Simulation Hybrid Fuzzy Control of SCARA Robot

Transcription

1 3rd WSEAS Internationa Conference on APPLIED and THEORETICAL MECHANICS, Spain, Decemer 4-6, Simuation Hyrid Fuzzy Contro of SCARA Root MARIUS-CONSTANTIN POPESCU, ILIE BORCOŞI, ONISIFOR OLARU, LUMINIŢA POPESCU, FLORIN GROFU Department of Eectromechanics University of Craiova str. A. I. Cuza nr.3, Craiova Department of Automatics and Informatics University of C-tin Brancusi str. Geneva nr.3, Tg-Jiu, Gorj ROMANIA Astract: - This paper presents the simuation of a Hyrid Fuzzy Controer suitae for industria appications. The deveoped Hyrid Fuzzy Controer consists of a weighted comination of the Direct Fuzzy Controer and Indirect Fuzzy Controer and a graduay activated supervisory controer. The unique feature of the Hyrid Fuzzy Controer is that no mathematica modes of the pant are required and the proposed controer is ae to adaptivey estimate the ound functions on-ine, which are required for the determination of the supervisory controer. The supervisor controer in the sense that a signas are ounded guarantees goa staiity of the cosed-oop system. The approach of rapid prototyping is empoyed to impement the Hyrid Fuzzy Controer so as to contro a Seectivey Compiance Assemy Root Arm. Simuink, an interactive graphica software for simuating dynamic systems, is used to mode, simuate and anayse the dynamic system. Key-Words: - SCARA Root, Simuink, Hyrid Fuzzy Controer Introduction The non-inear dynamics governing root motion presents a chaenging contro proem in the fied of contro engineering. As the manipuators are usuay used to perform high-precision tasks, root controers are required to contro the motion of a root effectivey and accuratey. However, as the manipuator is a mutivariae non-inear couping dynamic system with uncertainties, it is difficut to otain an accurate mathematica mode so that cassica or modern contro aws can e appied. To accommodate system uncertainties and variations, earning or adaptive techniques must e incorporated. Whie conventiona controers cannot effectivey contro the motion of a root due to compications of these noninear effects, fuzzy ogic, however, coud offer a promising approach to circumvent the deficiency. This is ecause in contrast with conventiona controers, fuzzy ogic aows one to formuate the contro proem of a pant in terms of inguistic rues drawn from the ehaviour of a 'human operator' instead of an agorithm synthesised from a rigorous mathematica mode of the pant. In this paper, a Hyrid Fuzzy Controer (HFC) suitae for rea-time industria appications is designed, deveoped and simuation. The HFC is made up of two on-ine adaptive fuzzy controers, namey Direct Fuzzy Controer (DFC) and Indirect Fuzzy Controer (IFC) with adaptive aws that can guarantee goa staiity of the cosed-oop system. In IFC, the dynamics of the root manipuator are estimated onine and the controer is chosen assuming that the estimated dynamics represents the true dynamics of the system. In AFC, the parameters of the controer are adjusted directy in order to reduce some norm of the output error etween the pant and the reference mode. The HFC is ae to compensate for environmenta variations such as payoad mass and disturance torque during the operation process. It is found to e superior to conventiona adaptive controers primariy ecause oundaries of uncertainties are eing adaptivey estimated. However, one major drawack of the HFC is that it requires massive computing resources and speeds crucia for its impementation. Therefore, the main chaenge for rea-time execution of this controer is to minimise the amount of processing that need to e performed y the controer efore the next input data arrives. The approach of rapid prototyping is

2 3rd WSEAS Internationa Conference on APPLIED and THEORETICAL MECHANICS, Spain, Decemer 4-6, empoyed to impement the HFC so as to contro a Seectivey Compiance Assemy Root Arm (SCARA) Root in rea time. Simuink, interactive graphica software for simuating dynamic systems, is used to mode, simuate and anayse the dynamic system. 2 Proem Formuation 2.. Background on the SCARA root Each of the four axes provides a different motion and contriutes to one degree of freedom of the root (see Fig. ). The asic SCARA geometry is reaised y arranging two revoute joints (T and T2 axes) and one prismatic joint (Z-axis) in such a way that a axes of motion are parae. The acronym SCARA characterises the mechanica features of a structure offering high stiffness to vertica oads and compiance to horizonta oads. Motion contro is impemented ony for axes Z, T and T2 in this work, which are designated as Joint, 2 and 3 respectivey The root dynamic mode The dynamic equations of a rigid ody root are a is generay expressed in the form of [, [4: M(θ )θ +C(, θ θ )θ +G(θ )+F Vθ +F C =τ, where M(θ ) is the n n inertia matrix of the manipuator, C(, θ θ ) is the n n vector of centrifuga and Coriois terms, G(θ ) is an n vector of gravity terms, F V is the n vector of viscous friction terms, F C is the n vector of couom terms and τ is the n vector of the input torque (generated y the joint motor). Each eement of M(θ ) and G(θ ) is a compicated function which depends on θ, anguar positions of a joints of the manipuator. Simiary, each eement of C(, θ θ ) is a compicated function of oth θ and θ. The movement of the end-effector in a desired trajectory, at a particuar veocity, thus requires a compex set of torque functions to e appied to the actuating system of the root at each joint. The dynamic mode of the SCARA root arm has een deveoped in [2, [3, with most of its parameters determined and verified through experiments. Based on this known mathematica mode of the root, simuation anaysis and design of the HFC were carried out using MATLAB simuation toos. set of highy non-inear couped differentia equations. The dynamics (of a nth joint root arm) 2.3. Motion contro A two-joint contro structure is used in this work

