Robust Fuzzy Control for 2-DOF Manipulator System

Transcription

1 Amercan Journal of Artfcal Intellgence 07; (): do: 0648/jaja07007 Robust Fuzzy Control for -DOF Manpulator System Ayman A Aly,, Aloqla A Mechancal Engneerng Dept, College of Engneerng, Taf Unversty, Taf, Saud Araba Mechancal Engneerng Dept, Faculty of Engneerng, Assut Unversty, Assut, Egypt Emal address: draymanelnaggar@yahoocom (A A Aly) To cte ths artcle: Ayman A Aly, Aloqla A Robust Fuzzy Control for -DOF Manpulator System Amercan Journal of Artfcal Intellgence Vol, No, 07, pp 56-6 do: 0648/jaja07007 Receved: Aprl 6, 07; Accepted: July 5, 07; Publshed: November 8, 07 Abstract: Robot manpulators have become ncreasngly mportant n the feld of automaton So modellng and control of robots n automaton wll be very mportant Ths paper presents a study of robust control approach employng fuzzy logc control technque for two degree of freedom (-DOF) manpulator robot A learnng control system s desgned so that ts learnng mechansm has the ablty to mprove the performance of the closed-loop system by generatng command nputs to the plant and utlzng feedback nformaton from the plant It s well known that robotc manpulators are hghly nonlnear couplng dynamc systems A fuzzy logc rule base s desgned, usng the knowledge obtaned from the operator Smulaton s performed to demonstrate the effectveness of control strategy Furthermore, the parameters of the controllers were optmzed usng MATLAB and smulatons' result reveals that control scheme s workng satsfactorly Keywords: Fuzzy Control, Dynamc Model, Robotc, Two DOF Manpulator Introducton In recent years, control engneers have become ncreasngly nterested n the robot trackng problem As a result many controllers have been developed whch compensate for uncertanty n the nonlnear second-order dynamcs commonly used to represent rgd-lnk robots Most of the more rgorously developed nonlnear controllers for rgd-lnk robots fall nto two categores, ndrect adaptve control and robust nonlnear control The nterested reader s referred to Abdallah et al, [] and K A Khall et al, [] for revew papers n these two areas Dynamcs of robot manpulators are hghly nonlnear and may contan uncertan elements such as frcton Many efforts have been made n developng control schemes to acheve the precse trackng control of robot manpulators [3-5] A robust fuzzy control desgn has been developed for a class of nonlnear systems, and the fuzzy control s robustly and globally stablsng The desgn assumes a general structure and needs no supervsory control In ths approach, a robust sub-control s desgned frst and fuzzfed for each rule to guarantee closed-loop stablty n each fuzzy set Indvdual robust controls are then blended nto the overall fuzzy controller, [6-7] Ths paper s organzed as follows: The next secton descrbes the system dynamcs to be controlled In the thrd secton, robust control system s brefly ntroduced The thrd secton descrbes the control strategy utlzng the proposed approach Smulaton results are presented n the fourth secton The concludng remarks on the system performance are gven n the last secton Manpulator Dynamcs The mechansm conssts of two ntersectng axes at the shoulder and elbow lnks wth revolute jonts The forward knematcs for open chan manpulator s computed by calculatng the ndvdual motons for each jont There are many methods for generatng the dynamcs of a mechancal system All methods generate equvalent sets of equatons, but dfferent forms of the equatons may be better suted for computaton or analyss A Lagrangan analyss for our dervaton wll be used, whch reles on the energy propertes of mechancal systems to compute the equaton of moton The resultng equatons can be computed n closed form, allowng detaled analyss of the propertes of the system For each lnk the frame L at the centre of mass and algned wth prncple nerta axes of the lnk

