初始化时需要求出的变量：相机和imu外参r t、重力g、尺度s、陀螺仪和加速度计偏置ba bg。

下面对两种算法初始化的详细步骤进行对比：

求陀螺仪偏置bg

  • 求解公式相同，求解方法不同。公式如下，VI ORB-SLAM使用图优化的方式。

　　　　

Vector3d Optimizer::OptimizeInitialGyroBias(const vector<cv::Mat>& vTwc, const vector<IMUPreintegrator>& vImuPreInt)

{

    int N = vTwc.size(); if(vTwc.size()!=vImuPreInt.size()) cerr<<"vTwc.size()!=vImuPreInt.size()"<<endl;

    Matrix4d Tbc = ConfigParam::GetEigTbc();

    Matrix3d Rcb = Tbc.topLeftCorner(,).transpose();

    // Setup optimizer

    g2o::SparseOptimizer optimizer;

    g2o::BlockSolverX::LinearSolverType * linearSolver;

    linearSolver = new g2o::LinearSolverEigen<g2o::BlockSolverX::PoseMatrixType>();

    g2o::BlockSolverX * solver_ptr = new g2o::BlockSolverX(linearSolver);

    g2o::OptimizationAlgorithmGaussNewton* solver = new g2o::OptimizationAlgorithmGaussNewton(solver_ptr);

    //g2o::OptimizationAlgorithmLevenberg* solver = new g2o::OptimizationAlgorithmLevenberg(solver_ptr);

    optimizer.setAlgorithm(solver);

    // Add vertex of gyro bias, to optimizer graph

    g2o::VertexGyrBias * vBiasg = new g2o::VertexGyrBias();

    vBiasg->setEstimate(Eigen::Vector3d::Zero());

    vBiasg->setId();

    optimizer.addVertex(vBiasg);

    // Add unary edges for gyro bias vertex

    //for(std::vector<KeyFrame*>::const_iterator lit=vpKFs.begin(), lend=vpKFs.end(); lit!=lend; lit++)

    for(int i=; i<N; i++)

    {

        // Ignore the first KF

        if(i==)

            continue;

        const cv::Mat& Twi = vTwc[i-];    // pose of previous KF

        Matrix3d Rwci = Converter::toMatrix3d(Twi.rowRange(,).colRange(,));

        //Matrix3d Rwci = Twi.rotation_matrix();

        const cv::Mat& Twj = vTwc[i];        // pose of this KF

        Matrix3d Rwcj = Converter::toMatrix3d(Twj.rowRange(,).colRange(,));

        //Matrix3d Rwcj =Twj.rotation_matrix();

        const IMUPreintegrator& imupreint = vImuPreInt[i];

        g2o::EdgeGyrBias * eBiasg = new g2o::EdgeGyrBias();

        eBiasg->setVertex(, dynamic_cast<g2o::OptimizableGraph::Vertex*>(optimizer.vertex()));

        // measurement is not used in EdgeGyrBias

        eBiasg->dRbij = imupreint.getDeltaR();

        eBiasg->J_dR_bg = imupreint.getJRBiasg();

        eBiasg->Rwbi = Rwci*Rcb;

        eBiasg->Rwbj = Rwcj*Rcb;

        //eBiasg->setInformation(Eigen::Matrix3d::Identity());

        eBiasg->setInformation(imupreint.getCovPVPhi().bottomRightCorner(,).inverse());

        optimizer.addEdge(eBiasg);

    }

    // It's actualy a linear estimator, so 1 iteration is enough.