3 3rd WSEAS Internationa Conference on APPLIED and THEORETICAL MECHANICS, Spain, Decemer 4-6, where the motion of the T-axis and T2-axis wi e controed simutaneousy. In this scheme, the Z-axis wi not e controed since the earier work is ased on a two-joint controer. The dynamics of the two joints can e represented y a mutipe-input mutipe-output (MIMO) system. The degree of difficuty wi increase with the numer of joints moving together. 3 Proem Soution 3.. Hyrid fuzzy controer The proposed contro agorithm is given y [5: ( α) τ D τ S τ = ατ + +, () where τ is the contro torque contriuted y the IFC, τ D is the contro torque contriuted y the DFC, τ S is the supervisory controer and α is a weighting factor. If fuzzy contro rues are more important and reiae than fuzzy descriptions, choose α to e sma; otherwise, choose α to e arge. The resuting certainty equivaent controer is: = τ ( θ / Φ) [ F( θ, θ / Φ) + θ + K E K E I G M V +, )+ G( θ ) ( θ / Φ) P, (2) θ = F( θ θ τ D, (3) where ( θ τ D / Φ) F, ( θ,θ G and ( θ / Φ) assume the foowing form: f ( x) = M = i= M y n n = i= e e 2 xi xi σ i 2 xi xi σ i, (4) where f ( x) is the fuzzy ogic system with Gaussian memership function, centre average defuzzifier and product-inference rue, and y, xi and σ i are adjustae parameters. Now adding and sutracting G(θ ) τ to Eq. (3) and after some manipuations, we otain the cosed-oop error equation due to the DFC which is given as: E =-K V Ė -K E p +G(θ ) [ τ ( θ / Φ ) τ D. (5) The cosed-oop error dynamics due to the IFC ecomes: E =-K v Ė -K E +[ ( ) p θ,θ / Φ [ ( θ / Φ) F -F(, θ θ ) + G -G(θ ) τ I. (6) Since the overa controer of Eq. () consists of a weighted comination of DFC and IFC, the respective weighted errors ecome: (-α ) E =-(-α )K v Ė -(-α )K p E + (-α )G[ τ ( θ / Φ) τ D (7) and α E =-α K v Ė -α K p E+α [ F( θ,θ - F(, θ θ ) + α [ ( θ / Φ) G -G(θ ) τ I. (8) The overa error governing the cosed-oop system is given y the sum of Eqs. (7) and (8), i.e. E Fθ,θ / Φ - F(, θ θ ) + =-K V Ė -K p E +α [ ( ) α [ ( θ / Φ) τ +(-α )G(θ )[ τ ( θ / Φ) G I τ D. (9) The overa error dynamics in the state-space form is: Ẋ =AX+α B I [( ( θ,θ G( θ ) I F -F(, θ θ )) + ( ( θ / Φ ) τ +(- α )B D [ τ ( θ / Φ) where X=( e e, e,, ) A= B I = 2 and p, v e n p2 G - τ D =AX+BU, () T, B D = 2 v2 2 22, B=[B I B D, = G(θ )=M - (θ ) α[ F( θ, θ / Φ) F( θ, θ) + α[ G( θ / Φ) G( θ) τ I U=, ( α) [ τ τ ( θ / Φ) D with K p and K v chosen such that A is a stae matrix,

