Position Domain PD Control: Stability and Comparison

Proceeding of the IEEE International Conference on Information and Automation Shenzhen, China June 2011

Position Domain PD Control: Stability and Comparison P.R. Ouyang and T. Dam Department of Aerospace Engineering Ryerson University Toronto, Canada [email protected] Abstract - In this paper, a position domain PD control for contour tracking is proposed to improve the contour tracking performance. To develop the position domain control, one motion is selected as a master (reference) motion and all other motions are viewed as slave motions. The reference motion is sampled equidistantly in the position domain and used as an independent variable. A dynamic model for the slave motions is developed in the position domain by transforming the original system dynamic equations from the time domain to the position domain. After that, a position domain PD control is proposed, and the stability analysis is conducted based on the Lyapunov function method. The developed position control is applied to linear motion tracking and circular motion tracking, and the comparison study is conducted. Simulation results demonstrate the effectiveness of the position PD control compared with traditional PD control and the crossed-coupled control.

certain degree. Including the contour error directly in the controller design could result in the improvement of the contour tracking performance compared to the independent axis control [4-7]. Since the objective of contour tracking is to make a controlled system follow the reference contour, the velocity of the manipulated system or the timing of the motion does not need to be strictly controlled. That is the starting point for the development of CCC. It should be mentioned that in CCC, each axis still need to be controlled, and there still have some tracking errors for each individual axis motion. In recent years, a new type of feedback controller called an event-driven controller was developed [8-11] as it is an event (e.g. the arrival of a new measurement), rather than the elapse of time, that triggers the controller to update the control action. The main advantage of event-driven control is to reduce the computing power and the communication between sensors, controller, and actuators in a control loop. Event-driven controllers better fit the new technology in software and computing platforms, which is focusing on the use of eventdriven systems to save computing utilization and communication load. A major reason that the event-driven control is scarce in applications is the difficulty involved in developing a system theory for event-driven system. Another drawback for the event-driven control is the tracking performance sacrifice. From the above discussion, we can see that the crosscoupled control and the event-driven control have the advantages and disadvantages, respectively. The motivation of this research is to take the advantages of both cross-coupled control and the event-driven control and to form a new control method called position domain control. The main objective of the position domain control is to improve the contour tracking performance by a simple and easy implementing way. The outline of this paper is as follows. In Section II we state the contour tracking control problem by adopting a simple two-axis control system. In section III we conduct the stability analysis based on the Lyapunov method. Section IV deals with a comparison study, among the position domain control, the traditional PD control in time domain control, and CCC. Some concluding remarks and future work are followed in Section V.

Keywords-contour tracking; robotics; position domain; PD control, cross coupled control.

I. INTRODUCTION For multi-axis motion control applications, contour tracking control is one of the most common controls encountered by industrial manipulators, CNC machine tools, and robots. There are many control strategies for improving single axial positioning accuracy through different feedback methods [1-8]. The traditional control for a multi-axial motion is an independent axis control where the motion of each axis is separately designed and tracked without regard to other axes. A great deal of efforts has been devoted to solving independent axial control problems for robotic manipulators and mechatronic systems [2, 3]. One can improve single axial positioning accuracy by applying various control strategies such as a high gain PID controller, a feed-forward controller, and a zero phase error tracking controller. The decoupled control design may be preferable if the disturbance in one axis does not affect the performance of the other axes. However, a good tracking performance for each individual axis does not guarantee the reduction of the contour error for multi-axis motion. Furthermore, poor synchronization of relevant motion axes results in diminished dimensional accuracy of the contour tracking [4-7]. An improved method to reduce the contour tracking error is the cross-coupled control (CCC) developed by Koren [4, 5] that can alleviate the defects of independent axis control in a

