[Learning Summary] IMU pre-integration derivation

This article is only used to record your own learning summary. Record the IMU pre-integration derivation process, without including specific principles.

symbolic representation

$R$ : represents the rotation matrix
$v$ : represents speed
$p$ : represents the displacement
$Exp$ : Exponential mapping, mapping the rotation vector to the rotation matrix
$\tilde{w}$: angular velocity observation value
$\tilde{f}$: Observed acceleration values
$b^g,b^a$ : Noise of angular velocity and acceleration
${the}^{gd},{the}^{ad}$ : Random walk of angular velocity and acceleration
angular index$i,j,k$
: corner indexwwat certain times (IMU coordinate system)$w$ : World coordinate system
$\Delta t,\Delta t_{ij}$: discrete time interval, the total time interval between ij

IMU direct integration

The direct integral of IMU is, according to iiAngle, velocity, and displacement at time$i$ , derive jjAngle, velocity, and displacement at time$j\; .$The formula is as follows:

The specific derivation process is as follows. The noise term is omitted in the derivation

(1) Derivation of rotation

(2) Derivation of velocity

(3) Integral of displacement :

All formulas for pre-integration:

The derivation process of pre-integration

(1) Pre-integration of rotation terms :

(2) Pre-integration of velocity term :

(3) Pre-integration of displacement term :

summary

I finally made up my mind to push it once, and finally pushed it out by myself. Although I don’t quite understand the specific meaning and reasons for the operation, I will take my time later.