IMU和GPS融合定位（ESKF）

说明

1.本文理论部分参考文章https://zhuanlan.zhihu.com/p/152662055和https://blog.csdn.net/brightming/article/details/118057262
ROS下的实践参考https://blog.csdn.net/qinqinxiansheng/article/details/107108475和https://zhuanlan.zhihu.com/p/163038275

理论

坐标系

error-state

ESKF GPS更新

初始化
初始位置为0,方向的roll和pitch可由加速度的重力方向确定,bias设置为0.

代码

代码请见github：https://github.com/ydsf16/imu_gps_localization

总体流程如下：接收IMU数据，进行积分，更新系统方程，接收GPS数据，计算K，更新状态量

代码整体框架说明
代码具体结构框架如下图所示：


主要函数介绍
1）状态定义：

struct State {
    
    
    double timestamp;
    
    Eigen::Vector3d lla;       // WGS84 position.
    Eigen::Vector3d G_p_I;     // The original point of the IMU frame in the Global frame.
    Eigen::Vector3d G_v_I;     // The velocity original point of the IMU frame in the Global frame.
    Eigen::Matrix3d G_R_I;     // The rotation from the IMU frame to the Global frame.
    Eigen::Vector3d acc_bias;  // The bias of the acceleration sensor.
    Eigen::Vector3d gyro_bias; // The bias of the gyroscope sensor.

    // Covariance.
    Eigen::Matrix<double, 15, 15> cov;

    // The imu data.
    ImuDataPtr imu_data_ptr; 
};

包含：

Eigen::Vector3d G_p_I;     // The original point of the IMU frame in the Global frame.
    Eigen::Vector3d G_v_I;     // The velocity original point of the IMU frame in the Global frame.
    Eigen::Matrix3d G_R_I;     // The rotation from the IMU frame to the Global frame.
    Eigen::Vector3d acc_bias;  // The bias of the acceleration sensor.
    Eigen::Vector3d gyro_bias; // The bias of the gyroscope sensor.

5个状态量，在协方差表示时，旋转也用三维的旋转角表示，所以，其协方差矩阵为15。

2） LocalizationWrapper构造函数
LocalizationWrapper构造函数主要实现加载参数、添加数据保存位置、订阅话题和发布话题

LocalizationWrapper::LocalizationWrapper(ros::NodeHandle& nh) {
    
    
    // Load configs.
    double acc_noise, gyro_noise, acc_bias_noise, gyro_bias_noise;
    nh.param("acc_noise",       acc_noise, 1e-2);
    nh.param("gyro_noise",      gyro_noise, 1e-4);
    nh.param("acc_bias_noise",  acc_bias_noise, 1e-6);
    nh.param("gyro_bias_noise", gyro_bias_noise, 1e-8);

    double x, y, z;
    nh.param("I_p_Gps_x", x, 0.);
    nh.param("I_p_Gps_y", y, 0.);
    nh.param("I_p_Gps_z", z, 0.);
    const Eigen::Vector3d I_p_Gps(x, y, z);

    std::string log_folder
        = "/home/sunshine/catkin_imu_gps_localization/src/imu_gps_localization";
    ros::param::get("log_folder", log_folder);

    // Log.
    file_state_.open(log_folder + "/state.csv");
    file_gps_.open(log_folder +"/gps.csv");

    // Initialization imu gps localizer.
    imu_gps_localizer_ptr_ = 
        std::make_unique<ImuGpsLocalization::ImuGpsLocalizer>(acc_noise, gyro_noise,
                                                              acc_bias_noise, gyro_bias_noise,
                                                              I_p_Gps);

    // Subscribe topics.
    imu_sub_ = nh.subscribe("/imu/data", 10,  &LocalizationWrapper::ImuCallback, this);
    //TODO 运行数据集需要修改的地方： gps的话题为/fix
    gps_position_sub_ = nh.subscribe("/fix", 10,  &LocalizationWrapper::GpsPositionCallback, this);
    
