Nonlinear Control Via Input-Output Feedback Linearization of a Robot Manipulator

Home , Nonlinear control

Advances in Science, Technology and Engineering Systems Journal Vol. 3, No. 5, 374-381(2018) ASTESJ www.astesj.com ISSN: 2415-6698

NonLinear Control via Input-Output Feedback Linearization of a Robot Manipulator

Wafa Ghozlane*, Jilani Knani Laboratory of Automatic Research LA.R.A, Department of Electrical Engineering, National School of Engineers of Tunis ENIT, University of Tunis ElManar, 1002 Tunis, Tunisia.

A R T I C L E I N F O A B S T R A C T Article history: This paper presents the input-output feedback linearization and decoupling algorithm for Received: 08 September, 2018 control of nonlinear Multi-input Multi-output MIMO systems. The studied analysis was Accepted: 13 October, 2018 motivated through its application to a robot manipulator with six degrees of freedom. The Online: 18 October, 2018 nonlinear MIMO system was transformed into six independent single-input single-output SISO linear local systems. We added PD linear controller to each subsystem for purposes Keywords: of stabilization and tracking reference trajectories, the obtained results in different Input-Output Feedback simulations shown that this technique has been successfully implemented. linearization Nonlinear system Robot manipulator MIMO nonlinear system SISO linear systems PD controller

1. Introduction the angular position of each joint of this robot arm for stabilization and tracking purposes. The obtained results in different In recent years, Feedback linearization has been attracted a simulations shown the efficiency of the derived approach [7]. great deal of interesting research. It's an approach designed to the nonlinear control systems, which based on the idea of This paper is organized as follows: It is divided into five transforming nonlinear dynamics into a linear form. The base idea sections. In Section 2, a description of the input-output feedback of this technique is to algebraically transform a nonlinear linearization approach is detailed. In Section 3, a simplified dynamics system into a totally or partially linear one, so that linear dynamic model of a robot manipulator with six degrees of control techniques can be applied. This notion can be used for freedom is presented, the input-output feedback linearization both stabilization and tracking control objectives of SISO or method is applicated to the above robot and the construction of linear PD controller is derived. In Section 4, the simulation results MIMO systems, and has been successfully applied to a number of are presented. Finally, the conclusion was elaborated in Section 5. practical nonlinear control problems such as [1-4]. 2. Input-Output feedback linearization for MIMO In fact, this technique has been successfully implemented in nonlinear system. several faisable applications of control, such as industrial robots, high performance aircraft, helicopters and biomedical dispositifs, In this section, we discussed the approach of input-output more tasks used the methodology are being now well advanced in feedback linearization of nonlinear systems, the central goal of industry [5-6]. feedback-linearization is to design a nonlinear-control-law as assumed that the inner-loop control is, in the most suitable case, In this case, we applied this technique to lead the control for precisely linearizes the nonlinear system after appropriate state each joint of a robot manipulator that is has six degrees of freedom, space modification of coordinates [1]. The developer can then which the equations of motion form a nonlinear, complex build an outer-loop-control in the new coordinates to obtain a linear relation between the output Y and the input V and to satisfy dynamic and multivariable system, then, we elaborated a PD the traditional control design specifications such as tracking, linear controller for each decoupled linear subsystem to control disturbance rejection, as shown in Figure 1. *Corresponding Author: Wafa Ghozlane, Email: [email protected] www.astesj.com 374 https://dx.doi.org/10.25046/aj030543 W . Ghozlane et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 3, No. 5, 374-381 (2018) Firstly, a diffeomorphism is a differentiable function whose inverse exists and is also differentiable. Second, we should assume that both the function and its inverse to be infinitely differentiable, such functions are usually referred to as ℂ∞ diffeomorphisms [3-4].

The diffeomorphism is used to transform one nonlinear system in another nonlinear system by making a change of variables of the form: Figure 1. Structure of Input-Output Feedback Linearization Approach. 푧 = 훷(푥) (5) The basic condition for using feedback linearization method is nonlinear dynamic MIMO of n -order with p number of inputs and Where Φ(x) represents n variables; outputs described in the affine form; 훷 푇 1 [ℎ 퐿 ℎ … 퐿 푟1−1ℎ ] 훷 1 푓 1 푓 1 푝 훷(푥) = [ 2] = [ ⋮ ], (6) ⋮ 푇 푋̇(푡) = 푓(푋(푡)) + ∑ 푔 (푋(푡))푈 (푡) 푟푝−1 푖 푖 [ℎ푝 퐿푓ℎ푝 … 퐿 ℎ ] 훷푛 푓 푝 푖=1 (1) 푌 (푡) = ℎ (푋(푡)) 푇 푖 푖 푥 = [푥1, 푥2 … 푥푛] { 푖 = 1,2, … 푝 The goal is to obtain a linear relation between the inputs and

Where, the outputs by differentiating the outputs 푦푗 until the inputs appear.

