Design of Controller for Robot Position Control
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1 eign of Conroller for Robo oiion Conrol Two imporan goal of conrol: 1. Reference inpu racking: The oupu mu follow he reference inpu rajecory a quickly a poible. Se-poin racking: Tracking when he reference inpu i conan ep inpu.. iurbance Rejecion: To reuce or eliminae he effec of iurbance inpu on he cloeloop conrol. 1
2 Conroller A imple, bu efficien conroller: roporional: Compenae proporionally for he error beween he reference inpu an he meaure oupu: U [ ] erivaive: ncreae he amping of he conrol yem, reuce overhoo an he eling-ime of he yem ime repone: U : U U U [ ] U U
3 Seay-Sae Error of he Conrol Syem Cloe-loop ranfer funcion: Error beween he e poin an he oupu: 1 1 E 3 Two ep inpu: Seay-ae error: f e if a loa exi, here exi a eay-ae error, E e lim
4 eign of conroller The characeriic equaion of he cloe-loop yem: The anar form of a econ orer yem: ζ : amping raio in roboic applicaion, ζ 1 criically-ampe i choen o have he fae non-ocillaory repone ω n : unampe naural frequency eermine he pee of he repone n n ω ζω 4 Equaing he coefficien of above equaion: n n n ζω ω ζω n ω φ φ ζ in
5 An example - conroller The conroller coefficien for a ample link wih phyical parameer: 1g. m ; 1N. m / ra / ωn, Chooing ζ 1 : ζωn ωn Sep repone for reference inpu 1 an no iurbance : e ω 1 n 5
6 The Effec of iurbance - conroller Sep repone for reference inpu 1 wih iurbance 4: The oupu oe no reach he e poin! eay-ae error exi. High conroller gain are require o reuce he eay-ae error. e 6
7 conroller f we a an inegraor, he eay-ae error will be remove, even in preence of he iurbance inpu. e 7
8 Seay-Sae Error wih Conrol Cloe-loop ranfer funcion: Error beween he e poin an he oupu: E Two ep inpu: Seay-ae error: No eay-ae error in he preence of iurbance, lim E e
9 Cloe-loop abiliy The roo of he characeriic equaion of he yem eermine he abiliy: Forming Rouh Array: Q 3 9 Therefore, large inegral gain reul in inabiliy. Sabiliy Coniion < <
10 Several racical ue in Robo Conrol n pracical implemenaion of a robo conrol yem, he following iue mu be coniere: Acuaor moor auraion Link an join flexibiliie Nonlineariie of he robo ynamic 1
11 Acuaor Sauraion The oupu of an acuaor of a robo a moor may reach i maximum poible value of orque, if he conroller oupu i high! n hi cae, he acuaor aurae an no more orque can be applie o he robo. A moor aurae when i reache i maximum allowable volage. The acuaor alo aurae when i reache i minimum poible value. While he acuaor aurae, he inegral conrol mu be omie o preven he buil-up of error. Some boun mu be pu on he oupu of he conroller o preven amage o he acuaor: 11
12 Effec of Acuaor Sauraion on Syem Repone Acuaor auraion low own he pee of he repone of he yem. Repone wih moor auraion ω 8 1
13 oin an Link Flexibiliy Since he robo link an join are mae from elaic maerial, hey a ome unwane flexibiliie o he yem. Thi reul in more flucuaion in he repone of he yem an low own he repone. The flexibiliy of he join can be repreene by a pring. Alo, he flexibiliy of he link can be ae a eparae pring. Thi reul in more egree of freeom, more ae variable an higher orer ranfer funcion. 13
14 Feeforwar Feeback Conrol Feeforwar Conrol: An inpu orque i irecly applie o he plan robo, bae on he value of he reference inpu. Aing feeforwar conrol o feeback conrol Arom an Murray, 8, chap. 11: o improve he repone o ime-varying reference inpu rajecory, o reuce he effec of iurbance. iavanage: The conrol yem i more eniive o variaion in proce moel. 14
15 Cloe-loop Tranfer Funcion Tranfer funcion of ifferen block raio of wo polynomial: Cloe-loop ranfer funcion from reference inpu o he oupu:,, p q G c H b a F ] [ ] [ c q p b c b a q T 15 For abiliy, boh enominaor polynomial mu be able: enominaor of feeforwar T.F.: b enominaor of he loop T.F.: 1 1 p c q p p q c H G
16 Chooing he Feeforwar Tranfer Funcion f he ranfer funcion of he feeforwar conroller i choen a: he cloe-loop ranfer funcion become: Therefore, he oupu will rack any ype of reference inpu ep, ramp,, 1 p q G G F 1 ] [ ] [ c q p b c b a q T E,, q b p a 16 Therefore, he oupu will rack any ype of reference inpu ep, ramp, polynomial, ec. wihou a eay-ae error. We mu know he plan ranfer funcion G prey well o have a goo feeforwar conroller. G mu have no zero in he lef half-plane minimum phae, o ha i invere F i able. E
17 Effec of iurbance on Error The relaion E in he previou page hol only if no iurbance exi. n a general cae, if here exi a iurbance, we have ome error: q E p q c The eay-ae error e can be remove by elecing proper ranfer funcion for he conroller H, for example a conroller. E 17
18 eigning a feeforwar conroller for an nepenen-oin Robo Conrol Syem Feeforwar conroller: lan: Conroller inpu o he plan: G F 1 ] [ 1 ] [ U U G 1 u & && 18 efining e -, from 1 an : ] [ ] [ u & & & & & e e e & && e
19 Seay-ae Error of he Conrol Syem ifferenial equaion for he error ignal: e&& e& e A he eay-ae iuaion, ime-erivaive are zero: e && e& Therefore: e e e f iurbance i zero : he eay-ae i zero e, e e - e 19
u(t) Figure 1. Open loop control system
Open loop conrol v cloed loop feedbac conrol The nex wo figure preen he rucure of open loop and feedbac conrol yem Figure how an open loop conrol yem whoe funcion i o caue he oupu y o follow he reference
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