Robots Autónomos

Robots Autónomos
Mapping and motion models
Domingo Gallardo
Depto. CCIA
http://www.rvg.ua.es/master/robots
Probabilistic Robotics
•
Slides taken from the book “Probabilistic Robotics” by
Thrun, Burgard y Fox
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References
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Probabilistic Robotics, cap. 5
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The general problem of mapping
What does the
environment look like?
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Localization and mapping
•
Localization: given a known map and a set of sensor
readings and robot actions, estimate the most likely
position of the robot in the map.
•
¿What if we don’t know the map? Mapping involves to
simultaneously estimate the pose of the vehicle and the
map.
•
The general problem is therefore denoted as the
simultaneous localization and mapping problem (SLAM)
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The general problem of mapping
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Formally, mapping involves, given the sensor data,
to calculate the most likely map
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Types of SLAM problems
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Grid maps or scans
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Landmark based
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Problems in mapping
•
•
Sensor interpretation
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How do we extract relevant information from raw sensor
data?
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How do we represent and integrate this information over
time?
Robot locations have to be estimated
•
How can we identify that we are at a previously visited
place?
•
This problem is the so-called data association problem.
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Robot Motion
•
•
Robot motion is inherently uncertain.
How can we model this uncertainty?
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Dynamic Bayesian Network for Controls,
State and Sensations
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Probabilistic motion models
•
To implement the Bayes Filter, we need the transition
model p(x | x’, u).
•
The term p(x | x’, u) specifies a posterior probability, that
action u carries the robot from x’ to x.
•
In this section we will specify, how
p(x | x’, u) can be modeled based on the motion equations.
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Coordinate Systems
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In general the configuration of a robot can be described by
six parameters.
•
Three-dimensional cartesian coordinates plus three Euler
angles pitch, roll, and tilt.
•
Throughout this section, we consider robots operating on a
planar surface.
•
The state space of such systems is three-dimensional
(x,y,θ).
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Typical Motion Models
•
•
In practice, one often finds two types of motion models:
•
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Odometry-based
Velocity-based (dead reckoning)
Odometry-based models are used when systems are
equipped with wheel encoders.
•
Velocity-based models have to be applied when no wheel
encoders are given.
•
They calculate the new pose based on the velocities and
the time elapsed.
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Example wheel encoders
These modules require
+5V and GND to
power them, and
provide a 0 to 5V
output. They provide
+5V output when they
"see" white, and a 0V
output when they "see"
black.
These disks are manufactured
out of high quality laminated
color plastic to offer a very crisp
black to white transition. This
enables a wheel encoder
sensor to easily see the
transitions.
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Dead reckoning
•
•
Derived from “deduced reckoning.”
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Achieved by calculating the current pose of the vehicle
based on its velocities and the time elapsed.
Mathematical procedure for determining the present
location of a vehicle.
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Reasons for motion errors
ideal case
bump
different wheel
diameters
carpet
and many more …
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Odometry models
•
•
Robot moves from
to
Odometry information
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The atan2 function
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Extends the inverse tangent and correctly copes with the
signs of x and y.
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Noise model for odometry
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The measured motion is given by the true motion corrupted
with noise.
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Typical distributions for probabilistics
motion models
Normal distribution
Triangular distribution
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Calculating the probability
•
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For a normal distribution
1.
Algorithm prob_normal_distribution(a,b):
2.
return
For a triangular distribution
1.
Algorithm prob_triangular_distribution(a,b):
2.
return
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Calculating the posterior
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Given x, x’ and u
1.
Algorithm motion_model_odometry(x,x’,u)
2.
odometry values (u)
3.
4.
5.
6.
values of interest (x,x’)
7.
8.
9.
10.
11. return
p1 · p2 · p3
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Application
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Repeated application of the sensor model for short
movements.
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Typical banana-shaped distributions obtained for 2dprojection of 3d posterior.
p(x|u,x’)
x’
u
x’
u
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Sample-based density representation
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How to sample from normal or triangular
distributions
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Sampling from a normal distribution
1.
Algorithm sample_normal_distribution(b):
2.
return
Sampling from a triangular distribution
1.
Algorithm sample_triangular_distribution(b):
2.
return
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Normally distributed samples
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Triangular distributions
103 samples
104 samples
105 samples
106 samples
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Rejection sampling
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Sampling from arbitrary distributions
1.
Algorithm sample_distribution(f,b):
2.
repeat
3.
4.
5.
until (
6.
return
)
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Example
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Sampling from
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Sample odometry motion model
1.
Algorithm sample_motion_model(u, x):
2.
3.
4.
5.
6.
sample_normal_distribution
7.
8.
Return
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Sampling from our motion model
Start
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Examples (odometry-based)
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Velocity-based model
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Equation for the velocity model
Center of circle:
with
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Posterior probability for the velocity model
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Sampling from the velocity model
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Examples (velocity-based)
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Map-consistent motion model
≠
Approximation:
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Summary
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We discussed motion models for odometry-based and
velocity-based systems
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We discussed ways to calculate the posterior probability p
(x| x’, u).
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We also described how to sample from p(x| x’, u).
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In practice, the parameters of the models have to be
learned.
•
We also discussed an extended motion model that takes
the map into account.
Typically the calculations are done in fixed time intervals
Δt.
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