It is based on linear parameter-varying controller

1. Contouring Control of CNC Machine Tools
Based on
Linear Parameter-Varying Controllers
1

2. CONTENTS
 INTRODUCTION
 THREE AXIS CNC MACHINE
 TRACKING AND CONTOURING ERROR
 CONTOURING ERROR APPROXIMATION
 CONTROLLER OBJECTIVES
 MIMO LPV CONTROLLER DESIGN
1. CONTROLLER STRUCTURE
2. CONTROLLER DESIGN
3. WEIGHTING FUNCTION TUNNING
 CONCLUSION
 REFERENCES
2

3. INTRODUCTION
 CNC or computer-numerical-control machines.
 High speed , high tolerance , precise surface finishing.
 CNC variants : Mills, lathes, EDM, 3d printer, etc.
 Contouring error : deviation of the cutting tool from the toolpath trajectory.
 Servo control : for achieving a satisfactory contouring error.
 tracking control : to minimize the tracking error in each axis individually.
 contouring control cross-coupling control (CCC)
task coordinate frame transformation.
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4.  CCC method
 Disadvantage : low effectiveness in dealing with nonlinear contours.
 Task coordinate frame transformation
 the axis coordinate system is transformed into tangential-normal-bidirectional
task frame.
 contouring and lag tracking errors : by varying PD and feed-forward
controller.
 Limited to SISO.
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Real time contour
error calculation
Signal from
compensator
Servo controller in
each axis

5.  The MIMO LPV feedback controller as a function of toolpath trajectory
1. toolpath trajectory direction
2. toolpath trajectory velocity.
 toolpath profiling precision is improved with high feed-rate .
 Controller design approach using
1. advanced gain scheduling control technique based on linear matrix
inequalities.
2. coordinate transformation matrices.
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6. THREE AXIS CNC MACHINE6
Fig. 1: An illustration of a three-axis CNC machine

7.  Table with workpiece mass : move in x and y directions.
 Table with machining tool : displaces in z direction.
 Three ball-screw feed-drive systems connected with DC motors for movement.
 Applied voltages current amplifiers DC motors
(𝑣 𝑥, 𝑣 𝑦 and 𝑣𝑧 [V]) (send current commands) (torques generated)
 Torque to the screws : linear displacements (positions) of the tables 𝑝 𝑥, 𝑝 𝑦 and 𝑝 𝑧.
 The positions measurement : linear encoders in real-time for feedback control.
 External forces 𝑓𝑥, 𝑓𝑦 and 𝑓𝑧 [N] : the resultant component of the cutting forces
and friction along the three axes.
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8. TRACKING AND CONTOURING ERROR
 The tracking error : difference between the
table position and its reference in each axis at
each time instant t:
𝑒 𝑡 = 𝑟 𝑡 − 𝑝 𝑡 (1)
 In 3 dimension,
𝑟 𝑡 = [𝑟𝑥(𝑡), 𝑟𝑦(𝑡), 𝑟𝑧(𝑡)] 𝑇
𝑝 𝑡 = [𝑝 𝑥(𝑡), 𝑝 𝑦(𝑡), 𝑝 𝑧(𝑡)] 𝑇 (2)
where 𝑟𝑖, i = x; y; z are the reference signals
in x, y and z directions.
 The contouring error : difference between the
displacement at each time instant and the
shortest-distance point on the reference
trajectory.
Fig. 2: Tracking error and
contouring error in 3axis CNC
machine
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9. CONTOURING ERROR APPROXIMATION
 The tracking error ,with encoder measurements 𝑝𝑖, i = x; y; z. is used directly for
feedback control.
 Mathematically, from fig.2 ,the tracking error can be represented as
𝑒 = 𝑒𝑡 𝑢 𝑡 + 𝑒 𝑛−𝑏 𝑢 𝑛−𝑏 , (3)
𝑒𝑖, i = t; n-b are scalars associated with the unit vectors.
 The contouring error is estimated as
𝟄 𝑒𝑠𝑡 = 𝑒 𝑛−𝑏 𝑢 𝑛−𝑏 , (4)
where
𝑒 𝑛−𝑏 𝑢 𝑛−𝑏 = 𝑒 𝑛−𝑏,𝑥 𝑢 𝑥 + 𝑒 𝑛−𝑏,𝑦 𝑢 𝑦 + 𝑒 𝑛−𝑏,𝑧 𝑢 𝑧 . (5)
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10. 10  The relation between measurable vector 𝑒 and unmeasurable scalars (𝑒𝑡; 𝑒 𝑛−𝑏) as
=
𝑇𝑡(α 𝑥, α 𝑦)
𝑇𝑛−𝑏(α 𝑥, α 𝑦)
𝑒, (6)
𝑒 𝑛−𝑏= 𝑒 𝑛−𝑏,𝑥
2
+ 𝑒 𝑛−𝑏,𝑦
2
+ 𝑒 𝑛−𝑏,𝑧
2
(7)
 The transformation matrices 𝑇𝑡 and 𝑇𝑛−𝑏 are given by
𝑇𝑡(α 𝑥, α 𝑦)= 𝑐 α 𝑥 𝑐 α 𝑦 𝑐 α 𝑧
𝑇𝑛−𝑏(α 𝑥, α 𝑦)=
𝑠 α 𝑥
2
−𝑐 α 𝑥 𝑐 α 𝑦 −𝑐 α 𝑥 𝑐 α 𝑧
−𝑐 α 𝑥 𝑐 α 𝑦 𝑠 α 𝑦
2
−𝑐 α 𝑦 𝑐 α 𝑧
−𝑐 α 𝑥 𝑐 α 𝑧 −𝑐 α 𝑦 𝑐 α 𝑧 𝑠 α 𝑧
2
(8)
for an angle, s() and c() denote sin and cos respectively.
𝑒𝑡
𝑒 𝑛−𝑏,𝑥
𝑒 𝑛−𝑏,𝑦
𝑒 𝑛−𝑏,𝑧

