Relative Reference Frames (Relative Coordinates Systems)

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Relative Reference Frames (Relative Coordinates
Systems)
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Two equivalent relative views 1. B has been
translated with transformation T( tx , ty ) away
from A. 2. A has been translated with
transformation T( -tx , -ty ) away from B.
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If PA is known and PB is unknown From As
perspective, P is fixed in Reference Frame A From
Bs perspective, P has been translated in the
opposite direction, therefore
PB T( -tx , -ty ) PA
If PB is known and PA is unknown From Bs
perspective, P is fixed in Reference Frame B From
As perspective, P has been translated with
Reference Frame B, therefore
PA T( tx , ty ) PB
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PB T( -tx , -ty ) PA
PA T( tx , ty ) PB
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Two equivalent relative views 1. B has been
rotated with transformation R(a) away from A. 2.
A has been rotated with transformation R(-a) away
from B.
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If PA is known and PB is unknown From As
perspective, P is fixed in Reference Frame A From
Bs perspective, P has been rotated in the
opposite direction, therefore
PB R( -a ) PA
If PB is known and PA is unknown From Bs
perspective, P is fixed in Reference Frame B From
As perspective, P has been rotated with
Reference Frame B, therefore
PA R( a ) PB
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PB R( -a ) PA
PA R( a ) PB
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Notation MAB Transformation Matrix that
transforms Reference Frame A to
a new Reference Frame B
PA T( tx , ty ) PB PB T( -tx , -ty ) PA
PA MAB PB PB MBA PA
MAB -1 MBA
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MAB T( 10,5 ) MBC R( 5o ) MCD T(
15,0 ) MDE R( 10o )
PA MAB PB MAB MBC PC
MAB MBC MCD PD MAB MBC MCD
MDE PE T( 10,5 ) R( 5o ) T( 15,0 ) R(
10o ) PE
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XM
Manipulator
YM
If h , k , L1 , L2 , a , b , d are parameters
for the robotic configuration as shown, what is
the relationship between PM and PW ?
Answer
PW T(h,k) R(a) T(L1,0) R(b) T(L2,0) R(d) PM
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PW T(h,k) R(a) T(L1,0) R(b) T(L2,0) R(d) PM
If MWM , h , k , L2 , a , b and d are given,
what is L1?
MWM T(h,k) R(a) T(L1,0) R(b) T(L2,0)
R(d) T(h,k) R(a) T(L1,0) R(b) T(L2,0) R(d)
MWM T(L1,0) R(a)-1 T(h,k) -1 MWM R(d)
-1 T(L2,0) -1 R(b) -1 T(L1,0) R(-a)
T(-h,-k) MWM R(-d) T(-L2,0) R(-b)
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Robot
Main Body
Transform
Transform
Transform
Transform
Transform
Head
Arm
Leg
Transform
Transform
Transform
Camera
Hand