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International Conference on Control, Automation and Systems 2007

Oct. 17-20,2007 in COEX, Seoul, Korea

Robust Nonlinear Observer for Flexible Joint Robot Manipulators

with Only Motor Position Measurement

Jaeyoung Lee l, Tae Jun Hal, Je Sung Yeonl, Sanghun Lee 2 and Jong Hyeon Park l

1Department of Mechanical Engineering, Hanyang University, Seoul,Korea (E-mail: [email protected])

2Electro-Mechanical Research Institute, Hyundai Heavy Industries Co., Ltd., Gyeonggi, Korea

Abstract: This paper proposes a design method of sliding mode observer for flexible joint robot manipulators. The

proposed observer structure is the conventional Luenberger observer with additional switching element that is a role to

attenuate system uncertainties and modeling errors. During sliding, the design method is that the nonlinear function due

to system uncertainty in sliding mode observer is divides into the parameter distinction and estimation error elements by

using the Lipschitz condition. From the design method, this paper proves relation of the asymptotical stability and robust

ness of overall system according to the positive nonlinear gain of switching element. The proposed observer accurately

estimates the velocity and position of link side in the flexible joint robot using only motor position measurement. In the

simulation result, the performance and robustness of proposed sliding mode observer is verified and compared with the

high gain observer.

Keywords: Robust Observer, Flexible Joint, Sliding Mode Observer, Nonlinear Observer.

1. INTRODUCTION

Many researchers have proposed observer design

methods. Wang and Gao presented a comparison study of

performances and characteristics of three advanced state

observers, including high-gain observers, sliding mode

observers and extended state observers [1]. These ob

servers were originally proposed to address the depen

dence of the classical observers, such as the Kalman Fil

ter and the Luenberger observer, on the accurate math

ematical representation of the plant. The extended stateobserver is much superior than others in dealing with dy

namic uncertainties, disturbances and sensor noise. Then,

several novel nonlinear gain functions are proposed to ad

dress the difficulty in dealing with unknown initial con

ditions. Especially, the gain modification method for the

nonlinear extended state observer is proposed to deal with

the unknown initial conditions.

As a representative observer for nonlinear system, the

high gain observer is introduced by Khalil for the first

time. The high gain observer robustly estimates the

derivatives of output signal. Also, it is easy to apply to

the observer structure for nonlinear systems and to prove

stability and robustness with only a linear gain term. Sim

ilarly, the sliding mode observer can be easily applied to

nonlinear systems in the presence of parameter uncertain

ties and initial condition difference of between the ob

server and the plant. Generally, the basic design method

for sliding mode observer has proposed by Slotine and

Hedrick [2]. It no clear guideline to apply to nonlin

ear systems. Also, there was no explanation about how

to determine the nonlinear gain function. Then, Misawa

and Hedrick propose a method for finding the nonlinear

and linear gain matrix for linear systems [3].

The nonlinear function of sliding mode observer was

designed from the model uncertainty and estimation error under the difference of between the observer and the

plant. The design of positive gain function using the non-

978-89-950038-6-2-98560/07/$15 @ICROS

56

linear function is important to prove the stability and ro

bustness of overall system. In the application case of non

linear system, for examples, each value of positive non

linear gain coefficients were selected by trial-and-error

[4]. As mathematical method in designing gain, Gand

and Kfoury proposed a concrete method for the positive

nonlinear gain function [5]. The positive nonlinear gain

function is defined as upper bounded value of parame

ter distinction and state error. They show the numerical

result regardless of the differences in the initial condi

tions of between the plant and the observer. However, theproposed method did not accurately estimate the flexible

motion in the presence of unstructured uncertainties of

the flexible link system.

Recently, Abdessameud and Khelfi proposed method

which is developed to dealt with the application of a vari

able structure observer for a class of nonlinear systems to

solve the trajectory tracking problem for rigid robot ma

nipulators [6]. They propose an assumption that nonlin

earities and uncertainties of system exists upper bounded

function. It satisfies some matching conditions. But, the

existence of bounded function which satisfies the match

ing condition is limited in application. This method isjustto estimate the velocity information in the rigid robot.

