Linear-Quadratic-Gaussian (LQG) Controller for Two link- robotic Manipulator Control

Transcription

1 Lnear-Quadrat-Gaussan (LQG) Controller for Two lnk- robot Manpulator Control Lela Fallah Aragh, M Habbnejad Korayem, Amn Nkoobn Abstrat Ths paper presents a new method for the ontrol of Two lnk- robot manpulator systems usng Lnear-Quadrat- Gaussan (LQG), n Ths method Lnear-Quadrat-Gaussan (LQG) that s a modern state-spae tehnque for desgnng optmal dynam regulators by kalman used for the ontrol of Two lnk- robot manpulator systems The smulaton result are very promsng Index Terms Two lnk- robot manpulator systems, usng Lnear-Quadrat-Gaussan ontroller, optmal dynam regulators I INTRODUCTION In the reent years usng ntellgee ontrol suh as fuzzy ontrol, Neural Network, Neuro Fuzzy and beause that they an ontrol nonlnear systems that would be dffult or mpossble to model mathematally In the reently years In Dynam ontrol of robot have been utlzed n many researhers work n ths area Suh as Lanfang Tan et al use a neural network approah for the moton ontrol of onstraned flexble manpulators robots [], Y, et al have nvestgated the robustness and stablty of a fuzzy log ontroller appled to a robot manpulator wth unertantes suh as frton, unmodeled dynams, and external dsturbane et [], Kumbla et al have mplemented herarhal ontrol on robot manpulator usng fuzzy log [3] Bannerjee, et al have used a Fuzzy Log Controller to aheve poston ontrol of a two-lnk manpulator [4] Adams, et al [5] have used GA to optmze the membershp funtons and rule bases of a mult-stage fuzzy PID ontroller wth a fuzzy swth ror robot ontrol Brudka, et al presented an ntellgent robot ontrol system whh employs ultrason dstane measurements, And for Conseutve stages of data proessng they used to neural networks applatons [6], Adamv, et al used neural networks applaton for moble robot ontrol on predetermned trajetory of the road [7], Ya- Chen, et al used an Fuzzy neural adaptve ontroller to multple-lnk robot ontrol [8], Devendra P, et al used the proportonal plus dervatve (PD) ontrol wth the PD ontroller gan parameters optmzed va Genet Algorthm Lela Fallah Aragh, PhD Student, Eletral Eng Dept, Tehran Researh Sene Azad Unversty, Tehran, Iran (orrespondng author to provde phone: e-mal: l_falah@srbauar) M Habbnejad Korayem, Professor, Eletral Eng Dept, Tehran Unversty of sene & tehnology, (e-mal: hkorayem@ustar) Amn Nkoobn, Assstant Professor, Eletral Eng Dept, Tehran Unversty of sene & tehnology (e-mal: ankoobn@ustar) (GA) And Fuzzy Log for ontrol of Two lnk- robot [9], ZG Zhang, et al report on the desgn and stablty analyss of a smple quadruped runnng ontroller that an autonomously generate steady runnng wth good energy effeny and suppress suh dsturbanes as rregulartes of terran [0], Dongbng Gu, et al presented a new path-trakng sheme for a ar-lke moble robot based on neural predtve ontrol, they employed A mult-layer bakpropagaton neural network to model non-lnear knemats of the robot nstead of a lnear regresson estmator n order to adapt the robot to a large operatng range [], Mathew L, et al studed on the mplementaton of several Intellgent ontrol tehnques as appled to the balanng of the nverted wedge problem These nluded a bas four-nput dret fuzzy ontroller (nludng the use of the nonlnear nput term) and an adaptve fuzzy ontrol tehnque known as the FMRLC [] Lawrene J Alder, et al developed a new approah to dentfaton s That utlzes a nomnal ontroller n order to solate the dynam effets related to the unertan parameters from the other known system dynams[], John L Meyery, et al presented the results of postve poston feedbak (PPF) ontrol and lnear quadrat Gaussan (LQG) ontrol for vbraton suppresson of a flexble struture usng pezoerams [3], In ths paper Two lnk- robot manpulator systems s ontrolled, The problem statement s gven n seton II In seton III LQG ontrollers s proposed Smulaton results are gven n seton IV Fnally seton V nludes onlusons, II DYNAMIC EQUATION OF TWO LINK- ROBOTIC MANIPULATOR SYSTEMS [] In ths seton we derve the equatons of moton for an ndvdual lnk based on the dret method has been derved, e Newton-Euler Formulaton The moton of a rgd body an be deomposed nto the translatonal moton wth respet to an arbtrary pont fxed to the rgd body, and the rotatonal moton of the rgd body about that pont The dynam equatons of a rgd body an also be represented by two equatons: one desrbes the translatonal moton of the entrod (or enter of mass), whle the other desrbes the rotatonal moton about the entrod The former s Newton's

