Decentralized Neural Network Control for Guaranteed Tracking Error ...

International Journal of Control, Automation, and Systems (2015) 13(4):906-915 DOI 10.1007/s12555-014-0132-2

ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555

Decentralized Neural Network Control for Guaranteed Tracking Error Constraint of a Robot Manipulator Seong-Ik Han and Jang-Myung Lee* Abstract: In this paper, a new constrained error variable similar to sliding mode surface (SMC) is proposed to ensure a prescribed position tracking performance of a robot manipulator. A decentralized controller using this constrained error variable and a radial basis function network (RBF) is designed. The proposed decentralized and constrained control system ensures a prescribed transient and steadystate time positioning performance of the decentralized manipulator components without violation of the prescribed performance. The effectiveness of the proposed decentralized and robust control scheme was determined by comparing the results of simulated and experimental evaluation with the conventional SMC and finite-time terminal SMC methods. Keywords: Constrained error variable, decentralized control, prescribed performance function, RBF network, robot manipulator.

1. INTRODUCTION In the last three decades, robot technologies have been rapidly developing in the industrial, military, and medical areas. The control technology for robotic motion and force has developed closely along with the development of control theories. In particular, beginning in the 1990s, the control issue of constrained robotic motion within a limited space, and the operating force or torque used in special robotic operations, was researched as a geometric constraint problem such as the holomonic and non-holonomic constraint of mobile robots, and the end-effector of a contacted manipulator [1,2]. However, the constraint issue of robotic motion or force in a free space, rather than in a limited space, has been relatively less touched upon because a systematic or general method for such an issue is difficult to obtain. The motion and force constraints in a free space are also very important in the prevention of unexpected external collisions, hazardous contact between robots, medical accidents in surgical robots, and so on. The issue of constraint is a more complicated problem in a free space than in a geometrically constrained space. Most traditional control systems have designed controller guaranteeing the stability of the control system in the infinite time to minimize a norm of error. Adjusting several control gains should therefore follow to __________ Manuscript received March 27, 2014; revised August 25, 2014; accepted October 27, 2014. Recommended by Associate Editor Seul Jung under the direction of Editor Fuchun Sun. This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (NRF-2013R1A1A2021174). Seong-Ik Han and Jang-Myung Lee are with the Department of Electronic Engineering, Pusan National University, Busandaehakro 63 Beon-gil, Geumjeon-gu, Busan 609-735, Korea (e-mails: {skhan, jmlee}@pusan.ac.kr). * Corresponding author. © ICROS, KIEE and Springer 2015

simultaneously satisfy the required transient and steadystate performance. However, for optimal control gain, only trial and error has been conducted to obtain the desired control performance. It is therefore very difficult to provide systematic and general constraint control performance using these conventional control schemes. Three new methods that consider a guaranteed timedomain performance in the design step without depending excessively on the control gain have recently been studied: a funnel control [3] as a high-gain, nonmodel-based constraint control; a transformation method using a tangent hyperbolic function for the tracking error [4]; and a state constraint method adopting a barrier Lyapunov function (BLF) [5]. However, there are some drawbacks to these methods. A funnel control application is limited in class S of linear or nonlinear systems with a relative degree of 1 or 2, stable zero dynamics, and the known sign of the high-frequency gain. A singularity problem may occur in the error transformation method [5] owing to the adoption of a tangent hyperbolic function in the prescribed function. In a BLF constraint method, because a piecewise smooth BLF was also adopted in an asymmetric BLF design, extra effort is required to ensure the continuity and differentiability of the piecewise smooth stabilizing functions. As described in this paper, a prescribed error constraint approach was developed by using a new tracking error constrained variable based on the basic concept of funnel control. A control scheme using this constrained error variable guarantees the prescribed tracking performance and can compensate for an unknown nonlinear function of a decentralized robotic manipulator [6] while considering the RBF network approximation process [7,8] without requiring information regarding an unknown nonlinear robotic dynamics. The SMC technique provides robust nonlinear control because it gives system dynamics with invariant

Decentralized Neural Network Control for Guaranteed Tracking Error Constraint of a Robot Manipulator

properties to uncertainties once the system dynamics are controlled in sliding mode surface [9-12]. The finite-time terminal sliding mode controllers (FSMC) [13-15] and conventional SMC were designed to obtain comparative simulated and experimental results with a proposed constrained control. The proposed control approach was applied successfully to the constrained free-spaced motion control of a robot manipulator in the presence of uncertainty. 2. PROBLEM FORMULATION

