common/svd22.c - RealtimeRoboticsGroup/test - Gitiles

 /* Copyright (C) 2013-2016, The Regents of The University of Michigan.
 All rights reserved.
 This software was developed in the APRIL Robotics Lab under the
 direction of Edwin Olson, ebolson@umich.edu. This software may be
 available under alternative licensing terms; contact the address above.
 Redistribution and use in source and binary forms, with or without
 modification, are permitted provided that the following conditions are met:
 1. Redistributions of source code must retain the above copyright notice, this
    list of conditions and the following disclaimer.
 2. Redistributions in binary form must reproduce the above copyright notice,
    this list of conditions and the following disclaimer in the documentation
    and/or other materials provided with the distribution.
 THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
 ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
 WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
 DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
 ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
 (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
 LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
 ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
 SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 The views and conclusions contained in the software and documentation are those
 of the authors and should not be interpreted as representing official policies,
 either expressed or implied, of the Regents of The University of Michigan.
 */

 #include <math.h>

 /** SVD 2x2.

     Computes singular values and vectors without squaring the input
     matrix. With double precision math, results are accurate to about
     1E-16.

     U = [ cos(theta) -sin(theta) ]
         [ sin(theta)  cos(theta) ]

     S = [ e  0 ]
         [ 0  f ]

     V = [ cos(phi)   -sin(phi) ]
         [ sin(phi)   cos(phi)  ]


     Our strategy is basically to analytically multiply everything out
     and then rearrange so that we can solve for theta, phi, e, and
     f. (Derivation by ebolson@umich.edu 5/2016)

    V' = [ CP  SP ]
         [ -SP CP ]

 USV' = [ CT -ST ][  e*CP  e*SP ]
        [ ST  CT ][ -f*SP  f*CP ]

      = [e*CT*CP + f*ST*SP     e*CT*SP - f*ST*CP ]
        [e*ST*CP - f*SP*CT     e*SP*ST + f*CP*CT ]

 A00+A11 = e*CT*CP + f*ST*SP + e*SP*ST + f*CP*CT
         = e*(CP*CT + SP*ST) + f*(SP*ST + CP*CT)
     	= (e+f)(CP*CT + SP*ST)
 B0	    = (e+f)*cos(P-T)

 A00-A11 = e*CT*CP + f*ST*SP - e*SP*ST - f*CP*CT
         = e*(CP*CT - SP*ST) - f*(-ST*SP + CP*CT)
 	    = (e-f)(CP*CT - SP*ST)
 B1	    = (e-f)*cos(P+T)

 A01+A10 = e*CT*SP - f*ST*CP + e*ST*CP - f*SP*CT
 	    = e(CT*SP + ST*CP) - f*(ST*CP + SP*CT)
 	    = (e-f)*(CT*SP + ST*CP)
 B2	    = (e-f)*sin(P+T)

 A01-A10 = e*CT*SP - f*ST*CP - e*ST*CP + f*SP*CT
 	= e*(CT*SP - ST*CP) + f(SP*CT - ST*CP)
 	= (e+f)*(CT*SP - ST*CP)
 B3	= (e+f)*sin(P-T)

 B0 = (e+f)*cos(P-T)
 B1 = (e-f)*cos(P+T)
 B2 = (e-f)*sin(P+T)
 B3 = (e+f)*sin(P-T)

 B3/B0 = tan(P-T)

 B2/B1 = tan(P+T)
  **/
 void svd22(const double A[4], double U[4], double S[2], double V[4])
 {
     double A00 = A[0];
     double A01 = A[1];
     double A10 = A[2];
     double A11 = A[3];

     double B0 = A00 + A11;
     double B1 = A00 - A11;
     double B2 = A01 + A10;
     double B3 = A01 - A10;

     double PminusT = atan2(B3, B0);
     double PplusT = atan2(B2, B1);

     double P = (PminusT + PplusT) / 2;
     double T = (-PminusT + PplusT) / 2;

     double CP = cos(P), SP = sin(P);
     double CT = cos(T), ST = sin(T);

     U[0] = CT;
     U[1] = -ST;
     U[2] = ST;
     U[3] = CT;

     V[0] = CP;
     V[1] = -SP;
     V[2] = SP;
     V[3] = CP;