4 3rd WSEAS Internationa Conference on APPLIED and THEORETICAL MECHANICS, Spain, Decemer 4-6, there exists a unique 2n 2n positive definite symmetric matrix, P which satisfies the Lyapunov equation: A T P + PA=-Q, () process continues y using Simuink and toooxes to deveop the agorithm as we as anayse the resuts. where Q is an aritrary 2n 2n positive definite matrix. Let the Lyapunov function candidate e: V e = 2 X T PX. (2) Differentiating Eq. (2) with respect to time and using Eqs. () and () eads to: V e = 2 T Ẋ +α X T PB I [( ( θ,θ - G( θ ) I PX+ 2 X T P Ẋ =- 2 X T QX F -F(, θ θ ))+( ( θ / Φ ) τ +(- α )X T PB D [ τ ( θ / Φ) G (3) τ D. In order for the system to e ounded, we require V e to e ounded which means that V e < whenever V e is greater than a constant V specified y the designer. However, from Eq. (3), we see that it is difficut to design τ D ( θ / Φ) and τ I ( θ / Φ) such that the ast two terms of Eq. (3) are ess than zero. We can sove this proem y appending another contro term τ S ( θ) which is caed supervisory contro aw. The fina contro aw ecomes: τ θ / Φ = ατ θ / Φ + α τ θ / Φ τ θ (4) ( ) ( ) ( ) ( ) ( ) F I D + S 3.2. Simuation mode of HFC The simuation program for the HFC is written in Simuink to faciitate Rapid Prototyping, a process that aows one to conceptuaize soutions using ock diagram modeing environment and predict the system performance prior to aying out hardware, writing any production software, or committing to a fixed design [8. Fig. 2 shows the rapid prototyping process in greater detais. The contro system mode and dynamics of the SCARA root are created in Simuink. The first step in the design is to deveop the SCARA root mode prior to agorithm deveopment. Fig. 2 Rapid prototyping process If the resuts are not satisfactory, we iterate the modeing/anaysis process unti satisfactory resuts are otained. Fig. 3 shows a Simuink diagram for simuating the HFC system for controing Joint 2 (T-axis) and Joint 3 (T2-axis) of the SCARA root. This prototype is modeed cosey ased on the SCARA root mode and the HFC agorithm. The input variaes to the HFC are position error and ink veocity of T and T2 axis. The output variae is used to contro the SCARA root. Once a simuation mode, which is ae to mode the root with sufficient accuracy, has een deveoped, the rapid prototyping

5 3rd WSEAS Internationa Conference on APPLIED and THEORETICAL MECHANICS, Spain, Decemer 4-6, Fig. 3. Contro system mode in Simuink 3.3. Simuation resuts To evauate the performance of our HFC, a series of simuation runs were performed. The joint trajectory responses were potted against time. Overshoots are kept stricty to zero and errors of ess than % recorded for a 2 joints. a) a) ) Fig. 4. Position trajectory of joint 2, a; Position error of joint 2,. ) Fig. 5. Position trajectory of joint 3, a; Position error of joint 3,. Fig. 4a shows the position tracking trajectories and the corresponding position tracking errors for joint 2 using the HFC with α set to.95 i.e., there are joint contriutions from the DFC and the IFC, with the IFC having 95% of the tota contriution. Fig. 5a depicts those of joint 3. Desired position trajectories are indicated in dashed ines and actua trajectories after incorporating the HFC are indicated in soid ines. Figs. 4 and 5 show the position errors of joint 2 and joint 3 respectivey. Both resuts show that overshoots are kept stricty to zero and errors of ess than % recorded for oth joints. The HFC software consists of primariy 2 modues: () The main contro modue, and (2) The GUI for signa monitoring and parameters tuning. Modue () is written in Simuink whereas Modue (2) is deveoped using Contro-Desk [, [6. The overa Simuink mode for the HFC is shown in Fig. 6.

6 3rd WSEAS Internationa Conference on APPLIED and THEORETICAL MECHANICS, Spain, Decemer 4-6, 27 8 Fig. 6. Overa Simuink graphica modes for the HFC 4 Concusion Once encouraging system performances are achieved, the Hyrid Adaptive Fuzzy Controer is impemented in rea-time through Rea-Time Workshop (RTW). The performance of the Hyrid Fuzzy Controer was found to e superior and it matches favouray the simuation resuts. A HFC suitae for rea-time industria appications has een successfuy designed, deveoped and simuation in this work. The approach of rapid prototyping process was adopted to shorten the deveopment time, thus reducing the cost of deveopment. The amount of time saved is sustantia especiay when no reuiding of codes is invoved. The performance of the HFC was found to e superior and it matches favoray the simuation resuts. With the promising resuts eing accompished, the authors are confident that the proposed HFC wi e very usefu in many other rea-time industria appications [7. References: [ Adekarim A., Terra Z., Impémentation d'un système de communication FM par 'utiisation du DSP TMS32VC542, Editeur Ecoe Nationae Poytechnique, 25 [2 Nianzu Zhang, Experimenta Research on Root Contro, M.Eng. thesis, Singapore, Nanyang Technoogica University, 996. [3 Meng Joo Er ', Moo Teng Lim, Hui Song Lim, Rea-time hyrid adaptive fuzzy contro of a SCARA root, Microprocessors and Microsystems 25, 2. [4 Onisifor O., Ampificatoare integrate in echipamente de automatizare, Editura Universitaria Craiova, 23. [5 Popescu M.C., Estimarea şi identificarea proceseor, Editura Sitech, Craiova, 26. [6 Popescu M.C, Utiisation des ordinateurs, Editura Universitaria, Craiova, 24. [7 Texas Instruments TMS32C54. [8 *** User Guide Mata for Simuink