(prerequisite-Kalman Filter) Localization, Tracking

• Localization

 

Localization = Sensor + Control Input + (Noise of Sensor + Noise of Control Input)

 

It recognizes the kinematic information of the car, which uses the sensor and the control input assuming that noise can be generated in the sensor values and control inputs.

 

Control Input + (Noise of Control Input) : You can see how much the accelerator pedal was pressed, how many degrees the wheel was moved, and so on.

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• Formula of Localization

 

$$p(x_t|x_{0:t-<!--\; -\; -->1},z_{1:t-<!--\; -\; -->1},u_{1:t})=p(x_t|x_{t-<!--\; -\; -->1},u_t)$$

 

$x_t$ : The LOCATION in time 't'.
$z_t$ : The SENSOR in time 't'.
$u_t$ : The CONTROL INPUT in time 't'.

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• Decomposition of Formula of Localization

 

1. The left side

 

$$p(x_t|x_{0:t-<!--\; -\; -->1},z_{1:t-<!--\; -\; -->1},u_{1:t})$$

 

It estimates the position at the current time 't' using the position of the previous time(t-1), sensor, control input.

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2. The right side

 

$$p(x_t|x_{t-<!--\; -\; -->1},u_t)$$

 

All x0 to xt-2 were omitted because the current state is a simplification assuming that it is affected by the previous state.

 

2-1. Markov Property

 

$P(x_t|x_0,x_1,...,x_{t-<!--\; -\; -->1})\approx <!--\; \approx \; -->P(x_t|x_{t-<!--\; -\; -->1})$

 

The preceding state, P(xt-1), implies all previous states.

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3. Update sensor $z_t$

 

$$p(z_t|x_{0:t},z_{1:t-<!--\; -\; -->1},u_{1:t})=p(z_t|x_t)$$

 

Update the sensor $z_t$ based on the $x_t$ obtained earlier.

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4. Final position( $bel(x_t)$, belif)
$$bel(x_t)=p(x_t|z_{1:t},u_{1:t})$$

 

Current state, based on all sensor and control input data from the start of the algorithm to the current.

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• Tracking

 

Tracking = Sensor + (Noise of Control Input)

Remove control input $u$ from the Formula of Localization.

 

$$p(x_t|x_{0:t-<!--\; -\; -->1},z_{1:t-<!--\; -\; -->1})=p(x_t|x_{t-<!--\; -\; -->1})$$
$$p(z_t|x_{0:t},z_{1:t-<!--\; -\; -->1})=p(z_t|x_t)$$
$$bel(x_t)=p(x_t|z_{1:t})$$

 

https://gaussian37.github.io/autodrive-ose-localization_and_tracking/