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. 2013. . 9. 1. . 91100. http://nd.ics.org.ru

: 62.529 MSC 2010: 93B18, 93B52

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

, , , . Mecanum, [1]. [2] [3], [4], [2]. , , , . , (, 45 , «» . .) (, , ),

11 2013 1 2013 « ». karavaev_yury@istu.ru tsa@istu.ru . . . 426069, , . , . , . 71 . 2013. T. 9. 1. . 91100

92

. . , . .

. , . . , . - . , 1. . 45 . xOy , : Vx -- Ox, Vy -- Oy , -- , .

. 1. .

[5]. , x = Ax + Bu , (1)

A B -- , x -- , u -- . , t , . , [6], (1) x
k +1

= Ax k + Bu,

(2)

A = tA + E, B = tB. k k +1. A = E ( A = 0 -- ), (1) ,

. 2013. T. 9. 1. . 91100

93

1:
k +1 Vx k+1 Vy k+1

=

k 1 0 0 Vx 0 1 0 · Vyk k 001
T

1 4 -1 4 -1 2(c + e) 2(c 1 1 1 4 4 4 1 1 -1 4 4 4 1 1 -1 + e) 2(c + e) 2(c +

+2r tQ

1 2 · . (3) 3 e) 4
kT

k k Vx +1 ,Vyk+1 , k+1 -- k + 1, Vx ,Vyk , k,

--

k - sin k 0 cos Q = sin k cos k 0 0 0 1 xOy ; B = 2r
1 4 -1 4 -1 2(c + e) 2(c 1 1 1 4 4 4 1 1 -1 4 4 4 1 1 -1 + e) 2(c + e) 2(c + e

)

[1], r -- , c -- , e -- , [1 ,2 ,3 ,4 ]T -- . ( ) , , , , .

A B , x = fx (t), y = fy (t), = f (t). , T , :
B 2 2

S=
A

(x (t)) +(y (t)) dt.

(4)

( A) , V = 0 = 0, B , , 2.

. 2013. T. 9. 1. . 91100

94

. . , . .

. 2. (A -- , A1 -- , B 1 -- , B -- ).

( A A1) , -- , ( B 1 B ) . (a -- , a -- , V -- ):
I VA = 0, I aA = 0,

V
III B

III B

= 0, a
I A1

V

I A1

=V

II A1

,

V a

II B1

=V

III B1

,

a

= 0,

=a

II A1

= 0,

II B1

=a

III B1

= 0.

, , , = f (t). V II = S II /T II . . A1 B 1 N t . , , t, S II = V II · t (. 3).

. 3. .

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

. 2013. T. 9. 1. . 91100

95

: k +1 Vx Vyk+1 k+1 , [1],

V = k+1 Vy k+1

k +1 x

k Vx - k Vy k

;

1 1 1 2 = 1 2r 1 3 4 1

-1 - 1
c

1- -1
c

c+e 2 +e 2 c+e 2 +e 2

k +1 Vx · Q · Vyk+1 k+1

. (5)

, Vx Vy , (4). k -- k , V
k +1

= = =

k

+1

k - k - k - t t - xk xk - xk - t t - yk yk - y - t t

-1

, , .

(6) (7) (8)

k +1 x k +1 y

xk y

+1

-1

k +1

k -1

V

, (6), (7), (8), (5), , : c+e 1 -1 - 2 1 xk+1 - xk xk - xk-1 - t 1 1 c + e k+1 t k 2 y 1 -y y k - y k-1 . 2 ·Q· = (9) - t t 2r 1 1 - c + e k+1 3 2 - k k - k-1 - t t 4 1 -1 c + e
2

Vmax = (Vx ,Vy ) , . (t) = const, = = 0. (Vx ,Vy ). Vmax 1 = = 2 = 3 = 4 = max ( Vmax = (Vx , 0)), 1 = 3 = -max 2 = 4 = max

. 2013. T. 9. 1. . 91100

96

. . , . .

( Vmax = (0,Vy )). max -- . (9) , r = 0.16, c = e = 1, 4 0 2 .

. 4. 0 2 : () , (b) .

