Arab J Sci Eng DOI 10.1007/s13369-016-2221-4

RESEARCH ARTICLE - COMPUTER ENGINEERING AND COMPUTER SCIENCE

Robust Proportional Control for Trajectory Tracking of a Nonlinear Robotic Manipulator: LMI Optimization Approach Ali Hussien Mary1 · Tolgay Kara2

Received: 27 November 2015 / Accepted: 11 May 2016 © King Fahd University of Petroleum & Minerals 2016

Abstract This paper proposes a new control configuration that is simple, model free and robust for trajectory tracking control of a multi-input–multi-output nonlinear robotic manipulator system. The proposed controller consists of two terms. The first term is a linear controller in proportional (P) control structure, and the second term is a nonlinear robustness term in sliding mode structure. This combined nonlinear controller exploits the simplicity and easy implementation properties of proportional-integralderivative control and robustness properties of the sliding mode control (SMC) against system uncertainties and parameter variations. Important feature of the proposed controller is avoiding the need to determine the accurate dynamic model of the plant, which is a necessity in standard SMC. Stability analysis is performed, and stability in closed loop is proved for the proposed control method. A control problem is restated as a convex optimization problem based on linear matrix inequality technique, and optimal gain of P controller is obtained. A simulation model of the plant is built in MATLAB–Simulink environment for testing the proposed controller. Closed-loop system performances are observed for standard SMC, computed torque control, SMC with proportional-derivative control and proposed control. Simulation results reveal the effectiveness of proposed method in response to system uncertainties, random noise and external disturbance.

B

Ali Hussien Mary [email protected] Tolgay Kara [email protected]

1

Al-Khwarizmi College of Engineering, University of Baghdad, Baghdad, Iraq

2

Department of Electrical and Electronics Engineering, University of Gaziantep, Gaziantep, Turkey

Keywords Sliding mode control · Model-free control · Robotic control · Proportional control · Nonlinear robotic manipulator · Linear matrix inequality

1 Introduction Robotic manipulators have been widely and successfully used in various fields like surgical manipulators, military applications, process industries and many more [1]. The good performance and accurate tracking are important features of these systems. A robotic manipulator is a MIMO system with hard nonlinearities and strong coupling among joints. Therefore, control problem of a robotic manipulator with the aim of good error performance and accurate tracking has drawn an increasing attention in recent years [2]. In industrial applications standard PID controller is still widely used for reasons such as easy implementation, modelfree design, easy tuning and low cost. Proportional-derivative (PD) controller is applied successfully in CTC with guaranteed global and asymptotic stability. However, such design is based on prior knowledge of system dynamics. The variations of robotic manipulator and external disturbances make this controller unable to provide good tracking performance [3,4]. Some weak points of conventional PID controller can be eliminated by hybridizing it with new control strategies such as fuzzy control, neural network (NN) control or robust control [3]. Presented for the first time by Emelyanov in 1967, SMC is a variable structure control strategy commonly preferred in robotics for its robustness to system uncertainties and external disturbances [4]. SMC is among the most effective and efficient approaches in solving control problems of nonlinear systems. In addition to widespread successful applications to power converters, automatic flight control and chemical processes, many researchers used different types of

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SMC for tracking problems of robotic manipulators as well [4–8]. One important problem associated with SMC of robotic manipulators is the difficulty in determining equivalent control, which requires exact dynamic model of the plant. Sometimes it is impossible to obtain accurate dynamic model of the manipulator due to its high complexity. A number of approaches have been proposed to solve this problem [9]. Combining SMC with adaptive control strategy is one of suggested solutions [10–14]. Adaptive control system is utilized to estimate and compensate for variations in dynamics of the system, and SMC is used to overcome the uncertainties and external disturbance [15]. NN has also been used in combination with SMC for its ability to approximate nonlinear functions [16–19]. In literature a controller is proposed based on terminal SMC and fuzzy control in order to estimate dynamics of the plant [20]. Due to difficulties in estimating dynamic model of the controlled system, this paper aims to suggest a simple, model free and robust controller for trajectory tracking of a robotic manipulator. This controller essentially combines the simplicity of a P controller and robustness of SMC. Based on the literature that discuss robust stability of perturbed nonlinear systems and illustrate stabilizing linear/nonlinear interconnected systems based on LMI formulation, the proposed method uses LMI to design the linear part of controller [21–23]. On the other hand, the robustness term’s gain value is selected in order to ensure stability of controlled manipulator. This paper is organized as follows: Sect. 2 demonstrates the independent joint control and dynamic modeling of robotic manipulator. Section 3 presents the proposed control method and reviews the mathematical preliminaries required for LMI formulation, which is discussed in Sect. 4. Section 5 discusses the simulation results of proposed control and compares its performance with other standard methods under model uncertainty, noise signal and external disturbance. Conclusions are presented in Sect. 6.

