Introduction to Robotics cpsc - 460

INTRODUCTION TO ROBOTICSCPSC - 460

Lecture 3A – Forward Kinematics

DH TECHNIQUES

A link j can be specified by two parameters, its length aj and its twist αj

Joints are also described by two parameters. The link offset dj is the distance from one link coordinate frame to the next along the axis of the joint. The joint angle θj is the rotation of one link with respect to the next about the joint axis.

DH TECHNIQUES

•Link twist αi :the angle from the Zi-1 axis to the Zi axis about the Xi axis. The positive sense for α is determined from zi-1 and zi by the right-hand rule.

•Joint angle θi the angle between the Xi-1 and Xi axes about the Zi-1 axis.

DH TECHNIQUES

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DH TECHNIQUES

The four parameters for each linkai: link length

αi: Link twist

di : Link offset

θi : joint angle

With the ith joint, a joint variable is qi

associated where

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TRANSFORMATION MATRIX

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Each homogeneous transformation Ai is represented as a product of four basic transformations

, , , ,i i i ii z z d x a xA Rot Trans Trans Rot

1. Rotation of about current Z axis

2. Translation of d along current Z axis

3. Translation of a along current X axis

4. Rotation of about current X axis


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i i i i i i i

i i i i i i i

i i i

c -c s s s a c

s c c -s c a s

0 s c d

0 0 0 1

iA

A

1000

00

00

0001

1000

0100

0010

001

1000

100

0010

0001

1000

0100

00

00

ii

ii

i

i

ii

ii

i CS

SC

a

d

CS

SC

A

, , , ,i i i ii z z d x a xA Rot Trans Trans Rot


The matrix Ai is a function of only a single variable, as three of the above four quantities are constant for a given link, while the fourth parameter is the joint variable, depending on whether it is a revolute or prismatic link

i i i i i i i

i i i i i i i

i i i

c -c s s s a c

s c c -s c a s

0 s c d

0 0 0 1

iA

DH NOTATION STEPS

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DH NOTATION STEPS

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DH NOTATION STEPS

From , the position and orientation of the tool frame are calculated.

i i i i i i i

i i i i i i i

i i i

c -c s s s a c

s c c -s c a s

0 s c d

0 0 0 1

iA

0nT


11 12 13

21 22 23

31 32 33

0 0 0 1

x

y

z

r r r d

r r r dT

r r r d

EXAMPLE I - TWO LINK PLANAR ARM

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• Base frame O0

•All Z ‘s are normal to the page

EXAMPLE I - TWO LINK PLANAR ARM

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Where (θ1 + θ2 ) denoted by θ12

and

EXAMPLE 2

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FORWARD KINEMATICS OF EXAMPLE 2

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EXAMPLE 3 - THREE LINK CYLINDRICAL MANIPULATOR

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18


19


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EXAMPLE 4 – THE SPHERICAL WRIST

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22


23


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EXAMPLE 5 - CYLINDRICAL MANIPULATOR WITH SPHERICAL WRIST

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derived in Example 2, and

derived in Example 3.

03T

36T


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27


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Forward kinematics:

1. The position of the end-effector: (dx ,dy ,dz )

2. The orientation {Roll, Pitch, Yaw }Rotation about X axis{ROLL}

Rotation about fixed Y axis{PITCH}

Rotation about fixed Z axis{YAW}

ROTATION – ROLL, PITCH, YAW

The rotation matrix for the following operations:

X

Y

Z

axis{YAW} Zfixedabout Rotation

}axis{PITCH Y fixedabout Rotation

axis{ROLL} Xabout Rotation

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CCSCS

CSSSCSCSSCCS

SSCSCSSCSSCC

CS

SCCS

SC

xRotyRotzRotR

0

0

001

C0S-

010

S0C

100

0

0

),(),(),(

EXAMPLE 4THE THREE LINKS CYLINDRICAL WITH SPHERICAL WRIST

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How to calculate

Compare the matrix R

With the rotation part of

, ,and

06T

C C S S C S S C S C S S

R S C C S S C S C S S S C

S C S C C

31S r 32C S r 21S C r

131( )Sin r 1 32( )

rSin

C

1 21sin ( )

r

C

11 12 13

21 22 23

31 32 33

r r r

r r r

r r r

Introduction to Robotics cpsc - 460

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