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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. - PowerPoint PPT Presentation
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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
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