978-1-61284-4577-0270-9/11/$26.00 ©2011 IEEE

8

Motion commands of the X-Y plane Cartesian robot widely used in industry can be divided into two categories: motion along straight lines and motion along circular paths. Along straight lines, each axis can be planned individually by the above path planning methods. On the other hand, circular paths have been planned based on the angular trajectory which is determined by the well known s-curve method. In this paper, for a linear motion of the end-effector (X-Y plane) of the 2-DOF Cartesian robot in the time domain, the contour tracking motion is expressed as:

II. SYSTEM DYNAMIC MODEL A. Description and Dynamic Model in Time Domain In this paper, we use a two-link Cartesian robot [1, 12] shown in Fig. 1 as an example of a multi-axis motion system to discuss the contour tracking control in the position domain. The dynamic model, including the mechanical system and the actuator for each axis, can be expressed as a general secondorder differential equation [1, 12].

x

y = kx ® ¯ y = kx

For a circular motion, the X-Y plane contour motion can be described as: x = R cos θ + a (6) ® ¯ y = R sin θ + b

y

Fig. 1 A 2-DOF Cartesian robot

where: θ = θ ini

According to the configuration shown in Fig. 1, the dynamic model of the Cartesian robot can be written as:

x + c1 x + k1 x = Fx ° ® °¯ y + c2 y + k2 y = Fy

(1)

acceleration of x and y axes, respectively. ci and ki are the

damping and stiffness of the system, and Fx and Fy are the control input forces. From Eq. (1), one can see that the dynamic model for these two axes is decoupled and PD control can be applied for the control of each axis. It is demonstrated that a global stability of trajectory tracking in the joint level can be achieved [13, 14] under PD feedback control.

0 · § ex · § Kdx + K py ¸¹ ¨© e y ¸¹ ¨© 0

where ex , ey and ex , ey

0 ·§ ex · ¸ Kdy ¸¨ ¹© e y ¹

(2)

derivative of tracking errors, respectively. K pi and K di are the proportional gain and derivative gain, respectively. Traditionally, the feedback control is time driven system where the joint or end-effector trajectory tracking is represented as a function of time. To design a smooth motion in each axis direction, a fifth-order polynomial model for r(t) [15] widely used in the trajectory planning is used and can be represented as: 3

{

4

5

r ( t ) = 30 ( t / T ) − 60 ( t / T ) + 30 ( t / T ) 2

3

(3) 4

}/T

(

)

B. Dynamic Model and Controller in Position Domain The dynamic model of the Cartesian robotic system is developed in the position domain by transforming the original system dynamic equations from the time domain to the position domain where the position information of a master actuator (X-axis) is measured and used as a known variable. Therefore, the Y position is controlled to follow the X position measured by a sensor. In this control methodology, the developed controller updates are synchronous with respect to the master position information; therefore, there is no tracking error for the master motion. Only the slave motion tracking errors will affect the contour tracking errors. It is clear that the position domain control is a kind of event-driven control. In the position domain control, we assume X-axis is a reference (master) axis and Y-axis is the serve (slave) axis. The Y-axis dynamic model will be rewritten related to X-axis position. In this paper, we assume the X-axis motion direction is unchanged in the considered contour section. Without loss of generality, we assume the X-axis motion is in positive direction and the x position is a monotonically increasing function of time or the velocity of X-axis is positive ( x > 0 ). Also, the x position is set to equidistance through the sensor measurement such as an encoder.

are the tracking errors and the

r ( t ) = 10 ( t / T ) − 15 ( t / T ) + 6 ( t / T )

° x = − Rθ sin θ (7) ® °¯ y = Rθ cos θ + r ( t ) (θ fin − θ ini ) , θ = θini + r ( t ) θ fin − θini , R

is the radius, and (a, b) is the center of the circular arc. From Eqs. (5-7), we can see that, in order to get a high contour tracking performance, the independent control of each axis should be synchronized accordingly. But this requirement is very difficult to implement because of the different dynamic characteristics of each axis. In the next section, we will propose a new control method that can overcome the problems associated with the crosscoupled control and the event control.

