Trajectory Tracking Control of Robot Manipulators using Discrete

Time-varying Pole Placement Technique

Yasuhiko Mutoh, Masakatsu Kemmotsu and Lisa Awatsu

Department of Engineering and Applied Sciences, Sophia University, 7-1 Kiocho, Chiyoda-ku, Tokyo, Japan

Keywords:

Trajectory Tracking Control, Linear Time-varying System, Discrete System, Time-Varying Pole Placement

Control.

Abstract:

For the trajectory tracking control problem of nonlinear systems, the most basic and classic strategy may

be applying the linear control technique to a linear time-varying approximate model around some desired

trajectory. However, this method is not commonly used because the design of a linear time-varying controller

is not simple. The authors proposed the simple design method of the pole placement controller for linear

time-varying discrete systems. In this paper, to show the applicability of the proposed linear time-varying

discrete pole placement technique to the trajectory tracking control problem of nonlinear systems, we apply

this control method to actual 2-link robot manipulator and present the experimental results.

1 INTRODUCTION

For the trajectory tracking control problem of non-

linear systems, the most basic and classic strategy

may be applying the linear control technique to a lin-

ear time-varying approximate model around some de-

sired trajectory. This method can be applied to any

type of nonlinear systems. However, since, controller

design method for linear time-varying system is not

necessarily simple (Nguyen(1987)) (Valsek(1995))

(Valsek(1999)), gain scheduling strategy, the nonlin-

ear control strategy, or PID control is commonly used

for such a control design problem.

The author et.al. proposed the simple pole place-

ment controller design method for linear time-varying

discrete systems (Mutoh(2011)) (Mutoh and Hara

(2011)). Such controller is obtained by ﬁnding a new

output signal so that the relative degree from the input

to this new output is equal to the system degree.

In this paper, we apply this control method to the

tracking control of an actual 2-link robot manipulator

to show the applicability of the proposed linear time-

varying discrete pole placement technique to the tra-

jectory tracking control problem of practical nonlin-

ear systems. In the following, some basic properties

of linear time-varying discrete systems are stated in

Section 2. Section 3 summarizes the design procedure

of a pole placement controller for linear time-varying

discrete systems. In Section 4, this control method

is applied to the trajectory tracking control problems

of practical 2-link robot manipulator and experimen-

tal results are presented to show the validity of this

control system.

2 BASIC PROPERTIES OF

LINEAR TIME-VARYING

DISCRETE SYSTEMS

In this section, some basic properties of linear time

varying multi variable discrete systems are presented.

Consider the following system.

x(k+ 1) = A(k)x(k) + B(k)u(k) (1)

Here, x ∈ R

n

and u ∈ R

m

are the state variable and the

input. A(k) ∈ R

n×n

and B(k) ∈ R

n×m

are time-varying

coefﬁcient matrices. The state transition matrix of the

system (1) from k = j to k = i, Φ(i, j), is deﬁned as

follows.

Φ(i, j) = A(i− 1)A(i− 2)···A( j) i > j (2)

Deﬁnition 1. System (1) is called ”completely reach-

able in n steps” if and only if, for any x

1

∈ R

n

there

exists a bounded input u(l) (l = k, ··· , k+n−1) such

that x(k) = 0 and x(k + n) = x

1

for all k.

Lemma 1. System (1) is completely reachable in n

steps if and only if the rank of the reachability matrix

deﬁned below is n for all k.

U

R

(k) =

B

0

(k), B

1

(k), ··· , B

n−1

(k)

(3)

373

Mutoh Y., Kemmotsu M. and Awatsu L..

Trajectory Tracking Control of Robot Manipulators using Discrete Time-varying Pole Placement Technique.

DOI: 10.5220/0005533603730379

In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 373-379

ISBN: 978-989-758-122-9

Copyright

c

2015 SCITEPRESS (Science and Technology Publications, Lda.)

where,

B

0

(k) = B(k + n− 1)

B

1

(k) = Φ(k+ n, k+ n− 1)B(k + n− 2)

.

.

