Delta Robot Control Using Single Neuron PID Algorithms Based on Recurrent Fuzzy Neural Network Identifiers

International Journal of Mechanical Engineering and Robotics Research Vol. 9, No. 10, October 2020

Le Minh Thanh1, Luong Hoai Thuong1, Phan Thanh Loc2, Chi-Ngon Nguyen2 1 Vinh Long University of Technical Education, Vietnam 2 Can Tho University, Vietnam Email: [email protected], [email protected], [email protected], [email protected]

Abstract—Parallel robot control is a topic that many implemented. For example, a robot proposed by Stewart researchers are still developing. This paper presents an with two platforms ensures fixed stability in a stationary application of single neuron PID controllers based on facility [3]. In 1985, a delta robot was developed and recurrent fuzzy neural network identifiers, to control the built in the Ecole Polytechnique Federale de Lausanne trajectory tracking for a 3-DOF Delta robot. Each robot (EPFL) called delta robots focused on industrial work [4]. arm needs a controller and an identifier. The proposed controller is the PID organized as a linear neuron, that the Based on this robot, the new architecture is implemented neuron’s weights corresponding to Kp, Kd and Ki of the according to the necessary characteristics in industry and PID can be updated online during control process. That school. For example, it is a robot with high accuracy but training algorithm needs an information on the object's slow motion is widely used in 3D printers [5]. In the sensitivity, called Jacobian information. The proposed industrial sector the need to optimize production has been identifier is a recurrent fuzzy neural network using to a major challenge for robotics companies since the 1980. estimate the Jacobian information for updating the weights Delta robots have been successfully researched and of the PID neuron. Simulation results on MATLAB/ manufactured in many countries. However, the high cost Simulink show that the response of the proposed algorithm and operational control of delta robots has always been an is better than using traditional PID controllers, with the setting time is about 0.3 ± 0.1 (s) and the steady-state error interesting topic for many innovative studies. is eliminated.  So that, the neural networks and fuzzy logic are applied for improving adaptive PID controllers for Delta Index Terms—delta robot, single neural PID, recurrent robot [6]. A non-parametric identifier of each robot arm fuzzy neural network, trajectory tracking using recurrent fuzzy neural network is built and trained online during control to estimate the object's sensitivity to Symbol Unit Meaning the input signal, also known as Jacobian information.

1,,  2  3 Degrees Angle of the upper leg of robot Based on the Jacobian information, a single-neuron- R mm upper disc radius adaptive PID controller will be updated online with three R mm lower disc radius connected weighs respectively three parameters Kp, Ki L1 mm upper arm length and Kd of the controller. Thus, with this principle, the L2 mm lower arm length PID controller will be automatically adapted due to the variation of the robot that classical control solutions can Abbreviation not achieve. DOF Degrees of freedom In the process of making efforts for manufacturing PID Proportional Integral Derivative delta robots to meet the industrial needs, this paper aims DC Direct current to control and conduct analysis, comparison, and RFNN Recurrent fuzzy neural network evaluation of different algorithms. The comparison and evaluation of efficiencies between traditional PID I. INTRODUCTION controllers and fuzzy-neural based PID controllers are implemented and tested in MATLAB/Simulink which the With flexible mechanisms, advantages of speed and absolute error values are used to evaluate the force, and precision delta robots have become popular performances of the closed loop system. and widely used in industry [1]. The complex structure of this robot makes them an interesting in research focus. II. DELTA ROBOT Delta robots were proposed in 1939, when Pollard built a robot to control the position of a spray gun [2]. In this A. Parallel Robot Model context, other robots with the same structure have been The model of delta robot is presented in Fig. 1 [7]-[9].

