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## Homework 04 - Rigid motion as a coordinate transformation

#### Motivation

In this task, we will study the effect of motion (rotation and translation) on a rigid body. To study this, we construct a very simple rigid body $r_1$ consisting of only two points $r_1=\{O, X\}$. To describe the relative position of these two points, we introduce a coordinate system $(O, \beta)$ with the origin at one of these points - $O$ and the standard basis $\beta$. The second point $X$ can thus be expressed as a vector $\vec{x}_\beta$ in this coordinate system. We set $\vec{x}_\beta = [1,2,3]^\top$. Now we apply the given motion on this rigid body represented by the coordinate system. That will result in a new coordinate system $(O', \beta')$. We construct a new rigid body $r_2 = \{O', Y\}$ where $Y$ has the same relative pose as $X$ in $r_1$, i.e. $\vec{y}'_{\beta'} = [1,2,3]^\top$, where $\vec{y}'$ is a vector expressing $Y$ in $(O', \beta')$. Finally, we apply the motion on point $X$ denoting it as $Z$. What are the coordinates of point $Z$?

Use Python to solve the following problems related to rigid motion. Use different colors to display your results.

1. Simulate the rigid motion with matrix $R$ and translation $\overrightarrow{O'O}_{\beta'}$ prescribed by Equation 5.4 in PRO-Lecture.pdf.
# approximate rotation
R = np.array([[0.8047,   -0.5059,   -0.3106],
[0.3106,    0.8047,   -0.5059],
[0.5059,    0.3106,    0.8047]])
# less approximate rotation
u, s, vh = np.linalg.svd(R)
R = u.dot(vh)
# translation O'O_β'
o_beta_pr = np.array([[1], [1], [1]])
1. Basis $\beta$ equals the standard basis $\sigma$. $O$=[0, 0, 0]
2. Find the coordinates of vectors of $\beta'$ in $\beta$ and vice versa.
3. Plot vectors of $\beta$ and $\beta'$ in the standard basis.
4. Plot coordinate systems ($O$, $\beta$) and ($O'$, $\beta'$ ). i.e. plot the basic vectors as bound vectors originating from points $O$ and $O'$, respectively.
5. Plot the bound vector $\vec{x}_\beta$ = $[1,2,3]^\top$ representing point $X$ in ($O$, $\beta$ ).
6. Plot the position vector in ($O'$, $\beta'$ ) of point $Y$ represented in ($O'$, $\beta'$ ) by vector $\vec{y}_{\beta'}$ = $[1,2,3]^\top$.
7. Consider point $Z$, where $X$ moves by the motion given above. Plot the bound vector representing the point $Z$ w.r.t. ($O$, $\beta$).

Create an empty dictionary in Python:

solution = {}

The keys for this dictionary will be:

• “b1_beta_pr” (3×1), “b2_beta_pr” (3×1), “b3_beta_pr” (3×1) containing the coordinates of vectors of $\beta$ in $\beta'$
• “b1_pr_beta” (3×1), “b2_pr_beta” (3×1), “b3_pr_beta” (3×1) containing the coordinates of vectors of $\beta'$ in $\beta$
• “oz_beta” (3×1) containing the coordinates of $\overrightarrow{OZ}_\beta$.

Finally, save solution to hw04.json:

import json
with open("hw04.json", "w") as outfile:
json.dump(solution, outfile)
Upload a zip archive hw04.zip (via the course ware) containing the following files:

1. hw04.py - python script used for computation
2. hw04.pdf - report file describing your solution with all figures
3. hw04.json