Inverted pendulum virtual control laboratory

INVERTED PENDULUM VIRTUAL CONTROL LABORATORY Jos´e L. Lima, Jos´e C. Gon¸cal es, Paulo G. Cos a, A. Paulo Mo ei a Poly echnic Ins i u e o B agan¸ca, Po ugal {jllima, goncal es}@ipb.p Facul y o Enginee ing o he Uni e si y o Po o, Po ugal {paco,amo ei a}@ e.up.p Abs ac : This pape desc ibes a ool o in e ac i e lea ning ha can be used o imp o e con ol sys ems design. The de eloped sys em is eady o use and allows es ing di e en con ol me hods. I can be used by s uden s o p oblem sol ing and indi idual lea ning. The i ual con ol labo a o y was implemen ed as a eaching aid du ing lec u es on con ol sys ems. As he e is no need o do any special p og amming o debugging, he s uden s can ocus on he con ol i ems. Classical con ol me hods such as PID and S a e-Space app oaches a e a ailable and gains can be uned. A iendly appea ance based on openGL 3D shows a simula ion o he eal wo d: A ca wi h an in e ed pendulum is ”bumped” wi h a o ce. The dynamic equa ions o mo ion o he con ol sys em a e linea ized assuming ha he pendulum does no mo e mo e han a ew deg ees away om he e ical allowing o apply linea con ol me hods. Al hough, he simula ed sys em is ealis ic and based on a Dynamics Engine. Keywo ds: Con ol educa ion, In e ac i e p og ams, Simula ion 1. INTRODUCTION Tasks like modeling, iden i ica ion, analysis and simula ion a e common in con ol sys em design (K oumo e al., 2003). The de eloped applica ion suppo s he simula ion o an in e ed pendulum sys em wi h i s con olle . The simula ed wo ld beha io and g aphics a e based on open sou ce pla o ms. The objec i e o his p ojec is o allow he s uden s o lea n and es con ol sys ems in an easy way. S uden s need a ” eady o go” and iendly applica ion o lea n by playing con ol heo y (Saco e al., 2002). Some i ual labo a o ies exis on-line on he web (W. and Tzes, 1999), p esen ing ad an ages like low cos , iendly use and suppo ing simul a- neous mul iple use s, as examples: ’Labo a o io Vi ual: pendulo in e ido’ (Uni e sidad de Val- ladolid (UVA), Labo a o io i ual: pendulo in e - ido, 2006) and Fuzzy Pendulum Demo p oposed by he In eg a ed Reasoning G oup o he Ins i- u e o In o ma ion Technology o he Na ional Resea ch Council o Canada (Ins i u e, 2006). Tools like Ma lab a e powe ul bu some imes ha d o p og am. A s uden o ien ed applica ion is p esen ed in o de o selec and une he con olle ha closes he loop. The inal esul is a iendly 3D in e ace scene allowing he use o mo e a ound he wo ld. This pape is o ganized as ollows: Ini ially, he in e ed pendulum sys em and some applica ions CONTROLO 2006 7 h Po uguese Con e ence on Au oma ic Con ol Ins i u o Supe io Técnico, Lisboa, Po ugal Sep embe 11-13, 2006 a e desc ibed. Then, in sec ion 3, sys em modeling equa ions whe e a S a e-Space app oach is used is desc ibed. Sec ion 4 p esen s he wo ld cons uc- ion, beha io and i is also shown how s uden s can in e ac wi h he g aphical in e ace, being possible o con ol he sys em manually. Then, in sec ion 5, he eal ime closed loop simula ion whe e esul s o di e en con ol me hods a e shown is p esen ed. Finally, sec ion 6 ounds up wi h conclusions and u u e wo k. 2. INVERTED PENDULUM OVERVIEW An in e ed pendulum is a pendulum ha ing i s cen e o g a i y loca ed abo e i s pi o poin . An in e ed pendulum is he e o e inhe en ly uns a- ble. When i s cen e o g a i y is di ec ly abo e he pi o poin , i may emain s a ic. I he pi o is s a ic and he cen e o g a i y is sligh ly displaced om he e ical posi ion, he pendulum will no e u n o i s o iginal posi ion bu will end o ind a new equilib ium posi ion such ha i s cen e o g a i y is a i s lowes possible posi ion. A eedback con ol sys em ha posi ions he pi o may be used o balance an in e ed pendulum (K akow, 2005). Some applica ions o he in e ed pendulum a e