The modular design of trajectory compensation based on ATCF for precision motion control

Mechanical Systems and Signal Processing 135 (2020) 106393

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

The modular design of trajectory compensation based on ATCF for precision motion control Ze Wang a,b, Chuxiong Hu a,b,⇑, Yu Zhu a,b, Ming Zhang a,b, Chi Zhang c a

State Key Lab of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China Beijing Key Lab of Precision/Ultra-Precision Manufacture Equipment and Control, Tsinghua University, Beijing 100084, China c CAS Center for Excellence in Nanoscience, Beijing Key Laboratory of Micro-nano Energy and Sensor, Beijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Science, Beijing 100083, China b

a r t i c l e

i n f o

Article history: Received 5 December 2018 Received in revised form 31 July 2019 Accepted 23 September 2019

Keywords: Contouring motion control Reference trajectory compensation Task coordinate frame ATCF Contouring error

a b s t r a c t To achieve excellent contouring motion control without affecting the structure of original closed-loop controller, a modular design of trajectory compensation based on accurate task coordinate frame (ATCF) is proposed in this paper for precision multi-axis systems. Specifically, the contouring error point, i.e., the point located on the reference contour which is closest to the actual one, is obtained by the numerical computation method. Due to the accurate calculation through the above method, the calculated contouring error point can almost overlap with the actual one even under extreme contouring tasks with high speed, large curvature and sharp corner. Based on the estimated contouring error point, actual point and the desired point, ATCF is constructed. The coordinates of the desired point in ATCF represent the distance error and the contouring error, respectively. Through the trajectory compensators with reasonable gains adjustment, these two errors are fed back to the position loop of the system to modify the reference trajectory and to improve contouring control performance. Comparative experiments under various contouring control tasks are conducted to validate the practical effectiveness of the proposed ATCF scheme. The experimental results illustrate that the proposed scheme can achieve not only nearly perfect contouring error estimation but also obvious improvement of contouring accuracy compared with individual axial control and CCC. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Multi-axis motion systems with contouring following ability are widely used in the practical applications, e.g. machine tools and manipulators [1–3]. Contouring performance, i.e., the deviation between the actual contour and the reference one, is one of the most important indexes for multi-axis motion systems. Due to the lag characteristics of axial motion control [4,5] and the incompatibility of dynamics between different axes [6], contouring performance cannot be evaluated well by the individual tracking control error of each axis. Therefore, as a more intuitive indictor to evaluate the above deviation, contouring error is defined as the shortest distance between the actual position and the reference contour. Hereby, effective suppression of contouring error is an important research topic for multi-axis cooperation control. In conventional motion control methods, contouring error is suppressed indirectly by reducing the tracking errors of individual axes. To achieve this

⇑ Corresponding author. E-mail address: [email protected] (C. Hu). https://doi.org/10.1016/j.ymssp.2019.106393 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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Z. Wang et al. / Mechanical Systems and Signal Processing 135 (2020) 106393