    //发布融合后的轨迹
    state_pub_ = nh.advertise<nav_msgs::Path>("fused_path", 10);
}

3）滤波算法进行预测

void ImuProcessor::Predict(const ImuDataPtr last_imu, const ImuDataPtr cur_imu, State* state) {
    
    
    // Time.
    const double delta_t = cur_imu->timestamp - last_imu->timestamp;
    const double delta_t2 = delta_t * delta_t;

    // Set last state.
    State last_state = *state;
    // mid_point integration methods
    // Acc and gyro.
    const Eigen::Vector3d acc_unbias = 0.5 * (last_imu->acc + cur_imu->acc) - last_state.acc_bias;
    const Eigen::Vector3d gyro_unbias = 0.5 * (last_imu->gyro + cur_imu->gyro) - last_state.gyro_bias;

    // Normal state. 
    // Using P58. of "Quaternion kinematics for the error-state Kalman Filter".
    state->G_p_I = last_state.G_p_I + last_state.G_v_I * delta_t + 
                   0.5 * (last_state.G_R_I * acc_unbias + gravity_) * delta_t2;
    state->G_v_I = last_state.G_v_I + (last_state.G_R_I * acc_unbias + gravity_) * delta_t;
    const Eigen::Vector3d delta_angle_axis = gyro_unbias * delta_t;
    if (delta_angle_axis.norm() > 1e-12) {
    
    
        // std::cout << "norm" << delta_angle_axis.norm() << "normlized"
        //           << delta_angle_axis.normalized() << std::endl;
        state->G_R_I = last_state.G_R_I
            * Eigen::AngleAxisd(delta_angle_axis.norm(),
                                delta_angle_axis.normalized())
                  .toRotationMatrix();
    }
    // Error-state. Not needed.

    // Covariance of the error-state.   
    Eigen::Matrix<double, 15, 15> Fx = Eigen::Matrix<double, 15, 15>::Identity();
    Fx.block<3, 3>(0, 3)   = Eigen::Matrix3d::Identity() * delta_t;
    Fx.block<3, 3>(3, 6)   = - state->G_R_I * GetSkewMatrix(acc_unbias) * delta_t;
    Fx.block<3, 3>(3, 9)   = - state->G_R_I * delta_t;
    if (delta_angle_axis.norm() > 1e-12) {
    
    
        Fx.block<3, 3>(6, 6) = Eigen::AngleAxisd(delta_angle_axis.norm(), delta_angle_axis.normalized()).toRotationMatrix().transpose();
    } else {
    
    
        Fx.block<3, 3>(6, 6).setIdentity();
    }
    Fx.block<3, 3>(6, 12)  = - Eigen::Matrix3d::Identity() * delta_t;

    Eigen::Matrix<double, 15, 12> Fi = Eigen::Matrix<double, 15, 12>::Zero();
    Fi.block<12, 12>(3, 0) = Eigen::Matrix<double, 12, 12>::Identity();

    Eigen::Matrix<double, 12, 12> Qi = Eigen::Matrix<double, 12, 12>::Zero();
    Qi.block<3, 3>(0, 0) = delta_t2 * acc_noise_ * Eigen::Matrix3d::Identity();
    Qi.block<3, 3>(3, 3) = delta_t2 * gyro_noise_ * Eigen::Matrix3d::Identity();
    Qi.block<3, 3>(6, 6) = delta_t * acc_bias_noise_ * Eigen::Matrix3d::Identity();
    Qi.block<3, 3>(9, 9) = delta_t * gyro_bias_noise_ * Eigen::Matrix3d::Identity();
    //协方差预测
    state->cov = Fx * last_state.cov * Fx.transpose() + Fi * Qi * Fi.transpose();