푇 푛 Suppose that 푟푗 is the smallest integer such that fully one of the 푋 = [푥1, 푥2 … 푥푛] ∈ 푅 : is the state vector. (푟푗) inputs appears in 푦푗 using this expression: 푇 푝 푈 = [푢1, 푢2 … 푢푝] ∈ 푅 : is the control input vector. p r (r ) 푇 (rj) j ( ) j−1 ( ) 푝 yj = Lf hj x + ∑ Lgi(Lf hj x )ui (7) 푌 = [푦1, 푦2 … 푦푝] ∈ 푅 : is the output vector, 푓(푋),푎푛푑 푔푖(푋): i=1 are n-dimentional smooth vector fields. i, j = 1,2, … p ℎ푖(푋): 푖푠 smooth nonlinear functions,with i=1,2...n. Where, 퐿 푖ℎ and 퐿 푖ℎ : Are the 푖푡ℎ Lie derivatives of h (x) Theorem1: 푓 푗 푔 푗 j respectively in the direction of f and g. Let f: ℛn → ℛn represent a smooth vector field on ℛn and let 휕ℎ 휕ℎ 퐿 ℎ (푥) = 푗 푓(푥), 퐿 ℎ (푥) = 푗 푔 (푥) (8) h : ℛn → ℛn represent a scalar function. The Lie Derivative of h, 푓 푗 휕푥 푔 푗 휕푥 푖 r : is the relative degree corresponding to the y output, it's the with respect to f, denoted Lfh, is defined as [1-2]. j j number of necessary derivatives so that at least one of the inputs 푝 휕ℎ 휕ℎ appear in the expression [5]. 퐿푓ℎ = 푓(푥) = ∑ 푓푖(푥) (2) 휕푥 휕푥푖 푖=1 If expression 퐿푔ℎ푗(푥) = 0 , for all i, then the inputs have not appeared in the derivation and it's necessary to continu the The Lie derivative is the directional derivative of h in the direction derivation of the output y . of f(x), in an equivalent way, the inner product of the gradient of j 2 h and f. We defined by Lf h the Lie Derivative of Lfh with The system (1) has the relative degree (r) if it satisfies: respect to f: 퐿 퐿 푘ℎ = 0 0 < 푘 < 푟 , 0 ≤ 푖 ≤ 푛, 0 ≤ 푗 ≤ 푛 (9) 2 푔푖 푓 푗 푗−1 퐿푓 ℎ = 퐿푓 ( 퐿푓ℎ) (3) { 푘 퐿푔푖퐿푓 ℎ푗 ≠ 0 푘 = 푟푗−1 In general we define: The total relative degree (r) was considered as the sum of all 푘 푘−1 the relative degrees obtained using (7) and must be less than or 퐿푓 ℎ = 퐿푓(퐿푓 ℎ) 푓표푟 푘 = 1, . . . , 푝 (4) equal to the order of the system (10): 0 푛 with 퐿푓 ℎ = ℎ 푟 = ∑ 푟푗 ≤ 푛 (10) Theorem2: 푗=1 To find the expression of the nonlinear control law U that n n n The function Φ: ℛ → ℛ defined in a region Ω ⊂ ℛ he is allows to make the relationship linear between the input and the called difeomorphisme if it checks the following conditions: output [6], the expression (2) is rewritten in matrix form as:

www.astesj.com 375 W . Ghozlane et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 3, No. 5, 374-381 (2018)

푇 푟1 푟푝 0 1 … 0 [푦1 … 푦푝 ] =∝ (푥) + 훽(푥). 푈 (11) 0 0 … 0 0 A = [… … … …] ∈ ℛri×ri; B = [⋮] ∈ ℛri; 푇 푇 ri ri 푟1 푟푝 0 0 … 1 푉 = [푣1 푣2 … 푣푝] = [푦1 … 푦푝 ] (12) 1 0 0 … 0 Where: [ ] ri Cri = 1 0 … 0 ∈ ℛ 퐿 푟1ℎ (푥) 푓 1 . 2.2. Design of the new control vector V: . ∝ (푥) = . (13) The vector v is designed to according the control objectives, for 푟푝 퐿푓 ℎ푝(푥) the tracking problem considered, it must satisfy:

[ ] rj rj−1 rj−1 vj = ydj + Krj−1 (ydj − yj ) + ⋯ 퐿 (퐿 (푟1−1)ℎ (푥)) 퐿 (퐿 (푟1−1)ℎ (푥)) … 퐿 (퐿 (푟1−1)ℎ (푥)) 푔1 푓 1 푔2 푓 1 푔푝 푓 1 (푟2−1) (푟2−1) (푟2−1) + K1 (yd − yj) ; (18) 퐿푔 (퐿푓 ℎ2(푥)) 퐿푔 (퐿푓 ℎ2(푥)) … 퐿푔 (퐿푓 ℎ2(푥)) j 1 . 2 . 푝 . 훽(푥) = … (14) . . . 1 ≤ 푗 ≤ 푝 . . where, . (푟푝−1) ( ) (푟푝−1) ( ) (푟푝−1) ( ) [퐿푔1(퐿푓 ℎ푝 푥 ) 퐿푔2(퐿푓 ℎ푝 푥 ) … 퐿푔푝(퐿푓 ℎ푝 푥 )] 2 rj−1 rj {ydj, ydj , … . , ydj , ydj } denote the imposed reference If β (x) is not singular, then it is possible to define the input trajectories for the different outputs. If the 퐾푖 are chosen so that transformation "the nonlinear control law " which has this form: the polynomial [9-10]; 푟푗 푟푗−1 푠 + 퐾푟푗−1푠 + ⋯ + 퐾2푠 + 퐾1 = 0 are Hurwitz (has roots 푈 = 훽(푥)−1. (−∝ (푥) + 푉) (15) with negative real parts). Then it can be shown that the error 푇 ( ) 푒푗(푡) = 푦푑푗(푡) − 푦푗(푡), satisfied lim 푒푗 푡 = 0. 푉 = [푣1 푣2 … 푣푝] 푡→∞

푇 3. Input-Output feedback linearization approach applied 푈 = [푢 푢 … 푢 ] 1 2 푝 to a robot manipulator with six degrees of freedom

푟푗 푦푗 = 푣푗 3.1. Dynamic modeling of a robot manipulator where In this section, we have applied the proposed approach to a V: Is the new input vector. is called a decoupling control law dynamic multivariable system which represent a robot arm with six degrees of freedom "EPSON C4". This is an open chain β(x):Is the invertible (pxp) matrix. Is called a decoupling matrix kinematic manipulator robot consisting of seven rigid bodies of the system. interconnected by six revolute joints n = 6 as [11]. So, deriving the motion of robot is a complex task due to the nonlinearities 2.1. Non-linear coordinate transformation: present in this system and the large number of degrees of freedom z Φ 푇 [12-13]. Then, it is essential to understand exactly the dynamics 1 1 [ℎ 퐿 ℎ … 퐿 푟1−1ℎ ] z Φ 1 푓 1 푓 1 of this interconnected chain of rigid bodies, to detemine the 푍 = [ 2] = [ 2] = [ ⋮ ] (16) ⋮ ⋮ 푇 inverse dynamics model such as relation (19), we analyzed the [ℎ 퐿 ℎ … 퐿 푟푝−1ℎ ] z푛 Φ푛 푝 푓 푝 푓 푝 evolution of motion of this mechanical non linear system by using the Euler-Lagrange equations, which is represented by the By applying the linearizing law to the system,we can transforme equation (20). the nonlinear system into linear form [7-8]: Γ = f(q, q̇ , q̈ , f ) (19) ̇ e {Z = Az + BV (17) ∂Lj Y = CZ ( ) n ∂qi̇ ∂Lj Γi = ∑j=1 d − i, j = 1, … , n (20) With, dt ∂qi

where Γ, q, q̇ , q̈ and fe depicting Torques, articular positions, Ar1 … 0 Br1 … 0 A = [ … … … ] , B = [ … … … ], velocities, accelerations and the external force.

0 … Arp 0 … Brp th 퐿푗: Defines the lagrangian of the 푗 link, which is the difference of the kinetic and potential energy, equal to 퐸푗 - 푈푗. Cr1 … 0 C = [ … … … ] th 퐸푗and 푈푗: Define the kinetic and the potential energies of the 푗 0 … Cr p link. And, www.astesj.com 376

W . Ghozlane et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 3, No. 5, 374-381 (2018)

The calculation of the kinetic energy; Uj: Defines the potential energy of the link 퐶푗, which is expressed by the equation (27): In this section, we have calculated the total kinetic energy of the system which depends on the configuration and joint velocities, T 0 0 Uj = −g0 (Mj × P + A × MSj) such that [13], then, it was described by the equation (21). j j T MSj n = −[g0 0] × T0j × [ ] (27) E = ∑j=1 Ej (21) Mj

