# Chapter II IMU sensor

Course Code:

Reference blog:

Homework 2 of handwriting VIO from scratch of dark blue College

Using imu_utils tool generates the Allan variance calibration curve of IMU

Error analysis of inertial navigation sensor

## 1. Generate simulation data set and calibrate Allan variance

### 1.1. Noise analysis of inertial devices

#### Multisensor fusion location notes (ren Qian):

Composition of signal and noise;

1) Quantization noise

The inherent noise of all quantization operations is the inevitable noise of digital sensors;

Cause: in the process of collecting continuous time signals into discrete signals through AD acquisition, the accuracy will be lost. The size of accuracy loss is related to the step size of AD conversion. The smaller the step size is, the smaller the quantization noise is.

2) Angular random walk

Wide band angular rate white noise: the gyro output angular rate contains noise, which is the white noise component;

Cause: the essence of attitude calculation is to integrate the diagonal velocity, which must also integrate the noise. The integration of white noise is not white noise, but a Markov process, that is, the error of the current time is obtained by accumulating a random white noise on the basis of the error of the previous time.

The error of Markov property contained in the angle error is called angle random walk.

3) Angular rate random walk

Similar to angular random walk, the Markov error contained in angular rate error is called angular rate random walk. The error of Markov property is the result of white noise accumulation of broadband angular acceleration rate.

4) Zero bias instability noise

Zero bias: bias, as it is often said, is generally not a fixed parameter, but a slow random drift within a certain range.

Zero bias instability: the zero bias changes slowly with time, and its change value cannot be estimated. It is necessary to assume a probability interval to describe how likely it is to fall within this interval. The longer the time, the larger the interval.

5) Rate ramp

This error is a trend error, not a random error.

Random error means that you can't use deterministic model to simulate and eliminate it. At most, you can only use probabilistic model to describe it. In this way, the prediction result is also approximate

Rate nature.

Trend error can be eliminated by direct fitting. The most common reason for this error in gyro is the change of zero position caused by temperature, which can be corrected by temperature compensation

Eliminate.

6) Zero bias repeatability

There will be an error when the start is not equal to zero. In actual use, it needs to be re estimated every time the power is on.

Allan analysis of variance does not include the analysis of zero bias repeatability.

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#### VIO notes (he Yijia)

Error classification:

The errors of accelerometer and gyroscope can be divided into deterministic error and random error.

Deterministic errors can be calibrated and confirmed in advance, including bias, scale

Random error usually assumes that the noise obeys Gaussian distribution, including Gaussian white noise and bias random walk

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### 1.2 generate imu data set of ROS

mkdir vio_sim_ws/src #Will Vio_ data_ simulation-ros_ Put version into src catkin_make rosrun vio_data_simulation vio_data_simulation_node # Generate IMU Bag dataset

rqt_bag info imu.bag # View the contents of the current packet path: imu.bag version: 2.0 duration: 3hr 59:59s (14399s) start: Feb 24 2022 13:04:10.04 (1645679050.04) end: Feb 24 2022 17:04:10.04 (1645693450.04) size: 1.0 GB messages: 2880001 compression: none [1344/1344 chunks] types: sensor_msgs/Imu [6a62c6daae103f4ff57a132d6f95cec2] topics: imu 2880001 msgs : sensor_msgs/Imu

### 1.3 imu_utils completes allan calibration

### 1.3 imu_utils completes allan calibration

Compile and install imu_utils toolkit, main reference website:

Using imu_utils tool generates the Allan variance calibration curve of IMU

Note: code_utils and imu_utils has sequence and cannot be compiled together

FILE : imu_utils_ws/src/imu_utils/launch new sim_imu.launch

<launch> <node pkg="imu_utils" type="imu_an" name="imu_an" output="screen"> <param name="imu_topic" type="string" value= "/imu"/> <!--imu ROS topic of conversation /imu --> <param name="imu_name" type="string" value= "sim_imu"/> <param name="data_save_path" type="string" value= "$(find imu_utils)/data/"/> <!--Save the data path after calibration of the table--> <param name="max_time_min" type="int" value= "230"/> <!--Maximum time to read data--> <param name="max_cluster" type="int" value= "100"/> </node> </launch>

roslaunch imu_utils sim_imu.launch rosbag play imu.bag -r 500 # Play at 500X

Calibration result: File: IMU_ utils_ ws/src/imu_ utils/data

Note: Sim_ The preset noise unit of IMU is gyro: rad / (s * sqrt (Hz)) ACC: M / (s ^ 2 * sqrt (Hz)), while imu_utils preset noise unit is gyro: rad/s acc: m/s^2. When unifying units, IMU needs to be_ The results of utils are divided by sqrt (Hz), sim_ The sampling frequency of IMU is 200 Hz, so IMU_ The results of utils need to be divided by sqrt(200) = 14.14

