Here's a new interactive tutorial on Euler angles:
http://www.mecademic.com/ressources/Eul…es/Euler-angles
Interactive tutorial on Euler angles
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ilian.bonev -
July 13, 2017 at 7:08 PM -
Thread is marked as Resolved.
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Here's a new interactive tutorial on Euler angles:
http://www.mecademic.com/ressources/Eul…es/Euler-anglesIs this right? The Staubli of my location tells me his robot is XYZEuler angle.
In addition, I want to know where is different between fixed XYZEuler and mobile ZYXEuler.I am very fuzzy.
BR!
Jonson -
You can either rotate with respect to the original (fixed) frame or the moved (movil frame)
Now, as convention you stick to rotate ALLWAYS with respect to the fixed (reference) frame or the movil frame.
When you rotate with respect to the movil frame you post-multiply the transformations
Whe you rotate with respect to the fixed frame you pre- multiply the transformations (rotations)
So if you rotate first in X, then in Y, finally in Z with respect to the movil frame the rotation is XYZ
If you rotate in X then in Y then in Z with respect to the fixed frame the compound rotation is ZYXa compound transformation XYZ can be seen either as a X then Y then Z rotation on the movil frame
or as a Z then Y then X rotation with respect to the fixed frame. -
An Euler angle is simply a means of representing the orientation of a point in space, relative to another point in space, in 3 dimensions.
The way an Euler works is that you rotate around each axis in sequence. This means that you rotate around one axis, then the second, then the third. The tricky bit is that, when you rotate around one axes, the other two move with it -- this is a solid body.
There are also many valid Euler sequences: XYZ, ZYZ, XYX, ZYX, etc -- they are all mathematically valid, but different brands of robots use different sequences. I'm going to assume an XYZ sequence for this example.
The classical example is an airplane, where the axes are titled "Roll, Pitch, and Yaw" -- X, Y, and Z for most of us robot types. Roll is the nose-tail axis, Pitch is the axis through the wings, and Yaw is the vertical axis.
Now, let's say that the World coordinate system has Z pointing straight up (gravity), X pointing North, and Y pointing West (right-hand rule). And we start with the airplane aligned perfectly -- heading due North, wings straight and level. This would have a Roll/Pich/Yaw Euler representation of 0,0,0, relative to World -- all 3 axes of both coordinate systems are aligned.
Now, imagine the pilot banks the aircraft 10deg to port -- this is an Euler of -10,0,0. But while the aircraft's X axis is still parallel with the World's X axis, the Y and Z axes are no longer parallel. This is where things get a bit tricky, because if the pilot pulls back on the stick and begins pitching the aircraft into the turn, the aircraft rotates around it's "new," tilted Y axis, rather than the World Y axis. And if the pilot then starts using the rudder, the aircraft rotates around a Z axis that has already been "tilted" twice -- first by the bank, then by the pitch maneuver.
Of course, in the real world, pilots perform all three rotations at once as a matter of course. But as an analogy to Euler angles, one would imagine each rotation taking place one at a time. So, first the "bank", rotating around the X axis. Then, some "pitch," rotating around the Y axis after the Y axis has been affected by the X axis rotation. Then, some "yaw," rotating the aircraft around its own Z axis, but only after that Z axis has been affected by both the prior X and Y rotations.
To put it another way: arrange your hand like this:
Then, rotate your hand 45deg around your index finger. Now rotate 45deg around your middle finger. Then 45deg around your thumb. You can see how each rotation changes the axis of the next rotation. That is why the sequence of an Euler angle is critical. A Fanuc uses XYZ, while KUKAs use ZYX -- putting both robots at identical orientations in space would require completely different Rx, Ry, and Rz values, because they use different sequences.
This is why people who do this for a living resort to matrix algebra. For every Euler sequence, there is a method for converting it into a 3x3 matrix. While the Euler angles for a Fanuc and a KUKA might be very different for the same point in space, the matrix equivalent for both robots would be identical. So a piece of software built to convert coordinates from one Euler sequence into another works by converting from Euler to matrix, then converting from the matrix to the other Euler sequence. Likewise, when performing math on Euler angles, one generally converts to matrix notation, performs the math, then converts the resulting matrix back into the desired Euler sequence.
Most robot programming languages contain utilities to do this math natively: pose multiplication, frame shift, geometric operator, etc -- every language uses its own term.
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Yes is correct, the algorithm is very well known .....
For your Staübli robot pick X, Y, Z form left to right
since Staublies use rotations in the movil frame double pick on either letter to switch to movil frame
It is very small but you will see: X' Y' Z'The compound transformation SHOULD read Rx()Ry()Rz()=Rotation Matrix
Now, just type the angle for each rotation or drag the bar using the mouse to play with it
There is an for App version called eulerangles (PASCAL BIGRAS, ILIAN BONEV Thank you! )
The app gives two additional results: the equivalent quaternion and the axis, angle rotationVERY USEFULL
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I got it!!!
Thank you SkyeFire and robotero!
BR!
Jonson