, ν C = ν 2 + ν 1 a. ω = ψ, ds dt = ν Ck(s)., ν C = (ω 2 + ω 1 )R. R + ν Ck(s)a. k = dt, ϕ 2 = x = t, y = t 2, 0 t 5, 0 1+4t 2 dt.

ÂÅÑÒÍÈÊ ÓÄÌÓÐÒÑÊÎÃÎ ÓÍÈÂÅÐÑÈÒÅÒÀ ÌÀÒÅÌÀÒÈÊÀ. ÌÅÕÀÍÈÊÀ. ÊÎÌÏÜ ÒÅÐÍÛÅ ÍÀÓÊÈ ÓÄÊ 531.1 c Ñ. À. Áå åñòîâà, Í. Å. Ìèñ à, Å. À. Ìèò îâ ÊÈÍÅÌÀÒÈ ÅÑÊÎÅ ÓÏÐÀÂËÅÍÈÅ ÄÂÈÆÅÍÈÅÌ ÊÎËÅÑÍÛÕ ÒÐÀÍÑÏÎÐÒÍÛÕ ÑÐÅÄÑÒÂ Â àáîòå àññìàò èâàåòñß âûâîä çàêîíîâ êèíåìàòè åñêîãî óï àâëåíèß äâèæåíèåì ò åõêîëåñíîãî è åòû åõêîëåñíîãî êèïàæåé ñ æåñòêèìè êîëåñàìè âäîëü ï îèçâîëüíîé ãëàäêîé ò àåêòî èè. Ïà àìåò- àìè óï àâëåíèß äëß ò åõêîëåñíîãî êèïàæà âûá àíû íåçàâèñèìûå óãëû â àùåíèß âåäóùèõ êîëåñ. Ïà àìåò îì óï àâëåíèß åòû åõêîëåñíîãî êèïàæà âûá àí óãîë ïîâî îòà ïå åäíåãî êîëåñà â äâóõêîëåñíîé ìîäåëè àâòîìîáèëß, îï åäåëßåìûé óãëàìè ïîâî îòà ïå åäíèõ êîëåñ ïî ï èíöèïó óëåâîãî óï àâëåíèß Àêêå ìàíà. Óñòàíîâëåíî, òî ï îèçâåäåíèå ñêî îñòè ë áîé òî êè êî ïóñà àâòîìîáèëß íà î èåíòè îâàííó ê èâèçíó åå ò àåêòî èè ßâëßåòñß êèíåìàòè åñêèì èíâà èàíòîì,îï åäåëß ùèì óãëîâó ñêî îñòü àâòîìîáèëß. Ï èâåäåíû åçóëüòàòû èñëåííîãî ìîäåëè îâàíèß è àíèìàöèè äâèæåíèß ò åõêîëåñíîãî è åòû åõêîëåñíîãî êèïàæåé, äåìîíñò è ó ùèå àäåêâàòíîñòü ï åäëàãàåìîé ìîäåëè êèíåìàòè åñêîãî óï àâëåíèß. Îáñóæäà òñß âîçìîæíîñòè ï èìåíåíèß óñòàíîâëåííûõ çàêîíîâ êèíåìàòè åñêîãî óï àâëåíèß äâèæåíèåì ï è óòî íåíèè àëãî èòìîâ ïà àëëåëüíîé ïà êîâêè, ï è å åíèè íàâèãàöèîííûõ çàäà óï àâëåíèß ìåõàíè åñêèìè ò àíñïî òíûìè ñ åäñòâàìè ï è ïîìîùè íàâèãàöèîííûõ ñèñòåì ÃËÎÍÀÑÑ è GPS, ï è å åíèè çàäà óï àâëåíèß ìîáèëüíûìè îáîòàìè ñ ïîìîùü äàò èêîâ ñëåæåíèß,à òàêæå ï è ï îåêòè îâàíèè àâòîäî îã,ò àíñïî òíûõ àçâßçîê,ïà êèíãîâ,àâòîçàï àâîê,äî îæíûõ ïóíêòîâ ïèòàíèß è ï è ñîçäàíèè ò åíàæå îâ. Êë åâûå ñëîâà: êèíåìàòè åñêîå óï àâëåíèå,ò åõêîëåñíûé êèïàæ,ìîáèëüíûé îáîò,ò àåêòî èß äâèæåíèß ò àíñïî òíîãî ñ åäñòâà,óãëû ïîâî îòà óï àâëßåìûõ êîëåñ,ï èíöèï óëåâîãî óï àâëåíèß Àêêå ìàíà,íàâèãàöèß,ìàíåâ è îâàíèå. Ââåäåíèå Òåõíè åñêèå âîçìîæíîñòè óï àâëåíèß ìåõàíè åñêèìè ñèñòåìàìè, îñíîâàííûå íà èñïîëüçîâàíèè ñîâ åìåííûõ èíôî ìàöèîííûõ òåõíîëîãèé, âûçâàëè ïîßâëåíèå â ïîñëåäíèå ãîäû äîñòàòî íî áîëü îãî êîëè åñòâà àáîò ïî òåî èè óï àâëåíèß ò àíñïî òíûìè ñ åäñòâàìè. Çàäà è íàâèãàöèè [1, 2], êîíò îëß ò àåêòî èè äâèæåíèß [3 7], àâòîíîìíîãî âîæäåíèß è ìàíåâ è îâàíèß ñ ìàëûìè àäèóñàìè àçâî îòà [8, 9], íàï àâëåííûå íà ïîâû åíèå áåçîïàñíîñòè è óâåëè- åíèß êîìôî òà, ñòèìóëè îâàëè ïîßâëåíèå â ïîñëåäíèå ãîäû íîâûõ ìàòåìàòè åñêèõ ìîäåëåé êîëåñíûõ ò àíñïî òíûõ ñ åäñòâ. è îêîå èñïîëüçîâàíèå ìîáèëüíûõ îáîòîâ âî ìíîãèõ ï èëîæåíèßõ ï èâåëî ê ñîçäàíè ìíîæåñòâà àëãî èòìîâ, ïîçâîëß ùèõ îñóùåñòâëßòü óï àâëåíèå èõ äâèæåíèåì [10 15]. Ìíîãèå èññëåäîâàíèß â å åíèè çàäà óï àâëåíèß ò àíñïî òíûìè ñ åäñòâàìè áàçè ó òñß íà ï èíöèïå óëåâîãî óï àâëåíèß Àêêå ìàíà μ âåëèêîì èçîá åòåíèè, ñäåëàííîì îêîëî 200 ëåò íàçàä. Ðóäîëüô Àêêå ìàí (20 àï åëß 1764 ã., òîëüáå ã, Ñàêñîíèß 30 ìà òà 1834 ã., Ëîíäîí) μ àíãëî-íåìåöêèé êíèãîòî ãîâåö, èçîá åòàòåëü, ëèòîã àô, èçäàòåëü è áèçíåñìåí ( èñ. 1). Â 1818 ãîäó çàïàòåíòîâàë ãåîìåò è óëåâîãî óï àâëåíèß [16]. Ðóëåâîé ï èâîä Àêêå ìàíà íå áûë íà ñàìîì äåëå èçîá åòåíèåì Àêêå ìàíà, õîòß îí ïîëó èë á èòàíñêèé ïàòåíò [17] íà ñâîå èìß è ñïîñîáñòâîâàë âíåä åíè åãî â õîäîâîé àñòè êà åò ( èñ. 2). Ôàêòè åñêèì èçîá åòàòåëåì áûë ä óã Àêêå ìàíà Äæî äæ Ëàíêåí ïå ãå èç Ì íõåíà, êà åòíûé ìàñòå êî îëß Áàâà èè [18]. Î åâèäíûì ï åèìóùåñòâîì òîãî èçîá åòåíèß áûëà âîçìîæíîñòü îñóùåñòâëßòü áåçîïàñíûé ïîâî îò â îã àíè åííîì ï îñò àíñòâå. Ïî ñëîâàì ñàìîãî Àêêå ìàíà: Áåçîïàñíîñòü, äîëãîâå íîñòü, êîíîìè íîñòü è êîìôî òõà àêòå èçó ò òî ïîëåçíîå èçîá åòåíèå.

