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ardupilot/libraries/AP_Math/matrix_alg.cpp

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/// -*- tab-width: 4; Mode: C++; c-basic-offset: 4; indent-tabs-mode: nil -*-
/*
* matrix3.cpp
* Copyright (C) Siddharth Bharat Purohit, 3DRobotics Inc. 2015
*
* This file is free software: you can redistribute it and/or modify it
* under the terms of the GNU General Public License as published by the
* Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* This file is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
* See the GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License along
* with this program. If not, see <http://www.gnu.org/licenses/>.
*/
#pragma GCC optimize("O3")
#include <AP_Math/AP_Math.h>
#include <AP_HAL/AP_HAL.h>
#include <stdio.h>
extern const AP_HAL::HAL& hal;
//TODO: use higher precision datatypes to achieve more accuracy for matrix algebra operations
/*
* Does matrix multiplication of two regular/square matrices
*
* @param A, Matrix A
* @param B, Matrix B
* @param n, dimemsion of square matrices
* @returns multiplied matrix i.e. A*B
*/
float* mat_mul(float *A, float *B, uint8_t n)
{
float* ret = new float[n*n];
memset(ret,0.0f,n*n*sizeof(float));
for(uint8_t i = 0; i < n; i++) {
for(uint8_t j = 0; j < n; j++) {
for(uint8_t k = 0;k < n; k++) {
ret[i*n + j] += A[i*n + k] * B[k*n + j];
}
}
}
return ret;
}
static inline void swap(float &a, float &b)
{
float c;
c = a;
a = b;
b = c;
}
/*
* calculates pivot matrix such that all the larger elements in the row are on diagonal
*
* @param A, input matrix matrix
* @param pivot
* @param n, dimenstion of square matrix
* @returns false = matrix is Singular or non positive definite, true = matrix inversion successful
*/
void mat_pivot(float* A, float* pivot, uint8_t n)
{
for(uint8_t i = 0;i<n;i++){
for(uint8_t j=0;j<n;j++) {
pivot[i*n+j] = (i==j);
}
}
for(uint8_t i = 0;i < n; i++) {
uint8_t max_j = i;
for(uint8_t j=i;j<n;j++){
if(fabsf(A[j*n + i]) > fabsf(A[max_j*n + i])) {
max_j = j;
}
}
if(max_j != i) {
for(uint8_t k = 0; k < n; k++) {
swap(pivot[i*n + k], pivot[max_j*n + k]);
}
}
}
}
/*
* calculates matrix inverse of Lower trangular matrix using forward substitution
*
* @param L, lower triangular matrix
* @param out, Output inverted lower triangular matrix
* @param n, dimension of matrix
*/
void mat_forward_sub(float *L, float *out, uint8_t n)
{
// Forward substitution solve LY = I
for(int i = 0; i < n; i++) {
out[i*n + i] = 1/L[i*n + i];
for (int j = i+1; j < n; j++) {
for (int k = i; k < j; k++) {
out[j*n + i] -= L[j*n + k] * out[k*n + i];
}
out[j*n + i] /= L[j*n + j];
}
}
}
/*
* calculates matrix inverse of Upper trangular matrix using backward substitution
*
* @param U, upper triangular matrix
* @param out, Output inverted upper triangular matrix
* @param n, dimension of matrix
*/
void mat_back_sub(float *U, float *out, uint8_t n)
{
// Backward Substitution solve UY = I
for(int i = n-1; i >= 0; i--) {
out[i*n + i] = 1/U[i*n + i];
for (int j = i - 1; j >= 0; j--) {
for (int k = i; k > j; k--) {
out[j*n + i] -= U[j*n + k] * out[k*n + i];
}
out[j*n + i] /= U[j*n + j];
}
}
}
/*
* Decomposes square matrix into Lower and Upper triangular matrices such that
