Dynamic Behaviour of Differentially Steered Mobile Robot

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1 nstons on Eet Engneeng, Vo (5), No 5 Dynm eho of Dffeenty Steee oe oot Fntše Dše ), Dne Hon ), n h Shm 3) Detment of oess Conto, Fty of Eet Engneeng n Infomts, nesty of e, Ceh e ) fntsese@e, ) 3) hshm@stente Astt hs te s me t otnng eston of the eho of n e ffeenty steee e system (moe oot) n n moton he mthemt ynm moe esng the moton (see n eton) of one oot s ont s ete se on the fst nes oh he tjetoy of ths ont s onete to tjetoes of whee ontt onts wth the ne of moton he ynm eho of motos n hsss, fom of ong etween motos n whees n s geomet mensons e ten nto ont he estng tjetoy eens on the sy otge of oth e motos he ynm moe w e se fo esgn n efton of oot s moton onto n AA / SIIN smton enonment eywos Dynm system moeng, ntegent onto, moe oot I INODCION he e es wth ynm moe of n e moe oot wth ffeenty steee e system n n moton A snge-xe hsss o te hsss s mosty se n se of sm moe oots [] A ste whee s e to snge-xe to ense stty hs soton togethe wth the neenent whee tton ows exeent moty on the onty to ss hsss see ommey e oot n Fg he ee mthemt moe omes fom y-ot, nomn geomet mensons n othe fetes of tht oot wth ew of e eho of n omonents n some smfyng ssmtons he m s to ete moe se on foes se y moto toqes of neenent whee es he moe w onsst of ynm eho eston of the hsss n DC sees motos he esente moton moe se on ente of mss (my eement) ynms s ffeent fom moes efetng nemts ony n ommony se n tete she eg n [] o [3] Stn moes ese oot s tjetoy tme eton eenng on nown whee see (nfomton fom whee see sensos) n hsss geomety oomety she eg n [] O moe extens stn moe wth ynm t esng whee see eeneny on the moto sy otge y esetng ynms, onstton, geomety n othe metes of the hsss n motos he moto sy otge ttng the whee ses ng toqe n theey whee otton Inet n esstne foes t gnst the ng toqe oth ng toqes nfene eh othe ese of these foes he n ne moton of the oot s est of os tme nt whee otton sees he n ne moton n e eomose to sm of ne (tnston) n otton motons Foe ne s the sttng ont fo the eton of moton eqtons If F s t foe tng to mss ont wth mss m n stne fom the xs of otton then t hos fo gene ne moton tht eto sm of foes tng to seete ont s eo see tete [5] F m m 3 3 net foe m m 3 3 Coos foe Ee's foe ( ) entfg foe Aton of ths gene eqton eqes sefton of n foes ong to t ontons n/o eenty mementng othe tng foes We w onse foes ognte y the moton of e oy ne wth esstnes (osses) n ton to ne moton foes Fg Dffeenty steee moe oot We w oxmte these foes n the smest mnne to e ooton to see Eqtons esng eenenes of ne n ng eotes of the efeene hsss ont to t whee moto otges w e est of the ynm t Seeton of the ont whee t tnston n otton sees w e ete nfenes sgnfnty nt eqtons n hene omexty of the estng () EEN5

2 nstons on Eet Engneeng, Vo (5), No 5 moe If the efeene hsss ont s t the ente of gty then nt eqtons of the ynm t e the smest t eqtons esng eenenes etween whee sees n ne n ng eotes e moe omte he ente of the xs jonng the whees s the most ommon efeene hsss ont n tete Sh hoe es to the smest eton of t whee sees to moton eqtons of tht ont e-off etween these two ohes s hosen n o e ont t the ente of gty ojeton to the xs jonng the whees s seete A tjetoy (tme ose) omtton of nothe hsss onts (onts whee whees meet the gon) sements the ynm t of the moe II AHEAICA ODE he ese moe oot s en y two DC motos wth ommon otge soe n neenent onto of eh moto he motos e onnete wth the ng whees thogh ge-ox wth onstnt ge to An e ge-ox mens tht t ees ney ng eoty n oosts the toqe (nonnetes e not onsee) osses n the moto n so n the geox e ooton to the