goal, various advanced control methods have been proposed such as data-driven control [7], learning adaptive robust control (LARC) [8,9], adaptive control and so on. Nonetheless, there inevitably is difference between the dynamical characteristics of each motion axis, purely improving single-axis tracking accuracy may not guarantee the reduction of contouring error. In order to achieve better contouring performance of biaxial systems, besides individual axial control, the coordination between two different axes should be considered. Therefore, contouring control has attracted wide attention and a series of contouring control methods have been proposed by researchers. cross coupled control (CCC) structure is firstly proposed by Koren [10]. In CCC, coordination control signal is generated according to the real-time estimated contouring error, and is distributed into the individual axial control loops to improve contouring performance. Since then many contouring control strategies have been reported, which can be classified into two different categories [11]. One category of these methods inherits basic structure of CCC [10] in which both additional contouring controller and individual axial controllers are necessary. In this structure, due to that the axial controllers and the contouring controller are designed independently, conventional CCC can be well combined with many axial tracking control methods such as ZPETC [12], ILC [13], fuzzy logic control [14,15], etc. As another kind of method, coordinate transformation, instead of additional contouring controller, is used to deal with contouring error. For instance, the moving task coordinate frame (TCF) was proposed by Chiu and Tomizuka [16], where the contouring error and the tangential error were estimated by projecting the tracking error to the normal direction and the tangential direction of the TCF, respectively. Similarly, tangential-contouring control (TCC) was also proposed [17], in which motion controllers were designed in normal direction and tangential direction of the reference contour. Meanwhile, contouring error could be directly controlled according to the dynamics along the normal direction. Following the concept of TCF and TCC, many control laws have been subsequently proposed [18]. The main issues of the existing contouring control methods can be divided into the following two aspects. The first one is how to realize the contouring control algorithm in industrial applications. For the existing packaged CNC systems, the contouring control algorithm is difficult to embed into the underlying servo motors with solid state motion controller which cannot be further modified. In order to promote the utility of contouring control algorithms, the cross-coupled pre-compensation method (CCPM) is proposed [19], in which the real-time estimated contouring error is used to modify the feedrate parameters and to further reduce contouring error. And then the concept of contouring error transfer function is presented in [20] to lay the theoretical foundation of contouring error pre-compensation structure. On the basis of the concept of contouring error transfer function, the model reference adaptive compensator is designed in [21] to obtain the relationship between the pre- and post-compensation contouring errors and to further improve the contouring accuracy. To realize a more effective compensation, not only estimated contouring error but also the curvature of the reference contour is further considered in [22]. In the above trajectory pre-compensation structure, contouring control acts on the position loop but not the underlying current loop or the torque loop of control systems. Consequently, this kind of compensation can be flexibly combined with any kind of axial tracking controllers such as sliding mode control [23,24] and model predictive control [25]. Accurate contouring error estimation is another issue for contour following tasks. Geometrical based approach is one of the most commonly used contouring error estimation method. Specifically, the tangent-line or the circle of curvature [10,26] at the current desired point can be utilized to estimate contouring error in real-time. However, the approximation accuracy of both tangent-line and circle of curvature method deteriorates obviously in some extreme cases such as high feed rate, large contour curvature and sharp corners. To deal with this problem, many advanced contouring error estimation methods have been presented. For free-form contours, an effective contouring error estimation method was presented [27], in which the estimation accuracy would not be influenced by the shape of the reference contour in theory. However, its validity can only be guaranteed under the assumption that the feed rate between any two adjacent desired points remains constant. Clearly, it is unrealistic in some practical applications. In [28], the estimated contouring error was defined as the shortest distance from the actual point to the circle of curvature at the desired position. This method outperforms conventional tangent-line approximation method with respect to estimated accuracy. However, it cannot do well if the tracking error is obviously larger than the radius of the contour curvature. Considering the effectiveness in practical application, a point-by-point comparison method is utilized to estimate contouring error [29]. This method was easy to be realized in real time for any contouring tasks, but excellent estimation accuracy can not be guaranteed. In order to avoid the inaccurate estimation of contouring error caused by geometrical approximation, an equivalent errors based method has been presented in [30] for the contouring control of NURBS curve. However, this method faced the difficulty of obtaining analytical path equation for complicated paths. Based on the coordinate rotation transformation, global task coordinate frame (GTCF) is proposed [31,32] to achieve the globally effective contouring error estimation. In GTCF, the first-order approximation of the actual contouring error can be obtained even in the contouring tasks with high speed and large curvature [33]. Nevertheless, in GTCF, rigorous contour shape function is indispensable which is usually hard to obtain for complicated free-form contours. In order to simultaneously tackle the above two problems widely existing in free-form contouring tasks, an accurate task coordinate frame (ATCF) based trajectory compensation method is proposed in this paper. Firstly, the Newton extremum seeking algorithm is used to seek the contouring error point. Benefit by the high accuracy of this numerical calculation method, nearly perfect contouring error point can be found which is impossible in traditional contouring error estimation method. Secondly, the moving ATCF is constructed based on the actual point, the desired point and the calculated contouring error point. The equivalent desired point is then defined in ATCF, and its X and Y coordinates mean the tangential error and the contouring error, respectively. Finally, the reasonable weight distribution of above two errors is determined according to the specific contouring tasks. And then the weighted errors can be utilized to modify the reference trajectory and to further improve contouring performance. Essentially, the proposed ATCF method can be viewed as a separate module independent