movement q˙ . Dynamic model of the manipulator in (1) shows highly nonlinear coupling between joint motions, which results in an increased control problem difficulty depending on the complexity of robot configuration. MIMO structure of the model can be handled as a combination of subsystems where each joint represents a system that interacts with other systems. In this independent joint control scheme the MIMO robotic system is handled as a set of single-input– single-output (SISO) systems maintaining the coverage of interaction among joints and treating the coupling effect as a disturbance. The dynamic model of each joint is expressed in scalar terms as follows:

2 Independent Joint Control

3 Proposed Control

Based on Euler–Lagrange equations of motion, the generic dynamics of an n-link robotic manipulator are expressed as

The objective of tracking control is to design a simple robust controller such that the joint angles can track the desired trajectories. Let the tracking error vector and its derivative for joint i be given by:

M (q) q¨ + N (q, q) ˙ + G (q) + F (q) ˙ =τ

(1)

where q ∈ R n is the joint angular displacement vector, M (q) ∈ R n×n is the symmetric positive definite inertia matrix, N (q, q) ˙ ∈ R n is the vector of coriolis and centrifu˙ ∈ R n is gal forces, G (q) ∈ R n is the gravity vector, F (q) n the frictional force vector, and τ ∈ R is the applied torque vector. Matrix M and vectors N, G and F are dependent nonlinearly upon the manipulator positions q and velocities of

123

m ii (q) q¨i (t) +

n 

m i j (q) q¨ j (t) + n i (q, q) ˙ + gi (q) ˙

j=1 j=i

+ f i (q) ˙ = τi

(2)

where n is the total number of joints in the manipulator, i is a positive integer with 1 ≤ i ≤ n that denotes the joint number, qi is the displacement of joint i, m ii is the inertia of the link connected to joint i, m i j is the inertia of the link between joints i and j, ηi is the total coriolis and centrifugal force, gi is the gravitational force, f i is the frictional force, and τi is the external torque acting on joint i. Defining di (q, q, ˙ q) ¨ as a time-varying disturbance torque representing the coupling effects between joints including the centrifugal, coriolis, friction and gravitational forces associated with joint i: ˙ q) ¨ = di (q, q,

n 

m ii (q) q¨ j (t) + n i (q, q) ˙

j=1 j=i

+gi (q) ˙ + f i (q) ˙

(3)

the model in (2) reduces to: m ii (q) q¨i (t) + di (q, q, ˙ q) ¨ = τi

(4)

ei = qdi − qi

(5)

e˙i = q˙di − q˙i

(6)

where qdi is the desired position and q˙di is the desired velocity. The sliding manifold is si = e˙i + ci ei

(7)

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with ci ∈ R + being a positive scalar. The proposed control law that combines P control with SMC is then given by: τi = k pi ei + h i sign (e˙i + ci ei )

(8)

where k pi is the proportional gain, and h i is the SMC gain of joint i. Control design procedure mainly consists of selecting  the proper values of control parameters k pi , h i for ensuring stability of robotic manipulator in closed loop. Resulting error dynamics under proposed control are given by the following closed-loop differential equation model for a single joint: m ii e¨i +

n 

m i j e¨ j + k pi ei + h i sign (e˙i + cei )

= m ii q¨di +

m i j q¨d j + n i + gi + f i

Vi = (9)

j=1 j=i

Remark 1 The proposed control design procedure is model free, which means there is no need to determine the model of the manipulator in contrast with standard SMC. Remark 2 Proposed control law is based only on the tracking error signal and its derivative, and it is a combination of linear proportional control and nonlinear robust control. As depicted in (8), P control term is used instead of equivalent control of standard SMC. Assumption 1 The square symmetric positive definite matrix M(q) in (1), which has the following structure: ⎡

m 11 (q) · · · ⎢ .. M (q) = ⎣ ... . m 1n (q) · · ·

⎤ m 1n (q) ⎥ .. ⎦ .