where x and y , x and y , x and y are position, velocity, and

§ Fx · § K px ¨F ¸ = ¨ 0 © y¹ ©

(5)

(4)

9

Fy ( x ) = K py ( yd ( x ) − y ( x ) ) + K dy ( yd′ ( x ) − y ′ ( x ) )

If we define the first and second derivative of y respect to reference x (the master motion) as:

dy dx dy ′ y ′′ = dx y′ =

K dy § dyd ( x ) dx dy ( x ) dx · − ¨ ¸ (16) x © dx dt dx dt ¹ K = K py ( yd ( x ) − y ( x ) ) + dy ( y d ( x ) − y ( x ) ) x = K py ( yd ( x ) − y ( x ) ) +

(8) (9)

It is clear shown that the position domain linear PD control is equivalent to a nonlinear PD control [16] in the time domain. Therefore, the proposed PD control in the position domain has the same stability property as the PD control developed in the time domain.

where y ′ is called the relative speed of Y-axis respect to Xaxis, and y ′′ the relative acceleration, all are in the position domain. It is clear that y ′ is the tangent of the contour at point of the reference x. To develop the dynamic model in the position domain, a relationship between the absolute velocity and the relative velocity relationship can be expressed as:

y =

dy dy dx = = y ′x dt dx dt

III. STABILITY OF POSITION DOMAIN PD CONTROL

A. Preparation and Lemma Consider a dynamic system described in the position domain by: y ′ ( x ) = f ( x, y ( x ) ) (17)

(10)

where x > 0 . Similarly, the following equation holds between the absolute acceleration and the relative acceleration: dy d ( y ′x ) dy ′ dx = = y= x + y ′ = x 2 y ′′ + xy ′ dt dt dt dt

where x ∈ R is the “position” of the master motion and y ( x ) ∈ R n is the state. Lemma 1: Let D ⊂ R n be a domain that contains the origin and V: [ 0, ∞ ) × D → R be a continuously differentiable

(11)

function such that

Applying Eq. (10) and (11) to the dynamic equation in Eq. (1), the dynamic model for the Y-axis motion in Eq. (1) can be rewritten in the position domain as: ′ ( x ) + k2 y ( x ) = Fy ( x ) x 2 y ′′ ( x ) + xy ′ ( x ) + c2 xy

γ1 ( y ) ≤ V ( y ) ≤ γ 2 ( y

V ′ ( y ) ≤ −W ( y ) , ∀ y ≥ μ > 0

(12)

r > 0 such that Br ⊂ D and suppose that

μ < γ 2−1 ( γ 1 ( r ) )

= K py ( yd ( x ) − y ( x ) ) + K dy ( yd′ ( x ) − y ′ ( x ) )

(20)

Then, there exists a class KL function α and for every

initial state y ( x0 ) , satisfying y ( x0 ) ≤ γ 2−1 ( γ 1 ( r ) ) , there is

(13)

X ≥ 0 such that the solution of dynamic equation satisfies y ( x) ≤ α

( y ( x ) , x − x ), 0

0

∀ x0 ≤ x ≤ x0 + X

y ( x ) ≤ γ 1−1 ( γ 2 ( μ ) ) , ∀x ≥ x0 + X

(14)

(21) (22)

Moreover, if D = R n and γ 1 belongs to class K, then (21) and (22) hold for any initial state y ( x0 ) , with no restriction on

Applying Eq. (14) to Eq. (13), the dynamic model based on the position domain PD control can be expressed as: x 2 y ′′ ( x ) + ( x + c2 x ) y ′ ( x ) + k2 y ( x )

(19)

W ( y ) is a continuous positive definite function. Take

The tracking error and the tracking error relative derivative in the position domain are defined as: °e y ( x ) = yd ( x ) − y ( x ) ® °¯e′y ( x ) = yd′ ( x ) − y ′ ( x )

(18)

