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

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Nonlinear Control Via Input-Output Feedback Linearization of a Robot Manipulator 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: 푘 퐿 퐿 ℎ = 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.
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