     // C0 = e+f. There are two ways to compute C0; we pick the one
     // that is better conditioned.
     double CPmT = cos(P-T), SPmT = sin(P-T);
     double C0 = 0;
     if (fabs(CPmT) > fabs(SPmT))
         C0 = B0 / CPmT;
     else
         C0 = B3 / SPmT;

     // C1 = e-f. There are two ways to compute C1; we pick the one
     // that is better conditioned.
     double CPpT = cos(P+T), SPpT = sin(P+T);
     double C1 = 0;
     if (fabs(CPpT) > fabs(SPpT))
         C1 = B1 / CPpT;
     else
         C1 = B2 / SPpT;

     // e and f are the singular values
     double e = (C0 + C1) / 2;
     double f = (C0 - C1) / 2;

     if (e < 0) {
         e = -e;
         U[0] = -U[0];
         U[2] = -U[2];
     }

     if (f < 0) {
         f = -f;
         U[1] = -U[1];
         U[3] = -U[3];
     }

     // sort singular values.
     if (e > f) {
         // already in big-to-small order.
         S[0] = e;
         S[1] = f;
     } else {
         // Curiously, this code never seems to get invoked.  Why is it
         // that S[0] always ends up the dominant vector?  However,
         // this code has been tested (flipping the logic forces us to
         // sort the singular values in ascending order).
         //
         // P = [ 0 1 ; 1 0 ]
         // USV' = (UP)(PSP)(PV')
         //      = (UP)(PSP)(VP)'
         //      = (UP)(PSP)(P'V')'
         S[0] = f;
         S[1] = e;

         // exchange columns of U and V
         double tmp[2];
         tmp[0] = U[0];
         tmp[1] = U[2];
         U[0] = U[1];
         U[2] = U[3];
         U[1] = tmp[0];
         U[3] = tmp[1];

         tmp[0] = V[0];
         tmp[1] = V[2];
         V[0] = V[1];
         V[2] = V[3];
         V[1] = tmp[0];
         V[3] = tmp[1];
     }

     /*
     double SM[4] = { S[0], 0, 0, S[1] };

     doubles_print_mat(U, 2, 2, "%20.10g");
     doubles_print_mat(SM, 2, 2, "%20.10g");
     doubles_print_mat(V, 2, 2, "%20.10g");
     printf("A:\n");
     doubles_print_mat(A, 2, 2, "%20.10g");

     double SVt[4];
     doubles_mat_ABt(SM, 2, 2, V, 2, 2, SVt, 2, 2);
     double USVt[4];
     doubles_mat_AB(U, 2, 2, SVt, 2, 2, USVt, 2, 2);

     printf("USVt\n");
     doubles_print_mat(USVt, 2, 2, "%20.10g");

     double diff[4];
     for (int i = 0; i < 4; i++)
         diff[i] = A[i] - USVt[i];

     printf("diff\n");
     doubles_print_mat(diff, 2, 2, "%20.10g");

     */

 }


 // for the matrix [a b; b d]
 void svd_sym_singular_values(double A00, double A01, double A11,
                              double *Lmin, double *Lmax)
 {
     double A10 = A01;

     double B0 = A00 + A11;
     double B1 = A00 - A11;
     double B2 = A01 + A10;
     double B3 = A01 - A10;

     double PminusT = atan2(B3, B0);
     double PplusT = atan2(B2, B1);

     double P = (PminusT + PplusT) / 2;
     double T = (-PminusT + PplusT) / 2;

     // C0 = e+f. There are two ways to compute C0; we pick the one
     // that is better conditioned.
     double CPmT = cos(P-T), SPmT = sin(P-T);
     double C0 = 0;
     if (fabs(CPmT) > fabs(SPmT))
         C0 = B0 / CPmT;
     else
         C0 = B3 / SPmT;

     // C1 = e-f. There are two ways to compute C1; we pick the one
     // that is better conditioned.
     double CPpT = cos(P+T), SPpT = sin(P+T);
     double C1 = 0;
     if (fabs(CPpT) > fabs(SPpT))
         C1 = B1 / CPpT;
     else
         C1 = B2 / SPpT;

