We have built an ultrasonic anemometer measuring 4 components of air velocity along the 4 space diagonals of a cube. The space diagonals can be characterized by vectors (1,1,1), (1,-1,1), (-1,1,1) and (-1,-1,1) in Cartesian coordinates. The measuring instrument delivers for each space diagonal direction a positive value once the air speed is upwards or negative once the air speed is directed downwards.
The mathematical challenge I am facing is to calculate the x, y and z components of air velocity from the set of 4 space diagonal components. There must be some sort of transformation from any three space diagonals to the right angle system. Actually there should be 4 of them which would allow for median values to be calculated. I am searching for this transformation and any help is highly appreciated.
For the technically interested: we have positioned 8 HR-SR04 sensors in the 8 corners of a cubic frame.
My solution is based on the following:
(x,y,z) be the air velocity vector. The space diagonals a,b,c,d are represented by a:(1,1,1), b:(1,-1,1), c:(-1,1,1) and d:(-1,-1,1).
The cosine of any projection angle of any axis to any space diagonal is 1/SQRT(3).
The air speed value along any given space diagonal is the matrix product of the space diagonals and the wind vector times the cosine of the projection angle.
For back calculation I just do a matrix multiplication of the inverted Matrix from the used three space diagonals with the vector composed of the three air speeds divided by the cosine of the projection angle.
At least in EXCEL this reconstructs the original wind vector. Can anyone approve?