As the antenna wheels follow the various ups and downs of the track, the tops of the A-frames move both up & down and back & forth by amounts that would be calculable exactly if the A-frames were rigid. In practice some of this translation is lost in the flexibility of the structures. Yaw is the lateral motion of the ends of the elevation axis. To extract it from the currently available inclinometry configuration we must use the pitch of the bottom beams of the A-frames, the geometry of the A-frames, and some factor representing the loss of translation due to the flexibility.

The A-frames are essentially triangles, with :

• a base of 7320mm, and
• a height of the apex above the base of 8150mm.
• The length of the elevation axis is (no rounding here !) 8000mm.

These formulae depend not only upon the rigidity of the A-frames, but also on perfect coupling between the A-frames and the elevation axis. Since the physical structures of the ends of the elevation axis are quite different we might expect in reality that the contributions of the A-frame motions to that of the elevation axis may differ, and that neither coupling will be perfect. The right end, for instance, houses the elevation encoder , so its performance may also impact this scaling factor. The left end is more passive, and comprises a screw joint - whose effect upon the sway of the left A-frame has been seen in inclinometry in times past.

We estimate the flexibility loss factor by comparing the pitch and roll observed using the TMU inclinometers with the pitch and roll that might be predicted using the A-frame pitches, rigid A-frame geometry and perfect coupling.

The relationships between the pitch of the elevation axis (as measured by the Y-axis of the inclinometer on the TMU, TY), and those of the A-frames (LY, RY) are shown in the figures below, where a traverse through 450 degrees of azimuth is followed :

Clearly, TY seems independent of LY, and strongly correlated with RY (via the encoder) with a flexibility factor of 0.80, determined by least squares fitting to all the data and shown in red.

The roll relationships (TX -vs- LX and RX) are shown below :

For TX -vs- LX the best-fitting straight line is not drawn (although the formal slope is seen to be 1.09). The data appear in sections to have a slope of about 0.90 or 0.95. The green lines of slope 0.90 are there for guidance. For TX -vs- RX, the data almost all fall on one straight line of formal slope 0.83, but clearly the bulk of the data satisfy a steeper slope - and again the green lines of slope 0.90 are meant to guide the eye to this conclusion also.

Conclusions

• If we had to calculate TX from LX and RX, we would include a flexibility factor of 0.90 - i.e. we would adopt a formula of
            TX = 0.90 * (RX + LX)/2
• The antenna appears to move en masse in the X-direction - i.e. in the roll direction.
• LX seems to suffer from more random changes than do either RX or TX.
• If we had to calculate TY from LY and RY, we would probably ignore LY and adopt
            TY = 0.80 * RY.
• The mean flexibility factor appears to be about 0.85 - in accordance with our current usage.
• To calculate yaw from that expected from the rocking of the A-frames (LY & RY), we should therefore use
            yaw = 0.85 * 0.46 * (LY - RY)
  i.e.           yaw = 0.39 * (LY - RY).
• Although we have confirmed the flexibility factor, it appears that through the years we (I) have forgotten to include the geometrical factor in the calculation of yaw, so I changed this factor in the appropriate track model programs on 970911.
  • 970912 - last night's pointing data was horrible : rms scatters in (daz,del) of (3.5,3.1) - since the elevation data was also poor I can't believe that the azimuth scatter was due to this new model, but I have withdrawn it - just in case , but also on the basis of the `yaw-factor' analysis below.
• (980403) Alternately, if, like pitch, both roll and yaw are disconnected from the left A-frame, 0.90*RX would represent roll (TX) equally well, while yaw would be -0.39*RY. The effects of this formulation are explored in the report examining the tracking data of 980225.

Yaw-Factor
The data of 970630_20 were corrected from the model with which they were taken (model1-970630.dat) onto models created from the same inclinometry but using various yaw factors. The results are shown in the table below :

       yaw factor   0.5  0.6  0.7  0.8  0.9  1.0  1.1  1.2
       rms in raz   2.0  1.9  1.9  1.8  1.8  1.9  1.9  2.0

So while the curve of rms residuals against yaw factor is rather shallow, the previously adopted value of 0.83 seems (coincidentally ?) useful.