2008 Sazawa - Robust Fast Tracking Control for Multi-Degrees-of-Freedom Motion System Considering Torque Saturation.pdf

Fast Continuous Path Tracking Control Considering
High Precision and Torque Saturation
Masaki Sazawa
Nagaoka University of Technology
1603-1, Kamitomioka, Nagaoka,
Niigata, 940-2188, JAPAN
Email: sazawa@vos.nagaokaut.ac.jp
Kiyoshi Ohishi
Nagaoka University of Technology
1603-1, Kamitomioka, Nagaoka,
Niigata, 940-2188, JAPAN
Email: ohishi@vos.nagaoaut.ac.jp
Abstract—Continuous path tracking control is an important
technology for the position control system such as factory
automation field. Particularly, large acceleration and/or deceleration
torque is required for continuous path tracking control
at its start position and its goal position. Each AC servo motor
of continuous path tracking control has limitation of current
and voltage. Therefore, in controlling a multi-degree-of-freedom
continuous path tracking control system, even if only the motor
torque of one axis has the current limitation, the actual position
response is not often equal to the desired trajectory reference.
In fast continuous path tracking control of multi-degree-offreedom
position control system, the position error sometimes
increases according to the unknown disturbance torque and
inertia variation.
In order to overcome these problems, this paper proposes a
new continuous path tracking control algorithm by considering
both the torque saturation and the coordinated motion. The
proposed method assures the coordinated motion by considering
the torque limitation. The effectiveness of the proposed method
is confirmed by the experimental results in this paper.
I. INTRODUCTION
Fast continuous path tracking control is demanded for the
multi-degrees-of-freedom position control system in order to
improve the productivity [1]. Moreover, fast continuous path
tracking control system such as X-Y tables is constructed by
AC servo motor and ball screw. Especially, large acceleration
and/or deceleration torque is necessary for fast continuous path
tracking control at its start position and its goal position. As
a result, the actual position response sometimes has the large
overshoot and the oscillated response. Moreover, the position
response of each axis cannot coordinate and follow to the
desired path tracking reference [2]–[10].
We have already proposed a new anti-windup algorithm
for motor drive control and fast potion control. This control
method realizes the fast and smooth positioning control
without overshoot and the oscillated response [11], [12]. This
paper applies the proposed anti-windup algorithm to the multidegrees-of-freedom
position control system such as X-Y table.
In multi-degrees-of-freedom position control system, when
the servo motor on Y-axis has torque saturation, the servo
motor on X-axis often increases the tracking error. In order
to overcome this problem, this paper proposes a new torque
limitation algorithm considering coordinated motion control
for robust fast continuous path tracking control system [13].
The proposed method is the on-line real time algorithm, which
assures the coordinated multi-degrees-of-freedom motion control
considering the torque saturation.
II. FAST CONTINUOUS PATH TRACKING CONTROL
Fig.1 shows the fast continuous path tracking control system.
The position control system is constructed by the minor
control loop of speed PI controller and current PI controller
with each output limitation [6]. In this paper, the limiter of
the speed controller output is constructed by the proposed
output limitation algorithm based on the priority feedback
compensation of proportional output, which is a new antiwindup
algorithm for PI controller [11], [12]. Since this
limitation algorithm of PI controller output has the robustness
on parameter variation [11], [12], this paper applies the proposed
anti-windup algorithm to the multi-degrees-of-freedom
position control system. When the bandwidth of position
controller becomes high enough, the transfer function from
the position command θ cmd to the actual position response
θ res can be treated as show in (1).
θ res
θ
ω
ωres αres
= =
cmd cmd
A. Braking Mode Algorithm
≃ 1 (1)
αcmd In order to realize the desired fast continuous path tracking
control, near to the goal position, the maximum torque deceleration
is often requested. Fig.2 shows the ideal deceleration
response to the target goal position. The fast continuous path
tracking control system decelerates by using the maximum
torque during the proposed decelerating time t d , which is the
time from the braking start time t brake to the braking end time
t end as shown in Fig.2. The moving distance ΔΘ err for the
decelerating time t d is defined as (2). The braking start time
t brake is determined by the load torque and the target goal
position. Hence, the moving distance ΔΘ err is treated as the
position error until the target goal position. In this method,
the decelerating speed ω m (t) and the decelerating time t d
are obtained as shown in (3) and (4). The moving distance
ΔΘ err is calculated by using the estimated load torque and
978-1-4244-4649-0/09/$25.00 ©2009 IEEE 3089

α cmd J n
I FF
K tn
ω cmd
i ref d
+
Position P + + ω ref e
+
+
u +
+
+ - Controller
+
ksiTs z-1
i ref
-
- + q
x ~ x +
Speed
- Current + -
ksp
Limiter
+
Limiter
-
-
Δup Δu
+
Δui
Speed Controller
ω res with proposed new anti-windup algorithm
target position
I
ω STOP
STOP =0
Braking Mode
+ - Algorithm
Braking Mode Algorithm
s
θ
cmd
θ
res
+
-
id res
iq res
kcp
kci
s
+
-
1
kcp
Current Controller
with anti-windup algorithm
+
+
kcp
kci
s
1 ΔVd
kcp
ΔVq
Vd
d-q
+
α-β
+ Vq
θre
d-q
t1,
t2
d-q
u,
v,w
iv
iu
p
α-β
t1,
t2
t1
t2
θrm
~
t1
+
-
+ -
~
t2
t1,
t2
Suvw
Inverter
PM
Enc.
