Homogeneous transforms Rotation matrices assume that the origins of - - PowerPoint PPT Presentation

SLIDE 1

Homogeneous transforms

Rotation matrices assume that the origins of the two frames are co-located.

• What if they’re separated by a translation?

x

A

p

y

A

A Bd

y

B

x

B

SLIDE 2

Homogeneous transform

x

A ˆ

y

A ˆ

B Ad

y

B ˆ

x

B ˆ

B A B B A A

d p R p + =

(same point, two reference frames)

p

B

p

A

SLIDE 3

Homogeneous transform

x

A

p

y

A

b ad

y

B

x

B

                = 1 1 p d R

B B A B A

        =                       = 1 1 1

33 32 31 23 22 21 13 12 11

p T p d r r r d r r r d r r r

B B A B z A y A x A

always one always zeros

B A B B A A

d p R p + =

SLIDE 4

Example 1: homogeneous transforms

What’s ?

A BT

x

A

y

A

x

B

y

B

l θ z

A

z

B

SLIDE 5

Example 1: homogeneous transforms

What’s ?

A BT

x

A

y

A

x

B

y

B

l θ z

A

z

B

          − = 1 ) cos( ) sin( ) sin( ) cos( θ θ θ θ

B AR

         − = l d A

B

x

A

y

A

z z B

A ,

x

B

y

B

θ

        = 1

A B A B A B

d R T               − − = 1 1 ) cos( ) sin( ) sin( ) cos( θ θ θ θ l TA

B

SLIDE 6

Example 2: homogeneous transforms

What’s ?

b aT

a

x

a

y l

θ

a

z

b

x

b

y

b

z l

( ) ( )

              − + + − = 1 1 2 2

4 4 π π

θ θ θ θ θ θ s l c s c l s c Tb

a

This arm rotates about the axis. a

z

SLIDE 7

Example 3: homogeneous transforms

x

A ˆ

y

A ˆ

z

A ˆ

x

c ˆ

y

c ˆ

z

c ˆ

θ φ l

          − − =           −           − = =

θ φ θ φ θ φ φ θ θ φ φ θ φ φ φ φ θ θ θ θ

c s s c s c s s c s c c c s s c c s s c R R R

c b b a c a

1 1

SLIDE 8

              − − − =         = 1 1

φ θ θ φ θ φ θ φ φ φ φ θ θ θ φ φ θ

c ls c s s c s ls c s c lc s c s c c d R T

a c a c a

Example 3: homogeneous transforms

x

A ˆ

y

A ˆ

z

A ˆ

x

c ˆ

y

c ˆ

z

c ˆ

θ φ l

         − = l d

c

          − =          −           − − − = =−

φ θ φ φ θ θ φ θ φ θ φ φ θ θ φ φ θ

c ls ls c lc l c s s c s c s s c s c c d R d

c c a a

SLIDE 9

Inverse of the Homogeneous transform         − =

−

1

1 A B T A B T A B A B

d R R T

A B A A B B

d p R p + =

( )

A B B T A B A

d p R p − =

Can also derive it from the forward Homogeneous transform:

        =

−

1

1

p T p

B A B A

where

SLIDE 10

Inverse of the Homogeneous transform

x

A ˆ

y

A ˆ

B Ad

y

B ˆ

x

B ˆ

( )

B A A A B B A A B A A B B

d p R d R p R p − = − =

(same point, two reference frames)

p

B

p

A

SLIDE 11

Example 1: homogeneous transform inverse

What’s ?

B AT

x

A

y

A

x

B

y

B

l θ z

A

z

B

              − − = 1 1 ) cos( ) sin( ) sin( ) cos( θ θ θ θ l TA

B

SLIDE 12

Example 1: homogeneous transform inverse

          − = 1 ) cos( ) sin( ) sin( ) cos( θ θ θ θ

B AR

         − = l d A

B

              − = =

−

1 1 ) sin( ) cos( ) sin( ) cos( ) sin( ) cos(

1

θ θ θ θ θ θ l l T T

B A A B

        − =

−

1

1 A B T A B T A B A B

d R R T

          =          −           − = − ) sin( ) cos( 1 ) cos( ) sin( ) sin( ) cos( θ θ θ θ θ θ l l l d R

A B B A

SLIDE 13

Example 2: homogeneous transform inverse

What’s ?

a bT

a

x

a

y l

θ

a

z

b

x

b

y

b

z l

( ) ( )

              − + + − = 1 1 2 2

4 4 π π

θ θ θ θ θ θ s l c s c l s c Tb

a

( ) ( )

              − − − + − =

+ + + +

1 2 1 2

4 4 4 4 π π π π

θ θ θ θ θ θ θ θ

θ θ θ θ s c c s l c s s s c c l s c Ta

b

SLIDE 14

Forward Kinematics

• Where is the end effector w.r.t. the “base” frame?