Êèíåìàòè åñêîå óï àâëåíèå äâèæåíèåì êîëåñíûõ ò àíñïî òíûõ ñ åäñòâ 255 Ðèñ. 1. Ðóäîëüô Àêêå ìàí ìåæäó 1810 1814 ãã. Ðèñ. 2. Ðóëåâîé ï èâîä Àêêå ìàíà (1818 ã.), â íàñòîßùåå â åìß èñïîëüçóåòñß â àâòîìîáèëßõ

256 Ñ. À. Áå åñòîâà, Í. Å. Ìèñ à, Å. À. Ìèò îâ Ðèñ. 3. Ýëåêò îìîáèëü êîíñò óêöèè. Æàíòî (1898 ã.) Ðèñ. 4. Ë áèòåëüñêèé îáîò Pololu 3pi. Ï îèçâîäèòåëè Pololu Robotics and Amherst, USA. Electronics, Las Vegas, USA Â íåêîòî ûõ èñòî íèêàõ [19] ï èíöèï óëåâîãî óï àâëåíèß ñâßçûâà ò ñ èìåíåì à ëß Æàíòî (1848 1906 ãã.). Ñ èòàåòñß, òî îí èçîá åë óëåâó ò àïåöè â 1878 ãîäó.. Æàíòî òàêæå èçâåñòåí ñâîèìè ëåêò îìîáèëßìè, êîòî ûå íà àë äåëàòü íà ñâîåé ôàá èêå â Ëè-Ìîæå ñ 1893 ãîäà. Íà îäíîì èç ëåêò îìîáèëåé ìà êè Æàíòî ô àíöóçñêèé àâòîãîíùèê ã àô Ãàñòîí äå àñëó-ëîáà 18 äåêàá ß 1898 ãîäà óñòàíîâèë ïå âûé ìè îâîé åêî ä ñêî îñòè íà áåçëî àäíîì (êàê òîãäà ãîâî èëè) êèïàæå μ 63 êì/ ( èñ. 3). 1. Êèíåìàòè åñêîå óï àâëåíèå ò åõêîëåñíûì êèïàæåì Ï è âûâîäå çàêîíà êèíåìàòè åñêîãî óï àâëåíèß äâèæåíèåì ò åõêîëåñíîãî êèïàæà â íàñòîßùåé àáîòå â êà åñòâå åàëüíîé åãî ìîäåëè âûáè àåòñß ìîáèëüíûé îáîò-òåëåæêà ( èñ. 4), êîòî ûé îñóùåñòâëßåò ê èâîëèíåéíîå äâèæåíèå çà ñ åò àçíîñòè óãëîâûõ ñêî îñòåé âåäóùèõ êîëåñ, â àùà ùèõñß ï è ïîìîùè ìîòî îâ- åäóêòî îâ ( èñ. 5). Ï è çàäàííîì çàêîíå äâèæåíèß ñ åäíåé òî êè îñè âåäóùèõ êîëåñ x = x(t), y = y(t) è ãåîìåò è åñêèõ ïà àìåò àõ ìîáèëüíîãî îáîòà-òåëåæêè íàõîäèòñß çàêîí êèíåìàòè åñêîãî óï àâëåíèß â âèäå çàêîíîâ â àùåíèß âåäóùèõ êîëåñ. Òàê êàê äâèæåíèå ïëàòôî ìû ìîáèëüíîãî îáîòà-òåëåæêè ßâëßåòñß ïëîñêî-ïà àëëåëüíûì è ìîæåò áûòü çàäàíî äâèæåíèåì îò åçêà C 1 C 2 = a ( èñ. 6), òî â êà åñòâå ï èíöèïèàëüíîé êèíåìàòè åñêîé ñõåìû îáîòà-òåëåæêè âûáè àåòñß ñõåìà, ï åäñòàâëåííàß íà èñóíêå 6. Óãëîâàß ñêî îñòü ïëàòôî ìû è ñêî îñòü ñ åäíåé òî êè C îñè òåëåæêè íàõîäßòñß ñ ïîìîùü àâåíñòâ

Êèíåìàòè åñêîå óï àâëåíèå äâèæåíèåì êîëåñíûõ ò àíñïî òíûõ ñ åäñòâ 257 Ðèñ. 5. Êîëåñî ñ ìîòî îì- åäóêòî îì Ðèñ. 6. Êèíåìàòè åñêàß ñõåìà äëß âûâîäà çàêîíà óï àâëåíèß ò åõêîëåñíûì êèïàæåì Ðèñ. 7. Êàä û àíèìàöèè (t =1c, t =2c, t =3c, t =4c)

258 Ñ. À. Áå åñòîâà, Í. Å. Ìèñ à, Å. À. Ìèò îâ ω = ν C CO = ν 2 ν 1, ν C = ν 2 + ν 1. (1.1) a 2 Âåäóùèå êîëåñà îáîòà-òåëåæêè ï è åãî ê èâîëèíåéíîì äâèæåíèè ó àñòâó ò â äâóõ â àùàòåëüíûõ äâèæåíèßõ μ âîê óã âå òèêàëüíîé îñè è âîê óã îñè C 1 C 2.Ñêî îñòè öåíò îâ êîëåñ ñâßçàíû ñ óãëîâûìè ñêî îñòßìè èõ â àùåíèß âîê óã îñè C 1 C 2 àâåíñòâàìè ω 1 = ν 1 R, ω 2 = ν 2 R. (1.2) Óãëîâàß ñêî îñòü îáîòà â åãî â àùåíèè âîê óã âå òèêàëüíîé îñè îï åäåëßåòñß àâåíñòâîì ω = ψ, ãäå ψ μ óãîë ïîâî îòà ïëàòôî ìû. Ñ ó åòîì îï åäåëåíèß ê èâèçíû ò àåêòî èè ìîæíî çàïèñàòü ω = dψ dt = dψ ds ds dt = ν Ck(s). Ï èíèìàß âî âíèìàíèå àâåíñòâà (1.1) è (1.2), íàõîäèì Îòêóäà ν C k(s) = (ω 2 ω 1 )R a, ν C = (ω 2 + ω 1 )R. 2 ω 1 = ν C R ν Ck(s)a 2R, ω 2 = ν C R + ν Ck(s)a 2R. (1.3) Î èåíòè îâàííàß ê èâèçíà ê èâîé ï è ïà àìåò è åñêîì çàäàíèè îï