MODELING WITH CURRENT DYNAMICS AND VIBRATION

CONTROL OF TWO PHASE HYBRID STEPPING MOTOR IN

INTERMITTENT DRIVE

Ryota Mori, Yoshiyuki Noda, Takanori Miyoshi, Kazuhiko Terashima

Department of Production Systems Engineering, Toyohashi University of Technology

Hibarigaoka 1-1, Tempaku, Toyohashi, 441-8580, Japan

Masayuki Nishida, Naohiko Suganuma

Tokyo Weld Co., Ltd.

Ashitaka 292-50, Numazu, Shizuoka, 410-0001, Japan

Keywords:

Stepping motor, phase plane analysis, vibration control.

Abstract:

This paper presents modeling of stepping motor and control design of input pulse timing for the suppression

control of vibration. The stepping motor has the transient response of electric current for the pulse input.

Therefore, the motor model considering the transient response of the current is built. The validity of the

proposed model is veriﬁed by comparing the model considering the transient response of the current with

the one without its consideration. Design of the pulse input timing in the method of the four pulse drive is

realized to achieve the desired angle without vibration and overshoot using an optimization method. Finally,

the effectiveness of the proposed method is demonstrated by comparing simulation results with experiments.

1 INTRODUCTION

The stepping motor has been widely used for factory

automation (FA) and ofﬁce automation (OA) equip-

ment, because it is able to realize high-accuracy posi-

tioning by an open loop control. It is used also for the

production process of electronic parts, and then the

settling time of the stepping motor is directly linked

to the productivity. Therefore, the high speed and the

low vibration are strongly desired in the production

line. However, the stepping motor vibrates around the

neighborhood of the equilibrium points owing to the

step-wise drive from the viewpoints of the motor char-

acteristics. To dampen the vibration of motor, (i) mi-

cro step drive method that makes changes the exciting

current change in details, and (ii) a inverse phase exci-

tation dumping, and (iii) a delay damping method, are

proposed(D.Ebihara and T.Iwasa, 1984). Especially,

the micro step drive is possible to drive with the low

vibration. Because it is made to drive by changing

at smaller angle than the basic step angle by making

the exciting current change in details (D.Ebihara and

T.Iwasa, 1984)D It is necessary to give the excitation

instruction considering the dynamic characteristic of

the system in the transient state of the start and the

stop times. As an adjustment method of the excita-

tion sequence, the method of applying a lowpass ﬁlter

as a pre-compensator (T.Miura et al., 2000), and the

method that uses the genetic algorithm (T.Miura and

T.Taniguchi, 1999) are proposed. The vibration con-

trol considering robustness for the vibration of the in-

ertia load is studied by these methods. Moreover, the

technique for decreasing the resonance using the posi-

tion and the speed feedback estimated by the observer

is given (S.M.Yang and E.L.Kuo, 2003).

The stepping motor has a strong nonlinearity.

Therefore, when the linear control theory is applied, it

is often linearized around the equilibrium point. How-

ever, the operation area of the stepping motor is wide,

so variable control gain is necessary to keep an excel-

lent control performance. Whereas, the method us-

ing an exact linearization by means of nonlinear feed-

back and coordinate transformation of state space is

proposed (M.Bodson et al., 1993). Moreover, the ap-

plication of the control algorithm of the artiﬁcial in-

telligence system such as fuzzy theory (F.Betin and

D.Pinchon, 1998) and neural network (K.Laid et al.,

2001) are provided.

Those control methods are mostly discussed con-

cerning with a step drive and a continuous drive.

However, the high speed and the high accurate po-

sitioning system by the ﬁne drive might be requested

in the FA equipment. In this case, the positioning by

a few number of pulse order is performed.

388

Mori R., Noda Y., Miyoshi T., Terashima K., Nishida M. and Suganuma N. (2007).

MODELING WITH CURRENT DYNAMICS AND VIBRATION CONTROL OF TWO PHASE HYBRID STEPPING MOTOR IN INTERMITTENT DRIVE.

In Proceedings of the Fourth International Conference on Informatics in Control, Automation and Robotics, pages 388-393

DOI: 10.5220/0001651303880393

Copyright

c

SciTePress

Therefore, in this paper, the vibration suppression

is studied when the stepping motor is intermittently

driven by a predetermined few number of step, assum-

ing FA application. The response of the stepping mo-

tor is varied by the inﬂuence of transient response of

the current, when the command pulse interval is rather

short for the excitation phase. Therefore, a mathe-

matical model comprised of the torque equation with

the dynamics of the current in the generated torque

is presented. By means of the phase plane analysis

and the optimization for the obtained mathematical

model, the pulse control timing with a low vibration is

obtained. The validity of the proposed model is con-

ﬁrmed through the experiments and the simulations.

