Transformation Matrix

Hi,

I read Vougaris and Trowbridge some time ago and Lohrmann, et. al. 1994 (N4000-702) last night. I am confused as to what the elements of the transformation matrix really are. I derived what I thought are relevant to the vector as described in the manual and on this board. Basically the projections of the measured velocity vector onto planes and an orhtogonal coordinate system. N4000-702 has some equations but I believe that they are wrong. I am hoping that someone out there can give me either the definition of the angles used in N4000-702 (I am assuming that they are the angle from the z-axis wertically downward to the center of the measurement beam). If this is the case, then the transformation martrix as stated gives the wrong values. The first problem that I see is that it is transposed, assuming standard linear algebra rules. The second problem is the existance of the 60 deg offsett in T_21 and T_22 rather than a 120 deg offset. The third problem that I see is the lack of negative signs or the ability of the offset angle to produce a negative value through the properties of the Sin or Cos.

If I use angles as they define, from the z-axis pointing from the measurement volume to the transmitter, and define the x-axis (probe coordinates) as positive along the sensor for beam 1, then I get the following in my derivation. Note that I am neglecting the heading angle and only transforming from the Measured to the probe coordinates, i.e. theta = 0.

T_11 = sin phi_1
T_21 = sin phi_1 cos 90
T_31 = cos phi_1
T_12 = sin phi_2 cos 120
T_22 = sin phi_2 cos 30
T_32 = cos phi_2
T_13 = sin phi_3 cos 240
T_23 = sin phi_3 cos 150
T_33 = cos phi_3

I did this before I read N4000-702 so my angles to rotate about the z - axis use all cos and disregard theta.

Constructing this gives

T = | T_11 T_12 T_13 |
   | T_21 T_22 T_23 |
   | T_31 T_32 T_33 |

and

V_probe = T * V_measured

So my question is is this correct or can someone send me a complete description of the transformation matrix.

Cheers,

Dave

Hi,

I would like to re-open this question if possible. Not really with regard to Atle's paper, but more operationally.

In my transformation matrix, I get T = ...

| 10925  -5442  -5486 |
|        28    9424  -9447 |
|   1414    1402    1414 |

Sine and Cosine are limited to -1 to +1. How do I relate the values reported to the instrument back to the angles? I want to do a rough check. Is there a decimal point ommitted in what is reported?

Thanks,
Dave

Dear David

The values in the T-matrix provided in the head configration file scales with 4096.  In other words, 10925 is really 2.67

If we describe the Vector beam geometry like this:

r1 = sin(alpha1) * ex + cos(alpha1) * ez
r2 = sin(alpha2) * sin(120) * ex + sin(alpha2) * cos(120) *ey + cos (alpha2) *ez
r3 = sin(alpha3) * sin(240) * ex + sin(alpha3) * cos(240) *ey + cos (alpha32) *ez

where ex, ey, ez are the unity vectors

then

V1,V2,V3 = T (r1,r2,r3) * U

where U is the velocity vector, Vi are the measured velocity and T is the transformation matrix above.

What we normally measure is Vi, so the matrix in the head configuration file is actually the inverse of (T).  Tough to do in the old days but easy to find with Matlab " />'>

The nominal inclination angles of the three receiver beams 1,2, and 3 is 15 degrees but they vary between 14 and 15.5 degrees and must be calibrated prior to shipping. This is why the matrix is little different for each Vector probe.

- Atle Lohrmann