Simulation imu data, preset imu_noise unit

Simulation imu data, preset imu_noise unit

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imu_utils calibrated units

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Error type | Truth rad/(s * sqrt(hz)) m/(s^2*sqrt(hz)) | imu_utils rad/s m/s^2 | imu_utils unit aligned rad/(s * sqrt(hz)) m/(s^2*sqrt(hz)) |
---|---|---|---|

gyro white noise | 0.015 | 0.21165531 | 0.014968 |

gyro bias random walk | 0.00005 | 0.00095079 | 0.0000672 |

acc white noise | 0.019 | 0.23441296 | 0.016578 |

acc bias random walk | 0.0005 | 0.00349127 | 0.00024691 |

Using matlab to describe the allan variance curve of gyro acc

FILE: imu_utils_ws/src/imu_utils/scripts/draw_allan.m

gyro_allan curve

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acc_allan curve

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## 2. Generate motion imu data and use median method and Euler method for inertial integration

Use herb's vio here_ data_ The simulation code generates the trajectory

### 2.1 compilation and operation

cd vio_data_simulation mkdir build cd buid cmake .. make -j # compile cd .. cd bin ./data_gen #After running, the imu track data set will be generated in the current directory cd .. cd python_tool python draw_trajcory.py # Draw trajectory curve

### 2.2 Euler method for inertial calculation of imu

FILE : vio_data_simulation/src/imu.cpp

Hebo's original code is used to calculate the inertia of Euler method. The formula is as follows

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The corresponding code is as follows: testImu_euler()

//Read the generated imu data, calculate the data with the imu dynamic model, and finally save the trajectory after imu integration, //Used to verify the effectiveness of data and model. void IMU::testImu_euler(std::string src, std::string dist) { std::vector<MotionData>imudata; LoadPose(src,imudata); std::ofstream save_points; save_points.open(dist); double dt = param_.imu_timestep; Eigen::Vector3d Pwb = init_twb_; // position : from imu measurements Eigen::Quaterniond Qwb(init_Rwb_); // quaterniond: from imu measurements Eigen::Vector3d acc_w_last ; Eigen::Vector3d Vw = init_velocity_; // velocity : from imu measurements Eigen::Vector3d gw(0,0,-9.81); // ENU frame Eigen::Vector3d temp_a; Eigen::Vector3d theta; for (int i = 1; i < imudata.size(); ++i) { MotionData imupose = imudata[i]; //delta_q = [1 , 1/2 * thetax , 1/2 * theta_y, 1/2 * theta_z] Eigen::Quaterniond dq; Eigen::Vector3d dtheta_half = imupose.imu_gyro * dt /2.0; dq.w() = 1; dq.x() = dtheta_half.x(); dq.y() = dtheta_half.y(); dq.z() = dtheta_half.z(); dq.normalize(); ///Euler integral of imu dynamic model Eigen::Vector3d acc_w = Qwb * (imupose.imu_acc) + gw; // aw = Rwb * ( acc_body - acc_bias ) + gw Qwb = Qwb * dq; Pwb = Pwb + Vw * dt + 0.5 * dt * dt * acc_w; Vw = Vw + acc_w * dt; //It is stored in the format of imu position, imu quaternion, cam position and cam quaternion. Since there is no cam, imu is saved twice save_points<<imupose.timestamp<<" " <<Qwb.w()<<" " <<Qwb.x()<<" " <<Qwb.y()<<" " <<Qwb.z()<<" " <<Pwb(0)<<" " <<Pwb(1)<<" " <<Pwb(2)<<" " <<Qwb.w()<<" " <<Qwb.x()<<" " <<Qwb.y()<<" " <<Qwb.z()<<" " <<Pwb(0)<<" " <<Pwb(1)<<" " <<Pwb(2)<<" " <<std::endl; } std::cout<<"test end"<<std::endl; }