. (4)

B

n−1

(k) = Φ(k+ n, k+ 1)B(k)

Let b

l

i

(k) be the i-th column of B

l

(k), then, the reach-

ability matrix U

R

(k) can be written as

U

R

(k) =

b

0

1

(k)···b

0

m

(k)|···|b

n−1

1

(k)···b

n−1

m

(k)

(5)

Note that b

r

i

(k) also satisﬁes the same equation as

(4), i.e.,

b

0

i

(k) = b

i

(k+ n − 1)

b

1

i

(k) = Φ(k + n, k+ n− 1)b

i

(k+ n − 2)

.

.

. (6)

b

n−1

i

= Φ(k + n, k+ 1)b

i

(k) (i = 1, ··· , m)

where b

i

(k) is the i-th column of B(k). Suppose that

the system (1) is completely reachable in n steps.

Then, the reachability indices, µ

i

(i = 1, ··· , m), can

be deﬁned such that

m

∑

i=1

µ

i

= n (7)

and the n× n truncated reachability matrix

R(k) =

h

b

0

1

(k), ··· , b

µ

1

−1

1

(k)|···|b

0

m

(k), ··· , b

µ

m

−1

m

i

(8)

is non-singular. It is assumed that µ

1

≥ µ

2

≥ ·· · ≥ µ

m

without loss of generality.

Finally, the vector relative degree of a linear

MIMO system is deﬁned. Let the following η(k) ∈

R

m

be the output vector of the system (1).

η(k) = H(k)x(k), H(k) ∈ R

m×n

(9)

Deﬁnition 2. System (1), (9) has the vector relative

degree, r

1

, r

2

, ···, r

m

from u to η, if and only if there

exist some matrix D(k) ∈ R

m×n

and some nonsingular

matrix Λ(k) ∈ R

m×m

that satisfy the following equa-

tion.

α

1

(z)

.

.

.

α

m

(z)

η(k) = D(k)x(k)+Λ(k)u(k)

(10)

Here, α

k

(z) is an arbitrary monic polynomial of de-

gree r

k

and z is a forward shift operator.

3 DESIGN OF DISCRETE

TIME-VARYING POLE

PLACEMENT CONTROLLER

In this section, the design procedure of the pole place-

ment controller for linear time-varying multi input

discrete systems is summarized. Suppose that the

system (1) is completely reachable with its reacha-

bility indices, µ

1

, ··· , µ

m

. The problem is to design a

state feedback for the system (1) so that the resulting

time-varying closed-loop system becomes equivalent

to some linear time-invariant system with arbitrarily

stable poles. For this purpose, we ﬁrst deﬁne a new

output signal y(k) ∈ R

m

of the system (1) by

y(k) = C(k)x(k) (11)

so that the total relative degree from u(k) to y(k) is

equal to the system degree n. Here,

y(k) =

y

1

(k)

y

2

(k)

.

.

.

y

m

(k)

∈ R

m

, C(k) =

c

1

(k)

c

2

(k)

.

.

.

c

m

(k)

∈ R

m×n

(12)

where y

i

(k) ∈ R and c

i

(k) ∈ R

1×n

. We have the fol-

lowing Theorem (Mutoh and Hara (2011)).

Theorem 1. If the system (1) is completely reachable

in n steps, there exists a new output y(k) such that

the vector relative degree from u(k) to y(k) becomes

µ

1

, ··· , µ

m

, which implies that the total relative degree

from u(k) to y(k) is n. And, such C(k) can be calcu-

lated by the following equation.

C(k) = WR

−1

(k− n) (13)

where

W = diag(w

1

, w

2

, ··· , w

m

)

w

i

=

0 ··· 0 1

∈ R

1×µ

i

(14)

(i = 1, ··· , m)

From this, the pole placement state feedback is ob-

tained in the following procedure.

Let q

i

(z) be the desired stable characteristic poly-

nomial of z as

q

i

(z) = z

µ

i

+ α

i

µ

i

−1

z

µ

i

−1

+ · ·· + α

i

1

z+ α

i

0

. (15)

(i = 1, ··· , m)

Since, the vector relative degree from u(k) to y(k) is

µ

1

, µ

2

, ···, µ

m

, we have the following equation.

q

1

(z)

.

.

.

q

m

(z)

y(k) = D(k)x(k) + Λ(k)u(k)

(16)

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

374

Here,

D(k) =

D

1

(k)

D

2

(k)

.

.

.

D

m

(k)

, Λ(k) =

Λ

1

(k)

Λ

2

(k)

.