Z. Wang et al. / Mechanical Systems and Signal Processing 135 (2020) 106393

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from the original closed-loop control system. Accordingly, the modular design of trajectory compensation suppress contouring error effectively without changing the structure of original controller. The main contribution of this paper can be summarized as follows:
Based on the numerical method for the estimation of contour error, ATCF is constructed in this paper for biaxial coordination motion system. In ATCF, the contouring error and the related distance error can be decoupled effectively even under the extreme cases such as high speed, large curvature and sharp corner.
According to the reasonable gain adjustment in ATCF, the novel optimal trajectory compensation for linear motion systems is determined to improve contouring performance without affecting the structure of original closed-loop controller. The rest of this paper is organized as follows. The brief introduction of the Newton algorithm and the definition of ATCF are provided in Section II. The proposed reference trajectory modification method and motion control scheme are discussed in Section III. The effectiveness of the proposed method is validated by the abundant simulations and experiments shown in Section IV. Section V gives the conclusion. 2. Numerical calculation based contouring error estimation For the free form contours, contouring error may not be estimated accurately through geometrical shape based methods. Furthermore, computational capability is no longer a problem for the controller nowadays. Therefore, an approach based on optimization and iteration instead of line approximation is desirable and practical. Suppose the reference contour cðtÞ is a parameterized planar curve which satisfies the following assumption: _ Assumption 1. cðtÞ is at least two order derivable, and cðtÞ – 0. Define the parameterized planar trajectory with respective to the fixed Cartesian coordinate frame XOY as




cðtÞ ¼

cx ðtÞ cy ðtÞ

 ; t 2 Rþ

ð1Þ

where t is the time. The actual position is described as aðtÞ ¼ ½ax ðtÞ; ay ðtÞT . actual position aðtÞ and the reference point at time ^t, i.e.,

ec ðt; ^tÞ is defined as the distance between the

ec ðt; ^tÞ ¼ kcð^tÞ  aðtÞk

ð2Þ

where k  k represents the Euclidean norm. Define t ¼ argmin ec ðt; ^tÞ, so ec ðt; t Þ is the minimum value of ec ðt; ^tÞ. The contour

ing error

ec ðtÞ can be expressed as



^t

ec ðtÞ ¼ kcðt Þ  aðtÞk

ð3Þ

where cðt Þ is the contouring error point at time t. A cost function at time t is defined as

Jðt; ^tÞ ¼

1 2 ec ðt; ^tÞ 2

ð4Þ

The minimum value of the cost function is associated with t . To calculate the minimum value of the cost function and to complete the contouring error estimation, the Newton algorithm is employed in this paper. We assume that there exists nonzero second derivation of the cost function J. For the derivation of J, by Taylor series expansion on ^t, one obtains

  @J @J  @ 2 J  þ ð^t  ~tÞ   @^t @^t ^t ¼ ~t @^t 2  ^t ¼ ~t

ð5Þ

where @J=@^t and @ 2 J=@^t 2 are the partial derivation and the second partial derivation of J, respectively, ~t is an arbitrary constant. In the extremum seeking of contouring error at arbitrary time t, i.e., t cam be viewed as the constant, the minimum value of Jðt; ^tÞ means @J=@^tj^t¼t ¼ 0. Let ^t ¼ t ; f ^t ð~tÞ ¼ @J=@^tj^t¼~t and f ^t^t ð~tÞ ¼ @ 2 J=@^t 2 j^t¼~t . Thus (5) can be rewritten as

t  ~t 

f ^t ð~tÞ f ^t^t ð~tÞ

ð6Þ

According to the above approximate expression, the recursion formula of the Newton algorithm is presented as

tiþ1 ¼ t i 

f ^t ðti Þ f ^t^t ðt i Þ

where i means the iteration of the Newton algorithm.

ð7Þ

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Z. Wang et al. / Mechanical Systems and Signal Processing 135 (2020) 106393

Remark 1. In the actual application of the proposed Newton algorithm, f ^t^t ðt; ^tÞ is calculated as

_ ^tÞk2 f ^t^t ðt; ^tÞ ¼ ðcð^tÞ  aðtÞÞ€cT ð^tÞ þ kcð Assuming that during the contouring control, _ ^tÞk2 , one can approximate (7) as f ^t^t ðt; ^tÞ  kcð

tiþ1 ¼ t i 

ð8Þ

ec is sufficiently small, and the reference contour is a smooth curve, thus

ðcðt i Þ  aÞc_ T ðt i Þ : _ i Þk2 kcðt

ð9Þ

According to Assumption 1, the recursive algorithm can be conducted just with the knowledge of position, velocity along the reference contour and the actual position. The optimal estimation of time is denoted as t, and the optimal contouring point cðtÞ can be obtained through the algorithm. The iterative termination criterion can be described by the following inequation:

  ðcðti Þ  aÞc_ T ðti Þ < f

ð10Þ

where the criterion constant f is a very small constant. If this criterion is satisfied, vector from the actual point a and the estimated contouring error point cðt i Þ is almost orthogonal to the tangential vector at cðt i Þ. Therefore, cðt i Þ can be viewed as the accurate contouring error point. Theorem 1. If the iterative initial value t 0 is close enough to t , i.e., t 0 2 Uðt  ; cq Þ where cq is a small positive number, the local convergence of the proposed Newton algorithm is guaranteed and the iterative result will converge to t  . Proof 1. According to (7):

        1     1 f ^ðti Þ j½f ^ðt i Þ  f ^^ðt i Þðt i  t  Þj ½f ^t ðti Þ  f ^t^t ðt i Þðti  t Þ ¼   t  ¼  jt iþ1  t j ¼ t i  t tt f ^t^t ðt i Þ f ^t^t ðti Þ f ^t^t ðt i Þ t

ð11Þ

Because f ^t ðt  Þ ¼ 0, (11) can be rewritten as

   1  jf ^ðt i Þ  f ^ðt Þ  f ^^ðt  Þðti  t Þ þ ðf ^^ðt  Þ  f ^^ðti ÞÞðt i  t  Þj jt iþ1  t j ¼  t tt tt tt f ^t^t ðt i Þ t    1  ½jf ^ðti Þ  f ^ðt  Þ  f ^^ðt  Þðti  t Þj þ jðf ^^ðt  Þ  f ^^ðti ÞÞðt i  t Þj 6  t tt tt tt f ^t^t ðti Þ t

ð12Þ

For 8q > 0; 9c1 > 0; t i 2 Uðt ; c1 Þ subject to jf ^t ðti Þ  f ^t ðt  Þ  f ^t^t ðt Þðt i  t Þj 6 qjðt i  t  Þj, and 9c2 > 0; ti 2 Uðt  ; c2 Þ subject to     jðf ^t^t ðt  Þ  f ^t^t ðt i ÞÞðt i  t  Þj 6 qjðt i  t  Þj. Define cq ¼ minðc1 ; c2 Þ. If t i 2 Uðt ; cq Þ, then jtiþ1  t  j 6 f ^^1ðti Þ2qjt i  t j, which means

  jt iþ1  t j  1  2q 6  f ^t^t ðti Þ jt i  t j

tt

ð13Þ

    According to Assumption 1 and Remark 1, f ^^1ðt Þ is bounded. Considering q can be chosen as arbitrary positive number. 9q tt i     subject to f ^^1ðti Þ2q  1. Obviously, if the iterative initial value t 0 is close enough to t  , i.e., t 0 2 Uðt ; cq Þ, each iteration i will sattt

isfy the condition t i 2 Uðt ; cq Þ. Therefore, the final result will converge to the accurate value t* and the theorem is proved. h

Remark 2. The result of the proposed Newton algorithm is local, and the global convergence cannot be guaranteed. However, this characteristic is not bad for contouring control. On one hand, the local result is depending on the initial value of the recursive algorithm. The initial value is consistent with the desired point which is close enough to the contouring point. According to Theorem 1, even the cost function has multiple extremum points, the expected result adjacent to the desired point can also be obtained through the algorithm. On the other hand, the local result is even better than the global one in some special cases. For example, the reference contour has sharp corners or loops as shown in Fig. 1. In Fig. 1(a), the local result is the distance between the actual position to Line 1, while the global one is the distance between the actual position to Line 2, which is shorter than the local result. Even so, the local result is the expected one due to that the desired point is located on Line 1. If the global result far away from the original trajectory is used in ATCF to be mentioned below, the modified trajectory will deviate from the reference contour, and then the contouring control task cannot be finished completely. Similar conclusion can be drawn through Fig. 1(b).

3. ATCF based trajectory compensation scheme In this section, the Accurate Task Coordinate Frame (ATCF) is constructed, and the trajectory compensation scheme appropriate weight adjustment is proposed on the basis of ATCF.

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Fig. 1. Special contours with (a) sharp corner; (b) loop.

3.1. Design of ATCF To get the accurate expression of contouring error and distance error, ATCF is developed in this paper. ATCF LO0R is constructed as depicted by the red axes in Fig. 2. In view of the reality of biaxial motion control that the contouring error point is close enough to the desired one, the following assumption can be made: Assumption 2. The reference contour possesses the same curvature at the contouring error point and the desired one. In other word, the reference contour between these two point can be viewed as a circular arc with radius r which can be calculated as the radius of curvature at the desired point.

