PRECIZIA ROBOŢILOR INDUSTRIALI

. PRECIZIA ROBOȚILOR INDUSTRIALI
Tipul matricei Expresia de definiţie
( x − y − z )
( z − y − x )
R γ β α
R α β γ
R
R
Restricţii
49
Tabelul 2.1
⎡cαz ⋅cβy cαz ⋅sβy ⋅sγ x −sα z ⋅cγ x cαz ⋅sβy ⋅ cγ x + sαz ⋅sγ
x ⎤
⎢ ⎥
R( z, αx ) ⋅R( y, βy ) ⋅ R( x, γx ) = ⎢sαz ⋅cβy sαz ⋅sβy ⋅ sγ x + cαz ⋅cγ x sαz ⋅sβy ⋅cγ x −cα z ⋅sγ
x ⎥
⎢ ⎥
⎢⎣ −sβy cβy ⋅sγ x cβy ⋅cγ
x ⎥⎦
⎛ β
γ ix ∈( − 1, 1)
tan ix α ⎞
A 2 , ix
⎜ ⎟
⎜ cβy cβ
⎟
⎝ y ⎠
γ = ± 1
0
ix
α z
Expresiile unghiurilor de orientare
β y
γ x
⎛ γ iy γ ⎞
Arc sin( − γ ix )
Atan 2 , iz
⎜ ⎟
⎜ cβy cβ
⎟
⎝ y ⎠
π
⎛ α ⎞
∓ Atan 2 , iz
⎜ −βiz
⎟
2
⎜ sβ
⎟
⎝ y ⎠
Tipul matricei Expresia de definiţie
( x − y − x )
( x − y − x )
R γ β α
R α β γ
Restricţii
( , ) ( , ) ( , )
Tabelul 2.2
⎡ cβy sβy ⋅sγ x sβy ⋅cγ
x ⎤
⎢ ⎥
α ⋅ β ⋅ γ = ⎢ α ⋅ β − α ⋅ β ⋅ γ + α ⋅ γ − α ⋅ β ⋅ γ − α ⋅ γ ⎥
⎢ ⎥
⎢⎣ −cα x ⋅sβy cαx ⋅cβy ⋅ sγ x + sαx ⋅cγ x cαx ⋅cβy ⋅cγ x −sα x⋅sγ x ⎥⎦
R x x R y y R x x s x s y s x c y s x c x c x s x c y c x c x s x
⎛ β
αix ∈( − 1, 1)
tan ix γ ⎞
A 2 , ix
⎜ − ⎟
⎜ sβy sβ
⎟
⎝ y ⎠
α ix = ± 1
tan ( iy , iy )
α x
Expresiile unghiurilor de orientare
β y
γ x
Arc cos( α ix )
⎛ αiy α ⎞
Atan 2 , iz
⎜ ⎟
⎜ sβy sβ
⎟
⎝ y ⎠
A 2 γ β ± π
0
Tipul matricei Expresia de definiţie
( γ x − βz −α
x )
( α − β −γ
)
x z x
Restricţii
Tabelul 2.3
⎡ cβz −sβz ⋅cγ x sβz ⋅sγ
x ⎤
⎢ ⎥
R( x, αx ) ⋅R( z, βz ) ⋅ R( x, γx ) = ⎢cαx ⋅sβz cαx ⋅cβz ⋅cγ x −sα x ⋅sγ x −cαx ⋅cβz ⋅sγ x −sα x ⋅cγ
x ⎥
⎢ ⎥
sα ⋅sβ sα ⋅cβ ⋅ cγ + cα ⋅sγ −sα ⋅cβ ⋅ sγ + cα ⋅cγ
⎛ γ
αix ∈( − 1, 1)
tan ix β
A 2 , ix ⎞
⎜ ⎟
⎝ sβz sβz
⎠
ix
⎣ x z x z x x x x z x x x ⎦
Expresiile unghiurilor de orientare
α x
β z
γ x
⎛ α αiy
Arc cos( α ix )
Atan 2 iz ⎞
⎜ , − ⎟
⎝ sβz sβz
⎠
α = ± 1
Atan 2( − β , γ )
± π
0
iz iz