2 Amercan Journal of Artfcal Intellgence 07; (): g sl 0 0 I 0 (0) = I r r and gsl (0) = o l o 0 0 Usng the Lagrangan formulaton, the equatons of moton of a two degrees-of-freedom rgd manpulator may be expressed by: T b T b T sl sl () T ( θ, θ) = ( V ) M V = θ J ( θ) M J ( θ) θ, () where M s the generalzed nerta matrx for the th lnk Now the total knetc energy can be wrtten as n T T( θ, θ) = T ( θ, θ) = θ M( θ) θ (3) = M = 0, M = Ix + mr To complete the dervaton of the Lagrangan, the potental energy of the manpulator must be calculated Let h (θ) be the heght of the centre of mass of the th lnk (heght s the component of the poston of the centre mass opposte the drecton of gravty) The potental energy for the th lnk s n V( θ) = V ( θ) = m gh ( θ) = = n (7) where m s the mass of the th lnk and g s the gravtatonal constant Combnng ths wth the knetc energy, then T L( θ, θ) = θ M( θ) θ V( θ) (8) Substtute n Lagrange s equatons, The matrx M (θ) R nxn s the manpulator nerta matrx In terms of lnk Jacobans, J, the manpulator nerta matrx s defned as d L L = τ dt θ θ (9) n T M( θ) = J ( θ) M J ( θ) (4) = Wth ths choce of lnk frames, the lnk nerta matrces have the general form M m m m = Ix Iy Iz where m s the mass of the object and I x, I y, and I z are the moments of nerta about the x-, y-, and z- axes of the th lnk frame Computaton the body Jacobans for each lnk frame J = and J rc ϕ r = 0 Sϕ 0 Cϕ 0 The nerta matrx for the system s gven by where M M M( θ) = = J M J + J M J M M T T y z z (5) (6) M = I S ϕ + I + I C ϕ + m r C ϕ, M = 0, where τ represent the actuator torque Then t can arrve to M( θ) θ + C( θ, θ) θ + N( θ, θ) = τ (0) where τ s the vector of actuator torques and N (θ, θ) ncludes gravty terms and others forces whch act at the jonts Ths s a second-order vector dfferental equaton for the moton of the manpulator as a functon of the appled jont torques The matrces M and C, whch summarze the nertal propertes of the manpulator, have some mportant propertes The Corols and centrfugal forces are computed drectly from the nerta matrx va the formula M M Cj( θ, θ) θ θ n n j Mk kj = Γ jk k = + k θ k k k θ j θ = = () A very messy calculaton shows that the nonzero values of г jk are gven CϕSϕ Iy Iz mr Γ = ( ), CϕSϕ Iy Iz mr Γ = ( ), CϕSϕ Iz Iy mr Γ = ( + ) Fnally, the effect of gravtatonal forces on the V manpulator are wrtten as N( θ, θ) =, θ where V: R n R s the potental energy of the manpulator V( θ) = m gh ( θ) + m gh ( θ) () where h s the heght of the centre of mass for the th lnk These can be found usng the forward knematcs map

3 58 Ayman A Aly and Aloqla A: Robust Fuzzy Control for -DOF Manpulator System ζ θ ζ θ sl (0) (0) sl g = e e g (3) whch gves h ( α ) = ro, h( ϕ) = lo r Sϕ Substtutng these expressons nto the potental energy and takng the dervatve gves V 0 N( θ, θ) = = θ m grcϕ 3 Fuzzy Model Reference Learnng Control (FMRLC) Desgn (4) The fuzzy controller by tself has lttle knowledge about how to control the manpulator As the algorthm executes, the output membershp functons are rearranged by the learnng mechansm, fllng up the rule base For nstance, once a slew s commanded, the learnng mechansm descrbed below wll move the centres of the output membershp functons of the actvated rules away from zero and begn to synthesze the fuzzy controller In ths case, t developed an FMRLC for automatcally syntheszng and tunng a fuzzy controller for the system The FMRLC structure shown n Fgure was used, whch tunes the coupled drect fuzzy controller Next, Each component of the FMRLC for the two-lnk system wll be descrbed, [9-0] The unverse of dscourse for the poston error nput e to the shoulder lnk controller was chosen to be [-00, +00] degrees, and the unverse of dscourse for the endpont acceleraton a s [-0, +0] g For the elbow lnk controller, the unverse of dscourse for the poston error e s [-80, +80] degrees, and the unverse of dscourse for the endpont acceleraton a s [-8, +8] g The output unverse of dscourse for v and v by gettng g v = 05 and g v = 0 was chosen The desred performance s acheved f the learnng mechansm forces ye (kt) 0, ye (kt) 0, for all k 0 It s mportant to make a proper choce for a reference model so that the desred response does not dctate unreasonable performance requrements for the plant to be controlled through smulaton, t was determned that 3 s a good s +3 choce for the reference models for both the shoulder and the elbow lnks Fgure FLMRC for the two lnk robot

