From 477f540ec64a07ad144deaf0c657f4d027ae364b Mon Sep 17 00:00:00 2001
From: Charlene Leong <charleneleong84@gmail.com>
Date: Thu, 30 May 2024 01:49:20 +0000
Subject: [PATCH] feat: clean

---
 _site/blog/index.html                         |  2 +-
 _site/blog/posts/mamba/index.html             |  2 +-
 _site/projects/index-python.xml               |  2 +-
 _site/projects/index.html                     |  4 ++--
 .../posts/bigT_ros_control/index.html         | 21 ++++++++++++-------
 _site/search.json                             |  8 +++----
 _site/sitemap.xml                             |  4 ++--
 projects/posts/bigT_ros_control/index.qmd     | 11 +++++-----
 8 files changed, 31 insertions(+), 23 deletions(-)

diff --git a/_site/blog/index.html b/_site/blog/index.html
index 47f6a33..2c544cd 100644
--- a/_site/blog/index.html
+++ b/_site/blog/index.html
@@ -215,7 +215,7 @@ <h5 class="quarto-listing-category-title">Categories</h5><div class="quarto-list
 
 <div class="quarto-listing quarto-listing-container-default" id="listing-listing">
 <div class="list quarto-listing-default">
-<div class="quarto-post image-right" data-index="0" data-categories="state space models,s4,mamba,sequence models,long range,modelling" data-listing-date-sort="1715558400000" data-listing-file-modified-sort="1716939327504" data-listing-date-modified-sort="NaN" data-listing-reading-time-sort="28" data-listing-word-count-sort="5573">
+<div class="quarto-post image-right" data-index="0" data-categories="state space models,s4,mamba,sequence models,long range,modelling" data-listing-date-sort="1715558400000" data-listing-file-modified-sort="1716962069058" data-listing-date-modified-sort="NaN" data-listing-reading-time-sort="28" data-listing-word-count-sort="5583">
 <div class="thumbnail">
 <p><a href="../blog/posts/mamba/index.html" class="no-external"></a></p><a href="../blog/posts/mamba/index.html" class="no-external">
 <p><img src="../blog/posts/mamba/feature.gif" class="thumbnail-image"></p>
diff --git a/_site/blog/posts/mamba/index.html b/_site/blog/posts/mamba/index.html
index ae134db..1a06770 100644
--- a/_site/blog/posts/mamba/index.html
+++ b/_site/blog/posts/mamba/index.html
@@ -1843,7 +1843,7 @@ <h1 data-number="5"><span class="header-section-number">5</span> References</h1>
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 </div> <!-- /content -->
-<script>var lightboxQuarto = GLightbox({"closeEffect":"zoom","openEffect":"zoom","descPosition":"bottom","selector":".lightbox","loop":false});
+<script>var lightboxQuarto = GLightbox({"openEffect":"zoom","selector":".lightbox","loop":false,"descPosition":"bottom","closeEffect":"zoom"});
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diff --git a/_site/projects/index-python.xml b/_site/projects/index-python.xml
index fc8fec8..c9b10db 100644
--- a/_site/projects/index-python.xml
+++ b/_site/projects/index-python.xml
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 <description></description>
 <generator>quarto-1.4.554</generator>
-<lastBuildDate>Fri, 24 May 2024 06:41:47 GMT</lastBuildDate>
+<lastBuildDate>Thu, 30 May 2024 01:42:01 GMT</lastBuildDate>
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diff --git a/_site/projects/index.html b/_site/projects/index.html
index f4bf4b6..a5688a3 100644
--- a/_site/projects/index.html
+++ b/_site/projects/index.html
@@ -243,7 +243,7 @@ <h5 class="quarto-listing-category-title">Categories</h5><div class="quarto-list
     </div>
 </div>
 <div class="list grid quarto-listing-cols-3">
-<div class="g-col-1" data-index="0" data-categories="robotics,control,navigation,ros" data-listing-date-sort="1715558400000" data-listing-file-modified-sort="1716531149801" data-listing-date-modified-sort="1716531149000" data-listing-reading-time-sort="10" data-listing-word-count-sort="1839">
+<div class="g-col-1" data-index="0" data-categories="robotics,control,navigation,ros" data-listing-date-sort="1715558400000" data-listing-file-modified-sort="1717033313627" data-listing-date-modified-sort="1717033313000" data-listing-reading-time-sort="10" data-listing-word-count-sort="1839">
 <a href="../projects/posts/bigT_ros_control/index.html" class="quarto-grid-link">
 <div class="quarto-grid-item card h-100 card-left">
 <p class="card-img-top"><img src="posts/bigT_ros_control/feature.gif" style="height: 150px;"  class="thumbnail-image card-img"/></p>
@@ -265,7 +265,7 @@ <h5 class="no-anchor card-title listing-title">
 </div>
 <div class="card-footer">
 <div class="card-file-modified listing-file-modified">
-<p>May 24, 2024</p>
+<p>May 30, 2024</p>
 </div>
 </div>
 </div>
diff --git a/_site/projects/posts/bigT_ros_control/index.html b/_site/projects/posts/bigT_ros_control/index.html
index 64bec4e..2b853ec 100644
--- a/_site/projects/posts/bigT_ros_control/index.html
+++ b/_site/projects/posts/bigT_ros_control/index.html
