A COEFFICIENT DIAGRAM METHOD CONTROLLER WITH BACKSTEPPING METHODOLOGY FOR ROBOTIC MANIPULATORS

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1 Journal of ELECTRICAL ENGINEERING, VOL. 66, NO. 5, 215, A COEFFICIENT DIAGRAM METHOD CONTROLLER WITH BACKSTEPPING METHODOLOGY FOR ROBOTIC MANIPULATORS Foua Haouari Bali Nourine Mohame Segir Boucherit Mohame Tajine A new robust control proceure for robot manipulators is propose in this paper. Coefficients iagram metho controllers CDM an Backstepping methoology are combine to create the novel control law. Two steps of backstepping on the resulting system are use to esign a nonlinear CDM-Backstepping controller. Simulations on a PUMA robot incluing external isturbances, parametric uncertainties an noises are performe to show the effectiveness an feasibility of the propose metho. K e y w o r s: manipulators, backstepping approach, coefficients iagram metho controller, robustness 1 INTRODUCTION At present, robot manipulators are the most important instruments use in manufacturing inustry. One of the most important challenges in the fiel of robot manipulators is to esign robusts controllers [1], in particular when manipulators are require to maneuver very quickly uner various isturbances. These systems are multivariable, nonlinear, strongly couple, an its highly nonlinear ynamics changes rapily an some ynamic parameters are uncertainty, e.g., unknown loas an isturbance. As a consequence, it is har to fin an exact mathematical moel. The traitional proportional an erivative (PD controller is very simple an oes not require any knowlege of the robot ynamics. However, it requires very large actuation to achieve precise control, which is not practical but highly emane in many cases. This is ue to the fact that robotic arms constantly move among wiely separate regions of their workspace such that no linearization vali for all regions can be foun. Compute torque metho utilizes mathematical moel an parameters of the robot manipulator to cancel the non linearities. Howevers, u to the requirement of precise knowlege of the systeme structure an parameters [1], the computational task is very extensive. Although aaptive controllers can realize fine control an compensate for partially unknown manipulator ynamics [3], they often suffer from heavy computational buren an this hiners their real time applications. Another technique is calle variable structure control with this, the system state are riven to a switching surface esigne to make the state convergeto the origin. As the system state cross the switching surface, the state become insensitive to system parameters variations, this metho oes not require knowlege of exact system parameters; it only requires the possible uper boun of uncertainty. A isavantage of this metho is that ue to the iscontinuous control activity, it may excite the unmoele ynamic, an has the possibility of chattering problem[7]. The sliing moe control metho share the common feature of using a iscontinus control law although the esign apppraoch is quite ifferent. As a result, the chattering of the control signal is a common rawback. A control system with severe chattering is impractical because it stresses actuators even to a point of istruction an it may excite unmoelle plant ynamic[8]. Furthermore, many mathematical theories are use in new control methoologies to esign nonlinear robust controller for robot manipulators. The success of the CDM control is attribute to its simplicity, stability, an robustness in presence of external isturbance, parametric uncertainties an noises. Different CDM controllers have been propose for linear system [4 6]. But, CDM controllers essential shortage is her limitation to linear system an the neeing 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 applie to robot manipulators. The controller is synthesize by joining a backstepping proceure with a CDM composition. In particular the controllers are esigne by imposing the positions tracking with exact gains that are nonlinear functions of the system state. As a result, the propose nonlinear backstepping control esign 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 use previously an this controller summarize the performance of CDM an Backstepping. Automatic Control Department, Ecole Nationale Polytechnique, Algeria Electrical Engineering an Computing faculty, Université Houari Boumeiene e Sciences et e la Technology, Algeria, haouari foua@yahoo.fr DOI: /jee , Print ISSN , On-line ISSN X c 215 FEI STU