* A*P = L*U, where P is the pivot matrix
* ref: http://rosettacode.org/wiki/LU_decomposition
* @param U, upper triangular matrix
* @param out, Output inverted upper triangular matrix
* @param n, dimension of matrix
*/
void mat_LU_decompose(float* A, float* L, float* U, float *P, uint8_t n)
{
memset(L,0,n*n*sizeof(float));
memset(U,0,n*n*sizeof(float));
memset(P,0,n*n*sizeof(float));
mat_pivot(A,P,n);
float *APrime = mat_mul(P,A,n);
for(uint8_t i = 0; i < n; i++) {
L[i*n + i] = 1;
}
for(uint8_t i = 0; i < n; i++) {
for(uint8_t j = 0; j < n; j++) {
if(j <= i) {
U[j*n + i] = APrime[j*n + i];
for(uint8_t k = 0; k < j; k++) {
U[j*n + i] -= L[j*n + k] * U[k*n + i];
}
}
if(j >= i) {
L[j*n + i] = APrime[j*n + i];
for(uint8_t k = 0; k < i; k++) {
L[j*n + i] -= L[j*n + k] * U[k*n + i];
}
L[j*n + i] /= U[i*n + i];
}
}
}
free(APrime);
}
/*
* matrix inverse code for any square matrix using LU decomposition
* inv = inv(U)*inv(L)*P, where L and U are triagular matrices and P the pivot matrix
* ref: http://www.cl.cam.ac.uk/teaching/1314/NumMethods/supporting/mcmaster-kiruba-ludecomp.pdf
* @param m, input 4x4 matrix
* @param inv, Output inverted 4x4 matrix
* @param n, dimension of square matrix
* @returns false = matrix is Singular, true = matrix inversion successful
*/
bool mat_inverse(float* A, float* inv, uint8_t n)
{
float *L, *U, *P;
bool ret = true;
L = new float[n*n];
U = new float[n*n];
P = new float[n*n];
mat_LU_decompose(A,L,U,P,n);
float *L_inv = new float[n*n];
float *U_inv = new float[n*n];
memset(L_inv,0,n*n*sizeof(float));
mat_forward_sub(L,L_inv,n);
memset(U_inv,0,n*n*sizeof(float));
mat_back_sub(U,U_inv,n);
// decomposed matrices no loger required
free(L);
free(U);
float *inv_unpivoted = mat_mul(U_inv,L_inv,n);
float *inv_pivoted = mat_mul(inv_unpivoted, P, n);
//check sanity of results
for(uint8_t i = 0; i < n; i++) {
for(uint8_t j = 0; j < n; j++) {
if(isnan(inv_pivoted[i*n+j]) || isinf(inv_pivoted[i*n+j])){
ret = false;
}
}
}
memcpy(inv,inv_pivoted,n*n*sizeof(float));
//free memory
free(inv_pivoted);
free(inv_unpivoted);
free(P);
return ret;
}
/*
* fast matrix inverse code only for 3x3 square matrix
*
* @param m, input 4x4 matrix
* @param invOut, Output inverted 4x4 matrix
* @returns false = matrix is Singular, true = matrix inversion successful
*/
bool inverse3x3(float m[], float invOut[])
{
float inv[9];
// computes the inverse of a matrix m
float det = m[0] * (m[4] * m[8] - m[7] * m[5]) -
m[1] * (m[3] * m[8] - m[5] * m[6]) +
m[2] * (m[3] * m[7] - m[4] * m[6]);
if (is_zero(det)){
return false;
}
float invdet = 1 / det;
inv[0] = (m[4] * m[8] - m[7] * m[5]) * invdet;
inv[1] = (m[2] * m[7] - m[1] * m[8]) * invdet;
inv[2] = (m[1] * m[5] - m[2] * m[4]) * invdet;
inv[3] = (m[5] * m[6] - m[3] * m[8]) * invdet;
inv[4] = (m[0] * m[8] - m[2] * m[6]) * invdet;
inv[5] = (m[3] * m[2] - m[0] * m[5]) * invdet;
inv[6] = (m[3] * m[7] - m[6] * m[4]) * invdet;
inv[7] = (m[6] * m[1] - m[0] * m[7]) * invdet;
inv[8] = (m[0] * m[4] - m[3] * m[1]) * invdet;
for(uint8_t i = 0; i < 9; i++){
invOut[i] = inv[i];
}
return true;
}
/*
* fast matrix inverse code only for 4x4 square matrix copied from
* gluInvertMatrix implementation in opengl for 4x4 matrices.
