Contour Control for Ball Screw Systemwith Asymmetric Friction
M2014SC024 Takayuki YAMAMOTO
Supervisor：Isao TAKAMI

Abstract
This paper proposes the method of the contour control. Tracking reference is achieved by the Perfect tracking control(PTC). Friction is compensated by the disturbance observer(DOB). By the experiments, the asymmetric of the friction is observed. In simulation anddesign of the DOB, the asymmetric of the friction isconsidered. The eﬀectiveness of the proposed method isveriﬁed by the simulations and experiment.

1The notation of Anm stands for the elements of n rowsm columns in A.

2

Modeling of The Ball Screw System

The overview of the ball screw system is shown inFigure 1. This system consists of the motors, the en-Introduction

In servo system, point to point(PTP) control or contour control are required. PTP control is making theobject move to the other point without considering theroute. Contour control is making the object move along
the predeﬁned route. Many methods are proposed forthe contour control of x-y ball screw system[1][2][3]. Perfect Tracking Control(PTC) is also used for the contourcontrol[4]. PTC is a kind of the two degree-of-freedom
and consists of the feedback controller and the feedforward controller. It achieves tracking control at eachsampling time. If the model used in design of PTC issimilar to the real system, the feedforward controller
is eﬀective and the contour control is achieved by thePTC. However, the real system is aﬀected by the disturbance. In this study, ball screw system is used as
the servo system. The ball screw system is widely usedin positioning control. It is not easily aﬀected by thefriction. However, the eﬀect of the friction is not negligible in the precise control. Even if the PTC is designed
completely, tracking control may not be achieved due tothe eﬀect of the friction. Therefore, the friction must
be compensated. Although many friction compensation
methods are proposed, disturbance observer(DOB) is akind of the eﬀective methods[5][6].
In this study, PTC is adopted to achieve the contourcontrol. PTC is designed based on [4]. However, since
PTC is designed for the linear system, the performanceof the PTC deteriorates for the system aﬀected by thefriction. If DOB compensates the disturbance, the behavior of the real system is similar with that of the linear system. Therefore, DOB based on LuGre model[5]is adopted to compensate the friction. To design theDOB, the parameters of the friction are identiﬁed. In
identiﬁcation, the asymmetric of the friction is observed.
In other words, some parameters of the friction varydepending on the sign of the velocity. Therefore, design of the DOB and simulation are conducted basedon the friction model, which depends on the sign of thevelocity[6][7]. It is diﬃculty to implement this DOB because it is necessary to solve the Riccati equation at theeach sampling time. In this study, the observer gain
depending on the velocity is derived and it is approximated to the polynomial before implementing the controller. The polynomial depending on the velocity isimplemented instead of the real observer gain. The effectiveness of the control system including the PTC andthe DOB is veriﬁed by simulation and experiments.

Figure 1 The picture of the ball screw systemcoders, ball screw and table. The table is mounted on
the ball screw. The table position is controlled by converting the rotational motion of the motor into linear
motion through the ball screw. The motors are AC servo
motors, which are controlled by the servo ampliﬁers.
Thanks to the servo ampliﬁers, the arbitrary torques are
given for the motors. The encoder measures the angleof the motor. The resolution of it is 2.5 × 105 [plus/rev].
In this research, the lower ball screw system and theupper one are regarded as X axis and Y axis, respectively. Although it is necessary to obtain the model foreach axis, the motion equation is the same as each other.
Therefore, the model for X axis is derived. For X axis,
let the angle of the motor, input torque for the motor
and the state space vector be θ(t)[rad], T (t)[Nm] andxm (t) = [θ(t) θ̇(t)]T . The state space representation isobtained as Eq. (1).