.

.

Λ

m

(k)

(17)

and D

i

(k) ∈ R

1×n

and Λ

i

(k) ∈ R

1×n

are deﬁned by

D

i

(k) =

α

i

0

, α

i

1

, ··· , α

i

µ

i

−1

, 1

c

0

i

(k)

c

1

i

(k)

.

.

.

c

µ

i

i

(k)

(18)

Λ

i

(k) = [0, ·· · , 0, 1, γ

i(i+1)

, ··· , γ

im

].

In the above equation, c

l

i

(k) is deﬁned by the follow-

ing recursive equations, using c

i

(k),

c

0

i

(k) = c

i

(k)

c

(l+1)

i

(k) = c

l

i

(k+ 1)A(k) (19)

(l = 0, 1, 2, ···)

and,

γ

ij

= c

µ

i

−1

i

(k+ 1)b

j

(k) (20)

for i = 1, ·· · , m. Then, by applying the state feedback

u(k) = −Λ

−1

(k)D(k)x(k) (21)

to the system (1), the closed loop system becomes as

follows.

q

1

(z)

.

.

.

q

m

(z)

y(k) = 0 (22)

This system is time-invariant and has the following

state representation.

w(k+ 1) = A

∗

w(k) (23)

where w(k) ∈ R

n

is the new state variable. The matri-

ces A

∗

∈ R

n×n

is written by

A

∗

=

A

∗

1

0

.

.

.

0 A

∗

m

(24)

and A

∗

i

∈ R

µ

i

×µ

i

is deﬁned as follows.

A

∗

i

=

0 1 0

.

.

.

.

.

.

.

.

.

.

.

. 1

−α

i

0

. . . . . . −α

i

µ

i

−1

(i = 1, . . . , m) (25)

From this, the characteristic polynomial of A

∗

is writ-

ten as follows using q

i

(z) deﬁned by (15).

q(z) =

m

∏

i=1

q

i

(z) (26)

(19) and (22) imply that w(k) is written as follows.

w(k) :=

y

1

(k)

.

.

.

y

1

(k+ µ

1

− 1)

.

.

.

y

m

(k)

.

.

.

y

m

(k+ µ

m

− 1)

=

c

0

1

(k)

.

.

.

c

µ

1

−1

1

(k)

.

.

.

c

0

m

(k)

.

.

.

c

µ

m

−1

m

(k)

x(k)

= P(k)x(k) (27)

On the other hand, from (1) and (21), the time-

varying state equation of the closed loop system be-

comes

x(k+ 1) = (A(k) − B(k)Λ

−1

D(k))x(k). (28)

Thus, the system (28) is equivalentto the system (23),

with the transformation matrix P(k). It is then obvi-

ous that the following equation holds.

P(k+ 1)(A(k) − B(k)Λ

−1

D(k))P

−1

(k) = A

∗

(29)

This implies that the state feedback (21) makes the

closed loop system equivalent to the system (23) that

has an arbitrarily stable characteristic polynomial,

q(z).

Note that the transformation matrix P(k) and

P

−1

(k) must be bounded functions, in other words,

P(k) must be a Lyapunov transformation, to ensure

the stability of the closed-loop system.

The procedures to obtain the state feedback gain

is summarized below.

Pole Placement Design Procedure

STEP 1. Calculate the reachability matrix U

R

(k− n)

and the reachability indices µ

i

.

STEP 2. Calculate C(k) = WR

−1

(k − n) for the new

output signal, y(k), using the truncated reachabil-

ity matrix R(k).

STEP 3. Determine the desired stable closed-loop

characteristic polynomials as follows for i =

1, ··· , m.

q

i

(z) = z

µ

i

+ α

i

µ

i

−1

z

µ

i

−1

+ · ·· + α

i

1

z+ α

i

0

STEP 4. Using (17) ∼ (20), calculate D(k) and Λ(k).

Then, the state feedback for the pole placement is

u(k) = −Λ

−1

(k)D(k)x(k)

TrajectoryTrackingControlofRobotManipulatorsusingDiscreteTime-varyingPolePlacementTechnique

375

4 TRAJECTORY TRACKING

CONTROL OF 2-LINK

MANIPULATORS

In this section, discrete time-varying pole placement

technique is applied to the trajectory tracking control

of a two-link robot manipulator.

