ScienceDirect. Dynamic model of a mobile robot

Transcription

1 Avalable onlne at ScenceDect Poceda Engneeng 96 (014 ) Modellng of Mechancal and Mechatonc Systems MMaMS 014 Dynamc model of a moble obot Ján Kadoš* Faculty of Electcal Engneeng and Infomaton Technology, Slovak Unvesty of Technology, Ilkovčova 3, Batslava, Slovak Republc Abstact The contbuton pesents the dynamc model of a dffeental two-wheeled moble obot n the ont fame,.e. n the plane of angula poston of both the ght-hand sde wheel and the left-hand sde one. The am s to ceate an appopate MIMO plant model fo the pupose of the moble obot moton contol synthess. Based on the Lagangan fomalsm, the complete descpton of the second ode matx dffeental equaton (the dynamc model of the moble obot) s povded, ncludng the nfluence of netal couplng foces. A pa of wheel dvng toques epesents the nput of gven model. Addtonally, a decoupled dynamc model fo the obust moton contol mplementaton s pesented The The Authos. Authos. Publshed Publshed by Elseve by Elseve Ltd. Ths Ltd. s an open access atcle unde the CC BY-NC-ND lcense ( Pee-evew unde esponsblty of oganzng commttee of the Modellng of Mechancal and Mechatonc Systems MMaMS Pee-evew 014. unde esponsblty of oganzng commttee of the Modellng of Mechancal and Mechatonc Systems MMaMS 014 Keywods: Eule-Lagange s equaton; knetc enegy; nstantaneous cente of otaton; hozontal plane; decouplng 1. Intoducton The challengng ambton of a moble obot contol s the fast and accuate moton along the desed taectoy n the task plane gven by an actual applcaton (ndusty, agcultue, tanspot, household, posthetcs etc.). To keep both the hgh velocty of the moton and the elablty of the appled contol, t s nevtable to take the dynamcs of the movng mass of the moble obot nto account. The moton of the dffeental two-wheeled moble obot s pefomed by the wheel otaton descbed va the angula poston and t s tme devatves n the ont fame. On the othe hand, fom the use s pont of vew, the poston and velocty of the moble obot moton n the task fame (fo * Coespondng autho. Tel.: E-mal addess: an.kados@stuba.sk The Authos. Publshed by Elseve Ltd. Ths s an open access atcle unde the CC BY-NC-ND lcense ( Pee-evew unde esponsblty of oganzng commttee of the Modellng of Mechancal and Mechatonc Systems MMaMS 014 do: /.poeng

2 04 Ján Kadoš / Poceda Engneeng 96 ( 014 ) the smplcty, assume the moton wthn the hozontal plane) usually descbed by the plana Catesan coodnates s pefeed. The homogeneous tansfomaton [1] between the ont and task fames counts to tme ndependent (non-dynamc) tansfomatons esultng n non-lnea fowad and nvese knematcs (FKM fowad knematc model, IKM nvese knematc model) of the obot. Ths mples that the stuctue of the moble obot moton contol (as well as the moton contol of any knematc chan n geneal) can consst of the nne (dynamc) open contol loop (compsng the contol algothm and the contolled plant) pefomed n the ont fame, and the oute feedback loop wth the non-dynamc, non-lnea fowad and nvese tansfomatons between the task and ont fames (Fg. 1) []. In fgue, pr and p dr denote the poston and the desed poston vectos n the Catesan task fame, qr and q dr stand fo the angula poston and the desed angula poston vectos n the ont fame, and e qr epesents the eo vecto of the angula poston n the ont fame. Theefoe, the dynamc model of the moble obot n the ont fame (esultng n a contol-affne dynamc model [3, 4]) athe than n the task fame (non-lnea dynamc model) has to be syntheszed. Ths appoach allows avodng the poblem of nonunqueness and sngulaty [5] of the nvese knematcs soluton n the nne contol loop, tansfomng t to less complcated poblem of the desed taectoy fomng. Nomenclatue ICR nstantaneous cente of otaton of the moble obot C cente of gavty (cente of the obot body) 0 nstantaneous tunng adus of the obot taectoy θ C obot angula poston wth espect to ICR (obot oentaton) C angula velocty of the obot oentaton m C mass of the obot body C adus of the obot (cylndcal) body J C moment of neta of the obot body m mass of each of the obot wheels adus of each of the obot wheels J moment of neta of each of the obot wheels L dstance between the wheels (wheel spacng, bas) l C poston of the obot n the task plane v C velocty of the obot n the task plane l poston of the -th obot wheel n the task plane v velocty of the -th obot wheel n the task plane q angula poston of the -th obot wheel an element of the angula poston vecto q q angula velocty of the -th obot wheel an element of the angula velocty vecto q dvng toque fo the -th obot wheel an element of the dvng toque vecto τ τ p d q d e q Inne contol loop q p IKM FKM n the ont fame q IKM Fg. 1. Block dagam of the moton contol n the task fame.

