fittest survival theory, that is tness = (1=J ). In addition, the rate for an individual selected is proportional to its fitness, in order to decide which individual’s characteristics will be passed down to the next generation. Moreover, an Elitist Model was used in this study. In the case of the best individual created in the last generation being discarded, the worst individual in this generation was replaced with the best current individual. A crossover operator can be used to create new individuals in genetic algorithms. With respect to the real-coded optimization, a unimodal normal distribution crossover (UNDX) proved to be effective [10]. The UNDX generates two offspring using the normal distribution defined by three parents, details of which can be found in [10]. The crossover rate was selected as pm = 0:6. With respect to real-coded optimization, Gaussian mutation is more effective than those normally used in GA. Gaussian mutation is mainly used in

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Fig. 7. Controlling results using the BP neural network with genetic algorithms and suboptimal controller. Fig. 5. Fitness evolution results.

VIII. CONCLUSION A 9-DOF modular robot was subjected to experimental modal analysis, and the dynamic behavior of the robot was investigated. The robot’s joint stiffness parameters were identified by means of the method presented, which provides a necessary condition for the modeling of the whole robot. In addition to this, the 9-DOF modular redundant robot was modeled using the FEM and the joint stiffness parameters identified by the modeling of each module and the experimental modal values. A reduction was made in the model of the whole robot to simplify calculations. A controller was designed by combining the generic algorithm with a neural network with the suboptimal control. It was shown that vibration control in the simplified model of the robot can be realized through enough training of the controller. The neural network and genetic algorithm based suboptimal controller proposed by this study is more robust and converges more easily than the classic BP neural network controller.

Fig. 6. Controlling results of the simplified model of the robot using the classical BP network controller.

evolutionary strategies, and it was exploited here to improve the mutation operator. The mutation was performed independently on each gene of the selected individual by adding a random value with expectation zero. Due to page limitations, we will not discuss the Gaussian mutation in detail. The mutation rate selected in this paper is pm = 0:001. A fitness versus generation curve resulting from this process is shown in Fig. 5.

VII. SIMULATION RESULTS The state feedback points are the sensor locations, as shown in Fig. 3. The Wilson- method was used to calculate the response in the time domain. The sampling interval was 100 s and the number of sampling points was 200. The weights and thresholds were set to random numbers in the zone [01 1]. The control results using the classical BP NN are shown in Fig. 6. Clearly, the system was divergent and unsteady over 20 training runs. However, the controller developed in this study is more steady and converges more easily, as shown in Fig. 7. It overcomes the problem of local optimization effectively, and is more appropriate for vibration control than the classical BP NN.

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