11. CONTROLLER OBJECTIVES
 The mapping from the vector 𝑒 to the scalar 𝑒 𝑛−𝑏 is nonlinear because of (7).
 The mapping from 𝑒 to the vector in the left-hand side of (6) is linear.
 Feedback controllers
 minimizes the approximated contouring error 𝟄 𝑒𝑠𝑡 in (4) and
 also suppresses vibration of the machine due to external forces.
Fig. 3: Feedback structure for contouring error minimization.
where the disturbance force vectors 𝑓 := [𝑓𝑥; 𝑓𝑦 ; 𝑓𝑧] 𝑇,
voltage input vector 𝑣 := [𝑣 𝑥; 𝑣 𝑦; 𝑣𝑧] 𝑇.
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12. MIMO LPV CONTROLLER DESIGN
 SISO controller : each axis is actuated independently by a feed-drive system.
 A single MIMO controller : function of path direction and velocity.
 compensate for the contouring error and
 to reject the disturbance forces .
 A linear time-invariant (LTI) state-space model of ‘CNC machine’ block in Fig. 3
is expressed by
G:
𝑥 𝑚 𝑡 = 𝐴 𝑚 𝑥 𝑚 𝑡 + 𝐵 𝑚𝑓 𝑓 𝑡 + 𝐵 𝑚𝑓 𝑣 𝑡 ,
𝑝 𝑡 = 𝐶 𝑚 𝑥 𝑚 𝑡
(9)
where 𝑥 𝑚 : state vector,
𝐴 𝑚, 𝐵 𝑚𝑓, 𝐵 𝑚𝑣 and 𝐶 𝑚 : constant system matrices of compatible dimensions.
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13. 1. CONTROLLER STRUCTURE
Fig. 4: controller structure for contouring control
 The parameter Ɵ(the gain-scheduling parameter) is defined by
Ɵ = [α 𝑥, α 𝑦] 𝑇 (10)
 The first system T(Ɵ) is a parameter-varying matrix which is the coordinate
transformation matrix defined by
T(Ɵ) =
𝑇𝑡(Ɵ)
𝑇𝑛−𝑏(Ɵ)
(11)
13