4.1 The Model of the Manipulator

Fig. 1 and Fig. 2 show the picture and the model of

the 2-link robot manipulator for the experiment. All

links rotate in the horizontal plane.

Figure 1: Two-Link Manipulator(SR-402DDII.)

Its motion equation is described as follows.

M(θ(t))

¨

θ(t) +C(θ(t),

˙

θ(t))

˙

θ(t) + D(

˙

θ(t)) = τ(t)

(30)

where,

θ(t) =

θ

1

(t)

θ

2

(t)

M(θ(t)) =

J

1

+ J

2

+ 2m

2

r

2

l

1

cosθ

2

(t),

J

2

+ m

2

r

2

l

1

cosθ

2

(t),

J

2

+ m

2

r

2

l

1

cosθ

2

(t)

J

2

C(θ(t),

˙

θ(t)) =

−2m

2

r

2

l

1

˙

θ

2

(t)sinθ

2

(t),

m

2

r

2

l

1

˙

θ

1

(t)sinθ

2

(t),

−m

2

r

2

l

1

˙

θ(t)

2

sinθ

2

(t)

0

D(

˙

θ(t)) =

2sgn(

˙

θ

1

(t))

0.25sgn(

˙

θ

2

(t))

J

i

= J

l

i

+ m

i

r

2

i

(i = 1, 2).

Here, θ

i

(t) and τ

i

(t) are joint angle and input torque

of i-th joint, l

i

and r

i

are length of the i-th link and

the distance between the i-th joint and the center of

gravity of the i-th link, and J

l

i

is the moment of inertia

of the i-th link about its center of gravity (i = 1, 2).

D(

˙

θ(t)) is a friction term which is estimated from the

experimental data.

In the above, the values of the physical parameters

are shown in Table 1.

Table 1: Parameter of Manipulator.

variable unit link1 link2

(i = 1, 2)

i = 1 i = 2

m

i

[kg] 3.43 1.55

l

i

[m] 0.2 0.2

r

i

[m]

0.1 0.1

J

l

i

[kgm

2

]

0.208 0.03

4.2 Experimental Results

In this section, we show the experimental result of the

trajectory tracking control of the 2-link robot manip-

ulator using the time-varying discrete pole placement

controller.

To design the discrete controller, we discretize the

manipulator system (30) by Euler method as follows.

Here, T

s

is the sampling time.

x(k+ 1) = x(k) +

0 T

s

I

2

0 T

s

Γ(x(k))

x(k)

+

0

T

s

Φ(x(k))

u(k)

= f(x(k), u(k)) (31)

where

x(k) =

θ(k)

˙

θ(k)

∈ R

4

u(k) =

τ

1

(k)

τ

2

(k)

∈ R

2

(32)

I

2

=

1 0

0 1

Γ(x(k)) = −M(θ(k))

−1

C(θ(k),

˙

θ(k)) ∈ R

2×2

Φ(x(k)) = M(θ(k))

−1

∈ R

2×2

Figure 2: Two-Link Manipulator Model.

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

376

The sampling time T

s

is 10 [msec] for the experi-

ment.

Let the desired trajectory of the end portion of this

manipulator be the circle in the horizontal X-Y work

space as presented by the the following equation,

X(t) = 0.08cos

π

5

t + 0.3 (33)

Y(t) = 0.08sin

π

5

t + 0.05 (34)

which is described in Fig.3.

0.2 0.3 0.4 0.45

−0.05

0

0.1

0.2

x

[m]

y [m]

Figure 3: Desired Trajectory of the End Portion.

From the desired trajectory of the end portion, the

desired trajectory of the joint angles, θ

∗

(t), and their

speed,

˙

θ

∗

(t), can be calculated using the inverse kine-

matics. Which gives the desired state variable x

∗

(t) as

follows.

x

∗

(t) =

θ

∗

(t)

˙

θ

∗

(t)

(35)

The desired input signal u

∗

(t) is obtained from θ

∗

(t)

and

˙

θ

∗

(t) using (30). MAXIMA is used to calcu-

late the explicit function representations for x

∗

(t) and

u

∗

(t), which are omitted here because of the space

limitation. Instead of this, the graphs of x

∗

(t) and

u

∗

(t) are shown in Fig.4 and 5.