p esen ed in he nex subsec ions. 2.1 Segway An in e ed pendulum is p esen in de ices such as a unicycle and a Segway Human T anspo e (Figu e 1). In hese de ices, he pi o o he pendulum is he axle o a wheel o pai o wheels. In a unicycle he wheel is powe ed by he ide . In a Segway Human T anspo e , he wheel is powe ed by an elec ic mo o . Mo ion o he wheel, o pai o wheel, is con olled so ha he pendulum is dynamically balanced (Segway web page, 2006). Fig. 1. Segway Human T anspo e The e is also ano he e sion o Segway, he Seg- way Robo ic Mobili y Pla o m (RMP), ha is a modi ied e sion o he Segway Human T ans- po e (HT) designed o p o ide scien is s and enginee s a mobile base o use in obo ics esea ch (Segway Robo ic Mobili y Pla o m, 2006). I has been well es ablished wi hin he li e a u e abou agen s and obo ics ha simula ion can be a powe ul ool o speeding up he de elopmen cycle o obo con ol sys ems (Go e al., 2004). 2.2 Rocke Na iga ion The in e ed pendulum con ol model is simila o he ocke launch (Oga a, 2002). I was belie ed ha , in ligh , he ocke would ”hang” om he engine like a pendulum hanging om a pi o . The weigh o he uel ank would keep he ocke lying s aigh up as long as he uel las ed. Howe e , his belie is inco ec , such a ocke will ne e ly in a s aigh line and will always u n and c ash in o he g ound soon a e launch. This is wha happened o Godda d’s ocke (Wikipedia - The ee encyclopedia, 2006). In he p esen wo k, he pendulum is ee o mo e in plane xy and he ca is able o slide ac oss xaxle only. Nowadays, Rocke s use Ine ial Na iga ion o sol e his p oblem. Ine ial na iga ion employs accele ome e s and gy oscopes o de e mine speed and posi ion. (Japan Ae ospace Explo a ion Agency (JAXA), 2006). 3. SYSTEM MODELING Con olling he in e ed pendulum is a classical p oblem in con ol labo a o ies because he pen- dulum dynamics is bo h nonlinea and uns able (Samad and Balas, 2003). The dynamic equa ions o mo ion o he con ol sys em a e linea ized, al hough he simula ed sys em is ealis ic and based on a Dynamics Engine. The in e ed pendulum sys em consis s o wo mo ing pa s: •The pendulum •The ca The dynamic equa ions o mo ion o he sys em a e linea ized assuming ha pendulum does no mo e mo e han a ew deg ees away om he e i- cal. One impo an issue o emphasize is ha ca wheels dynamics is igno ed, in o he wo ds, hei mass and momen o ine ia a e conside ed null. The ac ua o and senso dynamics a e despised bu a sa u a ion nonlinea i y can be applied o he ac ua o as shown in he sec ion 5 whe e a as dynamic con olle is applied. The modeled Sys em is shown in Figu e 2. Fig. 2. In e ed pendulum sys em The cons an s and a iables o his case s udy a e de ined as ollows: •Mmass o he ca 3 kg •mmass o he pendulum 1.57 kg •b ic ion o he ca 0.1N/m/s •lleng h o pendulum cen e o mass 2 m •Iine ia o he pendulum 8.37 kgm2 •F o ce applied o he ca •N eac ion o ce applied o he ca •xca posi ion coo dina e •θpendulum angle om e ical •gg a i y accele a ion 9.8m/s2 Whe e he pendulum momen o ine ia is shown in equa ion (1). I=1 3ml2(1) The well known New on’s second law o mo ion, equa ions (2) and (3), whe e Fis he applied o ce, P he g a i a ional o ce and T he applied o que, allows o w i e he Lag angian equa ions o he sys em desc ibed in equa ions (4) and (5) (Uni e si y o Michigan, Ma lab con ol u o ial, 2006). XF=ma (2) XT=I¨ θ(3) F−N−b˙x=M¨x(4) −Pl sin(θ)−Nl cos(θ) = I¨ θ(5) In o de o apply linea con ol me hods he sys em equa ions should be linea ized, assuming ha θ=π+φ, whe e φ ep esen s a small angle om he e ical upwa d di ec ion. The e o e, cos(θ)≈ −1, sin(π+φ)≈ −φ, and ˙ θ2≈0 (Uni e si y o Michigan, Ma lab con ol u o ial, 2006). The linea ized sys em equa ions can be ep e- sen ed in S a e-Space o m (Ribei o, 2002), as shown in he nex equa ions:     ˙x ¨x ˙ θ ¨ θ     =A    x ˙x θ ˙ θ     +B