## Determining the Point and Angle of Rotation

I've an irregular box that is set on a fixture. The fixture is capable to translate and rotate the box. The original position of the box is known, by probes located at each side. If the box is translated and rotated to a new known location, with same probes. How do I determine the point and the angle of rotation that leads to this new location?.

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 Recognitions: Homework Help Welcome to PF. You know the initial and final positions of the probes don't you? I don't understand your difficulty.
 Yes, old and new point coordinates (x, y, z) are known. My problem is how to map or to relate old and new data in order to extract the point and angles of rotation and translation.

Recognitions:
Homework Help

## Determining the Point and Angle of Rotation

Yes I realize what you hope to achieve, I don't understand, given that you have a complete description of the state (position and orientation) before and after the motion, what it is you find difficult about this. Where are you stuck?

For instance, you may relate the two positions/orientations via translation and rotation matrixes.... or just by differences as in: the object has moved distance d and rotated A degrees about axis (x,y,z).

For instance - I could have an object with three rods sticking out which point to a common center ... plotting the ends of these rods will let me track the position and orientation of the object using coordinate geometry.

 The case is that body (box) is set, initially, randomly on a fixture where the cartesian coordinates (x, y, z) of certain points are recorded. According to these readings the final reading (the new position and orientation) are designed. To Achieve this goal (final position and orientation), a translation and rotation command is given to certain jacks in order to get the designed location. Note: Rotation point is not known. So, is it possible, mathematicaly to solve this problem? Thank you.
 Recognitions: Gold Member Science Advisor Staff Emeritus That's like asking "Pierre goes form Paris to Lyon. What route does he take?" There are a large number of possible routes and no way to determine one just knowing the first and last positions. There is no unique answer.
 May you give me one of these solutions in steps of symbol math?
 Here's how I'd try approaching the problem: break it into steps. 1) Find the rotation angle and plane (or axis, in 3D) You can do this by looking at a fixed-body reference frame before and after the rotation. Suppose the old frame is e, and the new frame is f. Each new basis vector is a linear combination of the old ones, $$f_i = R_i^{~j} e_j$$ and I believe the matrix elements are given by $$R_i^{~j} = f_i \cdot e^j$$ where $e^j$ is the j'th reciprocal basis vector. (Of course, you can choose an orthonormal frame so that $e^j = e_j$.) Once you have your rotation matrix, it should have one real eigenvalue of 1. Solve it for the corresponding eigenvector; this is your rotation axis. To get the angle, one way might be to find a vector orthogonal to the axis (i.e., in the plane of rotation). Then, rotate it by this matrix, and take the dot product with its unrotated self. This should give you cos theta. 2) Find your translation. The most general 3D motion consists of a rotation about an arbitrary point, plus a translation along the rotation axis. Now that you know the rotation axis, you can find that translation directly: it's the projection of the displacement of the center of mass along the rotation axis. So if your axis is $\hat{r}$, your displacement will be $t\hat{r} = ((x'-x)\cdot\hat{r})\hat{r}$, where $x' (x)$ is your final (original) center of mass. 3) Find the point of rotation. You now know the axis and angle. Find the distance (projected onto the plane of rotation!) that the center of mass has moved. The initial point $A=x$, the projected final point $B=x'-t\hat{r}$, and center of rotation C form an isosceles triangle. The angle ABC is the rotation angle, and the side AB is the projected distance your center of mass has moved. You can now solve for r, the distance of the center of mass from the center of rotation. DISCLAIMER: I don't really know if this approach is correct, since I haven't tried it. Anything I said could be flat-out wrong. But I hope it gives you some ideas. :)