Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://ics.org.ru/upload/iblock/25b/ND_2015_v11_n4_07.pdf Äàòà èçìåíåíèÿ: Tue Dec 29 14:42:49 2015 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:59:11 2016 Êîäèðîâêà:

. 2015. . 11. 4. . 721­734. http://nd.ics.org.ru



: 62.529 MSC 2010: 93B18, 93B52


. . , . .
, . . , . , . : , , ,

05 2015 02 2015 . . ( 2) 15-08-09261-. . . ( 3) 14-19-01303. aka@rcd.ru 426034, , . , . , . 1 karavaev_yury@istu.ru . . . 426069, , . , . , . 7

. 2015. T. 11. 4. . 721­734


722

. . , . .

1.
, . [1­4]. [4­9] [10­12]. , , . (, , ) , , . , , . , , [13]. . , .

2.
. Rs , ( ). , . (), , (. . 1). , ( ) , . .
g aO b

e3 e1 C e2

. 1. .

. O -- () , , . C e1 e2 e3 -- , e1 , e2 , e3 , ,

. 2015. T. 11. 4. . 721­734




723

e1 , e3 . C (. . 1). ( ) C e1 e2 e3 . r = (x, y , 0), e1 e3 Q, , , , C e1 e2 e3 , 1 1 1 Q = 2 2 2 . 3 3 3 , N = {(r , ,, Q)} = R2 â T2 â SO(3). (2.1) F = v - Rs â = 0, v , -- . , «» [14­16]. « ­ » :
2 T = 1 ms v 2 + 1 Is 2 + 1 mb vb + 1 (, Ibc ), 2 2 2 2

U = -mb Rb g( , e3 ),

ms , Is -- , mb , Ibc = diag(Ibc1 ,Ibc1 ,Ibc3 ) -- , vb vb = v - Rb â e3 , = + e1 + e3 , (2.2)

Rb -- . ­ ( . [13]). [13], , , , : ¨ e3 ,Ib3 ( + e3 ) = K , ¨ e1 , Ib ( + e1 ) - mb Rb Rs e3 â ( â + â ) - e1 ,mb Rb Rs ( â ) â ( + e1 ) â e3 + e1 , â mb Rb Rs ( â ) â e3 +(Is + Ib ) + e1 Ib1 + e3 Ib3 + mb Rb g (e1 , â e3 ) = K , + (2.3) +

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724

. . , . .

¨ mb Rb Rs ( â + â ) â e3 +(Is + Ib ) + e1 Ib1 + e3 Ib3 - mb Rb Rs ( â ) â ¨ â ( + e1 ) â e3 +( + e1 ) â mb Rb Rs ( â ) â e3 + +(Is + Ib ) + e1 Ib1 + e3 Ib3 + mb Rb g â e3 = =R +R
s

¨ (ms + mb )Rs ( â + â ) - mb Rb ( + e1 ) â e3 â + ( + e1 ) â (ms + mb )Rs â - mb Rb ( + e1 ) â e3 â ,

s

= â ( + e1 ),
2 2 Ib = diag(Ib1 ,Ib1 ,Ib3 ) = diag(Ibc1 + mb Rb ,Ibc1 + mb Rb ,Ibc3 ) -- , K , K -- ( ), . (2.3) (2.1) ,

Q = Q + AQ, A 3 - 2 0 = - 3 0 1 2 - 1 0 ,

r = Q v,

(2.4)





0 0 0 A = 0 0 1. 0 -1 0

, (K = K = 0), , 2 = 1, : ­ F1 = ( + e3 , e3 ) = 3 + ; ­
2 E = 1 ms v 2 + 1 Is 2 + 1 mb vb + 1 (, Ibc ) - mb Rb g( , e3 ). 2 2 2 2

(2.5)

(2.6)

(2.3) ­ 1 . , , , . , -, .
, x = v (x) (x), div ((x)v (x)) = 0.
1

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725

2.1.
[13] , . , . (), [4, 13]. (). , . [13] , -- . , . , . , (), , . 0 . , , . 1. . [13], Is + Ib3 0 . (2.7) 0 = - Ib3 0 0 . ¨ (2.8) K = -Is . 2. . . . = (0, sin , cos ), = (t), (2.9) (0) = (T ) = 0, (0) = (T ) = 0. (2.10)

: K =
22 2 ¨ ¨ m2 Rb Rs cos2 + I0 Ib1 + mb Rb sin mb Rb Rs cos + gI0 b

mb Rb Rs cos - I0 ¨ mb Rb sin (g + Rs 2 ) - (mb Rb Rs cos - Ib1 ) . 1 = mb Rb Rs cos - I0

, (2.11)

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726

. . , . .

3. . v0 (2.3), (2.4) K = 0 K = 0: = ±e3 , = v0 e, Rs 1 v =- 0, Rs = 0 , (2.12) r = ±v0 Qe2 ,

cos 0 - sin 0 0 Q = sin 0 cos 0 0 , 0 0 ±1

0 -- , 0 -- , , OX . (2.12), , . . , , . , . 4. . (2.3) K = 0 K = 0: v0 , = Rs 3 =- = (sin , 0, cos ), = v0 v0 - e1 , Rs 3 (2.13)

2 2 -Rs ((ms + mb ) + mb Rb 1 ) - Rs 3 (mb Rb + 1 (Ib1 - Ib3 )) + Is v0 + mb Rb gRs 2 1 , Rs v0 Ib3 1

, v0 , -- . v0 , . , , . K K , . ( ) [5]. . , ( ) , . , .