    // Time and imu.
    state->timestamp = cur_imu->timestamp;
    state->imu_data_ptr = cur_imu;
}

predict主要是对nominal state的运动学估计，以及对协方差的递推（为了在观测值来的时候，结合两者的协方差算出K值）。

协方差传递的Fx的计算，与《Quaternion Kinematics for the error-state KF》中的一致：
（误差的传递与nominal state的传递都是一样的，遵循的都是相同的运动学模型，只是误差会在之前的数值的基础上继续包括新增的各种运动误差，如imu的bias，随机游走等，而nominal则不管这些值，按照正常的运动学模型递推。）

accelerometer_noise_density: 0.012576 #continous 
accelerometer_random_walk: 0.000232  
gyroscope_noise_density: 0.0012615 #continous 
gyroscope_random_walk: 0.0000075  

代码中是这样设置的：

    Eigen::Matrix<double, 12, 12> Qi = Eigen::Matrix<double, 12, 12>::Zero();
    Qi.block<3, 3>(0, 0) = delta_t2 * acc_noise_ * Eigen::Matrix3d::Identity();
    Qi.block<3, 3>(3, 3) = delta_t2 * gyro_noise_ * Eigen::Matrix3d::Identity();
    Qi.block<3, 3>(6, 6) = delta_t * acc_bias_noise_ * Eigen::Matrix3d::Identity();
    Qi.block<3, 3>(9, 9) = delta_t * gyro_bias_noise_ * Eigen::Matrix3d::Identity();

这些参数，代码中是在初始化时设置的：

LocalizationWrapper::LocalizationWrapper(ros::NodeHandle& nh) {
    
    
    // Load configs.
    double acc_noise, gyro_noise, acc_bias_noise, gyro_bias_noise;
    nh.param("acc_noise",       acc_noise, 1e-2);
    nh.param("gyro_noise",      gyro_noise, 1e-4);
    nh.param("acc_bias_noise",  acc_bias_noise, 1e-6);
    nh.param("gyro_bias_noise", gyro_bias_noise, 1e-8);

<launch>
    <param name="acc_noise"       type="double" value="1e-2" />
    <param name="gyro_noise"      type="double" value="1e-4" />
    <param name="acc_bias_noise"  type="double" value="1e-6" />
    <param name="gyro_bias_noise" type="double" value="1e-8" />

    <param name="I_p_Gps_x"       type="double" value="0.0" />
    <param name="I_p_Gps_y"       type="double" value="0.0" />
    <param name="I_p_Gps_z"       type="double" value="0.0" />

    <param name="log_folder"      type="string" value="$(find imu_gps_localization)" />

    <node name="nmea_topic_driver" pkg="nmea_navsat_driver" type="nmea_topic_driver" output="screen" />
    <node name="imu_gps_localization_node" pkg="imu_gps_localization" type="imu_gps_localization_node" output="screen" />

    <node pkg="rviz" type="rviz" name="rviz" output="screen" 
      args="-d $(find imu_gps_localization)/ros_wrapper/rviz/default.rviz" required="true">
    </node>

</launch>

代码中：

state->cov = Fx * last_state.cov * Fx.transpose() + Fi * Qi * Fi.transpose();

也就是以下代码实现：

void GpsProcessor::ComputeJacobianAndResidual(const Eigen::Vector3d& init_lla,  
                                              const GpsPositionDataPtr gps_data, 
                                              const State& state,
                                              Eigen::Matrix<double, 3, 15>* jacobian,
                                              Eigen::Vector3d* residual) {
    
    
    const Eigen::Vector3d& G_p_I   = state.G_p_I;
    const Eigen::Matrix3d& G_R_I   = state.G_R_I;

    // Convert wgs84 to ENU frame.
    Eigen::Vector3d G_p_Gps;//测量值
    ConvertLLAToENU(init_lla, gps_data->lla, &G_p_Gps);