åäåëßåòñß àâåíñòâîì k = ẋÿ ẍẏ (ẋ 2 +ẏ 2 ) 3. Ïîäñòàâëßß òó ôóíêöè â àâåíñòâà (1.3), ïîëó àåì ω 1 = ẋ2 +ẏ 2 R a ẋÿ ẍẏ ẋ2 2R ẋ 2 +ẏ 2, ω +ẏ 2 = 2 + a ẋÿ ẍẏ R 2R ẋ 2 +ẏ 2, ẋÿ ẍẏ ω = ẋ 2 +ẏ 2. Ìàòåìàòè åñêàß ìîäåëü êèíåìàòè åñêîãî óï àâëåíèß äâèæåíèåì îáîòà-òåëåæêè ï èíèìàåòâèä ϕ 1 = t 0 ( ẋ2 +ẏ 2 R a ) ẋÿ ẍẏ 2R ẋ 2 +ẏ 2 dt, ϕ 2 = t 0 ( ẋ2 +ẏ 2 R + a ) ẋÿ ẍẏ 2R ẋ 2 +ẏ 2 dt. Âêà åñòâå ï èìå à àññìàò èâàåòñß äâèæåíèå ìîáèëüíîãî îáîòà, êîãäà òî êà C äâèæåòñß ïî ò àåêòî èè, çàäàííîé ó àâíåíèßìè x = t, y = t 2, 0 t 5, ãäå êîî äèíàòû çàäàíû â ñì, â åìß μ â ñ, ï è ñëåäó ùèõ ãåîìåò è åñêèõ ïà àìåò àõ: a =8ñì, R =3ñì. Ðàññòîßíèå äî òî êè D ê åïëåíèß àññè ñ ïàññèâíûì êîëåñîì CD =9ñì. Çàêîí êèíåìàòè åñêîãî óï àâëåíèß è çàêîí ïîâî îòà ïëàòôî ìû â òîì ñëó àå ï èíèìà ò âèä ϕ 1 = t 0 ( 1+4t 2 3 8 3(1 + 4t 2 ) ) dt, ϕ 2 = t t 1 ψ =2 0 1+4t 2 dt. 0 ( 1+4t 2 + 3 8 3(1 + 4t 2 ) ) dt,

Êèíåìàòè åñêîå óï àâëåíèå äâèæåíèåì êîëåñíûõ ò àíñïî òíûõ ñ åäñòâ 259 Ðèñ. 8. Êèíåìàòè åñêàß ñõåìà åòû åõêîëåñíîãî ò àíñïî òíîãî ñ åäñòâà Âèçóàëèçàöèß äâèæåíèß îáîòà-òåëåæêè îñóùåñòâëßëàñü â ïàêåòå ñèìâîëüíîé ìàòåìàòèêè MathCAD. Ñ èñïîëüçîâàíèåì ôóíêöèé ã àôè åñêîãî åäàêòî à áûëà ñîçäàíà ìîäåëü îáîòà, ïà àìåò èçîâàííàß ñîãëàñíî ïîëó åííîé ìàòåìàòè åñêîé ìîäåëè, è âûïîëíåíà àíèìàöèß åãî äâèæåíèß ïî çàäàííîé ò àåêòî èè ( èñ. 7). 2. Êèíåìàòè åñêîå óï àâëåíèå åòû åõêîëåñíûì êèïàæåì Ðå åíèå çàäà è î êèíåìàòè åñêîì óï àâëåíèè äâèæåíèåì åòû åõêîëåñíîãî ò àíñïî òíîãî ñ åäñòâà ï îâîäèòñß ñ èñïîëüçîâàíèåì êèíåìàòè åñêîé ñõåìû, èçîá àæåííîé íà èñóíêå 8. Ï è äâèæåíèè àâòîìîáèëß âûïîëíßåòñß êèíåìàòè åñêîå óñëîâèå, ïîçâîëß ùåå â àùàòüñß êîëåñàì àâòîìîáèëß áåç ï îñêàëüçûâàíèß. Òàêîå ñîñòîßíèå íàçûâàåòñß óñëîâèåì Àêêå ìàíà è âû àæàåòñß àâåíñòâîì ctg δ o ctg δ i = w l, (2.1) ãäå δ i μ óãîë ïîâî îòà âíóò åííåãî óï àâëßåìîãî êîëåñà è δ o μ óãîë ïîâî îòà âíå íåãî óï àâëßåìîãî êîëåñà. Óãëû ïîâî îòà âíóò åííåãî è íà óæíîãî êîëåñ îï åäåëß òñß ñ ó åòîì ïîëîæåíèß ìãíîâåííîãî öåíò à ñêî îñòåé O. Ï èíöèï óëåâîãî óï àâëåíèß Àêêå ìàíà ñ äîñòàòî íî âûñîêîé òî íîñòü åàëèçóåòñß â àâòîìîáèëßõ çà ñ åò ï èìåíåíèß óëåâîé ò àïåöèè ( èñ. 9). Öåíò ìàññ óï àâëßåìîãî àâòîìîáèëß äâèæåòñß ïî ê óãó àäèóñîì R: R = a 2 + l 2 ctg 2 δ, (2.2) ãäå óãîë δ îï åäåëßåòñß å åç ñ åäíåå çíà åíèå êîòàíãåíñîâ âíóò åííåãî è âíå íåãî óãëîâ ïîâî îòà óï àâëßåìûõ êîëåñ: ctg δ = ctg δ o +ctgδ i. (2.3) 2 Óãîë δ ßâëßåòñß êâèâàëåíòîì óãëà ïîâî îòà âåëîñèïåäà, èìå ùåãî òàêó æå êîëåñíó áàçó l è äâèæóùåãîñß ïî ê óãó àäèóñà R, è èñïîëüçóåòñß â äâóõêîëåñíîé ìîäåëè àâòîìîáèëß ( èñ. 10).

260 Ñ. À. Áå åñòîâà, Í. Å. Ìèñ à, Å. À. Ìèò îâ Ðèñ. 9. Ðóëåâîå óï àâëåíèå Àêêå ìàíà Ðèñ. 10. Äâóõêîëåñíàß ìîäåëü Óãîë δ âûáè àåòñß âêà åñòâå ïà àìåò à óï àâëåíèß. Òîãäà çàêîí óï àâëåíèß äâèæåíèåì àâòîìîáèëß ïî çàäàííîé ò àåêòî èè íàõîäèòñß èç ó àâíåíèß (2.2): ( δ = k(s) ) 1 / k(s) arcctg k 2 (s) a2 l 2. Ñ ó åòîì îï åäåëåíèß î èåíòè îâàííîé ê èâèçíû çàêîí óï àâëåíèß ï èíèìàåòâèä ( ẋÿ ẍẏ (ẋ δ = ẋÿ ẍẏ arcctg 2 +ẏ 2 ) 3 ) / (ẋÿ ẍẏ) 2 a2 l 2. Çàêîí èçìåíåíèß óãëîâ ïîâî îòà ïå åäíèõ êîëåñ àâòîìîáèëß íàõîäèòñß ñ ó åòîì ñîîòíî åíèé (2.1) è (2.3): ctg δ left =ctgδ w 2l, ctg δ right =ctgδ + w 2l. Óãëîâàß ñêî îñòü êóçîâà àâòîìîáèëß â åãî â àùåíèè âîê óã âå òèêàëüíîé îñè îï åäåëßåòñß àâåíñòâîì

