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KON 318E : Introduction to Robotics

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KON 318E : Introduction to Robotics

=

10

dRH

T

=

10

1 dRRHTT

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Basic transforms:

Three pure translation, three pure rotation

=

0100

0010

001

,

a

axTrans

=

00

00

0001

,

cs

scxRot

3

KON 318E : Introduction to Robotics

=

=

1000

100

0010

0001

1000

0100

010

0001

1000

,

,

c

b

cz

by

Trans

Trans

=

=

1000

0100

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1000

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0010

00

1000

,

,

cs

sc

cs

sc

z

y

Rot

Rot

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Euler angles: we have only discussed ZYZ Euler angles. What is the set of

all possible Euler angles that can be used to represent any rotation matrix?

XYZ, YZX, ZXY, XYX, YZY, ZXZ, XZY, YXZ, ZYX, XZX, YXY, ZYZ

ZZY cannot be used to describe any arbitrary rotation matrix since twoconsecutive rotations about the Z axis can be composed into one rotation

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Compute the homogeneous transformation representing a translation of 3

units along the x-axis followed by a rotation of /2 about the current z-axis

followed by a translation of 1 unit along the fixed y-axis

,1 ,3 , / 2y x zT T T T =

5KON 318E : Introduction to Robotics

1 0 0 0 1 0 0 3 0 1 0 0

0 1 0 1 0 1 0 0 1 0 0 0

0 0 1 0 0 0 1 0 0 0 1 0

0 0 0 1 0 0 0 1 0 0 0 1

1 1 0 3

1 0 0 1

0 0 1 0

0 0 0 1

=

=

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Challenge: given all the joint parameters of a manipulator,

determine the position and orientation of the tool frame

Tool frame: coordinate frame attached to the most distal link of themanipulator

Inertial frame: fixed (immobile) coordinate system fixed to the mostroximal link of a mani ulator

6KON 318E : Introduction to Robotics

Therefore, we want a mapping between the tool frame and theinertial frame

This will be a function of all joint parameters and the physical geometry

of the manipulator

Purely geometric: no need to worry about joint torques or dynamicsyet !!!

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A n-DOF manipulator will have njoints (either revolute or prismatic) and n+1

links (since each joint connects two links)

link i-1 link i

joint i

7KON 318E : Introduction to Robotics

We assume that each joint only has one DOF.

Although this may seem like it does not include things like sphericalor universal joints, we can think of multi-DOF joints as a combinationof 1DOF joints with zero length between them

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Frame o0 is the inertial frame

Frame on is the tool frame

Joint i connects link i-1 and link i

Frame oiis connected to link i

8KON 318E : Introduction to Robotics

Joint variables, qi

=

prismaticisjointif

revoluteisjointif

id

iq

i

i

i

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We said that a homogeneous transformation allowed us to express the

position and orientation of ojwith respect to oi what we want is the position and orientation of the tool frame with respect to the

inertial frame

An intermediate step is to determine the transformation matrix that gives positionand orientation of oiwith respect to oi-1: Ai ( Homogeneous Trans.)

zi-1

zi

9KON 318E : Introduction to Robotics

Now we can define the transformation ojto oias:( ) ij

ji

ji

T

I

AAAA

Tji

jijii

ij

>

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