/** * This file is part of ORB-SLAM3 * * Copyright (C) 2017-2021 Carlos Campos, Richard Elvira, Juan J. Gómez Rodríguez, José M.M. Montiel and Juan D. Tardós, University of Zaragoza. * Copyright (C) 2014-2016 Raúl Mur-Artal, José M.M. Montiel and Juan D. Tardós, University of Zaragoza. * * ORB-SLAM3 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. * * ORB-SLAM3 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 ORB-SLAM3. * If not, see . */ /****************************************************************************** * Author: Steffen Urban * * Contact: urbste@gmail.com * * License: Copyright (c) 2016 Steffen Urban, ANU. All rights reserved. * * * * Redistribution and use in source and binary forms, with or without * * modification, are permitted provided that the following conditions * * are met: * * * Redistributions of source code must retain the above copyright * * notice, this list of conditions and the following disclaimer. * * * 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. * * * Neither the name of ANU nor the names of its contributors may be * * used to endorse or promote products derived from this software without * * specific prior written permission. * * * * 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 ANU OR THE 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. * ******************************************************************************/ #include "MLPnPsolver.h" #include // MLPnP算法，极大似然PnP算法，解决PnP问题，用在重定位中，不用运动的先验知识来估计相机位姿 // 基于论文《MLPNP - A REAL-TIME MAXIMUM LIKELIHOOD SOLUTION TO THE PERSPECTIVE-N-POINT PROBLEM》 namespace ORB_SLAM3 { /** * @brief MLPnP 构造函数 * * @param[in] F 输入帧的数据 * @param[in] vpMapPointMatches 待匹配的特征点 * @param[in] mnInliersi 内点的个数 * @param[in] mnIterations Ransac迭代次数 * @param[in] mnBestInliers 最佳内点数 * @param[in] N 所有2D点的个数 * @param[in] mpCamera 相机模型，利用该变量对3D点进行投影 */ MLPnPsolver::MLPnPsolver(const Frame &F, // 输入帧的数据 const vector &vpMapPointMatches // 待匹配的特征点，是当前帧和候选关键帧用BoW进行快速匹配的结果 ) : mnInliersi(0), // 内点的个数 mnIterations(0), // Ransac迭代次数 mnBestInliers(0), // 最佳内点数 N(0), // 所有2D点的个数 mpCamera(F.mpCamera) // 相机模型，利用该变量对3D点进行投影 { mvpMapPointMatches = vpMapPointMatches; // 待匹配的特征点，是当前帧和候选关键帧用BoW进行快速匹配的结果 mvBearingVecs.reserve(F.mvpMapPoints.size()); // 初始化3D点的单位向量 mvP2D.reserve(F.mvpMapPoints.size()); // 初始化3D点的投影点 mvSigma2.reserve(F.mvpMapPoints.size()); // 初始化卡方检验中的sigma值 mvP3Dw.reserve(F.mvpMapPoints.size()); // 初始化3D点坐标 mvKeyPointIndices.reserve(F.mvpMapPoints.size()); // 初始化3D点的索引值 mvAllIndices.reserve(F.mvpMapPoints.size()); // 初始化所有索引值 // 一些必要的初始化操作 int idx = 0; for (size_t i = 0, iend = mvpMapPointMatches.size(); i < iend; i++) { MapPoint *pMP = vpMapPointMatches[i]; // 如果pMP存在，则接下来初始化一些参数，否则什么都不做 if (pMP) { // 判断是否是坏点 if (!pMP->isBad()) { // 如果记录的点个数超过总数，则不做任何事情，否则继续记录 if (i >= F.mvKeysUn.size()) continue; const cv::KeyPoint &kp = F.mvKeysUn[i]; // 保存3D点的投影点 mvP2D.push_back(kp.pt); // 保存卡方检验中的sigma值 mvSigma2.push_back(F.mvLevelSigma2[kp.octave]); // Bearing vector should be normalized // 特征点投影，并计算单位向量 cv::Point3f cv_br = mpCamera->unproject(kp.pt); cv_br /= cv_br.z; bearingVector_t br(cv_br.x, cv_br.y, cv_br.z); mvBearingVecs.push_back(br); // 3D coordinates // 获取当前特征点的3D坐标 Eigen::Matrix posEig = pMP->GetWorldPos(); point_t pos(posEig(0), posEig(1), posEig(2)); mvP3Dw.push_back(pos); // 记录当前特征点的索引值，挑选后的 mvKeyPointIndices.push_back(i); // 记录所有特征点的索引值 mvAllIndices.push_back(idx); idx++; } } } // 设置RANSAC参数 SetRansacParameters(); } // RANSAC methods /** * @brief MLPnP迭代计算相机位姿 * * @param[in] nIterations 迭代次数 * @param[in] bNoMore 达到最大迭代次数的标志 * @param[in] vbInliers 内点的标记 * @param[in] nInliers 总共内点数 * @return cv::Mat 计算出来的位姿 */ bool MLPnPsolver::iterate(int nIterations, bool &bNoMore, vector &vbInliers, int &nInliers, Eigen::Matrix4f &Tout) { Tout.setIdentity(); bNoMore = false; // 已经达到最大迭代次数的标志 vbInliers.clear(); // 清除保存判断是否是内点的容器 nInliers = 0; // 当前次迭代时的内点数 // N为所有2D点的个数, mRansacMinInliers为正常退出RANSAC迭代过程中最少的inlier数 // Step 1: 判断，如果2D点个数不足以启动RANSAC迭代过程的最小下限，则退出 if (N < mRansacMinInliers) { bNoMore = true; // 已经达到最大迭代次数的标志 return false; // 函数退出 } // mvAllIndices为所有参与PnP的2D点的索引 // vAvailableIndices为每次从mvAllIndices中随机挑选mRansacMinSet组3D-2D对应点进行一次RANSAC vector vAvailableIndices; // 当前的迭代次数id int nCurrentIterations = 0; // Step 2: 正常迭代计算进行相机位姿估计，如果满足效果上限，直接返回最佳估计结果，否则就继续利用最小集(6个点)估计位姿 // 进行迭代的条件: // 条件1: 历史进行的迭代次数少于最大迭代值 // 条件2: 当前进行的迭代次数少于当前函数给定的最大迭代值 while (mnIterations < mRansacMaxIts || nCurrentIterations < nIterations) { // 迭代次数更新加1，直到达到最大迭代次数 nCurrentIterations++; mnIterations++; // 清空已有的匹配点的计数,为新的一次迭代作准备 vAvailableIndices = mvAllIndices; // Bearing vectors and 3D points used for this ransac iteration // 初始化单位向量和3D点，给当前ransac迭代使用 bearingVectors_t bearingVecs(mRansacMinSet); points_t p3DS(mRansacMinSet); vector indexes(mRansacMinSet); // Get min set of points // 选取最小集,从vAvailableIndices中选取mRansacMinSet个点进行操作，这里应该是6 for (short i = 0; i < mRansacMinSet; ++i) { // 在所有备选点中随机抽取一个，通过随机抽取索引数组vAvailableIndices的索引[randi]来实现 int randi = DUtils::Random::RandomInt(0, vAvailableIndices.size() - 1); // vAvailableIndices[randi]才是备选点真正的索引值，randi是索引数组的索引值，不要搞混了 int idx = vAvailableIndices[randi]; bearingVecs[i] = mvBearingVecs[idx]; p3DS[i] = mvP3Dw[idx]; indexes[i] = i; // 把抽取出来的点从所有备选点数组里删除掉，概率论中不放回的操作 vAvailableIndices[randi] = vAvailableIndices.back(); vAvailableIndices.pop_back(); } // 选取最小集 // By the moment, we are using MLPnP without covariance info // 目前为止，还没有使用协方差的信息，所以这里生成一个size=1的值为0的协方差矩阵 // |0 0 0| // covs[0] = |0 0 0| // |0 0 0| // ? 为什么不用协方差的SVD分解，计算耗时还是效果不明显？ cov3_mats_t covs(1); // Result transformation_t result; // Compute camera pose // 相机位姿估计，MLPnP最主要的操作在这里 computePose(bearingVecs, p3DS, covs, indexes, result); // Save result // 论文中12个待求值赋值保存在mRi中，每个求解器都有保存各自的计算结果 mRi[0][0] = result(0, 0); mRi[0][1] = result(0, 1); mRi[0][2] = result(0, 2); mRi[1][0] = result(1, 0); mRi[1][1] = result(1, 1); mRi[1][2] = result(1, 2); mRi[2][0] = result(2, 0); mRi[2][1] = result(2, 1); mRi[2][2] = result(2, 2); mti[0] = result(0, 3); mti[1] = result(1, 3); mti[2] = result(2, 3); // Check inliers // 卡方检验内点，和EPnP基本类似 CheckInliers(); if (mnInliersi >= mRansacMinInliers) { // If it is the best solution so far, save it // 如果该结果是目前内点数最多的，说明该结果是目前最好的，保存起来 if (mnInliersi > mnBestInliers) { mvbBestInliers = mvbInliersi; // 每个点是否是内点的标记 mnBestInliers = mnInliersi; // 内点个数 cv::Mat Rcw(3, 3, CV_64F, mRi); cv::Mat tcw(3, 1, CV_64F, mti); Rcw.convertTo(Rcw, CV_32F); tcw.convertTo(tcw, CV_32F); mBestTcw.setIdentity(); mBestTcw.block<3, 3>(0, 0) = Converter::toMatrix3f(Rcw); mBestTcw.block<3, 1>(0, 3) = Converter::toVector3f(tcw); Eigen::Matrix eigRcw(mRi[0]); Eigen::Vector3d eigtcw(mti); } // 用新的内点对相机位姿精求解，提高位姿估计精度，这里如果有足够内点的话，函数直接返回该值，不再继续计算 if (Refine()) { nInliers = mnRefinedInliers; vbInliers = vector(mvpMapPointMatches.size(), false); for (int i = 0; i < N; i++) { if (mvbRefinedInliers[i]) vbInliers[mvKeyPointIndices[i]] = true; } Tout = mRefinedTcw; return true; } } } // 迭代 // Step 3: 选择最小集中效果最好的相机位姿估计结果,如果没有，只能用6个点去计算这个值了 // 程序运行到这里，说明Refine失败了，说明精求解过程中，内点的个数不满足最小阈值，那就只能在当前结果中选择内点数最多的那个最小集 // 但是也意味着这样子的结果最终是用6个点来求出来的，近似效果一般 if (mnIterations >= mRansacMaxIts) { bNoMore = true; if (mnBestInliers >= mRansacMinInliers) { nInliers = mnBestInliers; vbInliers = vector(mvpMapPointMatches.size(), false); for (int i = 0; i < N; i++) { if (mvbBestInliers[i]) vbInliers[mvKeyPointIndices[i]] = true; } Tout = mBestTcw; return true; } } // step 4 相机位姿估计失败，返回零值 // 程序运行到这里，那说明没有满足条件的相机位姿估计结果，位姿估计失败了 return false; } /** * @brief 设置RANSAC迭代的参数 * * @param[in] probability 模型最大概率值，默认0.9 * @param[in] minInliers 内点的最小阈值，默认8 * @param[in] maxIterations 