Scholarly article on topic 'A Coefficient Diagram Method Controller with Backstepping Methodology for Robotic Manipulators'

A Coefficient Diagram Method Controller with Backstepping Methodology for Robotic Manipulators Academic research paper on "Electrical engineering, electronic engineering, information engineering"

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Academic research paper on topic "A Coefficient Diagram Method Controller with Backstepping Methodology for Robotic Manipulators"

DE GRUYTER

Journal of ELECTRICAL ENGINEERING, VOL. 66, NO. 5, 2015, 270-276

A COEFFICIENT DIAGRAM METHOD CONTROLLER WITH BACKSTEPPING METHODOLOGY FOR ROBOTIC MANIPULATORS

Fouad Haouari * — Bali Nourdine ** Mohamed Segir Boucherit — Mohamed Tadjine

A new robust control procedure for robot manipulators is proposed in this paper. Coefficients diagram method controllers CDM and Backstepping methodology are combined to create the novel control law. Two steps of backstepping on the resulting system are used to design a nonlinear CDM-Backstepping controller. Simulations on a PUMA robot including external disturbances, parametric uncertainties and noises are performed to show the effectiveness and feasibility of the proposed method.

Keywords: manipulators, backstepping approach, coefficients diagram method controller, robustness

1 INTRODUCTION

At present, robot manipulators are the most important instruments used in manufacturing industry. One of the most important challenges in the field of robot manipulators is to design robusts controllers [1], in particular when manipulators are required to maneuver very quickly under various disturbances. These systems are multivariable, nonlinear, strongly coupled, and its highly nonlinear dynamics changes rapidly and some dynamic parameters are uncertainty, e.g., unknown loads and disturbance. As a consequence, it is hard to find an exact mathematical model. The traditional proportional and derivative (PD) controller is very simple and does not require any knowledge of the robot dynamics. However, it requires very large actuation to achieve precise control, which is not practical but highly demanded in many cases. This is due to the fact that robotic arms constantly move among widely separated regions of their workspace such that no linearization valid for all regions can be found. Computed torque method utilizes mathematical model and parameters of the robot manipulator to cancel the non linearities. Howevers, du to the requirement of precise knowledge of the systeme structure and parameters [1], the computational task is very extensive. Although adaptive controllers can realize fine control and compensate for partially unknown manipulator dynamics [3], they often suffer from heavy computational burden and this hinders their real time applications. Another technique is called variable structure control with this, the system state are driven to a switching surface designed to make the state converge to the origin. As the system state cross the switching surface, the state become insensitive to system parameters variations, this method does not require knowledge of exact system parameters; it only requires

the possible uper bound of uncertainty. A disadvantage of this method is that due to the discontinuous control activity, it may excite the unmodeled dynamic, and has the possibility of chattering problem [7]. The sliding mode control method share the common feature of using a discontinus control law although the design apppraoch is quite different. As a result, the chattering of the control signal is a common drawback. A control system with severe chattering is impractical because it stresses actuators even to a point of distruction and it may excite unmod-elled plant dynamic [8]. Furthermore, many mathematical theories are used in new control methodologies to design nonlinear robust controller for robot manipulators.

The success of the CDM control is attributed to its simplicity, stability, and robustness in presence of external disturbance, parametric uncertainties and noises. Different CDM controllers have been proposed for linear system [4-6]. But, CDM controllers' essential shortage is her limitation to linear system and the needing of exponential stability for a given nonlinear systems.

Our goal in this paper is to eliminate this insufficiency by proposing a non linear robust controller CDM-Backstepping applied to robot manipulators. The controller is synthesized by joining a backstepping procedure with a CDM composition. In particular the controllers are designed by imposing the positions tracking with exact gains that are nonlinear functions of the system state. As a result, the proposed nonlinear backstepping control design is not only to stabilize the robot system, but also to oblige the tracking errors to converge to zero exponentially, then the novelty scientific in this work is that non linear CDM has not used previously and this controller summarize the performance of CDM and Backstepping.