∀ x ≥ 0 ∀ y ∈ D , were γ 1 and γ 2 are class K functions and

Eq. (12) builds the dynamic relationship between two axes in the position domain by transferring the dynamic equation from the time domain to the position domain. To obtain an accurate contour performance, a high precision measurement is required for the master motion. For the position domain dynamic model Eq. (12), similar to the traditional PD control, we propose a new position domain PD control as follows: Fy ( x ) = K py e y ( x ) + K dy e′y ( x )

)

how large μ is. Proof: See reference [17]. This Lemma means that the dynamic system is globally uniformly exponentially convergent to a closed ball for any initial conditions.

(15)

Examining the control law for Y-axis in Eq. (15), we have the following equation:

10

§§ β · · V ≤ − ¨ ¨ 1 − ¸ x + c2 x + K dy − β x 2 ¸ e′y2 2 ¹ ©© ¹ x c x K + + § 2 dy · 2 − β ¨ K dy + k 2 − ¸ ey + ( e′y + β ey ) ρ max 2 © ¹ x + c2 x + K dy >0 γ 5 = K py + k 2 − 2 Define: β x + c2 x + K dy − β x 2 > 0 γ 6 = §¨ 1 − ·¸ 2¹ © Then we have: V ≤ − βγ 5 e 2y + β ρ max e y − γ 6 e′y2 + ρ max e′y

B. Stability Analysis The dynamic equation can be re-described in the error function format as follows: x 2 e′′y ( x ) + ( x + c2 x + K dy ) e′y ( x ) + ( K py + k2 ) e y ( x )

(22)

x + c2 x ) yd′ ( x ) + k2 yd ( x ) = x 2 yd′′ ( x ) + (

For the dynamic system described in the position domain, we define the following Lyapunov function: V ( e y ( x ) , e′y ( x ) ) =

1 ( ey 2

ªk + k e′y ) « py 2 2 ¬ β x

where 0 < β < 1 . If choose control gain so that:

β x 2 º § ey · »¨ ¸ x 2 ¼ © e′y ¹

(23)

K py > β 2 x − k2 2

Applying another inequality: a2 1 2 az − bz 2 ≤ − bz for a > 0, and b 4 We have:

(24)

Then it is easy to demonstrate that V is positive definite matrix. It is easy to demonstrate that: 1 1 ab ≥ − ( a 2 + b 2 ) , ab ≤ ( a 2 + b 2 ) 2 2 Therefore, applying the above inequalities to V, the following inequalities can be obtained:

β ρ max ey − βγ 5 ey2 ≤ − ρ max e′y − γ e′ ≤ − 2 6 y

γ6

4 Applying (31) to (29), we get:

βγ 5 4

b>0

β ρ max γ5

(27)

(28)

(29)

(30)

2

ey2 +

e′ + 2 y

ρ

2

(31)

max

γ6

§β 1 · 2 βγ γ V ≤ − 5 ey2 − 6 e′y 2 + ¨ + ¸ ρ max (32) 4 4 © γ5 γ6 ¹ Therefore, from Lemma 1, we can demonstrate that both the tracking error and the derivative of the tracking error are bounded as follows:

1 1 K py + k2 − β x 2 ) ey2 + (1 − β ) x 2 e′y2 ≤ V ( e y ( x ) , e′y ( x ) ) ( 2 2 (25) 1 1 2 2 2 2 ′ ≤ ( K py + k2 + β x ) ey + (1 + β ) x ey 2 2 From (25), we can see that the defined Lyapunov function satisfies (18). In the position domain control, the reference position of x motion is the independent variable that is the same meaning of t in the time domain. e y and e′y are functions of the independent variable x. Therefore, the derivative of V is related to variable x. ρ = x 2 yd′′ ( x ) + ( x + c2 x ) yd′ ( x ) + k2 yd ( x ) and Define

lim e x →∞

y

≤2

1

γ

2 5

+

1

βγ 5γ 6

ρ max

(33) β 1 ρ + e′y ≤ 2 lim γ 5γ 6 γ 62 max x →∞ It clearly shows that the maximum error can be reduced to a very small value by increasing control gain K py (related to constant γ 5 ) and K dy (related to constant γ 6 ).