2 Journal of ELECTRICAL ENGINEERING 66, NO. 5, y Link 1 we have [ ] c11 (xx h(x = 2 +c 12 (xx 4 +g 1 (x+b 1 x 2 c 21 (xx 2 +c 22 (xx 4 +g 2 (x+b 2 x 4 Link 2 x Fig. 1. Tow link rigi robot manipulator an the state space representation is ẋ 1 = x 2, ẋ 3 = x 4, 1 1 h 1 (x ẋ 2 = (1 β γ β(τ +( τ 2 +h 2 (x β γ ẋ 4 = 1 2 h 2 (x (1 β γ β(τ +( τ 1 +h 1 (x β γ (8 2 ROBOT STATE SPACE MODEL Consier the robot manipulator with rigi links an rotary joints. Furthermore, it is assume that each egree of freeom of the manipulator is powere by an inepenent torque source [1]. The equations of motion for n egree-of-freeom of manipulator are formulate by using the lagrangian formulation an may be expresse by M(q q +C(q, q q +g(q+f( q = τ(t. (1 Where q, q an q are n 1 vectors of joint positions, velocities an accelerations [2], M(q is a n n symmetric an positive efinite matrix function which is also calle generalize inertia matrix, C(q, q q is n 1 vector resulting from Coriolis an centripetal accelerations, moreover f( q is n +1 vector of friction, g(q is vector of generalize gravitational forces an τ(t is the n 1 vector of joint torque supplie by actuators. The robot manipulator that we are going to use for our application is calle PUMA robot as shown in Fig. 1, it is characterizeby tworotaryjoints ientifie by n = 2 variables an where M = q = (, τ = (τ 1 τ 2, (2,3 [ ] [ ] β c11 c ; C = 12, f = (b β γ c 21 c 1 b 2 (4 22 with = 1 3 l2( m 1 +4m 2 +3m 2 cos, β = m 2 l 2( cosq 2, γ = 13m2 l 2, c 11 = m 2 l 2 sin(, c 12 = c 12 /2, c 21 = 1 2 m 2l 2 sin(, c 22 =. g 1 = 1 ( 1 2 m 1glcos( +m 2 gl 2 cos(+q 3 +cos, g 2 = 1 2 m 2glcos( +, Denoting it comes out: τ(t = M(q q +h(q, q (5 (x 1,x 2,x 3,x 4 = (,,, (6 ẋ 1 = x 2, ẋ 2 = F 1 (x+g 1 (xτ, ẋ 3 = x 4 F 1 (x = h 1 (x (1 β F 2 (x = h 2 (x 1 β ẋ 4 = F 2 (x+g 2 (xτ (1 β γ γ β + h 2(x 1 β γ γ β + h 1(x 1 G 1 (x = (1 β γ β β γ β γ β γ β γ (1 β γ 1 G 2 (x = (1 β γ β (1 β γ 3 CDM CONTROL DESIGN (9 β, (1 β, (11 β, (12 β. (13 Coefficient iagram metho is an algebraic approach with polynomial form, it alow to esign easily the controller uner the conitions of stability, time omain performance an robustness. The performance specification, equivalent time constant an stability inex are specifie in the close loop transfer function an relate to the controller parameters algebraically. Habitually, the orer of the controller is less than the orer of the plant. The output of the controlle close-loop system is y = N(sF(s P(s r+ A(sN(s, (14 P(s where y is the output, r is the reference input, u is the control an is the external isturbance signal, N(s an D(s are the numerator an the enominator of the transfer function of the plant, respectively, A(s is the enominator polynomial of the controller transfer function, while F(s an B(s are calle the reference numerator an the feeback numerator polynomials of the controller transfer function. Also P(s is the characteristic polynomial an given by P(s = D(sA(s+N(sB(s = n µ i s i. (15 i=

3 272 F. Haouari B. Nourine M. S. Boucherit M. Tajine: A COEFFICIENT DIAGRAM METHOD CONTROLLER WITH... The nominal mathematical moel is R(s = N(s D(s = a ms m +a m 1 s m 1 + +a b n s n +b n 1 s n 1 + +b. (16 G(x = ( G1 (x, G 2 (x F(x = (F 1 (x F 2 (x. (26 The controller polynomials A(s an B(s are A(s = n l i s i B(s = i= n k i s i. (17 i= The equivalent time constant T inicate the time response spee an the stability inices γ i give the stability an the waveform of the time response. They are efine in terms of the coefficients of the characteristic polynomial in (15 as T = µ 1 µ, γ i = µ 2 i µ i 1 µ i+1, for i = 1...