In this paper, we propose a design method that is based

on the conventional Luenberger observer with additional

switching element. The switching element is a role for

attenuating system uncertainties and modeling errors to

guarantee the robustness. The positive gain function for

switching element is proposed by designing the upper

bounded function of the parameter uncertainty and esti

mation error. During sliding from the sliding condition,

the error dynamic is represented as the linear system with

the nonlinear function. The nonlinear matrix inequality

for system stability proof is designed as the positive nonlinear function which is consisted of the estimation error

and bounded parameter distinction. In case of the max-

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imum size of the nonlinear function, this paper proves

a relation of the asymptotically stable and robustness of

overall system according to the positive nonlinear gain of

switching element. The proposed method accurately esti-

mates the velocity and position of link side in the flexible

joint robot using only measured angle position of motor.

From the simulation result, the performance and robust-ness of proposed sliding mode observer is verified and

compared with the high gain observer.

The paper is organized as follows. Section 2 describes

the theory of sliding mode observer. Section 3 is the ap-

plication to the robot manipulators with joint flexibility.

The observer gain design method is defined in section 4.

Section 5 is proved the asymptotically stable of proposed

observer. Section 6 shows the estimation performance

and the robustness of proposed observer from the simu-

lation result and compared with the high gain observer.

Section 7 is the conclusion.

2. SLIDING-MODE OBSERVER

Consider the following nonlinear system:

x(t) = Ax(t) + f (x, t) + Bu(t), (1)

y(t) = C x(t), (2)

where x(t) ∈ Rn, y(t) ∈ R

p and u(t) ∈ Rm are the

state, the output, and the control input, respectively; B ∈Rn×m, A ∈ R

n×n and C ∈ R p×n are the input matrix,

linear matrix and output matrix, respectively. Nonlinear

function f (x, ·) is assumed to be continuous in x. Now, it

is desired to reconstruct state x from the measurement y

and u. The structure of sliding-mode observers proposed:˙x(t) = Ax(t) + f (x, t) + Bu − Ls − K ∗ sgn(s), (3)

where L and K are the linear gain and the positive non-

linear gain, f (·) is an estimate on f (·), and sliding variable

s is defined by

s := y − y = C x. (4)

with estimation error x defined by

x∆= x − x. (5)

sgn(s)∆= y =

sgn(s1) ... sgn(s p)T

∈ R p (6)

with sign function sgn(·).Then, from Eqs. (1) and (3), the error dynamics of the

observer becomes

˙x(t) = Ax(t) + ∆f − Ls − K ∗ sgn(s), (7)

where ∆f ∆= f (x, t) − f (x, t). For a positive-definite func-

tion of V = 1

2sT s, if

V = sT s < −η||s||, (8)

sliding occurs in a finite time. During the sliding, sliding

variable s remains zero, i.e., s = 0, and thus from Eq. (7),

s = C (Ax(t) + ∆f − K ∗ sgn(s)) = 0, (9)

from which

sgn(s) = (CK )−1C (∆f + Ax(t)). (10)

Therefore, once the system is on sliding surface, i.e., s =0, from Eqs. (7) and (10), the error dynamics becomes

˙x = (I − K (CK )−1C )(∆f + Ax(t)). (11)

3. APPLICATION TO THE FLEXIBLEJOINT ROBOT MANIPULATORS

The dynamic of N-link flexible joint robot manipula-

tors is represented by

M (q)q + C (q, q)q + g(q) + K (q − θ) = 0

J θ + K (θ − q) = u, (12)

where q, θ ∈ Rn denote the link positions and the motor

angle, respectively; M, K and J denote the inertia matrix

and the joint stiffness matrix and the motor inertia ma-

trix, respectively; C (q, q) denotes the matrix associated

with Coriolis and Centrifugal acceleration and vector g

denotes the gravitational term; and input u denotes the

motor torque.