By discretizing these signals using the sampling

time T

s

, the discrete desired state trajectory x

∗

(k) and

the discrete desired input u

∗

(k) are obtained. Note

that we use the same variable for continuous space

and discrete space, i.e., x(t) is continuous variable,

0 5 10 15

−1

0

1

2

2.5

time [sec]

x

( )

x

( )

x

( )

x

( )

x

( )

t

t

t

t

t

Figure 4: Desired Trajectory x

∗

(t).

0 5 10 15

−4

−2

0

2

4

time [sec]

u

1

( ) [Nm]

0 5 10 15

−0.5

0

0.5

time [sec]

u

2

( ) [Nm]

t

t

Figure 5: Desired Input u

∗

(t).

x(k) is a discrete variable of k-th step and x(kT

s

) is a

sampling variable in the t-axis.

To obtain the linear time-varying approximate

model around the desired trajectory, x

∗

(k) and u

∗

(k),

deﬁne ∆x(k) and ∆u(k) by

∆x(k) = x(k) − x

∗

(k)

∆u(k) = u(k) − u

∗

(k)

(36)

Then we have the following approximate model from

(31).

x(k+ 1) =

∂

∂x

f(x

∗

(k), u

∗

(k))∆x(k)

+

∂

∂u

f(x

∗

(k), u

∗

(k))∆u(k)

= A(k)∆x(k) + B(k)∆u(k) (37)

where,

A(k) =

0 0 1 0

0 0 0 1

0 a

32

(k) a

33

(k) a

34

(k)

0 a

42

(k) a

43

(k) a

44

(k)

(38)

B(k) =

0 0

0 0

β

31

(k) β

32

(k)

β

41

(k) β

42

(k)

(39)

Here, the explicit function representation of A(k) and

B(k) are obtained by using MAXIMA, which are de-

scribed in Appendix A, for reference.

Fig.6 shows the closed loop response of the ma-

nipulator end portion in the horizontal work space.

The initial position of the end portion is (0.4, 0) in

the coordinate of the horizontal work space. This ini-

tial condition corresponds to the initial condition of

state variable vector, x

1

(0) = x

2

(0) = x

3

(0) = x

4

(0) =

0. The response of the manipulator state variable

x(kT

s

) (joint angles and their speed), the state er-

ror ∆x(kT

s

) = x(kT

s

) − x

∗

(kT

s

), are shown in Fig.7

TrajectoryTrackingControlofRobotManipulatorsusingDiscreteTime-varyingPolePlacementTechnique

377

and Fig.8 respectively. The control input u(kT

s

) =

u

∗

(kT

s

) + ∆u(kT

s

) is shown in Fig.9. The desired sta-

ble poles of the closed loop system and the observer

are chosen as (−5, −90, −5, −90).

0.2 0.3 0.4 0.45

−0.05

0

0.1

0.2

x [m]

y [m]

Experimental Result

Desired Trajectory

Figure 6: Response of End Portion.

0 5 10 15

−1

0

1

2

2.5

time [sec]

x(kTs)

x

1

(kTs)

x

2

(kTs)

x

3

(kTs)

x

4

(kTs)

Figure 7: Response of State Variable x(k).

0 5 10 15

−1

0

1

2

2.5

time [sec]

∆x(kTs)

∆x

1

(kTs)

∆x

2

(kTs)

∆x

3

(kTs)

∆x

4

(kTs)

Figure 8: State Error ∆x(k) = x(k) − x

∗

(k).

0 5 10 15

−4

−2

0

2

4

time [sec]

u

1

(kTs) [Nm]

0 5 10 15

−0.5

0

0.5

time [sec]

u

2

(kTs) [Nm]

Figure 9: Input Torque u(k).

5 CONCLUSIONS

In this paper, the trajectory tracking control of non-

linear systems was considered. For this purpose, the

pole placement controller designed by the simple pro-

cedure was applied to the linear time-varying discrete

approximate model of the system around some de-

sired trajectory. This controller was applied to track-

ing control of the actual 2-link robot manipulator to

show the applicability of this type of controller. The

experimental results showed that this controller has

very good performance.

