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There are many "measurement free" informative-ness metrics, like the Fisher Information Matrix. All you need are the positions of the robot and the positions of the landmarks in the map to determine how much information about the robot's position would be obtained by measuring the landmark locations. (Or visa-versa, how much information about athe innovation from measurements is applied to both target you get given robot locations, just reverse $t$ and robot $r$ below(it's SLAM right?), so the same metric works for both).

There are many "measurement free" informative-ness metrics, like the Fisher Information Matrix. All you need are the positions of the robot and the positions of the landmarks in the map to determine how much information about the robot's position would be obtained by measuring the landmark locations. (Or visa-versa, how much information about a target you get given robot locations, just reverse $t$ and $r$ below).

There are many "measurement free" informative-ness metrics, like the Fisher Information Matrix. All you need are the positions of the robot and the positions of the landmarks in the map to determine how much information about the robot's position would be obtained by measuring the landmark locations. (Or visa-versa, the innovation from measurements is applied to both target and robot (it's SLAM right?), so the same metric works for both).

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There are many "measurement free" informative-ness metrics, like the Fisher Information Matrix. All you need are the positions of the robot and the positions of the landmarks in the map to determine how much information about the robot's position would be obtained by measuring the landmark locations. (Or visa-versa, how much information about a target you get given robot locations, just reverse $t$ and $r$ below).

Critically, this simplifies the planning problem as your state doesn't change except for the uncertainty and planned motion. Check the equations here for EKF, and note that the only two equations that are relevant to calculate the new covariance are $$P_{i|i-1}=F^T_iP_{i-1|i-1}F_i+Q$$ and $$P=P-PH^T (H P H^T + R) HP$$$$P=P-PH^T (H P H^T + R)^{-1} HP$$ ... (I expanded the kalman gain into the last equation).

we simplify that second one to $$P^{-1}=P^{-1}+H^TRH$$$$P^{-1}=P^{-1}+H^TR^{-1}H$$.

$$I=\sum_{i=1}^n H_i^TPH_i$$$$I=\sum_{i=1}^n H_i^TR^{-1}H_i$$

where $R$ is the measurement noise (e.g., the bearing error of measurement to each landmark), for each time step in your trajectory. Since we are looking at information (inverse of covariance) we want to maximize the trace, determinant, or something else of $\sum_{i=1}^n H_i^TRH_i$$\sum_{i=1}^n H_i^TR^{-1}H_i$

There are many "measurement free" informative-ness metrics, like the Fisher Information Matrix. All you need are the positions of the robot and the positions of the landmarks in the map to determine how much information about the robot's position would be obtained by measuring the landmark locations.

Critically, this simplifies the planning problem as your state doesn't change except for the uncertainty and planned motion. Check the equations here for EKF, and note that the only two equations that are relevant to calculate the new covariance are $$P_{i|i-1}=F^T_iP_{i-1|i-1}F_i+Q$$ and $$P=P-PH^T (H P H^T + R) HP$$ ... (I expanded the kalman gain into the last equation).

we simplify that second one to $$P^{-1}=P^{-1}+H^TRH$$.

$$I=\sum_{i=1}^n H_i^TPH_i$$

where $R$ is the measurement noise (e.g., the bearing error of measurement to each landmark), for each time step in your trajectory. Since we are looking at information (inverse of covariance) we want to maximize the trace, determinant, or something else of $\sum_{i=1}^n H_i^TRH_i$

There are many "measurement free" informative-ness metrics, like the Fisher Information Matrix. All you need are the positions of the robot and the positions of the landmarks in the map to determine how much information about the robot's position would be obtained by measuring the landmark locations. (Or visa-versa, how much information about a target you get given robot locations, just reverse $t$ and $r$ below).

Critically, this simplifies the planning problem as your state doesn't change except for the uncertainty and planned motion. Check the equations here for EKF, and note that the only two equations that are relevant to calculate the new covariance are $$P_{i|i-1}=F^T_iP_{i-1|i-1}F_i+Q$$ and $$P=P-PH^T (H P H^T + R)^{-1} HP$$ ... (I expanded the kalman gain into the last equation).

we simplify that second one to $$P^{-1}=P^{-1}+H^TR^{-1}H$$.

$$I=\sum_{i=1}^n H_i^TR^{-1}H_i$$

where $R$ is the measurement noise (e.g., the bearing error of measurement to each landmark), for each time step in your trajectory. Since we are looking at information (inverse of covariance) we want to maximize the trace, determinant, or something else of $\sum_{i=1}^n H_i^TR^{-1}H_i$

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  1. Write a path search algorithm that finds its way without considering uncertainty.
  2. Derive $H$, or look it up in any bearings localization paper.
  3. To determine the "goodness" of a path for use in the search algorithm, add in $trace(H^TRH)$ for each pose along the path.
  4. Over a glass of wine, noticeNotice that the result matches the FIM of the trajectory (exercise left to the reader), and you have correctly and in a theoretically sound way determined the most informative trajectory.
  1. Write a path search algorithm that finds its way without considering uncertainty.
  2. Derive $H$, or look it up in any bearings localization paper.
  3. To determine the "goodness" of a path for use in the search algorithm, add in $trace(H^TRH)$ for each pose along the path.
  4. Over a glass of wine, notice that the result matches the FIM of the trajectory (exercise left to the reader), and you have correctly and in a theoretically sound way determined the most informative trajectory.
  1. Write a path search algorithm that finds its way without considering uncertainty.
  2. Derive $H$, or look it up in any bearings localization paper.
  3. To determine the "goodness" of a path for use in the search algorithm, add in $trace(H^TRH)$ for each pose along the path.
  4. Notice that the result matches the FIM of the trajectory (exercise left to the reader), and you have correctly and in a theoretically sound way determined the most informative trajectory.
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