From 3c5ce19729901d994415817394d0b8972a040af6 Mon Sep 17 00:00:00 2001 From: Chris Padwick Date: Wed, 15 Jan 2025 07:47:20 -0800 Subject: [PATCH 1/2] initial commit of latex rotations doc --- doc/rotations.tex | 467 ++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 467 insertions(+) create mode 100644 doc/rotations.tex diff --git a/doc/rotations.tex b/doc/rotations.tex new file mode 100644 index 0000000..137362d --- /dev/null +++ b/doc/rotations.tex @@ -0,0 +1,467 @@ +\documentclass[11pt]{article} + +%----------------------------------------------------------------------- +% PACKAGES +%----------------------------------------------------------------------- +\usepackage[utf8]{inputenc} % Encodes input in UTF-8 +\usepackage[T1]{fontenc} % Better handling of diacritics +\usepackage{lmodern} % Latin modern font +\usepackage{amsmath,amssymb} % Math symbols and environments +\usepackage{amsthm} % Theorem-like environments +\usepackage{graphicx} % For including figures +\usepackage{hyperref} % Hyperlinks in PDF +\usepackage{caption} % Better captions for figures/tables +\usepackage{cite} % Better citation handling +\usepackage{geometry} % Adjust page margins +\geometry{margin=1in} + +%----------------------------------------------------------------------- +% TITLE, AUTHOR, AND DATE +%----------------------------------------------------------------------- +\title{\textbf{A Gentle Introduction to Coordinate Frame Conversions}} +\author{C. Padwick} +\date{\today} + +%----------------------------------------------------------------------- +% CUSTOM COMMANDS & ENVIRONMENTS (OPTIONAL) +%----------------------------------------------------------------------- +\newtheorem{theorem}{Theorem}[section] +\newtheorem{lemma}{Lemma}[section] +\newtheorem{proposition}{Proposition}[section] +\newtheorem{definition}{Definition}[section] + +% Example macro to quickly typeset a 2D rotation matrix +\newcommand{\RtwoD}[1]{% +\begin{pmatrix} +\cos #1 & -\sin #1 \\ +\sin #1 & \cos #1 +\end{pmatrix}% +} + +% Example macro to quickly typeset a 3D rotation matrix around the z-axis +\newcommand{\RthreeD}[1]{% +\begin{pmatrix} +\cos #1 & -\sin #1 & 0 \\ +\sin #1 & \cos #1 & 0 \\ +0 & 0 & 1 +\end{pmatrix}% +} + +%----------------------------------------------------------------------- +\begin{document} + +% MAKE TITLE AND TABLE OF CONTENTS +\maketitle +\tableofcontents +\clearpage + +%----------------------------------------------------------------------- +% ABSTRACT +%----------------------------------------------------------------------- +\begin{abstract} +Imagine you have a robot with a camera mounted on it. The camera measures something of interest, +like the distance to the nearest chair or the distance to the nearest wall, and you +want the robot to navigate to that object. Since it is usually impossible to locate the +camera exactly at the origin of the robot, it is necessary to convert the distance from the +camera to the object into a distance from the robot. This is called a coordinate frame conversion. +Generally speaking this coordinate frame conversion consists of rotation and translation from the +camera frame to the robot frame. This document is a short introduction to Euclidean coordinate frame conversions. +This document provides a gentle introduction to coordinate transformations (rotations and translations) +with the intent of giving the reader a solid understanding of the fundamentals without needing +an advanced mathematics or engineering course in linear algebra. +\end{abstract} + +%----------------------------------------------------------------------- +% 1. INTRODUCTION +%----------------------------------------------------------------------- +\section{Introduction} +\label{sec:intro} + +Image this: you are building a robot for a First Robotics competition and your vision mentor says "we need +to correct the apriltag vector detected by the vision system to align with the robot's coordinate frame." +If you don't know what this means, then you are reading the right document! + +You don't need a familiarity