(10)

m nn (q)

j ≤ n,

(11)

where m i−j is the lower bound and m i+j is the upper bound. Centrifugal and coriolis force vector, frictional force vector and gravity vector in the manipulator model in (2) are also bounded by the upper limits as follows: N (q, q) ˙ 2 ˙ q ˙ 2 ≤ N + q

(12)

F(q) ˙ 2 < F + q ˙ 2 + F0

(13)

G(q)2 < G +

(14)

+

where N + , F , F0 and G + are positive scalars.

1 2 s 2 i

(15)

Differentiating (15) and using (2), (5), (7) and (8), one can derive the following derivative of the suggested Lyapunov function candidate: ⎡

⎛

⎢ ⎜ ⎢ ⎜ ⎢ 1 ⎜ ⎜k p ei + h i sign (si ) c e ˙ + q ¨ − V˙i = si ⎢ di i ⎢ i i m ii ⎜ ⎢ ⎜ ⎣ ⎝ ⎞⎤ ⎟⎥ ⎟⎥ ⎟⎥ ⎥ m i j q¨ j − n i − f i − gi ⎟ − ⎟⎥ ⎟⎥ ⎠⎦ j =1 j = i n 

(16)

Using the assumed upper limits in (11–14) and boundedness of the desired values and their derivatives:

consists of bounded inertia values: m i−j ≤ m i j (q) ≤ m i+j , 1 ≤ i,

Theorem 1 Consider the nonlinear robotic manipulator system in (2) and the proposed control law in (8). If the assumptions addressed in (11–14) are true and desired angular positions and their derivatives up to order two are bounded,  is possible to select proper control para then it meters k pi , h i that guarantee stability of the closed-loop system with the dynamic model in (9). Moreover the final tracking error and its derivative are both convergent to zero. Proof The following positive definite Lyapunov function candidate is used to verify stability.

j=1 j=i n 

Desired angular positions and their first- and second-order derivatives are also bounded [24].

⎧ ⎡ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎪ ⎨ −h − k |e | ⎢ ⎢ i pi i V˙i ≤ |si | +⎢ ⎢ci |e˙i | + |q¨di | + ⎪ mi j ⎢ ⎪ ⎪ ⎪ ⎣ ⎪ ⎪ ⎩ ⎛ 1 + − mi j

⎜ ⎜ n  ⎜ +  ⎜n + f + + g + + m i+j q¨d j i i ⎜ i ⎜ ⎝ j =1 j = i

⎞⎤⎫ ⎪ ⎪ ⎪ ⎟⎥⎪ ⎪ ⎟⎥⎪ ⎟⎥⎬ ⎥ ⎟  − e¨ j ⎟⎥ ⎪ ⎟⎥⎪ ⎪ ⎠⎦⎪ ⎪ ⎪ ⎭ (17)

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Right-hand side of (17) is a negative definite function if the following condition is satisfied: h i + k pi |ei | > m i+j [ci |e˙i | + |q¨di | ⎛ 1 + − mi j

⎞⎤

⎟⎥ ⎜ ⎟⎥ ⎜ n  ⎟⎥ ⎜ +   + ⎥ ⎜η + f + + g + +  m i j q¨d j − e¨ j ⎟ i i ⎟⎥ ⎜ i ⎟⎥ ⎜ ⎠⎦ ⎝ j =1 j = i (18)

Consequently, the time derivative of the Lyapunov function candidate in (15) is guaranteed to be negative definite and the closed-loop   system in (9) is stable if the controller parameters k pi , h i are selected in accordance with the condition in (18). Furthermore, the tracking error in (5) and its derivative in (6) converge to zero according to Barbalat’s Lemma [25, 26]

This section discusses robust stability property of perturbed nonlinear systems with the following structure [22],

(20)

while g(t, x) is perturbed term due to modeling error, external disturbance and parameter variations, which exist in practical systems. Usually g(t, x) is not exactly known, but some information can be available such as the upper bound of g(t, x). Let v (t, x) be the Lyapunov function, which satisfies the following: c1 x2 ≤ v ≤ c2 x2 ∂v ∂v + f (t, x) ≤ −c3 x2 ∂t ∂ x    ∂v    ≤ c4 x2 ∂x 

+ c3 γ x2

(25)

If γ is sufficiently small with the following upper bound, γ <

c3 c4

(26)

it follows that v˙ (t, x) ≤ − (c3 − γ c4 ) x2 , (c3 − γ c4 ) > 0

(27)