ρ max = x yd′′ ( x ) + ( x + c2 x ) yd′ ( x ) + k2 yd ( x ) max . Then we 2

IV. CONTOUR TRACKING RESULTS

have: 2 V = ( K py + k2 ) e y e′y + x 2 e′y e′′y + β x 2 ( e′y ) + β x 2 e y e′′y

In this section, some simulation contour tracking examples are tested to verify the effectiveness of the proposed position domain PD control and to compare the performances between the new position domain PD control and the time domain PD control based on the same control gains. We assume the dynamic model of each axis in the Cartesian robot is the same and the system parameters are selected as follows.

= ( K py + k2 ) e y e′y + ( e′y + β e y ) x 2 e′′y + β x 2 e′y2 = ( K py + k2 ) e y e′y + β x 2 e′y2

(

+ ( e′y + β e y ) ρ − ( x + c2 x + K dy ) e′y − ( K py + k2 ) e y

= − ( x + c2 x + K dy − β x 2 ) e′y2 − β ( K py + k 2 ) e 2y

)

(26)

c1 = c2 = 10 , and k1 = k2 = 50

− β ( x + c2 x + K dy ) e y e′y + ( e′y + β e y ) ρ

In this paper, the CCC control laws are selected [4-7] as:

Using the following inequality: 1 − ab ≤ ( a 2 + b 2 ) 2 Eq. (26) can be rewritten as:

Fcx = K px ex + K dx ex − C x ( K pc ec + K dc ec ) Fcy = K py e y + K dy e y + C y ( K pc ec + K dc ec )

11

(34)

control are identical; it means that the contour tracking control part does not make any contribution for the final contour tracking performance, which can be concluded from the CCC control law (35) itself. This result shows the limitations of the CCC control.

(35)

For circular motion, the contour errors are defined as: ec = −Cx ex + C y ey

3.5

ec = −Cx ex + C y ey − C x ex + C y ey

-3 x 10 Tracking in position domain

ey e Cx = sin θ − x , C y = cos θ + 2R 2R ey e C x = θ cos θ − x , C y = −θ sin θ + 2R 2R

-3

5

Y axis

2.5 2 1.5 1

Tracking with CCC 5 X axis Y axis

2

0.2

0.4 0.6 0.8 X-axis position (m)

1

-1

1

-3

Tracking in time domain

X axis Y axis

3

2

1

0

0

x 10

4

3

0.5 0

x 10

4 Tracking errors (m)

Tracking errors (m)

3

Tracking errors (m)

For linear motion, the contour errors are defined as: ec = −Cx ex + C y ey Cx = sin θ , C y = cos θ ec = −Cx ex + C y ey

0

0

0.5

1 Time (Sec.)

1.5

-1

2

0

0.5

1 Time (Sec.)

Position domain control

A. Linear Motion For the linear contour tracking, two cases are considered in the simulations. The first case is a line with y=0.5x (m), and the second is y=x, all for T=2 sec., and x ∈ [ 0,1] .

50

45

45

40

40

40

35

35

30 25 20 15

35

30 X-axis Y-axis

25 20

30

20 15

10

10

10

5

5

0

0

0

0.2


1

X-axis Y-axis

25

15

5 0

0.5

1 Time (Sec.)

1.5

0

2

0

0.5

1 Time (Sec.)

B. Zigzag Motion Tracking Study In this example, a zigzag motion was controlled by three different control laws. Fig. 4 and 5 show the simulation results for two different motions in Y-axis ([0, 1], and[0, 2]). From these figures, one can see that the contour tracking error for the position domain control is much smaller than the CCC control and the traditional PD control. An amplified contour tracking results are also shown in Fig. 2(b). It clearly shows the significance of the position domain control. 4

x 10

-3

Tracking in position domain

x 10

14

-3

Tracking with CCC 20

x 10

-3


Y axis 12

X axis Y axis

15

X axis Y axis

4

1 0.8 0.6

Tracking errors (m)

