Abstract:

I performed actuator diagonalization for ETMX MN actuators.
After diagonalization, large coupling between L and T seems disappeared.

Detail:

First, I balanced H1/H2/H3 and V1/V3 actuator by using TM OpLevs.
Balancing method is just adding offset from MN_TEST_{L,T,V} and change the COILOUTF gain so that the TM can keep the same angle before and after adding offset (since V2 actuator is used only for rol actuation, we don't need balancing).
Then, we started actuator diagonalization.

For the actuator decoupling, I used the same resonances as I used for sensor decoupling (klog29743).
What we would like to know is coupling matrix (A), which can be defined as follows:

where F and F' represent diagonalized force vector and current force vector (force vector without decoupling), respectively.
Then, actuator decoupling matrix can be obtained as A^-1.

To measure coupling, matrix, I injected white noise around the targe resonances from TEST_{L,T,V,R,P,Y} and measured transfer functions from excitation signals to each sensor at resonant frequency.
If actuator coupling doesn't exist, L resonance (2.48 Hz) doesn't excited when injectiong white noise around 2.48 Hz from TEST_{T,V,R,P,Y}.
Therefore, if the 2.48 Hz peak can be seen in the transfer functions of T2L, V2L, and so on, those peaks should be excited by the actuator couplings.
So, each element of actuator coupling matrix can be obtained by calculating T2L/L2L, V2L/L2L, and so on.
However in practice, different from the sensor decoupling, it is not good to use the value at resonant frequencies for calculating each element because the time when measuring L2L, T2L, and so on, is different, and reproducibility of TF gain at resonant frequencies is not good.
So, I used a slope of resonances to calculate the matrix elements.

Figure 1 and 2 show examples of P2P and R2P transfer functions, respectively, around pitch resonance (7.4 Hz), and following table shows TF gain at 7.36 (lower slope), 7.4 (on resonance), and 7.43 (higher slope) Hz.

Though differemce of R2P/P2P value at lower and higher frequency is about 5%, the difference at slopes and on resonance is 40%, which is too large error to be used for actuator decoupling.

Also, there are several TF measurement results, which is not suitable for using both side slopes for matrix calculation (due to the coupling from the other DoFs, anti-resonance can be seen at very close frequency from the resonances).
Figure 3 and 4 show the examples of TFs of Y2Y and L2Y, respectively.
In this case, anti-resonance can be seen at the higher slope of L2Y TF, so only lower slope was used for calculating matrix element.

After the above calculation, coupling matrix is finally obtained as follows:

So, decopling matrix can be obtained as fig5 and implemented in ETMX_MN_ACTALIGN matrix.

After implementing ACTALIGN matrix, MN transfer function was measured.
Figure 6-11 show the measurement results.
Basically, coupling transfer function was reduced, so this decoupling method seems working well.

Note:

This time, I used photosensors for all DoFs but maybe it is better to use photosensors for translational DoFs and MN OpLevs for angular DoFs to obtain better SNR.
It takes about one day to check all measurement data (36 files) and make coupling matrix, so it is better to automate the process.
However, since there are some arbiterary choices to calculate coupling matrix element such as which (or both) slope and which frequencies should be used for calculation, it is hard to automate the matrix calculation at this moment.