2023 updates to orientation chapter. #129
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Vectors have a magnitude, direction, and sense (:math: |
moorepants
changed the title
I use "direction" to tell us what line in space the vector is parallel to and "sense" the positive or negative sense along that direction. I show it a bit more in the vectors chapter. |
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@wwolfie this is ready to be checked if you have time. Sorry that I'm late getting this updated. I'll try to do it earlier in the week next week. |
This wording seems to imply that vectors can have a displacement. So I don't think I'll add this. Maybe "even if vectors are drawn with an apparent parallel displacement they are the same". I show this in the figure though, maybe it just isn't as explicit as it could be. |
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I admit I did not look at the picture. Your wording -if the picture does not say it- of course is much better. Sense: clear now, thanks! |
| Write :math:`{}^N\mathbf{C}^A` for simple rotations about both the shared | ||
| :math:`\hat{n}_x` and :math:`\hat{a}_x` and shared :math:`\hat{n}_y` and | ||
| :math:`\hat{a}_y` axes, rotating :math:`A` with respect to :math:`N` through | ||
| angle :math:`\theta`. |
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The direction of the angle is never specified. Maybe the default direction is best explained at the spot where also the "right-handedness" of the frames is specified?
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| \begin{bmatrix} | ||
| 1 & 0 & 0 \\ | ||
| 0 & \cos{\theta} & \sin{\theta} \\ |
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It's early morning, but I have convinced myself that this is the inverse of the correct solution
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In [1]: import sympy as sm
In [2]: import sympy.physics.mechanics as me
In [3]: theta = sm.symbols('theta')
In [4]: A = me.ReferenceFrame('A')
In [5]: B = me.ReferenceFrame('B')
In [6]: B.orient_axis(A, theta, A.x); B.dcm(A)
Out[6]:
Matrix([
[1, 0, 0],
[0, cos(theta), sin(theta)],
[0, -sin(theta), cos(theta)]])
In [7]: B.orient_axis(A, theta, A.y); B.dcm(A)
Out[7]:
Matrix([
[cos(theta), 0, -sin(theta)],
[ 0, 1, 0],
[sin(theta), 0, cos(theta)]])
In [8]: B.orient_axis(A, theta, A.z); B.dcm(A)
Out[8]:
Matrix([
[ cos(theta), sin(theta), 0],
[-sin(theta), cos(theta), 0],
[ 0, 0, 1]])
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I think it is correct, at least per my definition of the rotation matrix notation.
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Yes, it is a change in notation that tripped me up
| Similarly to the simple example above, we can write these equations: | ||
| Similar to the simple example above, we can write these equations if the | ||
| :math:`\alpha_y` and :math:`\alpha_z` angles relate the :math:`\hat{a}_y` and | ||
| :math:`\hat{a}_z` unit vectors to those of :math:`N`: |
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Easiest to add this comment here rather than make new PR:
In the sentence "Since we are working with mutually perpendicular unit vectors ...", 'mutually perpendicular' and the second 'unit' are not needed, which might confuse readers who do not know the dot product <-> cosine rule by heart.
orientation.rst
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| successive simple orientations to go from :math:`A` to :math:`D`. We can | ||
| formulate the direction cosine matrices for the reference frames using the same | ||
| technique for the successive simple orientations shown in :ref:`Successive | ||
| Orientations`, but now we will have three dimensional orientation between |
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A bit unclear, maybe like this:
| Orientations`, but now we will have three dimensional orientation between | |
| Orientations`, but now we will have a sequence of three rotations between |
orientation.rst
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| formulate the direction cosine matrices for the reference frames using the same | ||
| technique for the successive simple orientations shown in :ref:`Successive | ||
| Orientations`, but now we will have three dimensional orientation between | ||
| :math:`A` and :math:`D` allowing :math:`D` to be oriented in any direction |
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Not sure that it is obvious that three is the right number, so would split:
| :math:`A` and :math:`D` allowing :math:`D` to be oriented in any direction | |
| :math:`A` and :math:`D`. With three rotations, :math:`D` can be oriented in any direction |
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I tried to reword this:
successive simple orientations to go from :math:`A` to :math:`D`. We can
formulate the direction cosine matrices for the reference frames using the same
technique for the successive simple orientations shown in :ref:`Successive
Orientations`, but now our sequence of three orientations will enable us to
orient :math:`D` in any way possible relative to :math:`A` in three dimensional
space.
orientation.rst
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| to the handgrip frame. These successive :math:`z\textrm{-}x\textrm{-}y` | ||
| orientations are a standard way of describing the orientation of two reference | ||
| frames and are often referred to as `Euler Angles`_ [#]_. | ||
| With these three successive orientations the camera can be oriented arbitrarily |
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Seems slightly redundant. Perhaps rotation is a better word here?
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I had a mixture of "rotation" and "orientation" last year when I first wrote this, but ended up changing everything to "orientation" to be consistent and to match the SymPy method names.
Co-authored-by: wwolfie <[email protected]>
Co-authored-by: wwolfie <[email protected]>
Co-authored-by: wwolfie <[email protected]>
Co-authored-by: wwolfie <[email protected]>
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Would this not be called an orthonormal matrix, if A * A.T = I and det(A) = 1 ? |
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Seems like they are the same thing here: https://en.wikipedia.org/wiki/Orthogonal_matrix |
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Thanks for the feedback here. Merging. |
Updates: