Identification of mimo dynamic system using inverse mimo neural narx model

This paper investigates the application of proposed neural MIMO NARX model to a nonlinear 2-axes pneumatic artificial muscle (PAM) robot arm as to improve its performance in modeling and identification. The contact force variations and nonlinear coupling effects of both joints of the 2-axes PAM robot arm are modeled thoroughly through the novel dynamic inverse neural MIMO NARX model exploiting experimental input-output training data. For the first time, the dynamic neural inverse MIMO NARX Model of the 2-axes PAM robot arm has been investigated.

TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 16, SỐ K2- 2013 IDENTIFICATION OF MIMO DYNAMIC SYSTEM USING INVERSE MIMO NEURAL NARX MODEL Ho Pham Huy Anh(1), Nguyen Thanh Nam(2) (1) Ho Chi Minh City University of Technology, VNU-HCM (2) DCSELAB, University of Technology, VNU-HCM (Manuscript Received on April 5th, 2012, Manuscript Revised May 15th, 2013) ABSTRACT: This paper investigates the application of proposed neural MIMO NARX model to a nonlinear 2-axes pneumatic artificial muscle (PAM) robot arm as to improve its performance in modeling and identification The contact force variations and nonlinear coupling effects of both joints of the 2-axes PAM robot arm are modeled thoroughly through the novel dynamic inverse neural MIMO NARX model exploiting experimental input-output training data For the first time, the dynamic neural inverse MIMO NARX Model of the 2-axes PAM robot arm has been investigated The results show that this proposed dynamic intelligent model trained by Back Propagation learning algorithm yields both of good performance and accuracy The novel dynamic neural MIMO NARX model proves efficient for modeling and identification not only the 2-axes PAM robot arm but also other nonlinear dynamic systems Keywords: dynamic modeling, pneumatic artificial muscle (PAM), 2-axes PAM robot arm, inverse identification, neural MIMO NARX model, back propagation (BP) algorithm INTRODUCTION Rehabilitation robots up to now begin to be applied for treatment of patients suffering from et al [3] added different constant external loads, by a robot in torque control mode Pneumatic Artificial Muscle (PAM) trauma or stroke Since the number of patients actuators are now used in the various fields of is large and the treatment is time consuming, it medical robots The modern robotics toward is a big advantage if rehabilitation robots can applications assist in performing treatment Noritsugu et al between robot actuator and human operator [1] designed an arm-like robot for treating PAM actuator has achieved increasing belief to patients with trauma, and developed four the ability of providing advantages such as high modes of linear motion with impedance control power/weight ratio, full of hygiene, easiness in to control the force during the movement preservation and especially the capacity of Krebs et al [2] designed a planar robot with human compliance which is the most important impedance control for guiding patients to make requirement in medical and human welfare movements along the specified trajectories Ju field Therefore PAM has been regarded during requires greater friendliness Trang 13 Science & Technology Development, Vol 16, No.K2- 2013 the recent decades as an interesting alternative approaches combining conventional methods to actuators with new learning techniques are required (Lin Consequently, PAM-based applications have and Lee, 1991)[13] Thanks to their universal been published increasingly Caldwell et al approximation capabilities, neural networks (2003) in [4] have developed and controlled of provide the implementation tool for modeling a PAM-based Soft-Actuated Exoskeleton for the complex input-output relations of the use in physiotherapy and training Kobayashi et multiple n DOF PAM manipulator which is al (2003) in [5] have applied PAM as to able to solve dynamic problems like variable- develop a Muscle suit for Upper Body coupling complexity and state-dependency Noritsugu et al (2005) in [6] have used PAM During the last decade several neural network for developing an Active Support Splint among models and learning schemes have been them applied to offline learning of manipulator hydraulic and electric principal dynamics (Karakasoglu et al., 1993)[14], difficulty inherent in PAM actuators is the (Katic et al., 1995)[15], (Lewis et al., problem of modeling and controlling them 1999)[16], (Boerlage et al., 2003)[17] In efficiently and precisely This is because they (Pham et al., 2005)[18], authors applied neuro- are highly nonlinear and time varying Since fuzzy the rubber tube and plastic sheath are manipulators for trajectory tracking Ahn and continually in contact with each other and the Anh in [19] have optimized successfully a PAM shape is continually changing, the PAM pseudo-linear ARX model of the PAM temperature varies with use, changing the manipulator using genetic algorithm These properties time authors in (Anh et al., 2007)[20] have Approaches to PAM modeling and control identified the highly nonlinear 2-axes PAM have included PID control, adaptive control manipulator (Lilly, 2003)[7], nonlinear optimal predictive networks Nevertheless, the drawback of all control [8], variable structure control [9], and these various soft computing approaches including manipulator as n independent decoupling intelligent model + phase plane switching joints Consequently, all intrinsic coupling control (Ahn et al., 2006)[10], neuro-fuzzy features of the n-DOF manipulator have not model and genetic control in (Carbonell et al., represented in its NN model respectively Unfortunately, of the up to now actuator over 2001)[11], (Lilly and Chang, 2003)[12] and so on modeling results based is and on control of robot recurrent considered the neural n-DOF To overcome this disadvantage, in this paper, a new approach of neural networks, Among such advanced modeling and proposed dynamic inverse neural MIMO control schemes, as to guarantee a good NARX model, firstly utilized in simultaneous tracking performance, robust adaptive control modeling and identification of the nonlinear 2- Trang 14 TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 16, SỐ K2- 2013 axes PAM robot arm system The experiment (also called “nodes” or “neurons”), and m results have demonstrated the feasibility and outputs units is shown in Fig good performance of the proposed intelligent inverse model which overcomes successfully external and internal disturbances such as contact force variations and highly nonlinear coupling effects of both joints of the 2-axes PAM robot arm The outline of this paper composes of the Figure Structure of feed-forward MLPNN section for introducing related works in PAM robot arm modeling and identification The section presents identification procedure of In Fig.1, w10, , wq0 and W10, ,Wm0 are weighting values of Bias neurons of Input Layer and Hidden Layer respectively an inverse neural MIMO NARX model using Consider an ARX model with noisy input, back propagation learning algorithm The section proves and analyses experimental studies and results considering the contact which can be described as A(q 1 ) y(t )  B(q 1 )u (t  T )  C (q 1 )e(t ) (1) force variations and highly nonlinear coupling A(q1) 1 a1q1  a2q2 effects of both joints of the nonlinear dynamic with system Finally, the conclusion belongs to the B(q 1 )  b1  b2 q 1 section IDENTIFICATION USING DYNAMIC INVERSE NEURAL MIMO C(q1)  c1  c2q1  c3q2 where e(t) is the white noise sequence with NARX zero mean and unit variance; u(t) and y(t) are MODEL 2.1 Dynamic Neural MIMO NARX Model Inverse Neural MIMO NARX model used input and output of system respectively; q is the shift operator and T is the time delay From equation (1), not consider noise in this paper is a combination between the Multi-Layer Perceptron Neural Networks (MLPNN) structure and the ARX model Due to this combination, Inverse MIMO NARX model possesses both of powerful universal approximating feature from MLPNN structure component e(t), we have the general form of the discrete ARX model in domain z (with the time delay T=nk=1) b1 z 1  b2 z 2   bnb z  nb y( z 1 )  u ( z 1 )  a1 z 1  a z    a na z  na and strong predictive feature from nonlinear ARX model (2) in which na and nb are the order of output -1 y(z ) and input u(z-1) respectively A fully connected 3-layer feed-forward MLP-network with n inputs, q hidden units