with robotics to read this document. The same concepts will +apply to any device which takes measurements in a given coordinate frame and those measurements +need to be converted to a different coordinate frame. Rotation matrices are essential in many +areas of mathematics, engineering, and computer science. In this doc I'm going to go over +some basics and give a gentle introduction so you can get the working knowledge you need and +move on to more advanced topics later. + +%----------------------------------------------------------------------- +% 2. Coordinate Frames and Vectors +%----------------------------------------------------------------------- +\section{Coordinate Frames and Vectors} +\label{sec:coordinateframes} + +A coordinate frame is a set of vectors, or axes, which are orthogonal to each other. Orthogonal +means the angle between any two axes is 90 degrees. A simple 2D coordinate frame is shown in figure \ref{fig:cf1}. + +\begin{figure}[h!] + \centering + \includegraphics{figures/cf1.png} + \caption{An 2D coordinate frame defined by X and Y.} + \label{fig:cf1} +\end{figure} + +The coordinate frame is useful because we can locate points in the coordinate frame by their X +and Y coordinates. For example, the point V in figure \ref{fig:cf1} has an x and a y coordinate +that can be determined by its position on the coordinate frame. In the figure these components +on the X and Y axes are shown by the dashed lines. + +In terms of expressing the position of a point in the coordinate frame, we can write a vector as follows: + +\begin{equation} +\vec{v} = +\begin{bmatrix} +v_x \\ +v_y +\end{bmatrix}. +\end{equation} + +where \(v_x\) is the x-component of the vector and \(v_y\) is the y-component of the vector. You can +compute the length of the vector easily too by taking the square root of the sum of the +squares of the components: + +\begin{equation} +|\vec{v}| = \sqrt{v_x^2 + v_y^2}. +\end{equation} + +The extension to 3-dimensions or even N-dimensions is straight forward - you just add one more +component for each dimension. + + +%----------------------------------------------------------------------- +% 3. 2D ROTATION MATRICES +%----------------------------------------------------------------------- +\section{2D Rotation Matrices} +\label{sec:2d-rotation} + +We’re going to work out some simple rotations in 2D so we can get some experience with +rotation matrices and how they work. This can all be extended to 3D but it's easier to +visualize in 2D and nicer to start with. +\noindent +Let’s draw an X-Y axis in 2D and a rotated version of the axis, A-B, shown in figure \ref{fig:2d1}. +The rotation angle is $\theta$. The rotation angle is positive, and is measured +counterclockwise. For simplicity let’s set the length of A and B both to 1, +.e.g \begin{equation} |A| = |B| = 1\end{equation}. + +\begin{figure}[h!] + \centering + \includegraphics{figures/2d1.png} + \caption{An 2D coordinate frame defined by X and Y, and a rotated version of the axis denoted by A-B.} + \label{fig:2d1} +\end{figure} + +Let's ask the following question: How do we get the coordinates of the points A and B +in the X-Y frame? We are given the angle of rotation $\theta$. We need to compute +the X-Y coordinates of each point A and B in the X-Y frame. + +\noindent + +Let's start with the point A. From trigonometry we can figure out the X-Y coordinates +of A by noticing $cos(\theta) = x / |A|$ or $x = cos(\theta) * |A|$. Since we said +that $|A| = 1$, we can just write $x = cos(\theta)$. Similarly, we can figure out +the y coordinate of A by noticing that $sin(\theta) = y / |A|$ or $y = sin(\theta) * |A|$, +and we can just write $y = sin(\theta)$. This gives us the X-Y +coordinates of A as: +\begin{equation} + \vec{A} = + \begin{bmatrix} + cos(\theta) \\ + sin(\theta) + \end{bmatrix}. +\end{equation} + +We can do the same for the point B. We can see that the X coordinate of B is going to be +negative since it interects with the negative X axis. We can do the same thing we did +for the A coordinate, except this