@@ -227,7 +227,7 @@ <h1 class="title">ROS Navigation and Control for a Triangular Holonomic Robot</h
       <div>
       <div class="quarto-title-meta-heading">Modified</div>
       <div class="quarto-title-meta-contents">
-        <p class="date-modified">May 24, 2024</p>
+        <p class="date-modified">May 30, 2024</p>
       </div>
     </div>
       
@@ -329,7 +329,14 @@ <h1 data-number="4"><span class="header-section-number">4</span> Control Algorit
 <section id="imports-and-global-variables" class="level2" data-number="4.1">
 <h2 data-number="4.1" class="anchored" data-anchor-id="imports-and-global-variables"><span class="header-section-number">4.1</span> Imports and Global Variables</h2>
 <p>We first import our initial <code>C++</code> <code>ROS</code> headers and instantiate our global variables from the robot’s physical characteristics.</p>
-<p>We include the following headers - [<code>ros/ros.h</code>] to include too headers necessary to use the most common public pieces of the ROS system. - <a href="http://docs.ros.org/en/kinetic/api/geometry_msgs/html/msg/Twist.html"><code>geometry_msgs/Twist.h</code></a> to publish linear and angular velocities to facilitate interoperability throughout the system. - <a href="http://docs.ros.org/en/kinetic/api/geometry_msgs/html/msg/Quaternion.html"><code>geometry_msgs/Quaternion.h</code></a> to publish orientation in quarternion form. - <a href="http://docs.ros.org/en/kinetic/api/sensor_msgs/html/msg/Imu.html"><code>sensor_msgs/Imu.h</code></a> to collect sensor data message type from the IMU. The IMU node <a href="http://wiki.ros.org/razor_imu_9do"><code>razor_imu_9dof</code></a> publishes messages to the <code>"imu"</code> topic in the ROS system to be used by motion and planning algorithms such as this <a href="http://wiki.ros.org/robot_pose_ekf"><code>robot_pose_ekf</code></a>. - <a href="https://cplusplus.com/reference/cmath/"><code>math.h</code></a> to perform common mathmetical operations and transformations.</p>
+<p>We include the following headers:</p>
+<ul>
+<li>[<code>ros/ros.h</code>] to include too headers necessary to use the most common public pieces of the ROS system.</li>
+<li><a href="http://docs.ros.org/en/kinetic/api/geometry_msgs/html/msg/Twist.html"><code>geometry_msgs/Twist.h</code></a> to publish linear and angular velocities to facilitate interoperability throughout the system.</li>
+<li><a href="http://docs.ros.org/en/kinetic/api/geometry_msgs/html/msg/Quaternion.html"><code>geometry_msgs/Quaternion.h</code></a> to publish orientation in quarternion form.</li>
+<li><a href="http://docs.ros.org/en/kinetic/api/sensor_msgs/html/msg/Imu.html"><code>sensor_msgs/Imu.h</code></a> to collect sensor data message type from the IMU. The IMU node <a href="http://wiki.ros.org/razor_imu_9do"><code>razor_imu_9dof</code></a> publishes messages to the <code>"imu"</code> topic in the ROS system to be used by motion and planning algorithms such as this <a href="http://wiki.ros.org/robot_pose_ekf"><code>robot_pose_ekf</code></a>.</li>
+<li><a href="https://cplusplus.com/reference/cmath/"><code>math.h</code></a> to perform common mathmetical operations and transformations.</li>
+</ul>
 <p>We then declare our ROS publishers <code>motor_control_pub</code> to publish the motor control signal to the wheels and <code>imu_pub</code> to publish and update the robot’s orientation.</p>
 <div data-add-from="control.cpp" data-start-line="1" data-end-line="18" data-code-line-numbers="true">
 <div class="sourceCode cell-code" id="cb1" data-startfrom="1"><pre class="sourceCode numberSource cpp number-lines code-with-copy"><code class="sourceCode cpp"><span id="cb1-1"><a href="#cb1-1"></a><span class="pp">#include </span><span class="im">"ros/ros.h"</span></span>
@@ -527,7 +534,7 @@ <h2 data-number="4.4" class="anchored" data-anchor-id="deriving-the-kinematic-eq
 \omega_2 = \frac{1}{r} ( \frac{\cos(\theta) v_y}{3} - \frac{\sin(\theta) v_x}{3} + \frac{\sqrt{3} \sin(\theta) v_y}{3} + \frac{\sqrt{3} \cos(\theta) v_x}{3}  + \omega_{\text{body}} h )
 \]</span></p>
 <p><span class="math display">\[
-\omega_3 = \frac{1}{r} ( -\frac{\sqrt{3} \sin(\theta) v_y}{3} + \frac{\cos(\theta) v_y}{3} - \frac{\sqrt{3} \cos(\theta) v_x}{3} - \frac{\sin(\theta) v_x}{3} + \omega_{\text{body}} * h )
+\omega_3 = \frac{1}{r} ( -\frac{\sqrt{3} \sin(\theta) v_y}{3} + \frac{\cos(\theta) v_y}{3} - \frac{\sqrt{3} \cos(\theta) v_x}{3} - \frac{\sin(\theta) v_x}{3} + \omega_{\text{body}} h )
 \]</span></p>
 <p>These equations relate the global motion commands (<span class="math inline">\(v_x\)</span>, <span class="math inline">\(v_y\)</span>, <span class="math inline">\(\omega_{\text{body}}\)</span>) to the angular velocities of the three wheels (<span class="math inline">\(\omega_1\)</span>, <span class="math inline">\(\omega_2\)</span>, <span class="math inline">\(\omega_3\)</span>) taking into account the robot’s orientation <span class="math inline">\(\theta\)</span>, wheel radius <span class="math inline">\(r\)</span> and distance from center to the wheels <span class="math inline">\(h\)</span>.</p>