2 EXPERIMENTAL SETUP

The experimental setup consists of motor with the in-

ertia load connected with a solid shaft as shown in

Figure 1. The shaft has been extended from both ends

of the motor, and the inertia load is installed in a one

end. The encoder has been installed to the shaft on

the other side.

The block diagram of the driving system of a step-

ping motor is shown in Figure 2. The motor driver

uses the commercial item, and the motor is driven up

to the ﬁxed current drive in a full step. The pulse in-

put is carried out by the designed timing in advance by

numeric calculation. The pulse is outputted according

to the timing speciﬁed from the Pulse Oscillator. The

excitation current is passed by excitation phases by

the motor driver, and the motor is driven. The excita-

tion phase of the motor can be switched by the pulse

being input from Pulse Oscillator to Motor Driver. In

this study, the motor is used with the excitation phase

comprised of Phase A, Phase B, Phase

¯

A, and Phase

¯

B. The excitation phase switches in order of AB →

B

¯

A →

¯

A

¯

B →

¯

BA → AB by two phase excitation. The

rotor angle is detected by the encoder of accuracy of

36000[pulse/rev]. The encoder signal is doubled by

up-down counter of four.

3 STEPPING MOTOR MODEL

3.1 Modeling of Stepping Motor

The stepping motor rotates stepwise by switching the

excitation phase by input pulse. Therefore, Eq.(1) can

be derived from the equation of motion of rotation

system (D.Ebihara and T.Iwasa, 1984).

J

¨

θ+ D

˙

θ+ T

L

= T

M

(1)

Stepping motor

Encoder

Inertia load

Coupling

Figure 1: Construction of stepping motor.

Controller

PIC 18F252

Pulse oscillator Motor driver

Encoder

Counter

PIC 18F252

Motor

iA,iB,iA,iB

θ

Figure 2: Block diagram of driving system for a stepping

motor.

,where J is an inertia moment, D is a damping coefﬁ-

cient, T

L

is a load torque, T

M

is the generated torque

of motor and θ shows a rotor angle. The generated

torque can be described as the sum of the generated

torque of each phase. The generated torque of each

phase of two phase hybrid stepping motor is repre-

sented by the following equations.

T

A

= −i

A

Ksin(N

R

θ) (2)

T

B

= −i

B

Ksin(N

R

θ−

π

2

) (3)

T

¯

A

= −i

¯

A

Ksin(N

R

θ−π) (4)

T

¯

B

= −i

¯

B

Ksin(N

R

θ−

3π

2

) (5)

,where K is a torque constant, N

R

is the number of a

rotor teeth and, i

A

, i

B

, i

¯

A

and i

¯

B

are excitation current

of each phase. The sufﬁx of torque T and current i

shows phase A, B,

¯

A and

¯

B respectively. Whenever

the pulse is input in the two phase stepping motor,

it is excited in order of AB → B

¯

A →

¯

A

¯

B →

¯

BA →

AB. Moreover, because i

A

= −i

¯

A

and i

B

= −i

¯

B

, then

it follows that T

A

= T

¯

A

and T

B

= T

¯

B

. Therefore the

generated torque T

M

becomes the sum of T

A

and T

B

. If

a magnetic axis is deﬁned as θ = 0 when phase A and

phase B are the excited states, the generated torque is

shown as follows.

T

M

= −i

A

Ksin(N

R

θ+

π

4

) −i

B

K sin(N

R

θ−

π

4

) (6)

Generally, it becomes i

A

= i

B

= i at the ﬁxed cur-

rent drive. And Eq.(7) is derived by the trigonometric

function formula.