퐸푗 : means the kinetic energy of the link 퐶푗 , which can be Then, we have followed the Euler-Lagrange formalism such formulated such that (24). Firstly, we have calcuated the linear that equation (20), we obtained the following relation (28): velocity and the angular velocity using the equations (22) and (23). Γ = A(q)q̈ + C(q, q̇ )q̇ + Q(q) (28) 푗푉 = 푗퐴(푗−1푉 + 푗−1푊 × 푃푗−1) + 휎 푞̇ 푎 (22) 푗 푗−1 푗−1 푗−1 푗 푗 푗 푗 where 푗 푗 푗−1 푗푊 = 푗−1퐴 × 푗−1푊 + 휎̅푗 × 푞̇푗 × 푎푗 (23) A(q): Represents the matrix of kinetic energy (n × n), these elements are calculated as follows: 푗−1 2 푉: The linear velocity, it is the derivative of the position q̇ i 푗−1 −퐴푖푖 is equal to the coefficient of located in the expression 푗−1 2 vector 푃푗 . of the kinetic energy. 푗−1 -퐴 is equal to the coefficient of 푞̇ 푞̇ 푗−1푊: The angular velocity. 푖푗 푖 푗 The initial conditions for a robot which the base is fixed, are 퐶(푞, 푞̇ )푞̇ : Defines the vector of coriolis and centrifugal forces/torques (n × 1), these elements are calculated from the 0 0 0푉 = 0 and 0푊 = 0. Christoffel symbol 푐푖,푗푘 such as system (29):

Then,we have expressed these relations (22) and (23) in Equation c = ∑n c q̇ (24) as. ij k=1 i,jk k { 1 ∂Aij ∂Aik ∂Ajk (29) ci,jk = [ + − ] 1 2 ∂qk ∂qj ∂qi 퐸 = [푗푊푇 푗퐽푗푊 + 푀 푗푉푇 푗푉 + 2푀푆푇(푗푉 ∧ 푗푊)] (24) 푗 2 푗 푗 푗 푗 푗 푗 푗 푗 푗 Q(q): Represents the vector of torques/forces of gravity, these where 휕푈 elements are calculated as: 푄푖 = 휕푞푖 푎푗 :is the unit vector along axis 푧푗. The elements of A, C and Q are according to the geometric and j j Mj, jMS et jJ:are the inertial standard parameters. inertial parameters of the mechanism. The dynamic equations of the robot form a system of n differential equations of the second Mj : is the mass of link 퐶푗. order, coupled and nonlinear. Then, since the inertia matrix A is invertible for q ∈ ℛ푛 we may solve for the acceleration q̈ of the j jMS :design the first moments of inertia of link 퐶푗 about the manipulator as [14]. origin of the frame 푅 It is equal to jMS =[ MX MY MZ ]T. 푗 j j j j q̈ = f(q, q̇ , Γ) (30) j −1 jJ: is the inertial tensor matrix (3x3) of link Cj with respect to the q̈ = −A(q) [C(q, q̇ )q̇ + Q(q) − Γ] (31) frame 푅 , it is expressed by the matrix (25). 푗 With, q = [q q q q q q ]T: The angular position vector (6x1). IXXj IXYj IXZj 1 2 3 4 5 6 j J == [IXZ IYY IYZ ] (25) 푇 j j j j 푞̇ = [푞̇1 푞̇2 푞̇3 푞̇4 푞̇5 푞̇6] : The angular velocity vector (6x1). I I I 푇 XZj YZj ZZj 푞̈ = [푞̈1 푞̈2 푞̈3 푞̈4 푞̈5 푞̈6] : The angular acceleration vector (6x1). 푇 훤 = [훤1 훤2 훤3 훤4 훤5 훤6] : The input torques vector(6x1). IXXj , IXYj , IXZj , IYYj , IYZj and IZZj represent the elements of the j 3.2. Application of the input-output feedback linearization inertial tensor the symetric matrix J of each link 퐶 which is j 푗 approach to a robot manipulator with six degrees of expressed by the matrix (25), we have defined all these inertial freedom: parameters values in our recent work [14]. The calculation of the potential energy; In this section, after we dermined the inverse dynamic model of the system [15-16], we used the equation of motion of the six- In this section, we have represented the potential energy for link of the rigid manipulator robot "EPSONC4" which a manipulator arm [14], which is written by the equation (26). represented by the relation (31) and we considered the state variables of the system defined in state space as; n U = ∑j=1 Uj (26) www.astesj.com 377 W . Ghozlane et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 3, No. 5, 374-381 (2018)