time we see that $sin(\theta) = -x / |B|$ or +$x = -sin(\theta) * |B|$ => $x = -sin(\theta)$. Take a moment and study the +diagram and convince yourself that this is true. The y-coordinate of B is going to be +$cos(\theta) = y / |B|$ or $y = cos(\theta) * |B|$ => $y = cos(\theta)$. Pause for a +few moments and study the diagram and convince yourself that this is true. + +Writing out the coordinates B, we get: +\begin{equation} + \vec{B} = + \begin{bmatrix} + -sin(\theta) \\ + cos(\theta) + \end{bmatrix}. +\end{equation} + +If you wanted to express these points as vectors in the X-Y frame, you could write them +as: + +\begin{equation} + \vec{A} = sin(\theta) \hat{x} + cos(\theta) \hat{y} +\end{equation} + +\begin{equation} + \vec{B} = -sin(\theta) \hat{x} + cos(\theta) \hat{y} +\end{equation} + +The $\hat{x}$ and $\hat{y}$ are unit vectors in the X-Y frame. They are aligned with the +X and Y axes respectively but have unit length. + +\subsection{Representing The Rotation As A Matrix} + +Now that we have the coordinates of A and B, we can represent the rotation as a matrix. This +is actually quite simple. We can write the rotation matrix as: +\begin{equation} + R(\theta) = + \begin{bmatrix} + cos(\theta) & -sin(\theta) \\ + sin(\theta) & cos(\theta) + \end{bmatrix}. +\end{equation} + +If you look at the entries of the matrix, you can recognize that we have taken the equation +for $\vec{A}$ and put the X-Y coordinates into the first column. The X coordinate goes in +the first row and the Y coordinate goes in the second row. For the second column, we have +taken the equation for $\vec{B}$ and put the X-Y coordinates into the second column. +The X coordinate goes in the first row and the Y coordinate goes in the second row. + +You might ask, why would we do this? Why not just use the equations for $\vec{A}$ and $\vec{B}$ above? +It is customary to write something like "it turns out that we can use the rotation matrix more +easily and more efficiently". It turns out that is true! + +\subsection{Primer on Matrix/Vector Multiplication} + +Ok, now let’s multiply a vector by a matrix. In case you haven’t done this before, here is +a quick primer on the rules of matrix multiplication: + +\begin{itemize} + \item When you multiply a vector by a matrix, the vector has to have the same number +of rows as the matrix has columns. In this case we’ll express the vector as a column vector +with 2 rows and 1 column. The matrix has 2 rows and 2 columns so we can multiply it by a 2x1 vector. + + \item When we multiply a vector by a matrix, as long as the vector has the same number of rows + as the matrix has columns, we get a vector of the same size as the input vector. Since our vector + has 2 elements, the result will have two elements as well. + + \item When we’re multiplying a matrix and a vector, we start at the first row of the matrix + and multiply the column elements of the matrix by the rows of the vector and add them. We put that + result in the first element of the result vector. We then move on to the second row of the matrix + and multiply the column elements of the second row of the matrix by the rows of the vector and add them. We put that + result in the second element of the result vector. We continue this process until we have computed + all the elements of the result vector. In this case, the result vector will have 2 elements since + we're only working with 2D rotation matrices. +\end{itemize} + +Here is an example: +\begin{equation} +\begin{bmatrix} a_1 & a_2 \\ +a_3 & a_4 \end{bmatrix} * \begin{bmatrix} v_1 \\v_2 \end{bmatrix} = \begin{bmatrix} a_1 v_1 + a_2 v_2 \\ +a_3 v_1 + a_4 v_2 \end{bmatrix} +\end{equation} + +\subsection{Applying The Rotation Matrix} + +Alright, now that that is out of the way, let’s do a rotation. Let’s say we have a vector in the A-B +frame of reference, and this vector is just the A axis. So the vector coordinates are just (1, 0) +in the A-B frame of reference. So let’s rotate this vector in the X-Y frame of reference: + +\begin{equation} + A_{xy-frame}= + \begin{bmatrix} + cos(\theta) & -sin(\theta) \\ + sin(\theta) & cos(\theta) + \end{bmatrix} + * + \begin{bmatrix} + 1 \\ + 0 + \end{bmatrix} = + \begin{bmatrix} + cos(\theta) + 0 \\ + sin(\theta) + 0 + \end{bmatrix} + = + \begin{bmatrix} + cos(\theta) \\ + sin(\theta) + \end{bmatrix} + \end{equation} + +This is the same answer we got before, so this is good news! +Let’s do it again, but for the B vector in the XY-frame: + +\begin{equation} + B_{xy-frame}= + \begin{bmatrix} + cos(\theta) & -sin(\theta) \\ + sin(\theta) & cos(\theta) + \end{bmatrix} + * + \begin{bmatrix} + 0 \\ + 1 + \end{bmatrix} + = + \begin{bmatrix} + 0 - sin(\theta) \\ + 0 + cos(\theta) + \end{bmatrix} + = + \begin{bmatrix} + -sin(\theta) \\ + cos(\theta) + \end{bmatrix} +\end{equation} + +This is the same result we got before, so this is good news! It means our matrix multiply +approach actually works! + +Ok, but why do we care? What does this buy us? This technique works for any vector in the A-B +frame of reference. So if someone gives us a vector in the A-B frame of reference, we can +multiply it by our 2D rotation matrix to get the vector in the X-Y frame of reference. +This is the first part of learning how to convert from one frame of reference to another! + +%----------------------------------------------------------------------- +% 4. 3D ROTATION MATRICES +%----------------------------------------------------------------------- +\section{Extending to 3D Rotation Matrices} +\label{sec:3d-rotation} + +Now that we have 2D rotation matrices, let’s extend them to 3D. We said before that all we need +to do is add another component for the extra dimension. Our 2D vectors now have 3 components, +and our 2D rotation matrices now have 3 rows and 3 columns, like this: + +\begin{equation} + \vec{v} = \begin{bmatrix} + v_x \\ + v_y \\ + v_z + \end{bmatrix} +\end{equation} + +\begin{equation} + \begin{pmatrix} + a_1 & a_2 & a_3 \\ + a_4 & a_5 & a_6 \\ + a_7 & a_8 & a_9 + \end{pmatrix} +\end{equation} + +\subsection{Rotations About Each Axis} + +It is possible to create a rotation matrix that rotates about the \(x\)-axis, the \(y\)-axis, +or the \(z\)-axis. These matrices are simple to construct, and look very similar to their 2D +counterparts. + +For instance, the rotation matrix around the \(z\)-axis is + +\begin{equation} + R_z(\theta) = + \begin{pmatrix} + cos(\theta) & -sin(\theta) & 0 \\ + sin(\theta) & cos(\theta) & 0 \\ + 0 & 0 & 1 + \end{pmatrix} +\end{equation} + +You can recognize this as just the 2D rotation matrix about X-Y with some extra elements. +If you think about why this is, it actually makes sense. In the 2D example we were +actually rotating about the z-axis, which faced out of the page. When we rotate about +the z-axis the vectors only change in the X-Y plane. So it makes sense that the matrix +would look this way in that the only components that change are the X-Y components. Another +way to think about this is that the third column of the matrix is (0, 0, 1), which also +happens to the be coordinates of the z-axis. Only the X-Y components change when whatever +rotate about the z-axis. + +The rotation matrix around the \(y\)-axis is + +\begin{equation} + R_y(\gamma) = + \begin{pmatrix} + cos(\gamma) & 0 & sin(\gamma) \\ + 0 & 1 & 0 \\ + -sin(\gamma) & 0 & cos(\gamma) \\ + 0 & 0 & 1 + \end{pmatrix} +\end{equation} + +Here I have introduced another angle $\gamma$. Notice that the second column of the matrix +is (0, 1, 0), which also happens to the be coordinates of the y-axis. When you rotate about +the y-axis the vectors only change in the X-Z plane. + +The rotation matrix around the \(x\)-axis is + +\begin{equation} + R_x(\alpha) = + \begin{pmatrix} + 1 & 0 & 0 \\ + 0 & cos(\alpha) & -sin(\alpha) \\ + 0 & sin(\alpha) & cos(\alpha) + \end{pmatrix} +\end{equation} + +Here I have introduced another angle $\alpha$. Notice that the first column of the matrix +is (1, 0, 0), which also happens to the be coordinates of the x-axis. When you rotate +about the x-axis the vectors only change in the Y-Z plane. + +\subsection{Worked