Manuscript received January 16, 2020; revised August 15, 2020. In this model, BiDi stitching is modeled into two material Corresponding author: Chi-Ngon Nguyen points located at Bi and Di. Each of them has a weight mb

and is connected by rigid, weightless rods. Thus, the cos sin  0Rl cos  1 1   1 1  dynamic model of this model consists of 4 solid objects,     rB sin11 cos 0 0 in which the AiBi (i = 1, 2, 3) stitches move around the 1    

0 0 1  l11 sin  axes perpendicular to the OAiBi plane. At Ai with mass (6) m and remaining solid objects (including three points Rlcos cos  cos  1 1 1 1 1 mounted at Di) with mass (mp+3mb), can be considered as Rlsin sin  cos  a moving table of linear motion. Three points located at 1 1 1 1 l sin Bi having mass mp, where mp is the operation mass. 11

The coordinates of vector rD1 are calculated as follows: r r u D11 P PD

xP   cos11 sin0   r  y   sin cos 00    (7) P  11    zP  0 0 1   0 

xP  cos1 r y sin r P 1 zP Combining equations (6) and (7) we have:

cos1 R r  l 1 cos  1 cos  1  xP Figure 1. Parallel robot model 3RUS [9].  rr sin R  r  l sin  cos   y (8) DBP111  1 1 1  The set of robot actuating includes: l11 sin zP

q 1  2  3 xPPP y z . Substituting (8) into (1), we get the equation linked to the first pin. Similarly, with the second and third legs, we B. Establishing the Linking Equations get the link equation of the robot as follows:

2 From Fig. 1, we have the linking equation for points B1 2 f1 = l 2 - cosα 1 R-r +l 1 cosα 1 cosθ 1 -x P  and D as follows: 1 2 - sinα R-r +l sinα cosθ -y - l sinθ +z2 =0 T  1 1 1 1 P 1 1 P 2 l r  r r  r  0 . (1) 2 2  DBDB1 1  1 1  f2 = l 2 - cosα 2 R-r +l 1 cosα 2 cosθ 2 -x P  (9) 2 where r , r are the positioning vectors of points B and - sinα R-r +l sinα cosθ -y - l sinθ +z2 =0 D1 B1 1  2  1 2 2 P  1 2 P  2 D1 in the Oxyz coordinate system, calculated according to 2 f3 = l 2 - cosα 3 R-r +l 1 cosα 3 cosθ 3 -x P  the vector equation: 2 2 - sinα3 R-r +l 1 sinα 3 cosθ3-y P - l 1 sinθ 3 +z P  =0 r r U . (2) BAAB1 1 1 1 C. The Kinetic Energy and Potential of the Robot In which, the coordinates vector r , U in the A1 AB11 The kinetic energy of the AiBi stages is calculated as coordinate system Ox1y1z1 have the forms: 1122 T TII (10) rR [ 00] Ai B i11 y A i B i y i A1 (3) 22 U AB [ l1 cos 1 0 l 1 sin 1 ] 11 The kinetic energy of mass mb is set at B1 as follow

Replace (3) with (2) and get the rB1 coordinate in 11 T m v2 m l 2 2 (11) Ox1y1z1 coordinate system mbi22 b B b1 i

rB1  R  l1cos 1 0  l 1 sin 1  (4) The kinetic energy of a moving table and mb masses is 11 The cosine matrix indicates the direction of the 2 2 2 2 (12) T3  mP 33 m b v P  m P  m b x P  y P  z P Ox1y1z1 coordinate system with the Oxyz coordinate 22 system as: Combining the kinetic expressions above, we get the

cos11 sin 0 model kinetic expression of the robot: A  sin  cos  0 (5) z  1  1 1 1  T I  m l 2 2  2  2   1yb 1 1 2 3 0 0 1 2  (13)

1 2 2 2 Infer the coordinates of the rB1 vector in the Oxyz  mP 3 m b x P  y P  z P coordinate system: 2 

The potential energy of a robot is calculated as follows: mmp3 b y p 1 21 sin  1R  r  l 1sin  1 cos  1  y p  (22)  gl1 m 1  mb  sin  1  sin  2  sin  3  2 (14) 22 sin  2R  r  l 1sin  2 cos  2  y p  g3 mb m P z P  2s3 in  3R  r  l 1sin  3 cos  3  y p  D. Set the Differential Equation of Motion of the Delta m3 mz   3 m  m g  2 z  l sin Robot  p b p b p 1 p 1 1  (23)

