From 05b13931ac6fc1b9bbf49fd43823c083d6e6ed5d Mon Sep 17 00:00:00 2001
From: "E. Joshua Rigler" <erigler@usgs.gov>
Date: Wed, 28 Aug 2019 10:54:13 -0600
Subject: [PATCH] Minor update to Notebook
Notebooks "change" just by opening them, which is unfortunate. Nothing substantial
actually changed, but I'm committing anyway, just to be safe.
---
docs/algorithms/Adjusted/AdjustedValidate.ipynb | 10 ++++------
1 file changed, 4 insertions(+), 6 deletions(-)
diff --git a/docs/algorithms/Adjusted/AdjustedValidate.ipynb b/docs/algorithms/Adjusted/AdjustedValidate.ipynb
index a536e1b0c..202511093 100644
--- a/docs/algorithms/Adjusted/AdjustedValidate.ipynb
+++ b/docs/algorithms/Adjusted/AdjustedValidate.ipynb
@@ -177,7 +177,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Note that order of operation is critical, but there is no single standard for composing a full affine transformation from its constituent matrices. The preceding example follows a norm used in the [Python Transforms3D](https://pypi.org/project/transforms3d/) package, that might better be represented algorithmically as $dot(\\textbf{T}, dot(\\textbf{R}, dot(\\textbf{S},\\textbf{H}) ) )$, where \"$dot$\" is a matrix dot product function. In words, a shear correction ($\\textbf{H}$ is first applied to the original $HEZ$ vector to orthogonalize it. This is followed by rescaling the orthogonalized $HEZ$ vectors with $\\textbf{S}$, which is followed by a rigid rotation $\\textbf{R}$ to align all axes with the desired $XYZ$ coordinate frame. Finally, the origin of the now rotated, scaled, and orthogonal vector is translated to make the coordinate frame origins coincide using $\\textbf{T}$.\n",
+ "Note that order of operation is critical, but there is no single standard for composing a full affine transformation from its constituent matrices. The preceding example follows a norm used in the [Python Transforms3D](https://pypi.org/project/transforms3d/) package, that might better be represented algorithmically as $dot(\\textbf{T}, dot(\\textbf{R}, dot(\\textbf{S},\\textbf{H}) ) )$, where \"$dot$\" is a matrix dot product function. In words, a shear correction ($\\textbf{H}$) is first applied to the original $HEZ$ vector to orthogonalize it. This is followed by rescaling the orthogonalized $HEZ$ vectors with $\\textbf{S}$, which is followed by a rigid rotation $\\textbf{R}$ to align all axes with the desired $XYZ$ coordinate frame. Finally, the origin of the now rotated, scaled, and orthogonal vector is translated to make the coordinate frame origins coincide using $\\textbf{T}$.\n",
"\n",
"Looking more closely at $\\textbf{R} = dot(\\textbf{R}_x, dot(\\textbf{R}_y,\\textbf{R}_z))$, it should be even more evident that order of operation is important. The rotation angles ($\\theta_x$, $\\theta_y$, and $\\theta_z$) that combine to form $\\textbf{R}$ are presented here in a manner that resembles typical yaw-pitch-roll rotations. This is 1 of 6 possible so-called Tait-Bryan rotation sequences (re-orientation by rotating around each unique axis once). Proper Euler angles have 6 additional possible sequences that involve a rotation about one axis, then a second, then again about the first. If that were not enough, both the Tait-Bryan and proper Euler angles can be with respect to fixed axes (extrinsic; as above), or with respect to the new axes after each rotation (intrinsic). This leads to 24 possible valid rotations! It is highly recommended that one convention is chosen and used consistently.\n",
"\n",
@@ -429,7 +429,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "...where $T_X$ and $T_Y$ are $H_{base}$ rotated rigidly into the absolute $XY$ frame through $D_{base}$, or as presented here, $\\theta_z$. But such a matrix cannot be solved for directly from the measurement matrices via least-squares. All is not lost, however. $\\theta_z$ can be found by invoking singular value decomposition (SVD) to obtain the eigenvectors that define an orthonormal rotation matrix between two vector spaces."
+ "...where $T_X$ and $T_Y$ are NOT independent offsets, but are geometrically linked to each other via $\\theta_z$ (that is, as $\\theta_z$ approaches 0, $T_Y$ approaches 0, and $T_X$ approaches $H_{base}$). But such a matrix cannot be solved for directly from the measurement matrices using least-squares. All is not lost, however. $\\theta_z$ can be found by invoking singular value decomposition (SVD) to obtain the eigenvectors that define an orthonormal rotation matrix between two vector spaces."
]
},
{
@@ -502,7 +502,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
- "Notice that there are now four degrees of freedom ($\\theta_z$, $T_X$, $T_Y$, and $T_Z$) instead of the three traditional (quasi)definitive baselines. This is because $T_X$ and $T_Y$ are allowed to float, rather than be trigonometrically locked to one another via $\\theta_z$. We might effectively do this with an affine transformation, and obtain a near perfect analog to (quasi)definitive processing, but one of the main problems with the traditional approach presented above is that it requires that the fluxgate sensor be aligned with the local magnetic meridian for the math/geometry to work out. Using affine transformations, this is no longer necessary. Of course by increasing the degrees of freedom from 3 to 4, the uncertainty of our final solution is slightly higher; but then the uncertainty *should be* higher, since a potential offset in the variometer's $E$ axis is now included in the calculations."
+ "Notice that there are now four free parameters ($\\theta_z$, $T_X$, $T_Y$, and $T_Z$) instead of just the three traditional (quasi)definitive baselines. This is because, unlike before, $T_X$ and $T_Y$ are independent of each other. Of course by increasing the degrees of freedom from 3 to 4, the uncertainty of our final solution is somewhat higher. However the uncertainty *should be* higher, since a potential offset in the variometer's $E$ axis is now permitted in the calculations."
]
},
{
@@ -1482,9 +1482,7 @@
"cell_type": "code",
"execution_count": 9,
"metadata": {
- "code_folding": [
- 0
- ]
+ "code_folding": []
},
"outputs": [],
"source": [
--
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