Trang 15 Science & Technology Development, Vol 16, No.K2- 2013 This paper investigates the potentiality of having only one hidden layer and using various simple MIMO NARX models in order sigmoid activation functions From Fig.1, to exploit them in modeling, identification and predictive output value yˆ (t ) is calculated as control as well Thus, by embedding a 3-layer follows: MLPNN (with number of neurons of hidden q  yˆi (w,W)  Fi WijOj (w) Wi0    j 1  layer = 5) in a 2nd order ARX model with its characteristic equation derived from (2) as follows: y1 (k )  b11u1 (k )  b12u2 (k )  a11 y1 (k  1)  a12 y2 (k  1) (3) y2 (k )  b21u1 (k )  b22u2 (k )  a21 y1 (k  1)  a22 y2 (k  1) We will design the proposed inverse MIMO Neural NARX11 model (na = 1, nb = 1, nk =1) with inputs (including u11 (t) and u12(t) identical to input value u1(t), u21(t) and u22(t) identical to input value u2 (t), and recurrent (4) q  n  Fi  Wij f j wjl zl  wj0  Wi0   l 1   j 1  The weights are the adjustable parameters of the network, and they are determined from a set of examples through the process called training The examples, or the training data as they are usually called, are a set of inputs, u(t), and corresponding desired outputs, y(t) delayed values y1(t-1), y2(t-1)), output values Specify the training set by: (y1hat(t), y2hat(t)) Its structure is shown in Fig Z N  u (t ), y(t ) t  1, , N  (5) The objective of training is then to determine a mapping from the set of training  ˆ data to the set of possible weights: Z N so produce that the network will predictions yˆ (t ) , which in some sense are Figure Structure of MIMO Neural NARX11 model “closest” to the true joint angle outputs y(t) of PAM robot arm By this way, the parameters a11, a12, b11, The prediction error approach, which is the b12 of linear ARX model now become strategy applied here, is based on the nonlinear and will be determined from the introduction of a measure of closeness in terms weighting values Wij and wjl of the nonlinear of a mean sum of square error (MSSE) MIMO Neural NARX model This feature criterion: makes MIMO Neural NARX model very powerful in modeling, identification and in model-based advanced control as well The class of MLPNN-networks considered in this paper is furthermore confined to those Trang 16   EN ,Z N  N y(t)  yˆ (t  ) y(t)  yˆ (t  ) 2N t 1 T (6) TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 16, SỐ K2- 2013 Based on the conventional error Back- Error to be minimized: Propagation (BP) training algorithms, the W ( k  1)  W ( k )    E (W k )  W k  m  yˆ i  y i 2  i 1 E weighting value is calculated as follows: (7) with k is kth iterative step of calculation and  is learning rate which is often chosen as a (10) Using chain rule method, we have: E  E  yˆ i  S i   W ij  yˆ i  S i  W ij (11) From equation (10), the following equation small constant value Concretely, the weights Wij and wjl of is derived neural NARX structure are then updated as: E   yˆ i  yi  yˆ i W ij k    W ij k    W ij k    W ij k      i O (12) (8) j q  i  yˆ i 1  yˆ i  y i  yˆ i  S i   W ij O j  bias i as with sum j 1 with  i is search direction value of ith neuron of output layer (i=[1  m]); Oj is the th output value of j neuron of hidden layer (j=[1  q]); yi and yˆ i are truly real output and predicted output of i th neuron of output calculation at ith node of output layer and yˆ i  , it gives  e  Si yˆi e Si 1 1  1    1   Si   S i Si  e  e   e Si      layer (i=[1  m]), and (13) yˆi 1  yˆi  w jl k  1  w jl k    w jl k  1  w jl k  1    j u l (9) m  j  O j 1  O j   iW ij S i  Oj Wij (14) i 1 in which  j is search direction value of jth Replace (12), (13), (14) to (11) and then put all to (7), the following equation is derived neuron of hidden layer (j=[1  q]); Oj is the W ij k    W ij k    W ij k   output value of jth neuron of hidden layer  W ij k      i O th (j=[1  q]); ul is input of l neuron of input layer (l=[1  n]) These results of equations (8) and (9) are demonstrated as follow in case of sigmoid being activate function of hidden and output (15) j  i  yˆ i 1  yˆ i  y i  yˆ i  Equation (8) has been demonstrated The same way for updating the weights of hidden layer, using the chain rule method, we have: layer Consider in case of output