Automatic Control Department, Ecole Nationale Polytechnique, Algeria Electrical Engineering and Computing faculty, Université Houari Boumediene de Sciences et. de la Technology, Algeria, haouari_fouad@yahoo.fr

DOI: 10.2478/jee-2015-0044, Print ISSN 1335-3632, On-line ISSN 1339-309X ©2015 FEI STU

Link 1

we have

Link 2

M(q)q + C(q, q)q + g(q) + f (q) = T(t) .

q = (qi q2)

= (T1 T2)1

a ß ß Y

cii C12 C21 C22

with a = —I2 (mi + 4m2 + 3i??2 cos <72),

2 (11 \ 2

/3 = mot + - cosq2), 7 = 13m.2^ ,

cii = -m2l2q2 sin(q2), C12 = C12/2,

C21 = ^m.2/2qi sin(q2), c22 = 0.

gi =^rn1glcos(q1) + ni2gl(^cos(q1 + q3) + cos qij,

T (t) = M (q)q+ h(q,q)

32 = -m2gl cos{qi + q2),

it comes out: Denoting

(X1,X2,X3,X4)T = (qi,qi,q2,q2)

cii(x)x2 + C12 (x)x4 + gi(x) + 61x2 C2i(x)x2 + C22 (x)x4 + g2 (x) + 62X4

and the state space representation is

Xi = X2, X3 = X4,

1 Tl-fol(x) ß

x2 = (1_ JL^)(-ä-} + ("T2 + /72(x))

Fig. 1. Tow link rigid robot manipulator

2 ROBOT STATE SPACE MODEL

Consider the robot manipulator with rigid links and rotary joints. Furthermore, it is assumed that each degree of freedom of the manipulator is powered by an independent torque source [1]. The equations of motion for n degree-of-freedom of manipulator are formulated by using the lagrangian formulation and may be expressed by

1 ,T2 — h2(x)) . , . M .

X4 = ---A--^^ + (-ri + M*)) —)

a7 (8)

(1 a-{ ß)

Xi = X2 , X2 = fi(x) + gi(x)t , X3 = X4 X4 = f2(x) + g2(x)t

a____ i

( ^ ck7 ft )

Where q, q and q are n x 1 vectors of joint positions, velocities and accelerations [2], M(q) is a n x n symmetric and positive definite matrix function which is also called generalized inertia matrix, C(q, q)q is n x 1 vector resulting from Coriolis and centripetal accelerations, moreover f (q) is n x +1 vector of friction, g(q) is vector of generalized gravitational forces and t(t) is the n x 1 vector of joint torque supplied by actuators.

The robot manipulator that we are going to use for our application is called PUMA robot as shown in Fig. 1, it is characterized by two rotary joints identified by n = 2 variables qi and q2 where

F2(x) =

+ 1-^/3'

g2(x) =

( CK'y ft ^ ^ CK'y ft ^

(9) (10) (11) (12) (13)

, f = (6iqi 62q2)T (4)

3 CDM CONTROL DESIGN

Coefficient diagram method is an algebraic approach with polynomial form, it alow to design easily the controller under the conditions of stability, time domain performance and robustness. The performance specification, equivalent time constant and stability index are specified in the closed loop transfer function and related to the controller parameters algebraically. Habitually, the order of the controller is less than the order of the plant.

The output of the controlled closed-loop system is

N(s)F(S) A(S)N(S) y = —~—"—1 h——" 1

where y is the output, r is the reference input, u is the control and d is the external disturbance signal, N(s) and D(s) are the numerator and the denominator of the transfer function of the plant, respectively, A(s) is the denominator polynomial of the controller transfer function, while F(s) and B(s) are called the reference numerator and the feedback numerator polynomials of the controller transfer function.

Also P(s) is the characteristic polynomial and given

P (s) = D(s)A(s) + N (s)B(s) = ]T M¿

Unauthentic

Downloa

The nominal mathematical model is

N (s) amsm + am_ism-1 + • • • + aq

G1(x) 0

Df \ _ V ^ _ "'m" ' "to—i"_I_I "'u flfi"!

lSj ~ £>(s) ~ 5„s" + + • • • + 60 ' 1 j

The controller polynomials A(s) and B(s) are

A(s) = £ 1isi B(s) = £ kjs4. (17)

i=Q i=Q

The equivalent time constant TQ indicate the time response speed and the stability indices Yj give the stability and the waveform of the time response. They are defined in terms of the coefficients of the characteristic polynomial in (15) as

-to = — Mo

Mi-iMi+i

for i = 1... (n — 1).

The settling time and the equivalent time constant is defined as

2.5 - 3 '

Yi = 2.5 , Yi = 2 , ; i = 2 - (n - 1), ; yo = Yn = to .