Fig. 2. Schematic diagram of the proposed ATCF.

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In ATCF, actual position O0 is the original point, and the connecting line between O0 and the estimated contouring error point is R axis. According to Assumption 2, the concentric circle of the reference contour passing through point O0 can be uniquely determined. L axis is defined as the arc of the above concentric circle, i.e., the blue arc shown in Fig. 2. Undoubtedly, the coordinates of the desired point in ATCF along L and R axis can be defined as the distance error and the contouring error, respectively. Contouring error point and the related contouring error ec can be calculated by the numerical method proposed in Section II. Define the equivalent error  e as the distance between the contouring error point and the desired one. According to the geometry, the lag angle b can be expressed as:



b ¼ 2 arcsin

e 2r



Hence, the distance error

ð14Þ

ed can be expressed as

ed ¼r0 b ¼ 2ðr  ec Þ arcsin 

r  ec e r

  e 2r

ð15Þ

For the contours with small curvature, the distance error can be directly approximated by e. 3.2. Reference trajectory compensation method Based on the contouring error and the distance error calculated in ATCF, we consider synthesizing a reasonable compensation to modify the reference trajectory in real-time and to improve contouring performance. To further analyze the relationship between the trajectory compensation and contouring error after compensation, the linear closed-loop systems are considered in the flowing analysis. According to Assumption 2, the reference contour during the contouring error point and the desired point can be approximated by uniform circular trajectory which can be expressed by the following functions of time t:

Rx ¼r cosðxt þ ux Þ

ð16Þ

Ry ¼r sinðxt þ uy Þ

where x is the angular frequency parameter, ux and uy are the phase of X and Y axis, respectively. Due to the characteristics of linear systems, the actual position output of these two axes posses the angular frequency totaly same to the reference trajectory, and can be expressed by

Px ¼r x cosðxt þ ux  hx Þ

ð17Þ

Py ¼r y sinðxt þ uy  hy Þ

where rx and r y represent the amplitude attenuation, and hx and hy are the tracking phase lag of X and Y axis, respectively. According to the geometry shown in Fig. 2, the following equations can be obtained:

Px ¼r x cosðxt þ ux  hx Þ ¼ r 0 cosðxt þ ux  bÞ

ð18Þ

Py ¼r y sinðxt þ uy  hy Þ ¼ r0 sinðxt þ uy  bÞ

If we just increase the amplitude of the reference trajectory by r=r 0 , the outputs P 0x and P0y are as follows:

P0x ¼ r  r x =r0  cosðxt þ ux  hx Þ P0y ¼ r  r y =r0  sinðxt þ uy  hy Þ

ð19Þ

Substitute (18) into (19), the outputs can be rewritten as:

P0x ¼r  cosðxt þ ux  bÞ

ð20Þ

P0y ¼r  sinðxt þ uy  bÞ

In other word, the actual position is exactly at the contouring error point shown in Fig. 2. Therefore, the optimal trajectory compensation should be

Rcomp ¼

r r  ðr  r 0 Þ r r ¼ ¼ ec r0 r0 r0

ð21Þ

The direction of the reference compensation vector is along the line between the desired point and the equivalent center of the circle. To summarize, the complete steps of trajectory compensation in ATCF can be conclude in algorithm 1.

Z. Wang et al. / Mechanical Systems and Signal Processing 135 (2020) 106393

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Algorithm 1 Steps of trajectory compensation in ATCF Steps of trajectory compensation in ATCF Input: The coordinates of the calculated contouring error point cðt i Þ, actual point O0 and desired point D; Output: The optimal reference trajectory compensation vector: Rcomp ; 1: L1 : the connection line between cðti Þ and O0; 2: L2 : the connection line between cðti Þ and D; 3: Equivalent center of the circle: intersection of L1 and the midperpendicular of L2 ; 4; Calculating r and;r 0 ; 5: Constructing ATCF as shown in Fig. 2 6: Calculating the coordinates of D in ATCF: ec and ed ; 7: Rcomp ¼ rr0 ec ; 8: Direction of the compensation vector: along the line between D and equivalent center; 9; Return Rcomp vector;

The calculated compensation vector can be directly fed back to the position input to achieve effective reference modification and to improve contouring performance. Remark 3. For the contour with no apparent change rate in curvature, r calculated in step algorithm 1 of algorithm 1. can be approximated by the radius of curvature at the desired point. So that the algorithm can be further simplified. In algorithm 1, only the contouring error is compensated. If the distance error ed is to be compensated, the corresponding coordinate along L axis of ATCF can be added to the compensation quantity. Remark 4. Essentially, algorithm 1. is a kind of simple algebraic operation which does not take up much computing time in real-time motion control. In order to further prove the above conclusion, the execution time of ATCF scheme is presented in the following real-time control experiments.

