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mmmarinhommmarinho
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[L3] Adding starting frame as zero to the SO(3) example.
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Lines changed: 157 additions & 34 deletions

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‎basic_lessons/lesson2_tutorial.ipynb‎

Lines changed: 157 additions & 34 deletions
Original file line numberDiff line numberDiff line change
@@ -53,14 +53,19 @@
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},
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{
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"cell_type": "code",
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"metadata": {},
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"metadata": {
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"ExecuteTime": {
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"end_time": "2026-02-08T11:48:53.982411Z",
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"start_time": "2026-02-08T11:48:52.996429Z"
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}
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},
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"source": [
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"%%capture\n",
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"%pip install numpy\n",
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"%pip install numpy --break-system-packages"
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],
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"outputs": [],
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"execution_count": null
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"execution_count": 1
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},
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{
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"cell_type": "markdown",
@@ -71,13 +76,18 @@
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},
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{
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"cell_type": "code",
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"metadata": {},
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"metadata": {
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"ExecuteTime": {
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"end_time": "2026-02-08T11:48:53.989102Z",
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"start_time": "2026-02-08T11:48:53.983066Z"
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}
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},
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"source": [
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"import numpy as np\n",
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"from math import pi, sin, cos"
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],
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"outputs": [],
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"execution_count": null
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"execution_count": 2
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},
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{
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"cell_type": "markdown",
@@ -127,15 +137,27 @@
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{
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"cell_type": "code",
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"metadata": {
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"trusted": true
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"trusted": true,
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"ExecuteTime": {
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"end_time": "2026-02-08T11:48:53.995284Z",
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"start_time": "2026-02-08T11:48:53.989652Z"
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}
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},
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"source": [
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"p = np.array([1.0, 2.0])\n",
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"\n",
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"print(f\"p={p}\")"
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],
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"outputs": [],
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"execution_count": null
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"p=[1. 2.]\n"
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]
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}
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],
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"execution_count": 3
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},
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{
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"cell_type": "markdown",
@@ -158,7 +180,11 @@
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{
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"cell_type": "code",
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"metadata": {
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"trusted": true
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"trusted": true,
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"ExecuteTime": {
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"end_time": "2026-02-08T11:48:54.008003Z",
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"start_time": "2026-02-08T11:48:54.002363Z"
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}
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},
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"source": [
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"p0 = np.array([1.0, 2.0])\n",
@@ -170,8 +196,16 @@
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"\n",
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"print(f\"p={p}\")"
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],
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"outputs": [],
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"execution_count": null
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"p=[10. 14.]\n"
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]
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}
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],
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"execution_count": 4
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},
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{
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"cell_type": "markdown",
@@ -191,7 +225,11 @@
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{
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"cell_type": "code",
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"metadata": {
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"trusted": true
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"trusted": true,
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"ExecuteTime": {
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"end_time": "2026-02-08T11:48:54.022625Z",
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"start_time": "2026-02-08T11:48:54.014821Z"
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}
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},
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"source": [
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"p0 = np.array([1.0, 2.0])\n",
@@ -204,8 +242,16 @@
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"\n",
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"print(f\"p_={p_}\")"
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],
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"outputs": [],
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"execution_count": null
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"p_=[0. 0.]\n"
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]
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}
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],
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"execution_count": 5
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},
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{
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"cell_type": "markdown",
@@ -234,7 +280,11 @@
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{
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"cell_type": "code",
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"metadata": {
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"trusted": true
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"trusted": true,
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"ExecuteTime": {
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"end_time": "2026-02-08T11:48:54.033749Z",
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"start_time": "2026-02-08T11:48:54.028105Z"
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}
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},
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"source": [
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"θ = pi/2\n",
@@ -244,8 +294,17 @@
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"\n",
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"print(f\"R={R}\")"
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],
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"outputs": [],
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"execution_count": null
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"R=[[ 6.123234e-17 -1.000000e+00]\n",
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" [ 1.000000e+00 6.123234e-17]]\n"
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]
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}
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],
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"execution_count": 6
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},
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{
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"cell_type": "markdown",
@@ -282,7 +341,11 @@
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{
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"cell_type": "code",
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"metadata": {
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"trusted": true
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"trusted": true,
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"ExecuteTime": {
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"end_time": "2026-02-08T11:48:54.053367Z",
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"start_time": "2026-02-08T11:48:54.039405Z"