layer: Trang 17 Science & Technology Development, Vol 16, No.K2- 2013 2.2 Experiment Set Up E E O j S j  w jl O j S j w jl (16) Then m E E S i  [ ] O j i 1 S i O j  E   q  W O  bias     ij j i   i 1   S i O j  j 1  m m i 1  m Wij      iWij i  i 1  E   S    (17) Figure Block diagram for working principle of n S j   w jl u l  bias j with as sum the 2-axes PAM robot arm l 1 calculation at jth node of hidden layer and Oj  S 1 e j O j e S j  S j 2-axes PAM robot arm shown through the , it gives schematic diagram of the 2-axes PAM robot arm and the photograph of the experimental 11  1 e  1 e  S j S j 1   1  S  1 e j  O j 1 O j  S j w jl A general configuration of the investigated (18) apparatus are shown in Fig.3 and Fig.4, respectively Both of joints of the 2-axes PAM robot arm are simultaneously  ul (19) modeled through and identified proposed neural MIMO NARX model The hardware includes an IBM compatible Replace (17), (18), (19) to (16) and then put all to (7), the following equation is derived w jl k  1  w jl k   w jl k  1 w jl k  1    j u l m  j  O j 1  O j   iWij i 1 Equation (9) has been demonstrated PC (Pentium 1.7 GHz) which sends the voltage signals u1(t) and u2(t) to control the two proportional valves (FESTO, MPYE-5-1/8HF- (20) 710B), through a D/A board (ADVANTECH, PCI 1720 card) which changes digital signals from PC to analog voltage u1(t) and u2(t) respectively The rotating torque is generated by the pneumatic pressure difference supplied from air-compressor between the antagonistic artificial muscles Consequently, the both of Trang 18 TAÏP CHÍ PHÁT TRIỂN KH&CN, TẬP 16, SỐ K2- 2013 joints of the 2-axes PAM robot arm will be Table The lists of experimental hardware rotated to follow the desired joint angle references (YREF1 (k) and YREF2(k)) respectively The joint angles, 1[deg] and 2 [deg], are detected by two rotary encoders (METRONIX, H40-8-3600ZO) and fed back to the computer through a 32-bit counter board (COMPUTING MEASUREMENT, PCI QUAD-4 card) which changes digital pulse signals to joint angle values y1 (t) and y2(t) Simultaneously, through an A/D board (ADVANTECH, PCI 1710 card) which will send to PC the external force value IDENTIFICATION USING DYNAMIC INVERSE NEURAL MIMO NARX MODEL which is detected by a force sensor CBFS-10 The pneumatic line is conducted under the pressure of 5[bar] and the software control algorithm of the closed-loop system is coded in C-mex program code run in Real-Time In general, the procedure which must be executed when attempting to identify a dynamical system consists of four basic steps (see Fig.5) Windows Target of MATLAB-SIMULINK  STEP (Getting Training Data) environment the  STEP (Select Model Structure) configuration of the hardware set-up installed  STEP (Estimate Model)  STEP (Validate Model: Table presents from Fig.3, and Fig.4 Force Sensor Figure Photograph of the experimental 2-axes Figure Neural MIMO NARX Model PAM robot arm Identification procedure Trang 19 Science & Technology Development, Vol 16, No.K2- 2013 To realize Step 1, Fig.6 presents the PRBS the hardware using the tested 2-axes PAM robot arm and the angle of both of PAM antagonistic pair The responding end-effector external force and joint experiment results of 2-axes PAM robot arm angle outputs collected from force sensor and force/position control prove that experimental rotary encoders This experimental PRBS control voltages u1 (t) and u2(t) applied to both input-output data is used for training and of PAM antagonistic pairs of the 2-axes PAM validating the Inverse neural MIMO NARX robot arm is to function well in these ranges model of the whole dynamic two-joint structure Likewise, the chosen frequency of PRBS-1(2) of the 2-axes PAM robot arm as illustrated in signals is also chosen carefully based on the Fig.7 working frequency of the 2-axes PAM robot arm will be used as an elbow and wrist 2-axes JOINT - PRBS TRAINING DATA [V ] PRBS input 5.5 PRBS input PAM-based rehabilitation robot in the range of 4.5 10 20 30 40 50 60 70 80 60 10 20 30 40 50 60 70 80 (0.025 – 0.2) [Hz] JOINT ANGLE output JOINT ANGLE output JOINT - INVERSE PRBS TRAINING DATA 40 20 JOINT - INVERSE PRBS TRAINING DATA 40 60 JOINT ANGLE input JOINT ANGLE input 20 40 20 [d e g ] -20 -20 20 0 -20 -40 10 20 30 40 50 F O R C E [N ] 60 40 -40 60 70 80 External FORCE 60 Filtered FORCE 40 -20 10 20 30 40 50 60 70 80 External FORCE Filtered FORCE -40 300 10 20 30 40 50 60 70 -40 80 300 20 20 10 20 30 40 50 60 70 80 10 20 30 40 50 60 experiment PRBS-1(2) inputs and Force/Joint Angle outputs during (40–80)[s] will be used for while PRBS-1(2) 70 20 30 40 50 10 20 30 40 50 inputs and 10 20 30 40 50 5.5 60 70 PRBS1 output 5.5 5 4.5 4.5 10 20 30 40 50 t [sec] 60 70 80 10 20 30 40 50 t [sec] Figure Inverse Neural MIMO NARX Model Training data obtained by experiment structure A nonlinear neural NARX model The range (4.4 – 5.6) [V] and the shape of structure is attempted The full connected st 60 70 80 PRBS2 output (0–40)[s] will be used for validation purpose PRBS-1 voltage input applied to the joint as Multi-Layer Perceptron (MLPNN) network well as the range (4.5 – 5.5) [V] and the shape architecture composes of layers with of PRBS-2 voltage input applied to rotate the neurons in hidden layer is selected (results joint of the 2-axes PAM robot arm is derived from Ahn et al., 2007 [24]) The final chosen carefully from practical experience structure of proposed Inverse neural MIMO Trang 20 80 FORCE INPUT 80 The 2nd step relates to select model 70 10 Force/Joint Angle outputs in the lapse of time nd 60 20 FORCE INPUT 80 [V ] Figure Input-Output training data obtained by training, 10 [N ] 20 10 set-up proportional valve to control rotating joint 5.5 4.5 [d e g ] on input applied simultaneously to the joints of JOINT - PRBS TRAINING DATA 40 based 60 70 80 TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 16, SỐ K2- 2013 NARX11 used in proposed neural MIMO voltage values u1(k-1), u2(k-1) respectively of NARX FNN-PID hybrid force/position control experimental modeling block diagram depicted scheme is shown in Fig.8 in Fig.9 The proposed neural MIMO NARX11 model structure is defined as a nonlinear neural MLPNN integrated a 1st order ARX model (with nA=1; nB=1 and nK=1) possessed neurons in hidden layer The activating function applied in neurons of hidden Layer and of output layer is hyperbolic tangent function and linear function respectively Fig.9 represents the experiment block diagram for modeling and identifying the Inverse neural MIMO NARX11 model of the 2-axes PAM Figure Block diagram for modeling of Inverse robot arm Neural MIMO NARX model of the 2-Axes PAM 2-AXES PAM ROBOT ARM - NEURAL MIMO INVERSE NARX MODEL robot arm u23(t) ESTIMATION of NEURAL INVERSE MIMO NARX -1 10 FITNESS CONVERGENCE u22(t) u21(t) yhat2(t) y2(t-1) u12(t) yhat1(t) FITNESS u13(t) -2 10 u11(t) y1(t-1) Figure Structure of proposed Inverse neural -3 10 MIMO NARX11 models of 2-axes PAM robot arm In Fig.8, input values u11(t)/ u21(t), u12(t)/ u22(t), u13(t)/ u23(t) and recurrent delayed input 10 20 30 40 50 Iteration 60 70 80 90 Figure 10 The fitness convergence of proposed Neural Inverse MIMO NARX11 Model values y1(t-1), y2(t-1) in neural structure of The 3rd step estimates trained Inverse proposed neural Inverse MIMO NARX11 neural MIMO NARX11 model A good model will be identical to input values Joint-1 minimized convergence is shown in Fig.10 Angle y1 (k), Joint-2 Angle y2(k), Force value with the minimized Mean Sum of Scaled Error yF(k) and desired recurrent delayed control (MSSE) value is equal to 0.002659 after Trang 21 100 Science & Technology Development, Vol 16, No.K2- 2013 number of training 100 iterations with the proposed Inverse neural MIMO NARX11 An nonlinear excellent estimating result, which proves the models Applying the same experimental perfect performance of resulted Inverse Neural diagram in Fig.6, an excellent validating result, MIMO NARX model, is also shown in Fig.11 which proves the performance of resulted JOINT ANGLE input JOINT ANGLE input [de g] MIMO NARX 20 minimized -20 -20 -40 10 15 20 25 35 -40 40 FORCE input 28 30 28 26 24 24 [N ] 26 6.5 10 15 20 [V ] 5.5 errors demonstrate the good performance of the Inverse neural MIMO 10 15 20 25 30 35 40 NARX11 Model (the excellent error < 0.01[V] FORCE input for both of Uh1/Uh2 control voltage values 30 35 406.5 PRBS1 reference Uh1 output 5.5 25 10 15 20 25 30 35 40 PRBS2 reference Uh2 output respectively applied to joints of the 2-axes PAM robot