Fui (qi , qi , qi ) = −M ni−1 (qi )[ΔM i qi + Ci (qi , qi )qi + Gi (qi ) + Ri (qi , qi , qi ) + F fi (qi , qi ) + Fdi (qi , qi )]

is a bounded nonlinear function such that Fui ≤ ρui and ρui is a positive constant. 2.2. Symmetric tracking error constrained variable By setting the joint tracking error as e = q − qd , a symmetric error constrained variable, rs(t), can be defined as T

2.1. Dynamics of robot manipulator A dynamic equation of an n rigid-link robot system can be described as follows: M (q )q + C (q, q )q + G (q ) + F f (q, q ) + Fd = u ,

(1)

where q, q , q ∈ R n×1 are the joint position, velocity and acceleration vectors, respectively, M (q) ∈ R n×n represents a positive definite symmetric moment of the inertia matrix, C (q, q ) ∈ R n×n is a centripetal Coriolis matrix, G (q) ∈ R n×1 is a gravity vector; F f (q, q ) ∈ R n×1 is a nonlinear friction torque vector, Fd ∈ R n×1 is an external disturbance torque, and u ∈ R n×1 is the control torque vector from the joint actuators. To apply decentralized control, equation (1) can be formulated in a joint space as follows: M i (qi )qi + Ci (qi , qi )qi + Gi (qi ) + Ri (qi , qi , qi ) + F fi (qi , qi ) + Fdi (qi , qi ) = ui , i = 1,..., n,

(2)

where an interconnection is defined as ⎡ n ⎤ Ri (q, q , q) = ⎢ ∑ M ij (q )q j + ( M ii (q ) − M i (qi ) ) qi ⎥ ⎢⎣ j =1, j ≠i ⎥⎦ ⎡ n ⎤ + ⎢ ∑ Cij (q, q )q j + Cij (q, q ) − Ci (qi , qi ) qi ⎥ , (3) ⎢⎣ j =1, j ≠i ⎥⎦

(

)

qi , qi , qi , Gi (q ), and ui are the ith element of the vectors q, q , q, G (q), and u, respectively, M ij (q ) and Cij (q, q ) are the ijth element of matrices M(q) and C (q, q ), respectively. The symmetric and positive definite inertia matrix, Mi(qi), is bounded by

ζ i ≤ M i (qi ) ≤ ζ i < ∞ for all qi ,

(4)

where ζ i and ζ i are positive constants. M i (qi ) −2Ci (qi , qi ) is a skew-symmetric matrix. In addition, Ci (qi , qi ) , Gi (qi ) and Ffi are bounded nonlinear functions. The disturbance, Fdi, is bounded by a positive constant ρdi : Fdi ≤ ρ di . As a consequence, (2) can be expressed as qi = Fui (qi , qi , qi ) + M ni−1ui ,

(5)

where Mi (qi ) = Mni (qi ) + ΔMi (qi ), M ni (qi ) and ΔM i (qi ) denote a nominal value and a perturbation of M i (qi ), respectively,

907

rs = [ rs1 (Fs1 ),..., rsn (Fsn ) ] ,

(6)

where rsi =

ki ei + ei , i = 1,..., n. Fsi − ei

(7)

ei = qi − qdi , ei = qi − qdi , qdi are the desired output trajectories for each joint, and ki > 0 is the design constant. A symmetric error boundary function, Fsi , is defined as follows: Fsi (t ) = ( ρ0i − ρ ssi )exp(− ai t ) + ρ ssi .

(8)

The error variable defined in (7) is employed to ensure the prescribed output performance of the control scheme. 2.3. Asymmetric error constrained variable An asymmetric prescribed error bound condition can alleviate an excessive temporal overshoot that appears in the symmetric error constraint although the error performance satisfies the error boundary function. An asymmetric error boundary function to describe the asymmetric performance can be defined as follows: Fapi (t ) = ( ρ0 pi − ρ sspi )exp (−a pi t ) + ρ sspi ,

(9)

Fani (t ) = ( ρ0 ni − ρ ssni )exp(−ani t ) + ρ ssni , Fai (t ) = Fapi (t )q + Fani (t )(1 − q ) ,

(10) (11)

where ⎧1 if ei (t ) ≥ 0 q(ei (t )) = ⎨ ⎩0 if ei (t ) < 0 .

