Adaptive Robust Control of a Linear Motor Driven Precision Industrial Gantry with Improved Cogging Force Compensation

This paper proposes a new model for cogging forces of linear motor systems. Sinusoidal functions of positions are used to capture the largely periodic nature of cogging forces with respect to position effectively while B-spline functions are employed to account for the additional aperiodic part of cogging forces. This model is experimentally demonstrated to be able to capture both the periodic and non-periodic characteristics of cogging force while having a linear parametrization form which makes effective on-line adaptive compensation of cogging forces possible. A discontinuous projection based desired compensation adaptive robust controller (DCARC) is then constructed for linear motors, which makes full use of the proposed cogging force model for an improved cogging force compensation. Comparative experimental results are obtained on both axes of a linear motor driven Anorad industrial gantry having a linear encoder resolution of 0.5 μm. Experiments are done with each axis running separately to compare the three algorithms: DCARC without THE WORK IS SUPPORTED IN PART BY THE US NATIONAL SCIENCE FOUNDATION (GRANT NO. CMS-0600516) AND IN PART BY THE NATIONAL NATURAL SCIENCE FOUNDATION OF CHINA (NSFC) UNDER THE JOINT RESEARCH FUND FOR OVERSEAS CHINESE YOUNG SCHOLARS(GRANT NO. 50528505) †THE STATE KEY LABORATORY OF FLUID POWER TRANSMISSION AND CONTROL cogging force compensation, DCARC with sinusoidal cogging force model compensation, and DCARC with the proposed cogging force model compensation. The results show that DCARC with proposed model compensation achieves the best tracking performance among the three algorithms tested, validating the proposed cogging force model. The excellent tracking performances obtained in experiments also verify the effectiveness of the proposed ARC control algorithms in practical applications. INTRODUCTION The problem of linear motor controls has received significant attentions in recent researches [1–5]. In controlling ironcore linear motors with permanent magnets, cogging forces, which arise due to the strong attraction forces between the ironcore and the permanent magnets, is a common phenomenon that cannot be ignored [4]. The control performances may deteriorate in presence of cogging forces when they are not appropriately accounted for. Thus significant research efforts have been devoted to the modeling and compensation of cogging forces [4, 6–9]. In [4], feedforward compensation terms, which are based on an off-line experimentally identified model of first-order approximation of cogging forces, are added to the position controller. Since not all magnets in a linear motor and not all linear motors Proceedings of DSCC2008 2008 ASME Dynamic Systems and Control Conference October 20-22, 2008, Ann Arbor, Michigan, USA 1 Copyright © 2008 by ASME DSCC2008-2182 of the same type are identical, feedforward compensation based on off-line identification model may be too sensitive and costly to be useful. In [6–8], the cogging force is assumed to be periodic functions with respect to position. Thus, Fourier expansion has been utilized, with the choice of first few significant terms, to represent the cogging force. Based on this, compensation algorithms have been designed and implemented. In another type of research such as [9], a neural-network-based learning feedforward controller is proposed to reduce positional inaccuracy due to cogging forces or any other reproducible and slowly varying disturbances. But this method simply assumes cogging force to be a general nonlinear function with respect to position, without considering its periodic nature. Furthermore, overall closed-loop stability is not guaranteed. In fact, it is observed in [9] that instability may occur at high-speed movements. Cogging forces have periodic nature, but due to the complicated physical interactions between different magnets, they may show some aperiodic characteristics. In this paper, we conduct explicit measurement of cogging forces on both axes of a linear motor driven Anorad industrial gantry. The