MODELING WITH CURRENT DYNAMICS AND VIBRATION CONTROL OF TWO PHASE HYBRID STEPPING

MOTOR IN INTERMITTENT DRIVE

389

T

M

= −

√

2iKsin(N

R

θ) (7)

This formulation is explained by the reference of

D.Ebihara and T.Iwasa, 1984. However, there is a

transient state between the time of the pulse input

and the switching time of the current of the excitation

phase, as shown in Figure 3. Here, Figure 3 shows the

excitation current transition of each phase after pulse

input when the stepping motor is driven, and Exp I

A

is a current of phase A and

¯

A, Exp I

B

is a excitation

current of phase B and

¯

B, Sim I

A

and Sim I

B

is the

assumed current in simulation. From Figure 3, when

the ﬁrst pulse is input, the Exp I

A

is switched from the

plus into the minus. It is meant to change the phase

¯

A from the phase A, because Exp I

A

is the excitation

current of phase A. Here, the change of the current

in Exp I

A

is almost linearly changed from the value

of +1.2[A] to the value of −1.2[A], when the excita-

tion phase is switched. Excitation current Exp I

B

of

the phase B is also similar. Therefore, the excitation

current is thought that the current is changed almost

linearly from the input of the pulse up to the switch-

ing the excitation phase as shown in Sim I

A

and Sim

I

B

. The current response of interval T

d

[s] is shown in

Eq.8, when the elapsed time after inputting the pulse

is deﬁned as T

i

[s], and the time from the input of the

pulse up to switching the excitation current is deﬁned

as T

d

[s].

i

A,B

= −(1−2

T

i

T

d

)I, (0 ≤ T

i

≤ T

d

) (8)

,where I is the rated current of the motor. From

Eq.(1), Eq.(6) and Eq.(8), the equation of motion is

shown in Eq.(9).

J

¨

θ+D

˙

θ+T

L

= −i

A

Ksin(N

R

θ+

π

4

)−i

B

Ksin(N

R

θ−

π

4

)

(9)

,where i

A

is shown by the following equation. As for

i

B

, it is omitted in this paper, due to the limitation of

paper, because it is similar equation.

i

A

=

I, (Phase A)

−(1−2

T

i

T

d

)I, (Phase A → Phase

¯

A)

−I, (Phase

¯

A)

(1−2

T

i

T

d

)I, (Phase

¯

A → Phase A)

(10)

3.2 Parameter Identiﬁcation

The parameter of Eq.(9) is identiﬁed from the step re-

sponse of stepping motor. The parameter is decided

to be minimized the error sum of square by a Sim-

plex Method(M.Hamaguchi et al., 1994). Here, J,

D and T

L

are the parameters including the encoder.

The step response result is shown in Figure 4. The

response corresponds with the experiments and the

simulations, and then the identiﬁcation is appropri-

ate, because simulation agrees well with experimen-

tal results. The model parameter is shown in Table 1,

where α is a step angle.

0 0.001 0.002 0.003 0.004

2

1.5

1

0.5

0

0.5

1

1.5

2

Time[s]

Current[A]

Exp I

A

Exp I

B

Sim I

A

Sim I

B

1st Pulse input

2nd Pulse input

3rd Pulse input

Td

Ti

Figure 3: Current response of stepping motor.

0 0.01 0.02 0.03 0.04 0.05

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time[s]

Angle[deg]

Simulation

Experiment

Figure 4: Step response of proposed model.

Table 1: Motor parameters.

Parameter Value

α 0.9 [deg]

I 1.2 [A/phase]

J 164.94 × 10

−7

[kg·m

2

]

D 0.001442 [kg·m·s]

T

L

0.00357 [N·m]

K 0.2662 [N·m/A]

T

d

0.0007 [s]

3.3 Comparison Between Proposed and

Conventional Model

The conventional model is written by the following

equation using Eq.(1) and Eq.(7).

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

390

J

¨

θ+ D

˙

θ+ T

L

= −

√

2IK sin(N

R

θ) (11)

The model parameters which did not consider the

current response can be also identiﬁed from the step

response in the same way with subsection 3.2. The

parameters obtained from the optimization are shown

in Table 2. The result of the step response is shown

in Figure 5. Experiments and simulations are almost

well in agreement as shown in Figure 5

The model which did not consider the current re-

sponse is shown in Eq.(11), and the proposed model

are compared. In comparison, the response of motor

in experiment and simulation are checked, when four

times pulses are inputted to motor. T

1

[s] is the interval

of input pulse from the ﬁrst pulse to the second pulse.

And T

2

[s] is the interval of input pulse from the sec-

ond pulse to the third pulse, T

3

[s] is the interval of in-

put pulse from the third pulse to the fourth pulse, and

then T

1

=400[µs]CT

2

=2331[µs]CT

3

=1428[µs]. The

pulse timing of T

1

, T

2

, and T

3

is respectively obtained

by using Simplex Method to minimize the evaluation

function as shown in Eq.(12) at Section 4. Experiment

and simulation results are shown in Figure 6.

