/****************************************************************************
*
* Copyright (C) 2012 PX4 Development Team. All rights reserved.
* Author: Lorenz Meier <lm@inf.ethz.ch>
*
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*
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* notice, this list of conditions and the following disclaimer in
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* distribution.
* 3. Neither the name PX4 nor the names of its contributors may be
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/**
* @file calibration_routines.c
* Calibration routines implementations.
*
* @author Lorenz Meier <lm@inf.ethz.ch>
*/
#include <math.h>
#include "calibration_routines.h"
int sphere_fit_least_squares(const float x[], const float y[], const float z[],
unsigned int size, unsigned int max_iterations, float delta, float *sphere_x, float *sphere_y, float *sphere_z, float *sphere_radius) {
float x_sumplain = 0.0f;
float x_sumsq = 0.0f;
float x_sumcube = 0.0f;
float y_sumplain = 0.0f;
float y_sumsq = 0.0f;
float y_sumcube = 0.0f;
float z_sumplain = 0.0f;
float z_sumsq = 0.0f;
float z_sumcube = 0.0f;
float xy_sum = 0.0f;
float xz_sum = 0.0f;
float yz_sum = 0.0f;
float x2y_sum = 0.0f;
float x2z_sum = 0.0f;
float y2x_sum = 0.0f;
float y2z_sum = 0.0f;
float z2x_sum = 0.0f;
float z2y_sum = 0.0f;
for (unsigned int i = 0; i < size; i++) {
float x2 = x[i] * x[i];
float y2 = y[i] * y[i];
float z2 = z[i] * z[i];
x_sumplain += x[i];
x_sumsq += x2;
x_sumcube += x2 * x[i];
y_sumplain += y[i];
y_sumsq += y2;
y_sumcube += y2 * y[i];
z_sumplain += z[i];
z_sumsq += z2;
z_sumcube += z2 * z[i];
xy_sum += x[i] * y[i];
xz_sum += x[i] * z[i];
yz_sum += y[i] * z[i];
x2y_sum += x2 * y[i];
x2z_sum += x2 * z[i];
y2x_sum += y2 * x[i];
y2z_sum += y2 * z[i];
z2x_sum += z2 * x[i];
z2y_sum += z2 * y[i];
}
//
//Least Squares Fit a sphere A,B,C with radius squared Rsq to 3D data
//
// P is a structure that has been computed with the data earlier.
// P.npoints is the number of elements; the length of X,Y,Z are identical.
// P's members are logically named.
//
// X[n] is the x component of point n
// Y[n] is the y component of point n
// Z[n] is the z component of point n
//
// A is the x coordiante of the sphere
// B is the y coordiante of the sphere
// C is the z coordiante of the sphere
// Rsq is the radius squared of the sphere.
//
//This method should converge; maybe 5-100 iterations or more.
//
float x_sum = x_sumplain / size; //sum( X[n] )
float x_sum2 = x_sumsq / size; //sum( X[n]^2 )
float x_sum3 = x_sumcube / size; //sum( X[n]^3 )
float y_sum = y_sumplain / size; //sum( Y[n] )
float y_sum2 = y_sumsq / size; //sum( Y[n]^2 )
float y_sum3 = y_sumcube / size; //sum( Y[n]^3 )
float z_sum = z_sumplain / size; //sum( Z[n] )
float z_sum2 = z_sumsq / size; //sum( Z[n]^2 )
float z_sum3 = z_sumcube / size; //sum( Z[n]^3 )
float XY = xy_sum / size; //sum( X[n] * Y[n] )
float XZ = xz_sum / size; //sum( X[n] * Z[n] )
float YZ = yz_sum / size; //sum( Y[n] * Z[n] )
float X2Y = x2y_sum / size; //sum( X[n]^2 * Y[n] )
float X2Z = x2z_sum / size; //sum( X[n]^2 * Z[n] )
float Y2X = y2x_sum / size; //sum( Y[n]^2 * X[n] )
float Y2Z = y2z_sum / size; //sum( Y[n]^2 * Z[n] )
float Z2X = z2x_sum / size; //sum( Z[n]^2 * X[n] )
float Z2Y = z2y_sum / size; //sum( Z[n]^2 * Y[n] )
//Reduction of multiplications
float F0 = x_sum2 + y_sum2 + z_sum2;
float F1 = 0.5f * F0;
float F2 = -8.0f * (x_sum3 + Y2X + Z2X);
float F3 = -8.0f * (X2Y + y_sum3 + Z2Y);
float F4 = -8.0f * (X2Z + Y2Z + z_sum3);
//Set initial conditions:
float A = x_sum;
float B = y_sum;
float C = z_sum;
//First iteration computation:
float A2 = A*A;
float B2 = B*B;
float C2 = C*C;
float QS = A2 + B2 + C2;
float QB = -2.0f * (A*x_sum + B*y_sum + C*z_sum);
//Set initial conditions:
float Rsq = F0 + QB + QS;
//First iteration computation:
float Q0 = 0.5f * (QS - Rsq);
float Q1 = F1 + Q0;
float Q2 = 8.0f * ( QS - Rsq + QB + F0 );
float aA,aB,aC,nA,nB,nC,dA,dB,dC;
//Iterate N times, ignore stop condition.
int n = 0;
while( n < max_iterations ){
n++;
//Compute denominator:
aA = Q2 + 16.0f * (A2 - 2.0f * A*x_sum + x_sum2);
aB = Q2 + 16.0f *(B2 - 2.0f * B*y_sum + y_sum2);
aC = Q2 + 16.0f *(C2 - 2.0f * C*z_sum + z_sum2);
aA = (aA == 0.0f) ? 1.0f : aA;
aB = (aB == 0.0f) ? 1.0f : aB;
aC = (aC == 0.0f) ? 1.0f : aC;
//Compute next iteration
nA = A - ((F2 + 16.0f * ( B*XY + C*XZ + x_sum*(-A2 - Q0) + A*(x_sum2 + Q1 - C*z_sum - B*y_sum) ) )/aA);
nB = B - ((F3 + 16.0f * ( A*XY + C*YZ + y_sum*(-B2 - Q0) + B*(y_sum2 + Q1 - A*x_sum - C*z_sum) ) )/aB);
nC = C - ((F4 + 16.0f * ( A*XZ + B*YZ + z_sum*(-C2 - Q0) + C*(z_sum2 + Q1 - A*x_sum - B*y_sum) ) )/aC);
//Check for stop condition
dA = (nA - A);
dB = (nB - B);
dC = (nC - C);
if( (dA*dA + dB*dB + dC*dC) <= delta ){ break; }
//Compute next iteration's values
A = nA;
B = nB;
C = nC;
A2 = A*A;
B2 = B*B;
C2 = C*C;
QS = A2 + B2 + C2;
QB = -2.0f * (A*x_sum + B*y_sum + C*z_sum);
Rsq = F0 + QB + QS;
Q0 = 0.5f * (QS - Rsq);
Q1 = F1 + Q0;
Q2 = 8.0f * ( QS - Rsq + QB + F0 );
}
*sphere_x = A;
*sphere_y = B;
*sphere_z = C;
*sphere_radius = sqrtf(Rsq);
return 0;
}