4 . , , n, .
2

(. 2) . A1. , , . : 1 : 67 . , , 1. , , . : . . W = WM WG , WM -- , WG = 1/67 -- . [7]: J dw(t) = M (t) - MH (t), dt (10)

M (t) = CM i(t) -- , MH (t) -- , w(t) = d -- , J --
dt

. 2013. T. 9. 1. . 91100

97

, CM -- , , i(t) -- . [7] di(t) = Ri(t)+ e(t) = u(t), (11) L dt L -- , R -- , e(t) = = CE · w(t) -- , u(t) -- , , CE -- , .
1. J L R U
H H

0.0002 2 15.7 1.48 12 0.11 60 10 000 / 0.0053 0.012 /

M

P n CE CM

k pk = d k [8], (10) (11)

, ( I I I . 2) , , , Mmax .
dt

(TA · TE · p2 + TE · p +1) · p · (p) = k1 · U (p) - k2 · (TA · p +1) · MH (p), TE =
J ·R , k = R , k1 = 1 , TA = L , (p), U (p), MH (p) -- 2 CE · CM CE · CM CE R

(t), u(t), MH (t). k2 · (TA · p +1) . WM = (TA · TE · p2 + TE · p +1) · p Matlab (. . 5). (M 2) ( ). 6 . (M 1) . 3 . (9) (. 4) ,

. 2013. T. 9. 1. . 91100

98

. . , . .

. 5. M 2 = M

max

, M 1 = 0.5M

max

.

. , w1 = w2 = w3 = w4 = wmax , V = Vx = 1 (. 4), Vy = = 0 -- , w1 = w3 = 0 w2 = w4 = wmax , V =
2 , Vx = Vy = 1 -- 2 2

( w1 = w3 = 0 , , ). , , 5. (. 2) . , ( ). (, , . .).

k (. . 6). k (xk ,yr ,k ). -r r , - . , ( ). , (3)
k Vx ,Vyk , kT

.

. 2013. T. 9. 1. . 91100

99

. 6. .

(, ) , , k +1 (. 6). ( ) ( ) . (x =
k +1

V

k +1 r

- xk )2 +(y r t

k +1

k - yr )2

.

(12)

T k (5), Vx +1 , Vyk+1 , k+1 (12) : V
k +1

= = =

k

+1

k - k - k r -r t t - xk xk - xk r -r t t
k yk - y - yr -r t t

-1

, , .

(13) (14) (15)

k +1 x k +1 y

xk y

+1

-1

k +1

k -1

V

Q : k - sin k 0 r cos r k cos k 0 Q = sin . r r 0 0 1 , (5) (13), (14), (15) . t , - . t ( ), .

. 2013. T. 9. 1. . 91100

100

. . , . .

. -, , t. , , k +2 k + n, k +1. Sk +1, Sk +2,... , Sk + n ( ) .

[1] . . : , 2012. 518 . [2] Liu Y., Zhu J., Williams R. L. II, Jianhua Wu. Omni-directional mobile robot controller based on tra jectory linearization // Robot. Auton. Syst., 2008, vol. 56, pp. 461479. [3] Purwin O., D'Andrea R. Tra jectory generation and control for four wheeled omnidirectional vehicles // Robot. Auton. Syst., 2006, vol. 54, pp. 1322. [4] . ., . ., . . // , 2011, . 7, 4, c. 785801. [5] . ., . . III. . : , 1982. 392 . [6] / . . . : , 1987. 712 . [7] - . Matlab & Simulink: . : -, 2011. 368 . [8] : 2- : . 1. / . . . : . , 1986. 387 .

Deviation based discrete control algorithm for omni-wheeled mobile robot
Yuri L. Karavaev1 , Sergey A. Trefilov
2

M. T. Kalashnikov Izhevsk State Technical University Studencheskaya st. 7, Izhevsk, 426069, Russia 1 karavaev_yury@istu.ru, 2 tsa@istu.ru

The pap er deals with deviation based control algorithm for tra jectory following of omni-wheeled mobile rob ot. The kinematic mo del and the dynamics of the rob ot actuators are describ ed. MSC 2010: 93B18, 93B52 Keywords: omni-wheeled mobile rob ot, discrete algorithm, deviation based control, linearization, feedback
Received March 11, 2013, accepted April 1, 2013 Citation: Rus. J. Nonlin. Dyn., 2013, vol. 9, no. 1, pp. 91100 (Russian)

. 2013. T. 9. 1. . 91100