x1 = q1, x2 = q̇ 1, x3 = q2, x4 = q̇ 2, x5 = q3, x6 = q̇ 3, 0 0 0 0 0 0 x7 = q4, x8 = q̇ 4, x9 = q5, x10 = q̇ 5, x11 = q6, x12 = q̇ 6, 0 0 0 After derivation of the above state variables, we written the 0 0 0 obtained system as; 0 0 0 x1̇ = x2 0 0 0 푔4(푥) = ; 푔5(푥) = ; 푔6(푥) = ẋ = q̈ = −A(x )−1[C(x , x )x + Q(x ) − Γ ] 0 0 0 2 1 1 1 2 2 1 1 −1 ẋ = x 퐴(푥7) 0 0 3 4 0 0 ẋ = q̈ = −A(x )−1[C(x , x )x + Q(x ) − Γ ] 0 4 2 3 3 4 4 3 2 퐴(푥 )−1 0 ẋ = x 0 9 5 6 0 0 0 −1 −1 x6̇ = q̈ 3 = −A(x5) [C(x5, x6)x6 + Q(x5) − Γ3] [ ] [ ] [퐴(푥 ) ] (32) 0 0 11 x7̇ = x8 −1 And, x8̇ = q̈ 4 = −A(x7) [C(x7, x8)x8 + Q(x7) − Γ4] x9̇ = x10 푇 푋 = [푥1, 푥2, 푥3, 푥4, 푥5, 푥6푥7, 푥8, 푥9, 푥10, 푥11, 푥12] ; ( )−1[ ( ) ( ) ] x10̇ = q̈ 5 = −A x9 C x9, x10 x10 + Q x9 − Γ5 푋̇ = [푥̇ , 푥̇ , 푥̇ , 푥̇ , 푥̇ , 푥̇ 푥̇ , 푥̇ , 푥̇ , 푥̇ , 푥̇ , 푥̇ ]푇; x ̇ = x 1 2 3 4 5 6 7 8 9 10 11 12 11 12 [ ]푇 [ ]푇 −1 푈 = 푢1, 푢2, 푢3, 푢4, 푢5, 푢6 = 훤1, 훤2, 훤3, 훤4, 훤5, 훤6 ; {x12̇ = q̈ 6 = −A(x11) [C(x11, x12)x12 + Q(x11) − Γ6] Then, the affine form of nonlinear, multivariable and dynamic 푦1 = ℎ1(푥) = 푥1 = 푞1; 푦 = ℎ (푥) = 푥 = 푞 ; model of the robot manipulator is appeared which given by the 2 2 3 2 푦 = ℎ (푥) = 푥 = 푞 ; following system (33): Links positions 3 3 5 3 (34) 푦4 = ℎ4(푥) = 푥7 = 푞4; p 푦5 = ℎ5(푥) = 푥9 = 푞5; ̇ X(t) = f(X(t)) + ∑ gi(X(t))Ui(t) {푦6 = ℎ6(푥) = 푥11 = 푞6; i=1 (33) So, we made the derivation of each output 푦 intel the inputs Yi(t) = hi(X(t)) 푖 { i = 1,2, … 6 appeared in the expression and we computed the relative degrees 푟푖 for each joint of the robot as follows [17-18]; where, 푦 = ℎ (푥) = 푥 1 1 1 푥2 푦̇ = 퐿 ℎ (푥) = 푥̇ = 푥 −1 1 푓 1 1 2 −퐴(푥1) [퐶(푥1, 푥2)푥2 + 푄(푥1)] (2) 2 푥 푦1 = 퐿푓 ℎ1(푥) + 퐿푔퐿푓ℎ1(푥)푢 4 −1 푟1 = 2 −퐴(푥3) [퐶(푥3, 푥4)푥4 + 푄(푥3)] 푦 = ℎ (푥) = 푥 푥6 2 2 3 −1 푦̇ = 퐿 ℎ (푥) = 푥̇ = 푥 −퐴(푥5) [퐶(푥5, 푥6)푥6 + 푄(푥5)] 2 푓 2 3 4 푓(푥) = 푥 ; (2) 2 ( ) ( ) 8 푦2 = 퐿푓 ℎ2 푥 + 퐿푔퐿푓ℎ2 푥 푢 (35) −1 −퐴(푥7) [퐶(푥7, 푥8)푥8 + 푄(푥7)] 푟 = 2 2 푥10 ⋮ −1 −퐴(푥9) [퐶(푥9, 푥10)푥10 + 푄(푥9)] 푦6 = ℎ6(푥) = 푥11 푥12 푦̇ = 퐿 ℎ (푥)푥̇ = 푥 6 푓 6 11 12 [−퐴(푥 )−1[퐶(푥 , 푥 )푥 + 푄(푥 )]] (2) 2 11 11 12 12 11 푦6 = 퐿푓 ℎ6(푥) + 퐿푔퐿푓ℎ6(푥)푢 0 0 0 { 푟6 = 2

퐴(푥 )−1 0 0 1 Therefore, the relative degree of each joint 푟 is well defined and 0 0 푖 0 ( )−1 0 is equal to 2, so we computed the nonlinear control law 푢푖(푡)of 0 퐴 푥3 0 0 0 each joint of the system as this relation (36); −1 0 0 퐴(푥5) 푟 푔1(푥) = ; 푔2(푥) = ; 푔3(푥) = 퐿 푖ℎ (푥(푡)) 푣 (푡) 0 0 0 푓 푖 푖 푢푖(푥(푡)) = − + (36) 푟푖−1 푟푖−1 0 0 0 퐿푔 퐿푓ℎ푖(푥(푡)) 퐿푔 퐿푓ℎ푖(푥(푡)) 0 0 0 푖 = 1 … 6 0 0 0 0 0 0 By using the nonlinear control law and diffeomorphic [ 0 ] [ 0 ] [ 0 ] transformation given above, the nonlinear dynamic system with six degrees of freedom is converted into the following

Brunovesky canonical form and simultaneously output decoupled [19].

www.astesj.com 378 W . Ghozlane et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 3, No. 5, 374-381 (2018) 푍̇ = 퐴푧 + 퐵푉 푦 (푡) − 푦 (푡) { (37) 푖 푖 푑푖 푖 푣푖(푡) = [퐾푝 퐾 ] [ ] (43) 푌 = 퐶푍 푑 (푟푖−1)( ) (푟푖−1)( ) 푦푑푖 푡 − 푦푖 푡 where, Then, we used the diffeomorphic transformation (16) presented

ℎ above, we considered these relations for each SISO subsystem: 1 퐿 ℎ 푓 1 푖 ( ) 푧1 = ℎ푖(푥 푡 ) − 푦푑푖 푧1 ℎ2 푥1 푧̇ 푥̇ v1 { (44) 1 1 푧푖 = 퐿 ℎ (푥(푡)) − 푦̇ 푧 퐿 ℎ 푥 v 2 푓 푖 푑푖 2 푓 2 2 ̇ 푧̇2 푥̇2 2 푍 = [ ⋮ ] = . = [ ⋮ ] ; 푍 = [ ] = [ ] ; V = [ ⋮ ] ⋮ ⋮ 푧 . 푥 푧̇ 푥̇ v 12 . 12 12 12 6 ℎ6 That is leads to the non-linear coordinate transformation given [퐿푓ℎ6] by:

0 1 ⋯ 0 푖 푖 0 0 푧1̇ = 푧2 = 퐿푓ℎ푖(푥(푡)) − 푦̇푑 ⋱ { 푖 (45) ⋱ ⋯ 푧̇푖 = 퐿 2ℎ (푥(푡)) + 퐿 퐿 ℎ (푥(푡))푢 (푥(푡)) − 푦̈ 0 1 2 푓 푖 푔 푓 푖 푖 푑푖 ⋮ ⋮ ⋯ 0 0 A = ; 푟푖−1 푟푖−1 (푗) (푗) ⋱ 푣푖(푡) = − ∑ 퐾푗푍푖 = − ∑ 퐾푗[퐿푓 ℎ푖(푥(푡)) − 푦푑 (푡)] (46) ⋱ ⋮ 푗=0 푗=0 푖 ⋮ ⋱ ⋮ ⋱ ⋮ ⋯ ⋱ ⋮ 0 1 0 ⋯ [ 0 0] So, we expressed the relation (46) in equation (36),we obtained:

푟 −1 0 1 0 푟푖 푖 (푗) (푗) ⋯ 0 ⋯ 0 퐿 ℎ (푥(푡)) − ∑ 퐾푗[퐿푓 ℎ푖(푥(푡)) − 푦푑 (푡)] 1 0 푢 (푥(푡)) = − 푓 푖 + 푗=0 푖 (47) 푖 푟푖−1 푟푖−1 ⋱ ⋯ ⋱ ⋯ 퐿푔 퐿푓ℎ푖(푥(푡)) 퐿푔 퐿푓ℎ푖(푥(푡)) ⋮ 0 ⋮ ⋮ 1 0 ⋮ B = ⋯ ; C = ⋯ 푖 = 1 … 6 1 ⋱ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ Next, we expressed the equation (47) in the subsystem (45), the ⋮ ⋯ ⋯ 0 ⋮ ⋯ ⋱ ⋱ [0 0 1] [ 0 0 1 0] non linear subsystem (45) was transformed to a linear subsystem which expressed as relation (48):

