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论文研究-带有初始误差的机械手轨迹跟踪的快速迭代学习控制.pdf
31 1
2011 1
Systems Engineering — Theory & Practice
Vol.31, No.1
Jan., 2011
: 1000-6788(2011)01-0165-07
: TP24; TP273
: A
Æ
,
( , 066004)
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Rapid ILC control of manipulator trajectory tracking
with initial error
(Key Lab of Industrial Computer Control Engineering of Hebei Province, Yanshan University, Qinhuangdao 066004, China)
WANG Hong-bin, WANG Yan
Abstract A discussion was made on the control of tracking for n joint manipulator systems with initial
error. First, corresponding systems were turned into lower order systems by reduction change and then
the lower order systems were redesigned, which could eliminate the limitation that the initial state of
each iterative must be equal to its ideal value in Kawamura’s method or fixed. Second, a new rapid
iterative learning algorithm was proposed to control the transformed system. It made the best of effective
information the system had saved and the output could converge to the desired output as quickly as
possible. The convergence speed of the algorithm is faster than the P type algorithm and the computation
process is easier than the PD type algorithm. Meanwhile, the instability due to differential operation is
avoided in the algorithm. Simulation results validate the validity of the scheme and real-time capability of
the system is improved.
Keywords manipulator system; initial error; rapid algorithm; iterative learning control; trajectory track-
ing
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n1, n3 ˙xk ], n2 ¨xk ].
(7)–(9) % (6),
R(t)¨xk(t) + ˆE(t) ˙xk(t) + ˆF (t)xk(t) + n ( ˙xk, ¨xk) + w(t) = − kpxk(t) − kv ˙xk(t) + µk(t)
R(t) = D (qd(t)) ,
ˆE(t) = [∂h (qk, ˙qk)/∂ ˙qk](qd, ˙qd) ,
ˆF (t) = [∂D (qk)/∂qk]qd
¨qd(t) + [∂h (qk, ˙qk)/∂qk](qd, ˙qd) + [∂ (c(qk) − ˆc(qk))/∂qk]qd
,
w(t) = D (qd) ¨qd(t) + h (qd, ˙qd) + c (qd) − ˆc (qd) ,
n ˙xk, ¨xk ].
cg (10) ] n, E
E(t) = ˆE(t) + kv, F (t) = ˆF (t) + kp, uk(t) = µk(t) − w(t).
R(t)¨xk(t) + E(t) ˙xk(t) + F (t)xk(t) = uk(t)
Kawamura h";B5$% 6851687, [
˙xk(0) = ˙qk(0) − ˙qd(0) = 0, xk(0) = qk(0) − qd(0) = 0
> 9;+ 6 PD C‘Aa\NQK, bQKJ8;
, 2QKBNALLQ, aN7 ;98bA. ] ek(t) = ˙yk(t), ?/2\
>$%:
uk+1(t) = uk(t) + Lp(t)ek(t) + Ld(t)∆ek(t), k = 1, 2,··· , t ∈ [0, T ]
(15)
, ∆ek(t) = ek−1(t) − ek(t), Lp(t) Ld(t) c?3, e62c P CA
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ˆyk(λ,ξ) ,
rU 1[6] 6
168
, ρ1 = I + B(t) [Lp(t) − Ld(t)], ρ2 = B(t)Ld(t).
vZ (14) ek(t) = ˙yk(t), k = 1, 2,··· ,
, * * 1 2 +
ek+1(t) = ek(t) + ˙yk+1(t) − ˙yk(t) = ek(t) + A(t)ˆyk(t) − B(t)F (t)ˆxk(t) + B(t)ˆuk(t)
ˆyk(t) = yk+1(t) − yk(t),
ˆxk(t) = xk+1(t) − xk(t), ˆuk(t) = uk+1(t) − uk(t).
f (15),
ˆuk(t) = Lp(t)ek(t) + Ld(t) [ek−1(t) − ek(t)] = [Lp(t) − Ld(t)] ek(t) + Ld(t)ek−1(t)
(17) % (16), E
ek+1(t) = [I + B(t) (Lp(t) − Ld(t))] ek(t) + B(t)Ld(t)ek−1(t) + A(t)ˆyk(t) − B(t)F (t)ˆxk(t)
]
A(t) ≤ ˆa,
B(t)F (t) ≤ ˆf ,
I + B(t) (Lp(t) − Ld(t)) ≤ ρ1,
B(t)Ld(t) ≤ ρ2.
