Euler Angles This means, that we can represent

an orientation with 3 numbers Assuming we limit ourselves to 3

rotations without successive rotations about the same axis:

Example

Gimbal Lock

Gimbal Lock Animation

Euler Summary Video

Quaternions Quaternions are a number system

that extends the complex numbers They were first described by Irish

mathematician William Rowan Hamilton in 1843

The quaternions H are equal to , a four-dimensional vector space over the real numbers

4R

Quaternions A quaternion has 4 components

Of the 4 components one is ‘real’ scalar number, and the other 3 form a vector in imaginary ijk space

0 1 2 3[ ]q q q q q

0 1 2 3q q iq jq kq

Quaternions Sometimes, they are written as the

combination of a scalar value s and a vector v

Where,q s v

0

1 2 3[ ]

s q

v q q q

Quaternions Algebra The quaternion group has 8

members:

Their product is defined by the equation:

, , , 1i j k

2 2 2 1i j k ijk

Quaternions - Algebra Using the same methods, we can

get to the following:

Quaternion AlgebraBy Euler’s theorem every rotation can be

represented as a rotation around some axis

with angle . In quaternion terms:

Composition of rotations is equivalent to quaternion multiplication.

K̂

1 2 3 42 2ˆ ˆ( , ) (cos( ) sin( ) ) ( , , , )Rot K K

ExampleWe want to represent a rotation around

x-axis by 90 , and then around z-axis by 90 :

31 1

2 2 2

(cos(45 ) sin(45 ) )(cos(45 ) sin(45 ) )( )( ) cos(60 )

3( ) ,120

3

o o o o

o

o

k ii j ki j k

i j kRot

Rotating with quaternionsWe can describe a rotation of a given

vector v around a unit vector u by angle :

this action is called conjugation.

* Pay attention to the inverse of q (like in complex numbers) !

Rotating with quaternionsThe rotation matrix corresponding to a rotation by the unit quaternion z = a + bi + cj + dk (with |z| = 1) is given by:

Its also possible to calculate the quaternion from rotation matrix:Look at Craig (chapter 2 p.50 )

Rotation Example If we want to do a rotation by x,

y ,z :

This is equal to:

( ) ( ) ( )z y xR R R

[cos( / 2) sin( / 2)][cos( / 2) sin( / 2)][cos( / 2) sin( / 2)]k j i

Denavit-Hartenberg Specialized description of

articulated figures Each joint has only one degree

of freedom rotate around its z-axis translate along its z-axis

What’s so interesting about 6 DOF ?

Denavit-Hartenberg1.Compute the link vector ai and the link length

2.Attach coordinate frames to the joint axes

3.Compute the link twist αi

4.Compute the link offset di

5.Compute the joint angle φi

6.Compute the transformation (i-1)Ti which transforms entities from linki to linki-1

Denavit-HartenbergThis transformation is done in several steps :

ixixizizi

i iiiiRaTdTRT 1

• Rotate the link twist angle αi around the axis xi

• Translate the link length ai along the axis xi

• Translate the link offset di along the axis zi

• Rotate the joint angle φi around the axis zi

17

Denavit-Hartenberg

10000cossin00sincos00001

ii

iiixi

R

100001000010

001 i

ix

a

aTi

1000100

00100001

iiz d

dTi

1000010000cossin00sincos

ii

ii

iziR

ixixizizi

i iiiiRaTdTRT 1

18

Denavit-Hartenberg

Multiplying the matrices :

In DH only φ and d are allowed to change.

1000cossin0

sinsincoscoscossincossinsincossincos

1

iii

iiiiiii

iiiiiii

ii d

aa

T

ixixizizi

i iiiiRaTdTRT 1

19

Denavit-Hartenberg Video

Example 1

Joint i ai i di i

1 a0 0 0 0

2 a1 -90 0 1

3 0 0 d2 2

D-H Link Parameter Table

: rotation angle from Xi-1 to Xi about Zi-1 i : distance from origin of (i-1) coordinate to intersection of Zi-1 & Xi along Zi-1

: distance from intersection of Zi-1 & Xi to origin of i coordinate along Xi

id

: rotation angle from Zi-1 to Zi about Xi

iai

a0 a1

Z0

X0

Y0

Z3

X2

Y1

X1

Y2

d2

Z1

X33O

2O1O0O

Z2

Joint 1Joint 2

Joint 3

http://opencourses.emu.edu.tr/file.php/32/lecture%20notes/Denavit-Hartenberg%20Convention.ppt

Example 1

1d

Joint i ai i di i

1 0 0 d1 0

2 0 -90 d2 0

3 0 0 d3 0

: rotation angle from Xi-1 to Xi about Zi-1 i : distance from origin of (i-1) coordinate to intersection of Zi-1 & Xi along Zi-1

: distance from intersection of Zi-1 & Xi to origin of i coordinate along Xi

id

: rotation angle from Zi-1 to Zi about Xi

iai