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Murilo Marinho
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[lesson2] Added exercise d.
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Lines changed: 169 additions & 23 deletions

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‎basic_lessons/lesson2_exercise_answers.ipynb‎

Lines changed: 112 additions & 14 deletions
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"source": "# Valid imports"
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},
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{
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"metadata": {},
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"metadata": {
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"ExecuteTime": {
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"end_time": "2026-01-29T22:31:23.725947Z",
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"start_time": "2026-01-29T22:31:23.709145Z"
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}
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},
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"cell_type": "code",
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"source": [
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"from math import pi, sin, cos\n",
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"import numpy as np"
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],
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"outputs": [],
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"execution_count": null
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"execution_count": 13
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},
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{
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"metadata": {},
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]
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},
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{
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"metadata": {},
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"metadata": {
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"ExecuteTime": {
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"end_time": "2026-01-29T22:31:23.749857Z",
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"start_time": "2026-01-29T22:31:23.733624Z"
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}
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},
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"cell_type": "code",
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"source": [
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"θ_a = pi/4.0\n",
@@ -74,16 +84,30 @@
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"# Printing the result is NOT a mandatory part of the answer.\n",
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"print(f'R_a = {R_a}')"
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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_a = [[ 0.70710678 -0.70710678]\n",
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" [ 0.70710678 0.70710678]]\n"
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]
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}
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],
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"execution_count": 14
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},
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{
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"metadata": {},
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"cell_type": "markdown",
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"source": "## Exercise b"
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},
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{
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"metadata": {},
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"metadata": {
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"ExecuteTime": {
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"end_time": "2026-01-29T22:31:23.765717Z",
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"start_time": "2026-01-29T22:31:23.750326Z"
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}
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},
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"cell_type": "code",
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"source": [
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"θ_b1 = pi/12.0\n",
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"\n",
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"print(f'R_b = {R_b}')"
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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_b = [[ 0.25881905 0.96592583]\n",
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" [-0.96592583 0.25881905]]\n"
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]
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}
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],
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"execution_count": 15
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},
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{
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"metadata": {},
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"cell_type": "markdown",
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"source": "## Exercise c"
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},
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{
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"metadata": {},
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"metadata": {
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"ExecuteTime": {
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"end_time": "2026-01-29T22:31:23.779702Z",
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"start_time": "2026-01-29T22:31:23.766189Z"
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}
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},
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"cell_type": "code",
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"source": [
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"θ_c = pi/3.0 # As given in the exercise\n",
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"x_c = 2.0 # As given in the exercise\n",
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"y_c = 5.0 # As given in the exercise\n",
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"θ_c = pi/3.0\n",
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"x_c = 2.0\n",
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"y_c = 5.0\n",
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"\n",
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"H_c1 = np.array([[cos(θ_c),-sin(θ_c), 0],\n",
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" [sin(θ_c), cos(θ_c), 0],\n",
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"\n",
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"print(f'H_c = {H_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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"H_c = [[ 0.5 -0.8660254 -3.33012702]\n",
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" [ 0.8660254 0.5 4.23205081]\n",
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" [ 0. 0. 1. ]]\n"
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]
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}
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],
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"execution_count": 16
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},
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{
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"metadata": {},
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"cell_type": "markdown",
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"source": "## Exercise d"
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},
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{
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"metadata": {
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"ExecuteTime": {
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"end_time": "2026-01-29T22:31:23.798367Z",
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"start_time": "2026-01-29T22:31:23.780110Z"
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}
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},
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"cell_type": "code",
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"source": [
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"θ_d = pi/3.0\n",
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"x_d = 2.0\n",
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"y_d = 5.0\n",
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"\n",
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"\n",
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"H_d1 = np.array([[1,0,x_d],\n",
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" [0,1,y_d],\n",
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" [0,0,1]])\n",
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"\n",
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"H_d2 = np.array([[cos(θ_d),-sin(θ_d), 0],\n",
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" [sin(θ_d), cos(θ_d), 0],\n",
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" [0, 0, 1]])\n",
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"\n",
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"\n",
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"H_d = H_d1 @ H_d2\n",
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"\n",
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"print(f'H_d = {H_d}')"
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],
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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_d = [[ 0.5 -0.8660254 2. ]\n",
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" [ 0.8660254 0.5 5. ]\n",
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" [ 0. 0. 1. ]]\n"
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]
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}
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],
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"execution_count": 17
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},
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{
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"metadata": {},
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"cell_type": "markdown",
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"source": "`H_c` is *not* the same as `H_d`. This indicates that the order of operations matter. That is, sequential pose transformations are not commutative."
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}
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],
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"metadata": {