An asymmetric error constrained variable, ra(t), can then be defined as T

ra = [ ra1 (Fa1 ),..., ran (Fan ) ] ,

(12)

where rai =

ki ei + ei , i = 1,..., n . Fai − ei

(13)

Remark 1: In (8) and (11), the constants error, ρssi, confines the size of the tracking errors under a steady state. The decreasing rate of ai regulates the required speed of convergence of the tracking errors, and the maximum overshoot and undershoot can be controlled by selecting ρ0i (0).

Seong-Ik Han and Jang-Myung Lee

908

Remark 2: In (7) and (13), if the tracking error | ei | approaches the prescribed error bounds, Fsi and Fai , i.e., | ei |→ Fsi and | ei |→ Fsi , then rsi → ∞ and rai → ∞. control gain components ui = −ki ei /(Fsi − | ei |) and ui = −ki ei /(Fai − | ei |), which are specified in (22) and (29), therefore increases, and conversely, as errors become small, gain components ui = −ki ei /(Fsi − | ei |) and ui = − ki ei /(Fai − | ei |) decrease. This affects the control actions such that the tracking error evolvements are suppressed using control actions such as ui = −ki ei /(Fsi − | ei |) and ui = −ki ei /(Fai − | ei |). Therefore, the prescribed tracking problem of the robot manipulator in the joint space can be solved using the constrained errors in (6) and (11) such that the error states remain within the prescribed constraint bound as the time approaches infinity. This property is shown in Fig. 2(i) and (j). 2.4. RBF network The RBF network has been widely applied in many engineering fields. The RBF network is fully connected three-layered feed-forward network. The response characteristics of the jth hidden unit j are assumed to be a Gaussian function ⎛ X −m j φ j = exp ⎜ − 2 ⎜⎜ 2σ j ⎝

2

⎞ ⎟, ⎟⎟ ⎠

(14)

where X denotes an input variable of the RBF network, mj and σj are the mean and standard deviation, respectively, of the jth unit receptive field, and the norm is Euclidean. The output of the RBF network is given by n

y( X ) =

∑w φ (X ) j j

= W fTΦ ( X )

,

(15)

1) Determine the control laws such that the system output q can track the desired continuously differentiable and uniformly bounded trajectory qd in the joint space while ensuring all closed-loop signals are bounded. 2) The prescribed constraints for the output tracking error, ei = qi − qdi , i = 1,..., n , are not violated despite the presence of an unknown function, Fui (qi , qi , qi ) , i = 1,..., n . Assumption 1: The state variables, qi , are available from the measurement and ei = qi − qdi and | ei | = | qi − qdi | can be constructed. Therefore, ei does not indicate d ei / dt but is an absolute value of ei . Remark 3: ei exists at t = 0 only for mechanical system such as a robotic system because ei means a velocity error and ei denotes the absolute of a velocity error. On the contrary, in a general case, ei is discontinuous at t = 0 and thus d ei / dt is impossible at t = 0 . Therefore, the application of a proposed constraint variable is limited to a mechanical system. The time derivative of (6) under assumption 1 can then be expressed as T

rs = [ rs1 (Fs1 ), , rsn (Fsn ) ]

⎡ e1 + Θ s1 (e1 , e1 , Fs1 ) + Λs1 (e1 , e1 , Fs1 , Fs1 ) ⎤ ⎢ ⎥ =⎢ ⎥, )⎥ ⎢⎣en + Θ sn (en , en , Fsn ) + Λsn (en , en , Fsn , Fsn ⎦

where Θ si (ei , ei , Fsi ) = ki

ei , Fsi − ei

Λsi (ei , ei , Fsi , Fsi ) = ki

j =1

T

where W f = [w1 ,..., wn ] and Φ = [φ1,...,φn ]T . The function f (x) can then be expressed as follows: f ( x) = W f*T Φ ( x) + ε * , ∀x ∈ Ω ⊂ R n ,

(16)

where | ε * |≤ ε m , ε * is the error of the RBF approximation and W f* is the optimal value of Wf that minimizes the RBF approximation error ε * . Therefore, W f* = arg min

W ∈R N ×1

{sup

x∈Ω

}

f ( x) − Wˆ fT Φ ( X ) .

(17)

* Because W f is unknown, it is replaced with Wˆ f , * which is an estimation of W f .