Fig. 1.
Continuous path tracking control system with anti-windup algorithm
Current Speed Position
0
ω 0
0
0
~
target position
ΔΘ err
brake point
time[s]
ω m (t)
time[s]
-I MAX
time[s]
Y-Axis
Table
AC Motor
Coupling
X-Axis
Encoder
Ball Screw
Coupling
AC Motor
Encoder
(a) tested X-Y table system
start, goal
(b) path reference
Fig. 3. System configuration of experimental system
tbrake
td
tend
Fig. 2.
Ideal deceleration & stop response
the maximum torque current I MAX , as shown in (5) [14].
ΔΘ err = ∫ t d
0 ω m(t)dt (2)
ω m (t) =ω 0 − K tnI MAX + ˆT dis
J
t (3)
t d =
2
1 Jω 0
K tn I MAX + ˆT (4)
dis
ΔΘ err = 1 2
Jω 2 0
K tn (I MAX + ˆT dis
K tn
)
B. System Configuration and Experimental System
Fig.3 shows the configuration of tested experimental system.
Fig.3(a) shows the tested X-Y table system. TABLE.I is the
specifications of tested motor and TABLE.II is the specifications
of tested experimental system. In the tested experimental
system, the axis inertia J such as X-axis is 6.21×10 −5 [kgm 2 ].
(5)
As the axis inertia of Y-axis is the total inertia including
the weight of X-axis, Y-axis inertia becomes 2.5 times value
of X-axis inertia. Hence, the tested motion control system
always has the inertia variation. The continuous path tracking
command is shown in the reference of Fig.3(b).
Fig.4 shows the experimental results of the tested continuous
path tracking control system using the conventional speed
PI controller using integrator with limiter. Fig.4(b) shows the
responses of position, speed and current of Y-axis on condition
that the driving time of path reference is 2 [sec]. In this
condition, the actual path response coincides with the path
reference completely, even if the continuous path tracking
control system has only limiter. However, on condition that
the driving time of path reference is 1 [sec], the actual
path response has large tracking error. In multi-degrees-offreedom
position control system, when the servo motor on
Y-axis has torque saturation, the servo motor on X-axis often
increases the tracking error. Therefore, the multi-degrees-offreedom
position control system must be required to have the
coordinated motion considering the torque limitation.
978-1-4244-4649-0/09/$25.00 ©2009 IEEE 3090

TABLE I
SPECIFICATIONS OF TESTED MOTOR
Rated power P [W] 200
Rated voltage v[V] 100
Torque constant K t[Nm/A] 0.1597
Actual inertia of joint X J x[kgm 2 ] 6.21 × 10 −5
Actual inertia of joint Y J y[kgm 2 ] 15.54 × 10 −5
Nominal inertia of joint X J xn[kgm 2 ] 6.21 × 10 −5
Nominal inertia of joint Y J yn[kgm 2 ] 6.21 × 10 −5
Rated maximum speed ω max[rad/s] 314
Rated current (d-q axis) i max[A] 4
Conversion factor K 0 [rad/m] 100π
Encoder pulse [pulse/rev] 2000
y[mm]
80
60
40
20
0
start, goal
P-reference
P-response(2s)
P-response(1s)
-20
TABLE II
SPECIFICATIONS OF EACH CONTROLLER
Bandwidth of current controller ω p[rad/s] 4000
Bandwidth of speed controller ω s[rad/s] 600
Bandwidth of position controller ω c[rad/s] 150
Current limitation values I limit [A] 4
Speed limitation value ω limit [rad/s] 300
III. TORQUE LIMITATION ALGORITHM FOR
MULTI-DEGREE-OF-FREEDOM SYSTEM
The torque current reference i ref should be divided into the
tracking control current i tra and the disturbance compensation
current i cmp , as shown in (6). The disturbance compensation
current i cmp is estimated by ordinary disturbance observer.