‎basic_lessons/lesson2_tutorial.ipynb‎

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@@ -253,16 +253,27 @@
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"source": [
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"# 2D poses (combined translation/orientation)\n",
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"\n",
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"2D poses can be represented using elements of SE(2). Translations and rotations can be combined into a single $\\mymatrix{H}\\in\\mathbb{R}^{3 \\times 3}$ with the following structure\n",
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"\n",
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"$$\\mymatrix{H}(x,y,\\theta) = \\left[\\begin{array}{ccc}\n",
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"2D poses can be represented using elements of SE(2). A translation followed by a rotation can be combined into a single $\\mymatrix{H}\\in\\mathbb{R}^{3 \\times 3}$ with the following structure\n",
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"\n",
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"$$\\mymatrix{H}(x,y,\\theta) =\n",
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"\\left[\\begin{array}{ccc}\n",
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" 1 & 0 & x \\\\\n",
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" 0 & 1 & y \\\\\n",
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" 0 & 0 & 1\n",
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" \\end{array}\\right]\n",
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"\\left[\\begin{array}{ccc}\n",
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" \\cos{\\theta} & -\\sin{\\theta} & 0 \\\\\n",
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" \\sin{\\theta} & \\cos{\\theta} & 0 \\\\\n",
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" 0 & 0 & 1\n",
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" \\end{array}\\right] =\n",
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"\\left[\\begin{array}{ccc}\n",
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" \\cos{\\theta} & -\\sin{\\theta} & x \\\\\n",
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" \\sin{\\theta} & \\cos{\\theta} & y \\\\\n",
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" 0 & 0 & 1\n",
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" \\end{array}\\right].$$\n",
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"\n",
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"<div class=\"alert alert-block alert-info\">\n",
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"The pose written as a homogenous matrix shown above represents a translation followed by a rotation. To see the difference, check the exercises at the end of this lesson.\n",
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"The translation and rotation order is extremelly important. Check the exercises at the end of this lesson.\n",
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"</div>\n",
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"\n",
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"For $\\theta = \\frac{\\pi}{2}$, $x = 0.1$, and $y = 0.2$, we have the following equivalent piece of code."
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"source": [
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"## Compositions of rotations in SO(3)\n",
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"\n",
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"Compositions in SO(3) follow the same rules as SO(2), where sequential frame transformations are represented by right multiplications. For instance,\n",
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"Compositions in SO(3) follow the same rules as SO(2), where sequential rotations are represented by right multiplications. For instance,\n",
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"\n",
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"$$\\mymatrix{R}_c = \\mymatrix{R}_a\\mymatrix{R}^a_b\\mymatrix{R}^b_c.$$\n",
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"\n",
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"source": [
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"# 3D poses\n",
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"\n",
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"3D poses are represented in SE(3) with matrices $\\mymatrix{H}\\in\\mathbb{R}^{4 \\times 4}$ with the following structure\n",
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"\n",
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"$$\\mymatrix{H}(\\myvec{t},\\mymatrix{R}) = \\left[\\begin{array}{ccc}\n",
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"A translation followed by a rotation in 3D is represented in SE(3) with matrices $\\mymatrix{H}\\in\\mathbb{R}^{4 \\times 4}$ with the following structure\n",
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"\n",
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"$$\\mymatrix{H}(\\myvec{t},\\mymatrix{R}) =\n",
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"\\left[\\begin{array}{ccc}\n",
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" \\mymatrix{I} & \\myvec{t} \\\\\n",
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" \\myvec{0} & 1\n",
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" \\end{array}\\right]\n",
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"\\left[\\begin{array}{ccc}\n",
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" \\mymatrix{R} & \\myvec{0} \\\\\n",
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" \\myvec{0} & 1\n",
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" \\end{array}\\right] =\n",
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"\\left[\\begin{array}{ccc}\n",
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" \\mymatrix{R} & \\myvec{t} \\\\\n",
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" \\myvec{0} & 1 \n",
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" \\end{array}\\right].$$"
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" \\end{array}\\right].$$\n",
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"\n",
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"<div class=\"alert alert-block alert-info\">\n",
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"The translation and rotation order is extremelly important.\n",
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"</div>"
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]
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},
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{
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],
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"outputs": [],
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"execution_count": null
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},
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{
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"metadata": {},
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"cell_type": "markdown",
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"source": [
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"## Exercise d\n",
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"\n",
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"Consider the same variables as in `Exercise c` by replacing the subscripts with `d`. Calculate, instead, the rotation followed by the translation. Store the result in the variable `H_d` shown in the cell below.\n",
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"\n",
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"Is `H_c` the same as `H_d`? What does that indicate?"
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]
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},
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{
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"metadata": {},
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"cell_type": "code",
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"outputs": [],
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"execution_count": null,
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"source": [
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"θ_d = pi/3.0 # As given in the exercise\n",
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"x_d = 2.0 # As given in the exercise\n",
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"y_d = 5.0 # As given in the exercise\n",
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"\n",
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"H_d = None # Replace None with your solution to this exercise."
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]
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}
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],
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"metadata": {

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