Robust Control of Dual-Linear-Motor-Driven Gantry Stage for Coordinated Contouring Tasks Based on Feed Velocity

Contouring control is one of the most important tasks for the dual-linear-motor-driven gantry system, whose control performance determines the in various precision machining equipment, mainly industrial machine tools. Nevertheless, there are some limitations in present results, including the insufficient consideration of system uncertainty and the approximation errors introduced by the computation of the path length. To overcome these issues, the effects of the rotational dynamic and uncertainties are studied and further refined in this article, and a robust control scheme with adaptive neural network identification is designed based on the desired contouring velocity. The method improves existing approaches and avoids the imperfection of the previous one, based on tracking control under global task coordinate frame. The effects of various environmental forces are considered and the B-spline wavelet neural network is employed to approximate unknown dynamic caused by the thrust ripples of the three motors. Some experiments on a real system verify the effectiveness and superiority of the proposed velocity-based contouring control method

Coordinates of the working head along the normal and tangential directions of the desired contour.

I. INTRODUCTION
A S ONE of the core subsystems of many high-end manufacturing equipment, dual-linear-motor-driven gantry system (DLMDGS) has always been a hot topic of study, such as surface mount assembly equipment, photolithography machines, computer numerical control machine tools, optical inspection systems, and so on. With the rapid development of digital and computer technology, the processing and production of traditional industrial products gradually tend to be refined, miniaturized, and high-quality quantification, which demands higher control performance of DLMDGS. In some specific production processes, such as 3-D printing, computer numerical control (CNC) drilling machine, CNC milling machine, laser engraving/cutting, etc., the contour accuracy is the core index to evaluate the performance of the equipment [1], [2]. Therefore, it is of great significance to investigate advanced control schemes for the contouring task of DLMDGS to improve their quality and efficiency.
The structure of typical DLMDGSs is shown in Fig. 1, where OXY is the inertial coordinate frame with the horizontal axis −→ OX and the longitudinal axis −→ OY ,õ represents the midpoint of the crossbeam, α and β refer to the directions along and perpendicular to the crossbeam, respectively, C 1 and C 2 are the centroid points of the crossbeam and the working head. For such a system, complex nonlinearities and uncertainties of the system are the two main factors which greatly affect the performance of the contouring control and have been investigated extensively, including inertia, friction, thrust ripple, external disturbance, etc. [3], [4], [5], [6]. In recent years, with the development of artificial neural network (ANN), in order to further reduce the impact of nonlinearities and uncertainties on system, many ANN-based control methods have been used to approximate the unknown dynamics and design the controller for linear motor system [7]. In [8], the authors employed radial basis function neural network (RBFNN) to fit the unknown function of the system and designed ARC law to improve the motion precision of the linear motor system. The idea has also been used in the biaxial system. Nevertheless, under these ANN-based control methods, better control effect depends on more neural network hidden layer nodes, and at the same time, it will introduce greater computational burden, which will affect the real-time control performance of the system. Moreover, the structure design of most neural networks is blind due to lacking theoretical basis. In [9], B-spline wavelet neural network was successfully utilized to identify the periodic unknown dynamic of the linear motor driven stage with lower calculation burden, which successfully overcomes the above two problems. Inspired by this, the impact of the complex nonlinearities and uncertainties on contour control performance of DLMDGS could be reduced further. In addition, the improvements of the coordination performance of the three motors also greatly increase the contouring control precision of the system. For a long time in the past, the coordination control problem of DLMDGS was simplified into two independent problems to be studied separately, i.e., the contouring control problem of planar biaxial system [10], [11], [12], [13], [14], [15], [16] and the synchronous control problem of two motors along the Y-axis [17], [18], [19], [20], [21], [22], [23], [24], [26]. In this way, although the contouring control performance can be improved to some extent, the motion coordination of the three linear motors is not fully taken into consideration, and therefore, the improvement is limited. The most impressed research for this topic in recent years is from [27]. Chen et al. [27] effectively investigated the structural characteristics of flexible components and their force coupling along Y-axis. Subsequently, a novel three degree-of-freedom (3-DOF) dynamic model was established by directly considering the rotational dynamics of the crossbeam, greatly improving the coordinated control design on the basis of global task coordinate frame (GTCF) approach [27]. For the details, the state of the system is composed of r c , r m , and , where r c represents the contouring error perpendicular to the direction of the contour; r m is the distance of movement along the contour; stands for the rotation angle of the crossbeam. The contouring control problem is transformed into a superposition of the tracking control problem along r m -axis and stabilization problems along other two axes, which improves the coordination of the three motors. However, there are still some improving potentials. On one hand, the uncertainty of the system, such as thrust ripple has not been fully considered, which makes it possible for the contouring control performance to be further improved. On the other hand, as shown in Fig. 2, P h (x h , y h ) is the actual position of the stage at time t; P a (x a , y a ) is the closest point on the contour to the point P h ; P d (x d , y d ) represents to the desired point on the contour. Under GTCF, the tracking error along the r m -axis should be calculated in real time for tracking control, i.e., the length of the curve P a P d in Fig. 2. In most cases, it is with almost no explicit solution, while numerical methods were often used to approximate it [28], which with the drawback of too much computational burden, moreover they are affected by approximation errors.
This article addresses the problem of the coordination contouring control problems of DLMDGS and designs a robust control scheme with adaptive neural network identification in order to break through the bottleneck of control precision. It is designed based on the desired velocity to avoid the calculation of tracking error along r m -axis, which spares the calculational cost to improve the performance of contouring control. For the purpose of higher precision of the contouring control, the complex environmental forces are considered and feed-forward compensated by a B-spline wavelet neural network (BSWNN) approximation. Based on the theory of wavelet analysis, the approximation effect of the designed BSWNN is guaranteed, which is also a guiding for the parameter setting of the approximator. A contrast experiment is carried out to validate the efficiency of the proposed control algorithm, where the superiorities of the velocity-field-based control scheme and the designed BSWNN are shown in the results.
The article is organized as follows: In Section II, the modeling and control objective are elaborated in detail. Proposed control scheme is introduced in Section III. The experimental guarantee appears in Section IV. Finally, Section V concludes this article.