2 equaton of moton for a mass partle, and the latter s alled Euler's equaton of moton Fgure shows all the fores and moments atng on lnk III Fg : Free body dagram of lnk n moton Let be V the lnear veloty of the entrod of lnk wth referene to the base oordnate frame O-xyz, whh s an nertal referene frame The nertal fore s then gven by m V, where m s the mass of the lnk and V s the tme dervatve of V Based on D Alembert s prnple, the equaton of moton s then obtaned by addng the nertal fore to the stat balane of fores n eq( ) so that, f, + + m g m V 0 f, 0,,, n f, and f, + are the ouplng fores appled to lnk by lnks - and +, respetvely, and g s the aeleraton of gravty Addng these terms to the orgnal balane of moments we have: N, N, + ( r, r, ) f, + ( r, ) f, + I ω ω ( I ω ) 0 () Smlarly, elmnatng f 0, yelds, N, and N, + are the moment appled to lnk by lnks - and +, respetvely If we onsder two ndvdual lnks robot, Let us obtan the Newton-Euler equatons of moton for the two ndvdual lnks () IV Fg: Mass propertes of two planar robot From eq () and (), the Newton-Euler equatons for lnk are gven by: f f + m g m V (3) 0,, 0 0,, 0, 0,,, N N ( r ) f + ( r ) f I ω 0 (4) Note that all vetors are x, so that moment N -, and the other vetor produts are salar quanttes Smlarly, for lnk, f m g m V (5), + 0,,, N ( r ) f I ω 0 (6) To obtan losed-form dynam equatons, we frst elmnate the onstrant fores and separate them from the jont torques, so as to expltly nvolve the jont torques n the dynam equatons For the planar manpulator, the jont torques τ and τ are equal to the ouplng moments: N, τ (7) Substtutng eq(7) nto eq(6) :, τ ( r ) ( m V m g) I ω 0 (8) τ τ ( r0, ) mv + ( r0, ) mv + ( r0, ) mg + ( r0, ) mg I ω 0 (9) Next, we rewrtev, ω and r, usng jont dsplaements, whh are ndependent varables Note

3 that ω s the angular veloty relatve to the base oordnate frame, whle s measured relatve to lnk Then, we have, ω ω + (0) The lnear velotes an be wrtten as l sn( ) V () l os( ) V τ () { l sn( ) + l sn( + ) } l sn( + ) { l os( ) + l os( + ) } l os( + ) H + H h h + G τ (3) H + H h+ G That: H m l + I + m l + l + ll os( )) + m l I ( I H + (4) dfferene between desred and Atual responses What s then requred s to desgn a ontroller so that the plant ontroller ombnaton satsfes the spefatons The LQG ontrol s a modern state spae tehnque for desgnng optmal dynam regulators It enables a trade off n performane and ontrol effort, and takes nto aount proess and measurement nose Lnear-Quadrat-Gaussan (LQG) ontrol s a modern statespae tehnque for desgnng optmal dynam regulators, The LQG regulator onssts of an optmal state-feedbak gan and a Kalman state estmator The frst desgn step s to seek a state-feedbak law that mnmzes the ost funton of regulaton performane, whh s measured by a quadrat performane rteron wth user-spefed weghtng matres, and to defne the tradeoff between regulaton performane and ontrol effort The next desgn step s to derve a state estmator usng a Kalman flter beause the optmal state feedbak annot be mplemented wthout full state measurement Sne the Kalman flter s an optmal estmator when dealng wth Gaussan whte nose, t mnmzes the G m l + H g os( ) + mg{ l os( + ) l os( )} m l + ll os( )) + ( I h m l sn( ) (5) l G m g l os( + ) (6) asymptot ovarane of the estmaton error In the optmal feedbak ontrol, a gan matrx s sought to mnmze a spefed quadrat performane rteron J (u) expressed as the ntegral of a quadrat form n the state vetor x plus a seond quadrat form n the ontrol vetor u, e III LQG ontroller [4, 5, 6, 7] J ( u) 0 T ( x Qx + u T Ru) dt ( 7) In a typal ontrol system desgn problem, nputs are a dynam system to be ontrolled (plant), a desrpton of the types of nputs to be enountered, and spefatons Regardng aeptable values of system errors, e, the The weghtng matres Q and R are user-spefed and defne the trade-off between Regulaton performane (how fast x(t) goes to zero) and ontrol effort [6,7] The frst

4 desgn step seeks a state feedbak law u kx that To form the LQG regulator, smply onnet the Kalman flter mnmzes the ost funton J (u) The mnmzng gan and LQ optmal gan K as shown n Fgure 5-3 matrx K s obtaned by solvng an algebra Rat equaton [4] Ths gan s usually alled the LQ-optmal gan Sometmes, the proess seekng the LQ state feedbak regulator, e LQ-optmal gan matrx k, s alled Lnear Quadrat Regulator (LQR) desgn It s lear that the LQ-optmal state feedbak u kx s not mplemental wthout full state measurement However, one Fgure4 Blok dagram of LQG regulator an derve a state estmate x suh that u kx remans Ths regulator has state spae equatons: optmal for output feedbak problem Ths state estmate s generated by Kalman flter [5] as shown below: x ( A LC + Bk) x + Ly0 (9) u kx (0) In addton to LQG desgn, a Dsturbane Aommodate Control (DAC) desgn and a hybrd LQG/DAC ontrol system Fg3: Blok dagram of the Kalman estmator Mathematally, the Kalman state estmator an be expressed by: x A x+ Bu + L( y0 C x) ( 8) wth two nputs, ontrols u and measurements y 0 Determnes the Kalman gan L n onjunton wth dynam systems through an are also studed n ths hapter Hene, nne ases wll have to be dsussed, whh are three ontrol systems, the LQG, the DAC and the hybrd LQG/DAC, n onjunton wth three dfferent ontrol models, the MCC model, the SCC model and the SSCC model, respetvely I n ths paper LQG regulator has been used for two lnk- robot manpulator systems Fgure 5 shows the struture of system that used n ths system [6] Algebra Rat equaton The Kalman flter s an optmal estmator when dealng wth Gaussan whte nose Spefally, t mnmzes the asymptot ovarane of the x estmaton error x