arm) 4.5 4.5 10 15 20 25 30 35 40 ERROR1 10 15 20 25 30 35 40 ERROR2 10 15 20 t [sec] 25 30 35 Finally, Table tabulates the resulting weighting values of proposed Inverse neural -1 MIMO NARX model which can be used not -1 40 10 15 20 t [sec] 25 30 35 40 only in modeling identification and simulation Figure 11 Estimation of 2-axes PAM robot arm offline but also can be applied effectively Inverse neural MIMO NARX11 Model online VALIDATION of INVERSE NEURAL MIMO NARX - JOINT 40 JOINT ANGLE input 20 [d e g ] Inverse in Fig.12 The experimental results of the 40 20 neural Inverse Neural MIMO NARX model, is shown ESTIMATION of INVERSE NEURAL MIMO NARX - JOINT ESTIMATION of INVERSE NEURAL MIMO NARX - JOINT 40 [V ] The last step relates to validate resulting VALIDATION of INVERSE NEURAL MIMO NARX - JOINT advanced control algorithms (Ahn and Anh, 2011)[21] The final JOINT ANGLE input designed structure of proposed Inverse MIMO -20 model-based 40 20 in NARX11 model is shown in Fig.8 -20 -40 300 10 15 20 25 30 35 40 30 10 15 20 25 30 FORCE input 40 CONCLUSIONS In this study, a new approach of recurrent 25 [N ] 25 35 FORCE input neural networks, proposed neural Inverse 20 6.50 10 15 20 25 20 30 35 40 6.5 PRBS1 reference Uh1 output 5.5 5 [V ] 5.5 4.5 10 15 20 25 30 35 40 PRBS2 reference Uh2 output 10 15 20 25 30 35 Error1 40 nonlinear 2-axes pneumatic artificial muscle 10 15 20 25 30 35 40 (PAM) system, has successfully overcome the Error -1 10 15 20 t [sec] 25 30 35 -1 40 MIMO NARX model firstly utilized in modeling and identification of the highly 4.5 [V ] contact force variations, coupled effect and 10 15 20 t [sec] 25 30 35 40 nonlinear characteristic of the 2-axes PAM Figure 12 Validation of 2-axes PAM robot arm robot arm system The 2-axes PAM robot Inverse neural MIMO NARX11 Model arm’s coupled dynamics was taken into account Results of training and testing on the complex dynamic systems such as 2-axes PAM Trang 22 TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 16, SOÁ K2- 2013 robot arm show that the newly proposed neural Acknowledgment Inverse MIMO NARX model presented in this This research was partially supported by study is quite suitable to be applied for the the NAFOSTED and the DCSELAB of Viet modeling and identification not only the 2-axes Nam PAM robot arm but also other nonlinear dynamic systems Table Resulted weights of Inverse neural MIMO NARX11 – Total Number of weighting values = 57 NHẬN DẠNG HỆ ĐỘNG HỌC MIMO SỬ DỤNG MƠ HÌNH MIMO NEURAL NARX NGƯỢC Hồ Phạm Huy Ánh(1), Nguyễn Thanh Nam(2) (1) Trường Đại học Bách Khoa, ĐHQG-HCM (2) ĐHQG-HCM TÓM TẮT: Bài báo khảo sát ứng dụng mơ hình neural MIMO NARX để cải thiện chất lượng nhận dạng hệ tay máy phi tuyến bậc dùng bắp thịt khí nén nhân tạo (PAM) Các yếu tố biến động lực tiếp xúc hay ảnh hưởng ghép cặp phi tuyến khớp nối tay máy nhận dạng đầy đủ mơ hình neural MIMO NARX ngược thơng qua liệu huấn luyện thực nghiệm đầu vào – đầu Lần đầu tiên, mơ hình động học nơ rôn MIMO NARX ngược tay máy bậc dùng bắp thịt khí nén nhân tạo (PAM) khảo sát hồn chỉnh Các kết cho thấy mơ hình động học thông minh đề xuất, huấn luyện thuật toán Lan Truyền Ngược (BP learning algorithm) cho chất lượng tốt với độ xác nhận dạng cao Mơ hình động học nơ rơn MIMO NARX ngược cho Trang 23 Science & Technology Development, Vol 16, No.K2- 2013 thấy chúng dùng hiệu nhận dạng không hệ tay máy PAM 2-bậc mà cho hệ động học phi tuyến đa biến khác Keywords: mơ hình động học, bắp thịt khí nén nhân tạo (PAM), tay máy PAM bậc, nhận dạng mơ hình ngược, mơ hình nơ rơn MIMO NARX ngược, thuật toán lan truyền ngược (BP) [7] REFERENCES Lilly, J H., Adaptive tracking for pneumatic muscle actuators in bicep and [1] Noritsugu, T., Tanaka, T., Application of tricep configurations, IEEE Trans Neural rubber artificial muscle manipulator as a Syst Rehabil Eng., , 11, 3, 333–339 rehabilitation robot, IEEE/ASME Trans (2003) Mechatronics, 2, 4, 259–267 (1997) [8] [2] Krebs, H I., Hogan, N., Aisen, M L., Volpe, B T., Robot-aided Nagaoka, T., 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Neural MIMO NARX model, is shown ESTIMATION of INVERSE NEURAL MIMO NARX - JOINT ESTIMATION of INVERSE NEURAL MIMO NARX - JOINT 40 [V ] The last step relates to validate resulting VALIDATION of