CS代写 Perception for Autonomous Systems 31392:

Perception for Autonomous Systems 31392:
State Estimation –
Lecturer: —PhD
15 Mar. 2021 DTU Electrical Engineering 2

Sum-UP so far
• State Estimation
• Markov Localization • Probability
• Total Probability
Coming UP:
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 •…
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Catching up
• Last part, we did state estimation specifically Localization
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Catching up
• Last part, we did state estimation specifically Localization
• Maximum confusion to Location estimation
15 Mar. 2021 DTU Electrical Engineering 5

Catching up
• Last part, we did state estimation specifically Localization
• Maximum confusion to Location estimation
• In other words:
“Used sensor information to manipulate an original belief (a uniform distribution) into a high confidence probability density function centered on our correct location“
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In this part..
• We will see how we can track objects.
• Not only their location as in the previous localization case, • But also infer their speed.
• In the case of autonomous driving it is quite important to track and predict the movement of objects. – Why?
– Which other examples can we find?
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Cases of Tracking
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Histogram filter

Differences between state estimation filters
Continuous

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Let’s see tracking as an example
• Assuming there is a point in space like this: • The object moves like this:
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Let’s see tracking as an example
• Assuming there is a point in space like this: • The object moves like this:
• What is the next point?
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Let’s see tracking as an example
• Assuming there is a point in space like this: • The object moves like this:
• What is the next point?
• For you it is easy! How about a machine?
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Kalman has a thing for Gaussians
• In our Markov model the world was divided
into discrete grids and each grid had a probability
• This is called a histogram:
• In we describe the distribution as a Gaussian:
– It is a continuous function and – The area under the curve is: 1
15 Mar. 2021 DTU Electrical Engineering

• 1D Gaussian is described by the pair: (μ, σ2)
• Which of the following are Gaussians?
• Which of the following have small, medium, larger (co)variance? • When doing state estimation which one do we prefer?
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• Kalman, as with the histogram filter involves the measurementmotion cycle:
• Which one requires convolution and which one a product?
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• Kalman, as with the histogram filter involves the measurementmotion cycle:
• Which one requires convolution and which one a product? – MeasurementProduct
– MotionConvolution
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• Kalman, as with the histogram filter involves the measurementmotion cycle:
• Which one applies Bayes Rule and which one Total Probability?
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• Kalman, as with the histogram filter involves the measurementmotion cycle:
• Which one applies Bayes Rule and which one Total Probability? – MeasurementBayes Rule
– MotionTotal Probability
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• In Kalman we call them “Measurement Update” and “Prediction” • Both of these involve the Gaussians
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• Assume we are localizing another robot with a prior as follows:
• Then we have a measurement which inform us that we have this location: • Where will the new mean be?
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• Where will the new peak be? r • The higher one -> as we gain information
• Let’s prove it
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• Assuming these gaussians: –μ 0σ2=4
– ν , r2= 4
• Assuming these gaussians: –μ 0σ2=8
– ν 3 , r2= 2
• Assuming these gaussians:
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Motion Update
• Called also prediction:
• As we move we lose some information:
• Assuming a gaussian before the prediction: –μ 8 σ2=4
• And a movement gaussian – ν 0, r2= 6
• What’s the Gaussian after the update?
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Let’s code the 1D Kalman
• Start with a initial belief of: – u=0,
– σ2=10000 • Motion:
– [1, 1, 2, 1, 1]
– Uncertainty: 2 • Measurement:
– [5, 6, 7, 9, 10] – Uncertainty: 4
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From 1D to Many D’s
• We just implemented a Full 1D Kalman filter.
• However the shines in Many D’s • Let’s see an example:
– A camera
– Or a pedestrian in front of a car
– Where should it be at t=3?
• That is the power of Kalman!!!
➔ AI and Control Theory
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Multivariate Gaussians
As promised we have married the Gaussians today Multi-Dimensional Gaussians
• The mean is a vector:
• The variance now is called covariance and is a matrix:
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Multivariate Gaussians
• Let’s start with an one dimensional motion example:
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Multivariate Gaussians – Space
t • Let’s go at the Kalman state space:
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Multivariate Gaussians – Space
• Let’s go at the Kalman state space: • Prior:
– Location: 1
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Multivariate Gaussians – Space
• Let’s go at the Kalman state space: • Prior:
– Location: 1 • Prediction:
– Velocity: 0
15 Mar. 2021 DTU Electrical Engineering

Multivariate Gaussians – Space
• Let’s go at the Kalman state space: • Prior:
– Location: 1 • Prediction:
– Velocity: 0 – Velocity: 1
15 Mar. 2021 DTU Electrical Engineering

Multivariate Gaussians – Space
• Let’s go at the Kalman state space: • Prior:
– Location: 1 • Prediction:
– Velocity: 0 – Velocity: 1 – Velocity: 2
15 Mar. 2021 DTU Electrical Engineering

Multivariate Gaussians – Space
• Let’s go at the Kalman state space: • Prior:
– Location: 1 • Prediction:
– Velocity: 0 – Velocity: 1 – Velocity: 2
Consider this Gaussian
15 Mar. 2021 DTU Electrical Engineering

Multivariate Gaussians – Space
• Let’s go at the Kalman state space: • Prior:
– Location: 1 • Prediction:
– Velocity: 0 – Velocity: 1 – Velocity: 2
Consider this Gaussian • Measurement:
15 Mar. 2021 DTU Electrical Engineering

Design of a
• Two types of States variables: – Observables
• State Transition Function: – Matrix
• Measurement Function:
– Vector (Usually be Matrix)
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Linear Algebra Formulation For For
(No need to memorize these:)
Prediction (Ingredients):
• X: State Vector (Including our Prior Info)
• P: Uncertainty Covariance (Incl. Prior Info)
• F: State Transition Matrix (we just discussed it) • u: External Motion (E.g. Deceleration from car) Measurement Update (Ingredients): :
• Z: Measurement
• H: Measurement Matrix
• R: Measurement Noise
• y: Error
• I : Identity Matrix
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Conclusion
• We’ve acquired an amazing skill!!!
• We now understand what state estimation is • We understand what a covariance represents • We know how to track objects in space
(We can handle even occlusions)
• We know how to estimate our position (attitude) over time
• We’ll come back on this (project) to do an end-to-end example!
15 Mar. 2021 DTU Electrical Engineering 52

Perception for Autonomous Systems 31392:
State Estimation –
Lecturer: —PhD
15 Mar. 2021 DTU Electrical Engineering 53

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