Lemma 1 Let x be an exponentially stable equilibrium point of the nominal system in (20). Let v (t, x) be the Lyapunov function of the nominal system which satisfies (21–23), and assume the perturbation term g (t, x) satisfies (24). Then the origin is an exponentially stable equilibrium point of the perturbed system in (19) [22].

x(t) ˙ = Ax(t) + g (t, x)

(28)

the equilibrium point is exponentially stable if the perturbed system satisfies the following two conditions:

(19)

where x ∈ R n , f (t, x), g(t, x) are continuous functions of t. The nominal system is x˙ = f (t, x)

   ∂v  2  v˙ (t, x) ≤ −c3 x2 +   ∂ x  g (t, x) ≤ −c3 x

For systems with the following structure:

4 Mathematical Preliminaries

x˙ = f (t, x) + g (t, x)

where γ is a nonnegative constant. Using (20–23),

(21)

i. A is a Hurwitz matrix , ii. g(t, x)2 ≤ γ x2

(29)

Let the quadratic Lyapunov function that satisfies (21) and (27) be v (t, x) = x T P x

(30)

with P being the solution of following equation: P A + AT P = −Q

(31)

Then, for a symmetric positive definite matrix Q = QT > 0

(32)

(22) There exists a unique solution P such that (23)

P = PT > 0

(33)

for positive values of c1 , c2 , c3 , c4 . Suppose that g(t, x) satisfies the following:

Then, derivative of the Lyapunov function for the nonlinear system in (19) satisfies:

g(t, x) ≤ γ x

v˙ (t, x) ≤ −λmin (Q) x22 + 2λmin (P) γ x22

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(24)

(34)

Arab J Sci Eng

      ˙ − G (q) − F(q)] ˙  g (x) ≤ k p q  +  M −1 [τ − N (q, q)

Therefore, the origin is globally exponentially stable if

(46)

γ < λmin (Q) /2λmin (P)

(35)

where δmin (Q) and δmin (P) are the minimum and maximum eigenvalues of the matrices Q and P, respectively.

5 LMI Formulation Based on LMI optimization technique, the control problem of robotic manipulator is considered as a convex optimization problem that minimizes the parameter γ and the objective is to find the linear part of controller, which is the proportional gain vector k p . Substituting the control law (8) into the manipulator dynamics in (1), the following equation is obtained:

  According to Theorem 1, k p q  and τ  are bounded because they depend on error signal, which is shown to be also bounded in the same theorem. Knowing that M is a positive definite, then it follows that [15],    −1  (47) M  ≤ r with the assumptions in (12–14); then, (46) becomes: g (x) ≤ γ x

(48)

The control parameter k p is selected so that the robotic manipulator is asymptotically stable with maximization of the γ parameter that satisfies (48). Theorem 2 If the matrix Anew is selected such that

˙ − G (q) − F (q)] ˙ q¨ = M −1 [τ − N (q, q)

(36)

(Anew )T P + P Anew + γ 2 P P + I < 0

(49)

In order to state the manipulator under control as a linear system with nonlinear perturbation, (36) is rewritten as follows:

where P is a positive definite symmetric matrix, then the controlled robotic manipulator is asymptotically stable [21].

˙ − G (q) − F(q)] ˙ q¨ = −k p q + k p q + M −1 [τ − N (q, q)

Proof Consider the candidate Lyapunov function that follows:

(37) q¨ = f (q) + g(q, q) ˙

(38)

f (q) = −k p q

(39) −1

˙ g (q, q) ˙ = k p q + M [τ − N (q, q)]     q x Let x = 1 = x2 q˙

(40) (41)

v (x) = x T P x

(50)

with P being a positive definite symmetric matrix. Then, v˙ (x) = x T [P Anew + Anew P] x + 2x T P g(x)

(51)

Using (24), Then the state representation of the closed-loop manipulator is  x˙ = Anew x + 

0 g(x)



(52)

By using the following algebraic inequality (42) 

0 In×n (−K p )n×n 0    x1 Kp 0 x g (x) = x2 0 0 Anew =

−1

2x T P g(x) ≤ 2γ P x x

(43) 2n×2n

+M [τ − N (x) − G (x) − F(x)] ⎤ ⎡ k p1 · · · 0 ⎢ . . .. ⎥ K P = ⎣ ... . . ⎦ 0 · · · k pn

ab <

a2 + b2 4

(53)

Then the inequality in (52) can be expressed as follows 2x T P g (x) ≤ γ 2 x T P P x + x T x

(54)

(44) (45)

Now, it is possible to apply Lemma 1 on controlled robotic manipulator with proposed controller to ensure the asymptotic stability of the closed-loop system.