3. DESIGN OF DECENTRALIZED CONSTRAINT CONTROLLER 3.1. Design of a decentralized symmetric constraint controller with the RBF network The control objectives of the robot manipulator are as follows:

(18)

ei (Fsi − ei

(Fsi − ei )

)

2

.

An unknown function of robotic dynamics can be expressed using the RBF network as follows: Fui = W fi*T Φ fi + ε fi ,

(19)

* where the approximation errors of | ε fi | ≤ σ i are * bounded. On the other hand, the estimates, Wˆ fi of W fi * and σˆ i of σ i* , are considered because W fi and σ i* * cannot be known in advance. Define W fi = W fi − Wˆ fi * and σ i = σ i − σˆ i . The Lyapunov function candidates are also defined as follows: n ⎛ ⎞ 1 1 1 V = ∑ ⎜ rsi2 + W fiT Γ −fi1W fi + σ i2 ⎟ , 2 2ησ i ⎠ i =1 ⎝ 2

(20)

where Γ fi = diag (η fi ) > 0, η fi > 0 , ησ i > 0 , i = 1,..., n , are the design constants. The time derivative of V can be written as n ⎛ ⎞ 1 V = ∑ ⎜ rsi rsi + W fiT Γ −f 1W fi + σ iσ i ⎟ ησ i ⎠ i =1 ⎝

Decentralized Neural Network Control for Guaranteed Tracking Error Constraint of a Robot Manipulator n

= ∑ rsi ⎡⎣ M ni−1ui + Θ si + Λ si + Fui − qdi ]

Θ ai (ei , ei , Fai ) = ki

i =1

n n 1 −∑ W fiT Γ −f 1Wˆ fi −∑ σ iσˆ i η i =1 σ i i =1

ei , Fai − ei

) = ki Λai (ei , ei , Fai , Fai

n

≤ ∑ rsi ⎡⎣ M ni−1ui + Θ si + Λ si +Wˆ fiT Φ fi + σˆ i − qdi ⎤⎦ i =1

(

n + ∑ W fiT rsi Φ fi − Γ −f 1Wˆ fi i =1 n

1 σ iσˆ i . η i =1 σ i

ui = M ni (qi ) ⎡⎣ −Wˆ fiT Φ fi − ki rsi + qdi − Θ si − Λ si σˆ r ⎤ − i si ⎥ , i = 1,..., n, rsi + ς i ⎥⎦

(23)

⎛ rsi2 ⎞ − ησ′ iσˆ i ⎟ , i = 1,..., n , σˆ i = ησ i ⎜ ⎜ r +ς ⎟ i ⎝ si ⎠

(24)

)

where η ′fi > 0 , ησ′ i > 0 , and ς i > 0 are the design constants. The following inequality can be easily obtained: rsi2 − ≤ − rsi + ς i , i = 1,..., n . rsi + ς i

(25)

n

n

n

n

i =1

i =1

i =1

i =1

V ≤ −∑ ki rsi2 + ∑η ′fiW fiT Wˆ fi + ∑ησ′ iσ iσˆ i + ∑ σ i*ς i

i =1

i =1

2

n ησ′ iσ i2

−

−∑ i =1

ρˆi rai ⎤ ⎥, rai + ς i ⎥⎦

i = 1,..., n,

(29)

Wˆ fi = Γ fi raiΦ fi − η ′fiWˆ fi , i = 1,..., n ,

(30)

⎛ rai2 ⎞ − ησ′ iσˆ i ⎟ , i = 1,..., n . σˆ i = ησ i ⎜ ⎜ r +ς ⎟ i ⎝ ai ⎠

(31)

(

)

3.3. Proof of stability of the closed-loop control system By selecting the following control gains ki = K i , i = 1,..., n , η ′fi = 2 K i , i = 1,… , n ,

(32)

2

+µ ,

where Ki = min[ki , η ′fi , ησ′ i ], (26) can then be expressed as n

n

n

i =1

i =1

i =1

V ≤ −∑ Ki rsi2 −∑ KiW fiT W fi −∑ Kiσ i2 + μ

(26)

where

⎛ η ′fiW fi*T W fi* η ′ σ *2 ⎞ + σ i i + σ i*ς i ⎟ . μ = ∑⎜ ⎜ ⎟ 2 2 i =1 ⎝ ⎠ n

3.2. Design of a decentralized asymmetric constraint controller with the RBF network qi can be obtained from a measurement based on assumption 1, the time derivative of (12) can be expressed as T

ra = [ rs1 (Fa1 ), , ran (Fan ) ]