    //optimizer.setVerbose(true);

    optimizer.initializeOptimization();

    optimizer.optimize();

    g2o::VertexGyrBias * vBgEst = static_cast<g2o::VertexGyrBias*>(optimizer.vertex());

    return vBgEst->estimate();

}
  • VINS中公式如下。使用LDLT分解，解方程组。

　　　

void solveGyroscopeBias(map<double, ImageFrame> &all_image_frame, Vector3d* Bgs)

{

    Matrix3d A;

    Vector3d b;

    Vector3d delta_bg;

    A.setZero();

    b.setZero();

    map<double, ImageFrame>::iterator frame_i;

    map<double, ImageFrame>::iterator frame_j;

    for (frame_i = all_image_frame.begin(); next(frame_i) != all_image_frame.end(); frame_i++)

    {

        frame_j = next(frame_i);

        MatrixXd tmp_A(, );

        tmp_A.setZero();

        VectorXd tmp_b();

        tmp_b.setZero();

        Eigen::Quaterniond q_ij(frame_i->second.R.transpose() * frame_j->second.R);

        tmp_A = frame_j->second.pre_integration->jacobian.template block<, >(O_R, O_BG);

        tmp_b =  * (frame_j->second.pre_integration->delta_q.inverse() * q_ij).vec();

        A += tmp_A.transpose() * tmp_A;

        b += tmp_A.transpose() * tmp_b;

    }

    delta_bg = A.ldlt().solve(b);

    ROS_WARN_STREAM("gyroscope bias initial calibration " << delta_bg.transpose());

    for (int i = ; i <= WINDOW_SIZE; i++)

        Bgs[i] += delta_bg;

    for (frame_i = all_image_frame.begin(); next(frame_i) != all_image_frame.end( ); frame_i++)

    {

        frame_j = next(frame_i);

        frame_j->second.pre_integration->repropagate(Vector3d::Zero(), Bgs[]);

    }

}

求重力加速度g、尺度s和外参平移量T

  • VINS中并没有求外参T，而是在紧耦合优化时以初值为0直接进行优化。两算法求取公式略有不同。VI ORB-SLAM公式如下，使用SVD分解求解方程的解：

　　　　

　　　　

// Solve A*x=B for x=[s,gw] 4x1 vector

    cv::Mat A = cv::Mat::zeros(*(N-),,CV_32F);

    cv::Mat B = cv::Mat::zeros(*(N-),,CV_32F);

    cv::Mat I3 = cv::Mat::eye(,,CV_32F);

    // Step 2.

    // Approx Scale and Gravity vector in 'world' frame (first KF's camera frame)

    for(int i=; i<N-; i++)

    {

        //KeyFrameInit* pKF1 = vKFInit[i];//vScaleGravityKF[i];

        KeyFrameInit* pKF2 = vKFInit[i+];

        KeyFrameInit* pKF3 = vKFInit[i+];

        // Delta time between frames

        double dt12 = pKF2->mIMUPreInt.getDeltaTime();

        double dt23 = pKF3->mIMUPreInt.getDeltaTime();

        // Pre-integrated measurements

        cv::Mat dp12 = Converter::toCvMat(pKF2->mIMUPreInt.getDeltaP());

        cv::Mat dv12 = Converter::toCvMat(pKF2->mIMUPreInt.getDeltaV());

        cv::Mat dp23 = Converter::toCvMat(pKF3->mIMUPreInt.getDeltaP());

        // Pose of camera in world frame

        cv::Mat Twc1 = vTwc[i].clone();//pKF1->GetPoseInverse();

        cv::Mat Twc2 = vTwc[i+].clone();//pKF2->GetPoseInverse();

        cv::Mat Twc3 = vTwc[i+].clone();//pKF3->GetPoseInverse();

        // Position of camera center

        cv::Mat pc1 = Twc1.rowRange(,).col();

        cv::Mat pc2 = Twc2.rowRange(,).col();

        cv::Mat pc3 = Twc3.rowRange(,).col();

        // Rotation of camera, Rwc

        cv::Mat Rc1 = Twc1.rowRange(,).colRange(,);

        cv::Mat Rc2 = Twc2.rowRange(,).colRange(,);

        cv::Mat Rc3 = Twc3.rowRange(,).colRange(,);

        // Stack to A/B matrix

        // lambda*s + beta*g = gamma

        cv::Mat lambda = (pc2-pc1)*dt23 + (pc2-pc3)*dt12;

        cv::Mat beta = 0.5*I3*(dt12*dt12*dt23 + dt12*dt23*dt23);

        cv::Mat gamma = (Rc3-Rc2)*pcb*dt12 + (Rc1-Rc2)*pcb*dt23 + Rc1*Rcb*dp12*dt23 - Rc2*Rcb*dp23*dt12 - Rc1*Rcb*dv12*dt12*dt23;

        lambda.copyTo(A.rowRange(*i+,*i+).col());

        beta.copyTo(A.rowRange(*i+,*i+).colRange(,));

        gamma.copyTo(B.rowRange(*i+,*i+));

