Mobile robotics II
Key points:
• Topological and geometric maps
• Varieties of spatial representations
• Relating sensing to space
• Localisation: determining and updating
• Learning maps
• Simultaneous localisation and mapping
Grid representation
• Can use ‘wave front’ to find route
Voronoi diagram
Geometric maps
Topological maps
• Can represent landmarks and the routes
between them as nodes and edges in a graph
• Edges can represent motor action needed to
get from one node to the next, or direction,
distance, path convenience etc.
• Can then use standard AI graph search
methods to find route from start to goal
Advantages of topological maps:
• Can describe location of robot and objects in its
world as a configuration space
• For mobile robots, usually collapse 6 d.o.f. to 2 d.o.f.
– assumes robot moves on ground plane
– can rotate on spot (so direction not important)
– only obstacle location matters
• Expand actual obstacles by robot size
• Navigation then involves finding routes through
freespace
Topological Raw sensor Store few
data
locations
• Only sparse data storage
• Representation matches problem description: e.g.
instruct robot to move between discrete locations
• Convert free space to graph by e.g. skeletonisation
– edges are equidistant locations from obstacles
• Recognition only requires consistency, not accuracy
• Can find routes using graph search, as for
topological map
• Can extrapolate between known locations
Extract
features
Store
continuous
links
Connect by
raw motor
action
Connect with
some metric
information
Make
Nodes have
Use sensor
inferences
global
model
between nodes position
Advantages of metric maps
• Can derive novel shortcuts
Figures from Murphy (2000)
Relating sensing to space
• Some sensors provide metric information
almost directly e.g. range finders
• Others are good for distinctive landmarks but
hard to convert to metric layout e.g. vision
• Main problems are
– aliasing (different places sometimes look the
same)
– variability (same place sometimes looks different)
• Common representation to fuse sensor/motor data
Metric
Absolute
Convert to Continuous
spatial data representation metric
Localisation
Relating sensing to space
• Tracking movement (commands or odometry)
to know where you are can help against
aliasing and variability problems
• Recognising where you are from external
cues can correct for the cumulative error of
dead-reckoning
• Note that metric approach fuses the two
sources of data in common representation
• ‘Lost robot problem’: can you recognise where
you are when switched on?
• Alternatively, given model of environment, what
position or node is the most probable, based on the
current sensory input?
• Essentially same as visual recognition problem in
previous lectures, and can use same Bayesian
approach i.e.
p( s | z ) =
p( z | s) p( s)
∑ p( z | s) p( s)
where s is robot location
and z is sensor input
and have been given or have learned p(z|s) and p(s)
s
Initial estimate of position and variance
Updating the localisation estimate
v1
Updating the localisation estimate
s1
• Other methods maintain multiple location
hypotheses with different probabilities
Measured position
• ‘Lost robot’ could also use active means to
confirm location – i.e. does movement produce the
expected sensory consequence?
• More generally, current position estimate can be
function of previous estimate, expected result
from moving, and current input
• Suggests a Kalman filter approach: estimate
current state of system based on previous state,
model and measurement, all of which are noisy
New estimate of
position and
variance merging
prediction and
measurement
vm2
v2
m2
s2
s2’
v2’
Predicted
position after
movement
v3
vm3
m3
s3’ s3
• Suggested interpretation of ‘place cells’
found in the rat brain
v3’
Learning the map
E.g. Occupancy grid approach:
Assuming robot knows where it is in grid, sensory input
provides noisy information about obstacles, e.g. for sonar
III I
r
R
II
probability of given sonar measurement (z)  R − r   β − α 
=
 + 
 2
if grid element in region I is occupied (O)
 R   β 
p ( z | O) p (O)
p ( z | O) p (O) + p ( z |~ O) p (~ O)
where p(O) will depend on previous measurements
Using Bayesian approach p (O | z ) =
Variants on SLAM:
Simultaneous localisation and mapping
• Usually not true that robot knows (with certainty)
its position while moving around to build the map
• But map built so far can be used to help correct
localisation estimate
t
Aim to merge all past sensor data: z = zt , zt −1 ,..., z0
t
and movement data: u = ut ,ut −1 ,...,u0
to obtain both current position st and map m
Use a Bayes filter sensor model motion model
p ( st , m | z t , u t ) =
p ( zt | st , m) ∫ p ( st | ut , st −1 ) p ( st −1 , m | z t −1 , u t −1 )dst −1
But
– Only weak convergence result
– Runs off-line, i.e. gather all data then process
– Only maintains single maximum likelihood map
Extended Kalman filter (e.g. Dissanayake et al 2001)
– Proven theoretically to converge on true map
– Maintains uncertainty estimates
– Can be operated incrementally
But
– Assumes gaussian noise (which is unlikely)
– Can’t deal with aliasing (which is common)
– Only tractable with limited number of features
normalisation factor
Further reading
Summary
Expectation maximisation (e.g. Thrun et al 1998)
– Deals explicitly with aliasing: uses hill-climbing to
find the most likely map based on the possible
paths of the robot given the sensor data
– Makes no assumptions about noise distributions
Variants on SLAM:
• There are sound theoretical approaches to
path planning, localisation, map building
and SLAM
• Problems lie in application to real, limited,
noisy, dynamic robots and environments
• Best approach is then dependent on task,
sensors and actuators
• ‘Hybrid’ solutions are often useful
Murphy, R. (2000) Introduction to AI Robotics MIT Press
Filliat, D. & Meyer, J-A (2003) Map-based navigation in mobile
robots: I. A review of localization strategies Cognitive
Systems Research 4:243-282
Thrun, S. (2002) Robotic Mapping: A survey
http://www-2.cs.cmu.edu/~thrun/papers/thrun.mapping-tr.html
Dissanayake, G. et al. (2001) A solution to the simultaneous
localisation and map building (SLAM) problem. IEEE
Transactions on Robotics and Automation 17:229-241
Thrun, S. Fox, D. and Burgard, W. (1998) A probabilistic
approach to concurrent mapping and localization for mobile
robots Autonomous Robots 5:253-271
Moser EI, Paulsen O (2001) New excitement in cognitive space:
between place cells and spatial memory. Curr Opin Neurobiol.
11:745-51.