 </section>
@@ -535,10 +542,10 @@ <h2 data-number="4.4" class="anchored" data-anchor-id="deriving-the-kinematic-eq
 <h2 data-number="4.5" class="anchored" data-anchor-id="converting-angular-velocities-to-pwm-signal-for-motor-control"><span class="header-section-number">4.5</span> Converting angular velocities to PWM signal for motor control</h2>
 <p>After we’ve calculated the angular velocities in terms on radians per second (<span class="math inline">\(rad/s\)</span>), we have to convert to a Pulse Width Modulation (PWM) signal to send updated speed and direction to the motors.</p>
 <p><strong>1. Angular Velocity to RPM</strong></p>
-<p>We first convert angular velocity (in radians per second) to revolutions per minute (RPM).</p>
-<p>We know that <span class="math inline">\(1 revolution=2\pi radians\)</span> and <span class="math inline">\(1 minute = 60 seconds\)</span>, therefore, the conversion factor is:</p>
+<p>We first convert angular velocity (<span class="math inline">\(rad/s\)</span>) to revolutions per minute (RPM).</p>
+<p>We know that <span class="math inline">\(1 \space revolution=2\pi \space radians\)</span> and <span class="math inline">\(1 \space minute = 60 \space seconds\)</span>, therefore, the conversion factor is:</p>
 <p><span class="math display">\[
-RPM = \omega\times(60/(2\pi)) \approx \omega\times 9.5493 \approx \omega\times 9.55
+RPM = \omega\times(60/2\pi) \approx \omega\times 9.5493 \approx \omega\times 9.55
 \]</span></p>
 <p><strong>2. Conversion from RPM to PWM</strong></p>
 <p>In order to measure and calibrate the relationship betwen PWM and RPM for the motor, an empirical experiment was performed in order to collect measurements and fit a linear regression model to the data.</p>
@@ -1046,7 +1053,7 @@ <h2 data-number="4.5" class="anchored" data-anchor-id="converting-angular-veloci
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diff --git a/_site/search.json b/_site/search.json
index d4a723d..6c4f427 100644
--- a/_site/search.json
+++ b/_site/search.json
@@ -11,7 +11,7 @@
     "href": "projects/index.html",
     "title": "Projects",
     "section": "",
-    "text": "Order By\n       Default\n         \n          Date - Oldest\n        \n         \n          Date - Newest\n        \n         \n          Updated - Oldest\n        \n         \n          Updated - Newest\n        \n     \n  \n    \n      \n      \n    \n\n\n\n\n\n\n\n\nROS Navigation and Control for a Triangular Holonomic Robot\n\n\n10 min\n\n\nAutonomous SLAM and control\n\n\n\nMay 13, 2024\n\n\n\n\n\nMay 24, 2024\n\n\n\n\n\n\n\nNo matching items"
+    "text": "Order By\n       Default\n         \n          Date - Oldest\n        \n         \n          Date - Newest\n        \n         \n          Updated - Oldest\n        \n         \n          Updated - Newest\n        \n     \n  \n    \n      \n      \n    \n\n\n\n\n\n\n\n\nROS Navigation and Control for a Triangular Holonomic Robot\n\n\n10 min\n\n\nAutonomous SLAM and control\n\n\n\nMay 13, 2024\n\n\n\n\n\nMay 30, 2024\n\n\n\n\n\n\n\nNo matching items"
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@@ -319,7 +319,7 @@
     "href": "projects/posts/bigT_ros_control/index.html#imports-and-global-variables",
     "title": "ROS Navigation and Control for a Triangular Holonomic Robot",
     "section": "4.1 Imports and Global Variables",
-    "text": "4.1 Imports and Global Variables\nWe first import our initial C++ ROS headers and instantiate our global variables from the robot’s physical characteristics.\nWe include the following headers - [ros/ros.h] to include too headers necessary to use the most common public pieces of the ROS system. - geometry_msgs/Twist.h to publish linear and angular velocities to facilitate interoperability throughout the system. - geometry_msgs/Quaternion.h to publish orientation in quarternion form. - sensor_msgs/Imu.h to collect sensor data message type from the IMU. The IMU node razor_imu_9dof publishes messages to the \"imu\" topic in the ROS system to be used by motion and planning algorithms such as this robot_pose_ekf. - math.h to perform common mathmetical operations and transformations.\nWe then declare our ROS publishers motor_control_pub to publish the motor control signal to the wheels and imu_pub to publish and update the robot’s orientation.\n\n#include \"ros/ros.h\"\n#include \"geometry_msgs/Twist.h\"\n#include \"geometry_msgs/Quaternion.h\"\n#include \"sensor_msgs/Imu.h\"\n#include &lt;math.h&gt;\n\nfloat omega1, omega2, omega3; // angular velocities\nfloat r = 0.0525; // Wheel radius (105mm/2)\nfloat h = 0.18;   // Distance from the center of the body to the wheel (180mm)\n\nfloat v_x = 0;    // Global translation speed in x (m/s)\nfloat v_y = 0;    // Global translation speed in y (m/s)\nfloat omega_body = 0;  // Rotational speed of the body (rad/s)\nfloat theta = 3.1415;  // Orientation of the body (rad)\n\nros::Publisher motor_control_pub;\nros::Publisher imu_pub;"
