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International Conference on Control, Automation and Systems 2007 Oct. 17-20,2007in COEX, Seoul, Korea New Controller Design for a Robot Manipulator A.Khodabakhshian (Ph.D) FacultyofEngineering, Isfahan University, Iran Abstract:Thispaperpresents anew systematic method to design a controller using the Quantitative Feedback Theory (QFT) for a robot identified as a non-minimum phase model.The robot manipulators arehighly nonlinear with an uncertain environment. A linearisedtransferfunction of a laboratory robot is developed using the off-line system identification. The effects of nonlinearitiesareaccounted by describing thelinearisedmodelparametersasstructured uncertainty. TheQFTdesign procedure is carried out to design a robust controllerthatsatisfies performance specifications for tracking.The designed fixed-gain controller is easy to implement. IndexTerms-Robot manipulator, identification I. INTRODUCTION In many industrial applications high speed and accuracy withgreatreliabilityareofthemostimportant requirements.In this regard, the robot manipulators are advantageousforsuchapplications.Theyare increasingly utilized and special care must be taken into account to design the robot controllers for achieving the above-mentioned goals. Robot manipulators are highly nonlinear and subject to parameter uncertainty; parameters change with time as a resultofvariationsinoperatingconditionsand component degradation.These changes in parameters significantly affect on the conventional feedback control strategies and the servo response and damping will be then reduced. Subsequently, manipulators move at slow speeds with unnecessary vibrations. In recent years, a wide varietyofcontrol theories such as adaptive and fuzzy techniques have been appliedto improve the performanceofthe robots considering the nonlinear dynamicsofmanipulators 1-5. Despite the existenceofa greatnumberofcontrol concepts, methods, and algorithms, there is still a large gap between theory and industrial practice.The reasons can be considered to the poor industrial control AminKhodabakhshianiswithIsfahanUniversity,Facultyof Engineering,Isfahan,Iran(telephone:+98-311-793-2771,e-mail: aminkheng.ui.ac.ir). 978-89-950038-6-2-98560107/$15ICROS 41 architecture, which does not allow the implementationof sophisticated algorithms 6.This problemis more complicated as shownin 7-9 that theseindustrial systems are, in fact, non-minimum-phase when sensing and actuation are considered.These real characteristics make the performance limitationofthe system occur 8. In this paper, the QFT technique is then used to design an explicitlead-lagcontrollerforalaboratoryrobot manipulator identified as a non-minimum phase model to solve the difficulties mentioned above.The goal is to arrive at a fixed-gain controller that: I) isoflow order and easy to implement, 2) is robust against uncertainties, and 3) does not require exact knowledgeofthe systems parameters. QFT is a robust controller design methodology aimed at plants with parametric and unstructured uncertainties. The concept was first introduced by Horowitz in the early sixties 10.This technique emphasizes the fact that feedback is only necessary becauseofuncertainty and that the amountoffeedback should therefore be directly related to the extentofplant uncertainty and unknown external disturbances.Minimising the costoffeedback, as measured by amountofcontroller bandwidth, is the mainobjectiveofQFTII.Therefore, theplant uncertainty and the closed loop tolerances are formulated quantitatively so that the costoffeedback can be assessed at each stageofthe design process. The method has been appliedtoa widerangeofengineeringproblems, including flight control 12, manufacturing systems 13, robots with minimum phase models 14 and power systems applications for both minimum-phase and non- minimumphasemodels15-16.Thesuccessful applicationsofthismethodprovide evidenceofits potential effectiveness for robot manipulators. The objectiveofthis study is to evaluate and to show the benefitsofQFTwhen applied to a laboratory robot, as an experimental case study, in which it has been identified as a non-minimum phase model. 2.SYSTEMIDENTIFICATION AND MODELLING In order to apply QFT technique, the robot manipulator model must be identified. Consider the block diagramof the system shown in Figure I in which P and G represent the robot manipulator and nonlinear proportional controller respectively. Figure 1: Block diagram of the robot manipulator For identifying the manipulator model (P) first the Pseudo Random Binary Sequence (PRBS) 17 is generated as the input signal to robot plant shown by (u) in Figure 1. The joint positions are measured as the output shown as (y). Both signals are received by the identification Toolbox of Matlab. Using the Box-Jenkins input-output model in the form of )()(/ )()()(/ )()(teqDqCnktuqEqBty+= leads to a parametric uncertain discrete transfer function. It should be mentioned that e(t) in the above equation represents the white noise existing in any practical experiment. Due to fact that the proposed QFT design being implemented in the analog domain, it is imperative that the equivalent s-transfer functions be obtained from z-transfer models. This has been done and shown as the following s-domain transfer function; )(20( )()(20( )( asss dscssb sP + + = where 0025,.0004.b, 8,40 c 17, 3d, 25. 