Iterative Learning Control of a Single-Link Flexible ... - IEEE Xplore

2013 Annual IEEE India Conference (INDICON)

Iterative Learning Control of a Single-Link Flexible Manipulator Based on an Identified Adaptive NARX Model Dinesh Mute

Subhojit Ghosh

Bidyadhar Subudhi

Industrial Electronics & Robotics Lab Department of Electrical Engineering NIT Rourkela, 769008, India. E-mail: [email protected]



Abstract—In this paper an iterative learning controller (ILC) is designed based on the identified model of a single-link flexible manipulator (SLFM). As the system is nonlinear and time-varying so to meet the demands of the control system design, an adaptive nonlinear autoregressive with exogenous input (NARX) model is identified using the input/output experimental data. Tuning of the ILC controller is carried out using least square method. Simulation results demonstrate the potential of the NARX model based ILC controller for precise rotation tracking of a single-link flexible manipulator with suppressing link vibration. Index Terms—NARX, Iterative learning control (ILC), Least square estimation, single-link manipulator, Identification.

I.

INTRODUCTION

Flexible manipulators have many advantages over rigid manipulator such as faster executable motions, higher payload carrying capacity, lower energy consumption and light weight. Because of above advantages flexible manipulators are widely used in various applications, like space exploration and nuclear plant. Hence the control design problem of flexible manipulator has attracted considerable attention. For designing a proper controller, the accurate model of flexible manipulator is necessary. Many existing literature provides the dynamic model of flexible manipulator based on the Bernoulli-Euler beam theory [1], in which the dynamics are represented in the form of a partial differential equation. The final dynamic model of flexible manipulator given by finite element method [1][2] and assumed mode method [3][4] is quite complex and require complete knowledge of flexible manipulator dynamics. So, to get round this difficulty, in the present work a NARX model of flexible manipulator is identified (i.e. parameter estimation from experimental data). The advantages of using NARX models are: there is no need of accurate knowledge of the flexible manipulator dynamics and physical parameters of the system. For estimation of a flexible manipulator model different identification methods have been applied. Rovner and Connon [5] used an autoregressive moving average (ARMA) model for represent the single-link flexible manipulator, and also in [6]

an ARMA model with weighted recursive least square (RLS) algorithm is adopted for parameter estimation. Yurkowich and Tzes [7] presented an identification and control of a single-link flexible manipulator using on-line frequency domain linear model. A major limitation with the above techniques is the assumption of linear dynamics in the model development. In this required, in the present work, the model is identified considering nonlinear and time-varying dynamics. The applicability of ILC for flexible manipulator has been exhibited in [10-13]. The identified adaptive NARX model has been used for tracking a desired trajectory using the incremental structure of ILC. The organization of the paper is as follows: Section II presents the dynamic model of a single-link flexible manipulator. In section III, the adaptive NARX model representation of system is described. In section IV, the incremental structure of ILC control is developed based on identified adaptive NARX model Section V contains the simulation results of both NARX model idenfication and ILC controller design. finally section VI provides the conclusion. II.

DYNAMIC MODEL OF A FLEXIBLE MANIPULATOR

The dynamics of a flexible manipulator at an arbitrary spatial point along the link at an instant of time can be represented using Euler-Bernoulli beam theory. A partial differential equation for deformation in the link due to flexibility can be written as [1]

(EI)i

∂ 4 ui (xi , t) ∂ 2 ui (xi , t) + (ρ) = 0, i = 1 i ∂x4i ∂t2

(1)

where, (ρ)i is the uniform mass density of the ith link and (EI)i is the flexural rigidity of the ith link. The solution of (1) is derived using the assumed mode method. The vibration along the link can be express as ui (xi , t) = φij (xi )∂ij (t), n = 2, i = 1

978-1-4799-2275-8/13/$31.00 ©2013 IEEE

(2)

where, φij (xi ) is the j th mode shape associated with link and it is a function of displacement along the length of the flexible manipulator and ∂ij (t) is the generalized co-ordinate of the beam. Final dynamic equations of the motion of manipulator using lagrangian equation can be written in the closed form as [4] ˙ q) h1 (θ, q, θ, ˙ 0 τ θ¨ M (θ, q) + +K = (3) ˙ q) q 0 q¨ h2 (θ, q, θ, ˙ where, M is a positive-definite symmetric inertia matrix, h is a vector of a centripetal forces, K is the diagonal stiffness matrix, τ is a column vector consisting of control torque at the joint location. III.