4 Amercan Journal of Artfcal Intellgence 07; (): The Fuzzy Inverse Models There are several steps nvolved n specfyng the fuzzy nverse models and these are outlned next Choce of Inputs: For our system there are two nverse models, each wth three nputs ye j (t), yc j (t), and a j (t) (j= correspondng to shoulder lnk and j= correspondng to elbow lnk) Several ssues dctated the choce of these nputs: () t s easy to specfy reference models for the shoulder and elbow lnk poston trajectores (as dscussed above) and hence the poston error sgnal s realty avalable; () t s found va smulaton that the rates of change of poston errors, yc j (t), j=,, and acceleraton sgnals aj (t), j=,, were very useful n decdng how to adjust the fuzzy controller; and (3) to mnmze the number of nputs to the fuzzy nverse models to ensure that t could mplement the FMLRC wth a short enough samplng nterval (n our case, 5 mllseconds) The drect use of the acceleraton sgnals a j (t), j=,, for the nverse models actually corresponds to choosng reference models for the acceleraton sgnals that say no matter what slew s commanded, the desred acceleratons of the lnks should be zero (why?) Whle t s clear that the lnks cannot move wthout acceleratng, wth ths choce the FMLC wll attempt to accelerate the lnks as lttle as possble to acheve the command slews, thereby mnmzng the amount of energy njected nto the modes of vbraton Next, rule base desgn for the fuzzy nverse models were dscussed 3 Choce of Rule Base For the rule bases of the fuzzy nverse models, rules smlar to those descrbed n [0] was used, for both the shoulder and elbow lnks except that the cubcal block of zeros s elmnated by makng the pattern of consequents unform These rules have premses that quantfy the poston error, the rate of change of the poston error, and the amount of acceleraton n the lnk The consequents of the rules represent the amount of change that should be made to the drect fuzzy controller by the knowledgebase modfer For example, fuzzy nverse model rules capture knowledge such as () f the poston error s large and the acceleraton s moderate, but the lnk s movng n the correct drecton to reduce ths error, then a smaller change (or no change) s made to the drect fuzzy controller than f the lnk were movng to ncrease the poston error; and () f the poston error s small but there s a large change n poston error and a large acceleraton, then the fuzzy controller must be adjusted to avod overshoot Smlar nterpretatons can be made for the remanng portons of the rule bases used for both the shoulder and elbow lnk fuzzy nverse models 33 Choce of Membershp Functons The membershp functons for both the shoulder and elbow lnk fuzzy nverse models are smlar to those used for the elbow lnk controller expect that the membershp functons on the output unverse of dscourse are unformly dstrbuted and there are dfferent wdths for the unverses of dscourse, as t was explaned next (these wdths defne the gans g yej, g ycj, g aj, and g pj for j=, ) The unverse of dscourse for ye to be [-80, +80] degrees for the shoulder lnk was chosen and [-50, +50] for the elbow lnk A larger unverse of dscourse for the shoulder lnk nverse model was chosen than for the elbow lnk nverse model than for the elbow lnk nverse model because t need to keep the change of speed of the shoulder lnk and [-50,+50] for the elbow lnk A larger unverse of dscourse for the shoulder lnk nverse model than for the elbow lnk nverse model was chosen because t need to keep the change of speed of the shoulder lnk gradual so as not to nduce oscllatons n the elbow lnk (the elbow lnk s mounted on the shoulder lnk and s affected by the oscllatons n the shoulder lnk) The unverse of dscourse for yc s chosen to be [-400, +400] degrees/second for the shoulder lnk and [ ] degrees/second for yc of the elbow lnk These unverses of dscourse were pcked after expermental determnaton of the angular veloctes of the lnks The output unverse of dscourse for the fuzzy nverse model outputs (p and p ) s chosen to be relatvely small to keep the sze of the changes to the fuzzy controller small, whch helps ensure smooth movements of the robot lnks In partcular, The output unverse of dscourse to be [ ] for the shoulder lnk nverse model was chosen, and [ ] for the elbow lnk nverse model Choosng the output unverse of dscourse for the nverse models to be [-, +] causes the learnng mechansm to contnually make the changes n the rule base of the controller so that the actual output s exactly equal to the reference model output, makng the actual plant follow the reference model closely Ths wll cause sgnfcant amounts of speed varatons n the motors as they try to track the reference models exactly resultng n chatterng along a reference model path, The choce of a smaller wdth for the unverse of dscourse keeps the actual output below the output of the reference model untl t reaches the set pont Ths ncreases the settlng tme slghtly but the response s much less oscllatory Ths completes the defnton of two fuzzy nverse models n Fgure 34 The Knowledge Base Modfer Gven the nformaton (from the nverse models) about the necessary changes n the nput needed to make ye 0 and ye 0, the knowledge base modfer changes the knowledge base of the fuzzy controller so that the prevously appled control acton wll be modfed by the amount specfed by the nverse model outputs p, =, To modfy the knowledge base, the knowledge base modfer shfts the centres of the output membershp functons (ntalzed at zero) of the rules that were on durng the prevous control acton by the amount p (t) for the shoulder controller and p (t) for the elbow controller Note that to acheve good performance, t was found va smulaton that certan enhancements to the FMRLC knowledge base modfcaton procedure were needed