When the fast continuous path tracking control system has
torque saturation, the amplitude of i tra should be adjusted
to the maximum tracking control current I traMAX as shown
in (7). Here, I MAX is the maximum torque current, k is the
sampling number.
I tra (k) = I ref (k) − I cmp (k). (6)
⎧
I MAX − I cmp (k)
⎪⎨
when I MAX >I refi
I traMAX (k) =
(7)
−I MAX − I cmp (k)
⎪⎩
when −I MAX

60
40
P-reference
P-response
position[rad]
20
0
-20
θ-cmd
θ-cmd-tilde
θ-res
0 0.5 1 1.5
20
y[mm]
0
Zoom up
speed[rad/s]
200
0
-200
ω-ref
ω-res
-20
0 0.5 1 1.5
-40
iq[A]
5
0
iq-ref
iq-res
Radius error[mm]
0.2
0.1
0
-0.1
-0.2
-60
-150 -100 -50 0 50 100 150
x[mm]
a) path response
0 0.5 1 1.5
Time[s]
b) tracking error
P-reference
P-response
10
-5
0 0.5 1 1.5
Time[s]
a) position, speed and current response of X axis
position[rad]
speed[rad/s]
θ-cmd
20
θ-cmd-tilde
θ-res
0
-20
0 0.5 1 1.5
ω-ref
200
ω-res
0
-200
0 0.5 1 1.5
5
5
iq-ref
iq-res
y[mm]
0
iq[A]
0
-5
-5
0 0.5 1 1.5
Time[s]
b) position, speed and current response of Y axis
-10
99.4 99.5 99.6 99.7 99.8 99.9 100 100.1 100.2
x[mm]
c) zoom up of path response
Fig. 5. Experimental results of the tested continuous path tracking control
using conventional torque current limitation algorithm
Ĩ refi (k) =ε min (k)I trai (k)+I cmpi (k) (11)
= Ĩtra i
(k)+I cmpi (k) (12)
The acceleration torque current adjustment ratio ε i is
defined as shown in (10). The acceleration torque current
adjustment ratio ε i shows the content ratio of I traMAX to
I trai in each axes. In this torque limitation algorithm, the
fast continuous path tracking control current I trai is adjusted
by using the minimum acceleration torque current adjustment
ratio ε min . When the fast continuous path tracking control
system has no torque saturation, the path tracking control
current I trai should be not adjusted. In this case, ε min
becomes equal to 1. The adjusted current reference Ĩref i
is
the sum of the adjusted acceleration torque current and the
disturbance compensation torque current as shown in (12).
Fig.5 shows the experimental results of the tested continuous
path tracking control system using this conventional
torque current limitation algorithm. When the continuous path
Fig. 6. Experimental results of the tested continuous path tracking control
system using conventional torque current limitation algorithm
tracking control system has torque saturation, the path tracking
error is small in comparison with that of Fig.5. The proposed
torque limitation algorithm has the validity on fast continuous
path tracking control. However, in the start position and the
goal position, the path tracking error has just large error as
shown in Fig.5-(b) and Fig.5-(c).
IV. TORQUE LIMITATION ALGORITHM FOR ROBUST
MULTI-DEGREES-OF-FREEDOM FAST POSITION CONTROL
In order to overcome this problem, this paper proposes a
new adjustment path backward torque limitation algorithm.
The new proposed sampling adjusted position command is
determined from the current sampling maximum path tracking
control current I traMAXi . The acceleration command α i (k) is
determined in (13). In this paper, the next sampling adjusted
cmd
position command ˜θ i (k +1) is defined as shown in (14).
The position adjustment ratio γ i (k +1) is defined as shown
in (15). Here, T s is sampling time.
This paper picks up the minimum value γ min (k +1) among
γ i (k +1)(i =1∼ N). The next sampling adjustment time
˜t(k +1) is obtained by the time function of position command
f i (t) as shown in (16) and (17). Since the proposed algorithm
calculates the next sampling corrected position command
978-1-4244-4649-0/09/$25.00 ©2009 IEEE 3092

α cmd i
ω cmd i
θ
cmd
i
Tracking Adjustment Algorithm of
Multi-Degrees-of-Freedom Motion System
~
I FF i
~
ω cmd i
~
+ + ω ref
i Position P
i
+
+
- Controller
-
Speed
Limiter
i
ω res i
θ
cmd
θ
res
Speed Controller
with proposed new anti-windup algorithm
e
+
+
ksiTs
-
ksp
i ref d i
+
+
z -1 u
i ref
+ q
x ~ i
x +
-
Current
+
-
Limiter
-
Δup Δu
+
Δui
target position
ω STOP =0
+ -
Braking Mode
Algorithm I STOP
Current Controller
with anti-windup algorithm
Inverter
PM
Enc.