3 Ján Kadoš / Poceda Engneeng 96 (014 ) l C, v C C,. C l, v l 1, v 1 ICR q,. q, τ 0 C m, J m C, J C m, J C q 1,. q 1, τ 1 L. Moble obot descpton Fg.. Schematc stuctue of the dffeental two-wheeled moble obot (top vew). Suppose the basc obot stuctue consstng of a cylndcal body and a pa of dffeentally dven sde wheels (Fg. ). Let = 1 denotes the ndex of the ght-hand sde wheel and = of the left-hand sde one. It s evdent, that the pa (l C, θ C) matches wth the numbe of degees of feedom of the moble obot n the task fame, and equally the pa (q 1, q ) descbes the obot DOFs n the ont fame. The uppe notaton yelds the basc expesson [6] fo the obot wheel poston L l q C, 0 C as well as the obot body poston and oentaton l C l l L l q (1) 1 1 q q, q q 1 l l C 1 () L L Consequently, we ae able to expess the coespondng veloctes of wheels v q 1, (3) and of the obot body v C v v 1 q q, 1 q q 1 v v C 1 (4) L L Undelne that expessons (3) and (4) ae the functons of elements q 1 and q of the angula velocty vecto q. Assumng the cylndcal shape of the moble obot body, ts moment of neta s gven by 1 JC mcc (5)

4 06 Ján Kadoš / Poceda Engneeng 96 ( 014 ) and smlaly s the moment of neta of each of the obot wheels 1 J m (6) We ae now eady to get the dynamc model of moble obot n the ont fame. 3. Dynamc model of the moble obot In MIMO dynamc system s analyss, the staghtfowad and poweful tool fo the dynamc model synthess epesents the Eule-Lagange method [1, 7] based on the system s total knetc and potental enegy concept. Lagange s enegy functon L stands fo the dffeence between the total knetc enegy E K and total potental enegy E P. Due to above mentoned smplfcaton moton wthn the hozontal plane (.e. the constant potental enegy) only the knetc enegy of a movng obot should be taken nto consdeaton. Futhemoe, the total knetc enegy E K of the dffeentally-dven moble obot s ndependent of ts poston n the task plane lkewse n the ont fame. Thus, fo the moble obot dynamcs descpton, the educed veson of the Eule-Lagange s equaton fo the -th ont coodnate q s gven by d dt E dq K Q (7) whee Q stands fo the genealzed non-consevatve (extenal) foce n the decton of the -th coodnate. Ths foce compses the dvng toque τ and the sum of load foces τ L (.e. dsspatve foces lke vscous fcton and ollng fcton). The esultant genealzed foce s then expessed by the dffeence Q (8) L The total knetc enegy of moble obot conssts of enegy of both the tanslatonal and the otatonal moton of the obot body and the obot wheels E K mcvc mv1 mv JC C Jq 1 Jq (9) Substtutng expessons (1) (6) to (9) yelds E K 3 m q q C m q q q q 1 C 1 1 (10) 4 L As mentoned above, the total knetc enegy of the moble obot s ndependent of the angula poston vecto q. Eule-Lagange s equaton (7) should be solved fo both degees of feedom,.e. fo q 1 and q, gvng the esultant system of two dffeental equatons of second ode 11q 1 1q 1 q q 1 1 L1 L (11) whee