⎡ e1 + Θ a1 (e1 , e1 , Fa1 ) + Λa1 (e1 , e1 , Fa1 , Fa1 ) ⎤ ⎢ ⎥ =⎢ ⎥ , (27) )⎥ ⎢⎣ en + Θ an (en , en , Fan ) + Λan (en , en , Fan , Fan ⎦

where

ui = M ni (qi ) ⎡⎣ −Wˆ fiT Φ fi − ki rai + qdi − Θai − Λ ai

ησ′ i = 2 K i , i = 1,… , n ,

Considering (22)-(25), (21) can be expressed as follows:

≤ −∑ ki rsi2 −∑

The control law and adaptive estimation laws for an asymmetric error-bound function can therefore be specified similarly like a symmetric case as follows:

(22)

Wˆ fi = Γ fi rsiΦ fi − η ′fiWˆ fi , i = 1,..., n ,

n

) = M ni−1 (qi )ui + Θ ai (ei , ei , Fai ) + Λai (ei , ei , Fai , Fai + Fui (qi , qi , qi ) − qdi , i = 1,..., n . (28)

(21)

The control law and adaptive estimation laws are specified as

T n η′ W fi fi W fi

.

) rai = ei + Θ ai (ei , ei , Fai ) + Λai (ei , ei , Fai , Fai

n

(

(Fai − ei )

)

2

Considering (5), (27) can be written as follows:

)

+ ∑ rsi σ i −∑ i =1

− ei ei (Fai

909

= − KV + µ ,

(33)

where K = min[2 K1 , , 2 K n ]. Integrating (33) over [0, t ] leads to µ⎞ µ µ ⎛ 0 ≤ V ≤ ⎜ V (0) − ⎟ e − Kt + ≤ V (0)e − Kt + . K⎠ K K ⎝

(34)

From (20), the constrained variables can be represented further as 1 n 2 µ rsi ≤ V (0) + , ∑ K 2 i =1

(35)

We can then have n

∑ rsi i =1

µ⎞ ⎛ ≤ 2 ⎜ V (0) e − Kt + ⎟ . K⎠ ⎝

(36)

* Next, from W fi = W fi − Wˆ fi and σ i = σ i* −σˆ i , we can similarly obtain


910

(

μ⎞ ⎛ ≤ 2λi max (Γ fi ) ⎜ V (0) e− Kt + ⎟ + W fi* , max K ⎝ ⎠ i = 1,..., n, (37)

μ⎞ ⎛ σˆ i ≤ 2ησ i ⎜ V (0) e − Kt + ⎟ + σ i* K⎠ ⎝ μ⎞ ⎛ , ≤ 2ησ i max ⎜ V (0) e − Kt + ⎟ + σ i* max K⎠ ⎝ i = 1,..., n. (38)

Therefore, rsi , Wˆ fi and σˆ i , i = 1,… , n , are bounded. It has been progressively shown that the control inputs, ui (rsi , Wˆ fi , σˆ i ), i = 1,..., n are bounded. It was therefore shown that all signals are bounded. Next, from V and (34), we can obtain the following expression: 2

⎞ µ ⎞ − Kt µ ⎛ ei 1 2 1⎛ + ei ⎟ ≤ ⎜ V (0) − ⎟ e + . rsi = ⎜ ki ⎟ K⎠ K ⎝ 2 2 ⎜⎝ Fsi − ei ⎠ (39) The following can then be obtained: 2

⎛ ei ⎞ μ ⎞ − Kt μ ⎛ ⎜⎜ ⎟⎟ ≤ ⎜ V (0) − ⎟ e + +Ψ (ei , ei , Fsi ) , K K ⎝ ⎠ ⎝ Fsi − ei ⎠ (40) ei ei 2 + ei . Some manipulating where Ψ (ei , Fsi ) = 2ki Fsi − ei (40) yields ki2 2

2 (Fsi − ei

)

μ ⎞ − Kt μ ⎛ ⎜ V (0) − ⎟ e + +Ψ (ei , ei , Fsi ) . ki K⎠ K ⎝ (41) From ei < Fsi , as t → ∞ , (41) can be expressed as follows: ei ≤

ei ≤ ≤

2 (Fsi − ei ki 2 ki

)

