PIC Microcontoller Basic Math Vector Math Methods

• Magnitude by Nikolai Golovchenko

Questions:

• clubkid49spam at hotmail.com asks:

  Hey there I have a pic16f882 and an MMA7360L accelerometer and i'm trying to convert the x and y axis readings into vectors to be able to calculate the accelerometers position from a given point at any given time. I already have the readings stored from the accelerometer but i'm not sure how i would now go about integrating these to reach position. Any help would be greatly appreciated, thanks in advance, ali

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Code:

• Nikolai Golovchenko shares this code:

  Method 2B. Accurate to -1.2%..+0.8%. Requires up to four constant multiplies.
  
  This method refines Levitt/Morris method.
  
   int magnitude2B(int x, int y)
   {
     int mx;
     int mn;
     x = abs(x);
     y = abs(y);
     mx = max(x, y);
     mn = min(x, y);
  
     if(mx >= 3*mn)
       return mx + mn/8;
     else if (mx >= 5/4*mn)
       return mx*7/8 + mn/2;
     else
       return mx*3/4 + mn*21/32;
   }
  
  

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• Nikolai Golovchenko shares this code:

  There are several approximate methods for finding magnitude. They vary in their
  complexity and accuracy. Here is a list of some options:
  
  Method 1. Accurate up to -0..+8%. Requires a multiply by a constant.
  
   int magnitude_Robertson(int x, int y)
   {
     int mx;
     int mn;
     x = abs(x);
     y = abs(y);
     mx = max(x, y);
     mn = min(x, y);
     return mx + mn*k;
   }
  
  The method is apparently due to:
  Robertson, G.H., "A Fast Amplitude Approximation for Quadrature Pairs", Bell Sys. Tech. J., 
  Vol. 50, Oct. 1971, pp.2849-2852.
  
  The constant k is chosen to minimize error in one of different ways (average error, absolute error, 
  peak-to-peak error, standard deviation, etc) and efficient to compute the multiply. 
  A few examples:
  
  1/2, 3/8, 1/4:          suggested by Robertson
  sqrt(2)-1 ~= 106/256:   minimizes p-p error (also, zero error at multiples of 45 deg)
  
  
  Method 2. Accurate to 2.8%. Requires three constant multiplies.
  
   int magnitude_Levitt(int x, int y)
   {
     int mx;
     int mn;
     x = abs(x);
     y = abs(y);
     mx = max(x, y);
     mn = min(x, y);
  
     if(mx >= 3*mn)
       return mx + mn/8;
     else
       return mx*7/8 + mn/2;
   }
  
  See B.K Levitt, G.A. Morris, "An Improved Digital Algorithm for Fast Amplitude
  Approximations of Quadrature Pairs", DSN Progress Report 42-40, May and June 1977, pp 97-101.
  <a href="http://tmo.jpl.nasa.gov/progress_report2/42-40/40L.PDF">http://tmo.jpl.nasa.gov/progress_report2/42-40/40L.PDF</a>
  
  
  Method 3. Accurate up to -1.5%...+0%. Requires a divide, multiply, and a constant multiply.
  
   int magnitude3(int x, int y)
   {
     int mx;
     int mn;
     x = abs(x);
     y = abs(y);
     mx = max(x, y);
     mn = min(x, y);
     if(mx == 0)
       return mx;
  
     return mx + mn*mn/mx*k;
   }
  
  The constant k is chosen similarly to method 1. For example, k=sqrt(2)-1~=106/256 again 
  minimizes the p-p error. The method is due to Nikolai Golovchenko. :)
  
  
  Method 4. Accurate up to -0.5%..+0%. Requires a divide, multiply, and two constant multiplies.
  
  This is a piecewise refinement of method 3. Vector (x,y) is assumed to be
  rotated to the range of 0 to 45 deg (i.e. x=mx and y=mn as in examples above)
  
     b = y/x
  
     if b <= 0.5
       b += b * 0.14      // ~5/32
     else
       b += (1-b) * 0.14 
  
     r = x + y * b * 106/256
  
  Method 5: Theoretically arbitrary accuracy. Requires a divide and two multiplies.
  
     r = x + y * lookup(y/x)
  
     lookup() - a function that represents a table with linear interpolation.
  
  
  

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