+    "text": "4.1 Imports and Global Variables\nWe first import our initial C++ ROS headers and instantiate our global variables from the robot’s physical characteristics.\nWe include the following headers:\n\n[ros/ros.h] to include too headers necessary to use the most common public pieces of the ROS system.\ngeometry_msgs/Twist.h to publish linear and angular velocities to facilitate interoperability throughout the system.\ngeometry_msgs/Quaternion.h to publish orientation in quarternion form.\nsensor_msgs/Imu.h to collect sensor data message type from the IMU. The IMU node razor_imu_9dof publishes messages to the \"imu\" topic in the ROS system to be used by motion and planning algorithms such as this robot_pose_ekf.\nmath.h to perform common mathmetical operations and transformations.\n\nWe then declare our ROS publishers motor_control_pub to publish the motor control signal to the wheels and imu_pub to publish and update the robot’s orientation.\n\n#include \"ros/ros.h\"\n#include \"geometry_msgs/Twist.h\"\n#include \"geometry_msgs/Quaternion.h\"\n#include \"sensor_msgs/Imu.h\"\n#include &lt;math.h&gt;\n\nfloat omega1, omega2, omega3; // angular velocities\nfloat r = 0.0525; // Wheel radius (105mm/2)\nfloat h = 0.18;   // Distance from the center of the body to the wheel (180mm)\n\nfloat v_x = 0;    // Global translation speed in x (m/s)\nfloat v_y = 0;    // Global translation speed in y (m/s)\nfloat omega_body = 0;  // Rotational speed of the body (rad/s)\nfloat theta = 3.1415;  // Orientation of the body (rad)\n\nros::Publisher motor_control_pub;\nros::Publisher imu_pub;"
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@@ -340,14 +340,14 @@
     "href": "projects/posts/bigT_ros_control/index.html#deriving-the-kinematic-equations-for-a-three-wheeled-omnidirectional-robot",
     "title": "ROS Navigation and Control for a Triangular Holonomic Robot",
     "section": "4.4 Deriving the Kinematic Equations for a Three-Wheeled Omnidirectional Robot",
-    "text": "4.4 Deriving the Kinematic Equations for a Three-Wheeled Omnidirectional Robot\nAssumptions and Setup\n\nRobot Configuration:\n\nThe robot is a triangle with three holonomic wheels.\nThe wheels are positioned 120 degrees apart from each other.\n\nVariables:\n\n\\(v_x\\): Linear velocity in the x-direction (global frame).\n\\(v_y\\): Linear velocity in the y-direction (global frame).\n\\(\\omega_\\text{body}\\): Rotational velocity of the robot around its center.\n\\(\\theta\\): Orientation of the robot (angle between the robot’s frame and the global frame).\n\\(r\\): Radius of each wheel.\n\\(h\\): Distance from the center of the robot to each wheel.\n\n\nDiagram\n\n\n\n\n\nDerivation\nWe aim to derive the angular velocities of the three holonomic wheels \\(\\omega_1\\), \\(\\omega_2\\), \\(\\omega_3\\) based on the robot’s linear velocities (\\(v_x\\), \\(v_y\\)) and rotational velocity (\\(\\omega_{\\text{body}}\\)) in the global frame, considering the robot’s orientation \\(\\theta\\).\n1. Transform Global Velocities to Local Velocities\nFirst, we need to transform The global velocities \\(v_x\\) and \\(v_y\\) into the local frame of the robot using its orientation \\(\\theta\\):\n\\[\nv_{x,\\text{local}} = v_x \\cos(\\theta) + v_y \\sin(\\theta)\n\\]\n\\[\nv_{y,\\text{local}} = -v_x \\sin(\\theta) + v_y \\cos(\\theta)\n\\]\n2. Express Local Velocities and Rotational Velocity in Terms of Wheel Velocities\nEach wheel contributes to the robot’s motion. The linear velocities of the wheels can be decomposed into components that affect the overall motion of the robot.\nFor a wheel positioned at an angle \\(\\alpha\\) with respect to the robot’s frame: \\[\n\\omega_{\\text{wheel}} = \\frac{1}{r} (v_{x,\\text{local}} \\cos(\\alpha) + v_{y,\\text{local}} \\sin(\\alpha) + \\omega_{\\text{body}} h)\n\\]\nWhere \\(\\omega_\\text{wheel}\\) is the angular velocity of the wheel, and \\(h\\) is the distance from the center to the wheel.\n3. Apply to Each Wheel\nEach wheel is positioned \\(120^\\circ\\) apart, so the angles \\(\\alpha\\) for the three wheels are \\(0^\\circ\\), \\(120^\\circ\\), and \\(240^\\circ\\).