SLIDE 15

Composition of homogeneous transforms

3 2 2 1 1 3

T T T T =

x y

1

q

z

2

q

3

q

x

1

y

1

y

2

x

2

x

3

y

3

1

l

2

l

3

l

Base to eff transform Transform associated w/ link 1 Transform associated w/ link 2 Transform associated w/ link 3

SLIDE 16

Forward kinematics: composition of homogeneous transforms

3 2 2 1 1 3

T T T T =

              − = 1 1

1 1 1 1 1 1 1 1 1

s l c s c l s c T               − = 1 1

2 2 2 2 2 2 2 2 2 1

s l c s c l s c T

x y

1

q

z

2

q

3

q

x

1

y

1

y

2

x

2

x

3

y

3

1

l

2

l

3

l

SLIDE 17

Forward kinematics: composition of homogeneous transforms

3 2 2 1 1 3

T T T T =

              − = 1 1

3 3 3 3 3 3 3 3 3 2

s l c s c l s c T

x y

1

q

z

2

q

3

q

x

1

y

1

y

2

x

2

x

3

y

3

1

l

2

l

3

l

SLIDE 18

Remember those double-angle formulas…

( ) ( ) ( ) ( ) ( )

φ θ φ θ φ θ sin cos cos sin sin ± = ±

( ) ( ) ( ) ( ) ( )

φ θ φ θ φ θ sin sin cos cos cos  = ±

SLIDE 19

3 2 2 1 1 3

T T T T =

              −               −               − = 1 1 1 1 1 1

3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 3

s l c s c l s c s l c s c l s c s l c s c l s c T

Forward kinematics: composition of homogeneous transforms

              + + + + − = 1 1

123 3 12 2 1 1 123 123 123 3 12 2 1 1 123 123 3

s l s l s l c s c l c l c l s c T

SLIDE 20

DH parameters

• There are a large number of ways that homogeneous transforms

can encode the kinematics of a manipulator

• We will sacrifice some of this flexibility for a more systematic

approach: DH (Denavit-Hartenberg) parameters.

• DH parameters is a standard for describing a series of transforms

for arbitrary mechanisms. x y

1

q

z

2

q

3

q

x

1

y

1

y

2

x

2

x

3

y

3

1

l

2

l

3

l

x z

1

q

y

2

q

3

q

x

1

z

1

z

2

x

2

x

3

z

3

1

l

2

l

3

l

SLIDE 21

Forward kinematics: DH parameters

( )

i i i i

d a θ α

These four DH parameters, represent the following homogeneous matrix:

              −                                           − = 1 1 1 1 1 1 1 1 1 1 1 1

i i i i i i i i

c s s c a d c s s c T

i i α α α α θ θ θ θ

Then, translate by along x axis

i

a

and rotate by about x axis

i

α

First, translate by along z axis

i

d

and rotate by about z axis

i

θ

SLIDE 22

Forward kinematics: DH parameters

              −                                           − = 1 1 1 1 1 1 1 1 1 1 1 1

i i i i i i i i

c s s c a d c s s c T

i i α α α α θ θ θ θ

              − − = 1

i i i

d c s s a s c c c s c a s s c s c

i i i i i i i i i i i i i i

α α θ α θ α θ θ θ α θ α θ θ

SLIDE 23

Forward kinematics: DH parameters x ˆ z ˆ y ˆ y ˆ

1

z ˆ

1

d

θ

x ˆ

1

y ˆ

1

z ˆ

1

a

x ˆ

1

α

y ˆ

2

z ˆ

2

Then, translate by along x axis

i

a

and rotate by about x axis

i

α

First, translate by along z axis

i

d

and rotate by about z axis

i

θ

( )

i i i i

d a θ α

Four DH parameters:

( ) ( ) ( ) ( )

i i i i

a x trans x rot d z trans z rot

T T T T T

, , , , α θ

=

SLIDE 24

Forward kinematics: DH parameters

• A series of transforms is written as a table:

xform 1 2

i

a

i

α

i

d

i

θ

1

a

1

α

1

d

1

θ

2

a

2

α

2

d

2

θ

SLIDE 25

Example 1: DH parameters

x y

1

q

z

2

q

3

q

x

1

y

1

y

2

x

2

x

3

y

3

1

l

2

l

3

l

i

a

i

α

i

d

i

θ

1

l

1

q

2

l

2

q

1 2 3

3

l

3

q

SLIDE 26

Example 1: DH parameters

x y

1

q

z

2

q

3

q

x

1

y

1

y

2

x

2

x

3

y

3

1

l

2

l

3

l

1 2 3

i

a

i

α

i

d

i

θ

1

l

1

q

2

l

2

q

3

l

3

q

              − = 1 1

1 1 1 1 1 1

1 1 1 q q q q q q

s l c s c l s c T

              − = 1 1

2 2 2 2 2 2

2 2 2 1 q q q q q q

s l c s c l s c T

              − = 1 1

3 3 3 3 3 3

3 3 3 2 q q q q q q

s l c s c l s c T

3 2 2 1 1 3

T T T T =

SLIDE 27

Example 2: DH parameters

1

q

2

q

3

q

x y z

1

z

1

x

2

z

2

x

2

y

3

z

3

y

3

x

1

l

2

l

3

l

SLIDE 28

Example 2: DH parameters

1

q

2

q

3

q

x y z

1

z

1

x

2

z

2

x

2

y

3

z

3

y

3

x

1

l

2

l

3

l

i

a

i

α

i

d

i

θ

1

l

2 π

−

1

q

2

l

2 2 π

− q

1 2 3

3

l

3

q

SLIDE 29

Example 3: DH parameters

1

q

x y z

1

x

1

l

2

q

3

q

4

q

1

z

1

y

2

x

2

y

2

x

2

y

2

x

2

y

SLIDE 30

Example 3: DH parameters

1

q

x y z

1

x

1

l

2

q

3

q

4

q

1

z

1

y

2

x

2

y

2

x

2

y

2

x

2

y

1 2 3 4

i

a

i

α

i

d

i

θ

1

l

1

q

2

q

3

q

π

4

q

2

l

3

l

4

l