从零手写VIO|第二节——imu.cpp代码解析

1.用欧拉角表示body坐标系到惯性系的旋转

• 下方公式表示的是用欧拉角表示从惯性系到body系的转换：
  
  

// euler2Rotation:   body frame to interitail frame
Eigen::Matrix3d euler2Rotation( Eigen::Vector3d  eulerAngles)
{
    double roll = eulerAngles(0);
    double pitch = eulerAngles(1);
    double yaw = eulerAngles(2);

    double cr = cos(roll); double sr = sin(roll);
    double cp = cos(pitch); double sp = sin(pitch);
    double cy = cos(yaw); double sy = sin(yaw);

    Eigen::Matrix3d RIb;
    RIb<< cy*cp ,   cy*sp*sr - sy*cr,   sy*sr + cy* cr*sp,
            sy*cp,    cy *cr + sy*sr*sp,  sp*sy*cr - cy*sr,
            -sp,         cp*sr,           cp*cr;
    return RIb;
}

2. 惯性系下的欧拉角速度转换到body坐标系

Eigen::Matrix3d eulerRates2bodyRates(Eigen::Vector3d eulerAngles)
{
    double roll = eulerAngles(0);
    double pitch = eulerAngles(1);

    double cr = cos(roll); double sr = sin(roll);
    double cp = cos(pitch); double sp = sin(pitch);

    Eigen::Matrix3d R;
    R<<  1,   0,    -sp,
            0,   cr,   sr*cp,
            0,   -sr,  cr*cp;

    return R;
}

3.addIMUnoise

高斯白噪声的离散时间的方差（IMU传感器获取数据为离散采样）（与连续时间相比较）：

Bias随机游走（其导数服从高斯分布）：

加速度的误差模型：

陀螺仪的误差模型

void IMU::addIMUnoise(MotionData& data)
{
    std::random_device rd;
    std::default_random_engine generator_(rd());
    std::normal_distribution<double> noise(0.0, 1.0);

    Eigen::Vector3d noise_gyro(noise(generator_),noise(generator_),noise(generator_));
    Eigen::Matrix3d gyro_sqrt_cov = param_.gyro_noise_sigma * Eigen::Matrix3d::Identity();
    data.imu_gyro = data.imu_gyro + gyro_sqrt_cov * noise_gyro / sqrt( param_.imu_timestep ) + gyro_bias_;

    Eigen::Vector3d noise_acc(noise(generator_),noise(generator_),noise(generator_));
    Eigen::Matrix3d acc_sqrt_cov = param_.acc_noise_sigma * Eigen::Matrix3d::Identity();
    data.imu_acc = data.imu_acc + acc_sqrt_cov * noise_acc / sqrt( param_.imu_timestep ) + acc_bias_;

    // gyro_bias update
    Eigen::Vector3d noise_gyro_bias(noise(generator_),noise(generator_),noise(generator_));
    gyro_bias_ += param_.gyro_bias_sigma * sqrt(param_.imu_timestep ) * noise_gyro_bias;
    data.imu_gyro_bias = gyro_bias_;

    // acc_bias update
    Eigen::Vector3d noise_acc_bias(noise(generator_),noise(generator_),noise(generator_));
    acc_bias_ += param_.acc_bias_sigma * sqrt(param_.imu_timestep ) * noise_acc_bias;
    data.imu_acc_bias = acc_bias_;

}

4. IMU::MotionModel

MotionData IMU::MotionModel(double t)
{

    MotionData data;
    // param
    float ellipse_x = 15;
    float ellipse_y = 20;
    float z = 1;           // z轴做sin运动
    float K1 = 10;          // z轴的正弦频率是x，y的k1倍
    float K = M_PI/ 10;    // 20 * K = 2pi 　　由于我们采取的是时间是20s, 系数K控制yaw正好旋转一圈，运动一周

    // translation
    // twb:  body frame in world frame
    Eigen::Vector3d position( ellipse_x * cos( K * t) + 5, ellipse_y * sin( K * t) + 5,  z * sin( K1 * K * t ) + 5);
    Eigen::Vector3d dp(- K * ellipse_x * sin(K*t),  K * ellipse_y * cos(K*t), z*K1*K * cos(K1 * K * t));              // position导数　in world frame
    double K2 = K*K;
    Eigen::Vector3d ddp( -K2 * ellipse_x * cos(K*t),  -K2 * ellipse_y * sin(K*t), -z*K1*K1*K2 * sin(K1 * K * t));     // position二阶导数

    // Rotation
    double k_roll = 0.1;
    double k_pitch = 0.2;
    Eigen::Vector3d eulerAngles(k_roll * cos(t) , k_pitch * sin(t) , K*t );   // roll ~ [-0.2, 0.2], pitch ~ [-0.3, 0.3], yaw ~ [0,2pi]
    Eigen::Vector3d eulerAnglesRates(-k_roll * sin(t) , k_pitch * cos(t) , K);      // euler angles 的导数

//    Eigen::Vector3d eulerAngles(0.0,0.0, K*t );   // roll ~ 0, pitch ~ 0, yaw ~ [0,2pi]
//    Eigen::Vector3d eulerAnglesRates(0.,0. , K);      // euler angles 的导数

    Eigen::Matrix3d Rwb = euler2Rotation(eulerAngles);         // body frame to world frame
    Eigen::Vector3d imu_gyro = eulerRates2bodyRates(eulerAngles) * eulerAnglesRates;   //  euler rates trans to body gyro p44：惯性系下欧拉角速度转到body坐标系下

    Eigen::Vector3d gn (0,0,-9.81);                                   //  gravity in navigation frame(ENU)   ENU (0,0,-9.81)  NED(0,0,9,81)
    Eigen::Vector3d imu_acc = Rwb.transpose() * ( ddp -  gn );  //  Rbw * Rwn * gn = gs

    data.imu_gyro = imu_gyro;
    data.imu_acc = imu_acc;
    data.Rwb = Rwb;
    data.twb = position;
    data.imu_velocity = dp;
    data.timestamp = t;
    return data;

}

5. 运动模型的离散积分

• 欧拉法
  

/// imu 动力学模型 欧拉积分
        Eigen::Vector3d acc_w = Qwb * (imupose.imu_acc) + gw;  // aw = Rwb * ( acc_body - acc_bias ) + gw
        Qwb = Qwb * dq;
        Vw = Vw + acc_w * dt;
        Pwb = Pwb + Vw * dt + 0.5 * dt * dt * acc_w;

• 中值法
  

        MotionData imupose = imudata[i];
        MotionData imupose_ = imudata[i-1];

        //delta_q = [1 , 1/2 * thetax , 1/2 * theta_y, 1/2 * theta_z]
        Eigen::Quaterniond dq;
        Eigen::Vector3d dtheta_half = 1.0/2.0* (imupose.imu_gyro +  imupose_.imu_gyro) * dt /2.0;
        // Eigen::Vector3d dtheta_half = (imupose.imu_gyro +  imudata[i-1].imu_gyro)/2 * dt /2.0;
        dq.w() = 1;
        dq.x() = dtheta_half.x();
        dq.y() = dtheta_half.y();
        dq.z() = dtheta_half.z();

        /// 中值积分
 	    Eigen::Vector3d acc_w =  (Qwb * (imupose_.imu_acc) + gw + Qwb*dq * (imupose.imu_acc) + gw )/2;  // aw = Rwb * ( acc_body - acc_bias ) + gw
        Qwb = Qwb * dq;
        Vw = Vw + acc_w * dt;
        Pwb = Pwb + Vw * dt + 0.5 * dt * dt * acc_w;