\nFor Wheel 1 where \\(\\alpha = 0^\\circ\\):\n\\[\n\\omega_1 = \\frac{1}{r} ( v_{x,\\text{local}} \\cos(0) + v_{y,\\text{local}} \\sin(0) + \\omega_{\\text{body}} h )\n\\]\nSince \\(cos(0^\\circ)= 1\\) and \\(\\sin(0^\\circ) = 0\\):\n\\[\n\\omega_1 = \\frac{1}{r} ( v_{x,\\text{local}} + \\omega_{\\text{body}} h )\n\\]\nSubstituting the local velocities:\n\\[\n\\omega_1 = \\frac{1}{r} ( v_x \\cos(\\theta) + v_y \\sin(\\theta) + \\omega_{\\text{body}} h )\n\\]\nFor Wheel 2 where \\(\\alpha = 120^\\circ\\):\n\\[\n\\omega_2 = \\frac{1}{r} ( v_{x,\\text{local}} \\cos(120^\\circ) + v_{y,\\text{local}} \\sin(120^\\circ) + \\omega_{\\text{body}} h)\n\\]\nSince \\(\\cos(120^\\circ) = -\\frac{1}{2}\\) and \\(\\sin(120^\\circ) = \\frac{\\sqrt{3}}{2}\\):\n\\[\n\\omega_2 = \\frac{1}{r} ( v_{x,\\text{local}}( -\\frac{1}{2} ) + v_{y,\\text{local}} ( \\frac{\\sqrt{3}}{2} ) + \\omega_{\\text{body}} h)\n\\]\n\\[\n\\omega_2 = \\frac{1}{r} ( (v_x \\cos(\\theta) + v_y \\sin(\\theta)) ( -\\frac{1}{2} ) + (-v_x \\sin(\\theta) + v_y \\cos(\\theta)) ( \\frac{\\sqrt{3}}{2} ) + \\omega_{\\text{body}} h )\n\\]\nExpanding and simplifying:\n\\[\n\\omega_2 = \\frac{1}{r} ( -\\frac{1}{2} v_x \\cos(\\theta) - \\frac{1}{2} v_y \\sin(\\theta) + \\frac{\\sqrt{3}}{2} (-v_x \\sin(\\theta)) + \\frac{\\sqrt{3}}{2} v_y \\cos(\\theta) + \\omega_{\\text{body}} h )\n\\]\n\\[\n\\omega_2 = \\frac{1}{r} ( -\\frac{1}{2} v_x \\cos(\\theta) - \\frac{1}{2} v_y \\sin(\\theta) - \\frac{\\sqrt{3}}{2} v_x \\sin(\\theta) + \\frac{\\sqrt{3}}{2} v_y \\cos(\\theta) + \\omega_{\\text{body}} h )\n\\]\nCombining terms and simplifying further:\n\\[\n\\omega_2 = \\frac{1}{r} ( \\frac{\\cos(\\theta) v_y}{3} - \\frac{\\sin(\\theta) v_x}{3} + \\frac{\\sqrt{3} \\sin(\\theta) v_y}{3} + \\frac{\\sqrt{3} \\cos(\\theta) v_x}{3}  + \\omega_{\\text{body}} h )\n\\]\nFor Wheel 3 where \\(\\alpha = 240^\\circ\\):\n\\[\n\\omega_3 = \\frac{1}{r} ( v_{x,\\text{local}} \\cos(240^\\circ) + v_{y,\\text{local}} \\sin(240^\\circ) + \\omega_{\\text{body}} h )\n\\]\nSince \\(\\cos(240^\\circ) = -\\frac{1}{2}\\) and \\(\\sin(240^\\circ) = -\\frac{\\sqrt{3}}{2}\\):\n\\[\n\\omega_3 = \\frac{1}{r} ( (v_x \\cos(\\theta) + v_y \\sin(\\theta)) ( -\\frac{1}{2} ) + (-v_x \\sin(\\theta) + v_y \\cos(\\theta)) ( -\\frac{\\sqrt{3}}{2} ) + \\omega_{\\text{body}} h )\n\\]\nExpanding and simplifying:\n\\[\n\\omega_3 = \\frac{1}{r} ( -\\frac{1}{2} v_x \\cos(\\theta) - \\frac{1}{2} v_y \\sin(\\theta) - \\frac{\\sqrt{3}}{2} (-v_x \\sin(\\theta)) - \\frac{\\sqrt{3}}{2} v_y \\cos(\\theta) + \\omega_{\\text{body}} h )\n\\]\n\\[\n\\omega_3 = \\frac{1}{r} ( -\\frac{1}{2} v_x \\cos(\\theta) - \\frac{1}{2} v_y \\sin(\\theta) + \\frac{\\sqrt{3}}{2} v_x \\sin(\\theta) - \\frac{\\sqrt{3}}{2} v_y \\cos(\\theta) + \\omega_{\\text{body}} h )\n\\]\nCombining terms and simplifying further:\n\\[\n\\omega_3 = \\frac{1}{r} ( -\\frac{\\sqrt{3} \\sin(\\theta) v_y}{3} + \\frac{\\cos(\\theta) v_y}{3} - \\frac{\\sqrt{3} \\cos(\\theta) v_x}{3} - \\frac{\\sin(\\theta) v_x}{3} + \\omega_{\\text{body}} h )\n\\]\nFinal Equations\nThe final simplified equations for the angular velocities of the wheels are:\n\\[\n\\omega_1 = \\frac{1}{r} ( v_x \\cos(\\theta) + v_y \\sin(\\theta) + \\omega_{\\text{body}} h )\n\\]\n\\[\n\\omega_2 = \\frac{1}{r} ( \\frac{\\cos(\\theta) v_y}{3} - \\frac{\\sin(\\theta) v_x}{3} + \\frac{\\sqrt{3} \\sin(\\theta) v_y}{3} + \\frac{\\sqrt{3} \\cos(\\theta) v_x}{3}  + \\omega_{\\text{body}} h )\n\\]\n\\[\n\\omega_3 = \\frac{1}{r} ( -\\frac{\\sqrt{3} \\sin(\\theta) v_y}{3} + \\frac{\\cos(\\theta) v_y}{3} - \\frac{\\sqrt{3} \\cos(\\theta) v_x}{3} - \\frac{\\sin(\\theta) v_x}{3} + \\omega_{\\text{body}} * h )\n\\]\nThese equations relate the global motion commands (\\(v_x\\), \\(v_y\\), \\(\\omega_{\\text{body}}\\)) to the angular velocities of the three wheels (\\(\\omega_1\\), \\(\\omega_2\\), \\(\\omega_3\\)) taking into account the robot’s orientation \\(\\theta\\), wheel radius \\(r\\) and distance from center to the wheels \\(h\\)."
+    "text": "4.4 Deriving the Kinematic Equations for a Three-Wheeled Omnidirectional Robot\nAssumptions and Setup\n\nRobot Configuration:\n\nThe robot is a triangle with three holonomic wheels.\nThe wheels are positioned 120 degrees apart from each other.\n\nVariables:\n\n\\(v_x\\): Linear velocity in the x-direction (global frame).\n\\(v_y\\): Linear velocity in the y-direction (global frame).\n\\(\\omega_\\text{body}\\): Rotational velocity of the robot around its center.\n\\(\\theta\\): Orientation of the robot (angle between the robot’s frame and the global frame).\n\\(r\\): Radius of each wheel.\n\\(h\\): Distance from the center of the robot to each wheel.\n\n\nDiagram\n\n\n\n\n\nDerivation\nWe aim to derive the angular velocities of the three holonomic wheels \\(\\omega_1\\), \\(\\omega_2\\), \\(\\omega_3\\) based on the robot’s linear velocities (\\(v_x\\), \\(v_y\\)) and rotational velocity (\\(\\omega_{\\text{body}}\\)) in the global frame, considering the robot’s orientation \\(\\theta\\).