We use the Lagrange factor to set the differential 222zpp  l 1 sin  2   3 z  l 1 sin  3  equation of motion of this model with helium links with the following form: 2 2 l1 cos 1 R  r  l 1 cos  1 cos  1  xp  2 d T T r f sin R  r  l sin  cos   y (24) i (15)  1  1 1 1 p    Qki   k 1,2,  ., m  2 dt qk  q ki1  q k l11 sin  z p   0 where qk is the extrapolation coordinates of the robot, fi is 2 l2  cos R  r  l cos  cos   x the linking equations, Qk is the extrapolation force, i is 2 2 1 2 2 p  the Lagrange factor. With this model, the vector of 2 (25) sin2 R  r  l 1 sin  2 cos  2  y p 6   extrapolation coordinates  R and the number of 2 associated equations is three, so m=6, r= 3. We divide the l12 sin  z p   0 forces acting on the robot into potential forces and forces 2 2 without potential energy, the extrapolation force Qk is l3 cos 3 R  r  l 1 cos  3 cos  3  xp  calculated as follows 2 (26) sin3 R  r  l 1 sin  3 cos  3  y p   QQ  np (16) 2 kk l13 sin  z p   0 qk From the parallel motion equations of robots, we get where Qnp are extrapolation forces corresponding to k the model of robot as shown in Fig. 2 and Fig. 3. forces without potential energy. Sum of extrapolation forces is

A  1  1   2  2   3  3 . (17) So, we have np np np Q11 , Q22 , Q3  3 and Figure 2. Model of Delta robot in Simulink. np Qk  0?with k = 4, 5, 6. Substituting kinetic and potential energies into (15), we get the motion equations of the robot as follows:

2 1 IIy m b l1 1  gl 1 m 1  m b cos  1   1 2  (18) sin1 R r cos  1 sin  1 xp 2 l  11 sin1 sin  1 ypp cos  1 z

2 1 IIy m b l12  gl 1 m 1  m b cos  2   2 2 (19) sin2 R r cos  2 sin  2 xp 2 l  Figure 3. Inside of Parallel robot model. 11 sin2 sin  2 ypp cos  2 z

2 1 III. CONTROLLER DESIGN IIy m b l13  gl 1 m 1  m b cos  3   3 2  (20) sin3 R r cos  3 sin  3 xp 2 l  3 1  sin3 sin  3 ypp cos  3 z

mp32 m b x p  1 cos  1 R  r  l 1 cos  1 cos  1  x p  (21) 22cos  2 R  r  l 1 cos  2 cos  2  xp 

 23cos  3 R  r  l 1 cos  3 cos  3  xp  Figure 4. Controller structure.

The structure of the control system for each robot arm where w1,i(k) |i=1,2,3 are gradients defined by (32-34), is presented in Fig. 4. successfully verified by Zhang et al. [11]:

A. Single Neural Adaptive PID Controller Ek  wk   Kp   1) Controller structure 11 wk 11   (32) The PID controller is built by a linear neuron with 03 E k  y k  u k  y k  inputs and zero trigger threshold [10] as shown in Fig. 5.  Kp  Kpe k  e1 y k  u k  w k  u k The input of neuron receives 03 corresponding values as     11     proportional, integral and differential components of the Ek  difference between response and reference signal as (27). wk  Ki  12    wk12   (33) E k  y k  u k  y k   Ki  Kie k  e2 y k  u k  w12  k  u k 

Ek  wk   Kd   13 wk 13   (34) E k  y k  u k  y k   Kd  Kd e k  e3 y k  u k  w13  k  u k 

k With  |k Kp , Ki , Kd are the learning rate constants; e1,

Figure 5. Structure of a PID controller adapted to a neuron. e1 and e3 determined by (27); yk  is the uk  response sensitivity to the control signal, also known as  de k e1  e k ;  e 2  e k dk ;  e 3  (27) Jacobian information, determined through recurrent fuzzy 0 dk neural network identifiers, which will presented in next The PID controller is set as follows: section. Three neuron-adaptive PID controllers are set up by

u k  u(k  1)  Kp  e 1  K i  e 2  K d  e 3 (28) MATLAB's three S-function, fully compatible with Simulink's standard library, a single-neuron-adapted PID where, e(k) is the difference between the reference signal controller depicted in Fig. 6 and two sets The remaining and the system response. is built in the same way as the SingleNeural_PID1. With the structure of a single neuron PID controller in Fig. 5, the output of the neuron is also the output of the PID controller, as shown in (29).

n w11  e1  w 12  e 2  w 13  e 3 du k  f n n (29) u( k ) u ( k -1) du ( k ) Figure 6. Single neural PID controller. In which, w1i | I = 1,2,3 are the weights of the neurons, which are the parameter sets (KP, Ki, Kd) of the PID controller and they are updated online during the control B. Recurrent Fuzzy Neural Newtwork Identifier process. 1) Identifier structure 2) Controller training The goal of training the single neuron PID controller is to adjust the network's weight set w1i (i=1,2,3) to minimize the cost function (30).