measured cogging forces exhibit certain periodic characteristics with respect to position, which can be represented by sinusoidal functions of positions with unknown weights. However, the amplitude of the weights changes significantly with position, as can be observed from the measurement. Based on this observation, a new model is proposed, considering both the periodic and non-periodic characteristics of cogging forces. We use sinusoidal functions of position to be part of basis functions. B-spline functions are then constructed to capture the changing amplitudes of sinusoidal functions with respect to position. The proposed model is demonstrated experimentally to be able to approximate the measured cogging forces very well, by using a least-squares curve fitting method. As opposed to the traditional neural-network-based blind modeling of cogging forces [9], the compactness and the linearparametrization form of the proposed model makes it a perfect choice for on-line adaptive cogging force compensation as well. To this end, a suitable model-based compensation algorithm should be designed. The idea of adaptive robust control (ARC) [10, 11] incorporates the merits of deterministic robust control (DRC) and adaptive control (AC), which guarantees certain robust performance in presence of uncertainties while having a controlled robust learning process for better control performance. The ARC has been extended into the desired compensation ARC (DCARC) in [12]. On the second half of this paper, a DCARC algorithm is designed making full use of the proposed cogging force model. The algorithm is then tested on a two-axes iron-core linear motor driven industrial gantry with severe cogging force effect. Comparative experimental results with each axis running separately show an improved performance over the previously obtained results with DCARCs [13] for both axes after using the proposed cogging force model, though the two axes have different measured cogging force patterns which are assumed to be unknown to users in the controller designs. These results validate the usefulness of the proposed cogging force model for linear motor controls and the excellent tracking performance of the proposed DCARC in practical applications. MODELING OF SYSTEMS AND PROBLEM FORMULATION The dynamics of 1-DOF linear motor systems can be represented by the following equation [14, 15]: Mẍ+Bẋ+Fc(ẋ)+Fr(x) = u+d (1) where x represents the position of linear motor, with its velocity and acceleration denoted as ẋ and ẍ respectively. M and B are the mass and viscous friction coefficient, respectively. Fc(ẋ) is the Coulomb friction term which is modeled by : S f (ẋ) = Af S f (ẋ) (2) where Af represents the unknown Coulomb friction coefficient and S f (ẋ) is a known continuous or smooth function used to approximate the traditional discontinuous sign function sgn(ẋ) for effective friction compensation in implementation. In Eq. (1), Fr(x) represents the position dependent cogging force. u is the control input force. d represents the lumped effect of external disturbances and various types of modeling errors. Traditionally, cogging forces are assumed to be continuous periodic functions with respect to position [14]. As a result, it can be represented by Fr(x) = ∑i=1(Si sin( 2iπ P x)+Ci cos( 2iπ P x)) where P is the pitch of magnet pairs and Si andCi are some constants. Practically, we can select the first few important terms and ignore all higher terms, i.e. i is from 0 to a positive integer n. This equation can well explain the periodic phenomena of cogging force and has been widely used in compensation algorithms [6–8,13]. However, due to many complicated physical effects, such as the differences among magnets, the actual cogging force may not be exactly periodic. In the experimental section of this paper, we measure the cogging forces explicitly using a force sensor. It can be observed that the amplitudes of sinusoidal functions vary with the change of position. Thus, using the periodic assumption may give an inaccurate model of cogging force and may deteriorate the resulting control performance. In order to capture the actual cogging force more precisely