### 2.3 inertia calculation of imu by median method

FILE : vio_data_simulation/src/imu.cpp

Compared with Euler method, the median method processes the mean value when obtaining the accel and gyro information of imu, which is closer to the real value

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The core modification code is to obtain the accel and gyro parts

Eigen::Vector3d gyro_middle = (imupose_last.imu_gyro + imupose.imu_gyro) / 2.0; // gyro after median integration Eigen::Vector3d dtheta_half = gyro_middle * dt /2.0; Eigen::Vector3d acc_w = (Qwb_last * (imupose_last.imu_acc) + gw + Qwb * (imupose.imu_acc) + gw) / 2.0 ;

The complete code is as follows: testImu_middle()

//Read the generated imu data, calculate the data with the imu dynamic model, and finally save the trajectory after imu integration, //Used to verify the effectiveness of data and model. Median integral method void IMU::testImu_middle(std::string src, std::string dist) { std::vector<MotionData>imudata; LoadPose(src,imudata); std::ofstream save_points; save_points.open(dist); double dt = param_.imu_timestep; Eigen::Vector3d Pwb = init_twb_; // position : from imu measurements Eigen::Quaterniond Qwb(init_Rwb_); // quaterniond: from imu measurements Eigen::Quaterniond Qwb_last(init_Rwb_); // Record the posture of the last moment Eigen::Vector3d acc_w_last ; Eigen::Vector3d Vw = init_velocity_; // velocity : from imu measurements Eigen::Vector3d gw(0,0,-9.81); // ENU frame Eigen::Vector3d temp_a; Eigen::Vector3d theta; for (int i = 1; i < imudata.size(); ++i) { MotionData imupose = imudata[i]; MotionData imupose_last = imudata[i-1]; ///Median integral Eigen::Quaterniond dq; Eigen::Vector3d gyro_middle = (imupose_last.imu_gyro + imupose.imu_gyro) / 2.0; // After the median value of gyro integral Eigen::Vector3d dtheta_half = gyro_middle * dt /2.0; dq.w() = 1; dq.x() = dtheta_half.x(); dq.y() = dtheta_half.y(); dq.z() = dtheta_half.z(); dq.normalize(); Eigen::Vector3d acc_w = (Qwb_last * (imupose_last.imu_acc) + gw + Qwb * (imupose.imu_acc) + gw) / 2.0 ; Qwb_last = Qwb; Qwb = Qwb * dq ; Pwb = Pwb + Vw * dt + 0.5 * dt * dt * acc_w; Vw = Vw + acc_w * dt; //It is stored in the format of imu position, imu quaternion, cam position and cam quaternion. Since there is no cam, imu is saved twice save_points<<imupose.timestamp<<" " <<Qwb.w()<<" " <<Qwb.x()<<" " <<Qwb.y()<<" " <<Qwb.z()<<" " <<Pwb(0)<<" " <<Pwb(1)<<" " <<Pwb(2)<<" " <<Qwb.w()<<" " <<Qwb.x()<<" " <<Qwb.y()<<" " <<Qwb.z()<<" " <<Pwb(0)<<" " <<Pwb(1)<<" " <<Pwb(2)<<" " <<std::endl; } std::cout<<"test end"<<std::endl; }

### 2.4 comparison between Euler method and median method

By comparing the trajectory curves in the figure below, it can be seen that the trajectory obtained by imu inertia integration using the middle median method is closer to the real value.

blue: GroundTruth orange: imu _int

Euler Euler inertial integral | Middle median method inertia integral |
---|---|

[the external chain picture transfer fails, and the source station may have an anti-theft chain mechanism. It is recommended to save the picture and upload it directly (IMG pptrjz34-16463743668) (PIC / euler_trajectory. PNG)] | [the external chain image transfer fails. The source station may have an anti-theft chain mechanism. It is recommended to save the image and upload it directly (img-zjkuits9-16463743669) (PIC / middle_trajectory. PNG)] |

edited by kaho 2022.2.24