     // e and f are the singular values
     double e = (C0 + C1) / 2;
     double f = (C0 - C1) / 2;

     *Lmin = fmin(e, f);
     *Lmax = fmax(e, f);
 }
	/* Copyright (C) 2013-2016, The Regents of The University of Michigan.
	All rights reserved.
	This software was developed in the APRIL Robotics Lab under the
	direction of Edwin Olson, ebolson@umich.edu. This software may be
	available under alternative licensing terms; contact the address above.
	Redistribution and use in source and binary forms, with or without
	modification, are permitted provided that the following conditions are met:
	1. Redistributions of source code must retain the above copyright notice, this
	list of conditions and the following disclaimer.
	2. Redistributions in binary form must reproduce the above copyright notice,
	this list of conditions and the following disclaimer in the documentation
	and/or other materials provided with the distribution.
	THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
	ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
	WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
	DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
	ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
	(INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
	LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND
	ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
	(INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
	SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
	The views and conclusions contained in the software and documentation are those
	of the authors and should not be interpreted as representing official policies,
	either expressed or implied, of the Regents of The University of Michigan.
	*/

	#include <math.h>

	/** SVD 2x2.

	Computes singular values and vectors without squaring the input
	matrix. With double precision math, results are accurate to about
	1E-16.

	U = [ cos(theta) -sin(theta) ]
	[ sin(theta) cos(theta) ]

	S = [ e 0 ]
	[ 0 f ]

	V = [ cos(phi) -sin(phi) ]
	[ sin(phi) cos(phi) ]


	Our strategy is basically to analytically multiply everything out
	and then rearrange so that we can solve for theta, phi, e, and
	f. (Derivation by ebolson@umich.edu 5/2016)

	V' = [ CP SP ]
	[ -SP CP ]

	USV' = [ CT -ST ][ eCP eSP ]
	[ ST CT ][ -fSP fCP ]

	= [eCTCP + fSTSP eCTSP - fSTCP ]
	[eSTCP - fSPCT eSPST + fCPCT ]

	A00+A11 = eCTCP + fSTSP + eSPST + fCPCT
	= e(CPCT + SPST) + f(SPST + CPCT)
	= (e+f)(CPCT + SPST)
	B0 = (e+f)*cos(P-T)

	A00-A11 = eCTCP + fSTSP - eSPST - fCPCT
	= e(CPCT - SPST) - f(-STSP + CPCT)
	= (e-f)(CPCT - SPST)
	B1 = (e-f)*cos(P+T)

	A01+A10 = eCTSP - fSTCP + eSTCP - fSPCT
	= e(CTSP + STCP) - f(STCP + SP*CT)
	= (e-f)(CTSP + ST*CP)
	B2 = (e-f)*sin(P+T)

	A01-A10 = eCTSP - fSTCP - eSTCP + fSPCT
	= e(CTSP - STCP) + f(SPCT - ST*CP)
	= (e+f)(CTSP - ST*CP)
	B3 = (e+f)*sin(P-T)

	B0 = (e+f)*cos(P-T)
	B1 = (e-f)*cos(P+T)
	B2 = (e-f)*sin(P+T)
	B3 = (e+f)*sin(P-T)

	B3/B0 = tan(P-T)

	B2/B1 = tan(P+T)
	**/
	void svd22(const double A[4], double U[4], double S[2], double V[4])
	{
	double A00 = A[0];
	double A01 = A[1];
	double A10 = A[2];
	double A11 = A[3];

	double B0 = A00 + A11;
	double B1 = A00 - A11;
	double B2 = A01 + A10;
	double B3 = A01 - A10;

	double PminusT = atan2(B3, B0);
	double PplusT = atan2(B2, B1);

	double P = (PminusT + PplusT) / 2;
	double T = (-PminusT + PplusT) / 2;

	double CP = cos(P), SP = sin(P);
	double CT = cos(T), ST = sin(T);

	U[0] = CT;
	U[1] = -ST;
	U[2] = ST;
	U[3] = CT;

	V[0] = CP;
	V[1] = -SP;
	V[2] = SP;
	V[3] = CP;