14.  The output of the system T(Ɵ) are given in the left-hand side of (6) .
 The vector 𝟄 𝑒𝑠𝑡 is given by
𝟄 𝑒𝑠𝑡 = [ 𝑒 𝑛−𝑏,𝑥 , 𝑒 𝑛−𝑏,𝑦 , 𝑒 𝑛−𝑏,𝑧] 𝑇 (12)
 The K(Ɵ; Ɵ) block is a linear parameter-varying (LPV) controller.
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15. 2. CONTROLLER DESIGN
 For the LTI model G in (9) and the parameter-varying matrix T(Ɵ) in (11), the
design of a gain-scheduling controller K(Ɵ, Ɵ) in Fig. 4 in an LPV form is
𝑥 𝑘
𝑣
=
𝐴 𝑘(Ɵ, Ɵ) 𝐵 𝑘(Ɵ, Ɵ)
𝐶 𝑘(Ɵ, Ɵ) 𝐷 𝑘(Ɵ, Ɵ)
𝑥 𝑘
𝑒𝑡
𝟄 𝑒𝑠𝑡
(13)
where 𝑥 𝐾 is a controller state vector.
 An LPV controller K, function of varying path direction Ɵ and velocity Ɵ.
 Auxiliary signals 𝑧𝑡 ∶ tangential error performance,
𝑧 𝟄 ∶ contouring control performance and
𝑧 𝑣 : control input energy.
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16.  The signal 𝑓𝑤 : to confine the frequency components of the signal 𝑓 .
 Weighting matrices 𝑊𝑡, 𝑊𝟄, 𝑊𝑉 and 𝑊𝑓 to
 take a tradeoff among these performances,
 improve performance at specific frequency ranges.
Fig. 5: Feedback system for LPV controller design
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17. WEIGHTING FUNCTION TUNNING
 Parametrizing the weighting functions by
W(s)=
𝑠 𝑀 𝐻
1/𝑘 𝑊 + 𝟂 𝑏
𝑠+ 𝟂 𝑏 𝑀 𝐿
1/𝑘 𝑊
𝑘 𝑊
(14)
where, tuning parameters are
𝑀𝐿 : low frequency gain of 𝑊−1
,
𝑀 𝐻 : high frequency gain of 𝑊−1,
𝟂 𝑏 : unit-gain crossing frequency of 𝑊−1,
𝑘 𝑊 ∶ degree of the function W.
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18. Weighting function Tuning parameter Effect
𝑊𝟄 Reduce 𝑀𝐿
Increase 𝟂 𝑏
Reduction in contouring
error and to improve the
controller bandwidth.
𝑊𝑡
𝑀𝐿 much larger than that
of 𝑊𝟄
𝟂 𝑏 much smaller than that
of 𝑊𝟄
Reduction in contouring
error and to improve the
controller bandwidth.
𝑊𝑓
Decrease 𝑀 𝐻
Increase 𝑀𝐿
𝟂 𝑏 below the first
principal resonant
frequency.
Improve the vibration
suppression by damping
out resonant modes.
𝑊𝑣 Increase the gain of 𝑊𝑣. Reduction in motor voltage
and to avoid saturation.
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19. CONCLUSION
 A novel linear parameter-varying (LPV) controller structure.
 The controller consists of two parts,
1. a parameter-varying coordinate transformation matrix which is a function of
machine tool’s reference trajectory angle,
2. a dynamic LPV system which depends on both the trajectory angle and its
velocity.
 The dynamic LPV system was designed by using the well-known advanced gain-
scheduling controller design technique based on linear matrix inequalities.
 Guidelines for tuning weighting functions in contouring controller design.
 Future work : contouring control method to five-axis machine cases.
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20. REFERENCES
 Masih Hanifzadegan and Ryozo Nagamune, “Contouring Control of CNC
Machine Tools Basedon Linear Parameter-Varying Controllers,” IEEE/ASME
Trans. Mechatronics, vol. 21, pp. 2522–2530, 2016.
 N. Khalick Mohammad, A.E. Uchiyama and S. Sano, “Energy saving in feed
drive systems using sliding-mode-based contouring control with a nonlinear
sliding surface,” IEEE/ASME Trans. Mechatronics, vol. 20, no. 2, pp. 572–579,
2015.
 M. R. Khoshdarregi, S. Tappe, and Y. Altintas, “Integrated five-axis trajectory
shaping and contour error compensation for high-speed CNC machine tools,”
IEEE/ASME Trans. Mechatronics, vol. 19, no. 6, pp. 1859–1871, 2014.
 D. Lam, C. Manzie, and M. C. Good, “Model predictive contouring control for
biaxial systems,” IEEE Trans. Control Systems Technology,vol. 21, no. 2, pp.
552–559, 2013.
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21. THANK YOU
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