Intro  

The general steps  to compute the tension from the sag surveys

• collect the p50 data
• exclude extraneous points (so you can manipulate the data).
• for the aug2020 surveys there were 14 to 255  million 3d points per data set. To exclude points:
• window in the xy, xz, or yz plane. This projection is easy since you just ignore the other 3d coordinate.
  
• select 2 points from a cable or set of cables and  use it to generate a line (in 2d or 3d space)
• Compute the perpendicular distance from each point to this line (dot product and then Pythagorean theorem).
• Use this  distance to exclude points
• These  things can typically reduce the points from 10's of million to a few 100 thousand.
  
• move 3d to 2d
• project the 3d points onto the xy place
• select 2 points on the  cables to generate a line
• Use the fit coef to rotate the xy line to the x axis. This leaves you with data points clustered around the x,z plane.
• select  points for each cable
• after rotating the 3d points to the xz plane, plot the 3d data in the y,z plane
• This will  separate the cables by their y distance.
• select 2 points on each cable, define a line, and compute the distance from all points to each line
• let the user iterate on the clipping distance.
  
• These points will be the starting point for fits to each cable.
  
• fit a catenary to the set of points for each cable.
• In the xz plane first fit a parabola to each cable set:
• zp=p0 + p1*x + p2*x^2
  
• Use the parabola coefs to compute the starting values for the catenary fit:
• zc= a + b*(cosh(x-c)/b -1)
  
• Use a Leven -marquardt non linear least squares fit  to fit  for the catenary coef a,b,c
  
• compute the tension from the catenary parameters and the cable slope.
• cableTension=b * cableWeight/linearDistance  * cos(cableZAngle)
• The cableZ angle was initially  taken from the linear coef of the xz parabolic fit.
• this gave xz cable angles that differed by up to a degree for the same set of main cables (pointed out by pierre ghisbain)
• i switched over to using the catenary fit and the actual cable length.
  
• For xmax use the x value for the maxium z value (this is at the tower)
• compute xmin=xmax -cable length (taking this from the drawings.
• This works since the platform side of the cable was usually the one blocked by the  platform or other cables.
  
• Evaluate the catenary fit at xmax,xmin,  get the slope and then the xz angle of the cable.
  
• projecting tensions to balance forces on platform and measure imbalances on the towers.
• The catenary fits is used to compute the xz vertical angle
• The linear fits in the x,y plane give the rotation angle to move the cables to the x,z plane
• The angles are biased by the y axis of the p50 scanner
• the difference in the angles tells the relative angle between the main and aux main cables.

Details of how the tension was computed

Collecting  the data

• leica xxx will move the project from the p50 to the windows computer
• cyclone can then be used to input the data files, plot the data, and then output the data as .las files.
• the .las files are then moved to Linux computers.
  

Idl software.

    Idl routines have been written to input the .las files and compute the tensions. The major processing steps are:

• Exclude points far from cables
  
• this can remove a large number of points
• select points that belong to each cable
  
• since the cables should hang vertically we can rotate the points to lie close to the xz plane
• Points are excluded by defining lines (via 2 points) and then computing the distance from all points for each line
• The final exclusion is by fitting a parabola to each set of cable points.
  
• fit a catenary to the points.

Inputting the data

• getxyzpnts() will input the data
• plcdinplas() will input the .las file
• an xyz[3,n] is then generated  by scaling the .las input data to floating point 

Localizing the cables:

    Most of this code is trying to exclude points that don't belong to the cable. findcables() tries to select points around the cables and reducing the # of points to a manageable number (typically going from 64e6 to around 2e5 points. preproccable() then does a more in depth attempt at selecting the points for each cable.