	// C0 = e+f. There are two ways to compute C0; we pick the one
	// that is better conditioned.
	double CPmT = cos(P-T), SPmT = sin(P-T);
	double C0 = 0;
	if (fabs(CPmT) > fabs(SPmT))
	C0 = B0 / CPmT;
	else
	C0 = B3 / SPmT;

	// C1 = e-f. There are two ways to compute C1; we pick the one
	// that is better conditioned.
	double CPpT = cos(P+T), SPpT = sin(P+T);
	double C1 = 0;
	if (fabs(CPpT) > fabs(SPpT))
	C1 = B1 / CPpT;
	else
	C1 = B2 / SPpT;

	// e and f are the singular values
	double e = (C0 + C1) / 2;
	double f = (C0 - C1) / 2;

	if (e < 0) {
	e = -e;
	U[0] = -U[0];
	U[2] = -U[2];
	}

	if (f < 0) {
	f = -f;
	U[1] = -U[1];
	U[3] = -U[3];
	}

	// sort singular values.
	if (e > f) {
	// already in big-to-small order.
	S[0] = e;
	S[1] = f;
	} else {
	// Curiously, this code never seems to get invoked. Why is it
	// that S[0] always ends up the dominant vector? However,
	// this code has been tested (flipping the logic forces us to
	// sort the singular values in ascending order).
	//
	// P = [ 0 1 ; 1 0 ]
	// USV' = (UP)(PSP)(PV')
	// = (UP)(PSP)(VP)'
	// = (UP)(PSP)(P'V')'
	S[0] = f;
	S[1] = e;

	// exchange columns of U and V
	double tmp[2];
	tmp[0] = U[0];
	tmp[1] = U[2];
	U[0] = U[1];
	U[2] = U[3];
	U[1] = tmp[0];
	U[3] = tmp[1];

	tmp[0] = V[0];
	tmp[1] = V[2];
	V[0] = V[1];
	V[2] = V[3];
	V[1] = tmp[0];
	V[3] = tmp[1];
	}

	/*
	double SM[4] = { S[0], 0, 0, S[1] };

	doubles_print_mat(U, 2, 2, "%20.10g");
	doubles_print_mat(SM, 2, 2, "%20.10g");
	doubles_print_mat(V, 2, 2, "%20.10g");
	printf("A:\n");
	doubles_print_mat(A, 2, 2, "%20.10g");

	double SVt[4];
	doubles_mat_ABt(SM, 2, 2, V, 2, 2, SVt, 2, 2);
	double USVt[4];
	doubles_mat_AB(U, 2, 2, SVt, 2, 2, USVt, 2, 2);

	printf("USVt\n");
	doubles_print_mat(USVt, 2, 2, "%20.10g");

	double diff[4];
	for (int i = 0; i < 4; i++)
	diff[i] = A[i] - USVt[i];

	printf("diff\n");
	doubles_print_mat(diff, 2, 2, "%20.10g");

	*/

	}


	// for the matrix [a b; b d]
	void svd_sym_singular_values(double A00, double A01, double A11,
	double Lmin, double Lmax)
	{
	double A10 = A01;

	double B0 = A00 + A11;
	double B1 = A00 - A11;
	double B2 = A01 + A10;
	double B3 = A01 - A10;

	double PminusT = atan2(B3, B0);
	double PplusT = atan2(B2, B1);

	double P = (PminusT + PplusT) / 2;
	double T = (-PminusT + PplusT) / 2;

	// C0 = e+f. There are two ways to compute C0; we pick the one
	// that is better conditioned.
	double CPmT = cos(P-T), SPmT = sin(P-T);
	double C0 = 0;
	if (fabs(CPmT) > fabs(SPmT))
	C0 = B0 / CPmT;
	else
	C0 = B3 / SPmT;

	// C1 = e-f. There are two ways to compute C1; we pick the one
	// that is better conditioned.
	double CPpT = cos(P+T), SPpT = sin(P+T);
	double C1 = 0;
	if (fabs(CPpT) > fabs(SPpT))
	C1 = B1 / CPpT;
	else
	C1 = B2 / SPpT;

	// e and f are the singular values
	double e = (C0 + C1) / 2;
	double f = (C0 - C1) / 2;

	*Lmin = fmin(e, f);
	*Lmax = fmax(e, f);
	}