ADAPTIVE NARX IDENTIFICATION MODEL

To identify the non-linear and time-varying dynamics of the flexible manipulator, an adaptive NARX model is used. The general form of a single-input single-output NARX model to represent the nonlinear system can be defined as [8-9] y(k) = f (y(k−1), ..., y(k−ny ), u(k−1), ..., u(k−nu ))+ξ(k) (4) where, the function f (.) represents a nonlinear function consisting of the cross product and higher order polynomial terms. The degree of the power terms in y(k) and u(k) is referred to as the degree of nonlinearity, and ny , nu are the maximum delay in the output and input respectively and ξ(k) is the white noise signal. Assuming f (.) as a polynomial of degree l gives the following representation of model y(k) = θ0 +

n X

θi1 xi1 (k) +

n n X X

1), ..., pn+ (k) = u2 (k − 1), ..., pM (k) = y(k − ny )u(k − nu ). The incorporation of time-varying dynamics demands an adaptive model with varying parameters. In this regard, in the present work, the parameter are iteratively updated by considering input-output data over different time intervals. Over each interval, the parameter estimation is carried out using least-square method. The identification accuracy, in terms of replicating the experimental dynamic is found to be much better for the adaptive model than the fixed structure model. IV.

ITERATIVE LEARNING CONTROL BASED ON ADAPTIVE NARX MODEL

In this section, Iterative learning based control technique is used to enhance tracking performance of the single-link robotic manipulator, using the error inputs obtained from each trial. The ILC has been implemented using the least square algorithm. It comprises feed-forward learning controller and a linear forward controller (Fig.1). In linear forward path P ID1 controller is used for stabilizing the system and the feedforward path incremental structure of ILC controller is used for tracking purpose. The advantage of using incremental structure is to avoid problems of filtering cause by measurement noise, obtained from sensor.

θi1 i2 xi1 (k)xi2 (k) + ...

i1 =1 i2 =i1

i1 =1

+ξ(k) (5) where n = ny + nu and x1 (k) = y(k − 1), x2 (k) = y(k − 2), ..., xny (k) = y(k − ny ), xny +1 (k) = u(k − 1), xny +2 (k) = u(k − 2), ..., xny +nu (k) = u(k − nu ) The polynomial coefficients θ0 are unknown parameters to be determined from the given input-output data and xi are delayed input and output terms. Above equation can be written in the linear regression function form as y(k) =

M X

pi (k)θi + ξ(k)

(6)

i=0

where the number of unknown parameter M depends on the value of ny , nu and l , whereas p0 and pi (k) are regressors as defined by xi . For a polynomial function representing quadratic non-linearity of order l , the M monomials are given as M =1+n+

n(n+1) 2

Fig. 1.

NARX model based iterative learning controller for SLFM.

The design of ILC controller is carried out in two phases. In the first phase, an ILC update law is carried out to yield the ideal input and output signals of the overall ILC augmented control system. In the second phase, signals ‘e’(error) and ‘∆u’(the change in reference input) are used to calculate the tuning parameters of P D2 controller by using a standard leastsquares (LS) algorithm. The update law is given as de(t) dt ∆yd,i+1 (t) = ∆yd,i (t) + λei (t + 1) ∆u(t) = kP e(t) + kD

where λ is the learning rate. p1 (k) = y(k − 1), ..., pny +1 (k) = u(k), ..., pn+1 (k) = y 2 (k +

(7) (8)

A PD-type update law [11] has been adopted for ILC. In this update law the change in the reference input is directly proportional to the error and its derivative term in each trial. Once the actual error (e) becomes very small (within bound of 0.002 m) the ILC stops updating. The ideal input ‘e’ and output ‘∆u’ for a cycle of the reference signal is now available for the next phase. Using e and ∆u, parameter of P D2 is determined by using least square method. ∆u(t) = ϕT (t)θ

(9)

where,

(a) T

T

θ = [kp2 kD2 ] and ϕ (t) = [e(t)

de(t) dt ]

kp2 and kD2 are the tuned gain parameters of the P D2 controller. Using LS algorithm, we have θ = (φT φ)−1 φT U, where, φ = [ϕT (1) ϕT (2) ... ϕT (N )]T (b)

U = [∆u(1) ∆u(2) ... ∆u(N )]T and N is the number of data used in estimation.