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Chi-Tsong Chen (1999) C Linear System Theory and De-

sign (Third edition). Oxford University Press

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APPENDIX

The followings are the explicit form of functioins of

the elements of A(t) and B(t) in equation (37) calcu-

lated by MAXIMA. We used this result with T = T

s

.

ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics

378

a

32

=

31

28830cos(x

2

(k))

3

x

4

(k)

2

+ 185340cos(x

2

(k)) x

4

(k)

2

T

(31cos(x

2

(k))− 90)

2

(31cos(x

2

(k))+ 90)

2

+

31

57660cos(x

2

(k))

3

x

3

(k) x

4

(k) +370680cos(x

2

(k)) x

3

(k) x

4

(k)

T

(31cos(x

2

(k))− 90)

2

(31cos(x

2

(k))+ 90)

2

+

31

28830cos(x

2

(k))

3

x

3

(k)

2

+ 472409cos(x

2

(k))

2

x

3

(k)

2

(31cos(x

2

(k))− 90)

2

(31cos(x

2

(k))+ 90)

2

+

31

185340cos(x

2

(k)) x

3

(k)

2

− 251100x

3

(k)

2

T

(31cos(x

2

(k))− 90)

2

(31cos(x

2

(k))+ 90)

2

a

33

= 1−

62sin(x

2

(k)) (30x

4

(k) +31cos(x

2

(k)) x

3

(k) +30x

3

(k)) T

(31cos(x

2

(k))− 90) (31cos(x

2

(k))+ 90)

a

34

= −

1860sin(x

2

(k)) (x

4

(k) +x

3

(k)) T

(31cos(x

2

(k))− 90) (31cos(x

2

(k))+ 90)

a

42

= −

31(28830cos(x

2

(k))

3

x

4

(k)

2

+ 472409cos(x

2

(k))

2

x

4

(k)

2

)T

(31cos(x

2

(k)) − 90)

2

(31cos(x

2

(k)) + 90)

2

−

31(185340∗ cos(x

2

(k))x

4

(k)

2

− 251100x

4

(k)

2

)T

(31cos(x

2

(k)) − 90)

2

(31cos(x

2

(k)) + 90)

2

−

31(57660cos(x

2

(k))

3

∗ x

3

(k)x

4

(k) +944818cos(x

2

(k))

2

x

3

(k)x

4

(k))T

(31cos(x

2

(k)) − 90)

2

(31cos(x

2

(k)) + 90)

2

−

31(370680cos(x

2

(k))x

3

(k)x

4

(k) −502200x

3

(k)x

4

(k))T

(31cos(x

2

(k)) − 90)

2

(31cos(x

2

(k)) + 90)

2

−

31(288300cos(x

2

(k))

3

x

3

(k)

2

+ 944818cos(x

2

(k))

2

x

3

(k)

2

)T

(31cos(x

2

(k)) − 90)

2

(31cos(x

2

(k)) + 90)

2

+

31(1853400cos(x

2

(k))

2

x

3

(k) −502200x

3

(k)

2

)T

(31cos(x

2

(k)) − 90)

2

(31∗ cos(x

2

(k)) + 90)

2

a

43

=

62sin(x

2

(k)) (31cos(x

2

(k)) x

4

(k) +30x

4

(k) +62cos(x

2

(k)) x

3

(k))T

(31cos(x

2

(k))− 90) (31cos(x

2

(k))+ 90)

+

31(300x

3

(k)) T

(31cos(x

2

(k))− 90) (31cos(x

2

(k))+ 90)

a

44

=

62 (31cos(x

2

(k))+ 30) sin(x

2

(k)) (x

4

(k) +x

3

(k)) T

(31cos(x

2

(k))− 90) (31cos(x

2

(k))+ 90)

+ 1

b

31

= −

30000T

961cos(x

2

(k))

2

− 8100

b

32

=

(31000cos(x

2

(k))+ 30000) T

961cos(x

2

(k))

2

− 8100

b

41

=

(31000cos(x

2

(k))+ 30000) T

961cos(x

2

(k))

2

− 8100

b

42

= −

(62000cos(x

2

(k))+ 300000) T

961cos(x

2

(k))

2

− 8100

.

TrajectoryTrackingControlofRobotManipulatorsusingDiscreteTime-varyingPolePlacementTechnique

379