5 IN LPF T q IN LPF T q LQG Controller NOISE Fg 7: the output of two lnk- robot manpulator systems LQG Controller NOISE Fg 5: struture of LQG regulator for Two lnk- robot manpulator systems As the frst method for losed loop ontrol of two lnk robot the LQG regulator ontroller has been used, fgure 8 shows the smulaton result IV SIMULATION RESULT In the frst the dynam model of two lnk- robot manpulator systems has been smulated usng Matlab software 007 as you see the system s unstable Fgure 6 shows the shemat of two lnk- robot manpulator systems and fgure 7 shows the smulaton result for step nput: Fg 6: shemat of the two lnk- robot manpulator systems Fg 8: the output of two lnk- robot manpulator systems usng LQG regulator,

6 V CONCLUSION Ths paper presents a new method for the ontrol of Two lnk- robot manpulator systems usng method Lnear-Quadrat- Gaussan (LQG) REFERENCES [] Lanfang Tan, Jun Wang, and Zongyuan Mao, "Constraned Moton Control of Flexble Robot Manpulators Based on Reurrent Neural Networks" IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 34, NO 3, JUNE 004 [] Y, SY and Chung, M, J, 997, A Robust Fuzzy Log Controller for Robot Manpulators wth Unertantes, IEEE Transatons on Systems, Man and Cybernets, 7, No 4, pp [3] K umbla, KK and Jamshd, M, 994, Control of Robot Manpulator Usng Fuzzy Log, Proeedngs of the Thrd IEEE Conferene on Fuzzy Systems, Vol, pp [4] Bannerjee, S and Woo, PY, 993, Fuzzy Log Control of Robot Manpulator, IEEE Conferene on Control Applatons, pp [5] Adams, JM and Rattan, KS, 00, Genet Mult-Stage Fuzzy PID Controller wth Fuzzy Swth, Proeedngs of the IEEE Internatonal Conferene on Control Applatons, pp [6] Brudka, M Paut, A, " Intellgent robot ontrol usng ultrason measurements" Proeedngs of the 6th IEEE Instrumentaton and Measurement Tehnology Conferene, 999 IMTC/99 [7] Adamv, O Koval, V Turhenko, I, Predetermned movement of moble robot usng neural networks" Proeedngs of the Seond IEEE Internatonal Workshop on Intellgent Data Aquston and Advaned Computng Systems: Tehnology and Applatons, 003 [8] Petropoulaks, L, " Intellgent ontrol usng agents and fuzzy behavoural strutures" Proeedngs of the 000 IEEE Internatonal Symposum on Intellgent Control, 000 [9] Ya-Chen Hsu Guanrong Chen Malk, HA, " Fuzzy neural adaptve ontroller desgn: wth applaton to multple-lnk robot ontrol" IEEE Internatonal Conferene on Neural Networks,997, [0] Dongbng Gu and Huosheng Hu, Neural Predtve Control for a Carlke Moble Robot Internatonal Journal of Robots and Autonomous Systems, Vol 39, No -3, May, 00 [] Mathew L Moore, John T Musaho, Kevn M Passno, Genet adaptve ontrol for an nverted wedge: experments and omparatve analyses Engneerng Applatons of Artfal Intellgene 00 [] Lawrene J Alder, Stephen M Rok, Adaptve Control of a Flexble- Lnk Robot Manpulator Presented at The Ameran Control Conferene, San Fransso, CA June 993 [3] John L Meyery, Wllam B Harrngtony, Brj N Agrawal and Gangbng Song, Vbraton suppresson of a spaeraft flexble appendage usng smart materal Smart Mater Strut 7 (998) Prnted n the UK [4] Dorato, P, Abdallah, C and Cerone V, LINEAR-QUADRATIC CONTROL AN INTRODUCTION, Prente-Hall, In, New Jersey, 995 [5] Fredland, B, CONTROL SYSTEM DESIGN, AN INTRODUCTION TO STATE-SPACE METHODS MGraw-Hll Book Company, 986 [6] Toolbox, Matlab 008 [7] ASHWIN DHABALE, R N BANAVAR and M V DHEKANE, LQG ontroller desgns from redued order models for a launh vehle S adhan Vol 33, Part, February 008, pp 4 Prnted n Inda