With the definition of a state

x∆=

x1 x2 x3 x4T

=θ θ q q

T ∈ R4n, (13)

the dynamic in Eq. (12) can be transform to a state-space

representation:

x(t) = Ax(t) + f (x) + Bu, (14)

where

A =

0 0 0 0−J −1K 0 J −1K 0

0 0 −I n 00 0 0 −I n

, (15)

f (x) =

x20

x3 + x4M (x3)−1K (x1 − x3) + x4

−M (x3)−1(C (x3, x4) + G(x3))

(16)

and

B =

0 J −1 0 0T

. (17)

Since only the measurement of the motor side angle,

θ, is available, y = x1. With

f (x) =

x2

(J −1K − J −1 ˆK )(x1 − x3)x3 + x4

M (x3)−1K (x1 − x3) + x4

−M (x3)−1(C (x3, x4) + G(x3))

, (18)

the nonlinear function is

∆f = f (x, t) − f (x, t) =

∆f 1 ∆f 2 ∆f 3 ∆f 4T

. (19)

Note s = y = x1, the dynamics of the error of the

observer is

˙x = Ax + ∆f − Ls − K sgn(s), (20)

where

K =K 1 K 2 K 3 K 4

T ∈ R4n×n,

L =L1 L2 L3 L4

T ∈ R4n×n. (21)

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Equation (8) result in

V = sT s = sT (∆f 1 − L1x1 − K 1sgn(s)). (22)

Suppose K i and Li for i = 1, 2,...,n are further re-

stricted by

K i = kiI n and Li = liI n (23)where I n is the identify matrix of n-dimension. Then,

Lyapunov stability condition of Eq. (8) is satisfied if

k1 ≤ η − L1x1 + |x2|. (24)

where η is a positive constant gain.

The switching term can be compelled the system to

remain on the sliding surface in the side of model dis-

tinction and disturbance. In order to do that, k1 must be

positive. Therefore, Eq. (24) has been represented

k1 ≤ η + L1|x1| + |x2|. (25)

When the system is located on the sliding surface, the Eq.(20) is represented by

s = C ˙x = C (Ax + ∆f − Ls − K sgn(s)) = 0. (26)

Generally, x1 must be zero on the sliding surface. But, the

system may be leave sliding surface because the system

has parameter uncertainty and disturbance. Therefore, x1has not zero. If ignoring the effects of l1x1, sgn(x1) can

be represented as

sgn(x1) = (CK )−1C (∆f + Ax). (27)

From the Eq. (26)and Eq. (27), the error vector equa-

tion becomes

˙x = (I − G(CG)−1C )(∆f + Ax)

=

0 0 0 00 −(k2/k1)I n 0 00 −(k3/k1)I n −I n 00 −(k4/k1)I n 0 −I n

x +

0∆f 2∆f 3∆f 4

.(28)

If the sate of the system is bounded, the positive gains can

be set for stability of the observer despite the fact that the

information on x2, x3 and x4 is not available.

4. GAINS

The pair(A,C )

is detectable, i.e., there exists a matrix

L of appropriate dimensions such that the eigenvalue of

A0 = A − LC is completely positioned in the open left

half-plane. Therefore, the linear gain matrix L can be

assigned. In case of the nonlinear gain, which is assumed

that system is stable and that the state variables are thus

bounded.

Propterty 1: With a stable controller for flexible joint

robot manipulators,

||xi|| ≤ γ i for i = 1, 2, 3, 4.Propterty 2: From Property 1, and due to the limit in

the torque generated by the motors and enough strength

of the springs to withstand any torque generated by the

motors, the twist angles at the spring elements and lim-ited:

||x1 − x3|| ≤ αtwist,

where αtwist is a positive constants.

Furthermore, the followings are assumed for flexible

joint robot manipulators.