μ +Ψ (ei , ei , Fsi ) K

μ +Ψ (ei , ei , Fsi ), i = 1,..., n . K

(42)

Therefore, the joint tracking errors ei (t ) can be arbitrarily decreased into smaller values by controlling the design parameters, ki and K. For an asymmetric constraint case, the same results for the stability analysis can be also obtained. Remark 4: If a conventional SMC with the RBF network is considered, the sliding surface is defined as

r = ke + e ,

)

ui = M ni (qi ) −Wˆ fi Φ fi − ki ri + qdi − γ i sat (ri ) ,

μ⎞ ⎛ Wˆ fi ≤ 2λi (Γ fi ) ⎜ V (0) e− Kt + ⎟ + W fi* K ⎝ ⎠

(43)

where r = [r1 ,..., rn ]T and k = diag (k1 ,..., ki ,..., kn ), ki > 0, is a design parameter. The controller can be specified using the adaptive laws given in (23) as follows:

i = 1,..., n, (44)

where γ i > 0, i = 1,..., n, are constants. Remark 5: The continuous finite-time SMC (FSMC) with the RBF network was considered to guarantee a rapid convergence time compared with a conventional SMC and the proposed constrained control. The finitetime sliding surface is defined as r = e + β sig (e)γ ,

(45)

where r = [r1 ,..., rn ]T , β = diag ( β1 ,… , β n ), 1 < γ 1 ,…γ n < 2,

and sig (e)γ = [ e1

γ1

sign(e1 ),..., en

γn

sign(en )]T .

The FSMC-type reaching law is defined as r = − k1r − k2 sig (e)ζ ,

(46)

where k1 = diag (k11 ,… , k1n ) , k2 = diag (k21 ,… , k2 n ) , k1i, k2i > 0, 0 < ζ = ζ 1 = = ζ n < 1 . For a rigid n-link manipulator in (2), if the FSMC manifold is chosen as (45), the model-based continuous control law of FSMC is designed as

(

ui = Fui (qi , qi , qi ) − M ni (qi ) qdi + βi−1γ i−1sig (ei )2−γ i

)

+ k1i ri + k2i sig (ri )ζ ,

i = 1,..., n.

(47)

If the RBF approximation is considered, (47) can be expressed as follows:

(

ui = Wˆ fiΦ fi − M ni (qi ) qdi + βi−1γ i−1sig (ei )2−γ i

)

+ k1i ri + k2i sig (ri )ζ , i = 1,..., n.

(48)

4. APPLICATION In this section, a proposed control scheme is evaluated with application to the Scorbot robot system described in Fig. 1 via simulation and experiment. Among the four links of the Scorbot robot manipulator, only two links (upper arm = link1 and forearm=link2) were selected. From (1), the dynamics and parameters for two DOF (degree-of-freedom) links of the Scorbot robot manipulator are described in [16]. 4.1. Simulation for a sine-wave joint position tracking In simulation, three controllers were deigned to evaluate the proposed control system: a conventional decentralized RBF sliding mode controller (SMC) given in (44), decentralized finite-time RBF sliding mode controller (FSMC) given in (48), and the proposed decentralized constrained RBF controller (PCC) given in (22) for the symmetric error constraint. The selected sine -wave qd (t ) = 0.1sin(π t ) (rad) was as the desired trajectory command. The chosen controller parameters are


911

The sine-wave joint motion was chosen as the desired trajectory. The selected sine-wave position command was qdi (t ) = 0.1sin(2.5132t ) (rad). The chosen controller parameters for SMC and FSMC are k1 = 100, k2 = 100, k11 = 1, k12 = 1, η f 12 = 1, η f 22 = 1, η ′f 12 = 0.001, η ′f 22 = 0.001, ησ 12 = 0.2, ησ 22 = 0.2, ησ′ 12 = 0.1, and ησ′ 22 = 0.1, β1 = β 2 = 1, γ 1 = γ 2 = 5 / 3, ζ 1 = ζ 2 = 0.5, and for PCC are k1 = 10, k2 = 10. The remaining parameters are the same as SMC and FSMC. The tracking-error performance functions selected for a symmetric constraint are Fs1 (t ) = (0.2 − 0.004)exp (−t ) + 0.004 if e11 (0) < 0 ,