The rising time of the conventional model is

shorter than proposed model, and the residual vibra-

tion remains close to the desired value. However, sim-

ulation result of the proposed model agrees well with

the experimental result. As a result, we conﬁrmed that

the more highly accurate model can be obtained by

the proposed model considering the current response.

Table 2: Parameters of conventional model.

Parameter Value

J 168.56 × 10

−7

[kg·m

2

]

D 0.001505 [kg·m·s]

T

L

0.001902 [N·m]

K 0.2582 [N·m/A]

4 DESIGN OF PULSE INPUT

TIMING

In this paper, the time that the rotor reaches the de-

sired angle without vibration, when input using four

pulses are given to the stepping motor, is determined

based on the proposed model. The time of the ﬁrst in-

put pulse is 0[s]. The one step angle is α = 0.9[deg],

so the desired angle by four input pulse is 3.6[deg].

The pulse interval T

1

, T

2

, and T

3

denotes the same

meaning with the preceding section. When T

1

are

changed from 300[µs] up to 2200[µs] every 50[µs] in-

terval, total 39 patterns, optimal T

2

and T

3

are decided

0 0.01 0.02 0.03 0.04 0.05

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Time[s]

Angle[deg]

Simulation

Experiment

Figure 5: Step response of conventional model.

0 0.002 0.004 0.006 0.008 0.01

0

0.5

1

1.5

2

2.5

3

3.5

Time[s]

Angle[deg]

Reference

Proposed

Conventional

Experiment

Figure 6: Comparison result of the conventional and pro-

posed model.

in the same way. The optimization problem of mini-

mizing the following evaluation function is solved as

a decision method.

min

T

2

,T

3

J

c

= (maxθ(t) −4α)

2

, (T

1

+ T

2

+ T

3

+ T

d

≤t)

(12)

,where T

d

is the settling time of the current for the

pulse input. To minimize the overshoot at the desired

angle, a Simplex Method is used as an optimization

technique. The evaluation function is decided to be

the error square of the desired angle and the maximum

angle after reaching at around the neighborhood of the

desired angle. If the overshoot with the desired angle

is small, the vibration then is also a little, because the

stepping motor is settled at the desired angle decided

according to the excitation phase.

The overshoot of the angle in the control time ob-

tained by a Simplex Method were almost 0.0009[deg]

or less. The overshoot was less than 0.1 % against

the step angle of α = 0.9[deg] by a Simplex Method.

Therefore, it is thought that there is little overshoot.

Simulation results using control timing obtained

MODELING WITH CURRENT DYNAMICS AND VIBRATION CONTROL OF TWO PHASE HYBRID STEPPING

MOTOR IN INTERMITTENT DRIVE

391

in a Simplex Method stated above are shown in

Figure 7 and Figure 8. Here, Figure 7 shows the

time response of rotor angle and angular velocity,

and Figure 8 shows the phase plane trajectory. The

point in the ﬁgure is control time of the pulse input,

and the arrows shows transitions of the pulse input

timing when T

1

changes from 300[µs] to 2200[µs].

VelocityErrorPlane method is commonly used in the

phase plane trajectory of the stepping motor to pre-

vent a transverse axis from becoming long (D.Ebihara

and T.Iwasa, 1984). Here, in VelocityErrorPlane

method, the phase difference

π

2

is subtracted when-

ever the excitation phase is switched. However, in this

paper, to see the response to control time of pulse four

times, the angle without subtracting the phase differ-

ence was drawn in a transverse axis.

There is a width of about 1[ms] to settling to the

desired angle using pulse timing in this study. Mo-

tion time can be shortened to use the pulse timing in

the trajectory with large angular velocity on the phase

plane.

When the current dynamics is not considered, the

changeover point on the phase plane without vibra-

tion by the desired angle is 3.6[deg] and 0[deg/s]

(D.Ebihara and T.Iwasa, 1984). However, it is con-

ﬁrmed that the last changeover point is the smaller

angle than 3.6 [deg], when there is the transient re-

sponse of the current. As seen from Figure 7, the time

to reach at the desired angle is about 700 [µs] after

the fourth input pulse is given, and it corresponds to

the transient response of the current for pulse input

assumed from Figure3. Consequently, it is conﬁrmed

that the excitation sequence with a little vibration can

be obtained only by considering the response of the

current. Therefore, to realize the vibration-free re-

sponse in the full step drive, the proposed model con-

sidering the current dynamics and the control design

based on the model are made clear through these anal-

yses.