Example} + +Let's see if our rotations still work in 3D. Let's take the same coordiate frame +as before in figure \ref{fig:2d1}. Let's imagine that we have a vector Z in the X-Y frame, +and that vector points straight out of the page. We'll call this frame X-Y-Z. The frame +A-B also has a vector C that is aligned with the Z-axis and points straight out of the page. +We'll call this frame A-B-C. So now we rotate A-B-C by angle $\theta$ around the Z-axis in the X-Y-Z frame, +and we want to compute the coordinates of the rotated A-B-C + +\subsection{Rotation About an Arbitrary Axis} +Discuss Rodrigues' rotation formula, quaternions, or Euler angles—whatever is relevant. + +%----------------------------------------------------------------------- +% 5. PROPERTIES OF ROTATION MATRICES +%----------------------------------------------------------------------- +\section{Properties of Rotation Matrices} +\label{sec:properties} + +Mention key properties like: +\begin{itemize} + \item Orthogonality: \( R^\top R = I \) + \item \(\det R = +1\) + \item Composing rotations + \item Relationship to \(\mathrm{SO}(n)\) +\end{itemize} + +%----------------------------------------------------------------------- +% 6. APPLICATIONS +%----------------------------------------------------------------------- +\section{Applications} +\label{sec:applications} + +Discuss practical uses: +\begin{itemize} + \item Robotics + \item Computer graphics + \item Aerospace + \item Image processing +\end{itemize} + +%----------------------------------------------------------------------- +% 7. CONCLUSION +%----------------------------------------------------------------------- +\section{Conclusion} +\label{sec:conclusion} + +Summarize key points. You may also mention limitations and open problems or suggest further reading. + +%----------------------------------------------------------------------- +% REFERENCES +%----------------------------------------------------------------------- +\begin{thebibliography}{99} + +\bibitem{ref:rotation-matrices} +Author Name, \emph{Title of Book or Paper}, Journal/Publisher, Year. + +\bibitem{ref:so3} +Another Author, ``Interesting Insights Into 3D Rotations,'' Conference Proceedings, Year. + +% Add more references as needed... + +\end{thebibliography} + +\end{document} From 64d5966efee811096d12ca297d3d79cd41ad4dfb Mon Sep 17 00:00:00 2001 From: Chris Padwick Date: Wed, 15 Jan 2025 16:03:40 -0800 Subject: [PATCH 2/2] add figures and update document --- doc/figures/2d1.png | Bin 0 -> 8464 bytes doc/figures/3d1.png | Bin 0 -> 11455 bytes doc/figures/cf1.png | Bin 0 -> 4322 bytes doc/figures/pinhole_camera_model.png | Bin 0 -> 17391 bytes doc/figures/robot.png | Bin 0 -> 17830 bytes 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z&8@F@mK8h;+-*&HV0g=0i=rty=@EUq5-m>+>-zaC6x`y6SfD>B+KL_~^sVFbebf2w zsBMMg{rbWA+Zye+YkIUWp*>|0u}pB6)O;cc?OVE7gI{2xhuWlj{$2on+h8RyxE^s2 zY%$FrsO-`g+~(92&D(E)9K&Oz_@zQIw+^<+fO*iw{*nbrs5Y2DXOq1w2m=2Y-(r`3Q%df2nHvu{Q+KUneon zo*o$zD+XHGqO9Wp{jpj}!tX38uUrFY7HNwj$>F(qPzp8|ddC1N#q{U|3%Q~vfg*U$qJ_yzb zi1$KotfpaGl-n8(0P@>8lIz*zJrq+$mzl{va$o1+2F{?gk?qG{vSC0s+j-) literal 0 HcmV?d00001 diff --git a/doc/rotations.tex b/doc/rotations.tex index 137362d..80f7ffc 100644 --- a/doc/rotations.tex +++ b/doc/rotations.tex @@ -377,8 +377,7 @@ \subsection{Rotations About Each Axis} \begin{pmatrix} cos(\gamma) & 0 & sin(\gamma) \\ 0 & 1 & 0 \\ - -sin(\gamma) & 0 & cos(\gamma) \\ - 0 & 0 & 1 + -sin(\gamma) & 0 & cos(\gamma) \end{pmatrix} \end{equation} @@ -401,45 +400,197 @@ \subsection{Rotations About Each Axis} is (1, 0, 0), which also happens to the be coordinates of the x-axis. When you rotate about the x-axis the vectors only change in the Y-Z plane. -\subsection{Worked Example} +These rotations are shown in figure \ref{fig:3d1}. -Let's see if our rotations still work in 3D. Let's take the same coordiate frame -as before in figure \ref{fig:2d1}. Let's imagine that we have a vector Z in the X-Y frame, -and that vector points straight out of the page. We'll call this frame X-Y-Z. The frame -A-B also has a vector C that is aligned with the Z-axis and points straight out of the page. -We'll call this frame A-B-C. So now we rotate A-B-C by angle $\theta$ around the Z-axis in the X-Y-Z frame, -and we want to compute the coordinates of the rotated A-B-C +\begin{figure}[h!] + \centering + \includegraphics[scale=0.75]{figures/3d1.png} + \caption{3D Rotations about each axis.