PRECIZIA ROBOŢILOR INDUSTRIALI

MODELAREA ERORILOR CINEMATICE .I.
2.
3.
4.
xy xy
ijmQ imjQ
Δ A =ΔA
⎛i = 1→ n,
⎞
⎜ j = 1→ i,
⎟
⎜m = j → i⎟
⎝ ⎠
( )
xy xy
( )
A ijQ R = Matrix AijQkl ; k = 1 → 3; l = 1 → 3 ;
( 3x3)
( )
xy xy
ijQ ijQk 4
A p = Matrix A k = 1 → 3
( ; )
( 3x1)
0
⎡
j ⎣
x
T ⎤
⎦yQ U j
j
[ T] m Q
Um m
[ T] i Q
0
[ T] j Q
U j
j
⎡
m⎣ x
T ⎤
⎦yQ
Um
m 0 j m
x
⋅ [ T] + [ T] ⋅U j ⋅ [ T] ⋅Um⋅ ⎡ T ⎤ ;
i Q j Q m Q i ⎣
Δ
⎦yQ
Δ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ Δ ⋅ ⋅
( )
132
( ) ( )
xy xy
⎡ΔAijmQ R ΔAijmQ
p ⎤
⎢ ⎥
⎢⎣ 000 0 ⎥⎦
xy xy
( )
Δ A ijmQ R = Matrix AijmQkl ; k = 1 → 3; l = 1 → 3
( 3x3)
xy xy
Δ A ( p) = Matrix A ; k = 1 → 4 , y = { p ; g}
ijmQ ijmQk 4
( )
( 3x1)
Pentru y=e , exprimând influenţa erorilor geometrice corespunzătoare fiecărui element al robotului, matricele
cinematice de erori sunt următoarele:
Δ A
xek
ijQ
⎛ k = 1 → n⎞
⎜ i = 1 → n ⎟
⎜ j = i → n ⎟
⎝ ⎠
xek xek
ijmQ im jQ
Δ A = ΔA
⎛k = 1→ n,
⎞
⎜ ⎟
⎜
i = 1→ n,
⎟
⎜ j = 1→ i,
⎟
⎜ ⎟
⎝m = j → i ⎠
0 ⎧ x k j
xek xek
⎡ ⎡
( ) ( ) T ⎤
ΔA [ ] j [ ] ,
ijQ R ΔAijQ p ⎤ Δ ⋅ T ⋅U ⋅ T dacak≤ j
⎪k⎣
⎦eQ
j Q i Q
⎢ ⎥= ⎨
000 0
0 j
x k
⎢⎣ ⎥⎦ ⎪ [ T] ⋅Uj⋅ ⎡ΔT ⎤ ⋅ [ T] , dacak≥ j+ 1
⎩ j Q k⎣ ⎦eQ
i Q
( )
xek xek
( )
Δ A ijQ R = Matrix Δ AijQlp ; l = 1 → 3; p = 1 → 3
( 3x3)
( )
xek xek
ijQ ijQl 4
Δ A p = Matrix Δ A l = 1 → 3
( ; )
( 3x1)
0 ⎧ x k j m
⎡ΔT ⎤ ⋅ [ T] ⋅U j ⋅ [ T] ⋅Um⋅ [ T] , k ≤ j
⎪ k ⎣ ⎦eQ
j Q m Q i Q
⎪ 0 j
x k m
⎨ [ T] ⋅U j ⋅ ⎡ T ⎤ [ ] m [ ] ,
j Q k ⎣
Δ
⎦
⋅ T ⋅U ⋅ T j+ 1≤ k ≤ m
eQ m Q i Q
⎪
0 j m
⎪ x k
[ T] ⋅U j ⋅ [ T] ⋅Um⋅ ⎡ΔT ⎤ ⋅ [ T] , m+ 1≤ k ≤ i
⎪⎩ j Q m Q k ⎣ ⎦eQ
i Q
( )
xek xek
( )
Δ AijmQ R = Matrix Δ AijmQlp ; l = 1→ 3; p = 1→ 3
( 3x3)
( )
xek xek
ijmQ ijmQl 4
Δ A p = Matrix Δ A l = 1 →
3
( ; )
( 3x1)
Q
(4.7)
(4.8)
(4.9)