5 EXPERIMENTAL RESULTS

Three patterns are selected based on the timing de-

signed in Section 4 and experiments are done. The

pulse control time used in the experiments is shown

in Table 3. Experimental results of No.1, No.2, and

No.3 are shown in Figure 9, Figure 10, and Figure 11,

respectively.

Simulation results of both the angles and the an-

gular velocity corresponds well to experimental re-

sults. Especially, the vibration of experimental num-

ber No.1 was suppressed within 0.01 [deg] as the

same as resolution of encoder.

0 0.001 0.002 0.003 0.004 0.005 0.006

0

1

2

3

Time[s]

Angle[deg]

0 0.001 0.002 0.003 0.004 0.005 0.006

0

500

1000

1500

Time[s]

Angular velocity[deg/s]

2nd Pulse

3rd Pulse

4th Pulse

Figure 7: Simulation result of time response using 4 pulse

input of various control time.

0 0.5 1 1.5 2 2.5 3 3.5

0

200

400

600

800

1000

1200

1400

1600

Angle[deg]

Angular velocity[deg/s]

2nd Pulse

3rd Pulse

4th Pulse

Experiment No.2

Experiment No.1

Experiment No.3

Figure 8: Phase plane trajectory of stepping motor.

Moreover, a vibration of about 810[Hz] is con-

ﬁrmed that is higher than eigenfrequency of the motor

in Figure 10 and Figure 11. To realize more highly

accurate vibration suppression, it is necessary to con-

sider the vibration in the present model. It is thought

that this higher harmonic vibration is due to the inﬂu-

ence of the harmonic component of the current, the

magnetic ﬁeld of the drive circuit and the stepping

motor.

Through these results, the proposed model consid-

ering the response of the current enabled us to achieve

the pulse control with little vibration.

Table 3: Pulse timing.

Experiment No. T

1

[µs] T

2

[µs] T

3

[µs] Total[µs]

No.1 1700 810 1810 4320

No.2 800 1722 1368 3890

No.3 2050 476 2144 4670

ICINCO 2007 - International Conference on Informatics in Control, Automation and Robotics

392

0 0.002 0.004 0.006 0.008 0.01

0

1

2

3

Time[s]

Angle[deg]

0 0.002 0.004 0.006 0.008 0.01

0

500

1000

1500

Time[s]

Angular Velocity[deg/s]

Reference

Simulation

Experiment

Simulation

Experiment

Figure 9: Experimental result of case No.1.

0 0.002 0.004 0.006 0.008 0.01

0

1

2

3

Time[s]

Angle[deg]

0 0.002 0.004 0.006 0.008 0.01

0

500

1000

1500

Time[s]

Angular Velocity[deg/s]

Reference

Simulation

Experiment

Simulation

Experiment

Figure 10: Experimental result of case No.2.

6 CONCLUSIONS

In this paper, a mathematical model by means of the

torque equation of the stepping motor considering the

dynamics of the current has been built. The relation

between the response of the current and the excitation

timing on the phase plane were discussed for a full

step drive of four pulses. From the simulation, in the

system that there is the transient response in the cur-

rent, it was conﬁrmed that the input timing of the last

pulse should be conducted before the end time, con-

sidering the duration time of current delay. The va-

lidity of the proposed model was shown by the com-

parison between the simulation and the experiment.

The pulse input timing without vibration was decided

by means of four pulses, and excellent vibration con-

trol was able to be realized based on the proposed

model using Simplex Method. The effectiveness of

the proposed approach was demonstrated through ex-

periments and simulations.

Robustness to the higher harmonic vibration and

0 0.002 0.004 0.006 0.008 0.01

0

1

2

3

Time[s]

Angle[deg]

0 0.002 0.004 0.006 0.008 0.01

0

500

1000

1500

Time[s]

Angular Velocity[deg/s]

Reference

Simulation

Experiment

Simulation

Experiment

Figure 11: Experimental result of case No.3.

model parameter variation by load variation, etc

should be investigated, and then robust control must

be achieved in near future.

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MODELING WITH CURRENT DYNAMICS AND VIBRATION CONTROL OF TWO PHASE HYBRID STEPPING

MOTOR IN INTERMITTENT DRIVE

393