II. MODELING AND PROBLEM STATEMENT
The concerned control problem is formulated, where the dynamic model of DLMDGS is introduced in Section II-A, and the control objective of the contouring task is shown in Section II-B. In order to simplify the analysis, the independent variables of some functions, function matrices, or function vectors are omitted in subsequent analysis, which does not cause confusion.

A. Dynamic Model of DLMDGS
In general, a DLMDGS is usually arranged in the configuration shown in Fig. 1, in which, one motor located on the crossbeam drives the working head moving along it, and the other two parallel motors fixed on both ends of the crossbeam cooperate to complete the motion tasks along Y -axis. Based on the mechanical analysis in [27], for a real DLMDGS, the rotational dynamics of the crossbeam and the translational movement of the moving stage can be captured by the following Euler-Lagrange dynamic equations: where M q , C q , K , A f , and B are the coefficient matrices of inertia, Coriolis/centrifugal force, stiffness, coulomb and viscous; T m is the transformation matrix from F m to u q ; their specific forms are shown in (2) shown at the bottom of this page; δ = [δ 1 , δ 2 , δ 3 ] represents the slow time-varying dynamics of the system; d(t) refers to the resulting force of other unknown disturbances and modeling errors; F e stands for the resultant vector of the environmental forces, mainly including the frictions and thrust ripples, i.e., Based on the theory of wavelet analysis ( [29]), the unknown functions f c i (x m i ) can be fitted by a wavelet series as follows: where W i,n are the weights of the neuron nodes; j i ∈ Z are the scale parameters; n ∈ Z is the translation parameter; ϕ(•) is chosen as a quadratic B-spline mother wavelet ( [9]). Subsequently, . For the studied system, l m , l g , and l e can be measured easily, and h is approximately zero, due to the almost symmetric structure. Subsequently, the uncertainty parameters and functions of the plant (1) are θ, W n , and d(t). For a real DLMDGS, bounds of the unknown parameters and functions can be obtained by preliminary identification ( [4], [5], [27]), which implies that the following assumption can be made reasonably.

