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Force Sensorless Control of Cutting Resistance forNC Machine Tools by Spindle Motor Control Using
Variable Pulse Number T-methodTeruaki Ishibashi and Hiroshi Fujimoto
Department of Advanced Energy, The University of Tokyo5-1-5, Kashiwanoha, Kashiwa, Chiba, 227-8561 Japan
Phone: +81-4-7136-3881, Fax: +81-4-7136-3881Email: [email protected], [email protected]
Abstract— This paper proposes a force sensorless control ofcutting resistance for numerical control (NC) machine tools byspindle motor. In conventional NC machine tools, speed of spindlemotor is set to a constant value during machining, and thespindle speed is determined according to operator’s experience.The proposed method estimates cutting resistance by disturbanceobserver, and the spindle speed is determined by controllingthe cutting resistance to follow the reference value. In addition,a novel method for velocity calculation which is referred asvariable pulse number T-method is proposed to reduce thevelocity measurement noise and to improve the robustness ofdisturbance observer.
I. INTRODUCTION
Recently, NC machine tools play an important role asmother machine in industry. A considerable amount of studyhas been done on the NC machine tools to realize high accu-racy and high speed of machining [1][2]. Cutting resistance,chatter vibration and thermal expansion have negative effecton cutting accuracy. Chatter vibration is the vibration of toolexcited by certain factor [3][4].
Cutting resistance is observed carefully because it causesmany problems during machining, such as chatter vibration,deflection of tool and so on. Usually, force sensor or accel-eration sensor are implemented to observe cutting resistance,which is not practical due to the cost and difficulty. Someprevious studies have proposed methods to predict and controlthe cutting resistance [5][6]. However, the methods are notpractical, because they need test cutting and identifying manyparameters of tool which are difficult to measure.
In this study. a model of cutting resistance is proposed,and control of the cutting resistance by spindle speed controlis proposed based on the model. This method does not needadditional sensor because cutting resistance is estimated bydisturbance observer [7][8]. Moreover, the proposed approachis not affected by the decreasing of inertia of stage duringmachining. In addition, a novel method to measure angularvelocity is proposed to suppress noise effects in disturbanceobserver.
(a) Equipment overview. (b) Schematic diagram of plant.
Fig. 1. Experimental equipment.
Fig. 2. Block diagram of plant.
II. CONTROLLED SYSTEM
This section will introduce the experimental plant andsimplified cutting resistance model.
A. Spindle motor
Fig. 1(a) shows an overview of experimental equipment,and Fig. 1(b) shows a schematic diagram of plant duringmachining. Machining is performed by feeding stage andpressing a work against tools. Cutting resistance arise duringmachining. In this paper, cutting resistance is controlled byspindle speed. The feed rate is set to a constant value duringmachining. Fig. 2 shows block diagram of plant. Here, vs isthe feed rate, i is current, Kt is torque coefficient, and T istorque. The torque is acquired by (1).
T = Kti. (1)
Cutting resistance is considered as a disturbance torque tospindle and denoted as Fcut. Other disturbance factors are nottaken into account because the volumes are quite smaller than
TABLE IPARAMETERS OF SPINDLE
Driver GPA40LInertia J 4.4 × 10−3 kg ·m2
Friction coefficient D 2 × 10−3 Nm · sTorque coefficient Kt 0.92 Nm/A
0 20 40 60 80 1000
0.5
1
1.5
2
ω [rad/s]
Fcut[N
m]
Experimental Data 1Experimental Data 2Model
(a) Relationship between cutting resistance and angu-lar velocity
1 2 3 4 50.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
vs [mm/s]
Fcut[N
m]
ω = 25 rad/sω = 40 rad/s
(b) Relationship between cutting resistance and feedrate
Fig. 3. Measurements of cutting resistance.
cutting resistance. Spindle speed is acquired by (2). Here, Jis inertia of spindle including tools, D is friction coefficientof spindle, ω is spindle speed. Table 1 shows parameters ofspindle and driver.
ω =1
Js+D(T − Fcut). (2)
B. Cutting resistance model
Simplified cutting resistance model is acquired by (3).This model assumes that cutting resistance is proportional tothickness of swarf. Here, k is a coefficient depending on axialdepth, radial depth, temperature, quality of work and so on.We denote this parameter as cutting resistance coefficient inthe following.