(n 1. (18 The settling time an the equivalent time constant is efine as t s T = 2.5 3, γ 1 = 2.5, γ i = 2, ;i = 2 (n 1, ;γ = γ n =. (19 The last values can be ajuste to assure the require performance, so that γ i > 1.5 for all i = 1 (n 1. Then the characteristic polynomial to be use to esign the parameters of a controller is P(s = µ [ { n i=2 (i 1 1 j=1 γ j i j (T s i} ] +T s+1. (2 Finally F(s whichisusuallyefineasthepre-filteruse for reucing the steay state error to zero an is selecte as a constant efine by F(s = P(s s= N(s 4 CONTROL OBJECTIVE. (21 In this section, we use the CDM-Backstepping algorithm to evelop the positions control law. These positions will converge exponentially to the reference value. The error positions are efine as. e 1 = q 1 = x 1 q 1, = q 2 = x 3 q 2 (22 an their erivatives are ė 1 = x 2 q 1, ė 2 = x 4 q 2. (23 Let ζ = (x 2 x 4. (24 then ζ = F(x+G(xτ (25 We can conclue that the positions errors e 1 an can be controlle using the auxiliaries variables ζ 1 an ζ 2 respectively, which can be controlle using the real control signal τ. Let ζ 1 an ζ 2 be the values of ζ 1 an ζ 2 respectively, which ensuring the stabilization of the positions tracking error e 1 an, also these esire values are etermine using Lyapunov approach by consiering the ynamic equation of e 1 an, consequently e 3 = ζ 1 ζ 1 an e 4 = ζ 2 ζ 2 with E = (e 3 e 4. Then ζ = (ζ 1 ζ 2, E = ζ ζ. (27 The control signal is written as follows A 1 (xτ +A 2 (x τ t = E c(t (28 E c (t = C (xζ B (xζ B 1 (x ζ, (29 A 1 (x, A 2 (x, C (x, B (x an B 1 (x are nonlinear matrix gains of multivariable nonlinear CDM controller introuce in (28 an (29. A backstepping proceure [9 2] is propose to eterminate the gains matrix assuring the exponential stability result for the links positions tracking errors. Step 1: Firstly we esign the virtual control law ζ 1(t then ζ 2(t, theirs positions error must asymptotically converge to zero. Step 2: seconly we choose the gains matrix A 1 (x, A 2 (x, C (x, B (x an B 1 (x by employing the augmente Lyapunov function that oblige the errors to track an exponential convergence. 5 CDM BACKSTEPPING CONTROL DESIGN Proposition 1. The positions tracking error e 1 an are exponentially stable with the following conition ζ 1 = λ 1 e 1 + q 1, ζ 2 = λ 2 + q 2. (3 Proof 1. The Lyapunov formulation can be written as V 1 = 1 2 e e2 2, (31 where its time erivative can be represente as V 1 = e 1 ė 1 + ė 2. (32 Since thevirtualcontrol ζ 1 an ζ 2 tracktheesirevalue specifie in (3, the erivative of the Lyapunov function become negative an takes the next form. V 1 λ 1 1 λ 1 2. (33

4 Journal of ELECTRICAL ENGINEERING 66, NO. 5, Then V 1. (34 As a result, the exponentially stability can be achieve for e 1 an. The control signal τ = (τ 1 τ 2 that oblige the errors e 3 an e 4 toconvergetozerowillbenoweucte. Let A 1 (x = K G(x t, A 2 (x = KG(x (35 with any positive efinite matrix K. Combining equations (27 with (29 gives An tacking Then E = ( B 1 C I ζ B 1 E c (36 C (x = B (x = C = Its secon erivative is [ ] c1. (37 c 2 E c = C E. (38 Ë c (t = C ζ (t C ζ(t. (39 Combining equations (28, (29 an (35 gives ζ(t = F(x+K 1 E c (4 K 1 = K 1. (41 Substituting equation (4 into (39, we obtain Then Ë c (t = C ζ (t C ( F(x+K 1 E c. (42 ( t Ė c (t = C ζ (t C F(x+K1 E c (ρρ (43 an using equation (38 with An tacking Ė(t = H(x K 2 E(ρρ (44 H(x = F(x ζ (t, K 2 = C K 1. (45 H(x = K 1 = [ ] [ H1 (x F1 (x = ζ ] 1 H 2 (x F 2 (x ζ 2 [ δ1 sign(z 1 ] δ 2 sign(z 2 z = e i+1 e i+1 (ρρ, i = 1, 2. (46 (47 Proposition 2. Consier the robot manipulator ynamic (8, in close-loop with the multivariable CDM control (28, suppose that the gains δ 1, δ 2, c 1 an c 2 are such that c 1δ 1 sign(z 1 e 3 (ρρ 1with 1 e 1 + H 1 (x, c 2δ 2 sign(z 2 e 4 (ρρ 2with 2 + H 2 (x. (48 Accoring to Lyapunov stability, it implies that the tracking errors e 1 (t, (t, e 3 (t an e 4 (t are exponentially stable an the close-loop system is internally stable. Proof. Consier the augmente Lyapunov function [12]. V 2 = V E E. (49 Its erivative along the plant trajectories is given by V 2 = V 1 +E top Ė. (5 Using the expressions of e 3 (t, e 4 (t an equation (32, we get V 2 = λ 1 1 λ 2 2 +e 1e 3 + e 4. (51 This gives V 2 λ 1 1 λ 2 2 +E [Ė +( e1 ]. (52 Changing the ynamics of E by (44 an the control signal by (28, then one has where v(t = E [( e1 If v(t < then Notice that v(t=e [( e1 V 2 λ 1 1 λ 2 2 +v(t ( ( ] H1 (x K H 2 (x 2 E(ρρ. (54 V 2 λ 1 1 λ 2 2. (55 ( H1 (x H 2 (x ( ] c1 δ 1 z 1 sign(z 1. c 2 δ 2 z 2 sign(z 2 (56 Furthermore, to guarantee the negativity of V2, the gains δ 1, δ 2, c 1 an c 2 must be chosen from inequality (48 for the reason that z i S(z i > an v(t. Therefore, it can be conclue that Backstepping V 2. (57 e 1 CDM Robot Fig. 2. CDM-Backstepping controller scheme