^
T dis i
Disturbance
Observer
ω res i
i ref q i
Braking Mode Algorithm
s
Fig. 7.
Proposed continuous path tracking control system considering torque current saturation and coordinated motion
˜θ cmd
i
(k +1) by using ˜t(k +1), ˜θcmd i (k +1) is determined
as shown in (18). Similarly, the corrected speed command
˜ω i
cmd (k +1) is calculated as shown in (19). Here, f ′ (t) is
the time function of speed command.
˜θ cmd
i
= θ cmd
i
α i (k) = K tn i
J ni
I traMAXi (k) (13)
(k +1)=θ cmd (k)+T s ω res (k)+ 1 2 T s 2 α i (k)
i
(k)+T s ωi
res (k)+ 1 2 T s
2
γ i (k +1)≡
cmd ˜θ i
θi
cmd
= T sωi
res
i
K tni
J ni
I traMAXi (k) (14)
(k +1)− θi
cmd (k)
(k +1)− θi
cmd (k)
(k)+ 1 2 T s 2 α i (k)
θi
cmd (k +1)− θi
cmd (k)
if I MAX > |I refi |, γ(k +1)=1
˜ω cmd
i (k +1)=
(15)
˜t(k +1)=f −1
i (˜θ i cmd (k + 1)) (16)
≃ t(k)+γ min (k +1)T s (17)
˜θ cmd
i (k +1)=f i (˜t(k + 1)) (18)
⎧
If γ min (k +1)=γ i (k +1)
⎪⎨ f i ′ (˜t(k + 1))
else γ min (k +1)≠ γ i (k +1) (19)
˜θ ⎪⎩ i
cmd (k +1)− θi
cmd (k)
T s
Using (14) and the proposed adjustment path backward
torque limitation algorithm, the next sampling corrected position
command ˜θ i (k +1) is defined as shown in (20). Here,
cmd
Ĩ FFi (k) is the corrected acceleration current reference based
on coordinated motion caused by torque saturation. As the
result, the acceleration current reference ĨFF i
(k) is obtained
as shown in (21).
˜θ i cmd (k +1)
= θ cmd
i
(k)+T s ωi
res (k)+ 1 2 T s
2
K tni
J ni
Ĩ FFi (k) (20)
Ĩ FFi (k) =
2J n i
Ts 2 {f i (t(k)+γ min (k +1)T s )
K tni
− θi
cmd (k) − T s ωi res (k)} (21)
Fig.7 shows the block diagram of the proposed fast continuous
path tracking control system considering torque current
saturation and coordinated motion. Fig.8 shows the experimental
results of the proposed continuous path tracking control
based on the proposed torque current limitation algorithm.
TABLE III shows the comparison of settling time and tracking
error. In fig.9(b), the continuous path tracking control of Y-
axis has torque saturation. The proposed method well realizes
the high accuracy path tracking control within radius error
0.02[rad], even if Y-axis acceleration torque is the maximum
torque.
V. CONCLUSION
This paper proposes a new continuous path tracking control
algorithm taking into account both the torque saturation and
the coordinated motion. This paper newly defines the new
position reference adjustment ratio γ by using disturbance
observer, which is the key for robust multi-degrees-of-freedom
fast position control. The proposed method well realizes the
coordinated motion considering the torque limitation, which
is the on-line real time algorithm. The effectiveness of the
proposed method is confirmed by the experimental results.
REFERENCES
[1] K. Sakai, M. Iwasaki, and N. Matsui :“High-speed and high-precision
positioning system by using mode switching control,”(in Japanese), IEEJ
Trans. IA, Vol. 118-D, No. 7/8, pp. 870–876, Jul./Aug. 1998.
