PRECIZIA ROBOŢILOR INDUSTRIALI

MODELAREA ERORILOR DINAMICE I.
0
j
v = ∑ A ⋅qɺ
; (6.23)
jQ jkQ k
k= 1
j j j
j 0 T ⎧ xyQT
[ ] xy ⎫ 0 ⎧ ⎫
Δ v jQ = j R ⋅ ⎨∑ ΔAjkQ ⋅ qɺ k + ∑ AjkQ ⋅Δ qɺ k ⎬ + j [ ΔR] ⋅ ⎨∑ AjkQ ⋅qɺ
k ⎬;
(6.24)
⎩k= 1 k= 1 ⎭ ⎩k= 1 ⎭
0
j j j
vɺ = A ⋅ qɺɺ + A ⋅qɺ ⋅qɺ
; (6.25)
∑ ∑∑
jQ jkQ k jklQ k l
k= 1 k= 1 l= 1
⎧ ⎧ j j j j
xy xy ⎫ ⎫
⎪ ⎪∑ ΔAjkQ ⋅ qɺɺ k + ∑ AjkQ ⋅ Δ qɺɺ k + ∑∑ ΔAjklQ ⋅qɺ k ⋅ qɺ
l + ⎪ ⎪
j 0 T
⎪Δ [ ] k 1 k 1 k 1 l 1
vɺ ⎪ = = = =
⎪
jQ = j R ⋅ ⎨ j j j j
⎬ + ⎪
⎪
⎨
⎪+ ∑∑ AjklQ ⋅ Δqɺ k ⋅ qɺ l + ∑∑ AjklQ ⋅qɺ k ⋅ Δqɺ
⎪ ⎪
l ⎬ . (6.26)
⎪
⎪⎩ k= 1 l= 1 k= 1 l= 1
⎪⎭
⎪
⎪ j j j
0 xyQT ⎧ ⎫ ⎪
⎪ + j [ ΔR] ⋅ ⎨∑ AjkQ ⋅ qɺɺ k + ∑∑ AjklQ ⋅qɺ k ⋅qɺ
l ⎬ ⎪
⎩
⎩k= 1 k= 1 l= 1 ⎭ ⎭
În expresiile prezentate anterior,
generalizată de inerţie iar
i
Q g şi
i
Q SU reprezintă forţele generalizate active,
170
i
Qiö forţa
i
Q fd forţele generalizate datorate frecării vâscoase sau uscate care apar în
cuplele motoare ale robotului. Expresia (6.6) mai poate fi scrisa în forma prezentata mai jos:
( ) ( ) ( ) ( ) ( ) ( 2
θ ;
ɺ
θ ;
ɺɺ
θ = θ ) Qk ⋅
ɺɺ
θ + θ
ɺ
Qk θ
ɺ
θ + θ
ɺ
Qk θ k + ( θ ) + ( θ )
Q M B C Q Q
m Qk k k
g SU
Pentru a obţine erorile forţei generalizate motoare, se diferențiază expresia (13.14), rezultând:
xy
( ) Qk ( ) Qk ( ) ( ) Qk ( )
(6.27)
⎧ xy
xy ΔMQk ⋅
ɺɺ
θk + M θ ⋅Δ
ɺɺ
θk + ΔB θ ⋅
ɺ
θ
ɺ
θ + B θ ⋅ Δ
ɺ
θ
ɺ
θ ⎫
⎪ +
k k ⎪
Δ QmQk
= ⎨ xy ⎬ ;(6.28)
( ) ( 2 ) ( ) ( 2 ) xy ( ) xy
+ ΔC θ Qk ⋅
ɺ
θ k + C θ Qk ⋅ Δ
ɺ
⎪ θ k + Δ Q ( )
⎩ g θ + ΔQSU
θ ⎪⎭
unde M ( Mij , i 1 n, j 1 n)
Δ = Δ = → = → ; (6.29)
⎧ ⎫
⎪ Δ = Δ = Δ ⋅ ⋅ +
Qk Qk
⎪
⎪ k = max ( i; j)
⎪
⎨ n n
⎬ ; (6.30)
⎪ k T k T
+ ∑ Trace { Aki ⋅ Δ Ipsk ⋅ Akj} + ∑ Trace { ΔAki ⋅ Ipsk ⋅ ΔAkj
} ⎪
⎪
⎩ k = max ( i; j) k = max ( i; j)
⎪
⎭
n
xy xy k T
( Mij ) ( M ji ) ∑ Trace { Aki Ipsk Akj}
⎡ i = 1 → n,
⎤
xy
Δ B( θ ) Qk = ⎢2⋅ Δ Vijm, j = 1 → n −1,
⎥ ; (6.31)
⎢
m = j + 1 → n
⎥
⎣ ⎦
k T
{ ki psk kjm}
⎧ n
⎫
⎪ ∑ Trace ΔA ⋅ I ⋅ A +
⎪
xy xy ⎪ k= max ( i; j)
⎪
Δ ( Vijm ) = Δ ( Vimj
) = ⎨ Qk Qk n n
⎬
⎪ k T k T
∑ Trace { Aki ⋅ Δ Ipsk ⋅ Akjm} + ∑
Trace { ΔAki ⋅ Ipsk ⋅ ΔAkjm}
⎪
⎪
⎩k = max ( i; j) k= max ( i; j)
⎪
⎭