11 2 E k  e2  k  y k  y k  (30) 22ref where yref(k) is the reference signal and y(k) is the system response. To adjust the weight set w1i | i = 1,2,3, the gradient descent method was applied:

Kp  w11 k 1  w 11 k   w 11  k 

Ki  w12 k 1  w 12 k   w 12  k  (31) Figure 7. Diagram of the RFNN identifier. Kd  w13 k 1  w 13 k   w 13  k 

The identifier is a recurrent fuzzy neural network Using back-propagation technique, RFNN's set of (RFNN - Fig. 7). The RFNN identifier has 4 layers, with connection weights will be adjusted according to the an input layer of 2 nodes, a fuzzy layer of 10 nodes, a following: fuzzy rule class of 25 nodes, and an output layer with 1 Ek  node. The structure of the RFNN identifier can be W k1  W k   W k  W k     (42) described as follows [12]: W Layer 1 - Input layer: This layer consists of 2 nodes that convey the input values to the next layer. Here In which,  is the learning rate constant and W is the feedback links are added to increase the responsiveness parameter to be adjusted during the training of the RFNN of the network. The output of layer 1 is described as (35): identifier. T Given e(k)=y(k)-ym(k) and W=[, m, σ, w] are the 1 1 1 k Oi k  x i k  i O i  k 1 , i  1,2 (35) error and the connection weight vector of the RFNN identifier, then the gradient of E(.) in (41) according to W With  1 is the connection weight at the current time k. i is determined as follows: The input of the corresponding RFNN identifier is the E k  y k  O4  k  current control signal and the past output of the response:  e k m   e k  1 (43) WWW   x1  k  u k  1 (36) With this principle, the weight of each RFNN network 1 x2  k  y k 1 layer is updated as follows [13]: Layer 2 - Fuzzy layer: This layer consists of (2x5) Ek  w4 k1  w 4 k ww   w 4 k  e k O 3 (44) nodes, each node representing a related function of the ij  ij  4 ij    i wij Gaussian form with mean value mij and standard  deviation σij, and defined as (37). m Ek  2 m k1  m k     1 ij ij  O k  m mij 2  i ij  (45) Oij  k exp  , i  1,2; j  1,2,...,5 (37) 2 2 O1 k m  m 43ij  ij  ij  m k e k w O ij     ik k 2 k  ij  At each node on the fuzzy layer, there are 2 parameters that are automatically adjusted during the online training  Ek  of the RFNN identifier, that is m and σ . ijkk1   ij       ij ij  Layer 3 - Law class: This layer consists of (5x5) nodes. ij (46) 2 The output of the q node in this class is determined as 2 O1  k  m k e k w43 O ij ij follows: ij     ik k 3 k  ij  O32 k O k, i  1,2,...,5; q  1,2,...,5 (38) q   iqi   i i Ek  11kk1        ii  1 Layer 4 - Output layer: This layer includes 1 linear i (47) neuron with the defined output as follows: 21 O11 k  m O k   1k  e k w 4 O 3 ij ij ij 4 4 3 i ik k 2 k  Oi k  w ij O j  k, i  1; j  1,2,...,25 (39)  ij  j where  s are the corresponding learning rate where w4 is the connecting weights from 3rd layer to 4th |s w , m , , ij constants. In addition, to estimate the output of the object layer. The output of this layer is also the output of the model y (k), the RFNN identifier must also estimate RFNN identifier: m yk Jacobian information   for online training of 4 ˆ uk  O1 k  ym  k f x 1 k, x 2  k  (40) ˆ the single neuron PID controller. Jacobian information is f u k,1 y k  determined as follows [14]-[19]:

2) Training the RFNN identitifiers 4 254 33 25 yk  O O OOqq    1 ..  w4 The goal of an online training algorithm for RFNN 3 ij   u k  uqq11O  u  u identifier is to adjust the weighting sets of the network q    32 and the parameters of the fuzzy class dependent functions 4 OOq qs (48)  w .. to achieve the minimum value of cost function (41): ij 2 qsOqs u