        // Tested the formulation in paper, -gamma. Then the scale and gravity vector is -xx

        // Debug log

        //cout<<"iter "<<i<<endl;

    }

    // Use svd to compute A*x=B, x=[s,gw] 4x1 vector

    // A = u*w*vt, u*w*vt*x=B

    // Then x = vt'*winv*u'*B

    cv::Mat w,u,vt;

    // Note w is 4x1 vector by SVDecomp()

    // A is changed in SVDecomp() with cv::SVD::MODIFY_A for speed

    cv::SVDecomp(A,w,u,vt,cv::SVD::MODIFY_A);

    // Debug log

    //cout<<"u:"<<endl<<u<<endl;

    //cout<<"vt:"<<endl<<vt<<endl;

    //cout<<"w:"<<endl<<w<<endl;

    // Compute winv

    cv::Mat winv=cv::Mat::eye(,,CV_32F);

    for(int i=;i<;i++)

    {

        if(fabs(w.at<float>(i))<1e-)

        {

            w.at<float>(i) += 1e-;

            // Test log

            cerr<<"w(i) < 1e-10, w="<<endl<<w<<endl;

        }

        winv.at<float>(i,i) = ./w.at<float>(i);

    }

    // Then x = vt'*winv*u'*B

    cv::Mat x = vt.t()*winv*u.t()*B;

    // x=[s,gw] 4x1 vector

    double sstar = x.at<float>();    // scale should be positive

    cv::Mat gwstar = x.rowRange(,);   // gravity should be about ~9.8
  • VINS求出尺度s、重力加速度g和速度v(这里求出滑窗内每帧的速度v意义在哪里？)，并没有求外参的平移，方法为LDLT分解。公式如下：

　　　　

　　　　

bool LinearAlignment(map<double, ImageFrame> &all_image_frame, Vector3d &g, VectorXd &x)

{

    int all_frame_count = all_image_frame.size();

    int n_state = all_frame_count *  +  + ;

    MatrixXd A{n_state, n_state};

    A.setZero();

    VectorXd b{n_state};

    b.setZero();

    map<double, ImageFrame>::iterator frame_i;

    map<double, ImageFrame>::iterator frame_j;

    int i = ;

    for (frame_i = all_image_frame.begin(); next(frame_i) != all_image_frame.end(); frame_i++, i++)

    {

        frame_j = next(frame_i);

        MatrixXd tmp_A(, );

        tmp_A.setZero();

        VectorXd tmp_b();

        tmp_b.setZero();

        double dt = frame_j->second.pre_integration->sum_dt;

        tmp_A.block<, >(, ) = -dt * Matrix3d::Identity();

        tmp_A.block<, >(, ) = frame_i->second.R.transpose() * dt * dt /  * Matrix3d::Identity();

        tmp_A.block<, >(, ) = frame_i->second.R.transpose() * (frame_j->second.T - frame_i->second.T) / 100.0;

        tmp_b.block<, >(, ) = frame_j->second.pre_integration->delta_p + frame_i->second.R.transpose() * frame_j->second.R * TIC[] - TIC[];

        //cout << "delta_p   " << frame_j->second.pre_integration->delta_p.transpose() << endl;

        tmp_A.block<, >(, ) = -Matrix3d::Identity();

        tmp_A.block<, >(, ) = frame_i->second.R.transpose() * frame_j->second.R;

        tmp_A.block<, >(, ) = frame_i->second.R.transpose() * dt * Matrix3d::Identity();

        tmp_b.block<, >(, ) = frame_j->second.pre_integration->delta_v;

        //cout << "delta_v   " << frame_j->second.pre_integration->delta_v.transpose() << endl;

        Matrix<double, , > cov_inv = Matrix<double, , >::Zero();