• findcables(xyz, p3, p3cbIar) will exclude most points that don't belong to a cable
• input xyz[3,n)
• output: p3[3,m] and p3cblIAr[ncables]
  
• plot all points in the  x,z plane. let the user exclude points using x and z ranges.
• user selects 2 points  in the x,z plane that lies along the middle of the cables.
• generate a line and compute the distance from all points to this line
• plot the distance from all points to this line. let the user exclude points by this distance.
• set p3[3,*] to the included points
• do a robust linear fit  to p3 points in the x,y plane.
• copy p3 to a rvec[3,m]  (a temp variable)
• rotate the rvec variable by the x,y slope (after removing the y offset)
• force the x,z slope to be positive.
• this should put the cables close to the x,z plane, only separated by their y offsets
• plot the rotated points y  vs z.  this will spread the cables out because of their y differences
• let the user select 2 points along each of the cables, generate a line for each cable
• loop through each cable
• plot the distance of all points to the line for this cable
• let the user specify the maximum distance to keep
• use this to select indices in p3[3,M] for this cable. store the indices in p3CblI[icable].pind (ptr to array of indices)
  
• return p3[*,m] and p3CblIar[ncables] holding the indices into p3 for each cable.
• preproccable(p3, p3cblIAr,index,fitI) This is called for each cable found in findcables
  
• refit the points for the cable
• x,y linear
• y,z, x,z 2nd order
• compute the distance of all points from each fit and let the user exclude  points
• rotate the points to the x,z plane
• fit a parabola to this and plot the residuals vs x,y, and z.. let the user exclude points
• When done refit the points in the x,y plane
• For each cable we end up with
• indices into p3[3,m] for this cable
• the x,y rotation angle needed to move this cable to the x,z plane
• the x,z angle giving the vertical slope of each cable
• the parabolic fit parameters that can be used to generate the initial values for the catenary fit.

Fitting a catenary and computing the tension.

• fitcatenary3dlm(p3,p3cblIar,index,fitI,fitcat)
• this uses a Leven -Marquardt non linear least squares fit a catenary to the data points.
• I ended up having to compute the derivatives for each parameter :).
  
• zc= a + b*(cosh(x-c)/b -1 )  is a catenary.
  
• a parabola is a close approximation to a catenary.
  
• computing the initial values from the parabolic fit
• zp=p0  + p1*x  + p2*x^2  is a parabolic fit
• for the catenary let u=(x-c)/b . since the cosh is exponential , u must be small and we can do a taylor series expansion in u.
• cosh(u)= 1 + u^2/2! + ...
• replacing u with (x-c)/b   and expanding gives
• zc=a + b*((1+(x-c)/b)^2/2 -1) = (1/b)*x^2 + (-2c/b)*x  + (a +c^2/b)
  
• equating each term with the parabola's coef gives:
• b=1/p2
• c=-(b*p1)/2 = -p1/(2*p2)
• a=(p0 -c^2/b)=p0 - (p1^2/(4*p2)
• These are used for the initial values of the fit
• Computing the tension.
• The cable tension is b * cableWeight/linearDistance  * cos(cableAngle)
• the cableAngle is computed from the catenary fit and the max,min x values (more info)
  
• The tension was computed in Newtons (since i used meters for distance in the p50) and then it was converted to lbs of force.
• the routine cmpcbltension() does this.
• The cable tension error is linearly related to the error in the cable linear density. The values used were:

• cable  diameter (inches)	linearDensity lbs/foot	locations for cable
  3"	18.9	platform mains catwalk to T4
  3 1/4"	22.18	platform aux backstay mains catwalk to T8
  3 5/8"	27.60	backstay aux
  1 7/8"	7.39	catwalk to t12?

  
  

Where the code is located

• /share/megs/phil/x101/p50/pro has:
• cmpcbltension.pro,findcables.pro,fitcatenary3dlm.pro,getxyzpnts.pro,pntlinedist2.pro, and preproccable.pro.
• cbl.h (defines lots of the structures used.
• /share/megs/phi/x101/p50/cables
• proccableall.pro .. this is a template that is located for each measurement location (eg t4 min,t8main, t12backstays..etc)
• /share/megs/phil/x101/p50/2008xx,2009xx/   has the code for each location (by measurement date).