112 4 2 3 2 O1 k m E k y k  y k  y k  O k (41) 4 Oq   ij  ij     m     1    w .. 22 ij 22 qsOqs   ij 

The RFNN identifier for each robot arm is built in TABLE III. PARAMETERS OF THREE RFNN IDENTIFIERS MATLAB's S-function as Fig. 8. Symbol Meaning Value

The learning T [;;]1  2  3 factor of the 0.01 0.01 0.01 neuron network c RFNN1 -1.5 -0.75 0 0.75 1.5 i1 network center Figure 8. The RFNN identifier in Simulink. -1.5 -0.75 0 0.75 1.5 vector c RFNN2 -1.35 -0.675 0 0.675 1.35 i2 network center yˆ  k  ymm( k )  x ( k ) f  x ( k  1), u ( k ) (49) -1.35 -0.675 0 0.675 1.35 vector RFNN3 -3.05 -1.525 0 1.525 3.05 ci3  network center -3.05 -1.525 0 1.525 3.05 IV. SPECIFICATIONS AND SIMULATION RESULTS vector Activation 0.5 0.5 0 0.5 0.5 A. Simulation Parameters bi1  threshold 1 0.5 0.5 0 0.5 0.5 The specifications of the robot are shown in Table I [6]. b Activation 0.5 0.5 0 0.5 0.5 Parameters of the single neuron PID controller are given i2 threshold 2 0.5 0.5 0 0.5 0.5 in Table II. And the parameters of RFNN identifiers are T presented in Table III. 0.1 0.1  b Activation 0.5 0.5 0 0.5 0.5 TABLE I. DELTA ROBOT MECHANICAL SPECIFICATIONS i3 threshold 3 0.5 0.5 0 0.5 0.5 Symbol Value Unit T Weighted T w 0.1 0.1 0.1 0.1 L1 0.3 m  1  vector 1 (25×1)   L2 0.8 m T Weighted T 0.15 0.15 0.15 0.15 R 0.26 m w2  vector 2 (25×1)   r 0.04 m T Weighted T 0.1 0.1 0.1 0.1 α1 0 rad/s w3  vector 3(25×1)   α 2π/3 rad/s 2 w Weighted 0.1 0.1 0 0.1 0.1 α 4π/3 rad/s i1 3 vector RFNN1 0.1 0.1 0 0.1 0.1 m1 0.42 kg mb 0.2 kg w Weighted 0.125 0.125 0.125 0.125 0.125 mp 0.75 kg i2 vector RFNN2 0.125 0.125 0.125 0.125 0.125 TABLE II. PARAMETERS OF ADAPTIVE NEURAL PID CONTROLLER Weighted 0.15 0.15 0.15 0.15 0.15 wi3  Symbol Meaning Value vector RFNN3 0.15 0.15 0.15 0.15 0.15 T Initial parameters of the T 800 100 150 Network inputs 2 w w w PID neuron   i KKKp I D T Learning rates of the T 0.2 0.1 0.1    PID neuron   KKKp I D

Figure 9. Delta parallel robot controller diagram in MATLAB / Simulink.

The response of the traditional PID and the single B. Simulation Results neuron PID to the reference trajectory are presented in The control system for the Delta robot is illustrated in Fig. 10, Fig. 11, Fig. 12, and Fig. 13 with changing load. Fig. 9 with the desired trajectory as (50). Simulation results show that single neural PID controllers x = 0.17sin(2 t)+0.3 are better than traditional PID controllers with the setting d times are about 0.3±0.1 seconds, the steady-state errors yd = 0.17cos(2 t)+0.2 (50) are eliminated.

zd = -0.7m

Figure 10. Comparison of traditional PID and single neuron PID Figure 11. Errors of responses. controllers.