\n1. Transform Global Velocities to Local Velocities\nFirst, we need to transform The global velocities \\(v_x\\) and \\(v_y\\) into the local frame of the robot using its orientation \\(\\theta\\):\n\\[\nv_{x,\\text{local}} = v_x \\cos(\\theta) + v_y \\sin(\\theta)\n\\]\n\\[\nv_{y,\\text{local}} = -v_x \\sin(\\theta) + v_y \\cos(\\theta)\n\\]\n2. Express Local Velocities and Rotational Velocity in Terms of Wheel Velocities\nEach wheel contributes to the robot’s motion. The linear velocities of the wheels can be decomposed into components that affect the overall motion of the robot.\nFor a wheel positioned at an angle \\(\\alpha\\) with respect to the robot’s frame: \\[\n\\omega_{\\text{wheel}} = \\frac{1}{r} (v_{x,\\text{local}} \\cos(\\alpha) + v_{y,\\text{local}} \\sin(\\alpha) + \\omega_{\\text{body}} h)\n\\]\nWhere \\(\\omega_\\text{wheel}\\) is the angular velocity of the wheel, and \\(h\\) is the distance from the center to the wheel.\n3. Apply to Each Wheel\nEach wheel is positioned \\(120^\\circ\\) apart, so the angles \\(\\alpha\\) for the three wheels are \\(0^\\circ\\), \\(120^\\circ\\), and \\(240^\\circ\\).\nFor Wheel 1 where \\(\\alpha = 0^\\circ\\):\n\\[\n\\omega_1 = \\frac{1}{r} ( v_{x,\\text{local}} \\cos(0) + v_{y,\\text{local}} \\sin(0) + \\omega_{\\text{body}} h )\n\\]\nSince \\(cos(0^\\circ)= 1\\) and \\(\\sin(0^\\circ) = 0\\):\n\\[\n\\omega_1 = \\frac{1}{r} ( v_{x,\\text{local}} + \\omega_{\\text{body}} h )\n\\]\nSubstituting the local velocities:\n\\[\n\\omega_1 = \\frac{1}{r} ( v_x \\cos(\\theta) + v_y \\sin(\\theta) + \\omega_{\\text{body}} h )\n\\]\nFor Wheel 2 where \\(\\alpha = 120^\\circ\\):\n\\[\n\\omega_2 = \\frac{1}{r} ( v_{x,\\text{local}} \\cos(120^\\circ) + v_{y,\\text{local}} \\sin(120^\\circ) + \\omega_{\\text{body}} h)\n\\]\nSince \\(\\cos(120^\\circ) = -\\frac{1}{2}\\) and \\(\\sin(120^\\circ) = \\frac{\\sqrt{3}}{2}\\):\n\\[\n\\omega_2 = \\frac{1}{r} ( v_{x,\\text{local}}( -\\frac{1}{2} ) + v_{y,\\text{local}} ( \\frac{\\sqrt{3}}{2} ) + \\omega_{\\text{body}} h)\n\\]\n\\[\n\\omega_2 = \\frac{1}{r} ( (v_x \\cos(\\theta) + v_y \\sin(\\theta)) ( -\\frac{1}{2} ) + (-v_x \\sin(\\theta) + v_y \\cos(\\theta)) ( \\frac{\\sqrt{3}}{2} ) + \\omega_{\\text{body}} h )\n\\]\nExpanding and simplifying:\n\\[\n\\omega_2 = \\frac{1}{r} ( -\\frac{1}{2} v_x \\cos(\\theta) - \\frac{1}{2} v_y \\sin(\\theta) + \\frac{\\sqrt{3}}{2} (-v_x \\sin(\\theta)) + \\frac{\\sqrt{3}}{2} v_y \\cos(\\theta) + \\omega_{\\text{body}} h )\n\\]\n\\[\n\\omega_2 = \\frac{1}{r} ( -\\frac{1}{2} v_x \\cos(\\theta) - \\frac{1}{2} v_y \\sin(\\theta) - \\frac{\\sqrt{3}}{2} v_x \\sin(\\theta) + \\frac{\\sqrt{3}}{2} v_y \\cos(\\theta) + \\omega_{\\text{body}} h )\n\\]\nCombining terms and simplifying further:\n\\[\n\\omega_2 = \\frac{1}{r} ( \\frac{\\cos(\\theta) v_y}{3} - \\frac{\\sin(\\theta) v_x}{3} + \\frac{\\sqrt{3} \\sin(\\theta) v_y}{3} + \\frac{\\sqrt{3} \\cos(\\theta) v_x}{3}  + \\omega_{\\text{body}} h )\n\\]\nFor Wheel 3 where \\(\\alpha = 240^\\circ\\):\n\\[\n\\omega_3 = \\frac{1}{r} ( v_{x,\\text{local}} \\cos(240^\\circ) + v_{y,\\text{local}} \\sin(240^\\circ) + \\omega_{\\text{body}} h )\n\\]\nSince \\(\\cos(240^\\circ) = -\\frac{1}{2}\\) and \\(\\sin(240^\\circ) = -\\frac{\\sqrt{3}}{2}\\):\n\\[\n\\omega_3 = \\frac{1}{r} ( (v_x \\cos(\\theta) + v_y \\sin(\\theta)) ( -\\frac{1}{2} ) + (-v_x \\sin(\\theta) + v_y \\cos(\\theta)) ( -\\frac{\\sqrt{3}}{2} ) + \\omega_{\\text{body}} h )\n\\]\nExpanding and simplifying:\n\\[\n\\omega_3 = \\frac{1}{r} ( -\\frac{1}{2} v_x \\cos(\\theta) - \\frac{1}{2} v_y \\sin(\\theta) - \\frac{\\sqrt{3}}{2} (-v_x \\sin(\\theta)) - \\frac{\\sqrt{3}}{2} v_y \\cos(\\theta) + \\omega_{\\text{body}} h )\n\\]\n\\[\n\\omega_3 = \\frac{1}{r} ( -\\frac{1}{2} v_x \\cos(\\theta) - \\frac{1}{2} v_y \\sin(\\theta) + \\frac{\\sqrt{3}}{2} v_x \\sin(\\theta) - \\frac{\\sqrt{3}}{2} v_y \\cos(\\theta) + \\omega_{\\text{body}} h )\n\\]\nCombining terms and simplifying further:\n\\[\n\\omega_3 = \\frac{1}{r} ( -\\frac{\\sqrt{3} \\sin(\\theta) v_y}{3} + \\frac{\\cos(\\theta) v_y}{3} - \\frac{\\sqrt{3} \\cos(\\theta) v_x}{3} - \\frac{\\sin(\\theta) v_x}{3} + \\omega_{\\text{body}} h )\n\\]\nFinal Equations\nThe final simplified equations for the angular velocities of the wheels are:\n\\[\n\\omega_1 = \\frac{1}{r} ( v_x \\cos(\\theta) + v_y \\sin(\\theta) + \\omega_{\\text{body}} h )\n\\]\n\\[\n\\omega_2 = \\frac{1}{r} ( \\frac{\\cos(\\theta) v_y}{3} - \\frac{\\sin(\\theta) v_x}{3} + \\frac{\\sqrt{3} \\sin(\\theta) v_y}{3} + \\frac{\\sqrt{3} \\cos(\\theta) v_x}{3}  + \\omega_{\\text{body}} h )\n\\]\n\\[\n\\omega_3 = \\frac{1}{r} ( -\\frac{\\sqrt{3} \\sin(\\theta) v_y}{3} + \\frac{\\cos(\\theta) v_y}{3} - \\frac{\\sqrt{3} \\cos(\\theta) v_x}{3} - \\frac{\\sin(\\theta) v_x}{3} + \\omega_{\\text{body}} h )\n\\]\nThese equations relate the global motion commands (\\(v_x\\), \\(v_y\\), \\(\\omega_{\\text{body}}\\)) to the angular velocities of the three wheels (\\(\\omega_1\\), \\(\\omega_2\\), \\(\\omega_3\\)) taking into account the robot’s orientation \\(\\theta\\), wheel radius \\(r\\) and distance from center to the wheels \\(h\\)."