[Am =00ẋm (t) = Am xm (t) + Bm u(t)y(t) = Cm xm p(t)
]
[]
102,
B=,
1m
R σ0− J+R2 M(1)

J+R2 M

Cm = [ 10 ]The physical parameters in Eq.(1) are shown in Table
1.
Table 1 Parameters of Ball Screw System
moment of inertia
J [Nms2 ]1.14 × 10−5
mass of table
M [kg]
2.988
ball screw constant R[m/rad]0.002/2π

3Modeling and Identification of TheFriction

The schematic diagram of the LuGre model is represented in Figure 2. In Figure 2, there are many bristles
on the contact surfaces between the two objects. Let theaverage displacement of the bristles and the velocity ofv

r[i]
r(t)

Su[i]

C 1 [z](Tr)+uc(t)

HM+yc (t)Pc (s)(Tu)

y[i]SM(Ty)z

C 2 [z]Figure 2 The schematic diagram of the LuGre model
Figure 3 The structure of the PTC
the contact surface be z [m] and v [m/s]. The friction
force FL is represented by the following equations[7].
FL = σ0 z + σ1 ż + σ2 v|v|ż = v − σ0zg(v)g(v) = (Fc + (Fs − Fc )e−|v/vs | )

(2)
(3)
(4)

The parameters for the LuGre model are shown in Table
2. To use the LuGre model in simulations and design
Table 2 Parameters of LuGre Modelspring constant of the bristleσ0 [N/m]viscous constant of the bristle
σ1 [Ns/m]
the viscous friction coeﬃcient
σ2 [Ns/m]
static friction force
Fs [N]
Coulomb friction force
Fc [N]
Stribeck velocityvs [m/s]of the controller, the parameters shown in Table 2 must
be obtained. As the results of the identiﬁcation, σ0 , σ1and σ2 are identiﬁed as 2733 × 104 [N/m], 95000[Ns/m]and 1083[Ns/m]. Fc , Fs and vs are identiﬁed as 11[N],
15[N] and 1083[Ns/m]. However, the Coulomb friction
force Fc and the static friction force Fs vary dependingon the sign of the velocity[6][7]. In [6], friction models
is described by Eq. (2), Eq. (3) and Eq. (5).
 Fc+ + (Fs+ − Fc+ )e−|v/vs | (v > 0)g(v) =F + (Fs− − Fc− )e−|v/vs | (v < 0) (5) c−(Fs+ + Fs− )/2(v = 0)
In this study, the asymmetric of the friction is observed.
In other words, Fc+ ̸= Fc− and Fs+ ̸= Fs− . Therefore,
let Fc and Fs be Fc+ and Fs+ . With using the samemethods for Fc+ and Fs+ , Fc− and Fs− are identiﬁed
as 25[N] and 27[N].

4Deign of The Controller System

In this chapter, the controller system is designed.
The controller system consists of PTC and disturbanceobserver based on LuGre model. Tracking control isachieved by the PTC. The structure of PTC is shown inFigure 3. As can be seen in Figure 3, the PTC consistsof the feedforward controller C1 [z] and the feedback controller C2 [z]. First, the feedback controller C2 [z] is designed to constitute the PTC. In this study, the feedbackcontroller C2 [z] is LQ controller. Next, the feedforward
controller C1 [z] is designed. However, the PTC is designed for the linear system which does not include thefriction FL . Therefore, the disturbance observer basedon LuGre model is used to compensate the friction.

First, discrete LQ controller is designed for Eq. (1).
To obtain discrete-time system, let the new state vector
be xmd (k) = [θd (k) θ̇d (k)]T . Eq. (1) is transformed into
Eq. (6).
xmd (k + 1) = Amd xmd (k) + Bmd u(k)(6)∫Here, Amd = eAm Ts , Bmd = 0Ts eAm τ Bm dτ . Ts is sampling time and k is the time index that indicates kth

sampling time. For Eq. (6), the controller is designed
to minimize the following cost function.
Jev =∞∑[xmd (k)T Qcd xmd (k) + u(k)T Rcd u(k)]

(7)