Fcut = kvsω. (3)
Fig. 3(a) shows the relation between spindle speed ω andcutting resistance Fcut estimated by disturbance observer.In experiment, a chemical wood is used as work, and vs,
axial depth and radial depth is set as 4 mm/s, 5 mm and14 mm, respectively. Experimental data 1 is measured atω = 15, 20, 25, 30, 40, 100 rad/s and Experimental data 2 is atω = 27 ∼ 88 rad/s. In experimental data 2, the measurementwas performed with a ramp speed signal. Fig. 3(b) showsthe relation between feed rate vs and cutting resistance Fcut
estimated by disturbance observer.These data show that cutting resistance has inverse relation
with spindle speed and proportional relation with feed rate.Control system is designed according to the simplified cuttingresistance model.
III. ANGULAR VELOCITY MEASUREMENT METHOD
This section presents conventional method and a novelmethod to measure angular velocity.
A. Conventional velocity estimation methods
In order to measure Angular velocity, M-method, T-method[9], S-method [10], M-method-based method using FPGA[11], and Average T-method [12] are commonly used. In theM-method, the angular velocity is calculated by counting thenumber of encoder pulses within one sampling. In contract, T-method calculates angular velocity by measuring the period ofone encoder pulse using base-clock of DSP. Accuracy of M-method and T-method is degraded in the low-speed area andhigh-speed area, respectively. S-method calculates the velocitysynchronized with the variation of number of the encoder pulsecounted in a sampling time. S-method can measure angularvelocity with higher accuracy in all-speed areas. In average T-method, angular velocity is calculated by measuring the periodof Np encoder pulses using base-clock of DSP. It can measureangular velocity with high accuracy in more higher-speed areathan T-method. Np is set as constant during measuring inaverage T-method, . So, it cannot be applied to the situationthat angular velocity varies due to the phase-lag and reducethe accuracy.
Variable pulse number T-method is novel method based onaverage T-method taking the variation of velocity into consider.In this method, Np changes during measuring according toangular velocity. The algorithm is described in subsection C.
B. Detection accuracy
Fig. 4 shows the principle of angular velocity measurement.Here, P is resolution of encoder, Ts is sampling period. Ns
is number of the pulse counted within one sampling period,Tclock is resolution of base-clock, T1p[i] is the actual periodof a encoder pulse, T1p[i] is measured T1p[i] by base-clock,TNp[i] is the actual period of Np encoder pulses, and TNp[i]is measured TNp[i] by base-clock.
Computing equations of each method are (4)∼(6). ωm, ωt
and ωa are angular velocity measured by M-method, T-method
Fig. 4. Principle of angular velocity detection.
Fig. 5. Angular velocity detection error.
and average T-method, respectively.
ωm[i] =2πNs[i]
P
1
Ts. (4)
ωt[i] =2π
P
1
T1p[i]. (5)
ωa[i] =2πNp[i]
P
1
TNp[i]. (6)
In M-method, detection error arises due to the resolutionof encoder. In contrast, in T-method and average T-method,detection error arises due to the resolution of base-clock. Fig. 5shows the detection error. Here, Em, Et and Ea are maximumdetection error using M-method, T-method and average T-method, respectively. Each value is acquired by (7)∼(9). Here,TNp is approximated by NpT1p. Assuming Tclock is muchsmaller than T1p, the detection error of average T-method isreduced to 1/Np compared with T-method.
Em[i] =2π
P · Ts. (7)
Et[i] =2π
P
Tclock
T1p[i](T1p[i] + Tclock). (8)
Ea[i] =2π
P
Tclock
T1p[i](NpT1p[i] + Tclock). (9)
C. Variable pulse number T-method
The procedure of variable pulse number T-method is shownas follows.
Fig. 6. Block diagram of angular velocity control
1) Count up the pulse number Np[i] synchronized withencoder signal.
2) Measure the TNp[i] synchronized with counting upNp[i].
3) Calculate angular velocity by (10) at every samplingperiod.
ωv[i] =2πNp[i]
P
1
TNp[i]. (10)
Here, ωv is angular velocity measured by variable pulsenumber T-method.
IV. CUTTING RESISTANCE CONTROL SYSTEM
This section will present the details of cutting resistancecontrol.
A. Speed control system
Fig. 6 shows the block diagram of angular velocity control.CPI is feedback controller which is designed to have abandwidth of ωp = 100 rad/s by pole placement method. Theparameters of CPI are represented as (11) (12). Disturbanceobserver is used to estimate and compensate the cuttingresistance. Here, τ denotes the time constant of LPF, which isassigned as 1/(2π · 200) s.
