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					 Piezoelectric Tool Actuator for Precision Machining on Conventional CNC
                                      Turning Centers

                             Andrew Woronko, Jin Huang, Yusuf Altintas
                Manufacturing Automation Laboratory, Department of Mechanical Engineering,
                      University of British Columbia, Vancouver, Canada, V6T 1Z4


Key words: Piezoelectric actuator; Precision turning; Sliding mode control

 A typical precision shaft machining operation may involve both turning and grinding operations; the latter
 required to meet part tolerances on form and surface quality. The positioning error of conventional
 turning centers, hindered by friction and backlash, may be as high as 10 microns depending on the
 design and wear of the machine. Clearly components requiring dimensional tolerances in the micron
 range will require subsequent finishing operations. Although high precision lathes are available to meet
 precision turning requirements, the need for thermal and vibration isolation and high capital cost may
 not be justifiable for some applications. For components requiring high dimensional tolerance only in
 local areas (e.g. bearing surfaces) it may be more cost effective to deliver precise positioning via a tool
 actuator mounted to a conventional machine. Although there has been extensive research in tool
 actuator design and control for ultra-precision diamond turning [1]~[5], few researchers have addressed
 the use of piezo based fast tool servos for precision shaft machining on conventional CNC turning
 machines. This paper presents a piezo based fast tool servo for precision shaft machining on
 conventional CNC turning machines. The mechanical design, structural properties, and controller
 design for high performance tracking and precision positioning of piezo actuator driven tool are
 presented with experimental results for shaft machining.

 An exploded solid model view of the actuator is presented in Fig. 1. A high voltage piezoelectric stack
 actuator is housed under preload within the monolithic guiding unit. Displacement in the radial (x)
 direction of the piezo stack is transmitted to the tool assembly via the use of four solid flexures, each
 with two circular hinges. The symmetric arrangement of the flexures is such that displacement is linear
 without parasitic error in the transverse direction. The tool adapter contains a Komet ABSN25 clamping
 system allowing for standard exchangeable tooling. Cutting force sensors are mounted at the fastener
 locations between the tool adapter and guiding unit. An alignment washer locates the piezo stack to the
 center of the assembly. The top and bottom plates support the guiding unit. A capacitive position
 sensor with 50 micron range and 10 nm resolution measures the relative displacement of the tool
 relative to the guiding unit mounted to the lathe turret. The sensor consists of a stationary probe
 fastened to the top plate, and the moving target mounted to the moving section of the guiding unit. Two
 clamping units, each containing a piezo stack and flexure assembly, serve to rigidly clamp the guiding
 unit if necessary for increased dynamic stiffness during roughing operations. The bottom plate provides
 a shank interface for standard lathe turrets. The overall envelope of the actuator body is 95.5 × 143 × 85
 mm, with a tool stick out of 67 mm in the radial direction and 20 mm in the feed direction. The main
 structural components are machined from a titanium alloy (Ti-Al6-V4) using wire EDM to reduce contour
 errors and residual stresses. The overall stroke is 36µm, and the natural frequency in the radial direction
 is 3200 Hz with a stiffness of 370 N/µm, which can be increased to 620 N/µm by activating the clamping
 units.
                                                       Guiding unit    Upper clamping unit
                                     Piezo stack
                               Sensor target




                                                                                     Turret
                                                                                     interface



                                                                      Tool adapter
                            Sensor probe
                                                                               Toolholder
                               Lower
                               Clamping unit    Hinge profile
                                                                                Y
                                                         r = 2.5 mm
                                                                           X
                                                          t = 3 mm                      Z




                              Fig.1: Exploded view of the actuator assembly


A sliding mode control design is implemented to reject cutting force disturbances and to compensate
nonlinearities of a piezoelectric tool actuator, which is based on the algorithms developed at UBC
MAL[10]. The piezo amplifier (Ga) and piezo stack (Gp) are modeled as constant gains, and the actuator
body and flexures are modeled as a single degree of freedom mechanical system with mass (m),
equivalent flexure spring constant (k) and damping (c) elements. The disturbance force (Fd) consists of
not only the cutting forces in the radial direction of cut but the influence of nonlinear piezo stack
hysteresis which deviates from the constant gain (Gp). The differential equation of the open loop system
which includes the amplifier, piezo stack, the disturbance force and flexure structure are given by:

                                        mx + cx + kx = GaG p u − Fd
                                         DD D                                                             (1)


Equation (1) can be expressed alternatively by considering the influence of the force disturbance at the
input:

                                           ADD + Bx + Cx = u − u d
                                            x     D                                                       (2)

where
                                             m                      c
                                     A=            ;        B=           ;
                                            Ga G p                Ga G p
                                              k                     1
                                    C=             ;       ud =          Fd
                                            Ga G p                Ga G p

A sliding surface S is designed to minimize both position and tracking errors of the piezo actuator as:

                                      S = λ ( xd − x ) + ( xd − x )
                                                           D    D                                         (3)

where the gain λ [1/s] specifies the bandwidth of the response; xd and x are the desired and actual tool
tip positions, respectively. As the sliding surface approaches to zero ( S → 0 ), the tool tip position
converges to the desired position command ( x → x d ) with zero velocity error ( x → x d ). The following
positive definite Lyapunov function, which is continuously differentiable, is selected for the control law
design:
                                                  1ˆ      (u − u ) 2 
                                                                ˆ                                          (4)
                               V [S, (ud − ud )] =  AS 2 + d d 
                                           ˆ
                                                  2          ρ       

      ˆ
where u d represents estimated disturbance and ρ is a disturbance observer gain. The observer
estimates the sum of all the disturbances in the system, including cutting forces, errors in parameter
estimates, and system nonlinearity caused by the piezo stack hysteresis. The derivative of the
              ˆ
disturbance ( u d ) is defined by:


                                             0 if ud ≤ d − and S ≤ 0
                                                   ˆ
                                                                                                          (5)
                        ud = ρSκ , where κ = 0 if ud ≥ d + and S ≥ 0
                        ˆ                          ˆ
                                             1   otherwise
                                             

where the disturbance estimate is bounded by upper and lower values, ud ∈ [d − , d + ] , and selected based
                                                                             ˆ
on the estimation of the cutting forces from the cutting mechanics and experience. According to
Layapunov stability theorem, the system is asymptotically stable if the derivative of the stability function
                   D
is negative, i.e. V < 0 . For a constant disturbance, the derivative of the Lyapunov function is given by:


                            D ∂V ∂S + ∂V         ∂(ud − ud ) ˆ D (ud − ud ) D
                                                        ˆ              ˆ                                   (6)
                            V=                              = ASS −         ˆ
                                                                           ud
                               ∂S ∂t ∂(ud − ud )
                                            ˆ        ∂t              ρ

                 D      ˆ ˆ D ˆ                   ˆ
                V = S ( B − A λ ) x + C Sx − Su + A S ( λ x d + DDd ) + S u d + S ( u d − u d )(1 − κ )
                                                          D     x         ˆ               ˆ                (7)


Since S (ud − ud )(1 − κ ) ≤ 0 from Equation (5), the derivative of the Lyapunov function is guaranteed to
              ˆ
be negative by the following expression:

                               ˆ ˆ D ˆ               ˆ D DD
                             S(B − Aλ)x + CSx − Su + AS(λxd + xd ) + Sud = −Ks S 2
                                                                      ˆ                                    (8)

where Ks > 0 is the selected positive feedback design gain of the control system. Equation (8) leads
directly to the control law:

                                  ˆ ˆ D ˆ          ˆ D x
                             u = (B − Aλ) x + Cx + A(λxd + DDd ) + ρκ ∫ Sdt + Ks S                         (9)


where K s , λ , and ρ are the parameters of the sliding mode controller which are to be tuned. A block
diagram of the controller is given in Fig. 2. The control law given in Equation (9) is in continuous time
domain, and is transformed into discrete time domain with control sampling interval Ts . The derivative
of the displacement and integral of the sliding surface can be evaluated at discrete time intervals as
follows, where low pass filter parameter α is set to 0.5 to avoid noise in the digital integration:

                                                    1−α                     
                              x(k ) = αx(k − 1) +
                              D        D                [x(k) − x(k − 1)]   
                                                     Ts                     
                                                                                                         (10)
                              ∫ S dt(k) = ∫ S dt(k − 1) + S(k)T s
                                                                            
                                                                            
                                                                            