Fig. 1. Photograph of the Scorbot manipulator. k1 = 5,10, 15, k2 = 5,10, 15, k11 = k12 = 1, k21 = k22 = 1, β1 = β 2 = 1, γ 1 = γ 2 = 5 / 3, ζ 1 = ζ 2 = 0.5, η f 12 = 0.1, η f 22 = 0.1, η ′f 12 = 0.001,η ′f 22 = 0.001, ησ 12 = 0.1, ησ 22 = 0.1, ησ′ 12 = 0.001, and ησ′ 22 = 0.001. The tracking error performance functions selected for a symmetric constraint are

Fs1 (t ) = (0.25 − 0.01)exp (−t ) + 0.01 if e11 (0) < 0, Fs 2 (t ) = (0.25 − 0.01)exp(−t ) + 0.01 if e21 (0) ≥ 0.

(49)

In Fig. 2, the simulation results are presented. In the PCC system, the tracking errors are insensitive to variations of the control gains and the prescribed error constraints are satisfied as shown in Fig. 3(e) and (f), while the magnitude of the control inputs of the PCC system is smaller than those of other systems as shown in Fig. 3(g) and (h) and the prescribed error constrains are violated. On the other hand, the tracking errors of the SMC and FSMC systems are varied according to the selected values of the control gain. As mentioned in Remark 2, the compensation effects of the error evolvement are presented in Fig. 3(i) and (j), where it is seen that the control components, −ki ei /(Fsi − ei ), suppress the tracking error by the reverse directional actions. 4.2. Experiment for a sine-wave joint position tracking In experiment, the three controllers were deigned to evaluate the proposed control system like simulation: SMC, FSMC, and PCC with the symmetric (22) and (29) asymmetric constraints. The designed controllers generated in the computer were implemented on the Matlab RTI system using an MF624 board [17]. The control signals were transferred to the DC servo motor of the Scorbot robot through the servo drive. The sample time was selected as 1 kHz. The torques of DC motor are obtained by the voltage as follows: nk nk τ i = i ti Vi − i ti kbi qi , Rmi Rmi

(50)

where Vi is the input voltage of the roll axis motor, ni = 65.5 is motor gear ratio, Rmi = 0.83Ω is the resistor coefficient of motor, kti = 0.0182 Nm/A denotes the torque constant, and kbi = 0.0182 V/rad/sec denotes the back emf constant.

Fs 2 (t ) = (0.2 − 0.0025)exp(−t ) + 0.0025 if e21 (0) ≥ 0 , (51) and for an asymmetric case are Fap1 (t ) = (0.05 − 0.0025)exp ( −t ) + 0.0025 ,

Fan1 (t ) = (0.2 − 0.01)exp (−t ) + 0.01 , Fap 2 (t ) = (0.2 − 0.008)exp ( −t ) + 0.008 ,

(52)

Fn 2 (t ) = (0.05 − 0.002)exp (−t ) + 0.002 .

The initial points of each state selected were q1(0) = – 0.05 rad and q2(0) = 0.05 rad. In the tracking outputs of link1 shown in Fig. 4(a), the tracking performance of FSMC was slightly improved compared to SMC. In Fig. 4(c), however, the tracking performance of both control systems violated the prescribed error bound, although the control gains selected for SMC and FSMC were 10-times that of PCC. On the other hand, the tracking errors of PCC fell within the prescribed error bound. For link 2, the constraint performance of PCC was satisfied as shown in Fig. 5(c). On the other hand, the tracking errors of the SMC and FSMC systems violated the prescribed error bound. For the asymmetric performance constraint, Fig. 6 shows that the conditions of the prescribed error performance given were satisfied. Therefore, the proposed PCC system can satisfy the given prescribed symmetric and asymmetric performance condition better than the conventional SMC and FSMC methods. 4.3. Experiment for the end-point circle trajectory tracking The performance constraint problem for an end-point trajectory tracking of the manipulator is next described. The direct kinematics for a circle trajectory in the task space can be given by ⎡ L sin(q1 ) + L2 sin(q1 + q2 ) ⎤ C r (q ) = ⎢ 1 ⎥. ⎣ L1 cos(q1 ) + L2 cos(q1 + q2 ) ⎦

(53)

In addition, the desired end-effector trajectory of the manipulator can be expressed as ⎡ x ⎤ ⎡ x + R cos(ω × t ) ⎤ Yd (t ) = ⎢ d ⎥ = ⎢ c ⎥, ⎣ yd ⎦ ⎣ yc + R sin(ω × t ) ⎦

(54)

where xc = yc = −0.1 m, R = 0.01 m and ω = 0.25 rad/ sec. This trajectory makes the manipulator tip trace a circle in the x0 − y0 plane with a radius of R = 0.01 m.