最大迭代次数，默认300 * @param[in] minSet 最小集，每次采样六个点，即最小集应该设置为6，论文里面写着I > 5 * @param[in] epsilon 理论最少内点个数，这里是按照总数的比例计算，所以epsilon是比例，默认是0.4 * @param[in] th2 卡方检验阈值 * */ void MLPnPsolver::SetRansacParameters(double probability, int minInliers, int maxIterations, int minSet, float epsilon, float th2) { mRansacProb = probability; // 模型最大概率值，默认0.9 mRansacMinInliers = minInliers; // 内点的最小阈值，默认8 mRansacMaxIts = maxIterations; // 最大迭代次数，默认300 mRansacEpsilon = epsilon; // 理论最少内点个数，这里是按照总数的比例计算，所以epsilon是比例，默认是0.4 mRansacMinSet = minSet; // 每次采样六个点，即最小集应该设置为6，论文里面写着I > 5 N = mvP2D.size(); // number of correspondences mvbInliersi.resize(N); // 是否是内点的标记位 // Adjust Parameters according to number of correspondences // 计算最少个数点，选择(给定内点数, 最小集, 理论内点数)的最小值 int nMinInliers = N * mRansacEpsilon; if (nMinInliers < mRansacMinInliers) nMinInliers = mRansacMinInliers; if (nMinInliers < minSet) nMinInliers = minSet; mRansacMinInliers = nMinInliers; // 根据最终得到的"最小内点数"来调整 内点数/总体数 比例epsilon if (mRansacEpsilon < (float)mRansacMinInliers / N) mRansacEpsilon = (float)mRansacMinInliers / N; // Set RANSAC iterations according to probability, epsilon, and max iterations // 根据给出的各种参数计算RANSAC的理论迭代次数,并且敲定最终在迭代过程中使用的RANSAC最大迭代次数 int nIterations; if (mRansacMinInliers == N) nIterations = 1; else nIterations = ceil(log(1 - mRansacProb) / log(1 - pow(mRansacEpsilon, 3))); mRansacMaxIts = max(1, min(nIterations, mRansacMaxIts)); // 计算不同图层上的特征点在进行内点检验的时候,所使用的不同判断误差阈值 mvMaxError.resize(mvSigma2.size()); // 层数 for (size_t i = 0; i < mvSigma2.size(); i++) mvMaxError[i] = mvSigma2[i] * th2; // 不同的尺度，设置不同的最大偏差 } /** * @brief 通过之前求解的(R t)检查哪些3D-2D点对属于inliers */ void MLPnPsolver::CheckInliers() { mnInliersi = 0; // 遍历当前帧中所有的匹配点 for (int i = 0; i < N; i++) { // 取出对应的3D点和2D点 point_t p = mvP3Dw[i]; cv::Point3f P3Dw(p(0), p(1), p(2)); cv::Point2f P2D = mvP2D[i]; // 将3D点由世界坐标系旋转到相机坐标系 float xc = mRi[0][0] * P3Dw.x + mRi[0][1] * P3Dw.y + mRi[0][2] * P3Dw.z + mti[0]; float yc = mRi[1][0] * P3Dw.x + mRi[1][1] * P3Dw.y + mRi[1][2] * P3Dw.z + mti[1]; float zc = mRi[2][0] * P3Dw.x + mRi[2][1] * P3Dw.y + mRi[2][2] * P3Dw.z + mti[2]; // 将相机坐标系下的3D进行投影 cv::Point3f P3Dc(xc, yc, zc); cv::Point2f uv = mpCamera->project(P3Dc); // 计算残差 float distX = P2D.x - uv.x; float distY = P2D.y - uv.y; float error2 = distX * distX + distY * distY; // 判定是不是内点 if (error2 < mvMaxError[i]) { mvbInliersi[i] = true; mnInliersi++; } else { mvbInliersi[i] = false; } } } /** * @brief 使用新的内点来继续对位姿进行精求解 */ bool MLPnPsolver::Refine() { // 记录内点的索引值 vector vIndices; vIndices.reserve(mvbBestInliers.size()); for (size_t i = 0; i < mvbBestInliers.size(); i++) { if (mvbBestInliers[i]) { vIndices.push_back(i); } } // Bearing vectors and 3D points used for this ransac iteration // 因为是重定义，所以要另外定义局部变量，和iterate里面一样 // 初始化单位向量和3D点，给当前重定义函数使用 bearingVectors_t bearingVecs; points_t p3DS; vector indexes; // 注意这里是用所有内点vIndices.size()来进行相机位姿估计 // 而iterate里面是用最小集mRansacMinSet(6个)来进行相机位姿估计 // TODO:有什么区别呢？答：肯定有啦，mRansacMinSet只是粗略解， // 这里之所以要Refine就是要用所有满足模型的内点来更加精确地近似表达模型 // 这样求出来的解才会更加准确 for (size_t i = 0; i < vIndices.size(); i++) { int idx = vIndices[i]; bearingVecs.push_back(mvBearingVecs[idx]); p3DS.push_back(mvP3Dw[idx]); indexes.push_back(i); } // 后面操作和iterate类似，就不赘述了 // By the moment, we are using MLPnP without covariance info cov3_mats_t covs(1); // Result transformation_t result; // Compute camera pose computePose(bearingVecs, p3DS, covs, indexes, result); // Check inliers CheckInliers(); mnRefinedInliers = mnInliersi; mvbRefinedInliers = mvbInliersi; if (mnInliersi > mRansacMinInliers) { cv::Mat Rcw(3, 3, CV_64F, mRi); cv::Mat tcw(3, 1, CV_64F, mti); Rcw.convertTo(Rcw, CV_32F); tcw.convertTo(tcw, CV_32F); mRefinedTcw.setIdentity(); mRefinedTcw.block<3, 3>(0, 0) = Converter::toMatrix3f(Rcw); mRefinedTcw.block<3, 1>(0, 3) = Converter::toVector3f(tcw); Eigen::Matrix eigRcw(mRi[0]); Eigen::Vector3d eigtcw(mti); return true; } return false; } // MLPnP methods /** * @brief MLPnP相机位姿估计 * * @param[in] f 单位向量 * @param[in] p 点的3D坐标 * @param[in] covMats 协方差矩阵 * @param[in] indices 对应点的索引值 * @param[in] result 相机位姿估计结果 * */ void MLPnPsolver::computePose(const bearingVectors_t &f, const points_t &p, const cov3_mats_t &covMats, const std::vector &indices, transformation_t &result) { // Step 1: 判断点的数量是否满足计算条件，否则直接报错 // 因为每个观测值会产生2个残差，所以至少需要6个点来计算公式12，所以要检验当前的点个数是否满足大于5的条件 size_t numberCorrespondences = indices.size(); // 当numberCorrespondences不满足>5的条件时会发生错误(如果小于6根本进不来) assert(numberCorrespondences > 5); // ? 用来标记是否满足平面条件，（平面情况下矩阵有相关性，秩为2，矩阵形式可以简化，但需要跟多的约束求解） bool planar = false; // compute the nullspace of all vectors // compute the nullspace of all vectors // step 2: 计算点的单位（方向向量）向量的零空间 // 利用公式7 Jvr(v) = null(v^T) = [r s] // 给每个向量都开辟一个零空间，所以数量相等 std::vector nullspaces(numberCorrespondences); // 存储世界坐标系下空间点的矩阵，3行N列，N是numberCorrespondences，即点的总个数 // |x1, x2, xn| // points3 = |y1, y2, ..., yn| // |z1, z2, zn| Eigen::MatrixXd points3(3, numberCorrespondences); // 空间点向量 // |xi| // points3v = |yi| // |zi| points_t points3v(numberCorrespondences); // 单个空间点的齐次坐标矩阵，TODO:没用到啊 // |xi| // points4v = |yi| // |zi| // |1 | points4_t points4v(numberCorrespondences); // numberCorrespondences不等于所有点，而是提取出来的内点的数量，其作为连续索引值对indices进行索引 // 因为内点的索引并非连续，想要方便遍历，必须用连续的索引值， // 所以就用了indices[i]嵌套形式，i表示内点数量numberCorrespondences范围内的连续形式 // indices里面保存的是不连续的内点的索引值 for (size_t i = 0; i < numberCorrespondences; i++) { // 当前空间点的单位向量,indices[i]是当前点坐标和向量的索引值， bearingVector_t f_current = f[indices[i]]; // 取出当前点记录到 points3 空间点矩阵里 points3.col(i) = p[indices[i]]; // nullspace of right vector // 求解方程 Jvr(v) = null(v^T) = [r s] // A = U * S * V^T // 这里只求解了V的完全解，没有求解U Eigen::JacobiSVD svd_f(f_current.transpose(), Eigen::ComputeFullV); // 取特征值最小的那两个对应的2个特征向量 // |r1 s1| // nullspaces = |r2 s2| // |r3 s3| nullspaces[i] = svd_f.matrixV().block(0, 1, 3, 2); // 取出当前点记录到 points3v 空间点向量 points3v[i] = p[indices[i]]; } // Step 3: 通过计算S的秩来判断是在平面情况还是在正常情况 // 令S = M * M^T，其中M = [p1,p2,...,pi]，即 points3 空间点矩阵 ////////////////////////////////////// // 1. test if we have a planar scene // 在平面条件下，会产生4个解，因此需要另外判断和解决平面条件下的问题 ////////////////////////////////////// // 令S = M * M^T，其中M = [p1,p2,...,pi]，即 points3 空间点矩阵 // 如果矩阵S的秩为3且最小特征值不接近于0，则不属于平面条件，否则需要解决一下 Eigen::Matrix3d planarTest = points3 * points3.transpose(); // 为了计算矩阵S的秩，需要对矩阵S进行满秩的QR分解，通过其结果来判断矩阵S的秩，从而判断是否是平面条件 Eigen::FullPivHouseholderQR rankTest(planarTest); // int r, c; // double minEigenVal = abs(eigen_solver.eigenvalues().real().minCoeff(&r, &c)); // 特征旋转矩阵，用在平面条件下的计算 Eigen::Matrix3d eigenRot; eigenRot.setIdentity(); // if yes -> transform points to new eigen frame // if (minEigenVal < 1e-3 || minEigenVal == 0.0) // rankTest.setThreshold(1e-10); // 当矩阵S的秩为2时，属于平面条件， if (rankTest.rank() == 2) { planar = true; // self adjoint is faster and more accurate than general eigen solvers // also has closed form solution for 3x3 self-adjoint matrices // in addition this solver sorts the eigenvalues in increasing order // 计算矩阵S的特征值和特征向量 Eigen::SelfAdjointEigenSolver eigen_solver(planarTest); // 得到QR分解的结果 eigenRot = eigen_solver.eigenvectors().real(); // 把eigenRot变成其转置矩阵，即论文公式20的系数 R_S^T eigenRot.transposeInPlace(); // 公式20： pi' = R_S^T * pi for (size_t i = 0; i < numberCorrespondences; i++) points3.col(i) = eigenRot * points3.col(i); } ////////////////////////////////////// // 2. stochastic model ////////////////////////////////////// // Step 4: 计算随机模型中的协方差矩阵 // 但是作者并没有用到协方差信息 Eigen::SparseMatrix P(2 * numberCorrespondences, 2 * numberCorrespondences); bool use_cov = false; P.setIdentity(); // standard // if we do have covariance information // -> fill covariance matrix // 如果协方差矩阵的个数等于空间点的个数，说明前面已经计算好了，表示有协方差信息 // 目前版本是没有用到协方差信息的，所以调用本函数前就把协方差矩阵个数置为1了 if (covMats.size() == numberCorrespondences) { use_cov = true; int l = 0; for (size_t i = 0; i < numberCorrespondences; ++i) { // invert matrix cov2_mat_t temp = nullspaces[i].transpose() * covMats[i] * nullspaces[i]; temp = temp.inverse().eval(); P.coeffRef(l, l) = temp(0, 0); P.coeffRef(l, l + 1) = temp(0, 1); P.coeffRef(l + 1, l) = temp(1, 0); P.coeffRef(l + 1, l + 1) = temp(1, 1); l += 2; } } // Step 5: 构造矩阵A来完成线性方程组的构建 ////////////////////////////////////// // 3. fill the design matrix A ////////////////////////////////////// // 公式12，设矩阵A，则有 Au = 0 // u = [r11, r12, r13, r21, r22, r23, r31, r32, r33, t1, t2, t3]^T // 对单位向量v的2个零空间向量做微分，所以有[dr, ds]^T，一个点有2行，N个点就有2*N行 const int rowsA = 2 * numberCorrespondences; // 对应上面u的列数，因为旋转矩阵有9个元素，加上平移矩阵3个元素，总共12个元素 int colsA = 12; Eigen::MatrixXd A; // 如果世界点位于分别跨2个坐标轴的平面上，即所有世界点的一个元素是常数的时候，可简单地忽略矩阵A中对应的列 // 而且这不影响问题的结构本身，所以在计算公式20： pi' = R_S^T * pi的时候，忽略了r11,r21,r31，即第一列 // 对应的u只有9个元素 u = [r12, r13, r22, r23, r32, r33, t1, t2, t3]^T 所以A的列个数是9个 if (planar) { colsA = 9; A = Eigen::MatrixXd(rowsA, 9); } else A = Eigen::MatrixXd(rowsA, 12); A.setZero(); // fill design matrix // 构造矩阵A，分平面和非平面2种情况 if (planar) { for (size_t i = 0; i < numberCorrespondences; ++i) { // 列表示当前点的坐标 point_t pt3_current = points3.col(i); // r12 r12 的系数 r1*py 和 s1*py A(2 * i, 0) = nullspaces[i](0, 0) * pt3_current[1]; A(2 * i + 1, 0) = nullspaces[i](0, 1) * pt3_current[1]; // r13 r13 的系数 r1*pz 和 s1*pz A(2 * i, 1) = nullspaces[i](0, 0) * pt3_current[2]; A(2 * i + 1, 1) = nullspaces[i](0, 1) * pt3_current[2]; // r22 r22 的系数 r2*py 和 s2*py A(2 * i, 2) = nullspaces[i](1, 0) * pt3_current[1]; A(2 * i + 1, 2) = nullspaces[i](1, 1) * pt3_current[1]; // r23 r23 的系数 r2*pz 和 s2*pz A(2 * i, 3) = nullspaces[i](1, 0) * pt3_current[2]; A(2 * i + 1, 3) = nullspaces[i](1, 1) * pt3_current[2]; // r32 r32 的系数 r3*py 和 s3*py A(2 * i, 4) = nullspaces[i](2, 0) * pt3_current[1]; A(2 * i + 1, 4) = nullspaces[i](2, 1) * pt3_current[1]; // r33 r33 的系数 r3*pz 和 s3*pz A(2 * i, 5) = nullspaces[i](2, 0) * pt3_current[2]; A(2 * i + 1, 5) = nullspaces[i](2, 1) * pt3_current[2]; // t1 t1 的系数 r1 和 s1 A(2 * i, 6) = nullspaces[i](0, 0); A(2 * i + 1, 6) = nullspaces[i](0, 1); // t2 t2 的系数 r2 和 s2 A(2 * i, 7) = nullspaces[i](1, 0); A(2 * i + 1, 7) = nullspaces[i](1, 1); // t3 t3 的系数 r3 和 s3 A(2 * i, 8) = nullspaces[i](2, 0); A(2 * i + 1, 8) = nullspaces[i](2, 1); } } else { for (size_t i = 0; i < numberCorrespondences; ++i) { point_t pt3_current = points3.col(i); // 不是平面的情况下，三个列向量都保留求解 // r11 A(2 * i, 0) = nullspaces[i](0, 0) * pt3_current[0]; A(2 * i + 1, 0) = nullspaces[i](0, 1) * pt3_current[0]; // r12 A(2 * i, 1) = nullspaces[i](0, 0) * pt3_current[1]; A(2 * i + 1, 1) = nullspaces[i](0, 1) * pt3_current[1]; // r13 A(2 * i, 2) = nullspaces[i](0, 0) * pt3_current[2]; A(2 * i + 1, 2) = nullspaces[i](0, 1) * pt3_current[2]; // r21 A(2 * i, 3) = nullspaces[i](1, 0) * pt3_current[0]; A(2 * i + 1, 3) = nullspaces[i](1, 1) * pt3_current[0]; // r22 A(2 * i, 4) = nullspaces[i](1, 0) * pt3_current[1]; A(2 * i + 1, 4) = nullspaces[i](1, 1) * pt3_current[1]; // r23 A(2 * i, 5) = nullspaces[i](1, 0) * pt3_current[2]; A(2 * i + 1, 5) = nullspaces[i](1, 1) * pt3_current[2]; // r31 A(2 * i, 6) = nullspaces[i](2, 0) * pt3_current[0]; A(2 * i + 1, 6) = nullspaces[i](2, 1) * pt3_current[0]; // r32 A(2 * i, 7) = nullspaces[i](2, 0) * pt3_current[1]; A(2 * i + 1, 7) = nullspaces[i](2, 1) * pt3_current[1]; // r33 A(2 * i, 8) = nullspaces[i](2, 0) * pt3_current[2]; A(2 * i + 1, 8) = nullspaces[i](2, 1) * pt3_current[2]; // t1 A(2 * i, 9) = nullspaces[i](0, 0); A(2 * i + 1, 9) = nullspaces[i](0, 1); // t2 A(2 * i, 10) = nullspaces[i](1, 0); A(2 * i + 1, 10) = nullspaces[i](1, 1); // t3 A(2 * i, 11) = nullspaces[i](2, 0); A(2 * i + 1, 11) = nullspaces[i](2, 1); } } // Step 6: 计算线性方程组的最小二乘解 ////////////////////////////////////// // 4. solve least squares ////////////////////////////////////// // 求解方程的最小二乘解 Eigen::MatrixXd AtPA; if (use_cov) // 有协方差信息的情况下，一般方程是 A^T*P*A*u = N*u = 0 AtPA = A.transpose() * P * A; // setting up the full normal equations seems to be unstable else // 无协方差信息的情况下，一般方程是 A^T*A*u = N*u = 0 AtPA = A.transpose() * A; // SVD分解，满秩求解V Eigen::JacobiSVD svd_A(AtPA, Eigen::ComputeFullV); // 解就是对应奇异值最小的列向量，即最后一列 Eigen::MatrixXd result1 = svd_A.matrixV().col(colsA - 1); // Step 7: 根据平面和非平面情况下选择最终位姿解 //////////////////////////////// // now we treat the results differently, // depending on the scene (planar or not) //////////////////////////////// // transformation_t T_final; rotation_t Rout; translation_t tout; if (planar) // planar case { rotation_t tmp; // until now, we only estimated // row one and two of the transposed rotation matrix // 暂时只估计了旋转矩阵的第1行和第2行，先记录到tmp中 tmp << 0.0, result1(0, 0), result1(1, 0), 0.0, result1(2, 0), result1(3, 0), 0.0, result1(4, 0), result1(5, 0); // double scale = 1 / sqrt(tmp.col(1).norm() * tmp.col(2).norm()); // row 3 // 第3行等于第1行和第2行的叉积（这里应该是列，后面转置后成了行） tmp.col(0) = tmp.col(1).cross(tmp.col(2)); // 原来是: // |r11 r12 r13| // tmp = |r21 r22 r23| // |r31 r32 r33| // 转置变成: // |r11 r21 r31| // tmp = |r12 r22 r32| // |r13 r23 r33| tmp.transposeInPlace(); // 平移部分 t 只表示了正确的方向，没有尺度，需要求解 scale, 先求系数 double scale = 1.0 / std::sqrt(std::abs(tmp.col(1).norm() * tmp.col(2).norm())); // find best rotation matrix in frobenius sense // 利用Frobenious范数计算最佳的旋转矩阵，利用公式(19)， R = U_R*V_R^T // 本质上，采用矩阵，将其元素平方，将它们加在一起并对结果平方根。计算得出的数字是矩阵的Frobenious范数 // 由于列向量是单列矩阵，行向量是单行矩阵，所以这些矩阵的Frobenius范数等于向量的长度（L） Eigen::JacobiSVD svd_R_frob(tmp, Eigen::ComputeFullU | Eigen::ComputeFullV); rotation_t Rout1 = svd_R_frob.matrixU() * svd_R_frob.matrixV().transpose(); // test if we found a good rotation matrix // 如果估计出来的旋转矩阵的行列式小于0，则乘以-1（EPnP也是同样的操作） if (Rout1.determinant() < 0) Rout1 *= -1.0; // rotate this matrix back using the eigen frame // 因为是在平面情况下计算的，估计出来的旋转矩阵是要做一个转换的，根据公式(21)，R = Rs*R // 其中，Rs表示特征向量的旋转矩阵 // 注意eigenRot之前已经转置过了transposeInPlace()，所以这里Rout1在之前也转置了，即tmp.transposeInPlace() Rout1 = eigenRot.transpose() * Rout1; // 估计最终的平移矩阵，带尺度信息的，根据公式(17)，t = t^ / three-party(||r1||*||r2||*||r3||) // 这里是 t = t^ / sqrt(||r1||*||r2||) translation_t t = scale * translation_t(result1(6, 0), result1(7, 0), result1(8, 0)); // 把之前转置过来的矩阵再转回去，变成公式里面的形态: // |r11 r12 r13| // Rout1 = |r21 r22 r23| // |r31 r32 r33| Rout1.transposeInPlace(); // 这里乘以-1是为了计算4种结果 Rout1 *= -1; if (Rout1.determinant() < 0.0) Rout1.col(2) *= -1; // now we have to find the best out of 4 combinations // |r11 r12 r13| // R1 = |r21 r22 r23| // |r31 r32 r33| // |-r11 -r12 -r13| // R2 = |-r21 -r22 -r23| // |-r31 -r32 -r33| rotation_t R1, R2; R1.col(0) = Rout1.col(0); R1.col(1) = Rout1.col(1); R1.col(2) = Rout1.col(2); R2.col(0) = -Rout1.col(0); R2.col(1) = -Rout1.col(1); R2.col(2) = Rout1.col(2); // |R1 t| // Ts = |R1 -t| // |R2 t| // |R2 -t| vector> Ts(4); Ts[0].block<3, 3>(0, 0) = R1; Ts[0].block<3, 1>(0, 3) = t; Ts[1].block<3, 3>(0, 0) = R1; Ts[1].block<3, 1>(0, 3) = -t; Ts[2].block<3, 3>(0, 0) = R2; Ts[2].block<3, 1>(0, 3) = t; Ts[3].block<3, 3>(0, 0) = R2; Ts[3].block<3, 1>(0, 3) = -t; // 遍历4种解 vector normVal(4); for (int i = 0; i < 4; ++i) { point_t reproPt; double norms = 0.0; // 计算世界点p到切线空间v的投影的残差，对应最小的就是最好的解 // 用前6个点来验证4种解的残差 for (int p = 0; p < 6; ++p) { // 重投影的向量 reproPt = Ts[i].block<3, 3>(0, 0) * points3v[p] + Ts[i].block<3, 1>(0, 3); // 变成单位向量 reproPt = reproPt / reproPt.norm(); // f[indices[p]] 是当前空间点的单位向量 // 利用欧氏距离来表示重投影向量(观测)和当前空间点向量(实际)的偏差 // 即两个n维向量a(x11,x12,…,x1n)与 b(x21,x22,…,x2n)间的欧氏距离 norms += (1.0 - reproPt.transpose() * f[indices[p]]); } // 统计每种解的误差和，第i个解的误差和放入对应的变量normVal[i] normVal[i] = norms; } // 