5 274 F. Haouari B. Nourine M. S. Boucherit M. Tajine: A COEFFICIENT DIAGRAM METHOD CONTROLLER WITH... position (ra position first joint esire position (a 9 1 (c 2.5 position (ra esire position position secon joint (b 9 1 ( y (m esire path path x (m (e Fig. 3. CDM-Backstepping control; test one (a Desire an positions of the first joint, (b Desire an positions of the secon joint, (c Actual torque of the first joint, ( Actual torque of the secon joint, (e Actual an esire path position (ra position esire position (a position (ra esire position (b y (m (e path (c 25 first joint position 9 1 secon joint ( esire path x (m Fig. 4. CDM-Backstepping control; test two (a Desire an positions of the first joint, (b Desire an positions of the secon joint, (c Actual torque of the first joint, ( Actual torque of the secon joint, (e Actual an esire path It implies that the ynamic system is exponentially stable accoring to Lyapunov stability theorem. The bouneness of state vector X = (x 1,x 2,x 3,x 4 is not guarantying by the asymptotically convergence of tracking errors. q 1 an q 2 are boune an the errors e 1 an are exponentially stable so that the state σ = (x 2,x 4 is boune, also the state η = (x 1,x 3 is boune; this proves that the origin of the subsystem σ = η is stable. Remark 2. Notice that strict knowlege of the limits i an functions F i is not require. Bouns can be employe on these variables to ensure a nonlinear robust

6 Journal of ELECTRICAL ENGINEERING 66, NO. 5, position (ra position esire position (a position (ra position (b first joint (c esire position 9 1 secon joint ( y (m.4.8 path esire path (e x (m 2. Fig. 5. CDM-Backstepping control; test three (a Desire an positions of the first joint, (b Desire an positions of the secon joint, (c Actual torque of the first joint, ( Actual torque of the secon joint, (e Actual an esire path controller uner conition of isturbance, parametric uncertainties an noises. The Integral gains must be satisfactorily large to realize(48. The CDM gains are selecte as esignate in Proposition 2. The following CDM gains have been use for all the simulate situations. (δ 1, δ 2 = (13, 11, (c 1, c 2 = (.7,.6. (58 6 SIMULATION RESULT In orer to evaluate the quality of the erive algorithms of nonlinear control, simulation tests were performe using Matlab for circular path an butterfly shape trajectory. Test one: External isturbances The externals isturbances that can be applie are isturbance in torques of τ 1 (t = 1 Nm an τ 1 (t = 13 Nm are applie for each joints of robotic system. The simulation plots shown in Fig. 3 inee verify that our CDM-Backstepping esign scheme can guarantee the best performance for each joint of the robotic manipulator to track its esire trajectory exponentially an eliminate the isturbance with no overshoot an with a negligible steay state error. Test two: Parametric uncertainties In the secon test for the robustness evaluation of the controllers, we introuce the following parametric uncertainties in the robot moels. Tool attache to en effector, then parametric uncertainties at secon link in mass m = 4 kg an in length l =.1 m. Coulomb friction an viscous friction are ae to each joint of robot manipulator an given by f cv1 = 2.5x 2 (t+1.8sign ( x 2 (t an f cv2 = 2x 4 (t+1.2sign ( x 4 (t. The simulation results in Fig. 4 show the strong robustness of the propose CDM-Backstepping control towars uncertainties affecting the robot mechanical parameters. Test three: Change in the esire path To test the controller s robustness the simulations have been execute using the same last externals isturbances an parametric uncertainties with noises applie for a butterfly shape trajectory, whish are use as a esire en-effector s path. The simulations results are epicte in Fig. 5, this?gure show that CDM-Backstepping control present a robust path following in 2D isplacement. 7 CONCLUSION This paper reveals a new approach of robust control systems CDM for robot manipulators using backstepping esign. The major istinctive of the propose approach is the application of the novel Lyapunov functions to construct the CDM-Backstepping controller. Global stability results are obtaine an the tracking errors converge to zeros with exponential forms. Simulations results have been given to emonstrate the theoretical analysis use in the controller. Further investigation can be irecte to the robustness of CDM-Backstepping controller.