Tracking errors (m)

Tracking errors (m)

1.2 3

2

1

3

10

2.5 2 1.5 1

1

0

0

-2

0

1

2 X-axis position (m)

3

4


-1

1

0

0.5

1 Time (Sec.)

1.5

-1

2

Cross coupled control

25

0.5

1 Time (Sec.)

1.5

2

Time domain control

60

50 45

50

15

10

5

40

30

20

X-axis Y-axis

0.2


1

0

4 Time (Sec.)

6

8

0

2

4 Time (Sec.)

6

8


Time domain control

1.5

1.5

1

1

1

0.5 0 Desired contour Real contour

-0.5

0

1

2 3 X-axis position (m)

4


-0.5 -1

0

1


4


-0.5 -1

0

1


4

(b) Contour tracking results with amplified errors by 5 times

25 20

Fig. 4 Contour tracking for zigzag motion with small motion in Y-axis.

15 10

10

0

2

-5

30

X-axis Y-axis

5 0

5

1.5

-1

35

40

Driving forces(N)

Driving forces (N)

Y-axis driving force(N)

20

0

Position domain control 0

(a). Contour tracking errors for three control systems Position domain control

X axis Y axis

10

(a) Contour tracking errors Y -ax is pos ition (m )

0.2

4

0.5

0

Y -ax is pos ition (m )

0

6

0

2

0.2 0

8

2

0.4 0

Tracking errors (m)

3


Y -ax is pos ition (m )

X axis Y axis

4

x 10

Tracking errors (m)

5

Y axis

1.4

-3

Tracking with CCC

Tracking errors (m)

-3

x 10

2

Fig. 3 Contour tracking errors and driving Forces for y=x

3.5

5

1.5

(b). Driving forces for three control systems

Fig. 2 shows the contour tracking errors and the driving forces for three different control methods. From this figure, one can see that the position domain control obtained the best result, the CCC control got a relative good result, and the PD control is the worst. It should be noticed that there is a price for good contour tracking using CCC, which is much large driving force (maximum value around 50N) in Y axis compared with position control and PD control (maximum value around 25N). The reason is that the CCC is a coupling control and tries to make the two axes motions synchronized so that the contour error becomes smaller. -3 x 10 Tracking in position domain

2

Time domain control

50

45

Driving forces(N)

K px = K py = K pc = 10000 , K dx = K dy = K dc = 6000


50

Driving forces (N)

Y-axis driving force(N)

For all the controllers, the control gains are set the same and selected as:

1.6

1.5

(a). Contour tracking errors for three control systems

0

0.5

1 Time (Sec.)

1.5

2

0

0

0.5

1 Time (Sec.)

1.5

Carefully check these two figures, one can see that the errors of the X-axis motion for time domain PD control are almost the same for two cases; while the errors of the Y-axis motion for CCC are almost the same. It means that the Y-axis motion dominates the contour error. In the position domain control, as the control gains are chosen the same in these two motion cases, the contour tracking error for fast motion obviously is larger than the slow motion. Such a conclusion is coincident with the stability analysis result.

