082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

7.3 Cartesian components 237
Engineeringapplication7.4
Robotpositions
Positionvectorsprovideausefulmeansofdeterminingthepositionofarobot.There
aremanydifferenttypesofrobotbutacommontypeusesaseriesofrigidlinksconnected
together by flexible joints. Usually the mechanism is anchored at one point.
Atypical example isillustratedinFigure 7.23.
b
c
d
Y
a
p
X
Figure7.23
Atypical robot configuration with vectors representing
the robot links.
TheanchorpointisXandthetipoftherobotissituatedatpointY.Thefinallink
is sometimes called the hand of the robot. The hand often has rotating and gripping
facilitiesanditssizerelativetotherestoftherobotisusuallyquitesmall.Eachofthe
robotlinkscanberepresentedbyavector(seeFigure7.23).Thevectordcorresponds
tothehand.Acommonrequirementinroboticsistobeabletocalculatetheposition
of the tip of the hand to ensure it does not collide with other objects. This can be
achieved by defining a set of Cartesian coordinates with origin at the anchor point
of the robot, X. Each of the link vectors can then be represented in terms of these
coordinates. For example, inthe case of the robot inFigure 7.23:
a=a 1
i+a 2
j+a 3
k
b=b 1
i+b 2
j+b 3
k
c=c 1
i+c 2
j+c 3
k
d=d 1
i+d 2
j+d 3
k
Thepositionofthetipofthehandcanbecalculatedbyaddingtogetherthesevectors.
So,
p=a+b+c+d
=(a 1
+b 1
+c 1
+d 1
)i+(a 2
+b 2
+c 2
+d 2
)j+(a 3
+b 3
+c 3
+d 3
)k
7.3.2 Linearcombinations,dependenceandindependence
Supposewehavetwovectorsaandb.Ifweformarbitraryscalarmultiplesofthese,that
isk 1
a andk 2
b, and add these together, we obtain a new vector c where c = k 1
a+k 2
b.
The vector c is said to be a linear combination of a and b. Note that scalar multiplication
and addition of vectors are the only operations allowed when forming a linear
combination. Vector c is said to depend linearly on a and b. Of course we could also
write
a = 1 k 1
c − k 2
k 1
b providedk 1 ≠ 0