otton see he hsss s eqe wth the ste whee wth no nfene on the hsss moton (ts nfene s ne n esstne oeffents tng gnst moton) he moe of the oot onssts of thee etey neenent ts Deston of the e DC sees motos s gen n seton A wo eqtons ese eeneny of the moto otton see n ent on the owe sy otge n ong toqe e ete to hssnms oton eqtons e esente n seton eeneny etween ne n ng eotes of the efeene hsss ont on toqes tng on ng whees Seton C s ete to eqtons esng how the moto see nfenes tnston n otton sees of the seete ont n to omete moe fomton In the st Seton D, the moe s tnsfome to sme fom whh s moe ste fo next sng n fo tjetoy of n ty ont ton Eqtons esng tjetoy oesonng to the ontt onts of the ng n ste whees wth the gon e fomte A DC Sees oto Dynms An eqent t of n e DC sees moto [6] s n Fg It onssts of esstne, ntne n mgnet fe of the moto he ommtto s not onsee he oto oes eet otge wth eese oty thn the soe otge eetomote foe, whh s ooton to the oto ng eoty he toqe of the oto s ooton to the ent not onse osses n the mgnet fe t ony eet osses n wnng n mehn osses ooton to the oto see Fst eqton eses moto eho thogh nng of otges (hhoff s ws) whee () [Ω] s moto wnng esstne, [H] s moto ntne, [gm s - A - ] s EF onstnt, [V] s soe otge, [s - ] s oto ng eoty n [A] s ent fowng thogh wnng Seon eqton s ne of toqes (eet enegy) moment of net s, otton esstne ooton to the otton see (mehn osses) o, o toqe of the moto x n toqe se y the mgnet fe whh s ooton to ent whee x s o x (3) x [gm ] s moment of net, [gm s - ] s oeffent of otton esstne, [gm s - ] s o toqe Chsss Dynms Chssnms s efne y the eto of ne eoty tng on the efeene hsss ont n wth otton of ths eto wth ng eoty (onstnt fo hsss onts) It s osse to te the tjetoy of ty hsss ont fom these es he ont fo whh the eqtons e ee s the ente of gty nom ojeton to the xs jonng the whees see Fg 3 hs es, ong to the thos, to the smest set of eqton fo the whoe moe We onse the gene ente of gty oston sy t s e to the ente of the jon etween whees Fg Eqent t of moto Ie eho mens tht whoe eet enegy se fo mgnet fe etng s tnsfome wthot ny osses to mehn enegy toqe of the moto We o Fg 3 Chsss gm n foes We onse foes nes s sttng eqtons It s osse to ee two foes F n F tng to hsss n eft () n ght () whee gon ontt onts wth one foe F n toson toqe tng n ont he EEN5

3 nstons on Eet Engneeng, Vo (5), No 53 hsss s htee wth s of the ng whees, tot mss m, moment of net wth eset to the ente of gty ote metes,, et s sefy eqton () fo o se he oston of the ente of gty s onstnt wth eset to the xs of otton so we o not nee to onse Coos foe Smy we o not onse the entfg foe hsss s sose to e so oy eesente s ont mss (ente of gty) ese the foe eto sng the moement ts n the ont n goes thogh the ente of gty t s enogh f we onse the net foe y ne moton At otton moton t s neessy to onse toqe se y Ee s foe ese the xs of otton oes not go thogh the ente of gty y the ne of foes sng ne moton we w onse foes F, F se y the es, net foe F S n esstne foe F O whh s ooton to see he ne of foes nfenng ne moton s F F F F whee m O S m () [g] s oot mss, [gs - ] s esstne oeffent gnst ne moton, [gm s - ] s toqe of the eft e, [gm s - ] s toqe of the ght e, [ms - ] s ne eoty n [m] s s of the whees he ne of toqes s sghty moe omte ese the otton xs oes not e n the ente of gty ht s why t s neessy to te nto onseton not ony hsss momentm, t so toqe E F E se y Ee s foe F E Smy s t ne moton we w onse toqe O se wth the esstne gnst otton to e ooton to the ng eoty O E (5) m whee [m] s stne of the ght whee fom ont, [m] s stne of the eft whee fom ont, [m] s stne of the ente of gty fom ont, [gm s - ] s esstne oeffent gnst otton moton, [gm ] s moment of net wth eset to otton xs n the ente of gty n [s - ] s ng eoty n ont he estng moment of net wth eset to the otton xs n the ont s gen y Eq (6) whh s the e xs theoem o Hygens-Stene theoem see eg [5] m (6) whee [gm ] s moment of net wth eset to the ente of gty n [m] s stne etween the ente of gty n ont C etonsh etween otton See of the oto n the Cente of ty Chsss oement (nemts) he eqton esng the eho of the two motos (ents n ng eoty) n the eho of the hsss (the see of the ne moement n see of the otton) e onnete ony thogh toqes of motos Eqtons of the w of onseton of enegy whh s oneson of eet enegy to mehn nng one tye of osses t eesent ony one etonsh etween the see of the two motos (ehe see of the e whees) n tes of moement n otton of the hsss An ton eton s gen y esgn of the e n hsss We exet tht oth e whees e fmy ne to otos of eent motos oe n e geox wth the ge to wthot nonnetes n ny fexe ts he geox eeses the ott ng eoty x wth eton to the nt ng see x ong to the tnsmsson to n smtneosy n the sme ooton neses ott toqe x wth eton to the nt toqe x (7) (7) Fthe we ssme tht oth e whees he the sme s n the ehe sees, een on the ng eoty of the geox ott, ong to etons (7) o etemne the e of the ne see n the ont n the ng eoty of otton et s stt fom Fg We exet tht oth e whees he the sme xs of otton n theefoe the ehe sees e wys e he stton shows the ostonng whee the ehe sees n ty oete (ng whees n ) n the ont We wnt to sefy sh ne n ng sees tht he the sme effet s the ton of the ehe see of the ng whees y sng the smty of tnges ete n Fg, we n ete the ehe sees of the whees, to the see n the ont ong to eton (8) n the ng eoty of otton ong to the eton (8) (8) (8) x ( ) EEN5

4 nstons on Eet Engneeng, Vo (5), No 5 y y y We n etemne fom the ne see of n ng see (moton eqtons) ent otton nge α of the hsss n the ent oston (the oonte, y ) of the ont [7] ong to etons α (9) x y os(α) (9) sn(α) (9) o etemne the ent oston of the ontts of thee hsss whees (onts, n ) wth gon we nee to now the oton of these onts n eton to the ont hs oton s shown n Fg 5 Fom geomet mensons we etemne eqton esng the ete oston of these onts n eton to the ont eenng on the nge of otton ete ostons x, y of the ont n x, y of the ont eenng on the nge of otton α e gen y x sn( α) y os( α ) () x sn( α) y os( α) () o etemne the ete oston x, y of the ont we se n xy ght tnge sefe y hyotense n thet n (see Fge 5) hen the eqtons fo ete oontes of the ont tng e ( ) γ tn x sn( α γ ) y os( α γ ) Fg ne n ng sees etons () he ynm t of the moe onssts of fo ffeent eqtons esng the eho of oth motos, two ffeent eqtons esng the ynms of the hsss n two ge eqtons wth eeneny of the ne n ng hsss sees on the ehe sees of the ng whees We n fn n these eqtons eght stte es esng the ent stte of the eft moto (ent, ng eoty of the oto, ong toqe ) n the ght moto (ent, ng eoty of the oto, ong toqe ) n the moement of the hsss (ne see n ng eoty of otton ) A the stte es e eenent on the tme oses of the owe of the eft n ght moto Eh moto hs ts own owe sy otge (, ) ten fom the ommon soe of the otge he sy otge onto of oth motos sng mfe wth the onto sgn x s shown n Fg 6 ese oth motos e owee fom the ommon soe t w e ten nto ont so effet of the nten esstne oth motos e onsee wth the sme metes We n wte wth sng the Eqs () n (3) n Fg 6 fo ffeent eqtons esng the eho of oth motos s ( ) () Fg 5 Aty hsss ont eton ( ) () Fg 6 otos wng () () Dffeent eqtons () n (5) esng the eho of the hsss omete the ynm moe We n ewte these eqtons wth eset to the eqtons (7) n ntoton of the "ee" s of the whee n tot moment of net (3) s m (3) Z y ( - )/ γtn(/ ) x EEN5

