How should I approach this (coordinate transformation) problem?

[Swulf]

I am starting to deal with optomechanical systems as part of my work, and am faced with what seems to be an uncomplicated problem, however I'm ashamed to admit that I am having great difficulty getting to grips with it. I'd like some pointers and/or advice as to how to go about solving these sorts of problems - please note, I want to *learn how to solve them*, I am not just looking for the 'answer'. It's the methods and approaches I want to understand so I can apply it to subsequent problems.

The system I am considering has a flat mirror with axis $X$, $Y$ and $Z$ as shown. Translations in the axial directions are allowed i.e. $\delta X$, $\delta Y$ and $\delta Z$. Rotations about the axes are also allowed, $\theta_X$, $\theta_Y$ and $\theta_Z$.

The articulated mirror assembly is to sit on another stage which can rotate by an angle $\phi_R$ and can tilt the entire mirror by $\phi_T$. Obviously $\theta_X = \phi_T$. The whole thing is intended to steer an incoming beam onto a surface by controlling $\phi_R$ and $\phi_T$ (i.e. $\theta_X$).

My difficulty in comprehension is how to determine the relationship between $\theta_Y$ and $\phi_R$. The system will not be able to move in $\theta_Y$, only in $\phi_R$, but some specifications on error have been given in terms of $\theta_Y$ and it is necessary to turn those into corresponding specifications on $\phi_R$. I am stumped as to how to properly transform between the two. It seems like it should be simple!

I would welcome any advice on how to properly determine the relationship between $\theta_Y$ and $\phi_R$.

Thank you,

Swulf

 

[Stephen Tashi]

. Obviously $\theta_X = \phi_T$.

I don't see that. If there is a rotation about the z-axis, the x-axis wil move. So it may no longer cooincide with the axis that the tilt is turning about.

The type of transformations you describe can written as the product of matrices. The order in which the transformations are applied will matter. You can't describe a a unique transformation just by giving the values of rotation angles and the translations because it matters which transformation is done first. If you have equipment that is simultaeneously making the rotations about several axes, I'll boldy say that this also can be handled - athough I haven't tried to this myself.

 

[Swulf]

Hi Stephen,

Thanks for replying!

I think I should have said that the $\phi_T$ angle is always about the $X$ axis that is defined on the mirror. So therefore it is the same angle; it is only defined as $\phi_T$ to give it similar nomenclature to $\phi_Z$.

I'm having trouble, though, seeing what the first step would be in constructing a transformation between a rotation about $\theta_Y$ and a rotation about $\phi_Z$. I guess I don't fundamentally understand the problem at all well... am I to start by considering how a point in the $X$, $Y$, $Z$ coordinate system moves when subjected to a rotation about $Y$, and then somehow map that into the coordinate system associated with $\phi_Z$, $\phi_T$? I'm lost.

 

[Stephen Tashi]

We need a better definition of what you are trying to accomplish. As I said, the values of the various angles and translations you listed do not define a specific transformation, so you must pick some order for them to be applied or you must say that you are trying to deal with a control system where all of them change simultaneously.

There is some ambiguity in your definitions of values. For example, if I perform the rotation $\theta_z$ then perform the rotation $\theta_x$, what axis do I use for the $\theta_x$ rotation? Do I used the current position of the x-axis where it landed after the rotation about the z-axis or do I use the original position of the x-axis before the rotation about the z-axis?

It sounds like what you are doing is similar to the problem of determining the coordinates of the wingtip of an aircraft given information about roll, pitch and yaw.

 

[CellCoree]

First of all, don't be ashamed to admit that you are having difficulty with this problem. As a scientist, it is natural to encounter challenges and obstacles in our work. The important thing is to approach the problem with a clear and systematic mindset, and to be open to learning new methods and approaches.

One approach to solving this problem is to use coordinate transformations. In optomechanical systems, it is common to use the Denavit-Hartenberg (DH) convention to define the coordinate frames and transformations between them. This convention is widely used in robotics and mechanics as well, so understanding it can be useful in various applications.

To start, you will need to define the coordinate frames for your system. In this case, you have three rotational axes (X, Y, and Z) and three translational axes (\delta X, \delta Y, and \delta Z). These axes should be defined in a way that is consistent with the DH convention.

Next, you will need to determine the transformation matrices between these frames. This can be done by defining the DH parameters for each frame and using them to calculate the transformation matrix. There are many resources available online that explain the DH convention and how to calculate the transformation matrices.

Once you have the transformation matrices, you can use them to determine the relationship between \theta_Y and \phi_R. Specifically, you can use the inverse kinematics equations to relate the two angles. This will allow you to convert any given specification on \theta_Y to an equivalent specification on \phi_R.

In summary, the key to solving this problem is to understand the DH convention and use it to determine the transformation matrices between the coordinate frames. From there, you can use inverse kinematics to relate the angles and convert between specifications. I hope this helps and best of luck in your work with optomechanical systems!