B. Problem Statement
Suppose that the desired contour is C (x, y) = 0, whose parametric equation by arc-length parameterization is r . The control objective of the contouring task for DLMGDS is to design input laws for u to drive the working head moving along the desired planar contour r d (t) with feed velocity ν md (t) = ẋ 2 d (t) +ẏ 2 d (t), and meanwhile, ensure the synchronous movement of the two motors in Y -direction. In actual industrial control, r d (t) is usually a piecewise smooth function with finite angular points, and therefore, the contour task can be divided into several subtasks where the desired contour of each subtask is a smooth function. Without loss of generality, r d (t) is assumed to be a second-order differentiable function vector.
Under GTCF ( [16], [27]), the state of DLMDGS is q r = [r c , r m , ] , where the Jacobi matrix from q to q r is in which C x and C y are the partial derivatives of C w.r.t x and y, respectively. Thus,q It should be noted that for an actual system, the rotation angle of the crossbeam is very small, so that J is invertible. Subsequently, in the task space, the dynamic model of the system can be rewritten as follows: It should be noted that a real DLMDGS is always assumed to be a rigid body with the finite load, and thus, from the definitions of M r , C r , and K r , the following properties hold for DLMDGS. Property 1: In any finite work space, M r is a symmetric positive definite matrix satisfying ζ 1 I ≤ M r ≤ ζ 2 I, with ζ 1 and ζ q being positive scalars.
Property 2:Ṁ r − 2C r is skew-symmetric. Property 3: M r , C r , and K r can be linearly parameterized by M x , M y , Jõ, and K .
As mentioned, r m is almost with nonexistent explicit solution. To avoid calculating r m , a control approach without the feedback of r m is required to avoid the above issue. For a contouring task, the working head are expected to be stabilized at zero along r c -and -axes and to move with desired feed speed ν md (t) along r m -axis, which implies that the tracking performance along r m -axis is not critical. Therefore, the control objective for the contouring task of DLMDGS can be described in the task space as follows.
Control objective: For a known desired contour r d (t) with the shape function C and feed velocity ν md (t), design law for u such that r c , andṙ m − ν md are as small as possible in spite of the complex uncertainties of the system.

III. MAIN RESULTS
In this section, an adaptive neural network control law for u is designed to achieve the control objective. First, the following definition is necessary for the subsequent analysis: where ν d = [−k c r c , ν md (t), −k ] ; k c and k are two positive constants. Then, from (5) and (6), σ satisfies where Subsequently, the robust control law for u is designed, whose structure is shown in Fig. 3 where u a and u nn refer to the feed-forward term to compensate the unknown forces F θ and F nnr , respectively; u z is the position feedback term to avoid the position deviation; K z > 0; u s1 and u s2 represent the terms of the state feedback compensation and robust feedback compensation, respectively; K σ > 0; K 0 = Diag(1, 0, 1); B(F nn , F) is a vector valued function, whose ith component is defined as F nni and F i are the ith component ofF nn and F respectively; S(•) represents the saturation function; the weightsŴ n and parameter estimationθ are updated bẏ in which 0 < Υ w,n ∈ R 3×3 and 0 < Υ θ ∈ R 13×13 are the step matrices of the updating laws; τ w,n and τ θ are the updating laws forŴ n andθ; P w (•) : R 3 → R 3 and P θ (•) : R 13 → R 13 are two projection mappings, whose ith element is defined as follows: According to (9) and (10), it can be obtainted that ifθ(0) ∈ [θ m , θ M ] and |u nn (0)| ≤ F(q,q, t), then for all t ≥ 0 Then, the robust law u s2 is designed satisfying whereF nnr = J − u nn − F nnr ;F θ = u a − F θ ; ε > 0 is a constant. An example for u s2 is shown as follows: in which H(q r ,q r , t) is a known function satisfying The stability of the closed-loop system consisting of the plant (5) and the control law (8)- (14) is illustrated in the following theorem.
Theorem 1: For any system with the dynamic model (1), if the Assumption 1 and Properties 1 to 3 hold, then, in any finite work space and feed velocity, the concerned system is ultimately uniformly bounded (UUB) under the control scheme (8)- (14) with arbitrary updating laws τ θ and τ w,n .
The following lemma is required to prove Theorem 1. Lemma 1: Consider the following two systems: are the solutions of the system (I) and (II) with the initial state Proof: Consider the derivative of V (x) = x x for (I) and (II) Noting that K > 0 and invoking the comparison theorem ( [30]), Lemma 1 holds. Proof of Theorem 1: Consider the system (1) under the following control law: where u s , u a , and u nn are the same as those in (8). Consider the following positive definite function: Invoking Property 2, (7), (12), and (18), the derivative of V s satisfieṡ where V s0 is the initial value of V s . Noting that e −2k min t converges to 0 as the time goes on, therefore, it can be obtained that lim t→∞ V s ≤ ε/(2k min ), which implies that the closed-loop system consisting of (1) and (18) is UUB. Comparing the control laws (8) and (18), and according to Lemma 1, Theorem 1 holds. Remark 1: Actually, u nn is in the form of neural network with four layer, as shown in Fig. 4, where the details are given as follows.
L1: The first layer is the input layer with three nodes whose inputs are x m 1 , x m 2 , and x m 3 , respectively. The activation function for the node with the input x m i is The second layer is the wavelet layer. There are three types of neuron nodes in this layer, whose inputs are j 1 x m 1 , j 2 x m 2 , and j 3 x m 3 , respectively. The number of neuron nodes is determined by the travel lengths of three motors and the compact support of the mother wavelet. The activation function for each node is ψ 2,i,n (•) = ϕ(• − n), where the subscript i, n refers to the nth node with the input j i x m i . The connection weight between the input and wavelet layers is constant and defaults to 1. L3: The third layer is the transition layer. There are three nodes: N T 1 , N T 2 , and N T 3 . The node N T i is connected to those with the activation function ψ 2,i,n in the wavelet layer, whose connection weight isŴ i,n . The activation function for all nodes is ψ 3,i (•) = •, i = 1, 2, 3.

