一、安装必要的软件

    1、 pip3  install   symforce

    2、 pip3  install   notebook==6.5.6

二、运行Jupyter NoteBook

   在Linux终端下执行:

    jupyter-notebook

    执行该命令后，会自动调起浏览器:

三、新建Python3(IPython)

四、输入Python代码并执行

import symforce
symforce.set_symbolic_api("symengine")
from symforce.notebook_util import display
import symforce.symbolic as sf


x = sf.Symbol("x")
y = sf.Symbol("y")
expr = x**2 + sf.sin(y) / x**2
display(expr)

Github:

https://github.com/symforce-org/symforce

市场上的传感器将有包括以下全部或部分规格的数据表。如果数据表上有这些规格，了解它们的含义很重要，因为是根据这些规格选择IMU。

Full Scale

它代表测量范围是传感器已校准（例如+/- 2g加速度计或+/-500°/s陀螺仪），它通常可以走得比最大规定值远一点，但测量的准确性不能保证。例如，下一个图表显示陀螺仪饱和在500°/s，这意味着任何更快的旋转将不会被传感器测量。


不建议选择小于预期运动的测量范围

Scale Factor Stability

比例因子误差是与所施加运动成比例的测量误差。先进的传感器热校准可以最大限度地减少这种误差。SBG系统的标度误差包括热稳定性误差和长期稳定性误差。


下图显示了一个理想的传感器（蓝色）将报告运动的实际输入，而一个现实的传感器（红色）将报告输出上的轻微偏移，与输入上的实际运动相比。

Non-linearity

非线性类似于比例因子：传感器的输出与它应该测量的实际输入不对应。然而，它的输出不是根据满量程（红色）产生比例误差，而是以一种更复杂的方式（绿色）变化。


非线性误差可以通过先进的校准程序最小化。

One Year Bias Stability

一年偏差规格是传感器数据表中非常重要的信息。它反映了传感器在其使用寿命期间会漂移多少。为了获得这样的规格，SBG系统运行了一个传感器加速老化过程，包括热循环、功率循环、冲击和振动应力。


这个参数反映了传感器偏差的哪一部分不能校准，以及导航滤波器需要补偿多少。在每次启动时，INS都会经过一个预热过程来估计传感器偏差。这个值越低，INS就越容易估计和补偿偏置，从而缩短预热时间。

Velocity/Angular Random Walk

加速度计的速度随机游走或陀螺仪的角度随机游走表示传感器的噪声。而零偏对应于长期噪声，随机游走对应于短期噪声。对这种噪声进行积分，将导致输出的随机变化。


速度随机游走的常用单位(VRW)：m/s/√hr或m/s/√s或g/√Hz


角度随机游走的常用单位(ARW): °/√hr或°/√s


这对于需要精确计算横滚和俯仰的角度应用尤其重要。

In Run Bias Instability

由于温度和机械应力等各种影响，IMU的偏置会随着时间的推移而变化，这就是为什么INS需要不断地通过外部辅助（如GNSS、里程表、DVL等）结合车辆动力学来重新估计偏置。


运行中偏置不稳定性是所有MEMS imu的一个神奇数字，总是以较低的值为目标。然而，这个值并不能提供有关实际传感器性能的很多见解。特别是因为:


该值通常在恒温下测量

偏置不稳定性从未与相关的时隙（tau）指定。如果在非常低的时隙（例如tau < 200s）获得偏置不稳定性，则该信息毫无意义，因为偏置将迅速发散。

Allan variance

The Allan Variance is a complex diagram, showing for various time scales how good is the sensor.

Allan variance is typically measured at ambient and control temperature so user should still be careful when comparing Allan variances.

When the high frequency noise is "Gaussian", which is typically the case, one can measure the Angular/Velocity random walk at a time scale of 1second.

The Bias Instability is the bottom of the curve.

What is more important is how the sensor behave after reaching its bias instability. While the best MEMS sensors will tend to stabilize, lower grade sensors or uncalibrated sensors will quickly ramp up, showing a sensor bias drift.

Vibration Rectification Error

It is generally provided on highly qualified systems, while most IMUs just do not specify what happens in vibrations.

The Vibration Rectification Error gives the expected bias for a given level of vibrations on accelerometers or gyroscopes.

For instance if the VRE is 1 °/h/g² this means we can expect a maximum of 1°/h of bias for 1g of random vibrations.

Sampling Rate

The Sampling Rate is the frequency at which the measurements are taken on the sensor's input. The sampling rate divided by 2 is the Nyquist frequency, which gives the maximum observable frequency of a signal we are measuring.