Propterty 3: The link inertia matrix M (q) is symmet-

ric, positive definite, and both M (q) and M −1(q) are

uniformly bounded as follows:||M (q)|| ≤ αM and ||M −1(q)|| ≤ αI

where αM and αI are positive constants.

Propterty 4: If suitably chosen, C (q, q) is uniformly

bounded such that

||C (q, q|| ≤ αc||q||

where αc is a positive constant.

Propterty 5: The gravitational term g(q) is uniformly

bounded such that

||g(q)|| ≤ αg

where αg is a positive constant.

Propterty 6: The stiffness matrix K is uniformlybounded as follows: where

||K || ≤ αK is a positive constant.

Consequently, the nonlinear observer gain can be de-

fined as

∆f 2 = (J −1K − J −1 ˆK )(x1 − x3)

≤ ||J −1K − J −1 ˆK || + ||x1 − x3||

≤ ξ 1||x1 − x3|| + ξ 2 := ρ1(x1, x3),

∆f 3 ≤ ξ 3||x3|| + ξ 4||x4|| + γ 3 + γ 4 := ρ2(x3, x4),

∆f 4 = M (x3)−1K (x1 − x3) − M (x3)−1(C (x3, x4)

+ g(x3)) + M (x3)−1K (x3 − x1)

+ M (x3)−1(C (x3, x4)x4 + g(x3)) + x4

≤ ||M (x3)−1K (x1 − x3) − M (x3)−1(C (x3, x4)

+ g(x3))|| + αI αK ||x3 − x1||

+ αI α||x4|| + αI αG + ||x4|| + γ 4

≤ ξ 5||x1 − x3|| + ξ 6||x4||2 + ξ 7

:= ρ3(x1, x3, x4), (29)

where ξ i for i = 1, ...,7 is a positive constant gain,

ρ1(x1, x3), ρ2(x3, x4) and ρ3(x1, x3, x4) are upper bound

function of ||∆f 2||, ||∆f 3|| and ||∆f 4||, respectively. Ac-

tually, the motor and link joint real velocity of system arebounded because the system is stable by the stable con-

troller. The gain for k1 is

k1 ≥ η + |l1x1| + |x2|desired upper bound. (30)

The unknown term x2 is selected to be the desired upper

bound of estimation error x2 from the assumption. Sim-

ilarly, nonlinear positive gain k2, k3 and k4 are can be

determined as follows

k2 = k1ρ1(x1, x3)

(x2)desired upper bound

,

k3 = k1ρ2(x3, x4)


,

k4 = k1ρ3(x1, x3, x4)


. (31)

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5. STABILITY

If the estimation state gain and constant gain in the

positive nonlinear gain function are choose small value

and large value, respectively, the overall system Eq. (28)

on sliding surface can be represented as

z(t) = Az(t) + ∆ f . (32)where

A =

−k2I n 0 0

−k3I n −I n 0−k4I n 0 −I n

,

z(t) =

x2(t)

x3(t)x4(t)

and ∆f =

∆f 2

∆f 3∆f 4

. (33)

The matrix A is stable, ki for i = 2, ...,4 is a constant

value and ∆f is the nonlinear function which represents

parameter uncertainty and estimation error. The stability

of overall system can be established by using V = zT P zas a Lyapunov function. It’s derivative function is

V = 2z(t)T P z(t)

= 2z(t)T P A + 2z(t)T P ∆f . (34)

The stability evaluation is presented in case of the

maximum size of uncertainty ∆f . Then, The nonlinear

function ∆f i for i = 2, ...,4 can be represented as

∆f 2 = (J −1K − J −1 ˆK )x3 + w1, (35)

∆f 3 = x2 + x3 + x4, (36)

∆f 4 = M (x3)−1 ˆK (x1 − x3) − M (x3)−1 ˆK (x1 − x3)

+ (M (x3)−1C (x3, x4)x4 − M (x3)−1C (x3, x4)x4)

