IMU Noise Model

December 12, 2022

The IMU Noise Model

continuous-time
- $E[n(t)] \equiv 0$
- $E\left[n\left(t_1\right) n\left(t_2\right)\right]=\sigma_g^2 \delta\left(t_1-t_2\right)$
  - $\sigma_g$ (noise strength)
    
    ⇒ 클수록 gyro measurements의 노이즈가 심함을 의미
    
    ⇒ $\sigma_g$ : gyroscope_noise_density in Kalibr
    
    ⇒ $\sigma_a$ : accelerometer_noise_density in Kalibr
discrete-time
- $n_d[k]=\sigma_{g d} w[k]$
  - $w[k] \sim \mathcal{N}(0,1)$
  - $\sigma_{g d}=\sigma_g \frac{1}{\sqrt{\Delta t}}$
  - $\Delta t$ : sampling time

Brownian motion process, Wiener process, Random walk 등으로 불림
continuous-time
- $\dot{b}_g(t)=\sigma_{b g} w(t)$
  - $w$ : white noise
    
    ⇒ 이 값은 bias의 strength와 결합되어 bias를 결정하게 됨
  - $\sigma_{bg}$ (gyro bias strength), $\sigma_{ba}$ (accel bias strength)
    
    ⇒ bias의 variations이 클수록 위 파라미터를 높게 설정해야 함
    
    ⇒ $\sigma_{bg}$ : gyroscope_random_walk in Kalibr
    
    ⇒ $\sigma_{ba}$ : accelerometer_random_walk in Kalibr
discrete-time
- $b_d[k]=b_d[k-1]+\sigma_{b g d} w[k]$
  - $w[k] \sim \mathcal{N}(0,1)$
  - $\sigma_{b g d}=\sigma_{b g} \sqrt{\Delta t}$

White Noise Model

$\begin{aligned}& E[\mathbf{n}(t)] \equiv \mathbf{0}_{3 \times 1} \\& E\left[\mathbf{n}\left(t_1\right) \mathbf{n}^T\left(t_2\right)\right]=\sigma_g^2 \mathbf{I}_{3 \times 3} \delta\left(t_1-t_2\right)\end{aligned}$

Parameter	Kalibr element	Symbol	Units
Gyroscope “white noise”	`gyroscope_noise_density`	$\sigma_{g}$	$\frac{\mathrm{rad}}{s} \frac{1}{\sqrt{H z}}$
Accelerometer “white noise”	`accelerometer_noise_density`	$\sigma_{a}$	$\frac{\mathrm{m}}{s^2} \frac{1}{\sqrt{H z}}$
Gyroscope “random walk”	`gyroscope_random_walk`	$\sigma_{bg}$	$\frac{\mathrm{rad}}{s^2} \frac{1}{\sqrt{H z}}$
Accelerometer “random walk”	`accelerometer_random_walk`	$\sigma_{ba}$	$\frac{\mathrm{m}}{s^3} \frac{1}{\sqrt{H z}}$
IMU sampling rate	`update_rate`	$\frac{1}{\Delta t}$	$H z$