How to Easily Estimate Vehicle Localization Errors from IMU Specs
Land vehicles operate on the ground where multi-path, signal interference, and full-signal loss present frequently occurring challenges to standalone GPS/GNSS navigation in urban environments. At the same time, autonomous vehicles seek to maintain 10 cm precision even in harsh conditions with a combination of IMU-based aiding and GNSS corrections.
Active GPS/GNSS spoofing and jamming threats are now commonplace in geo-politically contested regions such as the Ukraine, and reports of these dangerous signal disturbances in non-contested regions is also on the rise. Such conditions drive the need for longer-term GNSS-denied navigation as discussed in this related article where ANELLO tested its products during active GPS/GNSS spoofing and jamming.
Inertial navigation is a well-proven method that mitigates unreliable and unavailable GNSS using an inertial measurement unit (IMU) sensor to track position change from the last known reliable GNSS position. Inertial navigation is also frequently referred to as “dead-reckoning” or “sensor fusion”.
How good of an IMU is required?
The answer depends on the positional accuracy required and how long the signal disturbances are expected to last. Getting a decent estimate of expected cross-track localization errors is pretty easy and does not require a complex inertial navigation simulator such as described in some of my previous blogs.
A gyroscope rate error results in a cross-track horizontal error, as shown in the figure below. Cross-track error is of critical safety importance for autonomous vehicles — a one meter cross-track error would result in an autonomous vehicle believing it’s in the center of a lane but actually veering into the lane next to it.
Some IMU errors can NOT be compensated for no matter how advanced a sensor-fusion or AI approach is implemented. Two fundamental characteristics of an IMU’s yaw-gyroscope sensor generally set the performance limit:
- Angle Random Walk (ARW): The contribution from the sensor’s white noise
- Bias Instability (BI): The inherent random temporal instability of the sensor’s drift
These parameters are hard sensor physics that limit localization and positioning performance when GPS is unreliable or unavailable.
How to Determine an IMU’s Angle Random Walk and Bias Instability
A commercial IMU has many specifications including measurement range, noise, temperature drift, scale-factor accuracy, bandwidth, power consumption, and many others. While higher-end IMUs also directly specify the ARW and BI parameters, it is still important to understand how these specifications are obtained.
In the Allan Variance method, a long stationary data collection of the sensor (typically several hours) is obtained. The data is then separated into time bins of a given width (x-axis), the bias in each bin is averaged, and the statistical variation in the bias is plotted (y-axis). The Allan Deviation curve shows both the ARW and BI in one plot, and it helps conceptualize why these errors can not be removed or reduced by sensor fusion. An example plot for the ANELLO EVK is shown below.
Some low-end MEMS sensors do not specify the ARW and BI specifications, so they must be measured by the end user to understand the potential effects of their errors. A sample Python code to compute and plot an Allan Deviation curve and extract the ARW and BI can be found here, and requires an evenly spaced time series of data points preferably collected at 100 Hz or faster.
At ANELLO Photonics, we design products to meet aggressive specifications for 10 cm position accuracy needed for a variety of autonomous vehicle applications. As shown in the plot above, the ANELLO IMU+, based on the ANELLO Silicon Photonic Optical Gyro (SiPhOG) has an ARW of 0.05 deg/rt(hr) and BI of 0.5 deg/hr.
Translating IMU Specifications to Position Errors
For a land vehicle that has a good additional reference source for velocity (e.g., a wheel speed sensor or visual odometer), the equations for cross-track horizontal error growth due to both bias instability (BI) and angle random walk (ARW) are shown below and dictated by the yaw gyroscope alongside the time with unreliable or unavailable GPS.
The following plots show how ARW impacts error growth when driving at 30 m/s or about 65 mph for both shorter and longer periods without GPS.
Similarly, the following plots show how BI impacts position error:
Note that these position error models are optimistic as ARW and BI errors combine geometrically, and there are many other error sources such as scale factor error, misalignments, temperature, and vibration. Nonetheless, these graphs show that optical gyroscopes are required for high position accuracies even for short-term disturbances in GPS. The Python code used to generate these plots is available here.
As shown above low ARW and low BI combined together enable solutions to meet the strict performance requirements in the autonomous vehicle industry. In addition, the ANELLO Evaluation kit (EVK) is available today to demonstrate the performance capabilities of the ANELLO IMU+ sensors combined with GNSS and sensor fusion algorithms. Contact us for more information about these and other products.
Conclusions
This blog shows how to estimate positional drift due to gyroscope errors. It also introduces two fundamental performance parameters — angle random walk and bias instability — needed to estimate “dead-reckoning”-based localization and navigation performance. It shows the importance of low angle random walk and low bias instability in order to maintain high position accuracy during GNSS loss. These low-noise and low-drift characteristics are the natural characteristics of optical gyroscopes, like the ANELLO Silicon Photonics Optical Gyroscope.