+ (M (x3)−1G(x3) − M (x3)−1G(x3)) + w2, (37)

where the parameter distinction w1 and w2 are boundedfunctions as following that:

w1 = (J −1K − J −1 ˆK )(x1 − x3),

w2 = M (x3)−1 ˆK (x1 − x3) − M (x3)−1K (x1 − x3)

+ (M (x3)−1C (x3, x4) − M (x3)−1C (x3, x4))x4

+ (M (x3)−1G(x3) − M (x3)−1G(x3)). (38)

Each element of ∆f 4(t, x) is piecewise continuous in

t. Therefore, ∆f 4(t, x) satisfy the Lipschitz condition as

like

||f (t, x) − f (t, x)|| ≤ H ||x − x||, (39)

where H is the positive constant matrix. Therefore,

The parameter uncertainty of N-link flexible joint robotcan be consisted as following inequality

∆f ≤

f 2

f 3f 4

(40)

where f j =

||f ji ||, ..., ||f jn ||T

for i = 1, 2,...,n and

j = 2, 3, 4 is vector matrix and . Therefore, the each

norm is

||f 2i|| = h1I n||x3i|| + ||w1i||,

||f 3i|| = h2I n||x2i|| + h3I n||x3i|| + h4I n||x4i||,

||f 4i|| = h5I n||x3i|| + h6I n||x4i|| + ||w2i||. (41)

Then, Eq. (40) is arranged as

∆f ≤

0 h1I n 0h2I n h3I n h4I n

0 h5I n h6I n

||x2i||

||x3i||||x4i||

+

||w1i||

0||w2i||

. (42)

After all, the following inequality is satisfied

2zT PH

||x2i||

||x3i||||x4i||

+ 2zT PW (W :=

||w1i||

0||w2i||

)

≤ 2zT PH z + W T P T PW + zT z − (PW − z)2, (43)

whereH is the positiveconstant symmetric matrix. Now,the Lyapunov function Eq. (34) is represented as

V ≤ 2z(t)T P A + 2z(t)T PH z(t) + 2z(t)T PW

= z(t)T (P A + AT P + PH + I )z(t)

+ (PW )2 − (PW − z(t))2

≤ −λmin(Q)||z(t)||2 + λmax(P 2)W 2. (44)

LMI solver minimize following inequality and determine

positive constant matrix P and Q.

AT P + P A + PH + I + Q ≤ 0. (45)

Matrix A which is consisted of the positive nonlin-

ear constant gain. it relate with the optimal solution P and Q. The proper P and Q can be selected by chang-

ing the positive nonlinear constant gain value. The op-timal solution P does not effect to the observer system.

But, it just gives the confirmation for asymptotical sta-

bility whether the positive nonlinear constant gain is a

right or not. The disturbance vector is the parameter dis-

tinction and also bounded constant value. Therefore, the

proposed observer in during sliding becomes dissipative

system and asymptotically stable with W = 0. If the pos-

itive constant optimal matrix Q has the sufficiently large

value than the parameter distinction W , the proposed de-

sign method guarantees the robust performance.

6. SIMULATIONS

The performance and robustness of proposed sliding

observer are verified against the parameter uncertainty

and compared with the high gain observer. The simu-

lation executes the 2-DOF robot manipulators with joint

flexibility. The two links are paralleled to the surface of

land straightly at the initial pose. The first joint moves

180 degree in counterclockwise. The second joint moves

90 degree in counterclockwise. Table I shows the nomi-

nal values of the physical parameters of the 2-DOF robot

manipulators with joint flexibility. Table II indicates the

observer structure. SMO1 is the Sliding Mode Observer

Table 1 Physical parameters for the robot manipulatorswith joint flexibility

Symbol Dimension Symbol Dimension

m1 6 kg lc1 0.15 m

m2 4 kg lc2 0.15 m

l1 0.3 m k1 1500 Nm/rad

l2 0.3 m k2 1200 Nm/rad

which has the dynamic equation while SMO2 consider

only linear element in the observer structure. HGO is

the High Gain Observer which is consisted of the Luen-

herger observer. The estimation performance of proposed

observer is shown in the Fig. 1(A). The first, second, esti-mated first and estimated second link joint are dot, solid,

dash dot and dash line, respectively. The Fig. 1(B) shows

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Table 2 Observer Structure

Observers Structure

SMO1 ˙x = Ax + f (x) + Bu + LC x + K ∗ sgn(x1)