搜索容器中的最小值，并返回该值对应的指针 std::vector::iterator findMinRepro = std::min_element(std::begin(normVal), std::end(normVal)); // 计算容器头指针到最小值指针的距离，即可作为该最小值的索引值 int idx = std::distance(std::begin(normVal), findMinRepro); // 得到最终相机位姿估计的结果 Rout = Ts[idx].block<3, 3>(0, 0); tout = Ts[idx].block<3, 1>(0, 3); } else // non-planar { rotation_t tmp; // 从AtPA的SVD分解中得到旋转矩阵，先存下来 // 注意这里的顺序是和公式16不同的 // |r11 r21 r31| // tmp = |r12 r22 r32| // |r13 r23 r33| tmp << result1(0, 0), result1(3, 0), result1(6, 0), result1(1, 0), result1(4, 0), result1(7, 0), result1(2, 0), result1(5, 0), result1(8, 0); // get the scale // 计算尺度，根据公式(17)，t = t^ / three-party(||r1||*||r2||*||r3||) double scale = 1.0 / std::pow(std::abs(tmp.col(0).norm() * tmp.col(1).norm() * tmp.col(2).norm()), 1.0 / 3.0); // double scale = 1.0 / std::sqrt(std::abs(tmp.col(0).norm() * tmp.col(1).norm())); // find best rotation matrix in frobenius sense // 利用Frobenious范数计算最佳的旋转矩阵，利用公式(19)， R = U_R*V_R^T Eigen::JacobiSVD svd_R_frob(tmp, Eigen::ComputeFullU | Eigen::ComputeFullV); Rout = svd_R_frob.matrixU() * svd_R_frob.matrixV().transpose(); // test if we found a good rotation matrix // 如果估计出来的旋转矩阵的行列式小于0，则乘以-1 if (Rout.determinant() < 0) Rout *= -1.0; // scale translation // 从相机坐标系到世界坐标系的转换关系是 lambda*v = R*pi+t // 从世界坐标系到相机坐标系的转换关系是 pi = R^T*v-R^Tt // 旋转矩阵的性质 R^-1 = R^T // 所以，在下面的计算中，需要计算从世界坐标系到相机坐标系的转换，这里tout = -R^T*t，下面再计算前半部分R^T*v // 先恢复平移部分的尺度再计算 tout = Rout * (scale * translation_t(result1(9, 0), result1(10, 0), result1(11, 0))); // find correct direction in terms of reprojection error, just take the first 6 correspondences // 非平面情况下，一共有2种解，R,t和R,-t // 利用前6个点计算重投影误差，选择残差最小的一个解 vector error(2); vector> Ts(2); for (int s = 0; s < 2; ++s) { // 初始化error的值为0 error[s] = 0.0; // |1 0 0 0| // Ts[s] = |0 1 0 0| // |0 0 1 0| // |0 0 0 1| Ts[s] = Eigen::Matrix4d::Identity(); // |. . . 0| // Ts[s] = |. Rout . 0| // |. . . 0| // |0 0 0 1| Ts[s].block<3, 3>(0, 0) = Rout; if (s == 0) // |. . . . | // Ts[s] = |. Rout . tout| // |. . . . | // |0 0 0 1 | Ts[s].block<3, 1>(0, 3) = tout; else // |. . . . | // Ts[s] = |. Rout . -tout| // |. . . . | // |0 0 0 1 | Ts[s].block<3, 1>(0, 3) = -tout; // 为了避免Eigen中aliasing的问题，后面在计算矩阵的逆的时候，需要添加eval()条件 // a = a.transpose(); //error: aliasing // a = a.transpose().eval(); //ok // a.transposeInPlace(); //ok // Eigen中aliasing指的是在赋值表达式的左右两边存在矩阵的重叠区域，这种情况下，有可能得到非预期的结果。 // 如mat = 2*mat或者mat = mat.transpose()，第一个例子中的alias是没有问题的，而第二的例子则会导致非预期的计算结果。 Ts[s] = Ts[s].inverse().eval(); for (int p = 0; p < 6; ++p) { // 从世界坐标系到相机坐标系的转换关系是 pi = R^T*v-R^Tt // Ts[s].block<3, 3>(0, 0) * points3v[p] = Rout = R^T*v // Ts[s].block<3, 1>(0, 3) = tout = -R^Tt bearingVector_t v = Ts[s].block<3, 3>(0, 0) * points3v[p] + Ts[s].block<3, 1>(0, 3); // 变成单位向量 v = v / v.norm(); // 计算重投影向量(观测)和当前空间点向量(实际)的偏差 error[s] += (1.0 - v.transpose() * f[indices[p]]); } } // 选择残差最小的解作为最终解 if (error[0] < error[1]) tout = Ts[0].block<3, 1>(0, 3); else tout = Ts[1].block<3, 1>(0, 3); Rout = Ts[0].block<3, 3>(0, 0); } // Step 8: 利用高斯牛顿法对位姿进行精确求解，提高位姿解的精度 ////////////////////////////////////// // 5. gauss newton ////////////////////////////////////// // 求解非线性方程之前，需要得到罗德里格斯参数，来表达李群(SO3) -> 李代数(so3)的对数映射 rodrigues_t omega = rot2rodrigues(Rout); // |r1| // |r2| // minx = |r3| // |t1| // |t2| // |t3| Eigen::VectorXd minx(6); minx[0] = omega[0]; minx[1] = omega[1]; minx[2] = omega[2]; minx[3] = tout[0]; minx[4] = tout[1]; minx[5] = tout[2]; // 利用高斯牛顿迭代法来提炼相机位姿 pose mlpnp_gn(minx, points3v, nullspaces, P, use_cov); // 最终输出的结果 Rout = rodrigues2rot(rodrigues_t(minx[0], minx[1], minx[2])); tout = translation_t(minx[3], minx[4], minx[5]); // result inverse as opengv uses this convention // 这里是用来计算世界坐标系到相机坐标系的转换，所以是Pc=R^T*Pw-R^T*t，反变换 result.block<3, 3>(0, 0) = Rout; // Rout.transpose(); result.block<3, 1>(0, 3) = tout; //-result.block<3, 3>(0, 0) * tout; } /** * @brief 旋转向量转换成旋转矩阵，即李群->李代数 * @param[in] omega 旋转向量 * @param[out] R 旋转矩阵 */ Eigen::Matrix3d MLPnPsolver::rodrigues2rot(const Eigen::Vector3d &omega) { // 初始化旋转矩阵 rotation_t R = Eigen::Matrix3d::Identity(); // 求旋转向量的反对称矩阵 Eigen::Matrix3d skewW; skewW << 0.0, -omega(2), omega(1), omega(2), 0.0, -omega(0), -omega(1), omega(0), 0.0; // 求旋转向量的角度 double omega_norm = omega.norm(); // 通过罗德里格斯公式把旋转向量转换成旋转矩阵 if (omega_norm > std::numeric_limits::epsilon()) R = R + sin(omega_norm) / omega_norm * skewW + (1 - cos(omega_norm)) / (omega_norm * omega_norm) * (skewW * skewW); return R; } /** * @brief 旋转矩阵转换成旋转向量，即李代数->李群 * @param[in] R 旋转矩阵 * @param[out] omega 旋转向量 问题: 李群(SO3) -> 李代数(so3)的对数映射 利用罗德里格斯公式，已知旋转矩阵的情况下，求解其对数映射ln(R) 就像《视觉十四讲》里面说到的，使用李代数的一大动机是为了进行优化,而在优化过程中导数是非常必要的信息 李群SO(3)中完成两个矩阵乘法，不能用李代数so(3)中的加法表示，问题描述为两个李代数对数映射乘积 而两个李代数对数映射乘积的完整形式，可以用BCH公式表达，其中式子(左乘BCH近似雅可比J)可近似表达，该式也就是罗德里格斯公式 计算完罗德里格斯参数之后，就可以用非线性优化方法了，里面就要用到导数，其形式就是李代数指数映射 所以要在调用非线性MLPnP算法之前计算罗德里格斯参数 */ Eigen::Vector3d MLPnPsolver::rot2rodrigues(const Eigen::Matrix3d &R) { rodrigues_t omega; omega << 0.0, 0.0, 0.0; // R.trace() 矩阵的迹，即该矩阵的特征值总和 double trace = R.trace() - 1.0; // 李群(SO3) -> 李代数(so3)的对数映射 // 对数映射ln(R)∨将一个旋转矩阵转换为一个李代数，但由于对数的泰勒展开不是很优雅所以用本式 // wnorm是求解出来的角度 double wnorm = acos(trace / 2.0); // 如果wnorm大于运行编译程序的计算机所能识别的最小非零浮点数，则可以生成向量，否则为0 if (wnorm > std::numeric_limits::epsilon()) { // |r11 r21 r31| // R = |r12 r22 r32| // |r13 r23 r33| omega[0] = (R(2, 1) - R(1, 2)); omega[1] = (R(0, 2) - R(2, 0)); omega[2] = (R(1, 0) - R(0, 1)); // theta |r23 - r32| // ln(R) = ------------ * |r31 - r13| // 2*sin(theta) |r12 - r21| double sc = wnorm / (2.0 * sin(wnorm)); omega *= sc; } return omega; } /** * @brief MLPnP的高斯牛顿解法 * @param[in] x 未知量矩阵 * @param[in] pts 3D点矩阵 * @param[in] nullspaces 零空间向量 * @param[in] Kll 协方差矩阵 * @param[in] use_cov 协方差方法使用标记位 */ void MLPnPsolver::mlpnp_gn(Eigen::VectorXd &x, const points_t &pts, const std::vector &nullspaces, const Eigen::SparseMatrix Kll, bool use_cov) { // 计算观测值数量 const int numObservations = pts.size(); // 未知量是旋转向量和平移向量，即R和t，总共6个未知参数 const int numUnknowns = 6; // check redundancy // 检查观测数量是否满足计算条件，因为每个观测值都提供了2个约束，即r和s，所以这里乘以2 assert((2 * numObservations - numUnknowns) > 0); // ============= // set all matrices up // ============= Eigen::VectorXd r(2 * numObservations); Eigen::VectorXd rd(2 * numObservations); Eigen::MatrixXd Jac(2 * numObservations, numUnknowns); Eigen::VectorXd g(numUnknowns, 1); Eigen::VectorXd dx(numUnknowns, 1); // result vector Jac.setZero(); r.setZero(); dx.setZero(); g.setZero(); int it_cnt = 0; bool stop = false; const int maxIt = 5; double epsP = 1e-5; Eigen::MatrixXd JacTSKll; Eigen::MatrixXd A; // solve simple gradient descent while (it_cnt < maxIt && !stop) { mlpnp_residuals_and_jacs(x, pts, nullspaces, r, Jac, true); if (use_cov) JacTSKll = Jac.transpose() * Kll; else JacTSKll = Jac.transpose(); A = JacTSKll * Jac; // get system matrix g = JacTSKll * r; // solve Eigen::LDLT chol(A); dx = chol.solve(g); // dx = A.jacobiSvd(Eigen::ComputeThinU | Eigen::ComputeThinV).solve(g); // this is to prevent the solution from falling into a wrong minimum // if the linear estimate is spurious if (dx.array().abs().maxCoeff() > 5.0 || dx.array().abs().minCoeff() > 1.0) break; // observation update Eigen::MatrixXd dl = Jac * dx; if (dl.array().abs().maxCoeff() < epsP) { stop = true; x = x - dx; break; } else x = x - dx; ++it_cnt; } // while // result } /** * @brief 计算Jacobian矩阵和残差 * @param[in] x 未知量矩阵 * @param[in] pts 3D点矩阵 * @param[in] nullspaces 零空间向量 * @param[in] r f(x)函数 * @param[out] fjac f(x)函数的Jacobian矩阵 * @param[in] getJacs 是否可以得到Jacobian矩阵标记位 */ void MLPnPsolver::mlpnp_residuals_and_jacs(const Eigen::VectorXd &x, const points_t &pts, const std::vector &nullspaces, Eigen::VectorXd &r, Eigen::MatrixXd &fjac, bool getJacs) { rodrigues_t w(x[0], x[1], x[2]); translation_t T(x[3], x[4], x[5]); // rotation_t R = math::cayley2rot(c); rotation_t R = rodrigues2rot(w); int ii = 0; Eigen::MatrixXd jacs(2, 6); for (int i = 0; i < pts.size(); ++i) { Eigen::Vector3d ptCam = R * pts[i] + T; ptCam /= ptCam.norm(); r[ii] = nullspaces[i].col(0).transpose() * ptCam; r[ii + 1] = nullspaces[i].col(1).transpose() * ptCam; if (getJacs) { // jacs mlpnpJacs(pts[i], nullspaces[i].col(0), nullspaces[i].col(1), w, T, jacs); // r fjac(ii, 0) = jacs(0, 0); fjac(ii, 1) = jacs(0, 1); fjac(ii, 2) = jacs(0, 2); fjac(ii, 3) = jacs(0, 3); fjac(ii, 4) = jacs(0, 4); fjac(ii, 5) = jacs(0, 5); // s fjac(ii + 1, 0) = jacs(1, 0); fjac(ii + 1, 1) = jacs(1, 1); fjac(ii + 1, 2) = jacs(1, 2); fjac(ii + 1, 3) = jacs(1, 3); fjac(ii + 1, 4) = jacs(1, 4); fjac(ii + 1, 5) = jacs(1, 5); } ii += 2; } } /** * @brief 计算Jacobian矩阵 * @param[in] pt 3D点矩阵 * @param[in] nullspace_r 零空间向量r * @param[in] nullspace_r 零空间向量s * @param[in] w 旋转向量w * @param[in] t 平移向量t * @param[out] jacs Jacobian矩阵 */ void MLPnPsolver::mlpnpJacs(const point_t &pt, const Eigen::Vector3d &nullspace_r, const Eigen::Vector3d &nullspace_s, const rodrigues_t &w, const translation_t &t, Eigen::MatrixXd &jacs) { double r1 = nullspace_r[0]; double r2 = nullspace_r[1]; double r3 = nullspace_r[2]; double s1 = nullspace_s[0]; double s2 = nullspace_s[1]; double s3 = nullspace_s[2]; double X1 = pt[0]; double Y1 = pt[1]; double Z1 = pt[2]; double w1 = w[0]; double w2 = w[1]; double w3 = w[2]; double t1 = t[0]; double t2 = t[1]; double t3 = t[2]; double t5 = w1 * w1; double t6 = w2 * w2; double t7 = w3 * w3; // t8 = theta^2 = w1^2+w2^2+w3^2 double t8 = t5 + t6 + t7; // t9 = sqrt(t8) = sqrt(theta^2) = theta double t9 = sqrt(t8); // t10 = sin(theta) double t10 = sin(t9); // t11 = 1 / theta double t11 = 1.0 / sqrt(t8); // t12 = cos(theta) double t12 = cos(t9); // t13 = cos(theta) - 1 double t13 = t12 - 1.0; // t14 = 1 / theta^2 double t14 = 1.0 / t8; // t16 = sin(theta)/theta*w3 // 令 A = sin(theta)/theta = t10*t11 // t16 = A*w3 double t16 = t10 * t11 * w3; // t17 = (cos(theta) - 1)/theta^2 * w1 * w2 // 令 B = (cos(theta) - 1)/theta^2 = t13*t14 // t17 = B*w1*w2 double t17 = t13 * t14 * w1 * w2; // t19 = sin(theta)/theta*w2 // = A*w2 double t19 = t10 * t11 * w2; // t20 = (cos(theta) - 1) / theta^2 * w1 * w3 // = B*w1*w3 double t20 = t13 * t14 * w1 * w3; // t24 = w2^2+w3^2 double t24 = t6 + t7; // t27 = A*w3 + B*w1*w2 // = -r12 double t27 = t16 + t17; // t28 = (A*w3 + B*w1*w2)*Y1 // = -r12*Y1 double t28 = Y1 * t27; // t29 = A*w2 - B*w1*w3 // = r13 double t29 = t19 - t20; // t30 = (A*w2 - B*w1*w3) * Z1 // = r13 * Z1 double t30 = Z1 * t29; // t31 = (cos(theta) - 1) /theta^2 * (w2^2+w3^2) // = B * (w2^2+w3^2) double t31 = t13 * t14 * t24; // t32 = B * (w2^2+w3^2) + 1 // = r11 double t32 = t31 + 1.0; // t33 = (B * (w2^2+w3^2) + 1) * X1 // = r11 * X1 double t33 = X1 * t32; // t15 = t1 - (A*w3 + B*w1*w2)*Y1 + (A*w2 - B*w1*w3) * Z1 + (B * (w2^2+w3^2) + 1) * X1 // = t1 + r11*X1 + r12*Y1 + r13*Z1 double t15 = t1 - t28 + t30 + t33; // t21 = t10 * t11 * w1 = sin(theta) / theta * w1 // = A*w1 double t21 = t10 * t11 * w1; // t22 = t13 * t14 * w2 * w3 = (cos(theta) - 1) / theta^2 * w2 * w3 // = B*w2*w3 double t22 = t13 * t14 * w2 * w3; // t45 = t5 + t7 // = w1^2 + w3^2 double t45 = t5 + t7; // t53 = t16 - t17 // = A*w3 - B*w1*w2 = r21 double t53 = t16 - t17; // t54 = r21*X1 double t54 = X1 * t53; // t55 = A*w1 + B*w2*w3 // = -r23 double t55 = t21 + t22; // t56 = -r23*Z1 double t56 = Z1 * t55; // t57 = t13 * t14 * t45 = (cos(theta) - 1) / theta^2 * (w1^2 + w3^2) // = B8(w1^2+w3^2) double t57 = t13 * t14 * t45; // t58 = B8(w1^2+w3^2) + 1.0 // = r22 double t58 = t57 + 1.0; // t59 = r22*Y1 double t59 = Y1 * t58; // t18 = t2 + t54 - t56 + t59 // = t2 + r21*X1 + r22*Y1 + r23*Z1 double t18 = t2 + t54 - t56 + t59; // t34 = t5 + t6 // = w1^2+w2^2 double t34 = t5 + t6; // t38 = t19 + t20 = A*w2+B*w1*w3 // = -r31 double t38 = t19 + t20; // t39 = -r31*X1 double t39 = X1 * t38; // t40 = A*w1 - B*w2*w3 // = r32 double t40 = t21 - t22; // t41 = r32*Y1 double t41 = Y1 * t40; // t42 = t13 * t14 * t34 = (cos(theta) - 1) / theta^2 * (w1^2+w2^2) // = B*(w1^2+w2^2) double t42 = t13 * t14 * t34; // t43 = B*(w1^2+w2^2) + 1 // = r33 double t43 = t42 + 1.0; // t44 = r33*Z1 double t44 = Z1 * t43; // t23 = t3 - t39 + t41 + t44 = t3 + r31*X1 + r32*Y1 + r33*Z1 double t23 = t3 - t39 + t41 + t44; // t25 = 1.0 / pow(theta^2, 3.0 / 2.0) // = 1 / theta^3 double t25 = 1.0 / pow(t8, 3.0 / 2.0); // t26 = 1.0 / (t8 * t8) // = 1 / theta^4 double t26 = 1.0 / (t8 * t8); // t35 = t12 * t14 * w1 * w2 = cos(theta) / theta^2 * w1 * w2 // 令 cos(theta) / theta^2 = E = t12*t14 // t35 = E*w1*w2 double t35 = t12 * t14 * w1 * w2; // t36 = t5 * t10 * t25 * w3 = w1^2 * sin(theta) / theta^3 * w3 // 令 sin(theta) / theta^3 = F = t10*t25 // t36 = F*w1^2*w3 double t36 = t5 * t10 * t25 * w3; // t37 = t5 * t13 * t26 * w3 * 2.0 = w1^2 * (cos(theta) - 1) / theta^4 * w3 // 令 (cos(theta) - 1) / theta^4 = G = t13*t26 // t37 = G*w1^2*w3 double t37 = t5 * t13 * t26 * w3 * 2.0; // t46 = t10 * t25 * w1 * w3 = sin(theta) / theta^3 * w1 * w3 = F*w1*w3 double t46 = t10 * t25 * w1 * w3; // t47 = t5 * t10 * t25 * w2 = w1^2 * sin(theta) / theta^3 * w2 = F*w1^2*w2 double t47 = t5 * t10 * t25 * w2; // t48 = t5 * t13 * t26 * w2 * 2.0 = w1^2 * (cos(theta) - 1) / theta^4 * 2 * w2 = G*w1^2*w2 double t48 = t5 * t13 * t26 * w2 * 2.0; // t49 = t10 * t11 = sin(theta) / theta = A double t49 = t10 * t11; // t50 = t5 * t12 * t14 = w1^2 * cos(theta) / theta^2 = E*w1^2 double t50 = t5 * t12 * t14; // t51 = t13 * t26 * w1 * w2 * w3 * 2.0 = (cos(theta) - 1) / theta^4 * w1 * w2 * w3 * 2 = 2*G*w1*w2*w3 double t51 = t13 * t26 * w1 * w2 * w3 * 2.0; // t52 = t10 * t25 * w1 * w2 * w3 = sin(theta) / theta^3 * w1 * w2 * w3 = F*w1*w2*w3 double t52 = t10 * t25 * w1 * w2 * w3; // t60 = t15 * t15 = (t1 + r11*X1 + r12*Y1 + r13*Z1)^2 double t60 = t15 * t15; // t61 = t18 * t18 = (t2 + r21*X1 + r22*Y1 + r23*Z1)^2 double t61 = t18 * t18; // t62 = t23 * t23 = (t3 + r31*X1 + r32*Y1 + r33*Z1)^2 double t62 = t23 * t23; // t63 = t60 + t61 + t62 = (t1 + r11*X1 + r12*Y1 + r13*Z1)^2 + (t2 + r21*X1 + r22*Y1 + r23*Z1)^2 + (t3 + r31*X1 + r32*Y1 + r33*Z1)^2 double t63 = t60 + t61 + t62; // t64 = t5 * t10 * t25 = w1^2 * sin(theta) / theta^3 = F*w1^2 double t64 = t5 * t10 * t25; // t65 = 1 / sqrt((t1 + r11*X1 + r12*Y1 + r13*Z1)^2 + (t2 + r21*X1 + r22*Y1 + r23*Z1)^2 + (t3 + r31*X1 + r32*Y1 + r33*Z1)^2) double t65 = 1.0 / sqrt(t63); // t66 = Y1 * r2 * t6 = Y1 * r2 * w2^2 double t66 = Y1 * r2 * t6; // t67 = Z1 * r3 * t7 = Z1 * r3 * w3^2 double t67 = Z1 * r3 * t7; // t68 = r1 * t1 * t5 = r1 * t1 * w1^2 double t68 = r1 * t1 * t5; // t69 = r1 * t1 * t6 = r1 * t1 * w2^2 double t69 = r1 * t1 * t6; // t70 = r1 * t1 * t7 = r1 * t1 * w3^2 double t70 = r1 * t1 * t7; // t71 = r2 * t2 * t5 = r2 * t2 * w1^2 double t71 = r2 * t2 * t5; // t72 = r2 * t2 * t6 = r2 * t2 * w2^2 double t72 = r2 * t2 * t6; // t73 = r2 * t2 * t7 = r2 * t2 * w3^2 double t73 = r2 * t2 * t7; // t74 = r3 * t3 * t5 = r3 * t3 * w1^2 double t74 = r3 * t3 * t5; // t75 = r3 * t3 * t6 = r3 * t3 * w2^2 double t75 = r3 * t3 * t6; // t76 = r3 * t3 * t7 = r3 * t3 * w3^2 double t76 = r3 * t3 * t7; // t77 = X1 * r1 * t5 = X1 * r1 * w1^2 double t77 = X1 * r1 * t5; double t78 = X1 * r2 * w1 * w2; double t79 = X1 * r3 * w1 * w3; double t80 = Y1 * r1 * w1 * w2; double t81 = Y1 * r3 * w2 * w3; double t82 = Z1 * r1 * w1 * w3; double t83 = Z1 * r2 * w2 * w3; // t84 = X1 * r1 * t6 * t12 = X1 * r1 * w2^2 * cos(theta) double t84 = X1 * r1 * t6 * t12; // t85 = X1 * r1 * t7 * t12 = X1 * r1 * w3^2 * cos(theta) double t85 = X1 * r1 * t7 * t12; // t86 = Y1 * r2 * t5 * t12 = Y1 * r2 * w1^2 * cos(theta) double t86 = Y1 * r2 * t5 * t12; // t87 = Y1 * r2 * t7 * t12 = Y1 * r2 * w3^2 * cos(theta) double t87 = Y1 * r2 * t7 * t12; // t88 = Z1 * r3 * t5 * t12 = Z1 * r3 * w1^2 * cos(theta) double t88 = Z1 * r3 * t5 * t12; // t89 = Z1 * r3 * t6 * t12 = Z1 * r3 * w2^2 * cos(theta) double t89 = Z1 * r3 * t6 * t12; // t90 = X1 * r2 * t9 * t10 * w3 = X1 * r2 * theta * sin(theta) * w3 double t90 = X1 * r2 * t9 * t10 * w3; // t91 = Y1 * r3 * t9 * t10 * w1 = Y1 * r3 * theta * sin(theta) * w1 double t91 = Y1 * r3 * t9 * t10 * w1; // t92 = Z1 * r1 * t9 * t10 * w2 = Z1 * r1 * theta * sin(theta) * w2 double t92 = Z1 * r1 * t9 * t10 * w2; // t102 = X1 * r3 * t9 * t10 * w2 = X1 * r3 * theta * sin(theta) * w2 double t102 = X1 * r3 * t9 * t10 * w2; // t103 = Y1 * r1 * t9 * t10 * w3 = Y1 * r1 * theta * sin(theta) * w3 double t103 = Y1 * r1 * t9 * t10 * w3; // t104 = Z1 * r2 * t9 * t10 * w1 = Z1 * r2 * theta * sin(theta) * w1 double t104 = Z1 * r2 * t9 * t10 * w1; // t105 = X1 * r2 * t12 * w1 * w2 = X1 * r2 * cos(theta) * w1 * w2 double t105 = X1 * r2 * t12 * w1 * w2; // t106 = X1 * r3 * t12 * w1 * w3 = X1 * r3 * cos(theta) * w1 * w3 double t106 = X1 * r3 * t12 * w1 * w3; // t107 = Y1 * r1 * t12 * w1 * w2 = Y1 * r1 * cos(theta) * w1 * w2 double t107 = Y1 * r1 * t12 * w1 * w2; // t108 = Y1 * r3 * t12 * w2 * w3 = Y1 * r3 * cos(theta) * w2 * w3 double t108 = Y1 * r3 * t12 * w2 * w3; // t109 = Z1 * r1 * t12 * w1 * w3 = Z1 * r1 * cos(theta) * w1 * w3 double t109 = Z1 * r1 * t12 * w1 * w3; // t110 = Z1 * r2 * t12 * w2 * w3 = Z1 * r2 * cos(theta) * w2 * w3 double t110 = Z1 * r2 * t12 * w2 * w3; // t93 = t66 + t67 + t68 + t69 // + t70 + t71 + t72 + t73 + t74 // + t75 + t76 + t77 + t78 + t79 // + t80 + t81 + t82 + t83 + t84 // + t85 + t86 + t87 + t88 + t89 // + t90 + t91 + t92 - t102 - t103 // - t104 - t105 - t106 - t107 // - t108 - t109 - t110 // // = (Y1 * r2 * w2^2) + (Z1 * r3 * w3^2) + (r1 * t1 * w1^2) + (r1 * t1 * w2^2) // + (r1 * t1 * w3^2) + (r2 * t2 * w1^2) + (r2 * t2 * w2^2) + (r2 * t2 * w3^2) + (r3 * t3 * w1^2) // + (r3 * t3 * w2^2) + (r3 * t3 * w3^2) + (X1 * r1 * w1^2) + (X1 * r2 * w1 * w2) + (X1 * r3 * w1 * w3) // + (Y1 * r1 * w1 * w2) + (Y1 * r3 * w2 * w3) + (Z1 * r1 * w1 * w3) + (Z1 * r2 * w2 * w3) + (X1 * r1 * w2^2 * cos(theta)) // + (X1 * r1 * w3^2 * cos(theta)) + (Y1 * r2 * w1^2 * cos(theta)) + (Y1 * r2 * w3^2 * cos(theta)) + (Z1 * r3 * w1^2 * cos(theta)) + (Z1 * r3 * w2^2 * cos(theta)) // + (X1 * r2 * theta * sin(theta) * w3) + (Y1 * r3 * theta * sin(theta) * w1) + (Z1 * r1 * theta * sin(theta) * w2) - (X1 * r3 * theta * sin(theta) * w2) - (Y1 * r1 * theta * sin(theta) * w3) // - (Z1 * r2 * theta * sin(theta) * w1) - (X1 * r2 * cos(theta) * w1 * w2) - (X1 * r3 * cos(theta) * w1 * w3) - (Y1 * r1 * cos(theta) * w1 * w2) - () // - (Y1 * r3 * cos(theta) * w2 * w3) - (Z1 * r1 * cos(theta) * w1 * w3) - (Z1 * r2 * cos(theta) * w2 * w3) double t93 = t66 + t67 + t68 + t69 + t70 + t71 + t72 + t73 + t74 + t75 + t76 + t77 + t78 + t79 + t80 + t81 + t82 + t83 + t84 + t85 + t86 + t87 + t88 + t89 + t90 + t91 + t92 - t102 - t103 - t104 - t105 - t106 - t107 - t108 - t109 - t110; // t94 = sin(theta)/theta^3* w1 * w2 = F*w1*w2 double t94 = t10 * t25 * w1 * w2; // t95 = w2^2*sin(theta)/theta^3*w3 = F*w2^2*w3 double t95 = t6 * t10 * t25 * w3; // t96 = 2 * w2^2 * (cos(theta) - 1) / theta^4 * w3 = 2*G*w2^2*w3 double t96 = t6 * t13 * t26 * w3 * 2.0; // t97 = cos(theta) / theta^2 * w2 * w3 = E*w2*w3 double t97 = t12 * t14 * w2 * w3; // t98 = w2^2 * sin(theta) / theta^3 * w1 = F*w2^2*w1 double t98 = t6 * t10 * t25 * w1; // t99 = 2 * w2^2 * (cos(theta) - 1) / theta^4 * w1 = 2*G*w2^2*w1 double t99 = t6 * t13 * t26 * w1 * 2.0; // t100 = w2^2 * sin(theta) / theta^3 = F*w2^2 double t100 = t6 * t10 * t25; // t101 = 1.0 / pow(sqrt((t1 + r11*X1 + r12*Y1 + r13*Z1)^2 + (t2 + r21*X1 + r22*Y1 + r23*Z1)^2 + (t3 + r31*X1 + r32*Y1 + r33*Z1)^2), 3.0 / 2.0) // = (t1 + r11*X1 + r12*Y1 + r13*Z1)^2 + (t2 + r21*X1 + r22*Y1 + r23*Z1)^2 + (t3 + r31*X1 + r32*Y1 + r33*Z1)^3 double t101 = 1.0 / pow(t63, 3.0 / 2.0); // t111 = w2^2 * cos(theta) / theta^2 = E*w2^2 double t111 = t6 * t12 * t14; // t112 = sin(theta) / theta^3 *w2 * w3 = F*w2*w3 double t112 = t10 * t25 * w2 * w3; // t113 = cos(theta) / theta^2 * w1 * w3 = E*w1*w3 double t113 = t12 * t14 * w1 * w3; // t114 = w3^2 * sin(theta) / theta^3 * w2 = F*w3^2*w2 double t114 = t7 * t10 * t25 * w2; // t115 = t7 * t13 * t26 * w2 * 2.0 = 2*w3^2*(cos(theta) - 1) / theta^4*w2 double t115 = t7 * t13 * t26 * w2 * 2.0; // t116 = w3^2 * sin(theta) / theta^3 * w1 = F*w3^2*w1 double t116 = t7 * t10 * t25 * w1; // t117 = 2 * w3^2 * (cos(theta) - 1) / theta^4 * w1 = 2*G*w3^2*w1 double t117 = t7 * t13 * t26 * w1 * 2.0; // t118 = E*w3^2 double t118 = t7 * t12 * t14; // t119 = 2*w1*G*(w2^2+w3^2) double t119 = t13 * t24 * t26 * w1 * 2.0; // t120 = F*w1*(w2^2+w3^2) double t120 = t10 * t24 * t25 * w1; // t121 = (2*G+F)*w1*(w2^2+w3^2) double t121 = t119 + t120; // t122 = 2*G*w1*(w1^2+w2^2) double t122 = t13 * t26 * t34 * w1 * 2.0; // t123 = F*w1*(w1^2+w2^2) double t123 = t10 * t25 * t34 * w1; // t131 = 2*(cos(theta) - 1) / theta^2*w1 = 2*B*w1 double t131 = t13 * t14 * w1 * 2.0; // t124 = t122 + t123 - t131 = 2*G*w1*(w1^2+w2^2)+F*w1*(w1^2+w2^2)-2*B*w1 // = (2*G+F)*w1*(w1^2+w2^2)-2*B*w1 double t124 = t122 + t123 - t131; // t139 = t13 * t14 * w3 = B*w3 double t139 = t13 * t14 * w3; // t125 = -t35 + t36 + t37 + t94 - t139 = -E*w1*w2+F*w1^2*w3+G*w1^2*w3+F*w1*w2-B*w3 double t125 = -t35 + t36 + t37 + t94 - t139; // t126 = (-E*w1*w2+F*w1^2*w3+G*w1^2*w3+F*w1*w2-B*w3)*X1 double t126 = X1 * t125; // t127 = t49 + t50 + t51 + t52 - t64 = A + E*w1^2 + (2*G+F)*w1*w2*w3 - F*w1^2 double t127 = t49 + t50 + t51 + t52 - t64; // t128 = (A + E*w1^2 + (2*G+F)*w1*w2*w3 - F*w1^2)*Y1 double t128 = Y1 * t127; // t129 = (-E*w1*w2+F*w1^2*w3+G*w1^2*w3+F*w1*w2-B*w3)*X1 + (A + E*w1^2 + (2*G+F)*w1*w2*w3 - F*w1^2)*Y1 - ((2*G+F)*w1*(w1^2+w2^2)-2*B*w1)Z1 double t129 = t126 + t128 - Z1 * t124; // t130 = 2 * (t3 + r31*X1 + r32*Y1 + r33*Z1) // * ((-E*w1*w2+F*w1^2*w3+G*w1^2*w3+F*w1*w2-B*w3)*X1 + (A + E*w1^2 + (2*G+F)*w1*w2*w3 - F*w1^2)*Y1 - ((2*G+F)*w1*(w1^2+w2^2)-2*B*w1)Z1) double t130 = t23 * t129 * 2.0; // t132 = t13 * t26 * t45 * w1 * 2.0 = 2*w1*G*(w1^2+w3^2) double t132 = t13 * t26 * t45 * w1 * 2.0; // t133 = t10 * t25 * t45 * w1 = F*w1*(w1^2+w3^2) double t133 = t10 * t25 * t45 * w1; // t138 = t13 * t14 * w2 = B*w2 double t138 = t13 * t14 * w2; // t134 = -t46 + t47 + t48 + t113 - t138 = -F*w1*w3+F*w1^2*w2+G*w1^2*w2+E*w1*w3-B*w2 double t134 = -t46 + t47 + t48 + t113 - t138; // t135 = (-F*w1*w3+F*w1^2*w2+G*w1^2*w2+E*w1*w3-B*w2)*X1 double t135 = X1 * t134; // t136 = -t49 - t50 + t51 + t52 + t64 = -A-E*w1^2+2*G*w1*w2*w3+F*w1*w2*w3+F*w1^2 double t136 = -t49 - t50 + t51 + t52 + t64; // t137 = (-A-E*w1^2+2*G*w1*w2*w3+F*w1*w2*w3+F*w1^2)*Z1 double t137 = Z1 * t136; double t140 = X1 * s1 * t5; double t141 = Y1 * s2 * t6; double t142 = Z1 * s3 * t7; double t143 = s1 * t1 * t5; double t144 = s1 * t1 * t6; double t145 = s1 * t1 * t7; double t146 = s2 * t2 * t5; double t147 = s2 * t2 * t6; double t148 = s2 * t2 * t7; double t149 = s3 * t3 * t5; double t150 = s3 * t3 * t6; double t151 = s3 * t3 * t7; double t152 = X1 * s2 * w1 * w2; double t153 = X1 * s3 * w1 * w3; double t154 = Y1 * s1 * w1 * w2; double t155 = Y1 * s3 * w2 * w3; double t156 = Z1 * s1 * w1 * w3; double t157 = Z1 * s2 * w2 * w3; // t12 = cos(theta) double t158 = X1 * s1 * t6 * t12; double t159 = X1 * s1 * t7 * t12; double t160 = Y1 * s2 * t5 * t12; double t161 = Y1 * s2 * t7 * t12; double t162 = Z1 * s3 * t5 * t12; double t163 = Z1 * s3 * t6 * t12; // t9 = theta, t10 = sin(theta) double t164 = X1 * s2 * t9 * t10 * w3; double t165 = Y1 * s3 * t9 * t10 * w1; double t166 = Z1 * s1 * t9 * t10 * w2; double t183 = X1 * s3 * t9 * t10 * w2; double t184 = Y1 * s1 * t9 * t10 * w3; double t185 = Z1 * s2 * t9 * t10 * w1; // t12 = cos(theta) double t186 = X1 * s2 * t12 * w1 * w2; double t187 = X1 * s3 * t12 * w1 * w3; double t188 = Y1 * s1 * t12 * w1 * w2; double t189 = Y1 * s3 * t12 * w2 * w3; double t190 = Z1 * s1 * t12 * w1 * w3; double t191 = Z1 * s2 * t12 * w2 * w3; // t167 = t140 + t141 + t142 + t143 + t144 // + t145 + t146 + t147 + t148 + t149 // + t150 + t151 + t152 + t153 + t154 // + t155 + t156 + t157 + t158 + t159 // + t160 + t161 + t162 + t163 + t164 // + t165 + t166 - t183 - t184 - t185 // - t186 - t187 - t188 // - t189 - t190 - t191 // // = (X1 * s1 * t5) + (Y1 * s2 * t6) + (Z1 * s3 * t7) + (s1 * t1 * t5) + (s1 * t1 * t6) // + (s1 * t1 * t7) + (s2 * t2 * t5) + (s2 * t2 * t6) + (s2 * t2 * t7) + (s3 * t3 * t5) // + (s3 * t3 * t6) + (s3 * t3 * t7) + (X1 * s2 * w1 * w2) + (X1 * s3 * w1 * w3) + (Y1 * s1 * w1 * w2) // + (Y1 * s3 * w2 * w3) + (Z1 * s1 * w1 * w3) + (Z1 * s2 * w2 * w3) + (X1 * s1 * t6 * t12) + (X1 * s1 * t7 * t12) // + (Y1 * s2 * t5 * t12) + (Y1 * s2 * t7 * t12) + (Z1 * s3 * t5 * t12) + (Z1 * s3 * t6 * t12) + (X1 * s2 * t9 * t10 * w3) // + (Y1 * s3 * t9 * t10 * w1) + (Z1 * s1 * t9 * t10 * w2) - (X1 * s3 * t9 * t10 * w2) - (Y1 * s1 * t9 * t10 * w3) - (Z1 * s2 * t9 * t10 * w1) // - (X1 * s2 * t12 * w1 * w2) - (X1 * s3 * t12 * w1 * w3) - (Y1 * s1 * t12 * w1 * w2) // - (Y1 * s3 * t12 * w2 * w3) - (Z1 * s1 * t12 * w1 * w3) - (Z1 * s2 * t12 * w2 * w3) double t167 = t140 + t141 + t142 + t143 + t144 + t145 + t146 + t147 + t148 + t149 + t150 + t151 + t152 + t153 + t154 + t155 + t156 + t157 + t158 + t159 + t160 + t161 + t162 + t163 + t164 + t165 + t166 - t183 - t184 - t185 - t186 - t187 - t188 - t189 - t190 - t191; // t168 = t13 * t26 * t45 * w2 * 2.0 = 2*G*w2*(w1^2+w3^2) double t168 = t13 * t26 * t45 * w2 * 2.0; // t169 = t10 * t25 * t45 * w2 = F*w2*(w1^2+w3^2) double t169 = t10 * t25 * t45 * w2; // t170 = t168 + t169 = (2*G+F)*w2*(w1^2+w3^2) double t170 = t168 + t169; // t171 = t13 * t26 * t34 * w2 * 2.0 = 2*G*w2*(w1^2+w2^2) double t171 = t13 * t26 * t34 * w2 * 2.0; // t172 = t10 * t25 * t34 * w2 = F*w2*(w1^2+w2^2) double t172 = t10 * t25 * t34 * w2; // t176 = t13 * t14 * w2 * 2.0 = 2*B*w2 double t176 = t13 * t14 * w2 * 2.0; // t173 = t171 + t172 - t176 = (2*G+F)*w2*(w1^2+w2^2) - 2*B*w2 double t173 = t171 + t172 - t176; // t174 = -t49 + t51 + t52 + t100 - t111 = -A+2*G*w1*w2*w3+F*w1*w2*w3+F*w2^2-E*w2^2 double t174 = -t49 + t51 + t52 + t100 - t111; // t175 = X1 * t174 = (-A+2*G*w1*w2*w3+F*w1*w2*w3+F*w2^2-E*w2^2)*X1 double t175 = X1 * t174; // t177 = t13 * t24 * t26 * w2 * 2.0 = 2*w2*G*(w2^2+w3^2) double t177 = t13 * t24 * t26 * w2 * 2.0; // t178 = t10 * t24 * t25 * w2 = F*w2*(w2^2+w3^2) double t178 = t10 * t24 * t25 * w2; // t192 = t13 * t14 * w1 = B*w1 double t192 = t13 * t14 * w1; // t179 = -t97 + t98 + t99 + t112 - t192 = -E*w2*w3+F*w2^2*w1+2*G*w2^2*w1+F*w2*w3-B*w1 double t179 = -t97 + t98 + t99 + t112 - t192; // t180 = (-E*w2*w3+F*w2^2*w1+2*G*w2^2*w1+F*w2*w3-B*w1)*Y1 double t180 = Y1 * t179; // t181 = A+2*G*w1*w2*w3+F*w1*w2*w3-F*w2^2+E*w2^2 double t181 = t49 + t51 + t52 - t100 + t111; // t182 = (A+2*G*w1*w2*w3+F*w1*w2*w3-F*w2^2+E*w2^2)*Z1 double t182 = Z1 * t181; // t193 = 2*G*w3*(w2^2+w3^2) double t193 = t13 * t26 * t34 * w3 * 2.0; // t194 = F*w3*(w2^2+w3^2) double t194 = t10 * t25 * t34 * w3; // t195 = (2*G+F)*w3*(w2^2+w3^2) double t195 = t193 + t194; // t196 = 2*G*w3*(w1^2+w3^2) double t196 = t13 * t26 * t45 * w3 * 2.0; // t197 = F*w3*(w1^2+w3^2) double t197 = t10 * t25 * t45 * w3; // t200 = 2*B*w3 double t200 = t13 * t14 * w3 * 2.0; // t198 = (2*G+F)*w3*(w1^2+w3^2) - 2*B*w3 double t198 = t196 + t197 - t200; // t199 = F*w3^2 double t199 = t7 * t10 * t25; // t201 = 2*w3*G*(w2^2+w3^2) double t201 = t13 * t24 * t26 * w3 * 2.0; // t202 = F*w3*(w2^2+w3^2) double t202 = t10 * t24 * t25 * w3; // t203 = -t49 + t51 + t52 - t118 + t199 = -A+2*G*w1*w2*w3+F*w1*w2*w3-E*w3^2+F*w3^2 double t203 = -t49 + t51 + t52 - t118 + t199; // t204 = (-t49 + t51 + t52 - t118 + t199 = -A+2*G*w1*w2*w3+F*w1*w2*w3-E*w3^2+F*w3^2)*Y1 double t204 = Y1 * t203; // t205 = 2*t1 double t205 = t1 * 2.0; // t206 = 2*r13*Z1 double t206 = Z1 * t29 * 2.0; // t207 = 2*r11*X1 double t207 = X1 * t32 * 2.0; // t208 = 2*t1 + 2*r13*Z1 + 2*r11*X1 + 2*r12*Y1 double t208 = t205 + t206 + t207 - Y1 * t27 * 2.0; // t209 = 2*t2 double t209 = t2 * 2.0; // t210 = 2*r21*X1 double t210 = X1 * t53 * 2.0; // t211 = 2*r22*Y1 double t211 = Y1 * t58 * 2.0; // t212 = 2*t2 + 2*r21*X1 + 2*r22*Y1 + 2*r23*Z1 double t212 = t209 + t210 + t211 - Z1 * t55 * 2.0; double t213 = t3 * 2.0; double t214 = Y1 * t40 * 2.0; double t215 = Z1 * t43 * 2.0; double t216 = t213 + t214 + t215 - X1 * t38 * 2.0; // jacs(0, 0) = t14 * t65 * (X1 * r1 * w1 * 2.0 + X1 * r2 * w2 + X1 * r3 * w3 + Y1 * r1 * w2 + Z1 * r1 * w3 + r1 * t1 * w1 * 2.0 + r2 * t2 * w1 * 2.0 + r3 * t3 * w1 * 2.0 + Y1 * r3 * t5 * t12 + Y1 * r3 * t9 * t10 - Z1 * r2 * t5 * t12 - Z1 * r2 * t9 * t10 - X1 * r2 * t12 * w2 - X1 * r3 * t12 * w3 - Y1 * r1 * t12 * w2 + Y1 * r2 * t12 * w1 * 2.0 - Z1 * r1 * t12 * w3 + Z1 * r3 * t12 * w1 * 2.0 + Y1 * r3 * t5 * t10 * t11 - Z1 * r2 * t5 * t10 * t11 + X1 * r2 * t12 * w1 * w3 - X1 * r3 * t12 * w1 * w2 - Y1 * r1 * t12 * w1 * w3 + Z1 * r1 * t12 * w1 * w2 - Y1 * r1 * t10 * t11 * w1 * w3 + Z1 * r1 * t10 * t11 * w1 * w2 - X1 * r1 * t6 * t10 * t11 * w1 - X1 * r1 * t7 * t10 * t11 * w1 + X1 * r2 * t5 * t10 * t11 * w2 + X1 * r3 * t5 * t10 * t11 * w3 + Y1 * r1 * t5 * t10 * t11 * w2 - Y1 * r2 * t5 * t10 * t11 * w1 - Y1 * r2 * t7 * t10 * t11 * w1 + Z1 * r1 * t5 * t10 * t11 * w3 - Z1 * r3 * t5 * t10 * t11 * w1 - Z1 * r3 * t6 * t10 * t11 * w1 + X1 * r2 * t10 * t11 * w1 * w3 - X1 * r3 * t10 * t11 * w1 * w2 + Y1 * r3 * t10 * t11 * w1 * w2 * w3 + Z1 * r2 * t10 * t11 * w1 * w2 * w3) - t26 * t65 * t93 * w1 * 2.0 - t14 * t93 * t101 * (t130 + t15 * (-X1 * t121 + Y1 * (t46 + t47 + t48 - t13 * t14 * w2 - t12 * t14 * w1 * w3) + Z1 * (t35 + t36 + t37 - t13 * t14 * w3 - t10 * t25 * w1 * w2)) * 2.0 + t18 * (t135 + t137 - Y1 * (t132 + t133 - t13 * t14 * w1 * 2.0)) * 2.0) * (1.0 / 2.0); jacs(0, 1) = t14 * t65 * (X1 * r2 * w1 + Y1 * r1 * w1 + Y1 * r2 * w2 * 2.0 + Y1 * r3 * w3 + Z1 * r2 * w3 + r1 * t1 * w2 * 2.0 + r2 * t2 * w2 * 2.0 + r3 * t3 * w2 * 2.0 - X1 * r3 * t6 * t12 - X1 * r3 * t9 * t10 + Z1 * r1 * t6 * t12 + Z1 * r1 * t9 * t10 + X1 * r1 * t12 * w2 * 2.0 - X1 * r2 * t12 * w1 - Y1 * r1 * t12 * w1 - Y1 * r3 * t12 * w3 - Z1 * r2 * t12 * w3 + Z1 * r3 * t12 * w2 * 2.0 - X1 * r3 * t6 * t10 * t11 + Z1 * r1 * t6 * t10 * t11 + X1 * r2 * t12 * w2 * w3 - Y1 * r1 * t12 * w2 * w3 + Y1 * r3 * t12 * w1 * w2 - Z1 * r2 * t12 * w1 * w2 - Y1 * r1 * t10 * t11 * w2 * w3 + Y1 * r3 * t10 * t11 * w1 * w2 - Z1 * r2 * t10 * t11 * w1 * w2 - X1 * r1 * t6 * t10 * t11 * w2 + X1 * r2 * t6 * t10 * t11 * w1 - X1 * r1 * t7 * t10 * t11 * w2 + Y1 * r1 * t6 * t10 * t11 * w1 - Y1 * r2 * t5 * t10 * t11 * w2 - Y1 * r2 * t7 * t10 * t11 * w2 + Y1 * r3 * t6 * t10 * t11 * w3 - Z1 * r3 * t5 * t10 * t11 * w2 + Z1 * r2 * t6 * t10 * t11 * w3 - Z1 * r3 * t6 * t10 * t11 * w2 + X1 * r2 * t10 * t11 * w2 * w3 + X1 * r3 * t10 * t11 * w1 * w2 * w3 + Z1 * r1 * t10 * t11 * w1 * w2 * w3) - t26 * t65 * t93 * w2 * 2.0 - t14 * t93 * t101 * (t18 * (Z1 * (-t35 + t94 + t95 + t96 - t13 * t14 * w3) - Y1 * t170 + X1 * (t97 + t98 + t99 - t13 * t14 * w1 - t10 * t25 * w2 * w3)) * 2.0 + t15 * (t180 + t182 - X1 * (t177 + t178 - t13 * t14 * w2 * 2.0)) * 2.0 + t23 * (t175 + Y1 * (t35 - t94 + t95 + t96 - t13 * t14 * w3) - Z1 * t173) * 2.0) * (1.0 / 2.0); jacs(0, 2) = t14 * t65 * (X1 * r3 * w1 + Y1 * r3 * w2 + Z1 * r1 * w1 + Z1 * r2 * w2 + Z1 * r3 * w3 * 2.0 + r1 * t1 * w3 * 2.0 + r2 * t2 * w3 * 2.0 + r3 * t3 * w3 * 2.0 + X1 * r2 * t7 * t12 + X1 * r2 * t9 * t10 - Y1 * r1 * t7 * t12 - Y1 * r1 * t9 * t10 + X1 * r1 * t12 * w3 * 2.0 - X1 * r3 * t12 * w1 + Y1 * r2 * t12 * w3 * 2.0 - Y1 * r3 * t12 * w2 - Z1 * r1 * t12 * w1 - Z1 * r2 * t12 * w2 + X1 * r2 * t7 * t10 * t11 - Y1 * r1 * t7 * t10 * t11 - X1 * r3 * t12 * w2 * w3 + Y1 * r3 * t12 * w1 * w3 + Z1 * r1 * t12 * w2 * w3 - Z1 * r2 * t12 * w1 * w3 + Y1 * r3 * t10 * t11 * w1 * w3 + Z1 * r1 * t10 * t11 * w2 * w3 - Z1 * r2 * t10 * t11 * w1 * w3 - X1 * r1 * t6 * t10 * t11 * w3 - X1 * r1 * t7 * t10 * t11 * w3 + X1 * r3 * t7 * t10 * t11 * w1 - Y1 * r2 * t5 * t10 * t11 * w3 - Y1 * r2 * t7 * t10 * t11 * w3 + Y1 * r3 * t7 * t10 * t11 * w2 + Z1 * r1 * t7 * t10 * t11 * w1 + Z1 * r2 * t7 * t10 * t11 * w2 - Z1 * r3 * t5 * t10 * t11 * w3 - Z1 * r3 * t6 * t10 * t11 * w3 - X1 * r3 * t10 * t11 * w2 * w3 + X1 * r2 * t10 * t11 * w1 * w2 * w3 + Y1 * r1 * t10 * t11 * w1 * w2 * w3) - t26 * t65 * t93 * w3 * 2.0 - t14 * t93 * t101 * (t18 * (Z1 * (t46 - t113 + t114 + t115 - t13 * t14 * w2) - Y1 * t198 + X1 * (t49 + t51 + t52 + t118 - t7 * t10 * t25)) * 2.0 + t23 * (X1 * (-t97 + t112 + t116 + t117 - t13 * t14 * w1) + Y1 * (-t46 + t113 + t114 + t115 - t13 * t14 * w2) - Z1 * t195) * 2.0 + t15 * (t204 + Z1 * (t97 - t112 + t116 + t117 - t13 * t14 * w1) - X1 * (t201 + t202 - t13 * t14 * w3 * 2.0)) * 2.0) * (1.0 / 2.0); // = 1/2 * r1 * t65 // - t14 * t93 // * t101 * t208 // = 1/2 * r1 * 1 / sqrt((t1 + r11*X1 + r12*Y1 + r13*Z1)^2 + (t2 + r21*X1 + r22*Y1 + r23*Z1)^2 + (t3 + r31*X1 + r32*Y1 + r33*Z1)^2) // - 1/theta^2 * ((Y1 * r2 * w2^2) + (Z1 * r3 * w3^2) + (r1 * t1 * w1^2) + (r1 * t1 * w2^2) // + (r1 * t1 * w3^2) + (r2 * t2 * w1^2) + (r2 * t2 * w2^2) + (r2 * t2 * w3^2) + (r3 * t3 * w1^2) // + (r3 * t3 * w2^2) + (r3 * t3 * w3^2) + (X1 * r1 * w1^2) + (X1 * r2 * w1 * w2) + (X1 * r3 * w1 * w3) // + (Y1 * r1 * w1 * w2) + (Y1 * r3 * w2 * w3) + (Z1 * r1 * w1 * w3) + (Z1 * r2 * w2 * w3) + (X1 * r1 * w2^2 * cos(theta)) // + (X1 * r1 * w3^2 * cos(theta)) + (Y1 * r2 * w1^2 * cos(theta)) + (Y1 * r2 * w3^2 * cos(theta)) + (Z1 * r3 * w1^2 * cos(theta)) + (Z1 * r3 * w2^2 * cos(theta)) // + (X1 * r2 * theta * sin(theta) * w3) + (Y1 * r3 * theta * sin(theta) * w1) + (Z1 * r1 * theta * sin(theta) * w2) - (X1 * r3 * theta * sin(theta) * w2) - (Y1 * r1 * theta * sin(theta) * w3) // - (Z1 * r2 * theta * sin(theta) * w1) - (X1 * r2 * cos(theta) * w1 * w2) - (X1 * r3 * cos(theta) * w1 * w3) - (Y1 * r1 * cos(theta) * w1 * w2) - () // - (Y1 * r3 * cos(theta) * w2 * w3) - (Z1 * r1 * cos(theta) * w1 * w3) - (Z1 * r2 * cos(theta) * w2 * w3)) // * (pow(((t1 + r11*X1 + r12*Y1 + r13*Z1)^2 + (t2 + r21*X1 + r22*Y1 + r23*Z1)^2 + (t3 + r31*X1 + r32*Y1 + r33*Z1)^2), 3/2)) // * (2*t1 + 2*r11*X1 + 2*r12*Y1 + 2*r13*Z1) jacs(0, 3) = r1 * t65 - t14 * t93 * t101 * t208 * (1.0 / 2.0); jacs(0, 4) = r2 * t65 - t14 * t93 * t101 * t212 * (1.0 / 2.0); jacs(0, 5) = r3 * t65 - t14 * t93 * t101 * t216 * (1.0 / 2.0); jacs(1, 0) = t14 * t65 * (X1 * s1 * w1 * 2.0 + X1 * s2 * w2 + X1 * s3 * w3 + Y1 * s1 * w2 + Z1 * s1 * w3 + s1 * t1 * w1 * 2.0 + s2 * t2 * w1 * 2.0 + s3 * t3 * w1 * 2.0 + Y1 * s3 * t5 * t12 + Y1 * s3 * t9 * t10 - Z1 * s2 * t5 * t12 - Z1 * s2 * t9 * t10 - X1 * s2 * t12 * w2 - X1 * s3 * t12 * w3 - Y1 * s1 * t12 * w2 + Y1 * s2 * t12 * w1 * 2.0 - Z1 * s1 * t12 * w3 + Z1 * s3 * t12 * w1 * 2.0 + Y1 * s3 * t5 * t10 * t11 - Z1 * s2 * t5 * t10 * t11 + X1 * s2 * t12 * w1 * w3 - X1 * s3 * t12 * w1 * w2 - Y1 * s1 * t12 * w1 * w3 + Z1 * s1 * t12 * w1 * w2 + X1 * s2 * t10 * t11 * w1 * w3 - X1 * s3 * t10 * t11 * w1 * w2 - Y1 * s1 * t10 * t11 * w1 * w3 + Z1 * s1 * t10 * t11 * w1 * w2 - X1 * s1 * t6 * t10 * t11 * w1 - X1 * s1 * t7 * t10 * t11 * w1 + X1 * s2 * t5 * t10 * t11 * w2 + X1 * s3 * t5 * t10 * t11 * w3 + Y1 * s1 * t5 * t10 * t11 * w2 - Y1 * s2 * t5 * t10 * t11 * w1 - Y1 * s2 * t7 * t10 * t11 * w1 + Z1 * s1 * t5 * t10 * t11 * w3 - Z1 * s3 * t5 * t10 * t11 * w1 - Z1 * s3 * t6 * t10 * t11 * w1 + Y1 * s3 * t10 * t11 * w1 * w2 * w3 + Z1 * s2 * t10 * t11 * w1 * w2 * w3) - t14 * t101 * t167 * (t130 + t15 * (Y1 * (t46 + t47 + t48 - t113 - t138) + Z1 * (t35 + t36 + t37 - t94 - t139) - X1 * t121) * 2.0 + t18 * (t135 + t137 - Y1 * (-t131 + t132 + t133)) * 2.0) * (1.0 / 2.0) - t26 * t65 * t167 * w1 * 2.0; jacs(1, 1) = t14 * t65 * (X1 * s2 * w1 + Y1 * s1 * w1 + Y1 * s2 * w2 * 2.0 + Y1 * s3 * w3 + Z1 * s2 * w3 + s1 * t1 * w2 * 2.0 + s2 * t2 * w2 * 2.0 + s3 * t3 * w2 * 2.0 - X1 * s3 * t6 * t12 - X1 * s3 * t9 * t10 + Z1 * s1 * t6 * t12 + Z1 * s1 * t9 * t10 + X1 * s1 * t12 * w2 * 2.0 - X1 * s2 * t12 * w1 - Y1 * s1 * t12 * w1 - Y1 * s3 * t12 * w3 - Z1 * s2 * t12 * w3 + Z1 * s3 * t12 * w2 * 2.0 - X1 * s3 * t6 * t10 * t11 + Z1 * s1 * t6 * t10 * t11 + X1 * s2 * t12 * w2 * w3 - Y1 * s1 * t12 * w2 * w3 + Y1 * s3 * t12 * w1 * w2 - Z1 * s2 * t12 * w1 * w2 + X1 * s2 * t10 * t11 * w2 * w3 - Y1 * s1 * t10 * t11 * w2 * w3 + Y1 * s3 * t10 * t11 * w1 * w2 - Z1 * s2 * t10 * t11 * w1 * w2 - X1 * s1 * t6 * t10 * t11 * w2 + X1 * s2 * t6 * t10 * t11 * w1 - X1 * s1 * t7 * t10 * t11 * w2 + Y1 * s1 * t6 * t10 * t11 * w1 - Y1 * s2 * t5 * t10 * t11 * w2 - Y1 * s2 * t7 * t10 * t11 * w2 + Y1 * s3 * t6 * t10 * t11 * w3 - Z1 * s3 * t5 * t10 * t11 * w2 + Z1 * s2 * t6 * t10 * t11 * w3 - Z1 * s3 * t6 * t10 * t11 * w2 + X1 * s3 * t10 * t11 * w1 * w2 * w3 + Z1 * s1 * t10 * t11 * w1 * w2 * w3) - t26 * t65 * t167 * w2 * 2.0 - t14 * t101 * t167 * (t18 * (X1 * (t97 + t98 + t99 - t112 - t192) + Z1 * (-t35 + t94 + t95 + t96 - t139) - Y1 * t170) * 2.0 + t15 * (t180 + t182 - X1 * (-t176 + t177 + t178)) * 2.0 + t23 * (t175 + Y1 * (t35 - t94 + t95 + t96 - t139) - Z1 * t173) * 2.0) * (1.0 / 2.0); jacs(1, 2) = t14 * t65 * (X1 * s3 * w1 + Y1 * s3 * w2 + Z1 * s1 * w1 + Z1 * s2 * w2 + Z1 * s3 * w3 * 2.0 + s1 * t1 * w3 * 2.0 + s2 * t2 * w3 * 2.0 + s3 * t3 * w3 * 2.0 + X1 * s2 * t7 * t12 + X1 * s2 * t9 * t10 - Y1 * s1 * t7 * t12 - Y1 * s1 * t9 * t10 + X1 * s1 * t12 * w3 * 2.0 - X1 * s3 * t12 * w1 + Y1 * s2 * t12 * w3 * 2.0 - Y1 * s3 * t12 * w2 - Z1 * s1 * t12 * w1 - Z1 * s2 * t12 * w2 + X1 * s2 * t7 * t10 * t11 - Y1 * s1 * t7 * t10 * t11 - X1 * s3 * t12 * w2 * w3 + Y1 * s3 * t12 * w1 * w3 + Z1 * s1 * t12 * w2 * w3 - Z1 * s2 * t12 * w1 * w3 - X1 * s3 * t10 * t11 * w2 * w3 + Y1 * s3 * t10 * t11 * w1 * w3 + Z1 * s1 * t10 * t11 * w2 * w3 - Z1 * s2 * t10 * t11 * w1 * w3 - X1 * s1 * t6 * t10 * t11 * w3 - X1 * s1 * t7 * t10 * t11 * w3 + X1 * s3 * t7 * t10 * t11 * w1 - Y1 * s2 * t5 * t10 * t11 * w3 - Y1 * s2 * t7 * t10 * t11 * w3 + Y1 * s3 * t7 * t10 * t11 * w2 + Z1 * s1 * t7 * t10 * t11 * w1 + Z1 * s2 * t7 * t10 * t11 * w2 - Z1 * s3 * t5 * t10 * t11 * w3 - Z1 * s3 * t6 * t10 * t11 * w3 + X1 * s2 * t10 * t11 * w1 * w2 * w3 + Y1 * s1 * t10 * t11 * w1 * w2 * w3) - t26 * t65 * t167 * w3 * 2.0 - t14 * t101 * t167 * (t18 * (Z1 * (t46 - t113 + t114 + t115 - t138) - Y1 * t198 + X1 * (t49 + t51 + t52 + t118 - t199)) * 2.0 + t23 * (X1 * (-t97 + t112 + t116 + t117 - t192) + Y1 * (-t46 + t113 + t114 + t115 - t138) - Z1 * t195) * 2.0 + t15 * (t204 + Z1 * (t97 - t112 + t116 + t117 - t192) - X1 * (-t200 + t201 + t202)) * 2.0) * (1.0 / 2.0); jacs(1, 3) = s1 * t65 - t14 * t101 * t167 * t208 * (1.0 / 2.0); jacs(1, 4) = s2 * t65 - t14 * t101 * t167 * t212 * (1.0 / 2.0); jacs(1, 5) = s3 * t65 - t14 * t101 * t167 * t216 * (1.0 / 2.0); } } // End namespace ORB_SLAM3