978-1-4244-4649-0/09/$25.00 ©2009 IEEE 3093

60
40
P-reference
P-response
position[rad]
20
0
-20
θ-cmd
θ-cmd-tilde
θ-res
0 0.5 1 1.5
20
y[mm]
0
Zoom up
speed[rad/s]
200
0
-200
ω-ref
ω-res
-20
0 0.5 1 1.5
-40
iq[A]
5
0
iq-ref
iq-res
Radius error[mm]
0.2
0.1
0
-0.1
-0.2
-60
-150 -100 -50 0 50 100 150
x[mm]
a) path response
0 0.5 1 1.5
Time[s]
b) tracking error
P-reference
P-response
10
-5
0 0.5 1 1.5
Time[s]
a) position, speed and current response of X axis
position[rad]
speed[rad/s]
θ-cmd
20
θ-cmd-tilde
θ-res
0
-20
0 0.5 1 1.5
ω-ref
200
ω-res
0
-200
0 0.5 1 1.5
5
5
iq-ref
iq-res
y[mm]
0
iq[A]
0
-5
-5
0 0.5 1 1.5
Time[s]
b) position, speed and current response of Y axis
-10
99.4 99.5 99.6 99.7 99.8 99.9 100 100.1 100.2
x[mm]
c) zoom up of path response
Fig. 8. Experimental results of proposed continuous path tracking control
based on the proposed torque current limitation algorithm
TABLE III
COMPARISON OF SETTLING TIME AND TRACKING ERROR
settling time path inner error path outer error
only current limiter 1.106s -3.929rad 0.831rad
(-12.51mm) (2.65mm)
conventional torque 1.046s -0.057rad 0.042rad
limitation method (-0.18mm) (0.13mm)
proposed torque 1.041s -0.018rad 0.008rad
limitaion method (0.06mm) (0.03mm)
[2] K. Ohishi and T. Someno: “Robust motion control with autonomous
consideration algorithm of joint torque saturation,” in Proc. IEEE
IES/IECON’98, pp.1812–1817, 1998.
[3] K. Ohishi, H. Nozawa, and S. Ohtaki: “Robust motion control with consideration
algorithm of joint torque saturation for redundant manipulator,”
in Proc. IEEE IES/IECON 2000, pp.2255–2260, 2000.
[4] F. Suzuki and Y. Hori: “Anti-windup strategy for feedback control system
based on youla parameterization,” (in Japanese), IEEJ Trans. IA, Vol.121-
D, No. 6, pp. 683–688, Jun. 2001.
[5] R. Watanabe, K. Uchida, E. Shimemura, and M. Fujita: “A new synthesis
of anti-windup and bumpeless transfer for system with constraint on
control input,” (in Japanese), SICE, Vol. 30, No. 6, pp. 660–668, Jun.
Fig. 9. Experimental results of proposed continuous path tracking control
based on the proposed torque current limitation algorithm
1994.
[6] K. Ohishi, E. Hayasaka, T. Nagano, M. Harakawa, and T. Kanmachi:
“High-performance speed servo system considering voltage saturation of
a vector-controlled induction motor,” IEEE Trans. Ind. Ele., Vol.53, No.3,
pp.795–802, Jun. 2006.
[7] C. Bohn, and D.P.Atherton: “An analysis package comparing PID antiwindup
strategies,” IEEE Control Syst. Mag., Vol. 15, No. 2, pp. 34–40,
Apr. 1995.
[8] K.J.Åstrom, and T.Hagglund:“Automatic Tuning of PID Controllers,”
Instrument Society of America, 1988
[9] Y. Peng, D. Vrancic, and R. Hanus: “Anti-windup, bumpless, and conditioned
transfer techniques for PID controllers,” IEEE Control Syst, Mag.,
Vol. 16, No. 4, pp. 48–57, Aug. 1996.
[10] M.V.kothare, P.J.Campo, M. Morari, and C.N.Nett: “A Unified framework
for the study of anti-windup designs,” Automatica, Vol. 30, No. 12,
pp. 1869–1883, 1994.
[11] M. Sazawa, T. Yamada, K. Ohishi, and S. Katsura: “Robust high speed
positioning servo system considering saturation of current and speed,” in
Proc. IEEE IES/ICIT 2006, pp. 866–871, Dec. 2006
[12] M. Sazawa, K. Ohishi, and S. Katsura : “Robust high speed position
servo system considering current & voltage limitation and load inertia
variation,” in Proc. IEEE IES/IECON 2007, pp.322-327 Nov. 2007.
[13] M. Sazawa, K. Ohishi, S. Katsura, and S. Kato :“Robust fast tracking
control for multi-degrees-of-freedom motion system considering torque
saturation,” in Proc. IEEE IES/ICIT 2008, Apr. 2008
[14] T. Mashimo, K. Ohishi, and H. Dohmeki :“High speed positioning
system considering unknown coulomb friction and inertia variation,” in
Proc. IEEE IES/IECON 2004, Nov. 2004
978-1-4244-4649-0/09/$25.00 ©2009 IEEE 3094