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Bayes Filter BF

Prediction step: $ \overline{bel}(x_t) = \int \underbrace{ p(x_t | u_t, x_{t-1} )}{\text{motion model}} bel(x{t-1})dx_{t-1} $

Correction step: $ bel(x_t) = \eta \underbrace{ p(z_t | x_t)}_{\text{observation model}} \overline{bel}(x_t) $

Kalman Filter

• Everything is Gaussian: $ p(x) = det ( 2 \pi \Sigma )^{-1/2} exp(-\frac{1}{2}(x-\mu)^T\Sigma^{-1}(x-\mu)) $
• Linear transition & observation model: $ f(x)=Ax+b $
• Optimal solution for linear models + G distributions (maybe least square)

$ x_t = A_tx_{t-1}+B_tu_t+\epsilon_t $

$ z_t=C_tx_t+\delta_t $

$A_t$: (n$\times$n). How state evolve without controls or noises.
$B_t$: (n$\times$l). How control $u_t$ changes the state. Map from control command to state space.
$C_t$: (k$\times$n). How to map from state to observation space.
Gaussian mean = 0.
$\epsilon_t$, $\delta_t$: Random variables of independent and normally distributed process and measurement noise with covariance $R_t$, $Q_t$.

Linear Obs model:

$ p(z_t|x_t) = det ( 2 \pi Q_t )^{-1/2} exp(-\frac{1}{2}(z_t-C_tx_t)^TQ_t^{-1}(z_t-C_tx_t)) $

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Algorithm KF

Kalman_filter($\mu_{t-1}$, $\Sigma_{t-1}$, $u_t$, $z_t$):
\# prediction step
$\overline{\mu}t=A_t\mu{t-1}+B_tu_t$
$\overline{\Sigma}t=A_t\Sigma{t-1}A_t^T+R_t$
\# correction step
$K_t = \overline{\Sigma}_tC_t^T(C_t\overline{\Sigma}_tC_t^T+Q_t)^{-1}$ # Kalman gain
$\mu_t=\overline{\mu}_t+K_t(z_t-C_t\overline{\mu}_t)$
$\Sigma_t=(I-K_tC_t)\overline{\Sigma}_t$
return $\mu_t, \Sigma_t$

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Extended KF

The linear model is hard to satisfy.

Linearisation: First Order Taylor expansion

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Algorithm EKF

A<-G, C<-H

Extended_Kalman_filter($\mu_{t-1}$, $\Sigma_{t-1}$, $u_t$, $z_t$):
# prediction step
$\overline{\mu}t=g(u_t, \mu{t-1})$ # g is non-linear motion model
$\overline{\Sigma}t=G_t\Sigma{t-1}G_t^T+R_t$
# correction step
$K_t = \overline{\Sigma}_tH_t^T(H_t\overline{\Sigma}_tH_t^T+Q_t)^{-1}$ # Kalman gain
$\mu_t=\overline{\mu}_t+K_t(z_t-h(\overline{\mu}_t))$ # h is non-linear obs model
$\Sigma_t=(I-K_tH_t)\overline{\Sigma}_t$
return $\mu_t, \Sigma_t$

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