4. Simulation and experimental investigation 4.1. Numerical simulation A numerical simulation is conducted to test the performance of the proposed ATCF based reference compensation method. In addition, to provide a sufficient comparison, individual tracking control and CCC are also implemented as follows: C1: Individual tracking control—all the two axes are controlled by PID controllers. It should be noted that there is not any additional contouring control mechanism used in this scheme. C2: ATCF based trajectory compensation—compared with C1, the contouring error is calculated by the presented numerical calculation method in real time, and the additional ATCF based trajectory compensation (described by algorithm 1.) is applied. C3: CCC—compared with C1, the position-loop CCC method is added as a kind of feedforward compensation. A two-axis motion system is used in the simulation which can be described by the following transfer functions:

Gx ¼

1 ; 5s þ 0:3

Gy ¼

1 6s þ 0:4

ð22Þ

The simulation is added to verify the applicability of the proposed method to different motion systems, so the motion system chosen in the simulation has different model from the actual system. X and Y axis are controlled by PID controller with different controller parameters. In order to further show the comparison results under different closed-loop controllers, the following two groups of control parameters, with high and low gain, are used: It should be noted that the additional multi-axis coordination mechanisms are added in C2 and C3 based on the PID control, i.e., C1. To fairly compare the performance of the coordination mechanisms. The controller parameters of C2 and C3 are exactly the same as C1. Therefore, the selection of PID control parameters has no essential effect on the subsequent comparison results. The step size of simulation with Parameter 1 and Parameter 2 are 0.2 ms and 1 ms, respectively (Table 1). The above closed-loop motion system is commanded to track a butterfly shape Nurbs trajectory described in [25]. To evaluate the contouring control performances, the following performance indexes are used: 1. jec jrms : root mean square value of contouring error. 2. jec jmax : maximum absolute value of contouring error.

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Z. Wang et al. / Mechanical Systems and Signal Processing 135 (2020) 106393

Table 1 Control parameters in simulation (lm).

Parameter 1 Parameter 2

K px

K ix

K dx

K py

K iy

K dy

2200 3885

500 3560

2 0.5

1850 4165

450 3455

3.5 1.4

The contouring errors and magnified actual contours (Parameter 1) of C1-C3 are plotted in Fig. 3 and the performance indexes are listed in Table 2. The magnified contours at sharp corners reveal that the contour of C2 is closest to the reference one compared with the C1 and C2. Moreover, as shown in Table 2 and contouring error in Fig. 3. (b), for both Parameter 1 and Parameter 2, jec jrms and jec jmax of C2 is smaller than those of C1 and C3. All the simulation results consistently illustrate the excellent contouring performance of the proposed ATCF based reference compensation method in the free-form complicated contouring tasks. In the following content, the effectiveness of the proposed method will be further verified by a series of actual motion control experiments.

Fig. 3. Contouring control performances of C1-C3 in simulation. (a) Actual contours. (b) Contouring errors.

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Z. Wang et al. / Mechanical Systems and Signal Processing 135 (2020) 106393 Table 2 Simulated contouring control results (lm). Parameter 1

Parameter 2

Controller

jec jrms

jec jmax

jec jrms

jec jmax

C1 C2 C3

168.65 87.35 106.45

458.44 250.04 296.26

32.93 16.29 20.50

101.49 50.23 63.34

4.2. EXPERIMENTAL Setup The proposed ATCF based reference trajectory compensation method is tested on the X and Y axis of a CNC motion system as shown in Fig. 4. The X and Y axis are driven by the ball screw system with Panasonic servo motors which are controlled by a dSPACE DS1202 control system. The controller board executes algorithms at a sampling frequency of 5kHz. And the feedback signal of each axis is from the encoder of the corresponding servo motor. The position measurement resolution of each axis is 0.025 lm. The frequency identification has been carried out to obtain the open loop model of each axis. The identified transfer functions of two axes are shown as follows

