Robot Environment Interaction

1
Robot Environment Interaction

• Environment perception provides information
  about the environments state, and it tends to
  increase the robots knowledge.
• Motion (control date), on the other hand, tends
  to induce a loss of knowledge due to noise
  (uncertainty).
• The evolution of state and measurements is
  governed by probabilistic laws. (Probabilistic
  Robotics)

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Robot Environment Interaction

• For state variable
• If the state variable is complete
• This is an example of Conditional independence
  (CI).

3
Robot Environment Interaction

• For measurement data
• If the state variable is complete
• This is another example of Conditional
  independence (CI).

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Robot Environment Interaction

State transition probability
measurement probability
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Robot Environment Interaction

• The state transition probability and the
  measurement probability together describes the
  dynamic stochastic system of the robot and its
  environment.
• See Figure 2.2.

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Robot Environment Interaction

• Besides measurement, control, etc, another key
  concept in probabilistic robotics is that of a
  belief.
• A belief reflects the robots internal knowledge
  about the state of the environment, because the
  state of the environment, to the robot, is
  unobservable.
• How belief is probabilistically represented in
  probabilistic robotics?

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Robot Environment Interaction

• The belief of a robot is represented in the form
  of conditional probability distribution (CPD) as
• Sometimes, the following CPD is also of interest.

predication
8
Bayes Filter-The single most important algorithm
in the book

• It calculates the belief distribution bel from
  measurement and control date.
• It is a recursive algorithm. It is the basis of
  all other algorithms in the book.

9
Simple Example of Bayes Filter Algorithm

• Suppose a robot obtains measurement z
• What is P(openz)?

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Causal vs. Diagnostic Reasoning

• P(openz) is diagnostic.
• P(zopen) is causal.
• Often causal knowledge is easier to obtain.
• Bayes rule allows us to use causal knowledge

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Example

• P(zopen) 0.6 P(z?open) 0.3
• P(open) P(?open) 0.5

• z raises the probability that the door is open.

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Combining Evidence

• Suppose our robot obtains another observation z2.
• How can we integrate this new information?
• More generally, how can we estimateP(x z1...zn
  )?

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Recursive Bayesian Updating
Markov assumption zn is independent of
z1,...,zn-1 if we know x.
Whats going on here?
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Example Second Measurement

• P(z2open) 0.5 P(z2?open) 0.6
• P(openz1)2/3

• z2 lowers the probability that the door is open.

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Example

• The previous examples seems only concern with
  measurement. What about control data (or motion,
  action)?
• How does control data play its role?

16
Actions

• Often the world is dynamic since
• actions carried out by the robot,
• actions carried out by other agents,
• or just the time passing by
• change the world.
• How can we incorporate such actions?

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Typical Actions

• The robot turns its wheels to move
• The robot uses its manipulator to grasp an object
• Plants grow over time
• Actions are never carried out with absolute
  certainty.
• In contrast to measurements, actions generally
  increase the uncertainty.

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Modeling Actions

• To incorporate the outcome of an action u into
  the current belief, we use the conditional pdf
• P(xu,x)
• This term specifies the pdf that executing u
  changes the state from x to x.

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Example Closing the door
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State Transition (probability distribution)

• P(xu,x) for u close door
• If the door is open, the action close door
  succeeds in 90 of all cases.

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Integrating the Outcome of Actions
Continuous case Discrete case
Whats going on here?
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Example The Resulting Belief
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Bayes Filters Framework

• Given
• Stream of observations z and action data u
• Sensor model P(zx).
• Action model P(xu,x).
• Prior probability of the system state P(x).
• Wanted
• Estimate of the state X of a dynamical system.
• The posterior of the state is called Belief

measurement probability
New terms
State transition probability
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Bayes Filters The Algorithm

• Algorithm Bayes_filter ( )
• for all do
• endfor
• return

Action model
Sensor model
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Bayes Filters
z observation u action x state
What is it?
Sensor model
Action model
recursion
26
Bayes Filters An Example

• Page 28-31.

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Markov Assumption (the Complete State Assumption)

• Underlying Assumptions
• Static world
• Independent noise
• Perfect model, no approximation errors
• In practice, Bayes filters have been found to be
  surprisely robust to violations of Markov
  assumption.

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Representations and Computation

• There exist quite a variety of techniques and
  algorithms that are all derived from the Bayes
  filter.
• Each such technique relies on different
  assumptions regarding the measurement and state
  transition probabilities and the initial belief.
• These assumptions then give rise to different
  types of posterior distributions, and
• The algorithms for computing them have different
  computational characteristics.
• Exact techniques only exist for highly
  specialized cases,
• In general, many require approximate.

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Bayes Filters are Familiar!

• Kalman filters
• Particle filters
• Hidden Markov models
• Dynamic Bayesian networks
• Partially Observable Markov Decision Processes
  (POMDPs)

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Summary

• Bayes rule allows us to compute probabilities
  that are hard to assess otherwise.
• Under the Markov assumption, recursive Bayesian
  updating can be used to efficiently combine
  evidence.
• Bayes filters are a probabilistic tool for
  estimating the state of dynamic systems.

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Mobile Robot Localization

• Mobile robot localization is the problem of
  determining the pose of a robot relative to a
  given map of the environment. Because,
• Unfortunately, the pose of a robot can not be
  sensed directly, at least for now. The pose has
  to be inferred from data.
• A single sensor measurement is enough?
• The importance of localization in robotics.
• Mobile robot localization can be seen as a
  problem of coordinate transformation. One point
  of view.

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Mobile Robot Localization

• Localization techniques have been developed for a
  broad set of map representations.
• Feature based maps, location based maps,
  occupancy grid maps, etc. (what exactly are
  they?) (See figure 7.2)
• (You can probably guess What is the mapping
  problem?)
• Remember, in localization problem, the map is
  given, known, available.
• Is it hard? Not really, because,

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Mobile Robot Localization

• Most localization algorithms are variants of
  Bayes filter algorithm.
• However, different representation of maps, sensor
  models, motion model, etc lead to different
  variant.
• Here is the agenda.

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Mobile Robot Localization

• We want to know different kinds of maps.
• We want to know different kinds of localization
  problems.
• We want to know different localization problems.

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Mobile Robot Localization

• Different kinds of maps.

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Mobile Robot Localization A Taxonomy

• Different kinds of Localization problems.
• Through 4 dimensions
• Local versus Global (initial knowledge)
• Static versus Dynamic (environment)
• Passive versus active (control of robots)
• Single robot or multi-robot