5 nstons on Eet Engneeng, Vo (5), No 55 EEN5 m m (3) m (3) It s osse to ewte the st two ge eqtons, (8) (8) esng the eenene etween otton see of oth motos n hsss moement sng the ssttton (3) s () () hese sx ffeent eqtons (,), (,), (3,) n two ge eqtons (,) ontnng eght stte es eesentng mthemt eston of the ynm eho of n e ffeenty steee moe oot wth osses ney eenent on the eotons o see he onto sgns n tht onto the sy otges of the motos e nt es n the see of the moement n see of otton e ott es Fom them wth sng eqtons (9) (9) we n etemne the ent oontes of the ont n nge of the hsss otton In the foowng ton of stey-stte es fo onstnt moto owe otges s gen he ton of the stey-stte s sef oth fo the heng of ee eqtons n seony fo the exement etemnton of the es of the nnown metes ese eqtons () () e ne wth eset to the stte es the ton of the stey-stte es to system of eght ne eqtons whh we n wte n the mtx fom s (5) D Comtton Fom of the oe A mthemt moe w e se n t fo the esgn, smton n ton of the onto moement of the oot he moe n e e nto thee seesnoe ts s shown n Fg 7 Fom the onto ont of ew the ton es e sgns n tht onto the sy otge of the motos he nstntneos see n see of otton e ott es of the ne t of the moe hese es e the nts to the onseqent non-ne t of the moe (9,,), whose otts e ontoe es - the oontes of the seete ont oston x, y n otton nge of the hsss α he st t s the ton of oontes of the oston of ty onts of the hsss We n mofy the ne t of the moe nto sme fom fo onto esgn oses to ee nme of ffeent eqtons fom sx to fo If we ssttte eqtons (,) nto (3,) n emnte toqes n y ssttton of (,) to (3,) we e e to ee fo ffeent eqtons (,) (3,) nto two (7,) We ntoe ssttton of the metes ong to foowng foms (6) m m (6) (6) (6) he ee ne t of the moe onssts of the set of eqtons (7) (7) [ ] ( [ ]) (7) [ ] ( [ ]) (7) n the ott es e gen y ge eqtons (,) It s osse to wte the ee ne t of the moe s stn stte-se moe n the mtx fom s Cx y Ax x y x (8) wth onstnt mtes A, n C Fg 7 oe ttonng nto ne n nonne t

6 nstons on Eet Engneeng, Vo (5), No 56 A ( ) ( ) ( ) ( ) C III EXAE OF HE EHAVIO s efton of the oe ee moe ws me y ton fo sttons whee we n gess the eho of the oot Fst e of the stte es n stey sttes w e gen fo some omntons of metes n moto sy otges Fthe tme oses of the oot tjetoy w e etemne fo some omntons of the tme oses of sy otges when the oot s sttng fom eo see he es of the metes ste n the foowng tes e se n tons hese es e hosen so tht they t est oghy oeson to the es estmte fo the oot n Fg he es of the geomet n othe metes of the hsss e ste n e I AE I CHASSIS AAEES Notton Ve nt Deston m stne of the eft whee fom ont 6 m stne of the ght whee fom ont m stne of the ente of gty fom jon etween whees m stne of ste whee fom jon etween whees 5 m sem-mete of ng whee 5 g tot mss of the oot gs- 55 gm 35 gms- oeffent of the esstne gnst the oot ne moton moment of net of the oot wth eset to the ente of gty oeffent of the esstne gnst the oot ottng Neessy metes fo the DC motos wth ommon otge soe eston e gen n e II We onse ent motos wth ent metes AE II DC OOS AAEES Notton Ve nt Deston Ω moto wnng esstty 5 H moto ntne gm s - A - eetomoto onstnt Z Ω soe esstne V soe otge 5 gm tot moment of net of the oto n geox Notton Ve nt Deston gm s esstne gnst ottng - oeffent of the of the oto n geox geox tnsmsson to A Stey Stte fo Dffeent ostons of the ont n otos Votges he stey sttes e te s soton of the system of eght eqtons n mtx fom (5) es of the whees e shown ng the fst seons of moton fom eo nt ontons te fom stte-se moe (8) n fom the eqtons fo the tjetoes ton (9, ) eft whee AE III SEADY SAE A ght whee m A s - Nm 353 ms - s jetoy of whees,, jetoes e otte