Lecture 11

By substituting back:

This product is the matrix

required for problem 2 of HW4.

This matrix represents the

robot’s forward kinematics

Inverse of a homogeneous

transformation matrix

Inverse of a homogeneous

transformation matrix

Inverse of a homogeneous

transformation matrix

Inverse

Inverse

Take

transpose of

rotation

matrix.

Reverse

displacement

vector.

Reverse

displacement

vector.

Reverse

displacement

vector.

Reverse

displacement

vector.

Refer reversed

displacement

vector to B frame

Refer reversed

displacement

vector to B frame

Refer reversed

displacement

vector to B frame

Z-Y-X Euler Angles

Z-Y-X Euler Angles

- Just three numbers are

needed to specify the

orientation of one set of axes

relative to another.

Z-Y-X Euler Angles

-Just three numbers are needed to

specify the orientation of one set of

axes relative to another.

-One possible set of these numbers is

the Z-Y-X Euler angles

Consider the {A} and {B}

frames shown below.

How can we define just three quantities

from which we can express all nine

elements of the rotation matrix that

defines the relative orientations of

these frames?

Beginning with the {A} frame,

rotate a positive  about the Z

A

axis.

Call this new frame {B’}

Note the rotation matrix

between {A} and {B’}

Note the rotation matrix

between {A} and {B’}

Note the rotation matrix

between {A} and {B’}

Note the rotation matrix

between {A} and {B’}

Note the rotation matrix

between {A} and {B’}

Next consider just the

intermediate {B’} frame.

Consider a positive rotation 

about the Y

B’

axis.

… take the last rotation  to be

about the X

B”

axis.

Is there a systematic way to

build ?

Member i-
1

Denevit Hartenberg

parameters

Denevit Hartenberg

parameters

Denevit Hartenberg

parameters

Three constants …

… and one variable.

D-H Example:

D-H Example:

D-H Example: Puma 560

D-H Example: Puma 560

D-H Example: Puma 560

First three rotations of Puma

First three rotations of Puma

i-1=1

i-1=1

The first rotation 

1

occurs

about the Z

1

axis.

The second rotation 

2

occurs

about the Z

2

axis.

However, the Z

2

axis and the

Z

1

axis intersect one another.

Therefore the X

1

axis may be

oriented arbitrarily.

Therefore the X

1

axis may be

oriented arbitrarily.

Since the two frames share

their origin, a

1

=d

2

=0

Since the two frames share

their origin, a

1

=d

2

=0

But what about 

1

?

But what about 

1

?

But what about 

1

?