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 cs-677sp2010:filters-lab [2014/12/12 13:37]ryancha created cs-677sp2010:filters-lab [2014/12/12 13:37] (current)ryancha 2014/12/12 13:37 ryancha 2014/12/12 13:37 ryancha created 2014/12/12 13:37 ryancha 2014/12/12 13:37 ryancha created Line 9: Line 9: x, y coordinates. The robot moves by specifying a direction (in x, y coordinates. The robot moves by specifying a direction (in radians from vertical) and a distance. The robot senses its radians from vertical) and a distance. The robot senses its - position as the distance from 2 beacons (<​math>​d_{A} ​the distance from + position as the distance from 2 beacons ($d_{A}$ the distance from - beacon A and <​math>​d_{B} ​the distance from beacon B) + beacon A and $d_{B}$ the distance from beacon B) === Prior === === Prior === - <​math>​x_{0} \sim N(0,1) \!​ + $x_{0} \sim N(0,1) \!$ - <​math>​y_{0} \sim N(0,1) \!​ + $y_{0} \sim N(0,1) \!$ === System or Transition Model === === System or Transition Model === - The movement of the robot is random in <​math>​d​, the distance + The movement of the robot is random in $d$, the distance - travelled, and <​math>​\theta​, the angle chosen. The intended distance + travelled, and $\theta$, the angle chosen. The intended distance - (5) and direction ​<​math>​\left(\frac{\pi}{5}\right) ​are fixed. + (5) and direction ​$\left(\frac{\pi}{5}\right)$ are fixed. - <​math> ​d \sim N(5,1)\!​ + $d \sim N(5,1)\!$ --> - <​math> ​\theta \sim \mathit{Uniform} \left(\frac{\pi}{5}-\frac{\pi}{36},​\frac{\pi}{5}+\frac{\pi}{36}\right)​ + $\theta \sim \mathit{Uniform} \left(\frac{\pi}{5}-\frac{\pi}{36},​\frac{\pi}{5}+\frac{\pi}{36}\right)$ - <​math>​x_{t+1}|x_{t} = x_{t} + d \cdot \cos \left(\theta\right)​ + $x_{t+1}|x_{t} = x_{t} + d \cdot \cos \left(\theta\right)$ - <​math>​y_{t+1}|y_{t} = y_{t} + d \cdot \sin \left(\theta\right)​ + $y_{t+1}|y_{t} = y_{t} + d \cdot \sin \left(\theta\right)$ === Observation or Sensor Model === === Observation or Sensor Model === The robot senses its position as the distance from beacon A The robot senses its position as the distance from beacon A - (<​math>​d_{A}​) and the distance from beacon B(<​math>​d_{B}​). Both are + ($d_{A}$) and the distance from beacon B($d_{B}$). Both are imperfect measures as shown below: imperfect measures as shown below: - <​math>​d_{A} = \sqrt{(-100-x)^{2}+(100-y)^{2}} ​​ + $d_{A} = \sqrt{(-100-x)^{2}+(100-y)^{2}} ​$ - <​math>​d_{B} = \sqrt{(150-x)^{2}+(90-y)^{2}} ​​ + $d_{B} = \sqrt{(150-x)^{2}+(90-y)^{2}} ​$ - <​math>​r_{A} \sim N (d_{A}, 1)\!​ + $r_{A} \sim N (d_{A}, 1)\!$ - <​math>​r_{B} \sim N (d_{B}, 1)\!​ + $r_{B} \sim N (d_{B}, 1)\!$ === Sample data === === Sample data ===