This parameter matters if we wish to observe higher frequency motions, for instance to achieve vibration monitoring or fast motion compensation.

Bandwidth

The Bandwidth is maximum frequency you can take a reliable acceleration or angular velocity reading. This will directly represent which frequency of motion you will be able to measure, as any signal with a frequency higher than the bandwidth will be attenuated.

This point should be considered for any application that involves vibration or fast motion analysis.

Resolution

The Resolution is the smallest increment measurable on a sensor.

This parameter is of less importance, as usually much lower than the noise of the sensor. The sensor angular or velocity random walk will be the actual decisive factor as this parameter gives the actual system noise level during 1 second averages.

Orthogonality

The Orthogonality represents the physical misalignment between the sensor of different axis.

Cross-Axis Sensitivity

This is the sensitivity of a sensor to a vibrations applied to a perpendicular axle. Ideally it should be minimal as each axle should be independent from each other.

Gyro-G

Gyroscopes can have a bias when under acceleration, this error is called gyro-g effect. This is typically estimated and compensated during calibrations.

github:

https://github.com/p-ranav/alpaca

https://base.movella.com/s/article/Understanding-the-relationship-between-the-noise-characteristics?language=en_US

Answer

In our documentation, we use the (FFT) noise density parameter for the gyroscope and the accelerometer signals. The manufacturers might use other noise characteristics in their datasheets: angle/velocity random walk parameters, power spectral densities (PSD), or standard deviation at a certain bandwidth (sampling rate). All the mentioned parameters contain the same information and here we present the relationship between these quantities (for reference see IEEE Std. 952-1997 C.1.1).

Relationship between the noise characteristics

Gyroscope Noise Density:

Gyroscope noise density characterizes the random fluctuations in the gyroscope's output per square root of frequency bandwidth. (Unit: angular rate per square root of hertz, typically degrees per second per square root hertz or radians per hour per square root hertz.)

Noise density  is usually determined using the log-log Allan deviation plot reading the value of the  slope at the cluster time

Angle Random Walk (ARW):

ARW quantifies the rate at which the angle uncertainty grows over time due to sensor noise. It is a crucial metric in determining the long-term stability and accuracy of gyroscopes. (Unit: angle per square root of time, typically degrees per square root hour or radians per square root hour.)

Angle random walk parameter:  relates to the noise density as follows:

Power Spectral Density (PSD):

Power Spectral Density describes how the noise present in the sensor’s output signal varies with angular frequency.  

Power spectral density:  relates to the noise density as follows:

Noise Standard Deviation:

Noise Standard Deviation represents the average magnitude of deviations of individual noise samples from the mean noise value at a certain bandwidth.

Noise standard deviation at bandwidth:  relates to the noise density as follows:

With other unit conversions it is important to remember the following relations:

The relationships between the accelerometer noise characteristics are similar: accelerometer noise density , velocity random walk parameter , power spectral density  , and noise standard deviation at a certain bandwidth .   

Examples:

• Converting Gyroscope Noise Density to Angle Random Walk:

Gyroscope Noise Density (ND) is typically given in units of (°/s/√Hz) or (°/h/√Hz), and we want to convert it to ARW (°/√s or °/√h).

Given:

To convert to ARW, we need to remember that Hertz (Hz) is the inverse of seconds (s).

       

              ………………. i.e. Angle Random Walk

Now, to convert it to °/√h we need to remember that 1 h = 3600 s. Hence, 1 s = 1/3600 h.

Thus,

  

This conversion shows the direct relationship between gyroscope noise density and Angle Random Walk, highlighting their interchangeability in sensor specifications.

• Converting Accelerometer Noise Density to Velocity Random Walk:

Accelerometer Noise Density (ND) is typically given in units of (m/s2/√Hz), and we want to convert it to VRW (m/s/√s).

Given:

To convert to VRW, as discussed above we need to remember that Hertz (Hz) is the inverse of seconds (s).

        

         

       ………………. i.e. Velocity Random Walk

 

Now, to convert it to m/s/√h we need to remember that 1 h = 3600 s. Hence, 1 s = 1/3600 h.

Thus,

 

This conversion shows the direct relationship between accelerometer noise density and Velocity Random Walk, highlighting their interchangeability in sensor specifications.

Note: Units used for accelerometers can switch between the SI m/s² and the more commonly used milli-g (mg) or micro-g (µg) just like you can see in our documentation. So, it is important to understand the following conversion:

 

 

1mg ≈ 9.81*0.001 ≈ 0.01 m/s², 1µg ≈ 9.81*0.001 ≈ 0.01 m/s²

 

Thus, a noise density of 1 mg/√Hz is approximately equal to a VRW of 0.01 m/s/√s