Combining (54) with (51) yields the following upper bound for the derivative of candidate Lyapunov function, v˙ (x) ≤ x T [P Anew + ATnew P + γ 2 P P + I ] x

(55)

As a result one can conclude that v˙ (x) ≤ 0 if [P Anew + ATnew P + γ 2 P P + I ] < 0

(56)

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Theorem 3 Let Anew = A I − K C

(57)

X = PK

(58) 











0 0 0I I 0 , K = , C= KP 0 00 0 I Then based on Theorem 2 that guarantees the asymptotic stability of controlled robotic manipulator system, the control problem in (9) can be restated as a convex optimization problem for finding X and P and then find X.

with A I =

K = P −1 X

uncertainties, external disturbances and random noises are presented. Consider the robotic manipulator of concern whose schematic diagram on the x–y plane is presented in Fig. 1. The dynamic equation of the 2-DOF robotic system can be expressed as follows [27]: 

τ1 τ2



(59)

Proof Based on Schur complement 

A B C D



> 0 ⇔ A > 0, D > 0, A − B D −1 C > 0

(60)

The inequality in (56) can be rewritten in LMI form as follows:   P Anew + ATnew P + I P <0 (61) P − γ12 I

⎡

 ⎤ m 2 L 22 (m 1 + m 2 ) L 21 + m 2 L 22 ⎢ +2m 2 L 1 L 2 cosθ2 +m 2 L 1 L 2 cosθ2 ⎥ ⎥   =⎢ ⎦ ⎣ m2 L 2 2 m 2 L 22 +m 2 L 1 L 2 cosθ2   θ¨ × ¨1 θ2       −m 2 L 1 L 2 2θ˙1 θ˙2 + θ˙22 sinθ2 v1 θ˙ 1 + + v2 θ˙ 2 m 2 L 1 L 2 θ˙12 sinθ2   p sgn(θ˙ 1 ) (66) + 1 p2 sgn(θ˙ 2 )

Gravitational force term is not included in the formulation of the model for the robotic manipulator operates on horizontal plane. Table 1 lists the variables used in the model and their short definitions. For the simulation of proposed control, desired trajectories are selected as follows:

y

Then, substituting (57) in (61), 



P A I − P K C + ATI P − (K C)T P + I P P − γ12 I P A I − X C + ATI P − C T X T + I P P − γ12 I

 <0

<0

(63)

It can be reformulated the convex problem as a convex optimization problem with respect to γ . Let β = γ12 Finally the convex optimization problem in LMI form is expressed as the problem of Minimizing β such that 

P A I − X C + ATI P − C T X T + I P P −β I P > 0, β > 0.

 <0

(64) (65)

6 Simulation Result In this section, the proposed method is tested for trajectory tracking control of a two-degree-of-freedom (2-DOF) robotic manipulator system, and the responses to model

123

θ2

(62)



θ1

x Fig. 1 Schematic diagram of a 2-DOF robotic manipulator on horizontal plane Table 1 Definitions of variables in (19) Variable

Definition

θ1

Angular position of the first link

θ2

Angular position of the second link

τ1

Applied torque of the first link

τ2

Applied torque of the second link

m1

Mass of the first link

m2

Mass of the second link

v1

Viscous friction coefficient of the first link

v2

Viscous friction coefficient of the second link

p1

Dynamic friction coefficient of the first link

p2

Dynamic friction coefficient of the second link

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Parameter

Link 1

Link 2

Mass (kg)

10

1

Length (m)

1

1

Viscous friction coefficient (Nm s)

0.1

0.1

Dynamic friction coefficient (Nm)

0.1

0.1

position (rad)

Table 2 Parameters of simulation model of plant

desired position CTC Proposed SMC PD-SMC

0.4

0.3

(a)

0.2 0

1

2

3

4

5

0.05 position (rad)

0.35 0.3 0.25 0.2

error (rad)

desired position CTC Proposed SMC PD-SMC

0.4

(a) 0

0

-0.05 1

2

3

4

50

error (rad)

-0.01 -0.02 -0.03

0

-0.05 0

0.5

1

1.5

2

2.5

CTC Proposed SMC PD-SMC

-200

(c) 0

1.5

2

2.5

(c) 1

2

3

4

5

time(s)