Êèíåìàòè åñêîå óï àâëåíèå äâèæåíèåì êîëåñíûõ ò àíñïî òíûõ ñ åäñòâ 261 Ðèñ. 11. Ã àôèê çàêîíà óï àâëåíèß ẋÿ ẍẏ ẋ2 +ẏ ω = 2 ( sin arcctg ( (ctg δ )/a )). ẋÿ ẍẏ a Óãëîâàß ñêî îñòü êóçîâà àâòîìîáèëß ìîæåò áûòü íàéäåíà àëüòå íàòèâíûì ñïîñîáîì. Èçâåñòíî [20], òî ï è ïîâî îòå êîëåñíîé ìà èíû ìãíîâåííûé öåíò ñêî îñòåé ìà èíû O ßâëßåòñß öåíò îì ê èâèçíû ò àåêòî èé âñåõ òî åê êî ïóñà. Îòêóäà ñëåäóåò, òî ï îèçâåäåíèå ñêî- îñòè ë áîé òî êè êî ïóñà àâòîìîáèëß íà î èåíòè îâàííó ê èâèçíó åå ò àåêòî èè ßâëßåòñß êèíåìàòè åñêèì èíâà èàíòîì, îï åäåëß ùèì óãëîâó ñêî îñòü àâòîìîáèëß. Â àñòíîñòè, ω = ν C k(s) = (ẋÿ ẍẏ) ẋ 2 +ẏ 2. ẋÿ ẍẏ Çàêîí êèíåìàòè åñêîãî óï àâëåíèß δ = δ(t) ïîçâîëßåò óñòàíîâèòü àíàëèòè åñêó ñâßçü ìåæäó ïà àìåò îì óï àâëåíèß è äâèæåíèåì ë áîé ôèêñè îâàííîé òî êè àâòîìîáèëß. Íà ßäó ñ ïîâî îòîì óëß â âåëîñèïåäíîé ìîäåëè àâòîìîáèëß â êà åñòâå ïà àìåò à óï àâëåíèß ìîæåò áûòü âûá àíà àçíîñòü óãëîâ ïîâî îòà åãî ïå åäíèõ êîëåñ [21]. Äëß ï èìå à àññìàò èâàåòñß ò àåêòî èß äâèæåíèß öåíò à ìàññ àâòîìîáèëß, çàäàííàß ó àâíåíèßìè x =5t, y =5 5cos π 2 t, ãäå êîî äèíàòû çàäàíû â ì, â åìß μ â ñ, ï è ñëåäó ùèõ åãî ïà àìåò àõ: w=1, 5 ì, l =2ì, a =1ì. Âóñëîâèßõ àññìàò èâàåìîãî ï èìå à π2 ((5 25 4 δ = cos π 2 t 2 +( 5π 2 25 π2 4 cos π arcctg sin π 2 t)2 ) 3 ) / 2 t (25 π2 4 cos π a 2 l 2. 2 t)2 Ã àôèê çàêîíà óï àâëåíèß ï åäñòàâëåí íà èñóíêå 11. Âèçóàëèçàöèß äâèæåíèß àâòîìîáèëß îñóùåñòâëßëàñü â ïàêåòå ñèìâîëüíîé ìàòåìàòèêè MathCAD. Íà èñóíêå 12 ï èâåäåíû êàä û àíèìàöèè ñëåäîâ îò êîëåñ àâòîìîáèëß äëß ìîìåíòîâ â åìåíè t =1c, t =2c, t =3c, t =4c. Âûâîäû Äëß ï îñòåé èõ ìîäåëåé êîëåñíûõ êèïàæåé ïîëó åíû óíèâå ñàëüíûå çàêîíû êèíåìàòè- åñêîãî óï àâëåíèß, êîòî ûå ïîçâîëß ò å àòü àçëè íûå êèíåìàòè åñêèå è äèíàìè åñêèå çàäà è. Ââåäåíèå äîïîëíèòåëüíûõ óòî íß ùèõ ãåîìåò è åñêèõ õà àêòå èñòèê êîëåñíîãî êèïàæà íå âíîñèò ï èíöèïèàëüíûõ èçìåíåíèé â ï åäëîæåííó ìàòåìàòè åñêó ìîäåëü è ï è

262 Ñ. À. Áå åñòîâà, Í. Å. Ìèñ à, Å. À. Ìèò îâ Ðèñ. 12. Êàä û àíèìàöèè ñëåäîâ, îñòàâëßåìûõ àâòîìîáèëåì

Êèíåìàòè åñêîå óï àâëåíèå äâèæåíèåì êîëåñíûõ ò àíñïî òíûõ ñ åäñòâ 263 Ðèñ. 13. Ñëåäû ìàíåâ à àâòîìîáèëß â îã àíè åííîì ï îñò àíñòâå íåîáõîäèìîñòè ìîæåò áûòü îñóùåñòâëåíî. Îäíèì èç ï ßìûõ ï èìåíåíèé ï åäëîæåííîé ìîäåëè êèíåìàòè åñêîãî óï àâëåíèß åòû åõêîëåñíûì ò àíñïî òíûì ñ åäñòâîì ìîæåò áûòü ñóùåñòâåííîå óòî íåíèå àëãî èòìîâ ïà àëëåëüíîé ïà êîâêè, ï åäëîæåííûõ â àáîòàõ [22 25], à òàêæå å åíèå íàâèãàöèîííûõ çàäà óï àâëåíèß ìåõàíè åñêèìè ò àíñïî òíûìè ñ åäñòâàìè ï è ïîìîùè íàâèãàöèîííûõ ñèñòåì ÃËÎÍÀÑÑ è GPS [26] è çàäà óï àâëåíèß ìîáèëüíûìè îáîòàìè ñ ïîìîùü äàò èêîâ ñëåæåíèß. Àëãî èòìû àñ åòà ãàáà èòíûõ ò àåêòî èé ìîãóò áûòü âîñò åáîâàíû ï è ï îåêòè îâàíèè àâòîäî îã ( èñ. 13), ò àíñïî òíûõ àçâßçîê, ïà êèíãîâ, àâòîçàï àâîê, äî îæíûõ ïóíêòîâ ïèòàíèß è ï è ñîçäàíèè ñèìóëßòî îâ-ò åíàæå îâ. ÑÏÈÑÎÊ ËÈÒÅÐÀÒÓÐÛ 1. Weinstein A.J., Moore K.L. Pose estimation of Ackerman steering vehicles for outdoors autonomous navigation // Proceedings of 2010 IEEE International Conference on Industrial Automation. Valparaiso, Chile. March,2010. P. 541 546. 2. Aguia A.P.,Hespanha J.P.Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty // IEEE Trans. Autom. Control. 2007. Vol.52. 