L4:
The fourth layer is the output layer. There are three nodes which are fully connected with the nodes in transition layer with the weights T m,ı,j . T m,ı,j represents the element on ı row, j column of the matrix T m . The activation functions are ψ 4,i = B i (•, F i ). Remark 2: In theory, the control scheme (18) can also achieve good control performance. However, in actual contouring control task, it may happen that the stage only tracks part of the desired contour, especially at the end of the desired trajectory, due to the lack of compensation for tracking error along r m − axis. In addition, the position deviation along r m − axis also seriously affects the performance of the contouring error by reason of the coupling between the motions along X− and Y − axes. Therefore, u z is designed to compensate for the position deviation along the curve to overcome these problems.
Theorem 2: If there is only parametric uncertainty in the system after a while, i.e., there exists t a > 0 such that d(t) = 0 for all t ≥ t a ≥ 0, the closed-loop system is asymptotically stable under the control scheme (8) and the following updating laws: Proof: According to Lemma 1, it is only needed to prove that the closed-loop system is asymptotically stable under the control scheme (18).
Consider the following positive definite function: Taking the derivative of V a , it becomeṡ Substituting the control law (18) and the updating laws (22) into Based on (10) and (22), it is obvious that Subsequently, (30) becomeṡ which implies that the system is asymptotically stable under the control law (18). Remark 3: For a real system, it is a tough assumption that there is only parametric uncertainty in the system. However, it is of great significance to the theoretical research on high precision control of linear motor. The results show that arbitrary control precision can be achieved for the closed-loop system as long as the structure of the dynamic model is accurate enough, even though there are unknown paramters in the system. Therefore, the high precision control problem is transformed into how to approximate the unknown function in the model with high accuracy.

A. Experiment and Environment Setup
To validate the effectiveness of the proposed control structure and the designed B-spline wavelet neural network, tests of four control approaches were carried out on a real DLMDGS shown in Fig. 5. The control input voltages are calculated by programming at TwinCAT3.0 in real-time patched Microsoft Windows OS, after which the SPLC, as the motion controller, works out the real-time driving current commands for each motor driver to drive the moving platform. The maximal travel lengths of the X-and Y-axis are 700 and 650 mm, respectively. Each motor is equipped with a grating linear displacement sensor with the accuracy of 50 nm to measure the displacement. After preidentification of the equipment, the nominal parameter values and their ranges of the equipment can be obtained, where the details are listed in Table I. In addition, it should be emphasized that for an actual DLMDGS, the rotational angle of the crossbeam is pretty small, and therefore, it is reasonable that cos ≈ 1, sin ≈ . The details and parameter settings of the four control schemes are listed as follows.
C1: (PID) The proportional-integral-derivative (PID) control law in the following form was employed to drive the system: was the desired value of q r ; s t was approximate by the method in [16]. The parameters were set as K p = Diag(3000, 2000, 4000), K d = Diag (25,20,15), and K i = Diag(5000, 2000, 40000). C2: (ARC) The adaptive robust control approach proposed in [27] was tested on the equipment, which consists of feedforward term to compensate the nonlinear dynamics and robust feedback term to ensure the stability of the system. The parameters of the robust term were set as Λ r = Diag(120, 100, 270), K s1 = Diag (25,20,15), K re = (500, 500, 500), and K ra = (500, 500, 500). The parameters of the recursive least square method were set as μ = 0.02, κ = 0.1, and ρ M = 5000. The initial value of Γ l was set as Γ l (0) = I 13×13 .
C4: (Proposed method) Drive the equipment under the control scheme (8) with adaptive laws (9). The B-spline mother wavelet was chosen as The scale parameter was set as j 1 = j 2 = j 3 = 100. The step parameter of the weight updating was set as Υ w,n = Diag(0.02, 0.02, 0.02). In addition, other parameters of the control law and updating law were set as the same as C3. Actually, the above settings ensure that the corresponding coefficients of the state feedback are the same under the three methods. All testings are completed in the same surroundings so that ensure the fairness of the contrast experiment, and the control intervals of them are fixed as 0.25 ms. The desired contour is an ellipse with the major axis of a = 0.1 m and the minor axis of b = 0.075 m, i.e., the shape function of the desired contour is The feed angular velocity is set as ω = 4 rad/s, on which the desired trajectories for Xand Y -axes are defined as For smooth startup of the equipment, the actual feed angular velocity is set to start at 0 rad/s and gradually increase to 4 rad/s over time, i.e., ω = 4(1 − e −t ) rad/s. The concerned testing segment is the last 10 s of the testing, where the feed angular velocity is 4 rad/s and the performance of the system goes to steady state.