    // Compute residual.
    //I_p_Gps_在imu坐标系下的位移？是固定值？
    //G_p_I + G_R_I * I_p_Gps_预测的状态计算出来的坐标点
    *residual = G_p_Gps - (G_p_I + G_R_I * I_p_Gps_);

    // Compute jacobian.
    jacobian->setZero();
    jacobian->block<3, 3>(0, 0)  = Eigen::Matrix3d::Identity();
    jacobian->block<3, 3>(0, 6)  = - G_R_I * GetSkewMatrix(I_p_Gps_);
}

第二项，其实就是真值对误差项目的求导，看下表：

   // Compute jacobian.
    jacobian->setZero();
    jacobian->block<3, 3>(0, 0)  = Eigen::Matrix3d::Identity();
    jacobian->block<3, 3>(0, 6)  = - G_R_I * GetSkewMatrix(I_p_Gps_);

通过GPS位置测量数据更新系统的状态

bool GpsProcessor::UpdateStateByGpsPosition(const Eigen::Vector3d& init_lla, const GpsPositionDataPtr gps_data_ptr, State* state) {
    
    
    Eigen::Matrix<double, 3, 15> H;
    Eigen::Vector3d residual;
    ComputeJacobianAndResidual(init_lla, gps_data_ptr, *state, &H, &residual);
    const Eigen::Matrix3d& V = gps_data_ptr->cov;

    // EKF.
    const Eigen::MatrixXd& P = state->cov;
    //P：状态的协方差矩阵；V：gps数据对误差量的协方差矩阵；H：观测数据（gps数据）对误差状态的雅可比矩阵
    const Eigen::MatrixXd K = P * H.transpose() * (H * P * H.transpose() + V).inverse();
    const Eigen::VectorXd delta_x = K * residual;

    // Add delta_x to state.
    AddDeltaToState(delta_x, state);

    // Covarance.
    const Eigen::MatrixXd I_KH = Eigen::Matrix<double, 15, 15>::Identity() - K * H;
    state->cov = I_KH * P * I_KH.transpose() + K * V * K.transpose();
}
//计算雅克比矩阵以及残差项 P61 Quaternion kinematics for the error-state Kalman filter
void GpsProcessor::ComputeJacobianAndResidual(const Eigen::Vector3d& init_lla,  
                                              const GpsPositionDataPtr gps_data, 
                                              const State& state,
                                              Eigen::Matrix<double, 3, 15>* jacobian,
                                              Eigen::Vector3d* residual) {
    
    
    const Eigen::Vector3d& G_p_I   = state.G_p_I;
    const Eigen::Matrix3d& G_R_I   = state.G_R_I;

    // Convert wgs84 to ENU frame.
    Eigen::Vector3d G_p_Gps;
    ConvertLLAToENU(init_lla, gps_data->lla, &G_p_Gps);

    // Compute residual.
    *residual = G_p_Gps - (G_p_I + G_R_I * I_p_Gps_);

    // Compute jacobian.
    jacobian->setZero();
    jacobian->block<3, 3>(0, 0)  = Eigen::Matrix3d::Identity();
    jacobian->block<3, 3>(0, 6)  = - G_R_I * GetSkewMatrix(I_p_Gps_);
}
//添加误差项到状态
void AddDeltaToState(const Eigen::Matrix<double, 15, 1>& delta_x, State* state) {
    
    
    state->G_p_I     += delta_x.block<3, 1>(0, 0);
    state->G_v_I     += delta_x.block<3, 1>(3, 0);
    state->acc_bias  += delta_x.block<3, 1>(9, 0);
    state->gyro_bias += delta_x.block<3, 1>(12, 0);

    if (delta_x.block<3, 1>(6, 0).norm() > 1e-12) {
    
    
        state->G_R_I *= Eigen::AngleAxisd(delta_x.block<3, 1>(6, 0).norm(), delta_x.block<3, 1>(6, 0).normalized()).toRotationMatrix();
    }
}