The above matrices A,B and C are of dimension respectively : 푧̇푖 0 1 푧푖 [ 1] = [ ] [ 1] 푖 퐾푖 퐾푖 푖 (48) (12 × 12), (12 × 6) 푎푛푑 (6 × 12). 푧̇2 푝 푑 푧2 We note that, the obtained linear system (37) definied by six 푖 = 1 … 6 decoupled and linear subsystems that is has the following form (38), and i=1...6; The obtained linear subsystem consists of a second order system with linear output, that is can be computed the poles of the above subsystem as; 0 1 0 푧̇ = [ ] 푧 + [ ] 푣 푠 = −휉휔 ∓ 휔 1 − 휉2 (49) { 푖 0 0 푖 1 푖 (38) 1,2 푛 푛√ 푦 = [1 0]푧 푖 푖 Where, 휉: means damping ratio, 휔 :means natural frequency, for 푛 goal of stability we chosed the parameters of each PD controller where, applied to each subsystem as [19]: 푧2푖−1 푧푖 = [ 푧 ] 2푖 퐾푗 = 휔 2 { 푝 푛 푗 = 1 … 6 (50) For the objective of linear control, we added to each subsystem a 푗 퐾 = 휉휔푛 linear PD controller which has represented equation (39) as; 푑

푖 푖 푣푖(푡) = 퐾푝(1 + 퐾푑푠)푒푖(푡) ; 푖 = 1 … 6 (39) 4. Simulation Results

푒푖(푡) = 푦푑푖(푡) − 푦푖(푡) (40) In order to improve the efficiency of the proposed approach, we 푟푖 푣푖 = 푦푖 (41) applied the above method of linearization to a robot manipulator which represent a non linear, decoupled and multivariable system. we obtained the relative degree for each subsystem as : 푟 = 2 푟푖 푖 We imposed the sinusoidal signals as 푦푖푑(푡), 푦̇푖푑(푡) and 푦푖푑 (푡) which are represented inputs desired trajectories to each joint of 푖 (푟푖−1) (푟푖−1) 푖 푣푖 = 퐾푑(푦푑푖 − 푦푖 ) + 퐾푝(푦푑푖 − 푦푖) (42) the studied system. Therefore, the relative degrees 푟푖 is well www.astesj.com 379 W . Ghozlane et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 3, No. 5, 374-381 (2018) defined, we presented the simulation results depicting the output 푟푖 푦푖 (푡)of each joint i=1...6 as shown in Figures 2,3,4,5,6,7,8-9.

푟푖 Figure 5. The output 푦푖 (푡) plot of the joint 2, the relative degree 푟2 = 2.

푟푖 Figure 2. The outputs 푦푖 (푡) plot of each joint, the relative degrees 푟푖 = 0, i=1...6. Figure 6. The output 푦 푟푖 (푡) plot of the joint 3, the relative degree 푟 = 2. Firstly, we made the simulation of the outputs of each joint of 푖 3 the robot, 푦푖(푡) = ℎ푖(푥(푡)) = 푥푖(푡), the results are therefore given in Figure 2, we can see the non linearity of each subsystem. Second, we applied the Lie derivation to each output we shown the results presented in Figure 3 as.

푟푖 Figure 7. The output 푦푖 (푡) plot of the joint 4, the relative degree 푟4 = 2.

푟푖 Figure 8. The output 푦푖 (푡) plot of the joint 5, the relative degree 푟5 = 2.

푟푖 Figure 3. The outputs 푦푖 (푡) plot of each joint, the relative degrees 푟푖 = 1, i=1...6.