The last values can be adjusted to assure the required performance, so that Yi > 1.5 for all i = 1 — (n — 1).

Then the characteristic polynomial to be used to design the parameters of a controller is

[- n i—1 1

Pis) = MO {E(n —) ^o«)4} + + 1

i=2 j=1 Yi—j

Finally F(s) which is usually defined as the pre-filter used for reducing the steady state error to zero and is selected as a constant defined by

F (s) =

P (s)ls=

f(x) = ( f1(x) f2(x))'

We can concluded that the positions errors e1 and e2 can be controlled using the auxiliaries variables Zi and Z2 respectively, which can be controlled using the real control signal t .

Let Zd and Z2 be the values of Zi and Z2 respectively, which ensuring the stabilization of the positions tracking error e1 and e2 , also these desired values are determined using Lyapunov approach by considering the dynamic equation of e1 and e2, consequently e3 = Zi — Zi and e4 = Z2 — Zi with E = (e3 e4)T . Then

Zd = ( zd zd )

E = z — Zd

The control signal is written as follows

Aî{x)t + A2{x)^ =Ec{t) Ec(t) = Co(x)Zd — Bq(x)Z — B(x)Z,

A1(x), A2(x), Cq(x), Bq(x) and B1(x) are nonlinear matrix gains of multivariable nonlinear CDM controller introduced in (28) and (29).

A backstepping procedure [9-20] is proposed to determinate the gains matrix assuring the exponential stability result for the links positions tracking errors. Step 1: Firstly we design the virtual control law Zd(t) then Z2(t), theirs positions error must asymptotically converge to zero.

Step 2: secondly we choose the gains matrix A1(x), A2(x), Cq(x), Bq(x) and B1(x) by employing the augmented Lyapunov function that oblige the errors to track an exponential convergence.

5 CDM-BACKSTEPPING CONTROL DESIGN

Proposition 1. The positions tracking error e1 and e2 are exponentially stable with the following condition

4 CONTROL OBJECTIVE

Zd = —A1e1 + qd , Zd = —A2e2 + 4 .

In this section, we use the CDM-Backstepping algorithm to develop the positions control law. These positions will converge exponentially to the reference value. The error positions are defined as.

e1 = 91 — = X1 — qf, e2 = 92 — (2 = X3 — (2 (22) and their derivatives are

(51 = X2 — qf , <S2 = X4 — 92 .

Let Z = (x2 X4)T. then Z = F (x) + G(x)t

Proof 1. The Lyapunov formulation can be written

where its time derivative can be represented as

V1 = e1(51 + e2(52 .

Since the virtual control Zi and Z2 track the desired value specified in (30), the derivative of the Lyapunov function become negative and takes the next form.

V < — A1e2 — A1e2.

V < 0.

As a result, the exponentially stability can be achieved for ei and e2 .

The control signal t = ( t1 t2 )T that oblige the errors e3 and e4 to converge to zero will be now deducted. Let

Ai (x) = -K^j^-, A2 (x) = -R'G(x)

with any positive definite matrix K.

Combining equations (27) with (29) gives

E = (B0-1CO — I )Zd — B0-1EC

And tacking

Co(x) = Bo(x) = Co Then

Ec = —COE . Its second derivative is

Ec(t) = CoCd(t) — CoC(t).

Combining equations (28), (29) and (35) gives C(t) = + KiEc K1 = K-1.

Substituting equation (40) into (39), we obtain Ec(t) = CoCd(t) — Co(F(x) + KiEc).

Ec(t) = Co^d(t) — Co(F(x) + Ki / Ec(p)dp)

and using equation (38)

E(t) = H(x) - kJ E(p)dp

tacking

H (x) = F (x) - Cd(t), K2 = CoKi.

H (x) =

"Hi(x)' 'Fi(x) - zrdl

_H2(x)_ .F2(x) - &J

¿1 sign(zi) 0

0 ¿2 sign(z2 ) •t

z = ei+i / ei+i(p)dp, i = 1, 2 . o

Proposition 2. Consider the robot manipulator dynamic (8), in closed-loop with the multivariable CDM control (28), suppose that the gains ¿i, ¿2, ci and c2 are such that

ciJi sign(zi) e3(p)dp

c2Î2 sign(z2 )/ e4(p)dp

^ Aiwith Ai > |ei| + |Hi(x)

A2with A2 > |e2| + |H2(x)|.