SMO2 ˙x = Ax + Bu + LC x + K ∗ sgn(x1)

HGO

˙x = Ax + Bu + LC x

0 0.5 1 1.5 2 2.5 3−20

0

20

40

60

80

100

120

140

160

180

Time (s)

J o i n t A n g l e ( d e g r e e s )

Real JT #1Real JT #2Estimated JT #1Estimated JT #2

A. Estimation

0 0.5 1 1.5 2 2.5 3

−0.5

0

0.5

x 10−10

Time (s)

J o i n t A n g l e E s t i m a t i o n E r r o r ( d e g r e e s )

JT #1JT #2

B. Estimation ErrorFig. 1 Link Joint Angle (SMO1).

the link joint estimation error.It is close to real state. The

Fig. 2(A) shows the phase portrait of proposed observer.

It is converged to the zero and the sliding mode observer

is located on the sliding surface.In case of the Fig. 2(B),

the system maintains the first pose and disturbance input

100Nm at 0.05 sec. In the Fig. 2(B), The observer sys-

tem has a relatively small linear gain than the large pos-

itive nonlinear gain. Then, the observer system is hardly

any influenced by the disturbance. The Fig. 3 showsthe comparison of other observers. The sliding mode ob-

servers with the dynamic equation is shown the good es-

timation performance as compare with HGO and SMO2.

Relatively, considering the dynamic equation has better

estimation performance than SMO2. The Fig. 5 shows

the robustness comparison of observers about the distur-

bance input and the sliding mode observer is more con-

vergence than the high gain observer at the second link

joint. Because, the joint flexibility has more influence to

the end link. So, there are shown the similar convergence

at the first link joint. But, at the second link joint, the

sliding mode observer has shown roughly twice as much

as the faster converging and smaller amplitude than thehigh gain observer. The Fig. 4 shows the estimation per-

formance of proposed observer about the sine input. The

−4 −3 −2 −1 0 1 2 3 4

x 10−12

−3

−2

−1

0

1

2

3x 10

−15

s

s °

JT #1JT #2

A. Phase Portrait

0 0. 01 0.02 0.03 0. 04 0. 05 0.06 0.07 0.08 0. 09−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time (s)

J o i n t A n g l e

E s t i m a t i o n E r r o r ( d e g r e e s )

JT #1 (SMO1)JT #2 (SMO1)

B. Estimation Error

Fig. 2 Link Joint Angle (SMO1).

Fig. 6 shows the estimation performance in comparison

of observers about the sine input. In high speed level

of motor side, the proposed observer is better estimation

performance than others.

7. CONCLUSION

The paper proposes a design method of sliding-mode

observer for the flexible joint robot manipulators. The

proposed method proved to be asymptotical stability on

during sliding and the robust performance using the pos-

itive nonlinear gain function. From the simulation result,

the proposed sliding observer hasbetter performance than

the high gain observer.

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0 0.5 1 1.5 2 2.5 3−4

−3

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0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

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0.1

Time (s)

J o i n t A n g l e E s t i m a

t i o n E r r o r ( d e g r e e s )


(Joint 1)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time (s)



(Joint 2)

Fig. 5 Link Joint Angle Estimation Error.

0 0.5 1 1.5 2 2.5 3−0.1

−0.05

0

0.05

0.1

0.15

Time (s)


JT #1 (SMO1)JT #1 (SMO2)

JT #1 (HGO)

(Joint 1)

0 0.5 1 1.5 2 2.5 3−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

Time (s)



(Joint 2)Fig. 6 Link Joint Angle Estimation Error.

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