CPI =KPs+KI
s. (11)
KP =2Jωp −D
Ktn, KI =
Jω2p
Ktn. (12)
B. Estimation of cutting resistance coefficient
In proposal method, cutting resistance coefficient is esti-mated using least-square method. The output y is representedas (13) on the assumption that y is the product of unknownparameter η and known parameter θ.
y = θη. (13)
Here, y, η and θ correspond to cutting resistance Fcut, cuttingresistance coefficient k and vs/ω, respectively. In this way, kcan be estimated by least-square method.
Cutting resistance varies according to the angle of the tools.In order to average cutting resistance in the period in whichtools rotate nr times (nr:integer), the data used to estimation
Fig. 7. Block diagram of cutting resistance control.
is limited to those obtained in the period. k is updated in everycontrol period.
C. Spindle speed reference
Fig. 7 shows block diagram of cutting resistance control.Spindle speed reference is generated based on cutting resis-tance model (14).
ω∗ =k
F ∗cut
vs. (14)
V. SIMULATION RESULTS OF VARIABLE PULSE NUMBERT-METHOD
This section present simulation results of angular velocitydetection.
Resolution of encoder and base-clock are set to(8000 Pulse/Rev) and 20 ns, respectively. Samplingtime of M-method and variable pulse number T-method is setto 1 ms. Taking account of ripple of angular velocity, Np ofaverage T-method is conservatively set to Np = 100 so thatupdating cycle would lower than the sampling time.
Fig. 8(a) and Fig. 8(b) show detected angular velocity whenmotor rotates with a velocity of 104.825 rad/s. Fig. 8(c)shows Np of variable pulse number T-method. It is obviousthat variable pulse number T-method can reduce detectionerror compared with M-method, T-method and average T-method. It is concluded that variable pulse number T-methodcan use as much Np as possible, while average T-methodshould use conservatively set Np.
VI. EXPERIMENT
This section presents experimental result of angular velocitydetection and cutting resistance control.
A. Experimental results of variable pulse number T-method
This section compares M-method, T-method, average T-method, variable pulse number T-method. In the experiment,incremental encoder (2000 Pulse/Rev) is used and quadru-pled in M-method. Resolution of base-clock is 20 ns andsampling time of M-method and variable pulse number T-method is set to 1 ms. Np is set conservatively so that updatingcycle would lower sampling time 1 ms in average T-method.Experiment was performed by angular velocity control shown
TABLE IICUTTING CONDITION
End mill KOBELCO 4MC (2 flutes)End mill ϕ 12 mmWork piece Artificial timber MBO600
Free length of end mill 74 mmRadial depth of cut 6 mmAxial depth of cut 14 mm
in Fig. 6 with constant angular velocity reference, and angularvelocity measured by M-method was used as feedback.
Fig. 9(a) and Fig. 9(c) show experimental results of angularvelocity detection. Fig. 9(b) and Fig. 9(d) show Np of variablepulse number T-method. Reference value of angular velocitywas set to ω∗ = 15, 50 rad/s. In average T-method, Np wasset to 3, 12. Np of variable pulse number T-method is as largeas possible and it is larger than that of average T-method.
Fig. 10 show estimated cutting resistance measured by dis-turbance observer using M-method and variable pulse numberT-method. When machining, spindle speed was set as constantat ω∗ = 20 rad/s, and feed rate is set as vs = 2, 3 mm/s. Byapplying variable pulse number T-method, the noise caused bydetection error is reduced.
B. Experimental results of cutting resistance control
Experiment was performed under the condition showed inTable. 2. Sample data used to estimating cutting resistancecoefficient k is limited to the data obtained in a period inwhich spindle rotate 1, 2 times (nr = 1, 2). Reference valueof cutting resistance is F ∗
cut = 0.3 Nm. Machining starts att = 0 s.
Fig. 11 shows the experimental results of conventionalmethod. In conventional method, reference value of spindlespeed was obtained by cutting test. There is error betweenreference value and averaged cutting resistance. This error iscaused by inhomogeneous quality of work and some changesin cutting condition.
Fig. 12 and Fig. 13 show experimental results of proposalmethod. In this method, lower limit was set so that spindlespeed does not fall below 15 rad/s. By estimating cuttingresistance coefficient k during machining, spindle speed refer-ence ω∗ changes and cutting resistance follows the referencevalue. Convergence speed of estimating reduces with theincreasing of nr. Even when nr = 1, cutting resistance canfollow reference value. So, nr = 1 is adequate in this cuttingcondition.