fo the stton tht the ogn of the oonte system s n the ente etween the whees, whh s on the x-xs n the eft oentton of the oot s n the eton of the y xs he sttng n fn ostons of the oot e sye sng the tnge tht onnets thee whees he tjetoy of the ente of gty s sye (e oo) n ton to the tes of the whees he stey-stte A (e III) oesons to the geomet ngement the ont s mwy etween the whees n oth motos he the sme sy otge he est s tht the oot moes ony ney he foowng thee exements show the nfene of the ente of gty oston he stey-stte (e IV) hos gn fo the symmet geomet ngement t ony one moto s owee EEN5

7 nstons on Eet Engneeng, Vo (5), No 57 he stey stte C (e V) shows the stton n the se tht the ont s n the exteme oston tows the eft whee n ony the eft moto s owee he stey-stte D (e VI) oesons to the sme oston of the ont tows the eft whee t ony the ght moto s owee In thee ses the oot ottes n t the sme tme the ont hs some ne see oth whees oe tnston moement ese of the ntetons AE VI SEADY SAE D eft whee ght whee A 8 67 s Nm 5686 ms s - jetoy of whees,, AE IV SEADY SAE eft whee ght whee m I A s Nm 6757 ms s - jetoy of whees,, Dynm eho fo t Cses he ynm eho s emonstte on the tme oses of ents n ng sees of the motos sttng fom eo nt ontons hs n Fg 8 show oses of the sy otges, ents n ng sees fo the se tht the ont s n the me etween oth motos wth the sme onstnt otge V he stton oesons to exement wth the metes n e III Dng otge Dng otge AE V SEADY SAE C eft whee ght whee A 67 8 s Nm 9 ms s - 8 jetoy of whees,, o (m) 5 5 Cent See o tme (s) o (m) 5 5 Cent See o tme (s) 6 Fg 8 Dynm eho onstnt sy otge V fo oth motos A stton whee the ont s n the me etween oth motos wth the ght moto otge V ony oesons to the exement wth the metes n e IV n t s emonstte n Fg EEN5

8 nstons on Eet Engneeng, Vo (5), No 58 Dng otge 5 5 Cent - - Dng otge 5 5 Cent 5 (m/s) Chsss see tme (s) (eg/s) jetoy of whees,, otton see tme (s) See o See o o (m) o (m) tme (s) 5 5 tme (s) Fg 9 Dynm eho onstnt sy otge V fo ght moto An stte exme of eho n the stton when oth otges e eo n wth ffeent mtes s n Fgs n On the eft moto etng tn of ses wth eo s, ty ye 5 % n mte 3 V s e On the ght moto etng tn of ses of oe eo s n mte V s e he oesonng ehe moton whees tjetoes s n Fg o (m) Dng otge 6 8 Cent See o tme (s) o (m) Dng otge 6 8 Cent See o tme (s) Fg Dynm eho eo otges Fg Dynm eho eo otges sees n tjetoes IV CONCSIONS he eho of the ynm moe n the smte sttons gees wth the exete eho of the oot he oston of the ente of gty oes not ffet the eho n stey stte he mmete ne see n the ont eens on ts oston t the tjetoes of the whees e neenent on the oston of the ont he nteton of the two es ws onfme he whee wthot sy otge ottes, ese of the foes of net n the foes of esstne In the tnsent stte, ths n se hnge n eton of the whee otton hs stton s seen n Fg 9 oto ynms s negge ome wth the exete ynms of the hsss fo the estmte moto metes ese the metes of the moe he hys menng t w e osse to mese ety some metes on e ee he entfton of ton metes w e osse exementy fom the mese tme oses of owe otges n oesonng oses of ng see of the whees ACNOWEDEN hs eseh ws sote y Insttton sot of he nsty of Eton, Yoth n Sots of the Ceh e EFEENCES [] Noá, oní ooty (ohony, senoy, říení) EN 5, ISN [] F Stenge, oots n Integent Systems; A Vt textoo, htt://wwwnetone/~stenge/isvexthtm [te ] [3] W s, A to n Eementy jetoy oe fo the Dffeent Steeng System of oot Whee Attos, htt://ossmsoefogenet/es/dffstee/dffsteehtm [te ] [] Z Wne, Oomete; htt://oot/ge/oomety/s [te ] [5] Z Hoá, F, Fy, o SN h, 976, [6] F o, V Feá, Zoy, Eeté ohony, ts: Af, 987 [7] Šejt, ehná mehn II nemt část SN h, 95, 56 EEN5