i=0

Qcd is the weight matrix for the state and Rcd is theweight matrix for the control input. If there exists P >0 satisfying Eq. (8), the controller u(k) = F xmd (k) =TT−(Rcd + BmdP Bmd )−1 Bmd
Amd xmd (k) is obtained.
ATmd P Amd + Qcd − P − ATmd P Bmd
TT(Rcd + BmdP Bmd )−1 Bmd
P Amd = 0(8)Next, C1 [z] is designed to track reference. As canbeen seen in Figure 3, there are two sampler S , SMand a holder HM for reference r(t), output y(t) and control input u(t). Let the each sampling period be Tr , Tyand Tu , respectively. In this study, Ty = Tu and Tr isnTu . Here, n is the order of Pc (s). Let the longest sampling period Tr be the ﬂame sampling period Tf . To
design the the multirate sampling controller C1 [z], themethod proposed in [4] is used. Let the state vector bex[i] = xm (iTf ). The discrete-time system at the sampling period Tf is obtained by the following equation.
x[i + 1] = Ax[i] + Bu[i], y[i] = Cx[i] + Du[i](9)The multirate feedforward controller C1 [z] is obtainedby the following equation[4].
C1 [z] = (M − C2 N )K]
A + BF B
M== I + z −1 F B,FI[]
A + BF B
N== D + z −1 (C + DF )B,
C + DF D[(10)

F = −B −1 A, K = B −1

Finally, the disturbance observer proposed in [5] isadopted to compensate the friction. This disturbanceobserver is discrete time system. To design it, Eq. (3),the state equation about the bristle, is discretized. Eq.
(11) is obtained.
zd (k + 1) = e(−σ0 α(θ̇d (k))Ts zd (k)−e(−σ0 α(θ̇d (k)))Ts − 1Rθ̇d (k)σ0 α(θ̇d (k))(11)|Rθ̇(k)|g(Rθ̇d (k))(12)α(θ̇(k)) =Let the estimated vector be x̂Ld (k) = [θ̂d (k) θ̇ˆd (k) ẑ(k)]T .
From Eq. (2), Eq. (6) and Eq. (11), the disturbanceobserver is obtained by the following equation.
ˆx̂Ld (k + 1) = ALd (θ̇d (k))xLd (k) + BLd ud (k)ˆ+KLd (θ̇d (k))(θd (k) − θ̂d (k))
ŷLd = CLd x̂Ld (k)

(13)
(14)

• Symmetric controller: This controller does not depend on the sing of velocity. In other words, it uses
the DOB designed for Eq.(4).
• Asymmetric controller: This controller depend on
the sing of velocity. In other words, it uses the DOB
designed for Eq.(5).
In section 4, The observer gain KLd (θ̇ˆd (k)) is obtainedby ﬁnding PLd (θ̇ˆd (k)) satisfying with Eq. (16) at eachsampling time. Therefore, it is diﬃculty to implement
this disturbance observer. To avoid this diﬃculty, theobserver gain KLd (θ̇ˆd (k)) is approximated to polynomial
in velocity domain. In this study, the full speed is not
so high because the reference of the precise control isvery small. The velocity range is deﬁned as between 2[rad/s] and −2 [rad/s]. As a example, the observer gain
is approximated at the positive velocity region. Theobserver gain KLd (θ̇ˆd (k)) is shown in Figure 4. Let theˆALd11 (θ̇d (k)) = Amd + [ O −Bmgd1 σ0 ] ,ˆALd12 (θ̇d (k)) = −Bmgd2 (σ0 − σ0 σ1 α(vd (k))),ˆˆ
ALd21 (θ̇d (k)) = O + [ O −af ] , ALd22 (θ̇d (k)) = bf ,[]∫ Ts0Bmgd1 =eAm τ Bmg1 dτ, Bmg1 =,R/(J + R2 )
0[]
∫ Ts0
Bmgd2 =eAm τ Bmg2 dτ, Bmg2 =,22R /(J + R )0[]e(−σ0 α(θ̇d (k)))Ts − 1
Bmd
R,BLd =, af =Oσ0 α(θ̇d (k))
bf = e(−σ0 α(θ̇d (k))Ts , CLd = [ 100 ]Here, KLd (θ̇ˆd (k)) is observer gain. The estimated friction force is obtained by the following equation.
FˆL =[Oσ1ˆ
σ0 − σ0 σ1 α(θ̇d (k))]

From [5], the observer gain is deﬁned by Eq. (15).