8.P. 1362 1379. 3. Wang D.,Qi F. Trajectory planning for a four-wheel-steering vehicle // Proceedings of the 2001 IEEE International Conference on Robotics and Automation. Seoul,Korea,May,2001. P. 3320 3325. 4. Longoria R.G. Steering and turning vehicles. 2012. http://www.yumpu.com/en/document/view/4254403/04-steering-and-turning-vehicles-department-of- mechanical- 5. Gusev S.V.,MakarovI.A.Trajectory tracking control for maneuverable nonholonomic systems // arxiv: math/0507567v1 [math.oc]. 2005. http://arxiv.org/pdf/math/0507567v1.pdf 6. Rusu R.B., Borodi M. On computing robust controllers for mobile robot trajectory calculus: Lyapunov // Technical report. Technical University of Cluj-Napoca. Unpublished Papers Series. 2005. http://files.rbrusu.com/publications/rusu05robotuxlyapunov.pdf 7. Nagy B., Kelly A. Trajectory generation for car-like robots using cubic curvature polynomials // Proceedings of 3rd International Conference on Field and Service Robotics. Helsinki,Finland. 2001. 6 p. http://www.frc.ri.cmu.edu/ alonzo/pubs/papers/fsr2001.pdf 8. She J.H., Xin X., Ohyama Y., Wu M., Kobayashi H. Vehicle steering control based on estimation of equivalent input disturbance // Proceedings of the 16th IFAC World Congress. Prague,2005. Vol. 16. Part 1. P. 1902 1902. 9. Ackermann J., Bunte T. Automatic car steering control bridges over the driver reaction time // Kybernetika. 1997. Vol. 33. P. 61 74. 10. Áî èñîâ À.Â.,Ëóöåíêî Ñ.Ã.,Ìàìàåâ È.Ñ. Äèíàìèêà êîëåñíîãî êèïàæà íà ïëîñêîñòè // Âåñòíèê Óäìó òñêîãî óíèâå ñèòåòà. Ìàòåìàòèêà. Ìåõàíèêà. Êîìïü òå íûå íàóêè. 2010. 4. Ñ. 39 48.

264 Ñ. À. Áå åñòîâà, Í. Å. Ìèñ à, Å. À. Ìèò îâ 11. Åâã àôîâ Â.Â., Ïàâëîâñêèé B.E., Ïàâëîâñêèé Â.Â. Äèíàìèêà, óï àâëåíèå, ìîäåëè îâàíèå îáîòîâ ñ äèôôå åíöèàëüíûì ï èâîäîì // Èçâåñòèß ÐÀÍ. Òåî èß è ñèñòåìû óï àâëåíèß. 2007. 5. Ñ. 171 176. 12. Àíä èàíîâà Î.Ã., Êî îëüêîâà Ì.À., Êî åòêîâ Ñ.À., Ê àñíîâà Ñ.À. Êèíåìàòè åñêîå óï àâëåíèå ìîáèëüíûì îáîòîì ï è äâèæåíèè ïî ïîëèãîíó ñ îáõîäîì ï åïßòñòâèé // Ò óäû êîíôå åíöèè Óï àâëåíèå â òåõíè åñêèõ ñèñòåìàõ. Ñàíêò-Ïåòå áó ã,2010. Ñ. 356 359. 13. Kochetkov S.A.,Utkin V.A. A trajectory stabilization algorithm for mobile robot // The Proceedings of 11-th International Workshop on Variable Structure Systems. Mexico,2010. P. 121 127. 14. Abdalla T.Y., Hamzah M.I. Trajectory tracking control for mobile robot using wavelet network // International Journal of Computer Applications. 2013. Vol. 74. 3. P. 32 37. 15. Liu Y., Zhu J.J., Williams R.L., Wu J. Omni-directional mobile robot controller based on trajectory linearization // Robotics and Autonomous Systems. Vol. 56. 5. 2008. P. 461 479. 16. Eckermann E. Die Achsschenkellenkung und andere Fahrzeug-Lenksysteme. http://www.be-tech.bplaced.net/inhalt/02tew/techwerk/lenkung/sachinformation/lenkungssysteme.pdf 17. Ackermann R. Improvements on axletrees applicable to four-wheeled carriages // GB-Patent 4212. 27.01.1818. 18. Ueber die Lankenspergersche und Ackermannsche bewegliche Patent-Achsen // Dinglers Polytechnisches Journal. 1820. Band 1. Nr. XXVII. P. 296 311. http://dingler.culture.hu-berlin.de/journal/page/pj001?p=312 19. Òîëóáêî Â.Á.,Âàñèëüåâ Á.Ã.,Áå åçàí À.