Rotations and Translations. Representing a Point 3D A tri-dimensional point A is a reference...
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Transcript of Rotations and Translations. Representing a Point 3D A tri-dimensional point A is a reference...
Representing a Point 3D A tri-dimensional point
A is a reference coordinate system
here
z
y
xA
p
p
p
P
Representing a Point 3D (cont.)
Once a coordinate system is fixed, we can locate any point in the universe with a 3x1 position vector.
The components of P in {A} have numerical values which indicate distances along the axes of {A}.
To describe the orientation of a body we will attach a coordinate system to the body and then give a description of this coordinate system relative to the reference system.
Example
0
2
1
PB ?PA
30
134.05.02866.0)30sin()30cos( 00 YB
XB
XA PPP
232.2YAP
000.0
232.2
134.0
0
2
1
*
000.1000.0000.0
000.0866.0500.0
000.0500.0866.0
PA
000.0ZAP
AX
BX
AYBY
PB
30
Description of Orientation
ˆˆˆˆˆˆ
ˆˆ ˆˆ ˆˆ
ˆˆ ˆˆ ˆˆ
ˆˆˆ
ABABAB
ABABAB
ABABAB
BA
BA
BAA
B
ZZZYZX
YZYYYX
XZXYXX
ZYXR
BX
AX
BX
AYBY
is a unit vector in B
is a coordinate of a unit vector of B in coordinates system A (i.e. the projection of onto the unit direction of its reference)
BA X
BX
Example Rotating B relative to A around Z by30
000.1000.0000.0
000.0866.0500.0
000.0500.0866.0
RAB
AX
BX
AYBY
What is a Frame ? A set of four vectors giving position and
orientation information. The description of the frame can be
thought as a position vector and a rotation matrix.
Frame is a coordinate system, where in addition to the orientation we give a position vector which locates its origin relative to some other embedding frame.
Arrows Convention An Arrow - represents a vector drawn
from one origin to another which shows the position of the origin at the head of the arrow in terms of the frame at the tail of the arrow. The direction of this locating arrow tells us that {B} is known relative to {A} and not vice versa.
Mapping a vector from one frame to another – the quantity itself is not changed, only its description is changed.
Rotating a frame B relative to a frame A about Z axis by degrees and moving it 10 units in direction of X and 5 units in the direction of Y. What will be the coordinates of a point in frame A if in frame B the point is : [3, 7, 0]T?
Example
30
000.0
562.12
098.9
000.0
000.5
000.10
0.000
7.562
0.902-
000.0
000.5
000.10
000.0
000.7
000.3
000.1000.0000.0
000.0866.0500.0
000.0500.0866.0
PA
Extension to 4x4
110001
PPRP BBORG
AAB
A
We can define a 4x4 matrix operator and use a 4x1 position vector
Example
If we use the above example we can see that:
1000
000.0000.1000.0000.0
000.5000.0866.0500.0
000.10000.0500.0866.0
TAB
P in the coordinate system A
1
000.0
562.12
098.9
1
000.0
000.7
000.3
1000
000.0000.1000.0000.0
000.5000.0866.0500.0
000.10000.0500.0866.0
PA
Example
000.1000.0000.0000.0
000.0000.1000.0000.0
000.0000.0000.1000.0
000.3000.0000.0000.1
TAB
000.1000.0000.0000.0
000.2000.0000.0000.1
000.0500.0866.0000.0
000.0866.0500.0000.0
TBC