u (6) Y=C    x ˙x θ ˙ θ     (7) A=        0 1 0 0 0−(I+ml2)b λ m2gl2 λ0 0 0 0 1 0−mlb λ mgl(M+m) λ0        (8) Whe e λ=I(M+m) + Mml2. B=        0 (I+ml2)b λ 0 ml λ        (9) C=1 0 0 0 0 0 1 0 (10) Resul ing Ain ma ix (11) and Bin ma ix (12). A=    0 1 0 0 0−0.0257 1.6925 0 0 0 0 1.0000 0−0.0055 2.4632 0     (11) B=    0 0.2566 0 0.0550     (12) 4. WORLD CONSTRUCTION AND BEHAVIOR Dynamic engines like New on Dynamics, YADE- Ye Ano he Dynamic Engine and ODE-Open Dynamics Engine a e powe ul ools ha allow p og amme s o c ea e a physic wo ld composed by objec s connec ed h ough join s and simula e hem. The simula ion dynamics a e based on a - icula ed igid body dynamics and includes o ces and collision ea men . The ODE base objec s o he pendulum and he ca a e a pa allelepiped and a cylinde espec i ely. The pa allelepiped is connec ed h ough an hinge join o one end o he cylinde , allowing i o mo e eely in he xy plane. I is also connec ed h ough a slide join allowing mo emen s along xaxle. 4.1 In e ac i e en i onmen issues Some ime ago, simula ion ools we e mainly ex based. Nowadays, compu e g aphics de elopmen allow us o make g ea scena ios like 3D ende ing, came a posi ioning and eedom ex u es cap i a - ing people’s a en ion. An in e ac i e educa ional sys em should ha e he ollowing equi emen s (K oumo e al., 2003): •G aphical use in e ace: easy HMI (human machine in e ace) educing as much as pos- sible ex yping. •People wan o easily each esul s wi h- ou much manual eading. Mos s uden s ha e mo e success expe imen ing han ead- ing om books only. •Simple e ms a e a mus : o unde s and he complex con ol me hods, s uden s should ha e access o basic e ms and concep s. •The simula ion sys em should be easily ac- cessible. The p esen ed sys em aims o help s uden s in con ol enginee ing cou ses. Concep s like eed- back, s abili y, PID Tunning and S a e-Space ap- p oach a e applied. Tools like Simulink, make he simula ion possible bu hey a e designed o co e a wide enginee ing a ea and s uden s mus memo ize some commands in o de o use hese powe ul So wa e. In he de eloped sys em he e a e no commands bu bu ons ins ead wi h a iendly 3D g aphical en- i onmen , as shown in Figu e 3. The g aphical based in e ac i e in e ace, allows use s o change PID gains and closed loop poles pa ame e s o he S a e-Space eedback con- olle app oach. Random noise in he measu e is also in oduced in o de o e alua e he sys em obus ness. 5. REAL TIME CLOSED LOOP SIMULATION Assuming ha simula ion ime s ep is much as e han dynamics, i is possible o assu e ha i is a con inuous ime model. A ime s ep o 20 ms is enough o alida e his app oach. The e a e h ee closed-loop con ol me hods ha can be used in his p ojec : he use can manually con ol he applied o ce o he ca , he PID con ol Top iew Da a acquisi ion g aph Closed loop poles Se ling ime Feedback gain ec o  Se  poin  Noise addi ion Came a posi ioning S a e ec o  Con ol me hod Nonlinea  sa u a ion Fig. 3. Vi ual Con ol Labo a o y Sc een Sho whe e use can in oduce p opo ional, in eg al and de i a i e gains and he S a e-Space eedback con ol whe e use can apply a pole placemen ap- p oach o in oducing he ime se ling, esul ing in a eedback gain ec o . A g aphical ime e olu- ion shows he ca posi ion, speed and pendulum angle and a zoom ea u e is also implemen ed in o de o highligh some de ails. A op and a on al iew, whe e use can place he came a e e ywhe e, shows he 3D eal wo ld. Use can also de ine he se poin and noise addi ion. 5.1 PID The analog PID con ol has been used success ully in many indus ial con ol sys ems o o e hal a cen u y. The basic p inciple o he PID con ol scheme is o ac on he a iable o be manipu- la ed ough a p ope combina ion o h ee con ol ac ions: p opo ional con ol ac ion (whe e he con ol is p opo ional o he e o signal, which is he di e ence