VII. CONCLUSION
In this paper, Force sensorless cutting resistance control byspindle motor is proposed. By the experiment, it is confirmedthat proposal method can make cutting resistance followreference value. In order to improve accuracy of disturbanceobserver, a novel method of angular velocity measurement“variable pulse number T-method” was proposed. By thesimulation and the experiment, effectiveness of this proposalmethod was confirmed.
0 2 4 6 8 10104.4
104.6
104.8
105
105.2
105.4
Time [ms]
ω[r
ad/s
]
ω0ωmωa(N=100)ωv
(a) Detected angular velocity
0 2 4 6 8 10104.82
104.822
104.824
104.826
104.828
Time [ms]
ω[rad/s]
ω0
ωa(N = 100)ωv
(b) Detected angular velocity (enlargedview)
0 2 4 6 8 10
133
134
Time [ms]
Np
(c) Counted number of pulse
Fig. 8. Simulation results of angular velocity detection.
0 20 40 60 80 10014
14.5
15
15.5
16
Time [ms]
ω[r
ad/s
]
ωm
ωa (N = 3)ωv
(a) Detected angular velocity (ω∗ =15 rad/s).
0 20 40 60 80 100
4
5
Time [ms]
Np
(b) Counted number of pulse (VPNT-method) (ω∗ = 15 rad/s).
0 20 40 60 80 100
49.6
49.8
50
50.2
50.4
Time [ms]
ω[r
ad/s]
ωm
ωa (N = 12)ωv
(c) Detected angular velocity (ω∗ =50 rad/s).
0 20 40 60 80 100
15
16
Time [ms]
Np
(d) Counted number of pulse (VPNT-method) (ω∗ = 50 rad/s).
Fig. 9. Experiment results of angular velocity detection.
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
1.2
Time [s]
Fcut[N
m]
Fcut (M − method)
Fcut (VPNT − method)
(a) Estimated cutting resistance (vs = 2 mm/s).
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
1.2
Time [s]
Fcut[N
m]
Fcut (M − method)
Fcut (VPNT − method)
(b) Estimated cutting resistance (vs = 3 mm/s).
Fig. 10. Experiment results of estimated cutting resistance by dusturbance observer.
0 1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
Fcut[N
m]
F ∗
cut
vs = 2 mm/svs = 3 mm/s
(a) Cutting resistance Fcut.
0 1 2 314
16
18
20
22
24
26
28
Time [s]
ω[r
ad/s
]
ω∗ (vs = 2 mm/s)ω∗ (vs = 3 mm/s)vs = 2 mm/svs = 3 mm/s
(b) Angular velocity of spindle ω.
0 1 2 30
0.2
0.4
0.6
0.8
1
Time [s]
i ref[A
]
vs = 2 mm/svs = 3 mm/s
(c) Current command iref .
Fig. 11. Experimental results (conventional method).
0 1 2 3−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
Fcut[N
m]
F∗
cut
nr = 1nr = 2
(a) Cutting resistance Fcut.
0 1 2 3−0.5
0
0.5
1
1.5
2
2.5
3
Time [s]
k[k
N]
nr = 1nr = 2
(b) Cutting resistance coefficient k.
0 1 2 314
16
18
20
22
24
26
28
Time [s]
ω[r
ad/s
]
ω∗ (nr = 1)ω∗ (nr = 2)nr = 1nr = 2
(c) Angular velocity of spindle ω.
0 1 2 30
0.2
0.4
0.6
0.8
1
Time [s]
iref[A
]
nr = 1nr = 2
(d) Current command iref .
Fig. 12. Experimental results (proposed method vs = 2 mm/s).
0 1 2 3−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Time [s]
Fcut[N
m]
F∗
cut
nr = 1nr = 2
(a) Cutting resistance Fcut.
0 1 2 3−0.5
0
0.5
1
1.5
2
2.5
3
Time [s]
k[k
N]
nr = 1nr = 2
(b) Cutting resistance coefficient k.
0 1 2 314
16
18
20
22
24
26
28
Time [s]
ω[r
ad/s
]
ω∗ (nr = 1)ω∗ (nr = 2)nr = 1nr = 2
(c) Angular velocity of spindle ω.
0 1 2 30
0.2
0.4
0.6
0.8
1
Time [s]iref[A
]
nr = 1nr = 2
(d) Current command iref .
Fig. 13. Experimental results (proposed method vs = 3 mm/s).
VIII. ACKNOWLEDGMENT
This work was supported in part by the MORI SEIKICo.Ltd.. The authors gratefully acknowledge valuable sugges-tions from Shinji ISHII, Kouji YAMAMOTO and Yuki TERADA from the MORI SEIKI Co. Ltd..
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