In this task, we will study the effect of motion (rotation and translation) on a rigid body. The term rigid can be understood as the relative positions of individual parts do not change by this motion.

To study this, we construct a very simple rigid body r1 consisting of only two points r1={O, X}. To describe the relative position of these two points, we introduce a coordinate system (O, β) with the origin at one of these points - O and standard basis β. The second point X can thus be expressed as a vector X_β in this coordinate system. X_β = [1,2,3] in (O, β).

Now we apply the given motion on this rigid body represented by the coordinate system. That will result in a new coordinate system (O', β'). We construct a new rigid body r2 ={O', Y} where Y has the same relative pose as X in r1. Y_β' = [1,2,3] in (O', β').

Finally, we apply the motion on point X denoting it as Z.

What are the coordinates of point Z?

Use MATLAB 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 o_{\beta'} prescribed by Equation 5.4 in PRO-Lecture.pdf.

% approximate rotation
R = [0.8047   -0.5059   -0.3106
     0.3106    0.8047   -0.5059
     0.5059    0.3106    0.8047];

% less approximate rotation
[U,D,V] = svd(R);
R = U*V';

% translation o_β'
o = [1;1;1];

1. Basis β equals the standard basis σ. O=[0; 0; 0]
2. Find the coordinates of vectors of β' in β and vice versa.
3. Plot vectors of β and β' in the standard basis, list the numeric values.
4. Plot coordinate systems (O, β) and (O', β'). i.e. plot the basic vectors as bound vectors originating from points O and O', respectively, list the numeric values.
5. Plot the bound vector X_β = [1;2;3] representing point X in (0, β), list the numeric values.
6. Plot the position vector in (O', β') of point Y represented in (O', β') by vector Y_β' = [1;2;3], list the numeric values.
7. Consider point Z, where X moves by the motion given above. Plot the bound vector representing the point Z w.r.t. (O, β), list the numeric values.

Upload via the course ware the zip archive hw04.zip containing

1. hw04.pdf report file describing your solution with all figures
2. hw04.m MATLAB source code, which generates the results and figures for the report
3. all your additional MATLAB files required by hw04.m