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     "title": "ROS Navigation and Control for a Triangular Holonomic Robot",
     "section": "4.5 Converting angular velocities to PWM signal for motor control",
-    "text": "4.5 Converting angular velocities to PWM signal for motor control\nAfter we’ve calculated the angular velocities in terms on radians per second (\\(rad/s\\)), we have to convert to a Pulse Width Modulation (PWM) signal to send updated speed and direction to the motors.\n1. Angular Velocity to RPM\nWe first convert angular velocity (in radians per second) to revolutions per minute (RPM).\nWe know that \\(1 revolution=2\\pi radians\\) and \\(1 minute = 60 seconds\\), therefore, the conversion factor is:\n\\[\nRPM = \\omega\\times(60/(2\\pi)) \\approx \\omega\\times 9.5493 \\approx \\omega\\times 9.55\n\\]\n2. Conversion from RPM to PWM\nIn order to measure and calibrate the relationship betwen PWM and RPM for the motor, an empirical experiment was performed in order to collect measurements and fit a linear regression model to the data.\nFor series of PWM signals from a low duty cycle to (eg. 10%) to high duty cycle (eg. 90%) generated using an oscilloscope, the respective RPM for the motor was recorded using a tachometer.\nThe resulting linear relationship is:\n\\[\nPWM = a \\times RPM + b  = 2.4307 \\times RPM + 36.2178\n\\]\nThis means that for every unit increase in RPM, the PWM value increases by approximately 2.4307 units, and when the RPM is zero, the PWM value starts at approximately 36.2178.\n3. Handling Very Small Angular Velocities\nSmall angular velocities &lt;=0.05 are ignored, to smooth noise in the signal and avoid unnecessary motor activation.\n\n\tif(fabs(omega) &lt;= 0.05) return 0;\n\n4. Linear Equation of Angular velocity to PWM\nThrough substituting and simplifying the equations above, we can derive the following linear relationship for converting angular velocity to PWM.\n\\[\nPWM = 2.4307 \\times RPM + 36.2178 =  2.43 \\times (\\omega\\times 9.55) + 36.22\n\\]\nThe result is then returned as an integer PWM value which can be used to control motor speed. We later apply this to each wheel when we calculate the new angular velocities from sensor feedback. We make \\(\\omega\\) absolute \\(|\\omega_{\\text{wheel}}|\\) in order to ensure the angular velocity is never non-negative.\n\nint\tomega2pwm(float omega) {\n\t/*\n\t\tomega ... angular velocity ( in rad/s )\n\t\trpm = omega*9.5493; // conversion from rad/s to rpm\t ( 1/(2*pi)*60 = 9.5493 )\n\t\tpwm = 2.4307*rpm + 36.2178; // conversion of rpm to pwm values\n\t*/\n\tif(fabs(omega) &lt;= 0.05) return 0;\n\n\treturn (int)(2.43*(fabs(omega)*9.55) + 36.22);\n}\n\n5. Setting Direction Bit of Motor\nThe direction bit of the motor is set based on the sign of the angular velocity \\(\\omega_{\\text{wheel}}\\).\n\nint sign(float number){\n\tif(number&gt;=0) return 1; else return 0;\n}\n\n\nThis project was executed as part of the course - ECEN430: Advanced Mechatronic Engineering 2: Intelligence and Design at Victoria University of Wellington 2018."
