Perception for Autonomous Systems 31392:
State Estimation – Histogram Filter
Lecturer: —PhD
15 Mar. 2021 DTU Electrical Engineering 2
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Cases of State Estimation
15 Mar. 2021 DTU Electrical Engineering 3
Cases of State Estimation
15 Mar. 2021 DTU Electrical Engineering 3
Cases of State Estimation
15 Mar. 2021 DTU Electrical Engineering 3
What is State Estimation
• Given a State Vector of a system
• Estimate over time the state using input of external sensors
Useful for:
• Localization
• Tracking
• Prediction
• Sensor Fusion •…
15 Mar. 2021 DTU Electrical Engineering 4
Continuous vs Discrete State Unimodal vs Multimodal Distribution
• Distribution
Continuous
Multimodal
15 Mar. 2021 DTU Electrical Engineering 5
Simple State Estimation Example
• Assume a robot in an area like this one:
15 Mar. 2021 DTU Electrical Engineering 6
Simple State Estimation Example
• Assume a robot in an area like this one:
• We model the position of the robot as follows:
15 Mar. 2021 DTU Electrical Engineering 6
Simple State Estimation Example
• Assume a robot in an area like this one:
• We model the position of the robot as follows:
15 Mar. 2021 DTU Electrical Engineering 6
Simple State Estimation Example
• Assume a robot in an area like this one:
• We model the position of the robot as follows:
• The robot can distinguish (sense) a door vs “no-door”:
15 Mar. 2021 DTU Electrical Engineering 6
Simple State Estimation Example
• Assume a robot in an area like this one:
• We model the position of the robot as follows:
• The robot can distinguish (sense) a door vs “no-door”:
15 Mar. 2021 DTU Electrical Engineering 6
Simple State Estimation Example
• Assume a robot in an area like this one:
• We model the position of the robot as follows:
• The robot can distinguish (sense) a door vs “no-door”:
• The robot moves to the right:
15 Mar. 2021 DTU Electrical Engineering 6
Simple State Estimation Example
• Assume a robot in an area like this one:
• We model the position of the robot as follows:
• The robot can distinguish (sense) a door vs “no-door”:
• The robot moves to the right:
• The robot senses again:
15 Mar. 2021 DTU Electrical Engineering 6
Simple State Estimation Example
• Assume a robot in an area like this one:
• We model the position of the robot as follows:
• The robot can distinguish (sense) a door vs “no-door”:
• The robot moves to the right:
• The robot senses again:
15 Mar. 2021 DTU Electrical Engineering 6
Simple State Estimation Example
• Assume a robot in an area like this one:
• We model the position of the robot as follows:
• The robot can distinguish (sense) a door vs “no-door”:
• The robot moves to the right:
• The robot senses again:
15 Mar. 2021 DTU Electrical Engineering 6
Sense, Let’s build it ourselves!
• This state estimation problem is called localization • Assume a robot which can be in one of five blocks:
15 Mar. 2021 DTU Electrical Engineering 7
Sense, Let’s build it ourselves!
• This state estimation problem is called localization • Assume a robot which can be in one of five blocks:
• Without any info, What is the probability of the robot being in each cell?
15 Mar. 2021 DTU Electrical Engineering 7
Sense, Let’s build it ourselves!
• This state estimation problem is called localization • Assume a robot which can be in one of five blocks:
• Without any info, What is the probability of the robot being in each cell?
15 Mar. 2021 DTU Electrical Engineering 7
Sense, Let’s build it ourselves!
• This state estimation problem is called localization • Assume a robot which can be in one of five blocks:
• Without any info, What is the probability of the robot being in each cell?
15 Mar. 2021 DTU Electrical Engineering 7
Sense, Let’s build it ourselves!
• Now our robot is allowed to sense:
15 Mar. 2021 DTU Electrical Engineering 8
Sense, Let’s build it ourselves!
• Now our robot is allowed to sense:
15 Mar. 2021 DTU Electrical Engineering 8
Sense, Let’s build it ourselves!
• Now our robot is allowed to sense:
• What does that mean for our probability?
15 Mar. 2021 DTU Electrical Engineering 8
Sense, Let’s build it ourselves!
• Now our robot is allowed to sense:
• What does that mean for our probability?
• Let’s start by arbitrarily multiplying any correct “sensing” with 0.6 & any incorrect with 0.2
15 Mar. 2021 DTU Electrical Engineering 8
Sense, Let’s build it ourselves!
• Now our robot is allowed to sense:
• What does that mean for our probability?
• Let’s start by arbitrarily multiplying any correct “sensing” with 0.6 & any incorrect with 0.2 • This is close to the belief. However, it is wrong, why?
15 Mar. 2021 DTU Electrical Engineering 8
Sense, Let’s build it ourselves!
• Now our robot is allowed to sense:
• What does that mean for our probability?
• Let’s start by arbitrarily multiplying any correct “sensing” with 0.6 & any incorrect with 0.2 • This is close to the belief. However, it is wrong, why?
15 Mar. 2021 DTU Electrical Engineering 8
Sense, Let’s build it ourselves!