        //cov.block<6, 6>(0, 0) = IMU_cov[i + 1];

        //MatrixXd cov_inv = cov.inverse();

        cov_inv.setIdentity();

        MatrixXd r_A = tmp_A.transpose() * cov_inv * tmp_A;

        VectorXd r_b = tmp_A.transpose() * cov_inv * tmp_b;

        A.block<, >(i * , i * ) += r_A.topLeftCorner<, >();

        b.segment<>(i * ) += r_b.head<>();

        A.bottomRightCorner<, >() += r_A.bottomRightCorner<, >();

        b.tail<>() += r_b.tail<>();

        A.block<, >(i * , n_state - ) += r_A.topRightCorner<, >();

        A.block<, >(n_state - , i * ) += r_A.bottomLeftCorner<, >();

    }

    A = A * 1000.0;

    b = b * 1000.0;

    x = A.ldlt().solve(b);

    double s = x(n_state - ) / 100.0;

    ROS_DEBUG("estimated scale: %f", s);

    g = x.segment<>(n_state - );

    ROS_DEBUG_STREAM(" result g     " << g.norm() << " " << g.transpose());

    if(fabs(g.norm() - G.norm()) > 1.0 || s < )

    {

        return false;

    }

    RefineGravity(all_image_frame, g, x);

    s = (x.tail<>())() / 100.0;

    (x.tail<>())() = s;

    ROS_DEBUG_STREAM(" refine     " << g.norm() << " " << g.transpose());

    if(s < 0.0 )

        return false;

    else

        return true;

}

　　这里在求出g之后，还对其进行了优化，方法为LDLT分解，公式如下。

　　　　

void RefineGravity(map<double, ImageFrame> &all_image_frame, Vector3d &g, VectorXd &x)

{

    Vector3d g0 = g.normalized() * G.norm();

    Vector3d lx, ly;

    //VectorXd x;

    int all_frame_count = all_image_frame.size();

    int n_state = all_frame_count *  +  + ;

    MatrixXd A{n_state, n_state};

    A.setZero();

    VectorXd b{n_state};

    b.setZero();

    map<double, ImageFrame>::iterator frame_i;

    map<double, ImageFrame>::iterator frame_j;

    for(int k = ; k < ; k++)

    {

        MatrixXd lxly(, );

        lxly = TangentBasis(g0);

        int i = ;

        for (frame_i = all_image_frame.begin(); next(frame_i) != all_image_frame.end(); frame_i++, i++)

        {

            frame_j = next(frame_i);

            MatrixXd tmp_A(, );

            tmp_A.setZero();

            VectorXd tmp_b();

            tmp_b.setZero();

            double dt = frame_j->second.pre_integration->sum_dt;

            tmp_A.block<, >(, ) = -dt * Matrix3d::Identity();

            tmp_A.block<, >(, ) = frame_i->second.R.transpose() * dt * dt /  * Matrix3d::Identity() * lxly;

            tmp_A.block<, >(, ) = frame_i->second.R.transpose() * (frame_j->second.T - frame_i->second.T) / 100.0;

            tmp_b.block<, >(, ) = frame_j->second.pre_integration->delta_p + frame_i->second.R.transpose() * frame_j->second.R * TIC[] - TIC[] - frame_i->second.R.transpose() * dt * dt /  * g0;

            tmp_A.block<, >(, ) = -Matrix3d::Identity();

            tmp_A.block<, >(, ) = frame_i->second.R.transpose() * frame_j->second.R;

            tmp_A.block<, >(, ) = frame_i->second.R.transpose() * dt * Matrix3d::Identity() * lxly;

            tmp_b.block<, >(, ) = frame_j->second.pre_integration->delta_v - frame_i->second.R.transpose() * dt * Matrix3d::Identity() * g0;

            Matrix<double, , > cov_inv = Matrix<double, , >::Zero();

            //cov.block<6, 6>(0, 0) = IMU_cov[i + 1];

            //MatrixXd cov_inv = cov.inverse();

            cov_inv.setIdentity();

            MatrixXd r_A = tmp_A.transpose() * cov_inv * tmp_A;

