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. 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.


INTRODUCTION
Rehabilitation robots up to now begin to be applied for treatment of patients suffering from trauma or stroke.Since the number of patients is large and the treatment is time consuming, it is a big advantage if rehabilitation robots can assist in performing treatment.Noritsugu et al.
[1] designed an arm-like robot for treating patients with trauma, and developed four modes of linear motion with impedance control to control the force during the movement.Krebs et al. [2] designed a planar robot with impedance control for guiding patients to make movements along the specified trajectories.Ju Consequently, PAM-based applications have been published increasingly.Caldwell et al. (2003) in [4] have developed and controlled of a PAM-based Soft-Actuated Exoskeleton for use in physiotherapy and training.Kobayashi et al. (2003) in [5] have applied PAM as to develop a Muscle suit for Upper Body.Noritsugu et al. (2005) in [6] have used PAM for developing an Active Support Splint among them.
Unfortunately, up to now principal difficulty inherent in PAM actuators is the problem of modeling and controlling them efficiently and precisely.This is because they are highly nonlinear and time varying.Since the rubber tube and plastic sheath are continually in contact with each other and the PAM shape is continually changing, the PAM temperature varies with use, changing the properties of the actuator over time.
Among such advanced modeling and control schemes, as to guarantee a good tracking performance, robust adaptive control approaches combining conventional methods with new learning techniques are required (Lin and Lee, 1991)[13].Thanks to their universal approximation capabilities, neural networks provide the implementation tool for modeling the complex input-output relations of the multiple n DOF PAM manipulator which is able to solve dynamic problems like variablecoupling complexity and state-dependency.
During the last decade several neural network models and learning schemes have been applied to offline learning of manipulator dynamics (Karakasoglu et al., 1993)[14], ( Katic et al., 1995)[15], (Lewis et al., 1999)[16], (Boerlage et al., 2003)[17].In (Pham et al., 2005) A fully connected 3-layer feed-forward MLP-network with n inputs, q hidden units (also called "nodes" or "neurons"), and m outputs units is shown in Fig. 1.Consider an ARX model with noisy input, which can be described as where e(t) is the white noise sequence with zero mean and unit variance; u(t) and y(t) are input and output of system respectively; q is the shift operator and T is the time delay.
in which n a and n b are the order of output y(z -1 ) and input u(z -1 ) respectively.

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This paper investigates the potentiality of various simple MIMO NARX models in order to exploit them in modeling, identification and control as well.Thus, by embedding a 3-layer MLPNN (with number of neurons of hidden layer = 5) in a 2 nd order ARX model with its characteristic equation derived from (2) as follows: We will design the proposed inverse ) with 6 inputs (including u 11 (t) and u 12 (t) identical to input value u 1 (t), u 21 (t) and u 22 (t) identical to input value u 2 (t), and recurrent delayed values y 1 (t-1), y 2 (t-1)), 2 output values (y 1hat (t), y 2hat (t)).Its structure is shown in Fig. 2.

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Based on the conventional error Back-Propagation (BP) training algorithms, the weighting value is calculated as follows: with k is k th iterative step of calculation and   is learning rate which is often chosen as a small constant value.
Concretely, the weights W ij and w jl of neural NARX structure are then updated as: output value of j th neuron of hidden layer (j=[1  q]); y i and i y ˆare truly real output and predicted output of i th neuron of output in which j  is search direction value of j th neuron of hidden layer (j=[1  q]); O j is the output value of j th neuron of hidden layer These results of equations ( 8) and ( 9) are demonstrated as follow in case of sigmoid being activate function of hidden and output layer.Consider in case of output layer: Error to be minimized: Using chain rule method, we have: From equation ( 10), the following equation is derived. with as sum calculation at i th node of output layer Replace ( 12), ( 13), ( 14) to (11) and then put all to (7), the following equation is derived.
The same way for updating the weights of hidden layer, using the chain rule method, we have: Replace ( 17), ( 18), ( 19) to ( 16) and then put all to (7), the following equation is derived.9) has been demonstrated.Table 1 presents the configuration of the hardware set-up installed from Fig. 3, and Fig. 4.

MODEL
In general, the procedure which must be executed when attempting to identify a dynamical system consists of four basic steps (see Fig. 5)  STEP 4 (Validate Model: et al. [3] added different constant external loads, by a robot in torque control mode.Pneumatic Artificial Muscle (PAM) actuators are now used in the various fields of medical robots.The modern robotics toward applications requires greater friendliness between robot actuator and human operator.PAM actuator has achieved increasing belief to the ability of providing advantages such as high power/weight ratio, full of hygiene, easiness in preservation and especially the capacity of human compliance which is the most important requirement in medical and human welfare field.Therefore PAM has been regarded during Science & Technology Development, Vol 16, No.
[18], authors applied neurofuzzy modeling and control of robot manipulators for trajectory tracking.Ahn and   Anh in [19]  have optimized successfully a pseudo-linear ARX model of the PAM manipulator using genetic algorithm.These authors in(Anh et al., 2007)[20] have identified the highly nonlinear 2-axes PAM manipulator based on recurrent neural networks.Nevertheless, the drawback of all these results is considered the n-DOF manipulator as n independent decoupling joints.Consequently, all intrinsic coupling features of the n-DOF manipulator have not represented in its NN model respectively.To overcome this disadvantage, in this paper, a new approach of neural networks, proposed dynamic inverse neural MIMO NARX model, firstly utilized in simultaneous modeling and identification of the nonlinear 2-TAÏ P CHÍ PHAÙ T TRIEÅ N KH&CN, TAÄ P 16, SOÁ K2-2013 Trang 15 axes PAM robot arm system.The experiment results have demonstrated the feasibility and 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 section 1 for introducing related works in PAM robot arm modeling and identification.The section 2 presents identification procedure of an inverse neural MIMO NARX model using back propagation learning algorithm.The section 3 proves and analyses experimental studies and results considering the contact force variations and highly nonlinear coupling effects of both joints of the nonlinear dynamic system.Finally, the conclusion belongs to
From equation (1), not consider noise component e(t), we have the general form of the discrete ARX model in domain z (with the time delay T=n k =1)

Figure 2 .
Figure 2. Structure of MIMO Neural NARX11 model By this way, the parameters a 11 , a 12 , b 11 , b 12 of linear ARX model now become nonlinear and will be determined from the weighting values W ij and w jl of the nonlinear MIMO Neural NARX model.This feature 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 4) 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).
Specify the training set by: objective of training is then to determine a mapping from the set of training data to the set of possible weights: in some sense are "closest" to the true joint angle outputs y(t) of PAM robot arm.The prediction error approach, which is the strategy applied here, is based on the introduction of a measure of closeness in terms of a mean sum of square error (MSSE) criterion:

Figure 3 .
Figure 3. Block diagram for working principle of the 2-axes PAM robot arm.

Table 1 .
The lists of experimental hardware