1503 ð23Þ s2 þ 18:68s 1209 Gy ðsÞ ¼ 2 s þ 30:3s

Gx ðsÞ ¼

According to the identification result of each axis, the PID parameters are well tuned. Specifically, K px ¼ 62:5; K ix ¼ 500; K dx ¼ 0:1; K py ¼ 36; K iy ¼ 400 and K dx ¼ 0:1. All the following experiments are conducted using same PID control parameters for a fair comparison. 4.3. EXPERIMENTAL Results The experiments are conducted to verify the excellent contouring control performance of the proposed ATCF based reference compensation method. The methods used for comparison, i.e., the controller types of C1, C2 and C3 is the same as that used in simulation. But the parameters of PID controller have been changed. 4.3.1. Case I—parabolic motion To test the contouring performance of the proposed scheme, the gantry is firstly commanded to track a parabolic contour with high speed and large-curvature, which can be expressed by:


Rd ¼

xd ðtÞ yd ðtÞ

"

 ¼

r x cosð2ptÞ

# ð24Þ

ry cosð2ptÞ2

where r x ¼ 2 mm; ry ¼ 40 mm.

Fig. 4. The industrial multi-axis mechatronic motion system.

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Z. Wang et al. / Mechanical Systems and Signal Processing 135 (2020) 106393

Fig. 5. Contouring control performances of C1-C3 in Case I. (a) Actual contours. (b) Contouring errors.

The reference contour is shown in Fig. 5(a). Note that the depicted reference contour has been zoomed horizontally for a clear depiction. Actually, the reference contour possesses both the high speed and the large curvature (the smallest radius of curvature of the contour is just 50 lm). In order to show the contouring performance of different methods, the actual contours of C1-C3 near the vertex of parabola are also shown in Fig. 5(a) as a zoomed-in figure. As shown in the figure, Y-axis is commanded to change motion direction during the vertex, which leads to the performance deterioration. Due to the lack of axial coordination mechanism in individual tracking control, C1 shows the obvious overshoot. By contract, the performance of C3 is a little bit better than C1. However, traditional contouring error estimation method in CCC cannot guarantee good estimation accuracy in the case of high speed and large curvature. Furthermore, direct position feedback in C2 is not the optimal compensation scheme. Compared with C3, the proposed ATCF based position loop compensation possesses not only accurate contouring error estimation but also appropriate compensation gain. Therefore, C2 can achieve the best contouring performance near the vertex. In other word, the green line shown in the zoomed-in figure is closest to the black reference contour. In order to further validate the performance comparison between different methods in the whole contouring control process. The contouring errors of C1-C3 during 0–0.5 s are plotted in Fig. 5(b), and the related contouring control performances are given in Table 3. As shown in Table 3 and Fig. 5(b), the contouring errors during 0.25 s are consistent with the results shown in Fig. 5(a). jec jrms of C1 in the whole contouring process is almost 8 lm; jec jrms of C3 is slightly smaller than that of C1. While C2 achieves jec jrms of 3.51 lm which is obviously less than C1 and C3 (jec jrms 45.7% of C1 and 74.1% of C3). Discontinuous jumps occur in velocity and acceleration at the beginning and the end of the trajectory, which cause the sharp deterioration of contouring errors. jec jmax of C1 is almost 26 lm; jec jmax of C3 is more than 21 lm, and jec jmax of C2 is around 17 lm. In summary, all the results consistently demonstrate that C3 outperforms C1 obviously with the aid of coordinate control,

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Z. Wang et al. / Mechanical Systems and Signal Processing 135 (2020) 106393 Table 3 contouring results of case I (lm). Controller

jec jrms

jec jmax

C1 C2 C3

7.68 3.51 4.74

25.87 17.05 21.26

but cannot achieve satisfied result in the case of high speed and large curvature. However, the proposed ATCF method processes excellent contouring performance during the whole control process. 4.3.2. Case II—sharp corner To further test the performance of the proposed ATCF method under extreme contouring tasks with sharp corner, a variable speed curve with a discontinuous sharp corner is employed, i.e.,


Rd ¼

xd ðtÞ yd ðtÞ

"

 ¼

rcosð2pt þ 0:3pÞ3

#

rsinð2pt þ 0:3pÞ3

ð25Þ

where r ¼ 30 mm. The reference contour and the zoomed-in figure of the actual contours of C1-C3 are shown in Fig. 6(a). The contouring errors during 0–0.2 s are plotted in Fig. 6(b). The related performances indexes, i.e., jec jrms and jec jmax are listed in

Fig. 6. Contouring control performances of C1-C3 in Case II. (a) Actual contours. (b) Contouring errors.