The controller is implemented in digital form in an in house developed Fast Cyclic Executive operating
system running on a Digital Signal Processing board (TMS320C32) with a PC host. The output control
signal (0~10V) is sent to a high voltage piezo amplifier (100 W power, 100 mA average current, 500 mA
peak current), which outputs a signal of 0 to +1000 volts to the piezo stack. The feedback signal from
the capacitive sensor is processed by the sensor electronics card, which outputs a voltage signal to the
DSP board. The sensor was calibrated with a laser interferometer, and has 0.2 volt/micron sensitivity,
                                                                        -8      2              -8
and 3 KHz bandwidth. The identified plant parameters are A = 0.08×10 V/µm/s , B = 190×10 V/µm/s,
and C = 0.353V/µm. The control sampling frequency was 7.5 KHz. The control parameters, tuned to
                                                                                            -5
achieve a step response rise time of 0.010 s with less than 1% overshoot, are Ks = 2.3×10 V/µm/s, λ
=700 1/s, and ρ = 0.6 V/µm. The upper and lower bounds of the disturbance estimate are set to
 ˆ
 d ∈ [−∞,+∞] such that no limits are imposed on the control signal.




                               Fig. 2: Block diagram of the sliding mode controller


Hard turning trials were performed for two workpiece materials; AISI 4340 steel with 35-40 HRC
hardness, and AISI 4320 steel with hardness 58-62 HRC. The inserts used were 35 degree V-type with
0.4 mm nose radius. For AISI 4340 steel, a PVD coated carbide tool was used, and for the high
                                              o                                                            o
hardness steel a CBN tool with 0.1mm x 25 chamfer was used. The tool holder (MVJNL16) had –9
              o                  o
back rake, -5 side rake, and 3 approach angle. The tool position for the semi-finish pass was set by
the x-axis feed drive of the CNC with the actuator controlled to a reference position. At the end of the
semi-finish cut the CNC radial (x) position remains fixed, the actuator is retracted 10 microns to clear the
workpiece, and the CNC z-axis drive returns the tool to the start of the cut. For the finishing pass the
depth of cut is set solely by the actuator. For precision shaft machining of local areas such as bearing
surfaces the finishing depth of cut is determined by comparing the part diameter after the semi-finish
pass to the design value.

The tool position and the radial cutting forces during machining of the 4340 steel is presented in Fig.3
for both the open loop and controlled cases. The cutting speed was 125 m/min, the feed rate was 0.05
mm/rev, and the depth of cut was 5 microns over a length of cut of 10 mm. Clearly the static deflection
due to an average cutting force of 6 N as well as the piezo stack nonlinearities are compensated for
when the controller is active. Positioning resolution of 20 nm has been achieved during machining with
feedback sensor uncertainty of 10 nm. The average and maximum roughness values were obtained
from the filtered roughness profile (Gaussian filter with 50 percent amplitude transmittance at a cutoff
value of 0.8 mm) over a sampling length of 8 mm. The maximum and average roughness for the 4340
steel is less than 2.0 microns and 0.3 microns, respectively. For the hardened 4320 steel the maximum
and average roughness values are decreased to less than 0.85 and 0.15 microns, respectively. The
frequency response function for position tracking as well as a sensitivity function of the actuator system
is given in Fig.4. The experimentally obtained bandwidth is 200 Hz for position tracking.
                                                                   Radial Cutting Force




       Radial Force [N]
                                   10

                                    5

                                    0


                                          12      14      16        18      20       22         24   26   28       30
                                                                          Time (s)
                                           Start of Cut                                                        End of Cut
                                                                Tool Position - Controlled
Position [micron]


                          5.04
                          5.02
                             5
                          4.98
                          4.96
                              10          12      14      16        18      20       22         24   26   28       30
                                                                         Time (s)
d = 0.010 mm                                                                                    d = 0.020 mm


                                                                 Tool Position - Uncontrolled
 Position [micron]




                          5.34
                          5.32
                          5.3
                          5.28
                          5.26
                          5.24
                                    10    12      14      16        18      20       22         24   26   28       30
                                                                         Time (s)                         d = 0.040 mm

                                    Fig.3. Tool position during finish machining at 5 micron depth of cut

                                                                 Position Tracking


                                                                           - 3dB
                X/Xd [um/um]




                                                                 Frequency [Hz]

                                                               Disturbance Rejection
                     X/Fd [nm/N]




                                                                  Frequency [Hz]


                    Fig.4: System frequency response function and sensitivity function
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