5 60 Ayman A Aly and Aloqla A: Robust Fuzzy Control for -DOF Manpulator System 4 Smulaton of -DOF Model wth Drect FMRLC System The total number of rules used by the FMRLC s for the shoulder controller, plus 343 for the base axs controller, plus 343 for the shoulder fuzzy nverse model, plus 343 for the base axs fuzzy nverse model, for a total of 50 rules Even wth ths number of rules, The samplng tme of T= mllseconds was used for the drect fuzzy controller Smulaton results obtaned from the use of the FMRLC are shown n Fgures - (a & b) and 3- (a & b) for a slew of for each lnk The rse tme for the response s about 03 sec and 05 sec, the settlng tme s approxmately 0 sec and 04, the delay tme s 0005 sec and 0 wth zero steady state errors for the two axes Dfferent payloads change the model frequences n the lnk/payload combnaton (eg, heaver loads tend to reduce the frequences of the modes of oscllaton) and the shapes of the error and acceleraton sgnals e (t), e (t), and a (t) (eg, behavour loads tend to slow the plant responses) Hence, Changng the payload smply results n the FMRLC developng, rememberng, and applyng dfferent responses dependng on the type of the payload varaton that occurred Essentally, the FMRLC uses data from the closed-loop system that s generated durng the smulaton operaton of the plant Ths enables t to acheve better performance than the drect fuzzy controller descrbed above The cross pondng controller sgnals s cleared n lower part of each fgure Fgure 4-a shows the tme responses of step dsturbance reference nput, the recoverng had happened after 0 sec for the base axs wth zero steady state error In Fgure 5-a after 03 sec for the shoulder axs wth zero steady state error The cross pondng controller sgnal s shown n Fgures 4-b and 5-b The dfference whch had happened between the reference nput and the axes response because of the way the learnng mechansm modfed the rule base of the controller to keep the response below that of the reference model Fgure System step response based on fuzzy controller (Base Axs) Fgure 3 System step response based on fuzzy controller (Shoulder Axs)

6 Amercan Journal of Artfcal Intellgence 07; (): Fgure 4 System moton curve response based on fuzzy controller (Shoulder Axs) Fgure 5 System moton curve response based on fuzzy controller (Shoulder Axs) 5 Concluson In ths paper, a desgn approach that can be used to desgn robot trackng controller was developed whch compensate for nonlnear dynamcs n robot model One of the most mportant challenges n the feld of robotcs s robot manpulators control wth acceptable performance, because these systems are mult-nput mult-output (MIMO), nonlnear and uncertanty The control problem of a nonlnear system such as the two DOF system s studed n ths paper A fuzzy controller has been mplemented Although fuzzy logc control has a model-free feature, t stll needs tme-consumng work for the rules bank and fuzzy parameters adjustment Ths approach has a learnng ablty for respondng to the tme-varyng characterstc of the system Its control rules bank can be establshed and modfed contnuously by onlne learnng wth zero ntal fuzzy rules FMRLC desgned n ths case s found to be adoptve and robust wth strong learnng ablty References [] ABDALLAH, C, D Awso N, D M, Do RATO, P, and JAMrsmm, M, 99, Survey of the robust control of robots IEEE Control Systems Magazne,, (), 4-30 [] Khall A Khall, Ayman A Aly, and A Abo-Ismal, Adaptve Based Block Fuzzy Controller of Flexble Robot Arm, IASTED Internatonal Conference on Intellgent Systems and Control, Orlando, USA, November 6-8, 008 [3] Ayman A Aly, Farhan A Salem, "Desgn of Intellgent Poston Control for Sngle Axs Robot Arm", Internatonal Journal of Scentfc & Engneerng Research, IJSER, Volume 5, Issue 6, pp , 04 [4] BARNETT, S, 984, Matrces n Control Theory (Malabar, Fla: Robert E Kreger) [5] CORLESS, M, 989, Trackng controllers for uncertan systems: applcaton to a mantc R3 robot Journal of Dynamc Systems, Measurement and Control,, [6] CORLESS, M J, and LEITMANN, G, 98, Contnuous state feedback guaranteeng unform ultmate boundedness for uncertan dynamc systems IEEE Transactons on Automatc Control, 6, [7] Ayman A Aly and Aly S Abo El-Lal, Self Tunng PI Fuzzy Poston Control of A Hydraulc Manpulator Journal of Engneerng Scence, JES, Vol 33, No, pp 99-09, January 005 [8] Ayman A Aly, Moton Control Of A Mechatroncs System Usng Fuzzy Model Reference Learnng Technque, Multdscplnary Internatonal Peer Revewed Journal, SAUSSUREA, Vol 5 (6): PP 96-0, 05 [9] J J E Slotne and W L, Appled nonlnear control vol 46: Prentce hall Englewood Clffs, NJ, 99 [0] Ayman A Aly, Moton Control of A Mechatroncs System Usng Fuzzy Model Reference Learnng Technque, Multdscplnary Internatonal Peer Revewed Journal, SAUSSUREA, Vol 5 (6): PP 96-0, 05