Singular perturbation control for flexible2joint man ipulator

Transcription

1 EL ECTR ICMACH IN ESANDCON TROL Vol3 No3 May 29,,, 2 (., 5; 2.,8223) :,,,,,,, PD, : ; ; ; : TP24 : A : 7-449X (29) Singular perurbaion conrol for flexible2join man ipulaor L IU Ye2chao, L IU Yi2wei, L IU Hong, 2 (. Sae Key Laboraory of Roboics and Sysem, Harbin Insiue of Technology, Harbin, 5, China; 2. Insiue of Roboics and Mecharonics, German Aerospace Cener, Munich 8223, Germany) Absrac: The radiional singular perurbaion app roach can only be used in he robo sysem w ih weak join flexibiliy. In order o solve his p roblem, a join flexibiliy compensaor was designed, which can increase he equivalen join siffness. So he lim iaion for he singular perurbaion app roach was re2 moved and i can be exended o he robo sysem w ih norm al join flexibiliy. Besides hese, an adap ive conrol law was designed for he slow subsysem, which can guaranee he rajecory error asymp oical racking. The p roposed conrol sraegy can be app lied o he engineering sysem s convenienly w ihou any join flexibiliy lim iaion and does no need he link acceleraion or jerk informaion. Finally, an ex2 perim enal sudy on he flexible join manipulaor was execued and he resuls verify ha whaever he PD, compued orque or adap ive conrol are used in he slow subsysem, he p roposed singular perurba2 ion sraegy is more valid han he radiional one. Key words: flexibiliy compensaion; adap ive conrol; singular perurbaion conrol; flexible join, : : ; (26AA4z228) : (98 - ),,, ; (977 - ),,, ;, [ ] [ 2 ] [ 3 ] [ 4 ] (966 - ),,,,

2 3 437, [, 5, 6 ] [ 7, 8 ] :Spong [ ] ; Ge [ 5 ], [ 5, 6, 9, ],,, Huang [ ], : ;,,,,,;,, 2 Spong [ 2 ], m = J+, () =M ( q) q + C ( q, gq) gq + G ( q), (2) = K (- q) (3) : q ; J ; K; m ; ; C G,,,,,,,, ( ),,,,,, () (3) J K - += - J q (4) (4), u e m = m - J q (5) K -, u e,,, m = K a u a + u f, (6) : u a ; u f ; K a, (6)(4), J K - += K a ( u a - K - J q) + u f (7) a u l = u a - K - a J q, (8) u f = - K p, (9) K p (7), J K - + ( I + K p )= K a u l () ( ) ( 4 ) K a = I + K p, () J K - += u = u a - K - J q, (2) e a K e = K a K (3) (4) (2),,,, u a,,, u a = u as + u af, (4) u af u as

3 438 3,, K e = K e / 2 u af 2 J = - K ev g= -K e2 g +K e K e2 g+ K e = K e ( u as - K - a J q), (5) = M e ( q) q + C ( q, gq) gq + G ( q) = u as, (6) M e ( q) =M ( q) + K - a J ( 6 ), :, ge ( ), e ( ), e = q d - q Barbala, y ( ) M ( ), y T ( lim y ( ) = )M ( ) y ( ) d<, (7) y = ge +e, = g gq d +e (8) (6) y T (M e ) yd = y T [M e ( q +y) +C gq +G - u as ]d = - 2 yt M e y C+ g G - u as ]d = + 2 y () T M e () y () - y T [M e ( +y) + 2 yt M e y + y T (W x - u as ) d, (9) W x =M e ( +y) + C+ g G, (2) W y egeg, x y T M e gyd = 2 yt M e y y T 2 Mg e ( q) - C ( q, gq) - 2 y T g y = M e yd, x^ =W T y, (23) ( 24), y T (W x - u as ) d = y T [W ( x - ^x) - K d y ]d = y T W x - [ x - ^x () ] W T yd- ^x () ( y T W ) d - y T W d 2 - K d y d 4 - [ x - ^x () ] 2 < (24), 2,: ),,,, [ 2 ] ; 2),(23) x^+ P ^x =W T y, P (9), y T ( ) y ( ) <, y T (W x - u as ) d <, (2) lim y ( ) =,, u as =W ^x + K d y, (22)

4 , 3 F ig. 3 3The robo prooype 4,,,DSP FPGA : FPGA, DSP, DSPFPGA ( PPSeCo), 5kHz,, 4 4 Fig. 4The hardware archiecure for he arm s conroller 32, 2,9, Paul [ 3 ],, [ 4 ] 4,L 2 [ e ] e M L 2 [ u ]C u I l q +m glcosq =, J += m, = K (- q) (25) : I l ; J ; m ; l ( I l + J ) q +m glcosq = m s, (26) J I l + q +m glcosq = u + as (27) K p mt =- K v g, (28) m s mn = ( + K p ) ( u as - K ev g ) - K p (29) 32. PD,PD m s = u as = K p e + K d ge, e = q d - q (3), K d =, K p, 2, 3, 4, 8, L 2 [ e ] e M K p, C u,, C u 2 (K p = 8 ),, K p = 4, K d,k p ( K p = 4 ), K d 2K d ,,, L 2 [ u ] ;

5 44 3,, K d,,k p = 4, K d = 4, PD PD,, ( )( ),,, m s = u as =M^ e ( q) ( q d + K p e + K d ge) + g ( q), (3) F ig. 6 6The error plos w ih wo schem es ba sed on com pue orque con rol 323 e = q d - q, y = ge +e, = g gq d +e, ( 24) W x =M e ( +y) +m g lcosq, (32) W = +y cosq T = q d +ge +y cosq T, (33) x x T = I l + J m gl x N = I J l + + K p, (34) (35) m gl, (26) m s = u as =W ^x + K d y, (36) x^ = ( +y) y, x^ 2 = 2 ycosq (37) PD,, =, =, = 5, 2 = 2, K d = 5 7 M^ e ( q), (3) (3), 6 F ig. 7 7The error plos ba sed on adapive con rol

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