7 276 F. Haouari B. Nourine M. S. Boucherit M. Tajine: A COEFFICIENT DIAGRAM METHOD CONTROLLER WITH... Appenix Rate ata of the simulate robot manipulator b 1 = 75 N/r/s, b 2 = 1 N/r/s, l 1 = 1 m, l 2 = 1 m, m 1 = 1 kg, m 2 = 1 kg, k 1 = 4 Nm/v, k 2 = 2 Nm/v, g = 9.81ms 2. References [1] LEWIS, F. L. ABDALLAH, C. T, DAWSON, D. M.: Robot Manipulator Control Theory an Practice, Marcel Dekker Inc, USA, New York, 24. [2] ANGELES, J.: Funamentals of Robotic Mechanical Systems: Theory, Methos, an Algorithms, Springer-Verlag, New York, 23, pp [3] CHEN, Y. F. HUANG, A. C.: Controller Design for a Class of Uneractuate Mechanical Systems, IET Control Theory an Applications 6 No. 1 (212, [4] BUDIYONO, A. KARTIDJO, M. SUGAMA, A.: Coefficient Diagram Metho for the Control of an Unmanne Unerwater Vehicle, Inian Journal of Marine Sciences 38 No. 3 (29, [5] MITSANTISUK, CH. NANDAYAPA, M. OHISHI, K. 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He got the Engineer egree in automatic control in 21, the Magister egree in 26 in Electrical Engineering, from Automatic Control Department, of Ecole Nationale Polytechnique of Algiers, Algeria. He is currently preparing his octorate egree in automatic control. He is Associate Professor at Ecole nationale e Technologie, His research interests inclue robust an nonlinear control. Bali Noureine was born in July 1965, in Algeria. He got the Engineer egree in automatic control in 1992, the Magister egree in 1996 an the PhD in 27 in Electrical Engineering, from Automatic Control Department, of Ecole Nationale Polytechnique of Algiers. He joine the Houari Boumeiene University of Sciences an Technology (USTHB in Algiers. He is Associate Professor, Vice-Dean in the Electrical Engineering an Computing faculty, an Hea of the Inustrial Maintenance an Reliability research team, in the Inustrial an Electrical Systems Laboratory. His current research interests are in the fiels of preictive control applications, control systems an maintenance management. Mohame Seghir Boucherit was born in, 1954 in Algiers, Algeria. He receive the Engineer egree in Electrotechnics, the Magister egree an the Doctorat Etat (PhD egree in Electrical Engineering, from the Ecole Nationale Polytechnique, of Algiers, Algeria, in 198, 1988 an 1995 respectively. Upon grauation, he joine the Electrical Engineering Department of Ecole Nationale Polytechnique. He is a Professor, Hea of the Inustrial systems an Diagnosis research team of the Process Control Laboratory an his research interests are in the area of Electrical Drives, Process Control Applications, an Diagnosis. Mohame Tajine was born in 1966, in Algiers, Algeria. He receive the Engineering egreefrom the Ecole Nationale Polytechnique, Algiers, Algeria, in 199, the MSc an PhD egrees in automatic control from the National Polytechnic Institute of Grenoble, Grenoble, France, in 1991 an 1994, respectively. From 1995 to 1997, he was a Researcher at the University of Picarie, Amiens, France. 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 an nonlinear control.