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Robotics: Ertuğrul Çetinsoy

The document discusses coordinate transformations in robotics. It explains that coordinate frames are used to describe the position and orientation of objects in space. The rotation matrix represents the orientation of a new coordinate frame with respect to a fixed reference frame. It also discusses properties of rotation matrices including combinations of rotations about different axes and representations using Euler angles and homogeneous transformations.

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Bekir Açıkça
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52 views17 pages

Robotics: Ertuğrul Çetinsoy

The document discusses coordinate transformations in robotics. It explains that coordinate frames are used to describe the position and orientation of objects in space. The rotation matrix represents the orientation of a new coordinate frame with respect to a fixed reference frame. It also discusses properties of rotation matrices including combinations of rotations about different axes and representations using Euler angles and homogeneous transformations.

Uploaded by

Bekir Açıkça
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Robotics

Ertuğrul Çetinsoy
Mechatronics Engineering Dept.

ertugrul.cetinsoy@marmara.edu.tr
Coordinate Transformations:

Inertial frame
Attention!

Similarly:

Attention!
Always ask in which coordinate frame
There is no standard of notation in the literature
p0i0=p1i0 p0 = p1 = p All expressed in ox0y0z0 coordinate frame

p0 and p1: component sets in different coordinate


frames

ox0y0z0 ox1y1z1
Ex:

The rotation matrix


i0 k1 =0 represents a coordinate
j0 k1 =0 transformation.
k0 k1=1

i0 j1 =-sinθ
j0 j1 =cosθ
k0 j1=0

i0 i1 =cosθ
j0 i1 =sinθ
k0 i1=0

Rotation around z0 axis


Orthogonal matrix
Determinant = 1
Right handed coordinate systems
Rotation matrix
Basic rotation matrices:
From previous example:
Properties of


Definition
Ex.
The rotation matrix represents the orientation of the new frame.
Ex:

The rotation matrix rotates a vector


in a fixed coordinate frame.
Combinations of rotations:

1st coordinate transformation

2nd coordinate transformation

Combined coordinate transformation

Formula of the combined coordinate transformation


Ex:

Rotation about y0, fixed frame y0 Rotation about z1, not about z0

Note the order of rotation matrices!!!

Order of multiplication differs the result!!!


Ex: What if all rotations are about fixed axes?

Rotation about y0 followed by rotation about z0:

Wrong Correct

Order of rotation matrices reversed!!!

Rotation about fixed axes:

Rotation about current axes:


Derivation is based on
Rotation about an arbitrary axis:
rotations about fixed axes.
Euler Angles Representation

Rotations about the current axes


Roll, Pitch, Yaw Angles Representation

Rotations about the fixed axes


Homogenous Transformations:

• 4x4 matrices giving us orientation and position of coordinate frames wrt. each
other

In order to be multiplied with this matrix,


the vector must also have 4 rows!

Converts the representation of tip point


in ox1y1z1 coordinate frame into its
representation in ox0y0z0 coordinate
frame
N
The tip position and orientation of the robot arm are expressed with H 0 matrix
Skew Symmetric Matrices:

ST+S=0 S skew symmetric matrix sij+sji=0

(ax, ay, az) can be expressed with 3 parameters

For a and b vectors S(a) b=a x b

S(αa+βb)= α S(a)+β S(b)

For rotation matrix R R(axb)=(Ra)x(Rb)

dRq ,
 S (q) Rq ,
d
dRq , d dRq , dRq ,
   S (q) Rq ,  S ( ) Rq ,
dt dt d d

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