            VectorXd r_b = tmp_A.transpose() * cov_inv * tmp_b;

            A.block<, >(i * , i * ) += r_A.topLeftCorner<, >();

            b.segment<>(i * ) += r_b.head<>();

            A.bottomRightCorner<, >() += r_A.bottomRightCorner<, >();

            b.tail<>() += r_b.tail<>();

            A.block<, >(i * , n_state - ) += r_A.topRightCorner<, >();

            A.block<, >(n_state - , i * ) += r_A.bottomLeftCorner<, >();

        }

            A = A * 1000.0;

            b = b * 1000.0;

            x = A.ldlt().solve(b);

            VectorXd dg = x.segment<>(n_state - );

            g0 = (g0 + lxly * dg).normalized() * G.norm();

            //double s = x(n_state - 1);

    }

    g = g0;

}

求加速度计偏置

  • VINS中并没有计算该值，和外参T一样，是在后面直接进行优化。VI ORB-SLAM中则单独对该值进行了求取，求取方式同样为SVD公式如下：

　　　　

　　　　

　　　 

// Step 3.

    // Use gravity magnitude 9.8 as constraint

    // gI = [0;0;1], the normalized gravity vector in an inertial frame, NED type with no orientation.

    cv::Mat gI = cv::Mat::zeros(,,CV_32F);

    gI.at<float>() = ;

    // Normalized approx. gravity vecotr in world frame

    cv::Mat gwn = gwstar/cv::norm(gwstar);

    // Debug log

    //cout<<"gw normalized: "<<gwn<<endl;

    // vhat = (gI x gw) / |gI x gw|

    cv::Mat gIxgwn = gI.cross(gwn);

    double normgIxgwn = cv::norm(gIxgwn);

    cv::Mat vhat = gIxgwn/normgIxgwn;

    double theta = std::atan2(normgIxgwn,gI.dot(gwn));

    // Debug log

    //cout<<"vhat: "<<vhat<<", theta: "<<theta*180.0/M_PI<<endl;

    Eigen::Vector3d vhateig = Converter::toVector3d(vhat);

    Eigen::Matrix3d RWIeig = Sophus::SO3::exp(vhateig*theta).matrix();

    cv::Mat Rwi = Converter::toCvMat(RWIeig);

    cv::Mat GI = gI*ConfigParam::GetG();//9.8012;

    // Solve C*x=D for x=[s,dthetaxy,ba] (1+2+3)x1 vector

    cv::Mat C = cv::Mat::zeros(*(N-),,CV_32F);

    cv::Mat D = cv::Mat::zeros(*(N-),,CV_32F);

    for(int i=; i<N-; i++)

    {

        //KeyFrameInit* pKF1 = vKFInit[i];//vScaleGravityKF[i];

        KeyFrameInit* pKF2 = vKFInit[i+];

        KeyFrameInit* pKF3 = vKFInit[i+];

        // Delta time between frames

        double dt12 = pKF2->mIMUPreInt.getDeltaTime();

        double dt23 = pKF3->mIMUPreInt.getDeltaTime();

        // Pre-integrated measurements

        cv::Mat dp12 = Converter::toCvMat(pKF2->mIMUPreInt.getDeltaP());

        cv::Mat dv12 = Converter::toCvMat(pKF2->mIMUPreInt.getDeltaV());

        cv::Mat dp23 = Converter::toCvMat(pKF3->mIMUPreInt.getDeltaP());

        cv::Mat Jpba12 = Converter::toCvMat(pKF2->mIMUPreInt.getJPBiasa());

        cv::Mat Jvba12 = Converter::toCvMat(pKF2->mIMUPreInt.getJVBiasa());

        cv::Mat Jpba23 = Converter::toCvMat(pKF3->mIMUPreInt.getJPBiasa());

        // Pose of camera in world frame

        cv::Mat Twc1 = vTwc[i].clone();//pKF1->GetPoseInverse();

        cv::Mat Twc2 = vTwc[i+].clone();//pKF2->GetPoseInverse();

        cv::Mat Twc3 = vTwc[i+].clone();//pKF3->GetPoseInverse();