5 Ján Kadoš / Poceda Engneeng 96 (014 ) C m mc, 1, (1) L epesents the moment of neta of the -th degee of feedom, C m 1 C,, 1,, L (13) s the couplng moment of neta between the DOFs (the poduct netal couplng foces Eule s foces), and coesponds to the undesable mutual q dq q dt d q, dt 1, (14) stands fo the angula acceleaton n the -th DOF. The angula poston q and angula velocty q ae the elements of the -th DOF s phase vecto q. Denote J R (15) the netal matx of the moble obot. Now, the system of equatons (11) can be ewtten n the fom of matx dffeental equaton of second ode Jq L (16) System of dffeental equatons (11) o (16) stands fo the contol-affne [3, 4] dynamc model of the moble obot n the ont fame. The soluton of ths system yelds the behavo of the obot wheel s angula poston n tme. To assue the desed taectoy of the moble obot, any contol algothm should geneate the coespondng dvng toque vecto τ. 4. Decoupled dynamc model To guaantee the desed behavo (pefect accuacy and fast dynamcs) of moble obot n pesence of load foces and vaable netal couplng foces (sgnal dstubances), the obust contol technques should be taken nto account. In geneal, the am of the obust contol s to keep the qualty of the contolled pocess despte the paametc and sgnal dstubances [8], pefeably wthout any adaptaton o on-lne dentfcaton. The only nfomaton about the dstubances s the estmated o computed ampltude (maxmal value). In the case of moble obot, the combnaton of the load toque and the netal couplng foce acts aganst the dvng toque. Denote τ Lmax the maxmal value of ths combnaton n the moble obot wokng plane. Ths maxmal value s dentcal also n the ont space L max max q const. q, q, q L (17) The constant (ndependent of the system s ont vaables) maxmal value n (17) mples, that usng (11) and (17), the decoupled dynamc model [9, 10, 11] of the moble obot fo the pupose of the obust contol algothm synthess can be ceated

6 08 Ján Kadoš / Poceda Engneeng 96 ( 014 ) q 1 1 q L1max L max (18) povded that once desgned fo the decoupled dynamc model (18), the esultant obust contol [9, 10, 11] guaantees the desed qualty of the eal moton contol system fo any combnaton of sgnal dstubances unde the condton L q Lmax (19) egadless of the fequency specta o the tme hstoy of dstubances. We can conclude that fom the pont of vew of the obust contol desgn, the dynamc model of a moble obot can be teated as a system of the pa of mutually ndependent double ntegatos (18). Smla appoach to the dynamc system analyss has been appled also n [1] and [13]. 5. Conclusons The pape shows two specfc featues of the moble obot dynamc model synthess, the desgne of the moton contol algothm should take nto account. The fst featue s that fo the moton contol n the task fame, the dynamc model n the ont fame can be appled. Ths yelds the contol-affne MIMO dynamc model n the fom of the system of two dffeental equatons of second ode. The second featue deals wth the decouplng of ths MIMO model to the pa of ndependent (SISO) double ntegato models, povded that the obust contol algothm fo the moble obot moton s unde consdeaton. The outcome of ths contbuton can essentally smplfy the desgn of a elable obust contol algothm n ognally complex and ntcate non-lnea MIMO contol task. Acknowledgements The wok pesented n ths contbuton has been suppoted by the Slovak Reseach and Development Agency unde the Gant No. APVV Refeences [1] A.J. Kovo, Fundamentals fo Contol of Robotc Manpulatos, Wley & Sons, New Yok, [] J. Kadoš, Specfc poblems of the obot moton contol n the task fame (n Slovak), Jounal EE 18 (01) 6 7. [3] A. Isdo, Nonlnea Contol Systems, thd ed., Spnge, London, [4] S. Sasty, Nonlnea Systems: Analyss, Stablty, and Contol, Spnge, New Yok, [5] M,W. Spong, M. Vdyasaga, Robot Dynamcs and Contol, Wley & Sons, Sngapoe, [6] G. Cook, Moble Robots (Navgaton, Contol and Remote Sensng), Wley & Sons, New Jesey, 011. [7] J. Kadoš, The smplfed dynamc model of a obot manpulato, 18 th Intenatonal Confeence Techncal Computng, Batslava, 010. [8] V.I. Utkn, Fst Stage of VSS: People and events, n: X. Yu, J.X. Xu (Eds.), Vaable Stuctue Systems: Towads the 1 st Centuy, Spnge, Beln, 00, pp [9] J. Kadoš, Smplfed obust contol of an anthopomophc leg, Poceedngs of the 19 th Intenatonal Confeence on Pocess Contol, Štbské Pleso, 013, pp [10] J. Kadoš, Robust command followng contol n the obot ont space, Poceedngs of the Intenatonal Confeence on Innovatve Technologes, In-Tech 01, Reka, 01, pp [11] J. Kadoš, Robust contol of a manpulato n the task space, 10 th Intenatonal Confeence Pocess Contol, Kouty nad Desnou, 01, pp. C014a-1 C014a-9. [1] M. Huba, P. Bsták, Z. Skachová, K. Žáková, Pedctve antwndup PI and PID-contolles based on I1 and I models wth dead tme, Poceedngs of the 6 th IEEE Medteanean Confeence on Contol and Systems, Algheo, 1998, pp [13] J. Han, Fom PID to actve dstubance eecton contol, IEEE Tansactons on Industal Electoncs, 56 (009)