Ì. Ðå åíèå ï îáëåìû ìàíåâ åííîñòè óâåëè åíèåì ñòåïåíè ïîäâèæíîñòè ìà èí ñ íåãîëîíîìíûìè ñâßçßìè êîëåñíîãî òèïà // Ìåõàíiêà òà ìà èíîáóäóâàííß. 2009. 2. C. 96 102. 20. Ïîçèí Á.Ì.,Ò îßíîâñêàß È.Ï. Î ï èìåíåíèè ï èíöèïà Äàëàìáå à ê ñîñòàâëåíè ó àâíåíèé ê èâîëèíåéíîãî äâèæåíèß ò àíñïî òíûõ ìà èí // Âåñòíèê æíî-ó àëüñêîãî ãîñóäà ñòâåííîãî óíèâå ñèòåòà. Ñå èß: Ìà èíîñò îåíèå. 2006. 11 (66). Âûï. 8. Ñ. 37 39. 21. Mityushov E.A., Misyura F.D. Calculations of the shape trajectories of vehicles and the Ackermann principle of steering // Preprint. http://www.intellectualarchive.com/?link=item&id=1060 22. Huston D.C. The Geometry of parallel parking // Paper presented at the annual meeting of the The Mathematical Association of America MathFest. Pittsburgh,2010. http://www.allacademic.com/meta/p436841_index.html 23. Gupta A.,Divekar R. Autonomous Parallel Parking Methodology for Ackerman Configured Vehicles // ACEEE International Journal on Control System and Instrumentation. 2011. Vol. 2. 2. P. 34 39. 24. Blackburn S.R. The geometry of perfect parking. http://personal.rhul.ac.uk/uhah/058/perfect_parking.pdf 25. Hoffman J. Perfect Parallel Parking // New York Times. 2010. http://nyti.ms/hiahrp 26. Ïîääóáíûé Â.È., Ïåí êèí À.Ñ. Óï àâëåíèå äâèæåíèåì êîëåñíûõ ìîáèëüíûõ ìà èí ñ èñïîëüçîâàíèåì ñïóòíèêîâûõ àäèîíàâèãàöèîííûõ ñèñòåì // Ïîëçóíîâñêèé âåñòíèê. 2012. Ò. 1. Âûï. 1. Ñ. 239 242. Ïîñòóïèëà â åäàêöè 21.05.2015 Áå åñòîâà Ñâåòëàíà Àëåêñàíä îâíà, ä. ô.-ì. í., äîöåíò, êàôåä à òåî åòè åñêîé ìåõàíèêè, Ó àëüñêèé ôåäå àëüíûé óíèâå ñèòåò èì. ïå âîãî Ï åçèäåíòà Ðîññèè Á.Í. Åëüöèíà,620002,Ðîññèß,ã. Åêàòå èíáó ã, óë. Ìè à,19. E-mail: s.a.berestova@urfu.ru Ìèñ à Íàòàëüß Åâãåíüåâíà, ñòà èé ï åïîäàâàòåëü, êàôåä à òåî åòè åñêîé ìåõàíèêè, Ó àëüñêèé ôåäå àëüíûé óíèâå ñèòåò èì. ïå âîãî Ï åçèäåíòà Ðîññèè Á.Í. Åëüöèíà,620002,Ðîññèß,ã. Åêàòå èíáó ã, óë. Ìè à,19. E-mail: n_misura@mail.ru Ìèò îâ Åâãåíèé Àëåêñàíä îâè,ä. ô.-ì. í.,ï îôåññî,êàôåä à òåî åòè åñêîé ìåõàíèêè,ó àëüñêèé ôåäå àëüíûé óíèâå ñèòåò èì. ïå âîãî Ï åçèäåíòà Ðîññèè Á.Í. Åëüöèíà,620002,Ðîññèß,ã. Åêàòå èíáó ã, óë. Ìè à,19. E-mail: mityushov-e@mail.ru

Êèíåìàòè åñêîå óï àâëåíèå äâèæåíèåì êîëåñíûõ ò àíñïî òíûõ ñ åäñòâ 265 S. A. Berestova, N. E. Misyura, E. A. Mityushov Kinematic control of vehicle motion Keywords: kinematic control, three-wheeled carriage, mobile robot, trajectory of vehicle motion, steering angle,ackerman steering principle,navigation,maneuvering. MSC: 70B15 The derivation of laws of kinematic control of motion of three-wheeled and four-wheeled carriages with hard wheels along an arbitrary smooth trajectory is considered in this paper. The independent angles of rotation of driving wheels are chosen as parameters of control for a three-wheeled carriage. The angle of rotation of afront wheel in the two-wheeled car models defined by theangles of rotation of front wheels on the basis of Ackermann steering is chosen as a control parameter for a four-wheeled carriage. It is established that the product of the velocity of any point of the vehicle body and the oriented curvature of its trajectory is a kinematic invariant determining the angular velocityofavehicle. The paper presents the results of numerical modeling