2

(b). Driving forces for three control systems Fig. 2 Contour tracking errors and driving Forces for y=0.5x

Fig. 3 shows another simulation results where two axes are moving at the same speed (y=x). It clearly shows the effectiveness of the position domain control. From fig. 3, one can see that the results for both CCC control and the PD

12

x 10

-3

Tracking in position domain 16 Y axis

7

-3

Tracking with CCC 20

Tracking errors (m)

5 4 3

10 8 6 4

2 1

0

0

-2 0

1


3

-3


4

2

4 Time (Sec.)

6

precisely. In this paper, a position domain PD control is proposed as an alternative to the time domain PD control. In this position domain control, the master motion yields zero tracking error for the position tracking because it is sampled as a reference, and only the slave motion tracking errors will affect the final contour tracking errors. The properties of the original dynamic system represented in the time domain are preserved by the developed dynamic system described in the position domain as these two systems are isomorphic. Most importantly, the contour tracking performance is significantly improved using the position domain PD control. Simulation results for different motions demonstrate the effectiveness of the position domain control. The proposed position domain control has the potential to be used as an alternative for the time domain control.

X axis Y axis

10

5

X axis Y axis

2

0

x 10

15

12

6 Tracking errors (m)

x 10

14

Tracking errors (m)

8

0

-5

8

0

2

4 Time (Sec.)

6

8

(a) Contour tracking errors Cross coupled control

Time domain control 2.5

2

2

1.5

1.5

1.5

1 0.5 0

1 0.5 0

Desired contour Real contour

-0.5 -1 0

Y-axis position (m)

2.5

2 Y-axis position (m)

Y-axis position (m)

Position domain control 2.5

1 0.5 0


-0.5 -1

1


3

4


-0.5 -1

0

1


4

0

1


4

(b Contour tracking results with amplified errors by 2 times Fig. 5 Contour tracking for zigzag motion with large motion in Y-axis.

ACKNOWLEDGMENT

C. Circular Motion Tracking Study In this example, a contour for a half circular motion ( x ∈ [ −0.5, 0.5] , y ∈ [ 0, 0.5] , R=0.5m, T=5sec.) is tracked to

This research is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant to the first author.

examine the effectiveness of the proposed position domain control. The control gains are chosen one-fifth of the control gains applied in linear motion. Fig. 6 shows the contour tracking for three different control systems, respectively. It demonstrates the high contour tracking performance of the position domain control, compared with the CCC control and the time domain PD control. -3

Tracking in position domain 8 6

6

4

5 4

Y axis

3

x 10

-3

Tracking with CCC

X axis Y axis

-2

[5]

0 X axis Y axis

-0.005 -0.01

-6 0 X-axis position (m)

-8

0.5

[4]

0.005

-4

0 -0.5

[3]


0

1

[2]

0.01

2

2

[1]

0.015

Tracking errors (m)

x 10

7

Tracking errors (m)

Tracking errors (m)

8

REFERENCES

0

1

2 3 Time (Sec.)

4

-0.015

5

[6] 0

1

2 3 Time (Sec.)

4

5

(a) Contour tracking errors for different control laws Cross coupled control

0.5

0.5

0.5

0.4

0.4

0.2


0.3


0.2

0.1

Y-axis position (m)

0.6

0.3

0.1

0


0.3


0.2

[9] [10]

0

-0.1 -0.5

0.5

[8]

0.4

0.1

0

-0.1 -0.5

[7]

Time domain control

0.6

Y-axis position (m)

Y-axis position (m)

Position domain control


-0.1 -0.5

0.5


0.5

(b) Contour tracking results with amplified errors by 2 times Cross coupled control 30

20

30

20

10 5 0 -5 -0.5

20

Torque (N.m)

15

10 0 X-axis Y-axis

-10

-30

0.5

10

[12]

0 X-axis Y-axis

-10

[13]

-20

-20 0 X-axis position (m)

[11]

Time domain control

40

Torque (N.m)

Y-axis Torque (N.m)

Position domain control 25

0

1

2 3 Time (Sec.)

4

5

-30

0

1

2 3 Time (Sec.)

4

5

[14]

(c). Driving forces for three different control systems Fig. 6 Circular contour tracking results for three different control systems

[15] [16]

V.

CONCLUSIONS

Contour tracking control is to control the movement of the end-effector following a desired contour effectively and

[17]

13

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