0

-400

1

0

0

(b)

0.5

CTC Proposed SMC PD-SMC

-50

-0.04

torque (Nm)

torque (Nm)

CTC Proposed SMC PD-SMC

0

(b)

5

0.01

0.5

1

1.5

2

2.5

time(s)

Fig. 2 Angular position (a), tracking error (b) and input torque (c) of link 1 versus time in response to model uncertainty



CTC Proposed SMC PD-SMC

qd1 (t) = 0.3 + 0.1 sin (t) , qd2 (t) = 0.3 + 0.1 cos (t)

(67)

Reference values for feedback control that are presented in (64) are continuous bounded functions with infinitely many continuous bounded higher-order derivatives; consequently, they both satisfy the assumptions in the present study. Simulation tests are implemented on the nominal dynamic model expressed for two degrees of freedom in (66), which represents the open-loop plant model for feedback control. Numerical values for plant parameters are selected as given in Table 2. The signum function is replaced by a saturation function with boundary values of ±0.05 to avoid chattering.

Fig. 3 Angular position (a), tracking error (b) and input torque (c) of link 2 versus time in response to model uncertainty

Tests are carried out using MATLAB–Simulink tool on a personal computer of 2.40 GHz (8 CPUs) processor speed and 12 GB RAM capacity. For ensuring the validity of the proposed method, simulation results are compared with wellestablished strategies for 2-DOF robotic manipulator control, standard SMC, CTC [28] and PD-SMC [29]. Remark 3 The controller parameters are selected as follows: MATLAB LMI toolbox is used to solve the inequalities in (60) and (61) for variables P and k. Resulting control parameters are k p1 = 5, k p2 = 5. The values of robust gain h and slope of sliding surface c are selected to satisfy (18) as follows: (h 1 , c1 ) = (55, 5) , (h 2 , c2 ) = (55, 5)

(68)

Simulation tests are designed for comparing the performances of the three control systems under three different circumstances. First, model uncertainties are integrated in the model and tests are implemented for robustness. The second case takes random noise effect into consideration, and the last case tests the three controllers for robustness to external disturbances.

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0.3 0.25 1

2

3

4

5

0.25

Computed Torque Proposed SMC PD-SMC

0 -0.02

1

1.5

2

2.5

CTC Proposed SMC PD-SMC

0 -200

0.5

1

1.5

4

5

2

2.5

CTC Proposed SMC PD-SMC

20 0 -20

(c)

-40

0.5

(c) 0

3

(b) 0

0

-400

2

CTC Proposed SMC PD-SMC

0

40

0.5

1

0.02

(b) 0

(a)

0.04

error (rad)

error (rad)

0.3

0

0 0.02

-0.04

0.35

0.2

(a)

0.2

torque (Nm)

position (rad)

0.35

desired position CTC Proposed SMC PD-SMC

0.4

desired position CTC Proposed SMC PD-SMC

0.4

torque (Nm)

position (rad)

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1

1.5

2

2.5

time(s)

0.5

1

1.5

2

2.5

time(s)

Fig. 5 Angular position (a), tracking error (b) and input torque (c) of link 2 versus time in response to noise signal

Fig. 4 Angular position (a), tracking error (b) and input torque (c) of link 1 versus time in response to noise signal

0.03

6.1 Robustness to model uncertainties In this section, the performance of proposed scheme is investigated under parameter variations of robotic manipulator. The masses and lengths of link 1 and link 2 are increased by 10 % of actual values to test the robustness. Figures 2 and 3 show the tracking performances of proposed control and four alternatives, CTC, SMC and PD-SMC under model uncertainty. The figures depict the angular position, input torque and tracking error variations versus time for the four controllers. Figures 2a and 3a show the position tracking of link 1 and link 2, respectively. It is observed that the steady-state tracking performances have no significant difference for the four controllers. In transient, CTC has the quickest recovery toward the desired track despite a small undershoot in position of link 1. Proposed control and PD-SMC have slightly better transient responses than that of SMC. Angular position tracking errors for the two links given in Figs. 2b and 3b illustrate the above-mentioned result. Required torques to drive link 1 and link 2 to track the desired trajectories are presented in Figs. 2c and 3c, respectively. Torque input values for the three controllers hardly differ at the steady state. However, transient torque requirements by CTC and SMC are significantly