} + \label{fig:3d1} +\end{figure} -\subsection{Rotation About an Arbitrary Axis} -Discuss Rodrigues' rotation formula, quaternions, or Euler angles—whatever is relevant. -%----------------------------------------------------------------------- -% 5. PROPERTIES OF ROTATION MATRICES -%----------------------------------------------------------------------- -\section{Properties of Rotation Matrices} -\label{sec:properties} +\subsection{Order of Rotations Matters A Lot} + +Rotation matrices are not commutative. That is, \( R_z(\theta) * R_y(\gamma) \neq R_y(\gamma) * R_z(\theta) \). +This is because the order of rotations matters! You can see this pictorially by making +a 3D coordinate frame out of your thumb and fingers on your right hand. If you rotate about +the z-axis first, then rotate about the y-axis, your hand will end up in a different orientation +than if you rotated about the y-axis first, then rotate about the z-axis. + +You should be able to convince yourself mathematically that $R_z(\theta) * R_y(\gamma) \neq R_y(\gamma) * R_z(\theta)$ +by multiplying the matrices together both ways and comparing the results. + +\subsection{Roll Pitch Yaw} + +Another way to think about rotations is in terms of roll, pitch, and yaw. Roll is rotation +around the x-axis, pitch is rotation around the y-axis, and yaw is rotation around the z-axis. +This is often used in aeronautical applications to describe the orientation of an airplane. + + +\subsection{Combining the Rotation Matrices} + +In you know the individual rotation angles about each of the 3 axes, you can construct +each rotation matrix. You can then combine those matrices to get a single rotation matrix +by multiplying them together as follows: + +\begin{equation} + R(\theta, \gamma, \alpha) = R_x(\alpha) * R_y(\gamma) * R_z(\theta) +\end{equation} + +But you do have to be careful about the order of multiplication. If we include a vector +in the equation as follows: + +\begin{equation} + v' = R(\theta, \gamma, \alpha) * \vec{v} = R_x(\alpha) * R_y(\gamma) * R_z(\theta) * \vec{v} +\end{equation} + +hopefully you can see that the first rotation is around the z-axis, followed by a rotation +about the y-axis, followed by a rotation about the x-axis. The equation reads right +to left, and the first thing that gets multiplied by the vector is the z-axis rotation. -Mention key properties like: -\begin{itemize} - \item Orthogonality: \( R^\top R = I \) - \item \(\det R = +1\) - \item Composing rotations - \item Relationship to \(\mathrm{SO}(n)\) -\end{itemize} %----------------------------------------------------------------------- -% 6. APPLICATIONS +% 5. Coordinate Transformations %----------------------------------------------------------------------- -\section{Applications} -\label{sec:applications} +\section{Coordinate Transformations} +\label{sec:transformations} + +Now we have enough background to tackle a coordinate transformation on a robot. Let’s say +that have a robot with a camera mounted on it. The coordinate frame of the robot is denoted +by $X_R$, $Y_R$, and $Z_R$. The coordinate frame of the camera is denoted by $X_C$, $Y_C$, +and $Z_C$. The robot coorinate frame is positioned at the origin of the robot such that +$Z_R$ points upwards and $X_R$ points forwards on the robot. + +The camera coordinate frame follows the coordinate frame of OpenCV, such that $Z_C$ points +outwards from the center of the camera, $X_C$ points to the right, and $Y_C$ points down. The +camera coordinate frame is shwon in figure \ref{fig:pinhole_camera_model}. + +\begin{figure}[h!] + \centering + \includegraphics[scale=0.5]{figures/pinhole_camera_model.png} + \caption{The camera coordinate frame.