7 ,25. 2a As can be seen the identified model is non-minimum phase. Although theoretical dynamics of a manipulator does not show to have a non-minimum phase model but in such practical-computerized system, non-minimum phase factors appear in identified transfer function because of sensors, actuators and computer interface equipment in the system 8. For example, it can be seen from P(s) that the term: 20 20 )( + = s s sH shows the first order PADE approximation of the delay function (see: Control System Toolbox of MATLAB) s e s s 1 .0 20 20 + The right half plane zero severely limits the achievable closed-loop bandwidth and robustness 7-8. However, as will be seen in the next section this existing problem can be easily resolved by the proposed controller. 3. QFT DESIGN Quantitative Feedback Theory (QFT) initially proposed by Horowitz and further developed by him and others 18 is considered as an efficient method for designing the robust controllers for plants with parameter uncertainties, unstructured uncertainties and mixed uncertainties. This method was first proposed for minimum-phase and stable systems. Chen and coauthors then developed this method even for uncertain non- minimum phase and unstable plants 19. In QFT, the closed-loop transfer function needs to satisfy certain performance requirements for a set of discrete frequencies. These requirements are specified in terms of tolerance bands within which the magnitude response of the closed-loop transfer function should lie. The uncertainties in the plant (P) are transformed onto the Nichols chart resulting in bounds on the open-loop transfer function of the system. A compensator (G) is then chosen by manually shaping the loop transmission such that it satisfies the bounds at each of the frequency points. A pre-filter (F) is then used to ensure that the closed-loop transfer function lies within the specified bands (see Figure. 2 20, Demo6). Figure 2: Block diagram of the system with the controller in QFT design Now the design procedure is summarized as follows; 1- In QFT design there are three primary control objectives. The first is stability with reasonable margins of () () 2 . 1 1 + jPG jPG , 0 where P and G are the robot manipulator and the controller to be designed using QFT respectively. The second is robust tracking as )( )(1 )()( )( jb jPG jPGjF ja + 10 where law control R G PF Y Signal Controlled U V esdisturbanc input signal reference dynamics plant N noise sensor esdisturbanc output filterpre + 42 108217 10 )( 23 + = sss sa and 6507. 10825.1 )30(055263.0 )( 2 + + = s s sb are the output margins in response to the unit step as an input. It should be mentioned that these objectives have been obtained for the robot manipulator by trial and error method. The third here is the robust input disturbance rejection and is considered here as 05. )(1 )( + jPG jp 1 A continuous-time uncertain transfer function model can have parametric, non-parametric or mixed parametric and non-parametric structures. Parametric uncertainty which is the case in this study implies the knowledge of variations in parameters of the transfer function. This has been given for the robot manipulator in section 2 (a, b, c and d). The design procedure will be first similar to the one for minimum phase system and in the step of loop- shaping the method given in 19 will be then followed. One of the most important factors in control design is to use an accurate description for the plant dynamics. Because QFT involves frequency-domain arithmetics, its design procedure requires us to define plant dynamics only in terms of its frequency response. The term template is then used to denote the collection of an uncertain plants frequency responses at a given frequency. The frequency range must be chosen based on the performance bandwidth and shape of the templates. Margin bounds should be computed up to the frequency where the shape of the plant template becomes invariant to frequency. Hence, for the robot manipulator given in this paper at approximately =50 rad/sec, the template shape becomes fixed, a quadrangular shape. The plant templates at several frequencies (50,20,10, 1 , 5,.1 .=) are shown in Figure 3. Figure 3: Plant templates at several frequencies The robust stability bounds at these frequencies are depicted in Figure 4. Performance specifications are typically defined within a finite frequency bandwidth which is related to the closed- loop system bandwidth and spectrum of the disturbances. There is very little to be gained by specifying transfer function magnitudes up to frequency of infinity 11. Therefore, in a QFT design, performance is specified only up to a finite frequency whose value is always problem dependent. The bounds for the robust tracking and the robust input disturbance rejection are shown in Figures 5 and 6 respectively. The intersection of bounds for the actual non-minimum phase robot manipulator is shown in Figure 7. Figure 4: Robust stability bounds Figure 5: Robust tracking bounds After determining the bounds of the system in QFT theory proposed by Horowitz loop-shaping has been given for a minimum phase and stable system 11. However, since robot manipulator model, as mentioned before, is non-minimum phase there should be some changes to be able to design the controller. The followings are based on the method given in 19 to solve the problem. 