+    "text": "4.5 Converting angular velocities to PWM signal for motor control\nAfter we’ve calculated the angular velocities in terms on radians per second (\\(rad/s\\)), we have to convert to a Pulse Width Modulation (PWM) signal to send updated speed and direction to the motors.\n1. Angular Velocity to RPM\nWe first convert angular velocity (\\(rad/s\\)) to revolutions per minute (RPM).\nWe know that \\(1 \\space revolution=2\\pi \\space radians\\) and \\(1 \\space minute = 60 \\space seconds\\), therefore, the conversion factor is:\n\\[\nRPM = \\omega\\times(60/2\\pi) \\approx \\omega\\times 9.5493 \\approx \\omega\\times 9.55\n\\]\n2. Conversion from RPM to PWM\nIn order to measure and calibrate the relationship betwen PWM and RPM for the motor, an empirical experiment was performed in order to collect measurements and fit a linear regression model to the data.\nFor series of PWM signals from a low duty cycle to (eg. 10%) to high duty cycle (eg. 90%) generated using an oscilloscope, the respective RPM for the motor was recorded using a tachometer.\nThe resulting linear relationship is:\n\\[\nPWM = a \\times RPM + b  = 2.4307 \\times RPM + 36.2178\n\\]\nThis means that for every unit increase in RPM, the PWM value increases by approximately 2.4307 units, and when the RPM is zero, the PWM value starts at approximately 36.2178.\n3. Handling Very Small Angular Velocities\nSmall angular velocities &lt;=0.05 are ignored, to smooth noise in the signal and avoid unnecessary motor activation.\n\n\tif(fabs(omega) &lt;= 0.05) return 0;\n\n4. Linear Equation of Angular velocity to PWM\nThrough substituting and simplifying the equations above, we can derive the following linear relationship for converting angular velocity to PWM.\n\\[\nPWM = 2.4307 \\times RPM + 36.2178 =  2.43 \\times (\\omega\\times 9.55) + 36.22\n\\]\nThe result is then returned as an integer PWM value which can be used to control motor speed. We later apply this to each wheel when we calculate the new angular velocities from sensor feedback. We make \\(\\omega\\) absolute \\(|\\omega_{\\text{wheel}}|\\) in order to ensure the angular velocity is never non-negative.\n\nint\tomega2pwm(float omega) {\n\t/*\n\t\tomega ... angular velocity ( in rad/s )\n\t\trpm = omega*9.5493; // conversion from rad/s to rpm\t ( 1/(2*pi)*60 = 9.5493 )\n\t\tpwm = 2.4307*rpm + 36.2178; // conversion of rpm to pwm values\n\t*/\n\tif(fabs(omega) &lt;= 0.05) return 0;\n\n\treturn (int)(2.43*(fabs(omega)*9.55) + 36.22);\n}\n\n5. Setting Direction Bit of Motor\nThe direction bit of the motor is set based on the sign of the angular velocity \\(\\omega_{\\text{wheel}}\\).\n\nint sign(float number){\n\tif(number&gt;=0) return 1; else return 0;\n}\n\n\nThis project was executed as part of the course - ECEN430: Advanced Mechatronic Engineering 2: Intelligence and Design at Victoria University of Wellington 2018."
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diff --git a/_site/sitemap.xml b/_site/sitemap.xml
index 03c470a..28d6551 100644
--- a/_site/sitemap.xml
+++ b/_site/sitemap.xml
@@ -2,7 +2,7 @@
 <urlset xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
   <url>
     <loc>https://www.ai-intuition.com/blog/posts/mamba/index.html</loc>
-    <lastmod>2024-05-28T23:52:40.270Z</lastmod>
+    <lastmod>2024-05-29T05:54:29.058Z</lastmod>
   </url>
   <url>
     <loc>https://www.ai-intuition.com/blog/index.html</loc>
@@ -30,7 +30,7 @@
   </url>
   <url>
     <loc>https://www.ai-intuition.com/projects/posts/bigT_ros_control/index.html</loc>
-    <lastmod>2024-05-24T06:12:29.801Z</lastmod>
+    <lastmod>2024-05-30T01:47:38.297Z</lastmod>
   </url>
   <url>
     <loc>https://www.ai-intuition.com/projects/draft/crane_plus_motion_planning/index.html</loc>
diff --git a/projects/posts/bigT_ros_control/index.qmd b/projects/posts/bigT_ros_control/index.qmd
index 1cd6ef7..ec37ecb 100644
--- a/projects/posts/bigT_ros_control/index.qmd
+++ b/projects/posts/bigT_ros_control/index.qmd
@@ -72,7 +72,8 @@ The robot's velocity commands are published on the `cmd_vel` topic from either
 
 We first import our initial `C++` `ROS` headers and instantiate our global variables from the robot's physical characteristics.
 
-We include the following headers
+We include the following headers:
+
    - [`ros/ros.h`] to include too headers necessary to use the most common public pieces of the ROS system.
    - [`geometry_msgs/Twist.h`](http://docs.ros.org/en/kinetic/api/geometry_msgs/html/msg/Twist.html) to publish linear and angular velocities to facilitate interoperability throughout the system.
    -  [`geometry_msgs/Quaternion.h`](http://docs.ros.org/en/kinetic/api/geometry_msgs/html/msg/Quaternion.html) to publish orientation in quarternion form.
@@ -272,7 +273,7 @@ $$
 $$
 
 $$
-\omega_3 = \frac{1}{r} ( -\frac{\sqrt{3} \sin(\theta) v_y}{3} + \frac{\cos(\theta) v_y}{3} - \frac{\sqrt{3} \cos(\theta) v_x}{3} - \frac{\sin(\theta) v_x}{3} + \omega_{\text{body}} * h )
+\omega_3 = \frac{1}{r} ( -\frac{\sqrt{3} \sin(\theta) v_y}{3} + \frac{\cos(\theta) v_y}{3} - \frac{\sqrt{3} \cos(\theta) v_x}{3} - \frac{\sin(\theta) v_x}{3} + \omega_{\text{body}} h )
 $$
 
 These equations relate the global motion commands ($v_x$, $v_y$, $\omega_{\text{body}}$) to the angular velocities of the three wheels ($\omega_1$, $\omega_2$, $\omega_3$) taking into account the robot's orientation $\theta$, wheel radius $r$ and distance from center to the wheels $h$.
@@ -286,12 +287,12 @@ After we've calculated the angular velocities in terms on radians per second ($r
 
 **1. Angular Velocity to RPM**
 
-We first convert angular velocity (in radians per second) to revolutions per minute (RPM).
+We first convert angular velocity ($rad/s$) to revolutions per minute (RPM).
 
-We know that $1 revolution=2\pi radians$ and $1 minute = 60 seconds$, therefore, the conversion factor is:
+We know that $1 \space revolution=2\pi \space radians$ and $1 \space minute = 60 \space seconds$, therefore, the conversion factor is:
 
 $$
-RPM = \omega\times(60/(2\pi)) \approx \omega\times 9.5493 \approx \omega\times 9.55
+RPM = \omega\times(60/2\pi) \approx \omega\times 9.5493 \approx \omega\times 9.55
 $$