B. Experimental Results and Discussions
Figs. 6 to 8 plot the graph of the motion paths, contouring errors, and crossbeam rotation angles of the system under the considered three control approaches. Although Fig. 6 shows that the working head of the equipment moves along the desired contour with high precision under all four control methods, there are still some differences in performance for the differences of the feed-forward compensation for system nonlinearity.
Under the control approach C1, the contouring error and rotation angle of the system were restrained by the proportional, integral, and differential linear feedback of the errors without any compensation for the nonlinear dynamics of the system. Consequently, the results were unsatisfactory both on the precision of the contouring control and the motion coordination of three motors. Compared with C1, the method C2 further compensates for the friction forces feed-forward according to the parameters identified through the recursive least squares method, under which the precision of the contouring error is enhanced considerably compared with C1. Nonetheless, the estimation of the tracking error along r m -axis was required for both the two approaches, which introduced errors in numerical calculation to affect the contouring performance.
Compared with the results under the method C2, the approach C3 considered the velocity feedback along r m -axis, where the position compensation was carried out by the PID feedback of the tracking errors of three motors. Since the calculation of the tracking error along r m -axis was avoided under C3, the contouring precision was improved further, which was validated by the experimental results shown in Fig. 7. Therefore, the structure of the C3 control approach is effective in improving the coordination of the motion of three motors. However, as shown in Fig. 8, due to the lack of compensation for the thrust ripples of three motors, the results of the synchronization control under the three methods were unsatisfactory.
For the controlled equipment, the frictions caused by the guides G2 and G3 are virtually the same, while the rotation of the crossbeam is predominantly caused by thrust ripples of the motors M2 and M3. Under the proposed method, the thrust ripples of three motors were compensated by the designed B-spline wavelet neural network. Hence, the synchronization of the motion of the motors M2 and M3 was significantly enhanced, which is validated by the results shown in Fig. 8.   1) The maximum value of the contouring error over the concerned testing segment: 2) The root-mean-square value of the contouring error over the concerned testing segment: where N τ is the number of sampling over the concerned testing segment; t 0 represents the initial time of the concerned testing segment.
3) The maximum value of the rotation angle over the concerned testing segment 4) The root-mean-square value of the rotation angle over the concerned testing segment From the results, compared with the control algorithm proposed in [27], the contouring control precision of the system under the proposed velocity-based control algorithm was improved at least 62.51% and 61.61% for the maximum value and the root-mean-square value. In addition, with the designed B-spline wavelet neural work, the values of E max and E mean are reduced more than 18.96% and 38.16% compared with the method without neural network.

V. CONCLUSION
An adaptive neural networks control scheme was proposed for coordinate contouring tasks based on the feed velocity. On the basis of stability, the proposed method ensures the performances of both the contouring control and synchronization control. Under the proposed control law, only the velocity along the desired contour is required instead of the arc length, which avoids the calculation of the state r m . Therefore, it avoids the numerical approximation error and saves the calculation cost. For the complex uncertainties of the system, both the friction and thrust ripple are considered and compensated by a linear friction model and B-spline wavelet neural networks, respectively. The experimental results not only verified the theoretical results in Section III, but also showed that the proposed method is superior to the existing methods. Future research will focus on the integrated design of the optimization of feed speed and control algorithm, which may improve the coordination and control precision of the system further. He is currently a Postdoctoral Researcher with the Research Institute of Intelligent Control and Systems, Harbin Institute of Technology. His research interests focus on adaptive robust control and mechatronics.