to achieve better tracking control performances, it is necessary to assume Si and Ci to be functions of position, namely Si = fSi(x) and Ci = fCi(x). With such a varying amplitude modification to periodic functions, the cogging force model becomes: Fr(x) = n ∑ i=1 ( fSi(x)sin( 2iπ P x)+ fCi(x)cos( 2iπ P x)) (3) 2 Copyright © 2008 by ASME The selection of fSin and fCin should also make full use of available physical characteristics of cogging forces to ease the design of effective on-line adaptive cogging force compensation. To this end, B-spline functions [16] are utilized to give a mathematical model of fSi and fCi respectively. Namely, fSi and fCi are chosen as: fSi(x) = m ∑ j=1 Nj,k(x)Si j (4) fCi(x) = m ∑ j=1 Nj,k(x)Ci j (5) where ⎧⎪⎨ ⎪⎩ Nj,k(x) = { 1 when x ∈ [Xj, Xj+1) 0 else k = 1 Nj,k(x) = x−Xj Xj+k−1−Xj Nj,k−1(x)+ Xj+k−x Xj+k−Xj+1 Nj+1,k−1(x) k ≥ 2 (6) where k is the order of B-spline and m is the number of control points needed. [X1,X2, · · · ,Xm+k−1,Xm+k] is the knot vector, with Xj+1 ≥ Xj defined to be as follows. Let m be the number of magnet segments on the linear motor axis. If a kth order B-spline function is used, then Xk is defined to be the position of the first magnet and Xk+m the last magnet, with Xk+ j as the position of the j+ 1-th magnet. By the construction of linear motor, Xk+ j = Xk + jP, j = 1, . . .m. So define X1, · · · ,Xk−1 as Xk− j = Xk− jP, j= 1, . . .k−1. Fig. 1 illustrates the linear motor position x on the range defined by knot vector. Assuming a linear motor of 5 magnet segments, the shapes of fSi(x) for the order of k = 1,3 with the values of control points described by the green dots are shown in Figs. 2 to 3 respectively. X1 X2 X3 Xk Xk+1 Xm+k−1Xm+k x Figure 1. Illustration of Linear Motor Position Range on B-Spline Interval Using B-spline function has the following merits: 1: The increment of neighboring elements in knot vector of Bspline function is selected as the physical pitch of magnets on the linear motor axis, i.e., Xj+1 −Xj = P. This means that the value of B-spline function changes with a unit of magnets’ pitch. Physically, it can interpret the changing amplitude of sinusoidal function caused by difference of each magnet. 2: B-spline function is linear to the control points (Si j and Ci j). The resulting cogging force model described by Eqs. (3) to (4) is thus linearly parametrized by the control points (Si j X1 X2 X3 X4 X5 X6 1 1.5 2 2.5 3 3.5 4 x (m) A m p lit u d e (V ) B−spline of order 1 Value of B−spline function Control points and convex hull Figure 2. B-Spline of Order 1 X3 X4 X5 X6 X7 X8 1 1.5 2 2.5 3 3.5 4 x (m) A m p lit u d e (V ) B−spline of order 3 Value of B−spline function Control points and convex hull Figure 3. B-Spline of Order 3 and Ci j) with known basis functions. Such a model significantly simplifies the on-line estimate of unknown control points for adaptive compensation of cogging forces. 3: The basis function Nj,k(x) is only active when x ∈ [Xj,Xj+k) and is zero in other regions. Such a nice property is especially preferable for real-time adaptive controls because in every sampling period, we only have to fetch a small portion of parameter estimates from memory for compensation and update. For example, when x ∈ [Xl ,Xl+1) for some l, only Si(l−k+1), Ci(l−k+1), · · · , Sil , Cil are needed for update and compensation. All other coefficients need not be considered. Thus the algorithm is economic, especially when k is chosen small. Combining Eqs. (3) (5), the cogging force Fr(x) is put in a concise form as: Fr(x) = A T r Sr(x) (7) 3 Copyright © 2008 by ASME where Ar = [S11, C11, ..., S ji, Cji, ..., Smn, Cmn] T ∈ R2mn is the vector of unknown control points and Sr = [N1,k(x)sin( 2π P x), N1,k(x)cos( 2π P x), ..., Nj,k(x)sin( 2iπ P x), Nj,k(x)cos( 2iπ P x), ..., Nm,k(x)sin( 2nπ P x), Nm,k(x)cos( 2nπ P x)] is a vector of known basis functions. With this cogging force model, the linear motor dynamics (1) can be linear parameterized as: Mẍ+Bẋ+Af S f (ẋ)+A T r Sr(x)−dn = u+ d̃ (8) where dn denotes the nominal value of d and d̃ = d− dn represents the time-varying portion of the lumped uncertainties. (8) can also be put in a state space form of