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Z. Wang et al. / Mechanical Systems and Signal Processing 135 (2020) 106393 Table 4 contouring results of case II (lm). Controller

jec jrms

jec jmax

C1 C2 C3

26.91 11.57 13.66

43.35 20.97 26.31

Fig. 7. Contouring control performances of C1-C3 in Case III. (a) Actual contours. (b) Contouring errors.

Table 5 Contouring results of case III (lm). Controller

jec jrms

jec jmax

C1 C2 C3

42.42 21.13 27.48

94.84 43.51 70.83

Z. Wang et al. / Mechanical Systems and Signal Processing 135 (2020) 106393

13

Fig. 8. Execution time of ATCF in Case I, II and III.

Table 4. According to (25) and the reference contour shown in Fig. 6(a), the gantry slows down gradually, then passes through a sharp point, and finally accelerates. Consequently, the contouring errors shown in Fig. 6(b) show an upward trend. As shown in Fig. 6(b) and Table 4, jec jrms of C3 is 57% decreased compared with C1. Moreover, the performance of C2 is better than C2. This result is similar to Case I, and further illustrate the excellent of the proposed ATCF based position compensation scheme. It should be noted that, the sharp corner is a great challenge for contouring control. The zoomed-in figure in Fig. 6(a) validate that C2 possess the best contouring performance due to the accurate contouring error estimation and effective compensation. All the results shown in Table 4 as well as the contouring errors of C1-C3 plotted in Fig. 6 consistently demonstrate that compared with traditional method, the proposed ATCF based method can also achieve contouring performance even in the extreme case with sharp corner. 4.3.3. Case III—Heart shape contour A heart shape contour is utilized as the reference contour to further test the performance of the proposed compensation method. The reference contour can be described by:

xd ðtÞ ¼ r x sin ð2ptÞ p yd ðtÞ ¼ r y cos 12 ð13 cosð2ptÞ  5 cosð4ptÞ 3

ð26Þ  2 cosð6ptÞ  cosð8ptÞÞ

where r x ¼ 32 mm and r y ¼ 2 mm. The contouring control performance is listed in Table 5. The contour shape under actual control and the contouring errors are plotted in Fig. 7(a) and (b), respectively. Based on the presented experimental results, it is easy to find the excellent contouring performance of C2, especially at sharp corners. Specifically, the contouring performance at 0.5s, i.e., the zoomed-in figure shown in Fig. 7(a) demonstrates that C2 can not only estimate the contouring error at the sharp corner accurately but also compensate the contouring error effectively for the extreme situation of large curvature. All the above simulations and experimental results can demonstrate that the proposed method possesses excellent contouring error estimation and real-time control ability. 4.4. Execution time of ATCF scheme in real-time control To demonstrate the feasibility of the proposed ATCF scheme in real-time control, the execution time of the algorithm during each simple time is plotted in Fig. 8. In Fig. 8, we can find that the execution time of Case I, II and III is far less than the simple time of the control system i.e., 0.2 ms. The RMS value of the execution time of Case I, II and III is 0.0334 ms, 0.0325 ms and 0.0345 ms, respectively. Therefore, we can draw the conclusion that the proposed Newton-ATCF method is fast enough for the practical real-time control. It should be noted that, the large curvature in Case I (around 0.25 s), the sharp corner in Case II (around 0.2 s) and Case III (around 0.5 s) do not result in more algorithm execution time. These results illustrate that the above extreme cases do not affect the real-time performance of the proposed algorithm. 5. Conclusion To improve contouring control performance of biaxial systems especially under the extreme cases with high-speed, large curvature and sharp corner, the ATCF based position loop compensation scheme is proposed in this paper. Firstly, the accurate contouring error point is obtained through numerical calculation algorithm in real-time. Based on the contouring error

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Z. Wang et al. / Mechanical Systems and Signal Processing 135 (2020) 106393

point, ATCF is constructed in 2D plane to express the contouring error and the related distance error clearly. And then, the optimal compensation is determined according to the above two errors. Finally, the compensation is feedback to the original trajectory to improve contouring performance. The effectiveness of the proposed method has been demonstrated by theoretical analysis and experiments. In addition, ATCF proposed in this paper can be viewed as a module independent of the existing closed-loop systems and will not change the structure of motion controller. Therefore, this research can provide an important reference for the contouring control of existing packaged CNC systems. 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