43 Figure 6: Robust input disturbance rejection bounds Figure 7: Intersection of bounds for the actual non- minimum phase robot manipulator 2- The transfer function P is defined as P=P1A (P1 is the transfer function with uncertainty) and P0=P2A in which P0 is the nominal plant. Transfer functions P2 and A are given as follows; )25. 2( )3)(8(0004. )( 2 + + = ss ss sP and )8)(20( )8)(20( )( + = ss ss s Note that P2 (s) is a minimum phase transfer function and A(s) is all-passing with the following properties ), 0, 1)(=sA Horowitz and Sidi 9 show that the new robust bounds for the new plant (P2 (s), which is a minimum phase system, is obtained by shifting the previous robust bounds of the non-minimum phase system (Figure 7) to the right with )(arg(180)( 1 jA+= The intersection shifted bounds for the non-minimum phase plant (P) can be considered equivalently for the minimum phase plant (P2) (see Figure 8). Horowitz and Sidi 21 claim that the QFT controller design using loop-shaping for the minimum-phase plant P2 (s) is equivalent to that for the non-minimum phase plant P(s) with shifted bands. However, as shown in 19 this method is necessary but not enough and in some cases the instability may occur because the Nyquist criterion is not satisfied. Chen and Ballance 19 show that if L2(j)=P2(j)G(j) is going to satisfy the Nyquist criterion, the stability lines of L(j)= P(j)G(j) should be shifted to the right for each frequency with ) )8)(20( )8)(20( arg()(arg()( 1 + = jj jj j A This has been shown in Figure 8. Figure 8: Shifted bands and shifted stability lines Having computed stability and performance bounds, the next step in a QFT design involves the design (loop- shaping) of a nominal loop function that meets its bounds. The nominal loop is the product of the nominal plant and the controller (to be designed). The nominal open-loop has to satisfy all bounds. The result of loop-shaping using the program written by author and QFT toolbox of Matlab 20 is shown in Figure 9. The final interactively designed controller is given by 2.5470 s 69.1010 s 39.3978 )( + + =sG Since in this design the feedback system involves tracking reference signals, the best choice will be to use a pre-filter F(s) in addition to the controller G(s) embedded within the closed-loop system (see Figure 2). The Bode diagrams resulted from pre-filter shaping is shown in Figure 10. This gives the following transfer function of F(s). )57693472818110548126 54917300986( )5769347 72662922879()( 2 3456 2 + + + += ss ssss sssF 44 Figure 9: Final design (loop shaping) Figure 10: Pre-filter shaping As shown in Figure 11 the system outputs for unit step reference input considering G(s) and F(s) as the controller and pre-filter respectively will be in the acceptable predetermined area. Figure 11: Area of acceptable outputs in time domain 4. EXPERIMENTAL RESULTS In order to compare the designed controller performance to that of the nonlinear proportional controller used in the laboratory robot, manufactured by Feedback Company and named MENTOR (see Picture 12 ), the output signals in response to the command signal are shown in Figure 13. As illustrated in Figure 13 the tracking performance of the QFT designed controller is better than that of used by the manufacturer. The control effort is also superior. Picture 12: The picture of the laboratory robot Figure 13: The compared results of the QFT designed controller and the preliminary nonlinear proportional controller of the robot manipulator 45 5. CONCLUSIONS Applying the QFT and the off-line system identification, a robot controller design method has been developed for a robot with the non-minimum phase model. Employing system identification technique, the dynamics of the manipulator is linearised into a third order model. According to the variation of poles, zeros and gains of the model, the QFT is then applied to design a controller for robot manipulator. The results of the presented controller are compared to that of the controller given by the manufacturer. The illustrations clearly show the benefits of using the proposed controller and can be summarized as follows; 1) Plant uncertainty due to different robot configurations can be easily managed by the controller. 2) The implementation is much easier and less on-line computational effort than the complex on-line adaptive techniques. 3) The overshoot and settling time of the robot response can be assured. 6. REFERENCES 1 F. Claugi, A. Robertsson, R. Johansson, Output Feedback Adaptive Control of Robot Manipulators Using Observer Backstepping, Proceeding of the 2002 IEEE/RSJ Int. Conference on Intelligent Robots and Systems, Lausanne, Switzerland, Oct. 2002, pp: 2091- 2096 2 C. Sousa, E. M. Hemerly, R. 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