• Now our robot is allowed to sense:
• What does that mean for our probability?
• Let’s start by arbitrarily multiplying any correct “sensing” with 0.6 & any incorrect with 0.2
• This is close to the belief. However, it is wrong, why?
• It does not add up to one. How to do it?
15 Mar. 2021 DTU Electrical Engineering 8
Motion, Let’s build it ourselves!
• Assume that the space is circular (i.e when moving right in the last cell you go to the first):
15 Mar. 2021 DTU Electrical Engineering 9
Motion, Let’s build it ourselves!
• Assume that the space is circular (i.e when moving right in the last cell you go to the first):
15 Mar. 2021 DTU Electrical Engineering 9
Motion, Let’s build it ourselves!
• Assume that the space is circular (i.e when moving right in the last cell you go to the first): • What will happen with the posterior probability?
15 Mar. 2021 DTU Electrical Engineering 9
Motion, Let’s build it ourselves!
• Assume that the space is circular (i.e when moving right in the last cell you go to the first): • What will happen with the posterior probability?
• However, this doesn’t happen. Usually the distribution becomes more uncertain:
15 Mar. 2021 DTU Electrical Engineering 9
Motion, Let’s build it ourselves!
• Assume that the space is circular (i.e when moving right in the last cell you go to the first): • What will happen with the posterior probability?
• However, this doesn’t happen. Usually the distribution becomes more uncertain:
15 Mar. 2021 DTU Electrical Engineering 9
Motion, Let’s build it ourselves!
• Assume that the space is circular (i.e when moving right in the last cell you go to the first): • What will happen with the posterior probability?
• However, this doesn’t happen. Usually the distribution becomes more uncertain:
15 Mar. 2021 DTU Electrical Engineering 9
Motion, Let’s build it ourselves!
• Assume that the space is circular (i.e when moving right in the last cell you go to the first): • What will happen with the posterior probability?
• However, this doesn’t happen. Usually the distribution becomes more uncertain:
15 Mar. 2021 DTU Electrical Engineering 9
Motion, Let’s build it ourselves!
• Assume that the space is circular (i.e when moving right in the last cell you go to the first): • What will happen with the posterior probability?
• However, this doesn’t happen. Usually the distribution becomes more uncertain:
15 Mar. 2021 DTU Electrical Engineering 9
That’s very good!!!!!
• Localization is just a sense/move cycle:
15 Mar. 2021 DTU Electrical Engineering 10
That’s very good!!!!!
• Localization is just a sense/move cycle:
Information Gain
Information Loss
Information Gain
15 Mar. 2021 DTU Electrical Engineering 10
That’s very good!!!!!
• Localization is just a sense/move cycle:
Information Loss
Information Gain
Information Gain
15 Mar. 2021 DTU Electrical Engineering 10
Sum Up Global Localization
• Belief Probability
• Measurements Multiplication followed by Normalization
• Moving Convolution
15 Mar. 2021 DTU Electrical Engineering 11
Belief – Formal Definitions
• Probability:
15 Mar. 2021 DTU Electrical Engineering 12
Belief – Formal Definitions
• Probability:
• Assuming 2 states:
15 Mar. 2021 DTU Electrical Engineering 12
Belief – Formal Definitions
• Probability:
• Assuming 2 states:
15 Mar. 2021 DTU Electrical Engineering 12
Belief – Formal Definitions
• Probability:
• Assuming 2 states:
• Assuming 5 states:
15 Mar. 2021 DTU Electrical Engineering 12
Belief – Formal Definitions
• Probability:
• Assuming 2 states:
• Assuming 5 states:
15 Mar. 2021 DTU Electrical Engineering 12
Measurement – Formal Definitions
• Bayes Rule
• Assuming a grid cell and the measurements:
• The belief of the location given a measurement:
15 Mar. 2021 DTU Electrical Engineering 13
Measurement – Formal Definitions
• Bayes Rule
• Assuming a grid cell and the measurements:
• The belief of the location given a measurement:
• A product of the prior with the measurement probability
• The “probability of seeing a measurement independently of location” (normalizer…)
15 Mar. 2021 DTU Electrical Engineering 13
Movement – Formal Definitions
• This is a somewhat complicated formula: • Notice:
– Grid Location – Time
• Here are the components: – Prior
– Movement
15 Mar. 2021 DTU Electrical Engineering 14
Movement – Formal Definitions
• This is a somewhat complicated formula: • Notice:
– Grid Location – Time
• Here are the components: – Prior
– Movement
• This is what is called Total Probability
15 Mar. 2021 DTU Electrical Engineering 14
Histogram-based State Estimation
Main histogram-based global localization problem (Markov Localization): • Memory scaling is exponential
• So, it is unfeasible in large real world problems.
15 Mar. 2021 DTU Electrical Engineering 15
Sum-UP so far
• State Estimation
• Markov Localization • Probability
• Total Probability
Coming UP:
15 Mar. 2021 DTU Electrical Engineering 16
Perception for Autonomous Systems 31392:
State Estimation – Histogram Filter
Lecturer: —PhD
15 Mar. 2021 DTU Electrical Engineering 17
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