In this case, the relative degrees equal to 푟푖 = 1, 퐿푔퐿푓ℎ푖(푥(푡)) = 0, so no one input has been appeared, as shown in Figure 3, the system still non linear. Finally, we made the derivation again of each recent output 푦̇ the inputs appeared 푟푖 푖 Figure 9. The output 푦푖 (푡) plot of the joint 6, the relative degree 푟6 = 2. in the expression and we computed the final relative degrees 푟푖 = 2 for each joint of the robot, 푦 (2) = 퐿 2ℎ (푥) + These Figures 4,5,6,7,8-9 shown that the output of each SISO 푖 푓 푖 subsystem asymptotically tracks the sinusoidal input trajectory 퐿 퐿 ℎ (푥)푢, 퐿 퐿 ℎ (푥) ≠ 0, the results was illustrated in these 푔 푓 푖 푔 푓 푖 with minimum of oscillation and the efficiency of the studied Figures 4,5,6,7,8-9. approach. 5. Conclusion

In this paper, the input-output feedback linearization approach is presented which is a way of algebraically transforming nonlinear, multivaribles, complex and dynamics systems into linear ones, so we can be applied linear control, this technique has attracted lots of research in recent years. For that we applied this proposed approach to derive the control of a robot 푟푖 Figure 4. The output 푦푖 (푡) plot of the joint 1, the relative degree 푟1 = 2. manipulator with six degrees of freedom which followed two major steps. Firstly, by using the above approach and www.astesj.com 380 W . Ghozlane et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 3, No. 5, 374-381 (2018) diffeomorphic transformation, we converted the non linear and decoupled dynamics model of the robot to linear system. Then, we designed a linear PD control law for each decoupled subsystem to control the angular position of each joint of this robot for tracking purposes. Finally, the obtained results in different simulations illustrated the accuracy of the proposed approach. References

[1] MW. Spong, S.Hutchinson, and M. Vidyasagar. Robot Dynamics and Control , Second Edition. January 28, 2004. [2] J.-J. Slotine & W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, 1991. [3] A. Isidori. Nonlinear Control Systems. Springer Verlag, 1989. [4] H. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, Second Edition, 1996. [5] J.Hauser,S.Sastry, and P.Kokolovic,“Non linear control via approximate Input_Output Linearization,The Ball and Beam Exemple”. IEEE Transactions onAutomatic Control ,Vol 37,No.3 March 1992. [6] E.D. Sontag,M. Thoma,A. Isidori and J.H. van Schuppen. Nonlinear and Adaptive Control with Applications, 2008 Springer-Verlag London Limited. [7] H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990. [8] M. Vidyasagar. Nonlinear Systems Analysis. Prentice Hall,Second Edition, 1993. [9] K.M. Hangos, J. Bokor et G. Szederkényi, Analysis and Control of Nonlinear Process Systems, Springer-Verlag London Limited, 2004. [10] J.HAUSER, SH. SASTRY and G. MEYER. “Nonlinear Control Design for Slightly Non minimum Phase Systems: Application to V/STOL Aircraft”. International Federation of Automatic Control. Vol. 28, No. 4, pp. 665-679, 1992. [11] W.Ghozlane and J.Knani.“Modelling and Simulation Using Mathematical and CAD Model of a Robot with Six Degrees of Freedom,”CEIT-2016, 4thInternational Conference on Control Engineering & Information Technology on IEEE.16-18 December 2016-Hammamet, Tunisia. [12] B.Siciliano and O. Khatib, Handbook of Robotics: Springer, 2007. [13] W. Khalil and E. Dombre, Modeling Identification and Control of Robots, Hermes Penton London, 2002. [14] Thomas R. Kurfess, Robotics and Automation Handbook, CRC press, 2005. [15] Y . L . Chen: “ Nonlinear Feedback and Computer Control of Robot Arms”. D.Sc. Dissertation, Washington University, St. Louis 1984. [16] V.D. Yurkevich, Design of Nonlinear Control Systems with the Highest Derivative in Feedback,World Scientific Publishing, 2004. [17] Frank L.Lewis, Robot dynamics and control, in robot Handbook: CRC press, 1999. [18] L. Sciavicco and B. Siciliano, Modeling and control of robot manipulators, 2nd edition, Springer-Verlag London Limited, 2000. [19] C.T.Leondes. Control and Dynamic Systems,Advances In Theory and Applications, part 1 of 2,USA,1991. [20] Zakaria, M. Z., Jamaluddin, H., Ahmad, R. & Loghmanian, S. M. R. (2010). “Multiobjective Evolutionary Algorithm Approach in Modeling Discrete- Time Multivariable Dynamics Systems”. Computational Intelligence, Modelling and Simulation (CIMSiM), 2010 Second International Conference on, Bali, Indonesia. 28-30 Sept. 2010. 65-70. [21] Zakaria, M. Z., Jamaluddin, H., Ahmad, R. & Loghmanian, S. M. R. (2012). “Comparison between Multi-Objective and Single-Objective Optimization for the Modeling of Dynamic Systems”. Journal of Systems and Control Engineering, Part I. Proc. Instn. Mech. Engrs., 226 (7): 994-1005.