be ween he inpu and he eedback signal), in eg al con ol ac ion (whe e he con ol ac ion is p opo ional o he in eg al e o signal) and de i a i e con ol ac ion (whe e he con ol ac ion is p opo ional o he de i a i e o he e o signal). In i s mos simple o m, PID in ol es h ee ma hema ical con ol unc ions wo king o- ge he : P opo ional, In eg al and De i a i e, as shown in equa ion (13). m( ) = Kp[e( ) + 1 T i Z 0 e( ) + T dde( ) d ] (13) As an example, is shown in Figu e 4 he closed loop sys em esponse o a dis u bance in he pen- dulum, whe e he ca posi ion is no con olled. Fig. 4. PID closed loop sys em wi h Kp=8, Ki=0 and Kd=8 da a g aph 5.2 S a e-Space eedback con olle The o ce applied o he ca can be gi en by he s a e eedback ec o p esen ed in equa ion (14). F=−(K1K2K3K4    x ˙x θ ˙ θ     (14) Ha ing he sys em desc ibed in S a e-Space, he K alues (closed loop gains) can be ound by equa ion (15) (Vacca o, 1995). K=0 0 0 1 Q−1α(A) (15) Whe e α(s), he cha ac e is ic equa ion, p esen ed in equa ion (17) and Q he con ollabili y ma ix, gi en by equa ion (16). Q=B AB A2B A3B(16) α(s) = (s−p1)(s−p2)(s−p3)(s−p4) (17) The desi ed closed loop poles a e p esen ed in equa ions (18) and (19). poles1,2=a±ib (18) poles3,4=c±id (19) As esul , he S a e-Space eedback ec o is gi en by he nex equa ions: K1=−1.8553(c2+d2)(b2+a2) K2= (3.7106a)d2+ 3.7106c2a+ 3.7106 b2c+ 3.7106a2c−0.1 K3= (18.182 + 8.6560a2+ 8.6560b2)d2 +72.727 c a+18.182 c2+ 18.182b2+ 18.182a2 +8.6560 b2c2+ 8.6560a2c2+ 44.786 K4= (−17.312a)d2−36.364a−36.364c -17.312 c2a−17.312cb2−17.312ca2 A S a e-Space as dynamic con olle , based on a ou h o de Bessel p o o ype wi h a ime se ling o 4 seconds (Vacca o, 1995), is shown in Figu e 5, whe e a=−1.00, b=1.27, c=−1.38 and d=0.41 wi h he eedback gain ec o ep esen ed by: K=−10.09 −21.27 278.20 185.51 . An example o nonlinea sa u a ion ea u e o he inpu o ce is also shown. Fig. 5. S a e-Space as dynamic con olle wi h sa u a ion da a g aph A slowe dynamic con olle , based on a ou h o de Bessel p o o ype wi h a ime se ling o 10 seconds (Vacca o, 1995), p esen ed in Figu e 6, is implemen ed whe e a=−0.40, b=0.51, c=−0.55 and d=0.17, wi h he eedback gain ec o ep e- sen ed by: K=−0.26 −1.46 75.80 41.03 . 5.2.1. Noise dis u bance ea u e Da a acquisi- ion in Physical Sys ems su e s p oblems such as signal noise, cumula i e e o s, elec omagne ic in e e ence and empe a u e d i . Random noise addi ion in pendulum angle and ca posi ion measu emen allows o illus a e his Fig. 6. S a e-Space slow dynamic con olle da a g aph p oblem. The Da a acquisi ion is shown in Figu e 7. Fig. 7. S a e-Space con olle da a g aph wi h noise 6. CONCLUSIONS AND FUTURE WORK This pape has in oduced an in e ac i e lea ning ool o au oma ic con ol cou ses, whe e i is desi ed o con ol an in e ed pendulum. The de eloped sys em allow s uden s o ocus on he con ol heo y, helping hem o imp o e hei skills. The 3D isualiza ion and anima ion e ec s a e also an ad an age o s uden s o a be e unde s anding o he physical sys ems. The eal ime g aphic, whe e a e shown pendulum angle, ca posi ion and applied o ce, p esen s da a in a way ha can be easily decoded by s uden s. As u u e wo k, a wo dimensional in e ed pen- dulum, placed in an omnidi ec ional ca , is being de eloped. REFERENCES Go, Ja ed, B e B owning and Manuela Veloso (2004). Accu a e and lexible simula ion o dynamic, ision-cen ic obo s. In e na- ional Con e ence on Au onomous Agen s and Mul i-Agen Sys ems. Ins i u e, NRC (2006). Fuzzy pendulum demo. h p://www.ii .n c.ca/IR public/ uzzy /FuzzyPendulum.h ml. Japan Ae ospace Explo a ion Agency (JAXA) (2006). h p://spacein o.jaxa.jp/ no e/ ocke /e/ oc10 e.h ml. K akow, Kalman I. (2005). Sys em-speci ic PI Con ol Theo y o Fluid and Mo ion Sys- ems. Uni e sal publishe s. 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