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727

3.
. , 1, , . , . 2 .

(a)

(b)

. 2. (a) ; (b) .

( ): Rs = 0.150 , ms = 1.625 , Is = diag(25.27·10-3 , 20.73·10-3 , 25.27·10-3 ) · 2 . , , . , . Rr = 0.087 , mr = 2.46 Ir = 5.64 · 10-3 · 2 . ( ): mb = 3.25 , Ib = diag(31.88 · 10-3 , 30.59 · 10-3 , 8.76 · 10-3 ) · 2 . (t), (t), , , , (t), (t).

3.1.
, (t) .

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728

. . , . .



0, t < t1 = (t) = 1.57 · sin 3 1.57, t > t 1 0, t < t2 = (t) = 11.304 · sin 11.304, t >

0, t , t1 t t = 1.5, 1 , 2.6,
t - 13 2, t t t = 4.1, 2 2 3 15
2

(3.1)

(3.2)

t , 2

3. , ( ), .
1.5 1 0.5 0 0 1 1 2 3 4 2* 5 6 7 8 1* 2 2*

10 8 6 4 2 0 0 1 1 1* 2 2 3 4

5

6

7

8

. 3. (3.1), (3.2)

t [t1 ,t ] 1 . , (t) (3.1) . (t) . . t [t ,t2 ] () . 1 t [t2 ,t ] 2 , t > t . 2 , (3.1), (3.2) , 4 . , ( ) (2.13).

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729

660 600 500 2 400 300 1* 200 100 0 ­100 1 200 400 600 2* 660 640 620 600 580 560 540 520 500 480

­200
. 4. (3.1), (3.2); () -- , .

( ) . , , . (3.1), (3.2). , . , , 4 . , . , . 5. 200 ( ). 5, , ( = 73 ), , , . , 2 , .
.
2

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730

. . , . .

2000 1500 1000 500 0 500 2

2* 1000 1500 2000 2500 3000

. 5. .

, (2.13) . , . , , , , . .

3.2.
, (t) (t) , . , , , . : 0, t < t1 = 0, 2 (t) = 1.57 · sin t , t1 t t = 1.5, (3.3) 1 3 1.57, t > t , 1 0, t < t2 = 1.9, 2 11.304 · sin t - 19 , t2 t t = 3.4, 2 3 30 (3.4) (t) = 11.304, t < t t3 = 6.9, 2 2 11.304 · sin t - 23 , t t t = 8.4, 3 3 3 10 0, t > t . 3 6a. , (3.3), (3.4) -

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731

1.5 1 0.5 0 1

1*

2

2*

3

3*

1000 800 600 2 1* 400 y 200 0 ­200 1

2*

0

2

4 2*

6 3

8

10

10 5 0 1 0 1* 2 2

200 400 600 800 1000 1200 x 3 3*
(b)

3* 4
(a)

­400 10 ­600

6

8

. 6. (a) (b).

, 6b . : t [t1 ,t ], 1 , t [t ,t3 ], , 2 t > t . , 3 , t = t3 - t . 2 (3.3), (3.4), , 6b . , . 6b . t [t1 ,t3 ] (3.1), (3.2). (t [t3 ,t ]) ( 3 ), . t . 1. «» -- . , , . (3.3), (3.4), 6b. 2. «C» -- , t . . , . -

. 2015. T. 11. 4. . 721­734


732

. . , . .

700 600 500 400 300 200 100 0 ­100 ­200 1 0 2 1*

2* 3 3*

Y,

200

400 600 X,

800

1000

1200

. 7. t = 1 .

, t = 1 , 7. 3. «» -- . . ( ), ( ). t , 8.
1200 1000 800 y 600 400 200 0 c 200 b 400 x 600 a 800 1000

. 8. «» : (a) t = 0.1 , (b) t = 0.25 , () t = 0.4 .

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733

4.
, , [13], . 1. . , (, ). 2. . . 3. ( ), .

5.
. . , . . , . . .


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Experimental research of dynamic of spherical robot of combined type
Alexander A. Kilin1 , Yury L. Karavaev
1

2

Udmurt State University Universitetskaya 1, Izhevsk, 426034 Russia 2 M. T. Kalashnikov Izhevsk State Technical University Studencheskaya st. 7, Izhevsk, 426069, Russia 1 aka@rcd.ru, 2 karavaev_yury@istu.ru

This pap er presents the results of exp erimental investigations for the rolling of a spherical rob ot of combined typ e actuated by an internal wheeled vehicle with rotor on a horizontal plane. The control of spherical rob ot based on nonholonomic dynamical by means of gaits. We consider the motion of the spherical rob ot in case of constant control actions, as well as impulse control. A numb er of exp eriments have b een carried out confirming the imp ortance of rolling friction. MSC 2010: 93B18, 93B52 Keywords: spherical rob ot of combined typ e, dynamic mo del, control by means of gaits, rolling friction
Received November 05, 2015, accepted December 02, 2015 Citation: Rus. J. Nonlin. Dyn., 2015, vol. 11, no. 4, pp. 721­734 (Russian)

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