According to Lyapunov stability, it implies that the tracking errors e1(t), e2(t), e3(t) and e4(t) are exponentially stable and the closed-loop system is internally stable.

Proof . Consider the augmented Lyapunov function [12].

v2 = v1 + \eje.

Its derivative along the plant trajectories is given by

V2 = Vi + E|topE. (50)

Using the expressions of e3(t), e4(t) and equation (32), we get

V2 = -Aiei - A2e2 + eie3 + e2e4 .

This gives

V2 < -Aie2 - A2e2 + E1

Changing the dynamics of E by (44) and the control signal by (28), then one has

V2 < -Aie2 - A2e2 + v(t)

where v(t) = ET

H$) - E(p)dp

If v(t) < 0 then

V2 < -Aie2 - A2e2 .

(53) . (54) (55)

Notice that v(t) = ET

eA + A Hi(x) A - / ciJiZi sign(zi) A

eJ I H2(xW I c2¿2Z2 sign(z2) J

Furthermore, to guarantee the negativity of V2 , the gains ¿1, ¿2 , c1 and c2 must be chosen from inequality (48) for the reason that ziS(zi) > 0 and v(t) < 0. Therefore, it can be concluded that

V2 < 0.

Fig. 2. CDM-Backstepping controller scheme

0.8 0.6 0.4 0.2 0

600 500 400 300 200 100 0

position (rad)

des ired / posil ion

a ctua 91d ________i

Po 91 /

2 5 position (rad)

2.0 1.5 1.0 0.5 0

1 2 3 4 5 6 torque (Nm)

time (s)

9 10 a5

desir edp ositio \ m V

/ V

V / \

92 posit ion

first oint

1 2 3 4 5 6 9 10

time (s)

200 150 100 50 0

1 2 3 4 5 6 9 10

time (s)

torque (Nm)

V se( J'c ond int

1 2 3 4 5 6 , , 9 10

time (s) 10

y actua path l

------------ \

esirf patl d \

1.6 2.0

Fig. 3. CDM-Backstepping control; test one (a) — Desired and actual positions of the first joint, (b) — Desired and actual positions of the second joint, (c) — Actual torque of the first joint, (d) Actual torque of the second joint, (e) — Actual and desired path

position (rad)

0.8 0.6 0.4 0.2 0

position (rad)

/ 91d (a)

d esire / d pos ition

7i a ctual

r p° 91 \

1 2 3 4 5 6

1 2 3 4 5 6 ... 9 10

time (s)

torque (Nm)

2.5 2.0 1.5 1.0 0.5 0 0.5

de sirec posi tion

V \

...... \ a q2 ctual posi ion

.cond ljoin

S. se t

time (s)

1 2 3 4 5 6 ... 9 10

time (s)

torque (Nm)

1 2 3 4 5 6 ,, 9 10 time (s)

Fig. 4. CDM-Backstepping control; test two (a) — Desired and actual positions of the first joint, (b) — Desired and actual positions of the second joint, (c) — Actual torque of the first joint, (d) Actual torque of the second joint, (e) — Actual and desired path

It implies that the dynamic system is exponentially stable according to Lyapunov stability theorem.

The boundedness of state vector X = (xi, x2, x3, x4) is not guarantying by the asymptotically convergence of tracking errors. qf and qf are bounded and the errors e1 and e2 are exponentially stable so that the state

a = (x2,x4) is bounded, also the state n = (x1,x3) is bounded; this proves that the origin of the subsystem a = n is stable.

Remark 2 . Notice that strict knowledge of the limits Aj and functions F is not required. Bounds can be employed on these variables to ensure a nonlinear robust

position (rad) position (rad)

Fig. 5. CDM-Backstepping control; test three (a) — Desired and actual positions of the first joint, (b) — Desired and actual positions of the second joint, (c) — Actual torque of the first joint, (d) Actual torque of the second joint, (e) — Actual and desired path

controller under condition of disturbance, parametric uncertainties and noises. The Integral gains must be satisfactorily large to realize (48). The CDM gains are selected as designated in Proposition 2. The following CDM gains have been used for all the simulated situations.

(¿1, ¿2) = (103, 110), (c1, C2) = (0.7, 0.6). (58)

6 SIMULATION RESULT

In order to evaluate the quality of the derived algorithms of nonlinear control, simulation tests were performed using Matlab for circular path and butterfly shape trajectory.

