[SINS] Initial Alignment for High-Accuracy Strapdown INS

I recently learned some material about high-accuracy strapdown inertial navigation and summarize it here. I am still new to this area, so corrections are welcome.

High-accuracy gyroscopes can sense Earth’s rotation rate ($\sim15^\circ/\mathrm h$), while accelerometers measure gravity. Under stationary conditions, a high-accuracy IMU can therefore determine its own attitude without external aiding and perform initial alignment.

Coarse alignment on a stationary base

In the local navigation frame (ENU), Earth gravity is

$$\mathbf g^n= \begin{bmatrix}0\\0\\-g\end{bmatrix}.$$

Earth’s rotation rate is

$$\boldsymbol\omega^n_{ie}= \begin{bmatrix} 0\\ \omega_{ie}\cos L\\ \omega_{ie}\sin L \end{bmatrix}.$$

Here $\omega_{ie}$ is Earth’s rotation rate, approximately $15.041067^\circ/\mathrm h$, and $L$ is geographical latitude.

The inertial-navigation angular-rate measurement relation and the specific-force equation are

$$\mathbf C^n_b\boldsymbol\omega^b_{ib} =\boldsymbol\omega^n_{ib} =\boldsymbol\omega^n_{ie}+\boldsymbol\omega^n_{en}+\boldsymbol\omega^n_{nb},$$ $$\dot{\mathbf v}^n= \mathbf C^n_b\mathbf f^b_{sf} -2(\boldsymbol\omega^n_{ie}+\boldsymbol\omega^n_{en})\times\mathbf v^n +\mathbf g^n.$$

On a stationary base, acceleration and velocity are approximately zero and $\boldsymbol\omega^n_{ie}+\boldsymbol\omega^n_{en}$ is very small. With these approximations,

$$\tilde{\mathbf C}^n_b\tilde{\boldsymbol\omega}^b_{ib} =\boldsymbol\omega^n_{ie}, \qquad \tilde{\mathbf C}^n_b\tilde{\mathbf f}^b_{sf} =-\mathbf g^n.$$

Solving the two equations gives the attitude matrix:

$$\hat{\mathbf C}^n_b= \begin{bmatrix} -(\tilde{\mathbf f}^b_{sf}\times\tilde{\boldsymbol\omega}^b_{ib})^\mathsf T/ \lVert\tilde{\mathbf f}^b_{sf}\times\tilde{\boldsymbol\omega}^b_{ib}\rVert\\ (\tilde{\mathbf f}^b_{sf}\times\tilde{\boldsymbol\omega}^b_{ib}\times\tilde{\mathbf f}^b_{sf})^\mathsf T/ \lVert\tilde{\mathbf f}^b_{sf}\times\tilde{\boldsymbol\omega}^b_{ib}\times\tilde{\mathbf f}^b_{sf}\rVert\\ (\tilde{\mathbf f}^b_{sf})^\mathsf T/ \lVert\tilde{\mathbf f}^b_{sf}\rVert \end{bmatrix}.$$

The attitude matrix therefore has no explicit geographic-position term.

Experiment

Professor Gongmin Yan has open-sourced a high-accuracy integrated-navigation algorithm and provides high-accuracy inertial-navigation data. We use one of those data sets for the experiment.

The SPANISA vehicle test uses a NovAtel 100C fiber-optic IMU. It includes stationary data at both the beginning and end, and also provides an IE post-processing result that can serve as ground truth for evaluating static-alignment accuracy.

Starting position

Method	Yaw (degree)	Pitch (degree)	Roll (degree)
IE post-processing ground truth	116.4122	1.1547	-0.6703
Stationary-base coarse alignment	116.9759	1.2903	-0.5747
Error	0.5637	0.1355	0.0955

Ending position

Method	Yaw (degree)	Pitch (degree)	Roll (degree)
IE post-processing ground truth	-66.1869	1.2109	-0.0910
Stationary-base coarse alignment	-67.5176	1.3439	-0.0046
Error	-1.3307	0.1330	0.0864

The static-alignment accuracy for roll and pitch is very high; yaw is less accurate.

For factors that affect stationary-base alignment errors, see reference [1]. Its conclusion is reproduced below.

Source code

ydsf16/TinyGrapeKit — SINS