        // Position of camera center

        cv::Mat pc1 = Twc1.rowRange(,).col();

        cv::Mat pc2 = Twc2.rowRange(,).col();

        cv::Mat pc3 = Twc3.rowRange(,).col();

        // Rotation of camera, Rwc

        cv::Mat Rc1 = Twc1.rowRange(,).colRange(,);

        cv::Mat Rc2 = Twc2.rowRange(,).colRange(,);

        cv::Mat Rc3 = Twc3.rowRange(,).colRange(,);

        // Stack to C/D matrix

        // lambda*s + phi*dthetaxy + zeta*ba = psi

        cv::Mat lambda = (pc2-pc1)*dt23 + (pc2-pc3)*dt12;

        cv::Mat phi = - 0.5*(dt12*dt12*dt23 + dt12*dt23*dt23)*Rwi*SkewSymmetricMatrix(GI);  // note: this has a '-', different to paper

        cv::Mat zeta = Rc2*Rcb*Jpba23*dt12 + Rc1*Rcb*Jvba12*dt12*dt23 - Rc1*Rcb*Jpba12*dt23;

        cv::Mat psi = (Rc1-Rc2)*pcb*dt23 + Rc1*Rcb*dp12*dt23 - (Rc2-Rc3)*pcb*dt12

                     - Rc2*Rcb*dp23*dt12 - Rc1*Rcb*dv12*dt23*dt12 - 0.5*Rwi*GI*(dt12*dt12*dt23 + dt12*dt23*dt23); // note:  - paper

        lambda.copyTo(C.rowRange(*i+,*i+).col());

        phi.colRange(,).copyTo(C.rowRange(*i+,*i+).colRange(,)); //only the first 2 columns, third term in dtheta is zero, here compute dthetaxy 2x1.

        zeta.copyTo(C.rowRange(*i+,*i+).colRange(,));

        psi.copyTo(D.rowRange(*i+,*i+));

        // Debug log

        //cout<<"iter "<<i<<endl;

    }

    // Use svd to compute C*x=D, x=[s,dthetaxy,ba] 6x1 vector

    // C = u*w*vt, u*w*vt*x=D

    // Then x = vt'*winv*u'*D

    cv::Mat w2,u2,vt2;

    // Note w2 is 6x1 vector by SVDecomp()

    // C is changed in SVDecomp() with cv::SVD::MODIFY_A for speed

    cv::SVDecomp(C,w2,u2,vt2,cv::SVD::MODIFY_A);

    // Debug log

    //cout<<"u2:"<<endl<<u2<<endl;

    //cout<<"vt2:"<<endl<<vt2<<endl;

    //cout<<"w2:"<<endl<<w2<<endl;

    // Compute winv

    cv::Mat w2inv=cv::Mat::eye(,,CV_32F);

    for(int i=;i<;i++)

    {

        if(fabs(w2.at<float>(i))<1e-)

        {

            w2.at<float>(i) += 1e-;

            // Test log

            cerr<<"w2(i) < 1e-10, w="<<endl<<w2<<endl;

        }

        w2inv.at<float>(i,i) = ./w2.at<float>(i);

    }

    // Then y = vt'*winv*u'*D

    cv::Mat y = vt2.t()*w2inv*u2.t()*D;

    double s_ = y.at<float>();

    cv::Mat dthetaxy = y.rowRange(,);

    cv::Mat dbiasa_ = y.rowRange(,);

    Vector3d dbiasa_eig = Converter::toVector3d(dbiasa_);

    // dtheta = [dx;dy;0]

    cv::Mat dtheta = cv::Mat::zeros(,,CV_32F);

    dthetaxy.copyTo(dtheta.rowRange(,));

    Eigen::Vector3d dthetaeig = Converter::toVector3d(dtheta);

    // Rwi_ = Rwi*exp(dtheta)

    Eigen::Matrix3d Rwieig_ = RWIeig*Sophus::SO3::exp(dthetaeig).matrix();

    cv::Mat Rwi_ = Converter::toCvMat(Rwieig_);

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