and animation of three-wheeled and four-wheeled carriages motion demonstrating the adequacy of the proposed model of kinematic control. The use of the proposed model can be a significant refinement of algorithms of parallel parking as well as the solution of navigation problems of managementofmotorvehicles using GPS and GLONASS navigation systems and problems of control of mobile robots with the help of tracking sensors. Also the proposed model can be useful for designing the motor roads,road interchanges, single-level and multilevel Parking lots,gasoline stations,on-the-go fast food stations and for the creation of car-simulators. REFERENCES 1. Weinstein A.J., Moore K.L. Pose estimation of Ackerman steering vehicles for outdoors autonomous navigation, Proceedings of 2010 IEEE International Conference on Industrial Automation, Valparaiso, Chile,March,2010,pp. 541 546. 2. Aguia A.P.,Hespanha J.P. Trajectory-tracking and path-following of underactuated autonomous vehicles with parametric modeling uncertainty, IEEE Trans. Autom. Control,2007,vol. 52,no. 8,pp. 1362 1379. 3. Wang D., Qi F. Trajectory planning for a four-wheel-steering vehicle, Proceedings of the 2001 IEEE InternationalConference on Robotics and Automation,Seoul,Korea,May,2001,pp. 3320 3325. 4. Longoria R.G. Steering and Turning Vehicles,2012. http://www.yumpu.com/en/document/view/4254403/04-steering-and-turning-vehicles-department-of- mechanical- 5. Gusev S.V., Makarov I.A. Trajectory tracking control for maneuverable nonholonomic systems, arxiv: math/0507567v1 [math.oc],2005. http://arxiv.org/pdf/math/0507567v1.pdf 6. Rusu R.B.,Borodi M. On computing robust controllers for mobile robot trajectory calculus: Lyapunov, Technicalreport, Technical University of Cluj-Napoca, Unpublished Papers Series,2005. http://files.rbrusu.com/publications/rusu05robotuxlyapunov.pdf 7. Nagy B.,Kelly A. Trajectory generation for car-like robots using cubic curvature polynomials, Proceedings of 3rd InternationalConference on Field and Service Robotics,Helsinki,Finland,2001,6 p. http://www.frc.ri.cmu.edu/ alonzo/pubs/papers/fsr2001.pdf 8. She J.H., Xin X., Ohyama Y., Wu M., Kobayashi H. Vehicle steering control based on estimation of equivalent input disturbance, Proceedings of the 16th IFAC World Congress,Prague,2005,vol. 16,part 1, pp. 1902 1902. DOI: 10.3182/20050703-6-CZ-1902.01903 9. Ackermann J.,Bunte T. Automatic car steering control bridges over the driver reaction time, Kybernetika, 1997,vol. 33,pp. 61 74. 10. Borisov A.V.,Lutsenko S.G.,Mamaev I.S. Dynamics of a wheeled carriage on a plane,vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki,2010,no. 4,pp. 39 48 (in Russian). 11. Evgrafov V.V., Pavlovsky V.V., Pavlovsky V.E. Dynamics, control, and simulation of robots with differential drive, Journal of Computer and Systems Sciences International, 2007, vol. 46, no. 5, pp. 836 841. 12. Andrianova O.G.,Korol'kovaM.A.,Kochetkov S.A.,Krasnova S.A. Kinematic control of the mobile robot motion on range while avoiding obstacles, Proceedings of the conference Controlin TechnicalSystems, Saint Petersburg,2010,pp. 356 359 (in Russian). 13. Kochetkov S.A., Utkin V.A. A trajectory stabilization algorithm for mobile robot, The Proceedings of 11-th InternationalWorkshop on Variable Structure Systems,Mexico,2010,pp. 121 127.