Figure 12. Trajectory tracking of traditional PID and single neural PID Figure 13. Responses when changing load from 3.21 Kg to 4.11 Kg. controller.

responses better than using traditional PID controllers TABLE IV. SYSTEM QUALITY STANDARDS (Table IV). The proposed algorithm is stable and fast Quality indicators PID Neuron PID response when simulating on Delta robot. In next step, Settling time 0.4±0.001(s) 0.3±0.001(s) the controllers will be tested on a real robot system. Overshoot 1.991 (%) 1.97 (%) CONFLICT OF INTEREST Rising time 2.535 s 2.688 s The authors declare no conflict of interest. V. CONCLUSION AUTHOR CONTRIBUTIONS This paper has applied single neural PID controllers based on RFNN identifiers to control a 3-DOF delta robot, Mr. Le Minh Thanh, first author, is a PhD candidate a nonlinear MIMO system. Each robot arm is controlled under supervising of Prof. Chi-Ngon Nguyen (last and by a single neural PID controller that can be online corresponding author), who has prepared the manuscript. training with the Jacobian information of that arm given Mr. Luong Hoai Thuong, 2nd author, has helped on by a RFNN identifier. Simulation results show that the correcting the paper. Mr. Phan Thanh Loc, 3rd author, is a controllers and identifiers can be updated online during Master’s degree student, also supervised by Prof. Nguyen, control action. The quality standards of improved system has built the simulation model. And Prof. Nguyen is chef

of research group, who has supervised for this study and Reinforcement Learning, Prof. Abdelhamid Mellouk, InTech, finalized this paper. 2011. [18] A. Errachdi and M. Benrejeb, “Performance Comparison of Neural Network Training Approaches in Indirect Adaptive REFERENCES Control,” International Journal of Control, Automation and Systems, vol. 16, no. 3, pp. 1448–1458, 2018. [1] J. Merlet, Parallel Robots. P.O. Box 17, 3300 AA Dordrecht, The [19] X. Y. Zhou, C. Yang, and T. Cai, “A model reference adaptive Netherlands: Kluwer Academic Publishers, 2000. control/PID compound scheme on disturbance rejection for an [2] P. W. L. V, Position-Controlling Apparatus, Patent US2 286 571 aerial inertially stabilized platform,” Journal of Sensors, vol. 2016, A, June, 16, 1942. 2016. [3] D. Stewart, A platform with Six Degrees of Freedom, 1991. [4] R. Clavel, Conception D’Un Robot Parallele Rapide a 4 Degres Copyright © 2020 by the authors. This is an open access article de Liberte, Ph.D. dissertation, Ecole Polytechnique Federale de distributed under the Creative Commons Attribution License (CC BY- Lausanne, Lausanne. NC-ND 4.0), which permits use, distribution and reproduction in any [5] M. Bouri and R. Clavel (2010), The Linear Delta: Developments medium, provided that the article is properly cited, the use is non- and Applications, in ISR 2010 (41st Inter. Symposium on th commercial and no modifications or adaptations are made. Robotics and ROBOTIK 2010 (6 German Conference on Robotics), Munich, Germany, 2010, pp. 1-8. Le Minh Thanh received a Bachelor of [6] W. Widhiada, T. G. T. Nindhia, and N. Budiarsa, “Robust control Engineering degree in Electrical and Electronic for the motion five fingered robot gripper,” International Journal Engineering at Cuu Long University in 2006, a of Mechanical Engineering and Robotics Research, vol. 4, no. 3, Master's degree in Automation at the University of pp. 226-232, 2015. Transport in Ho Chi Minh City in 2011. He is now [7] O. T. Abdelaziz, S. A. Maged, M. I. Awad, “Towards dynamic a lecturer of the Faculty Electrical – Electronics, task/posture control of a 4dof humanoid robotic arm,” Vinh Long University of Technical Education. International Journal of Mechanical Engineering and Robotics Research, vol. 9, no. 1, pp. 99-105, 2020. [8] D. Elayaraja, R. Ramakrishnan, M. Udhayakumar, and S. Luong Hoai Thuong received a Bachelor of Ramabalan, Intelligent Control of Mobile Robot Using C++, Engineering in Control Engineering from Can Tho International Journal of Mechanical Engineering and Robotics University in 2009, a Master's degree in Electronic Research, vol. 3, no. 2, pp. 429-434, 2014. Engineering at Ho Chi Minh City University of [9] N. D. Dung, Reverse Dynamics of Parallel Delta Space Robots, Technical Education in 2015. He is currently a LATS Mechanical and mechanical engineering: 9.52.01.01, lecturer in the Faculty of Electrical - Electronics, Vietnam National Library, code: 629.892 / Đ455L, 2018. Vinh Long University of Technical Education.

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