(15)ˆPLd (θ̇d (k)) > 0 is satisfying with Eq. (16).
ˆˆˆ
ALd (θ̇d (k))PLd (θ̇d (k))ALd (θ̇d (k))T + QLdˆˆ
T−PLd (θ̇d (k)) − ALd (θ̇d (k))P CLd(RLdˆˆ
T −1
T+CLd PLd (θ̇d (k))CLd ) CLd P ALd (θ̇d (k)) = 0

ˆelements of the KLd (θ̇d (k)) be the following equation.
ˆKLd (θ̇d (k)) =x̂Ld(k)ˆˆˆ
TKLd (θ̇d (k)) = ALd (θ̇d (k))PLd (θ̇d (k))CLdˆ
T −1 T{(Rcd + CLd PLd (θ̇d (k))CLd) }Figure 4 The observer gain in the plus velocity domain[ˆKLd1 (θ̇d (k))ˆKLd2 (θ̇d (k))ˆKLd3 (θ̇d (k))]TIn Fig. 4, the gain for position, the gain for speed andthe gain for bristle represent KLd1 (θ̇ˆd (k)), KLd2 (θ̇ˆd (k))
and KLd3 (θ̇ˆd (k)), respectively. To implement the disturbance observer, KLd (θ̇ˆd (k)) is approximated to fourth
degree polynomial. The result of it is shown in in Figure 5. In Figure 5, the solid line, the dashed line and the(16)In Eq. (15) and Eq. (16), RLd and QLd are the weightmatrix for the estimated state and the weight matrix
for the observed output, respectively. Eq.(16) varies at
each sampling time. The observer gain KLd (θ̇ˆd (k)) isobtained by ﬁnding PLd (θ̇ˆd (k)) satisfying with Eq. (16)at each sampling time.

5

Simulations and Experiments

The conditions for the controller are described. The
sampling times Ty , Tu and Ts are 0.001[s]. The samplingtimes Tr is 0.002[s]. The weight matrices in LQ controller
and the weight matrices in DOB are deﬁned as follows.
Qcd = diag [ 10QLd = diag [ 111 ] , Rcd = 11 ] , RLd = 109

The following two controller are implemented in simulations and experiments.

Figure 5 The observer gain in the plus velocity domain
dotted line approximate KLd1 (θ̇ˆd (k)), KLd2 (θ̇ˆd (k)) andˆ
KLd3 (θ̇d (k)), respectively. As can be seen in Figure 4and Figure 5, the approximated gain represents the observer gain suﬃciently. Using the same way, the gain isapproximated at the negative velocity region.
In simulation, Eq.(5) is used. Two simulations areππconducted. In ﬁrst one, the reference is r(t) = 10sin( 15)tππ[rad]. In second one, the reference is r(t) = 100sin( 15)t

[rad]. The ﬁrst simulation results are shown in Figure
6. Figure 6 shows the motor angle. As can be seen

Figure 8 The ﬁrst experiment of output when the refππerence is r(t) = 10sin( 15)t [rad].
Figure 6 The ﬁrst simulation of output when the referππence is r(t) = 10sin( 15)t [rad].
in Figure 6, the symmetric controller does not achieve
tracking reference. The asymmetric controller achieves
tracking reference. The second simulation results areshown in Figure 7. Figure 7 shows the motor angle.

Figure 9 The ﬁrst experiment of output when the refππerence is r(t) = 100sin( 15)t [rad].

References
Figure 7 The ﬁrst simulation of output when the referππence is r(t) = 100sin( 15)t [rad].
As can be seen in Figure 7, the symmetric controller
does not achieve tracking reference. The asymmetriccontroller achieves tracking reference.
Next, Two experiments are conducted. In ﬁrst one,ππ
the reference is r(t) = 10sin( 15)t [rad]. In second one,ππ
the reference is r(t) = 100sin( 15)t [rad]. The ﬁrst experiment results are shown in Figure 8. Figure. 8 shows themotor angle. As can be seen in Figure 8, the symmetric controller does not achieve tracking reference. The
asymmetric controller achieves tracking reference. The
second experiment results are shown in Figure 9. Figure
9 shows the motor angle. As can be seen in Figure 9,
the symmetric controller does not achieve tracking reference. The asymmetric controller achieves tracking reference.

6Conclusion

In this study, the method for the precise contour control is proposed. The PTC is designed to track reference.
The disturbance observer is designed to compensate thefriction. Moreover, the asymmetric of the friction is considered in simulation and design of the DOB. The eﬀectiveness of the proposed control system is veriﬁed by thesimulations and the experiments.

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