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}
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},
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"source": [
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"θ = pi/2\n",
@@ -296,8 +359,18 @@
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"\n",
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"print(f\"H = {H}\")"
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],
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"outputs": [],
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"execution_count": null
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"H = [[ 6.123234e-17 -1.000000e+00 1.000000e-01]\n",
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" [ 1.000000e+00 6.123234e-17 2.000000e-01]\n",
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" [ 0.000000e+00 0.000000e+00 1.000000e+00]]\n"
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]
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}
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],
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"execution_count": 7
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},
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{
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"cell_type": "markdown",
@@ -341,7 +414,11 @@
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{
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"cell_type": "code",
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"metadata": {
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"trusted": true
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"trusted": true,
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"ExecuteTime": {
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"end_time": "2026-02-08T11:48:54.067078Z",
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"start_time": "2026-02-08T11:48:54.054258Z"
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}
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},
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"source": [
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"Rz = np.array([[cos(θ),-sin(θ), 0],\n",
@@ -365,8 +442,24 @@
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"# A rotation about x\n",
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"print(f\"Rx={Rx}\")"
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],
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"outputs": [],
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"execution_count": null
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Rz=[[ 6.123234e-17 -1.000000e+00 0.000000e+00]\n",
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" [ 1.000000e+00 6.123234e-17 0.000000e+00]\n",
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" [ 0.000000e+00 0.000000e+00 1.000000e+00]]\n",
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"Ry=[[ 6.123234e-17 0.000000e+00 1.000000e+00]\n",
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" [ 0.000000e+00 1.000000e+00 0.000000e+00]\n",
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" [-1.000000e+00 0.000000e+00 6.123234e-17]]\n",
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"Rx=[[ 1.000000e+00 0.000000e+00 0.000000e+00]\n",
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" [ 0.000000e+00 6.123234e-17 -1.000000e+00]\n",
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" [ 0.000000e+00 1.000000e+00 6.123234e-17]]\n"
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]
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}
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],
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"execution_count": 8
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},
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{
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"cell_type": "markdown",
@@ -376,20 +469,24 @@
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"\n",
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"Compositions in SO(3) follow the same rules as SO(2), where sequential rotations are represented by right multiplications. For instance,\n",
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"\n",
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"$$\\mymatrix{R}_c = \\mymatrix{R}_a\\mymatrix{R}^a_b\\mymatrix{R}^b_c.$$\n",
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"$$\\mymatrix{R}^0_c \\mymatrix{R}^0_a\\mymatrix{R}^a_b\\mymatrix{R}^b_c.$$\n",
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"\n",
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"This can be represented by the following equivalent piece of code using sample angles."
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]
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},
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{
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"cell_type": "code",
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"metadata": {
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"trusted": true
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"trusted": true,
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"ExecuteTime": {
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"end_time": "2026-02-08T11:48:54.082984Z",
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"start_time": "2026-02-08T11:48:54.068586Z"
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}
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},
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"source": [
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"θa = pi/4\n",
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"Ra = np.array([[cos(θa),-sin(θa), 0],\n",
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" [sin(θa), cos(θa), 0],\n",
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"θ0_a = pi/4\n",
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"R0_a = np.array([[cos(θ0_a),-sin(θ0_a), 0],\n",
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" [sin(θ0_a), cos(θ0_a), 0],\n",
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" [0, 0, 1]])\n",
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"\n",
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"θa_b = -pi/2\n",
@@ -402,11 +499,21 @@
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" [0, cos(θb_c), -sin(θb_c)],\n",
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" [0, sin(θb_c), cos(θb_c)]])\n",
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"\n",
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"Rc = Ra @ Ra_b @ Rb_c\n",
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"print(f\"Rc={Rc}\")"
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"R0_c = R0_a @ Ra_b @ Rb_c\n",
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"print(f\"R0_c={R0_c}\")"
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],
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"outputs": [],
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"execution_count": null
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"R0_c=[[ 4.32978028e-17 -9.23879533e-01 -3.82683432e-01]\n",
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" [ 4.32978028e-17 3.82683432e-01 -9.23879533e-01]\n",
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" [ 1.00000000e+00 2.34326020e-17 5.65713056e-17]]\n"
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]
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}
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],
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"execution_count": 9
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},
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{
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"cell_type": "markdown",
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{
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"cell_type": "code",
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"metadata": {
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"trusted": true
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"trusted": true,
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"ExecuteTime": {
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"end_time": "2026-02-08T11:48:54.180705Z",
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"start_time": "2026-02-08T11:48:54.083625Z"
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}
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},
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"source": [
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"Ra_ = Rc @ Rb_c.T @ Ra_b.T \n",
@@ -430,8 +541,20 @@
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"else:\n",
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" print('The results are too different.')\n"
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],
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"outputs": [],
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"execution_count": null
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"outputs": [
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{
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"ename": "NameError",
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"evalue": "name 'Rc' is not defined",
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"output_type": "error",
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"traceback": [
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"\u001B[31m---------------------------------------------------------------------------\u001B[39m",
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"\u001B[31mNameError\u001B[39m Traceback (most recent call last)",
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"\u001B[36mCell\u001B[39m\u001B[36m \u001B[39m\u001B[32mIn[10]\u001B[39m\u001B[32m, line 1\u001B[39m\n\u001B[32m----> \u001B[39m\u001B[32m1\u001B[39m Ra_ = \u001B[43mRc\u001B[49m @ Rb_c.T @ Ra_b.T \n\u001B[32m 3\u001B[39m \u001B[38;5;28;01mif\u001B[39;00m np.allclose(Ra, Ra_):\n\u001B[32m 4\u001B[39m \u001B[38;5;28mprint\u001B[39m(\u001B[33m'\u001B[39m\u001B[33mThe results are pretty much the same!\u001B[39m\u001B[33m'\u001B[39m)\n",
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"\u001B[31mNameError\u001B[39m: name 'Rc' is not defined"
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]
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}
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],
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"execution_count": 10
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},
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{
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"cell_type": "markdown",

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