Test one: External disturbances

The externals disturbances that can be applied are disturbance in torques of Td1(t) = 10 Nm and Td1(t) = 13 Nm are applied for each joints of robotic system.

The simulation plots shown in Fig. 3 indeed verify that our CDM-Backstepping design scheme can guarantee the best performance for each joint of the robotic manipulator to track its desired trajectory exponentially and eliminate the disturbance with no overshoot and with a negligible steady state error. Test two: Parametric uncertainties In the second test for the robustness evaluation of the controllers, we introduced the following parametric uncertainties in the robot models.

• Tool attached to end effector, then parametric uncertainties at second link in mass m = 4 kg and in length l = 0.1 m.

• Coulomb friction and viscous friction are added to each joint of robot manipulator and given by fcvi = 2.5x2 (t) + 1.8 sign(X2 (t)) and fcv2 = 2X4 (t) + 1.2 sign(X4(t)) . The simulation results in Fig. 4 show the strong robustness of the proposed CDM-Backstepping control towards uncertainties affecting the robot mechanical parameters.

Test three: Change in the desired path To test the controller's robustness the simulations have been executed using the same last externals disturbances and parametric uncertainties with noises applied for a butterfly shape trajectory, whish are used as a desired end-effector's path. The simulations results are depicted in Fig. 5, this ?gure show that CDM-Backstepping control present a robust path following in 2D displacement.

7 CONCLUSION

This paper reveals a new approach of robust control systems CDM for robot manipulators using backstepping design. The major distinctive of the proposed approach is the application of the novel Lyapunov functions to construct the CDM-Backstepping controller. Global stability results are obtained and the tracking errors converge to zeros with exponential forms. Simulations results have been given to demonstrate the theoretical analysis used in the controller. Further investigation can be directed to the robustness of CDM-Backstepping controller.

Appendix

Rated data of the simulated robot manipulator bi = 75 N/r/s, b2 = 10 N/r/s, 11 = 1 m, 12 = 1m, m1 = 10 kg, m2 = 10 kg, k1 = 40 Nm/v, k2 = 20 Nm/v, g = 9.81 ms-2 .

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Received 23 August 2014

Fouad Haouari was born in 1977, in Algeria. He got the Engineer degree in automatic control in 2001, the Magister degree in 2006 in Electrical Engineering, from Automatic Control Department, of Ecole Nationale Polytechnique of Algiers, Algeria. He is currently preparing his doctorate degree in automatic control. He is Associate Professor at Ecole nationale de Technologie, His research interests include robust and nonlinear control.

Bali Noureddine was born in July 1965, in Algeria. He got the Engineer degree in automatic control in 1992, the Magister degree in 1996 and the PhD in 2007 in Electrical Engineering, from Automatic Control Department, of Ecole Nationale Polytechnique of Algiers. He joined the Houari Boume-diene University of Sciences and Technology (USTHB) in Algiers. He is Associate Professor, Vice-Dean in the Electrical Engineering and Computing faculty, and Head of the Industrial Maintenance and Reliability research team, in the Industrial and Electrical Systems Laboratory. His current research interests are in the fields of predictive control applications, control systems and maintenance management.

Mohamed Seghir Boucherit was born in, 1954 in Algiers, Algeria. He received the Engineer degree in Electrotech-nics, the Magister degree and the Doctorat d'Etat (PhD degree) in Electrical Engineering, from the Ecole Nationale Polytechnique, of Algiers, Algeria, in 1980, 1988 and 1995 respectively. Upon graduation, he joined the Electrical Engineering Department of Ecole Nationale Polytechnique. He is a Professor, Head of the Industrial systems and Diagnosis research team of the Process Control Laboratory and his research interests are in the area of Electrical Drives, Process Control Applications, and Diagnosis.

Mohamed Tadjine was born in 1966, in Algiers, Algeria. He received the Engineering degreefrom the Ecole Nationale Polytechnique, Algiers, Algeria, in 1990, the MSc and PhD degrees in automatic control from the National Polytechnic Institute of Grenoble, Grenoble, France, in 1991 and 1994, respectively. From 1995 to 1997, he was a Researcher at the University of Picardie, Amiens, Fiance. Since 1997, he is with the Department of Automatic Control of the Ecole Nationale Polytechnique, Algiers, Algeria, where he is currently a Professor. His research interests are in robust and nonlinear control.