V.

Fig. 2. Experimental input (a) and output (b) data for the identification of adaptive NARX model.

RESULTS AND DISCUSSION

For identification of adaptive NARX model the SLFM is excited with band-limited white noise (noise power = 20 watt and sampling time = 0.001 sec) shown in figure 2(a) and the corrosponding output in term of rotation angle is shown in figure 2(b). The experiment is performed for 2 sec with sampling time 0.001 sec i.e a record of 2000 experimental samples are considered. The order and number of delayed terms in the input and output are considered as 2 and 3 (nu = ny ) respectively. Simulated results of the identified NARX model in figure 3 depicts that the estimated adaptive model gives the correct representation of SLFM dynamics. The error between experimental output and estimated model output is quite low (figure 3). The ILC controller is designed based on the identified adaptive NARX model. The proposed ILC controller is tested for sinusoidal signal. Figure 4 shows that the model and desired output for the first iteration. As the iteration increases the output of SLFM matches with the desired trajectory. The model output converges to the desired trajectory in the tenth iteration shown in figure 5. The effectiveness of the ILC in tracking the desired trajectory is depicted in the iterative variation of mean square error in figure 6. Figure 6 reports the reduction in mean square value of error with respect to iteration number.

Fig. 3. Experiment output versus the simulated output of the identified adaptive NARX model.

VI.

CONCLUSION

For designing a proper controller for tracking a desired trajectory, the single-link flexible manipulator system is first identified by using adaptive NARX model from experimental data. Simulation results shows that the identified adaptive NARX model gives good representation of SLFM and avoids complexity resulting from the use of partial differential equation based model.

Fig. 4.

Fig. 5.

Desired output versus the actual output in the first iteration.

Desired output versus the actual output in the tenth iteration.

Based on the identified adaptive NARX model, we propose the incremental structure of ILC controller for a single-link flexible manipulator. The proposed ILC based approach is found to be quite effective in tracking a desired trajectory over a definite time interval. R EFERENCES [1] P. W. Usoro, S. S. Mahil, and R. Nadira, “A Finite Element/Lagrange Approach to Modeling Lightweight Flexible Manipulators, part one: Onelink System”, Sensors and Controls for Automated Manufacturing and Robotics, 1984. [2] M. O. Tokhi and Z. Mohamed, “Finite element approach to dynamic modelling of a flexible robot manipulator: performance evaluation and computational requirements”, Commun Numer. Meth. Engg., vol.15, pp.669-678, 1999. [3] A. De Luca and B. Siciliano, “Trajectory control of a non-linear one-link flexible arm”, International Journal on Control, vol.50, no.5, pp.16991715, 1989. [4] A. De Luca and B. Siciliano, “Closed-form dynamic model planar multilink lightweight robots”, IEEE Trans. on System Man and Cybernetics, vol.21, no. 4, pp. 826-839, 1991. [5] D. M. Rovener and R. H. Cannon, “Experiment toward on-line identification and control of a very flexible one-link manipulator”, The International Journal of Robotics Research, vol.6, pp.3-19, 1987. [6] S. Yurkovich, K. L. Hillsley and A. P. Tzes, “Identification and Control for a Manipulator with Two Flexible-Links”, Proc. IEEE Conf. on Decision and Control, Honolulu, Hawaii, pp.1995-2002, Dec.1990. [7] S. Yurkovich and A. P. Tzes, “Experiments in Identification and Control of a Flexible-Link Manipulator”, IEEE Contr. Syst. Mag.,pp. 4147, Feb.1990. [8] L. Piroddi and W. spinelli, “An identification algorithm for polynomial NARX models based on simulation error minimization”, International Journal of Control vol.76, no.17, pp.283-295, 2003.

Fig. 6.

Iterative variation of mean square error (MSE) of the ILC.

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