} + \label{fig:pinhole_camera_model} +\end{figure} + +The camera is placed on a location on the robot such that the camera $Z_C$ points towards +the front of the robot. The camera is looking at something of interest, such as the apriltag, +which is placed somewhere in the camera's field of view. This setup is shown in figure \ref{fig:robot}. + +\begin{figure}[h!] + \centering + \includegraphics[scale=0.5]{figures/robot.png} + \caption{The robot and camera coordinate frames.} + \label{fig:robot} +\end{figure} + + +In this case the camera measures the location of the apriltag in the camera coordinate frame. +That is denoted by the vector called "Apriltag Distance" on the diagram. In order to get the +coordinates of the apriltag in the robot coordinate frame, we need to transform the vector +called "Aptriltag Distance" by the rotation matrix of the robot, and apply the translation +vector denoted by "T" in the diagram. The resulting vector is the location of the apriltag +in the robot coordinate frame, denoted by "Apriltag to Robot". + +The translation vector is just the mounting point of the camera in the robot coordinate frame. +We generally try to make it so that the camera is rigidly mounted on the robot, and +doesn't move around. If the camera is not rigidly mounted on the robot, then the translation vector +will be off and our results will be off. The translation vector is simply a 3-component vector +as follows: + +\begin{equation} + \vec{T} = \begin{bmatrix} + t_x \\ + t_y \\ + t_z + \end{bmatrix} +\end{equation} + +Generally this can be derived fairly accurately from CAD Drawings. The rotation matrix may also +be derived from CAD drawings, or it can be measured directly. We will work out how to +measure the rotation matrix in the next section. In this section let's work out how to +build a rotation matrix from a CAD drawing. + +The first thing to notice is that the camera coordinate frame is rotated from the robot +frame. The $Z_C$ axis aligns with the robot $X_R$ axis. So we need a rotation +about the Y axis by 90 degrees to get those aligned. After that we can rotate +by -90 degrees about the X axis to get the camera coordinate frame aligned with +the robot coordinate frame. That is the rotation matrix we need to build. +This is shown in figure \ref{fig:rotations}. + +\begin{figure}[h!] + \centering + \includegraphics[scale=0.5]{figures/rotations.png} + \caption{Two rotations to align the camera coordinate frame with the robot coordinate frame.} + \label{fig:rotations} +\end{figure} + +Putting this together we would get the following rotation matrix: + +\begin{equation} + R_x(-90) * R_y(90) * \vec{v} +\end{equation} + +which expands to: + +\begin{equation} + \begin{pmatrix} + 1 & 0 & 0 \\ + 0 & cos(\alpha) & -sin(\alpha) \\ + 0 & sin(\alpha) & cos(\alpha) + \end{pmatrix} + * + \begin{pmatrix} + cos(\gamma) & 0 & sin(\gamma) \\ + 0 & 1 & 0 \\ + -sin(\gamma) & 0 & cos(\gamma) + \end{pmatrix} + * + \begin{pmatrix} + v_x \\ + v_y \\ + v_z + \end{pmatrix} +\end{equation} + +where $\gamma$ is the angle of rotation about the Y axis equal to 90 degrees ($\pi/2$ radians), +and $\alpha$ is the angle of rotation about the X axis equal to -90 degrees ($-\pi/2$ radians). + +Putting this all together and including the translation vector we get: + +\begin{equation} + \vec{v_{robot-frame}} = R * \vec{v_{camera-frame}} + \vec{T} +\end{equation} + +or + +\begin{equation} + \vec{v_{robot-frame}} = + \begin{pmatrix} + 1 & 0 & 0 \\ + 0 & cos(\alpha) & -sin(\alpha) \\ + 0 & sin(\alpha) & cos(\alpha) + \end{pmatrix} + * + \begin{pmatrix} + cos(\gamma) & 0 & sin(\gamma) \\ + 0 & 1 & 0 \\ + -sin(\gamma) & 0 & cos(\gamma) + \end{pmatrix} + * + \begin{pmatrix} + v_x \\ + v_y \\ + v_z + \end{pmatrix} + + + \begin{pmatrix} + t_x \\ + t_y \\ + t_z + \end{pmatrix} +\end{equation} -Discuss practical uses: -\begin{itemize} - \item Robotics - \item Computer graphics - \item Aerospace - \item Image processing -\end{itemize} %----------------------------------------------------------------------- % 7. CONCLUSION