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noise signal

0.02 0.01 0 -0.01 -0.02

0

50

100

150

200

time(s)

Fig. 6 A noise signal

higher than proposed control in link 1, and in link 2 proposed control has a sudden torque requirement that lasts for a short duration. SMC torque input is similar to that of proposed control in link 2, while CTC has an oscillatory torque variation for a much longer duration of time and the torque of PD-SMC method suffers from high chattering specially in transient period in link 2. 6.2 Robustness to random noise In this section, a random noise signal is added to the controlled variable in order to examine the validity of proposed controller. A noise signal with amplitude of 15×10−3 is gen-

Arab J Sci Eng

desired position CTC Proposed SMC PD-SMC

0.2

1

2

3

-0.02

1

1.5

2

2.5

CTC Proposed SMC PD-SMC

0 -200 -400

0.5

1.5

1

2

3

0.02

40

2.5

0.5

0

erated and added at the feedback path of closed-loop system. The trajectories, tracking errors and control input variations versus time for the four tested controllers are given in Fig. 4 for link 1 and Fig. 5 for link 2. Noise signal waveform is given in Fig. 6. Figure 4 shows clearly the high ability of proposed controller scheme to suppress noise signal. As expected, the results show the high sensitivity of CTC and PD-SMC to noise signal which leads to high variation on control signal, while SMC and proposed controller exhibit similar tracking performances and torque input variations that are less sensitive to noise. It should be noted that torque input for each link has a much smoother variation under proposed control in comparison with CTC, SMC and PD-SMC as depicted in Figs. 4 and 5.

6.3 Robustness to external disturbances This section presents the effectiveness and robustness of proposed controller against an external disturbance. The trajectory tracking performance, position tracking error and control signal of each joint for proposed controller along with SMC, CTC and PD-SMC are presented in response to a disturbance input of:

1

1.5

5

2.5

2

CTC Proposed SMC PD-SMC

20 0 -20

(c) 0

0.5

1

1.5

2

2.5

time (s)

time(s)

Fig. 7 Angular position (a), tracking error (b) and input torque (c) of link 1 versus time in response to external disturbance

4

(b)

-40

(c) 0

2

CTC Proposed SMC PD-SMC

0

0.5

1

0.04

(b) 0

(a) 0

5

torque (Nm)

error (rad)

4

CTC Proposed SMC PD-SMC

0

-0.04

0.3

0.2

(a) 0

torque (Nm)

position (rad)

0.3

desired position CTC Proposed SMC PD-SMC

0.4

torque (Nm)

position (rad)

0.4

Fig. 8 Angular position (a), tracking error (b) and input torque (c) of link 2 versus time in response to external disturbance

d (t) = 5 sin 3t Nm

(69)

applied at each link. Results are shown in Figs. 7 and 8 for link 1 and link 2, respectively. Tracking error variations in Figs. 7b and 8b reveal that proposed control has good accuracy. In addition, torque input for the proposed approach is significantly smaller, especially for link 1 as in Fig. 7c. Disturbance rejection property of closed-loop system under proposed control with model-free control design is thus revealed by the presented simulation test results.

7 Conclusion In this paper, the trajectory tracking control problem of a twolink planar rigid nonlinear robotic manipulator is studied and a new robust control scheme based on P control and SMC is proposed. Important features of the proposed method are robustness and simplicity of control law. The proposed controller is model free, which eliminates the need to determine the dynamic model of controlled system, while prior knowledge of dynamic plant model is necessary in standard SMC. Simulation of a 2-DOF robotic manipulator is used to test the proposed method and compare it with CTC, SMC and

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PD-SMC in order to illustrate the robustness in terms of sensitivity to model uncertainty, noise suppression and external disturbance rejection. Simulation of a 2-DOF robotic manipulator is used to test the proposed method and compare it with CTC, SMC and PD-SMC in order to illustrate the robustness in terms of sensitivity to model uncertainty, noise suppression and external disturbance rejection. From simulation results, it is concluded that the proposed scheme can track the desired trajectory well, while it is more effective and robust than computed torque method and slightly better than SMC in terms of control effort required to get good tracking and steadystate error. Also, the proposed method is better than PD-SMC in terms of required torque because there is a considerable amount of chattering in control signal of PD-SMC method, while the control signal in proposed method is smooth. Moreover, proposed scheme is superior to SMC in presenting the desired control performance without the need for prior knowledge of dynamic plant model.

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