|
|
| Line 1: |
Line 1: |
| - | {| style="color:Gold;background-color:#500050;border: none" cellpadding="4" cellspacing="5" border="5" width="99%" align="center"
| + | delete this page please |
| - | !align="center";style="border: none;" |[[Image:WashingtonColorSeal-21-clip.gif|50px]]
| + | |
| - | !align="center"; style="border: #6b0c6a inset 3px;" |[[Team:University_of_Washington|<font color="gold">Home</font>]]
| + | |
| - | !align="center"; style="border: #6b0c6a outset 3px;" |[[Team:University_of_Washington/Team|<font color="gold">The Team</font>]]
| + | |
| - | !align="center"; style="border: #6b0c6a outset 3px;" |[[Team:University_of_Washington/Project|<font color="gold">The Project</font>]]
| + | |
| - | !align="center"; style="border: #6b0c6a outset 3px;" |[[Team:University_of_Washington/Modeling|<font color="gold">Modeling</font>]]
| + | |
| - | !align="center"; style="border: #6b0c6a outset 3px;" |[[Team:University_of_Washington/Notebook|<font color="gold">Notebook</font>]]
| + | |
| - | !align="center"; style="border: #6b0c6a outset 3px;" |[[Team:University_of_Washington/Protocols|<font color="gold">Protocols</font>]]
| + | |
| - | !align="center"; style="border: #6b0c6a outset 3px;" |[[Team:University_of_Washington/Parts|<font color="gold">Parts Submitted <br>to the Registry</font>]]
| + | |
| - | !align="center"; style="background-color:#03943d;border: #6b0c6a outset 3px;" |[[Team:University_of_Washington/Measurement Kit|<font color="gold">Measurement Kit</font>]]
| + | |
| - | !align="center"; style="background-color:#c42317;border: #6b0c6a outset 3px;" |[[Team:University_of_Washington/SeToB|<font color="gold">SeToB</font>]]
| + | |
| - | !align="center"; style="border: #6b0c6a outset 3px;" |[[Team:University_of_Washington/Safety|<font color="gold">Safety</font>]]
| + | |
| - | |}
| + | |
| - | | + | |
| - | {|style="color:Gold;background-color:#FFdd00; border: none" cellpadding="5" cellspacing="5" border="0" width="99%" align="center"
| + | |
| - | | + | |
| - | <html>
| + | |
| - | <head>
| + | |
| - | <style>
| + | |
| - | #globalWrapper {
| + | |
| - | margin: 0px;
| + | |
| - | padding: 0px;
| + | |
| - | background-color:#500050;
| + | |
| - | }
| + | |
| - | #content{
| + | |
| - | background:#FFdd00;
| + | |
| - | border-left-color:#FFdd00;
| + | |
| - | border-right-color:#FFdd00;
| + | |
| - | }
| + | |
| - | #footer-box {
| + | |
| - | border-top-color:#FFdd00;
| + | |
| - | border-right-color:#FFdd00;
| + | |
| - | border-bottom-color:#FFdd00;
| + | |
| - | border-left-color:#FFdd00;
| + | |
| - | background:#FFdd00;
| + | |
| - | }
| + | |
| - | </style>
| + | |
| - | </head>
| + | |
| - | </html>
| + | |
| - | | + | |
| - | ==Mathematical Modeling==
| + | |
| - | ===Alpha===
| + | |
| - | ====Problem====
| + | |
| - | Given the following construction and point ''p'', or (''x'',''y'',''z'') find the angles ''θ<sub>1</sub>'', ''θ<sub>2</sub>'', and ''θ<sub>3</sub>''.
| + | |
| - | | + | |
| - | Note that positive ''z'' is the down direction.
| + | |
| - | | + | |
| - | ====Constants====
| + | |
| - | *''TR''
| + | |
| - | **Top radius
| + | |
| - | *''BR''
| + | |
| - | **Bottom radius
| + | |
| - | *''L''
| + | |
| - | **Linkage
| + | |
| - | *''CA''
| + | |
| - | **Control Arm
| + | |
| - | *''φ<sub>1</sub>'' and ''φ<sub>2</sub>''
| + | |
| - | **Two angles
| + | |
| - | | + | |
| - | ====Construction====
| + | |
| - | From the top to the bottom:
| + | |
| - | *A circle centered at the origin with radius ''TR'', named ''O''
| + | |
| - | *Make three lines ''A<sub>1</sub>'', ''A<sub>2</sub>'', and ''A<sub>3</sub>'' such that:
| + | |
| - | **The lines ''A<sub>x</sub>'' are perpendicular to a tangent of ''O'' and a radius of ''O'',
| + | |
| - | **The angle between the radii of ''A<sub>1</sub>'' and ''A<sub>2</sub>'' is ''φ<sub>1</sub>'', and
| + | |
| - | **The angle between the radii of ''A<sub>2</sub>'' and ''A<sub>3</sub>'' is ''φ<sub>2</sub>''.
| + | |
| - | *Make a circle centered at point ''p'' with radius ''BR'', named ''P''
| + | |
| - | *Find three points ''P<sub>1</sub>'', ''P<sub>2</sub>'', and ''P<sub>3</sub>'' such that:
| + | |
| - | **They are on the circumference of ''P'',
| + | |
| - | **The angle between the radius which touches ''P<sub>1</sub>'' and the radius that touches ''P<sub>2</sub>'' is ''φ<sub>1</sub>'',
| + | |
| - | **The angle between the radius which touches ''P<sub>2</sub>'' and the radius that touches ''P<sub>3</sub>'' is ''φ<sub>2</sub>'',
| + | |
| - | **The ray from the center of ''P'' to ''P<sub>x</sub>'' is parallel to the ray from the center of ''O'' to the point which is on ''A<sub>x</sub>'' and ''O'' 's circumference, for all ''x''.