2 (18) gnf,
Y 1 Y 2 3g% (19),
ek+1(t) e
ek+1(t) ≤ ρ1 ek(t) + ρ2 ek−1(t) + ˆaˆyk(t) + ˆf ˆxk(t)
−λt
ξk ≤ ρ1 ek(λ,ξ) + ρ2 ek−1(λ,ξ) + q(˜lξ)k + sˆuk(λ,ξ)
ˆuk(λ,ξ) .
(˜lξ)k,
−1
λ
bh
ˆa + ˆfed
λ
(λ − a)
−1
q = 2r ˆf + mh
s =
ˆa + ˆfed
]
2 (17) gnf,
Lp(t) − Ld(t) ≤ l1,
Ld(t) ≤ l2,
ˆuk(λ,ξ) ≤ l1 ek(λ,ξ) + l2 ek−1(λ,ξ)
31
(16)
(17)
(18)
(19)
(20)
(21)
(22)
(21) % (20),
−λt
ek+1(t) e
ek+1(λ,ξ) ≤ (ρ1 + sl1)ek(λ,ξ) + (ρ2 + sl2)ek−1(λ,ξ) + q(˜lξ)k
ξk ≤ (ρ1 + sl1)ek(λ,ξ) + (ρ2 + sl2)ek−1(λ,ξ) + q(˜lξ)k
(22) ;Kh t .,
(23)
n (λ, ξ) f ξ > 1, 0 < ˜lξ < 1, F λ + s (l1 + l2) → 0, ;
≤ [ρ1 + ρ2 + s (l1 + l2)] max
ek(λ,ξ) ,ek−1(λ,ξ)
+ q(˜lξ)k
ρ1 + ρ2 < 1 4
f
h
lim
k→∞
ek+1(λ,ξ) = lim
k→∞
ek(t) e
−λt
eλtξ
ξk
ek(t) =
ek(λ,ξ) = 0
−k ≤ ek(λ,ξ) eλT ξ
−k
lim
k→∞
˙yk(t) = lim
k→∞
ek(t) = 0,
t ∈ [0, T ]
(24)
(25)
(26)
U 2 (15) ?3 Lp(t), Ld(t) jF ρ1 + ρ2 < 1, uk(t) k [0, T ] \D,
lim
k→∞
vZ (21)
qk(t) − qd(t) = lim
k→∞
ˆuk(λ,ξ) ≤ (l1 + l2) max
˙qk(t) − ˙qd(t) = 0,
ek(λ,ξ) ,ek−1(λ,ξ)
t ∈ [0, T ].
(27)
1
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0 ≤ lim
k→∞
ˆuk(t) ≤ (l1 + l2) max
lim
k→∞
ek(λ,ξ) , lim
k→∞
ek−1(λ,ξ)
ˆuk(t) = 0
t
t
0
0
h {uk(t)} k [0, T ] \D. J
lim
k→∞
yk(t) ≤ yk(0) +
˙yk(s)ds
yk(t) ≤ yk(0) +
˙yk(s)ds
yk(t) ≤ ˆrˆlk + T ek(t)
0 ≤ lim
k→∞
yk(t) ≤ lim
k→∞
ˆrˆlk + T ek(t)
2 (30) gnf,
Of (13) ek(t) = ˙yk(t),
<@
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169
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
(36)
(37)
(38)
lim
k→∞
˙xk(t) = lim
k→∞
yk(t) = 0
yk(t) = 0
lim
k→∞
e
− t
0 D(s)dsyk(t)
e
− t
0 D(s)ds
≤ M lim
≤ M.
