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Lecture 09: SLAM

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Lecture 09: SLAM Lecture 09: SLAM Presentation Transcript

  • Probabilistic State EstimationWith Application To Vehicle Navigation
    Matthew Kirchner
    Naval Air Warfare Center – Weapons Division
    Department of ECEE – University of Colorado at Boulder
    November 1, 2010
  • Topics
    Why?
    Review
    Background
    Kalman Filter
    Particle Filter
    SLAM
  • Why Probability?
    Real sensors have uncertainty
    May have multiple sensors
    Some states are not directly observable
    Ambiguous sensor observations
  • Importance of Bayes Rule
  • Bayes Rule
    Likelihood
    Prior
    Posterior
    Normalizing Constant
  • Equivalent Bayes Rule
  • Bayes Rule Example
    School: 60% boys and 40% girls
    All boys wear pants
    Half of girls wear skirts, half wear pants
    You see a random student and can only tell they are wearing pants.
    Based on your observation, what is the probability the student you saw is a girl?
  • Bayes Rule Example
    School: 60% boys and 40% girls
    All boys wear pants
    Half of girls wear skirts, half wear pants
    You see a random student and can only tell they are wearing pants.
    Based on your observation, what is the probability the student you saw is a girl?
    We want to find:
    P(Student=Girl | Clothes=Pants)
    Prior?
    P(Student=Girl) = 0.4
    Likelihood?
    P(Clothes=Pants | Student=Girl) = 0.5
    Normalizing Constant?
    P(Clothes=Pants) = 0.8
    Bayes Rule!
    P(Student=Girl | Clothes=Pants) = (0.5)*(0.4)/(0.8) = 0.25
  • Gaussian
    1D
    2D
  • Gaussian
    1D
    2D
  • Gaussian
    2D Probability density function described by mean vector and covariance matrix
    1D Probability density function described by mean and variance
  • Functions of Random Variables
  • Functions of Random Variables
    Linear function
    Mean:
    Covariance:
    Linear functions only!
  • Kalman Filter
    Introduced in 1960 by Rudolf Kalman
    Many applications:
    Vehicle guidance systems
    Control systems
    Radar tracking
    Object tracking in video
    Atmospheric models
  • Kalman Filter
    We want to find:
  • Kalman Filter
    Process
    Model
    Process
    Model
    Observation
    Model
    Observation
    Model
    Observation
    Model
  • Kalman Filter
    Process Model:
    Deterministic:
    Probabilistic:
    Used to calculate prior distribution
    Observation Model:
    Deterministic:
    Probabilistic:
    Used to calculate likelihood distribution
  • Kalman Filter: Assumptions
    Underlying system is modeled as Markov
    All beliefs are Gaussian distributions
    Additive zero mean Gaussian noise
    Linear process and observation models
    Process Model:
    Observation Model:
  • Kalman Filter Steps
    Using process model, previous state, and controls, find prior
    Sometimes called ‘predict’ step, or ‘a priori’
    Using prior and observation model, find sensor likelihood
    If I knew state, what should sensors read?
    Find observation residual
    Difference between actual sensor values and what was calculated in step 2. Also sometimes called ‘innovation’.
  • Kalman Filter Steps
    Compute Kalman gain matrix
    Using Kalman gain and prior, calculate posterior
    Sometimes called ‘correct’ step or ‘a posteriori’
  • Kalman Filter Steps
    Algorithm Kalman_Filter( )
    1a:
    1b:
    2:
    3:
    4:
    5a:
    5b:
    Return( )
  • Kalman Filter
  • Kalman Filter Example
  • Kalman Filter Example
  • Kalman Filter Example
    GPS Lost
    GPS Reacquired
  • Kalman Filter Example
  • Kalman Filter
    O(k^2.4+n^2)
    Many real systems are non linear
    Extended Kalman filter
    Unscented Kalman filter
    Particle filter
    Some systems are non-Gaussian
    Particle filter
  • Extended Kalman Filter
    Linearize process and observation models
    By finding Jacobian matrices
    Analytically or numerically
    Then use regular Kalman filter algorithm
    Sub-optimal
  • 29
    EKF Linearization
  • EKF Linearization
  • Unscented Kalman Filter
  • Unscented Kalman Filter
    EKF
    UKF
  • Particle Filter
  • Particle Filter
    Represent distribution as set of randomly generated samples, called ‘particles’.
    Functions can be nonlinear and non-gaussian
    Multi-hypothesis belief propagation
  • Particle Filter
    Sample the prior
    Compute likelihood of particles given measurement
    Also called particle ‘weights’
    Sample posterior: Sample from particles proportional to particle weights
    Also called ‘resampling’ or ‘importance sampling’
  • Simultaneous Localization and Mapping
    SLAM Problem:
    Need map to localize
    Need location to make map
    Brainstorming: How can we solve this problem?
    Map could be locations of landmarks or occupancy grid
    Kalman filter based: landmark positions part of state variables
    Particle filter based: landmark positions or occupancy map included in each particle
  • Feature-based SLAM
  • FastSLAM - Example
  • Other Probabilistic Applications
  • Other Probabilistic Applications