Lecture 8 Backlash and Quantization. Material. Linear and Angular Backlash. Example: Parallel Kinematic Robot. Backlash.

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1 Lecre 8 Backlash and Qanizaion Maerial Toda s Goal: To know models and compensaion mehods for backlash Lecre slides Be able o analze he effec of qanizaion errors Noe: We are sing analsis mehods from previos lecres (describing fncions, small gain heorem ec.), and hese have references o he corse book(s). Qanizer / Linear and Anglar Backlash x o F o Example: Parallel Kinemaic Robo Ganr-Ta robo: Need backlash-free gearboxes for high precision F in T in T o θ in θ o EU-projec: SMErobo TM Backlash ead-zone Model Backlash (glapp) is presen in mos mechanical and hdralic ssems increasing wih wear bad for conrol performance ma case oscillaions Noe: A gear box wiho an backlash will no work if emperare rises Force (Torqe) x o (θ in θ o ) Ofen easier o se model of he form ( ) x o ( ) Uses implici assmpion:f o =F in,t o =T in. Can be wrong, especiall when no conac. The Sandard Model Servo moor wih Backlash Assme insead ẋ o =ẋ in when in conac ẋ o =when no conac No model of forces or orqes needed/sed x o θ o P-conrol of servo moor θ θ ref in θ in θ b o + K +st s θ in How does he vales ofk andaffec he behavior?

2 Effecs of Backlash escribing Fncion for a Backlash.5 No backlashk=.5,, BacklashK=.5,, Oscillaions fork=b no fork=.5 ork=. Wh? Limi ccle becomes smaller ifis made smaller, b i never disappears d IfA>hen d N(A)= b +ia A b = A [ π π arcsin v wih Im ( ) d A N(A) Re 8 8 a = d ( ) d π A ( d A and ) ( d ) ] A elsen(a)=. mine exercise escribing Fncion Analsis 3 Nqis iagrams Inp and op of backlash.8.. Sd he plo for he describing fncion for he backlash on he previos slide. Where does he fncion end fora and wh? N(A) Imaginar Axis 3 K= /N(A) K= K=.5 Real Axis ForK=,=.: inersecion beweeng(jω) and /N(A) occrs fora=.33,ω=. Simlaion:A=.33, ω=π/5.=. escribing fncion predics oscillaion well! Limi ccles? Backlash Limi Ccles The describing fncion mehod is onl approximae. Can one deermine condiions ha garanee sabili? + K b θ in θ in θ o +st s Rewrie he ssem as θ in θ o θ in BL θ o Noe: θ in and θ o will no converge o zero Idea: Consider insead θ in and θ o G(s) Noe ha he block BL saisfies { θ θ o = in in conac oherwise G(s) Analsis b small gain heorem Analsis b circle crierion Backlash block has gain (from θ in o θ o ) Hence closed loop is sable ifg(s) asmpoicall sable and G(iω) <for all ω Backlash block has gain in he secor[,] (from θ in o θ o ) /k = and /k = Hence closed loop is sable ifreg(iω)> for all ω. (For or moor example his proves sabili whenk<)

3 Backlash compensaion Linear Conroller esign Inrodce phase lead o avoid he /N(A) crve: Mechanical solions Insead of onl a P-conroller we choosek(s)=k +st +st ead-zone Linear conroller design + k +st +st b +st θ in θ in θ o s Backlash inverse Backlash Inverse ConrollerK(s)=k +st +st Simlaion wiht =.5,T =. x o.5 8 Nqis iagrams.5, wih/wiho filer 5 5 Imaginar Axis wih filer x o, wih/wiho filer wiho filer 5 5 Real Axis No limi ccle, oscillaion removed! Idea: Le jmp± whenẋ o shold change sign. Works if he backlash is direcl on he ssem inp! Backlash Inverse Example Perfec compensaion If ()= + if()>( ) if()<( ) ( ) oherwise =henxo ()=() (perfec compensaion) <: Under-compensaion (decreased backlash) >: Over-compensaion, ofen gives oscillaions Moor wih backlash on inp, P-conroller Example Under compensaion Example Over compensaion

4 Backlash More advanced models Example: Parallel Kinemaic Robo Ganr-Ta robo: Need backlash-free gearboxes for ver high precision Warning: More deailed models needed someimes Model wha happens when conac is aained Model exernal forces ha inflence he backlash Model mass/momen of ineria of he backlash. EU-projec: SMErobo TM hp:// "Roaional o Linear moion" Backlash in gearbox and rails Gear box Rack-and-pinion (Swe. kggsång ) Moor connecs here Remed: Use wo moors, possible o moors in opposie direcions: One moor can ac as spring and brake o "redce" backlash. Need measremens on boh moor and arm-side. Backlash compensaion Qanizaion Qanizer / How accrae shold he converers be? (8- bis?) Wha precision is needed in compaions? (8- bis?) From maser hesis b B. Brochier, Conrol of a Ganr-Ta Srcre, LTH, See also maser heses b j. Schiffer and L. Hal, 9. Qanizaion in A/ and /A converers Qanizaion of parameers Rondoff, overflow, nderflow in operaions NOTE: Compare wih (differen) limis for qanizer/dead-zone rela in Lecre. Linear Model of Qanizaion escribing Fncion for eadzone Rela Model he qanizaion error as a sochasic signale independen ofwih recanglar disribion over he qanizaion size..8. e Works if qanizaion level is small compared o he variaions in e e Q Recanglar noise disribion over[, ] has he variance + Var(e)= / e f e de= e / de= Lecre N(A)= πa /A fora>and zero oherwise

5 ( ( ( escribing Fncion for Qanizer escribing Fncion for Qanizer Qanizer / N( A) A/ A< ( ) N(A)= n k πa A n <A< n+ (See exercise) k= The maximm vale is/π.7 fora.7. Predics limi ccle if Nqis crve inersecs negaive real axis o he lef of π/.79. Shold design for gain margin >/.79=.7! Noe ha redcingonl redces he size of he limi oscillaion, he oscillaion does no vanish compleel. 5 mine exercise Example Moor wih P-conroller. Rondoff a inp,=.. Nqis crve inersecs a.5k. Hence sable fork<wiho qanizaion. Sable oscillaion prediced fork>/.7=.57. N( A) A/ How does he shape of he describing fncion relae o he nmber of possible limi ccles and heir sabili. Wha if he Nqis plo inersecs he negaive real axis a.8? inersecs he negaive real axis a? inersecs he negaive real axis a? O p O p O p 5 5 K=.8 K=. K=. 5 Time Example oble inegraor wih nd order conroller Nqis crve Qanizaion a A/ converer oble inegraor wih nd order conroller,= Inp Op Op A/ conroller /A process Time escribing fncion:a /=., periodt=39 Simlaion:A =. andt=8 Qanizaion a /A converer Qanizaion Compensaion oble inegraor wih nd order conroller,=. Inp Unqanized Op Time Use improved converers, (small qanizaion errors/larger word lengh) Linear design, avoid nsable conroller, ensre gain margin>.3 Use he racking idea from ani-windp o improve /A converer conroller igial /A + Analog escribing fncion:a /=.5, periodt=39 Simlaion:A =.5 andt=39 Beer predicion, since more sinsoidal signals Use analog diher, oversampling and digial low-pass filer o improve accrac of A/ converer + A/ filer decim. 5

6 Toda s Goal To know models and compensaion mehods for backlash Be able o analze he effec of qanizaion errors Qanizer /