t
k→∞
xk(t) ≤ xk(0) +
˙xk(s)ds
0
xk(t) = 0
lim
k→∞
lim
k→∞
qk(t) − qd(t) = lim
k→∞
˙qk(t) − ˙qd(t) = 0,
t ∈ [0, T ]
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7Æ. in
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p7Y i!ig+, n? [6], .]jc. 2
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τ3 ˙q2
τ3 + τ2 cos q2
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l1 = l2 = 0.5m, m1 = m2 = 8kg, I1 = I 2 = 0.4kg · m2.
T, 6
170
(11) ?3L
R(t) =
, * * 1 2 +
˙xk(0) = ˙qk(0) − ˙qd(0) =
0, (−0.2)k
31
T
,
E(t) = ˆE(t) + kv =
F (t) = ˆF (t) + kp =
3.8 + 2 cos t3
0.9 + cos t3
−5.4t2 sin t3 −0.9
0.9 cos t sin t3
0.9 + cos t3
,
1
cos t + 6t2
0
sin t3
+ kv,
−12t sin t3 −6t sin t3 − 2.7
−6t sin t3
8.1t4 cos t3
t2 cos t + 3t4
cos t3
+ kp.
ln kv = 1.5I, kp = I.
n e
t
0 G(s)ds = R(t), Lp(t) = −0.2I, Ld(t) = 0.1I, @ ρ1 + ρ2 < 1, jFD;. 6
uk+1(t) = uk(t) − 0.2 (R(t) ˙xk(t))
= uk(t) − 0.3 (R(t) ˙xk(t))
(R(t) ˙xk−1(t))
+ 0.1
+ 0.1 (R(t) ˙xk−1(t))
− (R(t) ˙xk(t))
, k = 1, 2,··· , t ∈ [0, 2] .
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stu
[1] , uv, w. _‘d[ x01$ [J]. +" , 1999, 25(5): 716–718.
Huang B J, Sun M X, Zhang X Z. Iterative learning control algorithms with initial update action[J]. Acta
Automatica Sinica, 1999, 25(5): 716–718.
[2] lm, um , lxn. 01‘yorzq#)*rs [J]. s{#% +", 2001, 20(4): 1418.
Yang X F, Fan X P, Yang S Y. Initial value researching of iterative learning and its application in robot[J].
Computing Technology and Automation, 2001, 20(4): 1418.
[3] uv. ‘y01$ [J]. $t, 2007, 22(8): 848–852.
Sun M X. Iterative learning control with initial state learning[J]. Control and Decision, 2007, 22(8): 848–852.
[4] yu, u |. 2-D ,*1,‘dwxy,xw$ [J]. +"%rs, 2007, 26(7): 16–
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Zhong Z W, Hao X Q. An open and closed-loop iterative learning control of linear system based on the 2-D
system theory[J]. Techniques of Automation Applications, 2007, 26(7): 16–18.
[5] Kawamura S, Miyazaki F, Arimoto S. Realization of robot motion based on a learning method[J]. IEEE Trans
on Systems Man and Cybemetics, 1988, 18(3): 126–134.
[6] zxz, , z{x. { n yz+) n y|}#)*,$({ [J]. +" , 2002, 28(2): 177–
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Xie S L, Tian S P, Xie Z D. Learning control scheme for n joint robotic systems with n actuators[J]. Acta
Automatica Sinica, 2002, 28(2): 177–182.
[7] Sun M X, He X X, Chen B Y. Repetitive learning control for time-varying robotic systems: A hybrid learning
scheme[J]. Acta Automatica Sinica, 2007, 33(11): 1189–1195.
[8] Chi R H, Hou Z S. Dual-stage optimal iterative learning control for nonlinear non-affine discrete-time systems[J].
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