266 Ñ. À. Áå åñòîâà, Í. Å. Ìèñ à, Å. À. Ìèò îâ 14. Abdalla T.Y., Hamzah M.I. Trajectory tracking control for mobile robot using wavelet network, InternationalJournalof Computer Applications,2013,vol. 74,no. 3,pp. 32 37. 15. Liu Y., Zhu J.J., Williams R.L., Wu J. Omni-directional mobile robot controller based on trajectory linearization, Robotics and Autonomous Systems,2008,vol. 56,no. 5,pp. 461 479. 16. Eckermann E. Die Achsschenkellenkung und andere Fahrzeug-Lenksysteme. http://www.be-tech.bplaced.net/inhalt/02tew/techwerk/lenkung/sachinformation/lenkungssysteme.pdf 17. Ackermann R. Improvements on axletrees applicable to four-wheeled carriages, GB-Patent 4212, 27.01.1818. 18. Ueber die Lankenspergersche und Ackermannsche bewegliche Patent-Achsen, Dinglers Polytechnisches Journal,1820,Band 1,Nr. XXVII,pp. 296 311. http://dingler.culture.hu-berlin.de/journal/page/pj001?p=312 19. Tolubko V.B., Vasil'ev B.G., Berezan A.M. Decision of problem of manoeuvrability by an increase degrees of mobile of machines with the non-holonomic copulas of the wheeled type, Mekhanika ta mashinobuduvannya,kharkov: Kharkov Polytechnic Institute,2009,no. 2,pp. 96 102 (in Russian). 20. Pozin B.M., Troyanovskaya I.P. About application of the principle of d'alembert to the compilation of equations of curvilinear motion of transport vehicles, Vestnik Yuzhno-Ural'skogo Gosudarstvennogo Universiteta. Seriya Mashinostroenie,2006,no. 11 (66),issue 8,pp. 37 39 (in Russian). 21. Mityushov E.A., Misyura F.D. Calculations of the shape trajectories of vehicles and the Ackermann principle of steering, Preprint. http://www.intellectualarchive.com/?link=item&id=1060 22. Huston D.C. The geometry of parallel parking, Paper presented at the annual meeting of the The MathematicalAssociation of America MathFest,Pittsburgh,2010. http://www.allacademic.com/meta/p436841_index.html 23. Gupta A., Divekar R. Autonomous parallel parking methodology for Ackerman configured vehicles, ACEEE InternationalJournalon ControlSystem and Instrumentation,2011,vol. 2,no. 2,pp. 34 39. 24. Blackburn S.R. The geometry of perfect parking. http://personal.rhul.ac.uk/uhah/058/perfect_parking.pdf 25. Hoffman J. Perfect parallel parking, New York Times,2010. http://nyti.ms/hiahrp 26. Poddubny V.I.,Penyushkin A.S. Motion control of mobile wheeled vehicles using the satellite navigation systems, Polzunovsky Vestnik,2012,vol. 1,no. 1,pp. 239 242 (in Russian). Received 21.05.2015 Berestova Svetlana Aleksandrovna,Doctor of Physics and Mathematics,Associate Professor,Department of Theoretical Mechanics,Ural Federal University,ul. Mira,19,Yekaterinburg,620002,Russia. E-mail: s.a.berestova@urfu.ru Misyura Natal'ya Evgen'evna,Senior Lecturer,Department of Theoretical Mechanics,Ural Federal University, ul. Mira,19,Yekaterinburg,620002,Russia. E-mail: n_misura@mail.ru Mityushov Evgenii Aleksandrovich,Doctor of Physics and Mathematics,Professor,Department of Theoretical Mechanics,Ural Federal University,ul. Mira,19,Yekaterinburg,620002,Russia. E-mail: mityushov-e@mail.ru