| + | |
| - | *Construct three line segments ''CA<sub>1</sub>'', ''CA<sub>2</sub>'', and ''CA<sub>3</sub>'' such that:
| + | |
| - | **''CA<sub>x</sub>'' is in the same plane as ''A<sub>x</sub>'', for all ''x'', and
| + | |
| - | **The angle between ''CA<sub>x</sub>'' and ''A<sub>x</sub>'' is ''θ<sub>x</sub>'', for all ''x''.
| + | |
| - | *The distance between ''P<sub>x</sub>'' and the end point of ''CA<sub>x</sub>'' that is not on ''A<sub>x</sub>'' is ''L'', for all ''x''.
| + | |
| - | | + | |
| - | ====Solution====
| + | |
| - | #Note that the control arms can only move in a circle, while linkage can move in a sphere.
| + | |
| - | #We will calculate ''θ<sub>1</sub>'' first, which only involves the points, circles and lines ''p'', ''CA<sub>1</sub>'', ''A<sub>1</sub>'', and ''P<sub>1</sub>''.
| + | |
| - | #Find the plane where ''CA<sub>1</sub>'' 's circle resides in. Use it to cut the sphere around ''P<sub>1</sub>''. For future reference, call ''CA<sub>1</sub>'' 's circle ''C<sub>1</sub>'' and the circle resulting from the cut ''C<sub>2</sub>''
| + | |
| - | #Define ''x<sub>o<sub>1</sub></sub>'', for x offset, for the difference in the ''x'' coordinates of the center of ''C<sub>1</sub>'' and the point ''P<sub>1</sub>''.
| + | |
| - | #*''x<sub>o<sub>1</sub></sub> = TR - (BR + x)''
| + | |
| - | #*''x<sub>o<sub>1</sub></sub> = TR - BR - x''
| + | |
| - | #Define ''D<sub>1</sub>'' to be the distance between the center of ''C<sub>1</sub>'' and ''C<sub>2</sub>''. It will also be the line connecting the centers.
| + | |
| - | #*''D<sub>1</sub> = sqrt(x<sub>1</sub><sup>2</sup> + z<sup>2</sup>)''
| + | |
| - | #Obviously, ''D<sub>1</sub>'', ''CA<sub>1</sub>'', and linkage form a triangle. The angle between ''CA<sub>1</sub>'' and ''D<sub>1</sub>'' is a close approximation to ''θ<sub>1</sub>'', but it is not exact. We will call this angle ''θ<sub>1<sub>1</sub></sub>''. We use the law of cosines to calculate ''θ<sub>1<sub>1</sub></sub>''.
| + | |
| - | #*''L<sup>2</sup> = CA<sup>2</sup> + D<sup>2</sup> - CA * D * cos(θ<sub>1<sub>1</sub></sub>)''<br><br>
| + | |
| - | #*''θ<sub>1<sub>1</sub></sub> = cos<sup>-1</sup>((D<sup>2</sup> + CA<sup>2</sup> - L<sup>2</sup>)/(2*D*CA))''<br><br>
| + | |
| - | #One endpoint of ''D<sub>1</sub>'' is on ''A<sub>1</sub>''. Thus, we can make another triangle, and the angle between ''D<sub>1</sub>'' and ''A<sub>1</sub>'' is the difference between ''θ<sub>1</sub>'' and ''θ<sub>1<sub>1</sub></sub>''. We will call this angle ''θ<sub>1<sub>2</sub></sub>''. If we have the length of the side on ''A<sub>1</sub>'' be ''z'', then the last side will be ''x<sub>o</sub>'' and the triangle will be a right angle triangle. Since only ''x<sub>o<sub>1</sub></sub>'' changes sign in the good interval, we should use a trigonometric function that involves ''x<sub>o</sub>'' in the numerator. Thus,
| + | |
| - | #*''θ<sub>1<sub>2</sub></sub> = sin<sup>-1</sup>(x<sub>o<sub>1</sub></sub>/D<sub>1</sub>)''<br><br>
| + | |
| - | #*''θ<sub>1</sub> = cos<sup>-1</sup>((D<sup>2</sup> + CA<sup>2</sup> - L<sup>2</sup>)/(2*D*CA)) - sin<sup>-1</sup>(x<sub>o<sub>1</sub></sub>/D<sub>1</sub>)''
| + | |
| - | #You just have to rotate the model ''φ<sub>1</sub>'' degrees counterclockwise for ''θ<sub>2</sub>'', and another ''φ<sub>2</sub>'' degrees for ''θ<sub>3</sub>'' using the two dimensional rotational matrixes
| + | |
| - | {''cos(φ<sub>2</sub>),-sin(φ<sub>2</sub>)''}
| + | |
| - | ''R<sub>1</sub>''={ }
| + | |
| - | {''sin(φ<sub>2</sub>) cos(φ<sub>2</sub>)''}
| + | |
| - | | + | |
| - | {''cos(φ<sub>3</sub>) -sin(φ<sub>3</sub>)''}
| + | |
| - | ''R<sub>2</sub>''={ }
| + | |
| - | {''sin(φ<sub>3</sub>) cos(φ<sub>3</sub>)''}
| + | |
"