*
* @param m, input 4x4 matrix
* @param invOut, Output inverted 4x4 matrix
* @returns false = matrix is Singular, true = matrix inversion successful
*/
bool inverse4x4(float m[],float invOut[])
{
float inv[16], det;
uint8_t i;
inv[0] = m[5] * m[10] * m[15] -
m[5] * m[11] * m[14] -
m[9] * m[6] * m[15] +
m[9] * m[7] * m[14] +
m[13] * m[6] * m[11] -
m[13] * m[7] * m[10];
inv[4] = -m[4] * m[10] * m[15] +
m[4] * m[11] * m[14] +
m[8] * m[6] * m[15] -
m[8] * m[7] * m[14] -
m[12] * m[6] * m[11] +
m[12] * m[7] * m[10];
inv[8] = m[4] * m[9] * m[15] -
m[4] * m[11] * m[13] -
m[8] * m[5] * m[15] +
m[8] * m[7] * m[13] +
m[12] * m[5] * m[11] -
m[12] * m[7] * m[9];
inv[12] = -m[4] * m[9] * m[14] +
m[4] * m[10] * m[13] +
m[8] * m[5] * m[14] -
m[8] * m[6] * m[13] -
m[12] * m[5] * m[10] +
m[12] * m[6] * m[9];
inv[1] = -m[1] * m[10] * m[15] +
m[1] * m[11] * m[14] +
m[9] * m[2] * m[15] -
m[9] * m[3] * m[14] -
m[13] * m[2] * m[11] +
m[13] * m[3] * m[10];
inv[5] = m[0] * m[10] * m[15] -
m[0] * m[11] * m[14] -
m[8] * m[2] * m[15] +
m[8] * m[3] * m[14] +
m[12] * m[2] * m[11] -
m[12] * m[3] * m[10];
inv[9] = -m[0] * m[9] * m[15] +
m[0] * m[11] * m[13] +
m[8] * m[1] * m[15] -
m[8] * m[3] * m[13] -
m[12] * m[1] * m[11] +
m[12] * m[3] * m[9];
inv[13] = m[0] * m[9] * m[14] -
m[0] * m[10] * m[13] -
m[8] * m[1] * m[14] +
m[8] * m[2] * m[13] +
m[12] * m[1] * m[10] -
m[12] * m[2] * m[9];
inv[2] = m[1] * m[6] * m[15] -
m[1] * m[7] * m[14] -
m[5] * m[2] * m[15] +
m[5] * m[3] * m[14] +
m[13] * m[2] * m[7] -
m[13] * m[3] * m[6];
inv[6] = -m[0] * m[6] * m[15] +
m[0] * m[7] * m[14] +
m[4] * m[2] * m[15] -
m[4] * m[3] * m[14] -
m[12] * m[2] * m[7] +
m[12] * m[3] * m[6];
inv[10] = m[0] * m[5] * m[15] -
m[0] * m[7] * m[13] -
m[4] * m[1] * m[15] +
m[4] * m[3] * m[13] +
m[12] * m[1] * m[7] -
m[12] * m[3] * m[5];
inv[14] = -m[0] * m[5] * m[14] +
m[0] * m[6] * m[13] +
m[4] * m[1] * m[14] -
m[4] * m[2] * m[13] -
m[12] * m[1] * m[6] +
m[12] * m[2] * m[5];
inv[3] = -m[1] * m[6] * m[11] +
m[1] * m[7] * m[10] +
m[5] * m[2] * m[11] -
m[5] * m[3] * m[10] -
m[9] * m[2] * m[7] +
m[9] * m[3] * m[6];
inv[7] = m[0] * m[6] * m[11] -
m[0] * m[7] * m[10] -
m[4] * m[2] * m[11] +
m[4] * m[3] * m[10] +
m[8] * m[2] * m[7] -
m[8] * m[3] * m[6];
inv[11] = -m[0] * m[5] * m[11] +
m[0] * m[7] * m[9] +
m[4] * m[1] * m[11] -
m[4] * m[3] * m[9] -
m[8] * m[1] * m[7] +
m[8] * m[3] * m[5];
inv[15] = m[0] * m[5] * m[10] -
m[0] * m[6] * m[9] -
m[4] * m[1] * m[10] +
m[4] * m[2] * m[9] +
m[8] * m[1] * m[6] -
m[8] * m[2] * m[5];
det = m[0] * inv[0] + m[1] * inv[4] + m[2] * inv[8] + m[3] * inv[12];
if (is_zero(det)){
return false;
}
det = 1.0f / det;
for (i = 0; i < 16; i++)
invOut[i] = inv[i] * det;
return true;
}
/*
* generic matrix inverse code
*
* @param x, input nxn matrix
* @param y, Output inverted nxn matrix
* @param n, dimension of square matrix
* @returns false = matrix is Singular, true = matrix inversion successful
*/
bool inverse(float x[], float y[], uint16_t dim)
{
switch(dim){
case 3: return inverse3x3(x,y);
case 4: return inverse4x4(x,y);
default: return mat_inverse(x,y,dim);
}
}
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