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(a) Tracking errors in link 1 of SMC.

(b) Tracking errors in link 2 of SMC.

(c) Tracking errors in link 1 of FSMC.

(d) Tracking errors in link 2 of FSMC.

(e) Tracking errors in link 1 of PCC.

(f) Tracking errors in link 2 of PCC.

(g) Control inputs in link 1 of SMC, FSMC and PCC at k1 = 5.

(h) Control inputs in link 2 of SMC, FSMC and PCC at k2 = 5.

(i) e1 × 400 and −k1e1 /(Fs1 − | e1 |).

(j) e2 × 400 and −k2 e2 /(Fs 2 − | e2 |).

Fig. 2. Simulation results.


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Fig. 3. Diagram of the Scorbot robot control system.

(a) Tracking output of SMC, FSMC, and PCC.

(a) Tracking outputs of SMC FSMC, and PCC.

(b) Tracking errors of SMC, FSMC, and PCC.

(b) Tracking errors of SMC, FSMC, and PCC.

(c) Control inputs of SMC, FSMC, and PCC. Fig. 5. Position tracking results of the link2 for the symmetric constraint.

(c) Control inputs of SMC, FSMC, and PCC. Fig. 4. Position tracking results of link1 for a symmetric constraint. The controller parameters and selected tracking error performance functions chosen were the same as those used in sine-wave tracking. The initial points selected of each state for each state were q1 (0) = −4.7 rad and q2 (0) = 2.9 rad. The parameters of the prescribed performance functions were selected for two cases: Fa1 (t ) = (0.2 − 0.002)exp (−4t ) + 0.002 , Fa 2 (t ) = (0.2 − 0.002)exp (−4t ) + 0.002 .

(55)

The circle-tracking outputs and errors are presented in Fig. 7, where the circle-tracking performance of the proposed PCC system was satisfied for the prescribed performance constraint condition. On the other hand, the results in Figs. 7(b) and (c) show that SMC and FSMC violate the prescribed constraint, and the circle tracking performance deteriorates under the same control parameter conditions as the PCC system. 5. CONCLUSION A decentralized error constrained control scheme was developed to ensure the position-tracking performance of a robotic manipulator in the presence of an unknown uncertainty. A tracking error constrained variable was proposed to ensure the prescribed tracking error performance, as well as an effective compensation for the decentralizing uncertainty. The proposed control


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(a) Link1.

(b) Link2.

(a) Circle tracking outputs of SMC, FSMC, and PCC.

(b) Joint tracking errors of SMC, FSMC, and PCC in link1.

Fig. 6. Position tracking errors of PCC for the asymmetric constraint. scheme satisfies the prescribed tracking performance of a robotic system with the RBF approximation for an unknown nonlinear function. Therefore, the designed controller has a simple structure, and is robust to uncertainties in the robotic manipulator positioning. Simulation and experimental evaluations of the Scorbot manipulator for sine-wave and circle trajectory tracking commands highlight the satisfactory prescribed positiontracking performance of the proposed control scheme.

(c) Joint tracking errors of SMC, FSMC, PCC in link2. Fig. 7. Circle command tracking results.

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Seong-Ik Han received his B.S. and M.S. degrees in Mechanical Engineering from Pusan National University, Busan, Korea, in 1987 and 1989, respectively, and his Ph.D. in Mechanical Design Engineering from Pusan National University, Busan, in 1995. From 1995 to 2009, he was an associate professor of electrical automation of Suncheon First College, Korea. Now he is with the Department of electronic engineering, Pusan National University, Korea. His research interests include intelligent control, nonlinear control, robotic control, vehicle system control, and steel process control. Jang-Myung Lee received his B.S. and M.S. degree in electronic engineering from Seoul National University, Seoul, Korea, in 1980 and 1982, respectively, and his Ph.D. degree in computer engineering from the University of Southern California, Los Angeles, in 1990.Since 1992, he has been a Professor with Pusan National University, Busan, Korea. He was the Leader of the “Brain Korea 21 Project” of Pusan National University. His research interests include intelligent robotics, advanced control algorithm, and specialized environment navigation/localization. Prof. Lee was the former president of the Korean Robotics Society.