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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.

- 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];

- Basis β equals the standard basis σ. O=[0; 0; 0]
- Find the coordinates of vectors of β' in β and vice versa.
- Plot vectors of β and β' in the standard basis, list the numeric values.
- 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.
- Plot the bound vector X
_{β}= [1;2;3] representing point X in (0, β), list the numeric values. - 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. - 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

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

courses/pro/labs/hw03.txt · Last modified: 2019/12/10 15:47 by steidsta