Download Particle Filters in Robotics, Sebastian Thrun

Particle Filters in Robotics, Sebastian Thrun
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1) Introduction
1970's
model based approaches
doesn't account for uncertainty
assumes model is perfect
1980's
reactive techniques
i.e. behaviour based (Brooks)
generate control from sensor measurements
assumes sensors are perfect
mid 1990's
techniques that assume imperfect models and sensors
probabilistic robotics
integrates imperrfect models and sensors thru prob laws
such as Bayes rule
at all levels of perception and decision making
particle filters
easy: robot localization
harder problems
SLAM
variable number of dimensions
FastSLAM
can handle > 100,000 dimensions in real time
tracking
people moving near a robot
caveats
will always be wrong
false independence assumptions
lack state variables that systematically affect
sensor and actuator noise
? want state to affect noise?
must work in real time - pf's computationally expensiv
vanialla (exponential-time) pf's
2) Particle filters
approximate techniqes for calculationg posteriors
in partially observable (?) controllable (?) Markov chains (?)
with discrete time
state is not observable - measure with zt
zt = stochastic (?) projection of true state xt
Bayes filters compute posterior recursively
if states, control, measurements all discrete
Markov chain = hidden Markov models (HMM)
then Bayes equation (1) can be implemented exactly
pf's are applied to continuous state spaces
closed form solns for (1) are usually unknown
if p(x0) is gaussian, and motion/measurement models are linear
then (1) = Kalman filter
if nonlinear, can use linearize with 1st order taylor -> ekf
unscented filters
better linear model thru non-random sampling
still rely on gaussian-linear approximation
advantages to pf's
can be applied to any model
that can be formulated using a Markov chain
anytime
comp time can be changed w number of particles,
depending on resources
easy to implement
don't have to linearize
don't need closed form soln's of conditional p(xt|ut,xt-1)
disadvantage
d-dim state space --> exponentially many particles in d
best in low-dim spaces
conditional independences allow for higher dim apps (?)
3) Particle Filters in Low Dimensional Spaces
classical example: robot localization
state = x,y,theta
position tracking - if error is small at all times (?)
more general: global localization
localizing robot under global uncertainty
fig 1a - distribution over entire world
robot could be anywhere
kidnapped robot problem
well-localized robot is tele-ported to other location wo being told
i.e. - robo-cup - judge picking up and moving robot
multi-robot localization (observing each other)
fig 1 - global localization
a) global uncertainty
b) bimodal uncertainty - after going down symmetric corridor
c) unimodal uncertanity - after going into unique room
three figures represent successive posteriors (1)
each particle = x,y,theta
pf's for localization ==> Monte Carlo localization (MCL)
motivated by condensation aglorithm - popular in computer vision
more successful than parametric techniques such as KF's
aibo soccer robot - only 50 particles - limited computation
recent research
problem: next-state probability p(xt|ut,xt-1) sometimes insufficient
kidnapped robot
sensor accuracy much better than control accuracy (?)
hybrid sampling
subset of samples generated according to measurement model
? how different from general pf?
figure 3: compares standard and mixture MCL's
unmodeled environmental state
- artificially inflate unc to accomodate systematic noise
- filters for preprocessing sensor data
ex: range sensors corrupted by people
clustering particles
diff resampling rates for diff clusters
empirically increases robustness to errs in map used in loc
stratified (?) sampling techniqes
to increase variance in sampling step (?)
multi-robot
? are all these methods addressing problem with next-state
probability?
4) Particle Filters in High Dimensional Spaces
number of particles scales exponentially with state dimension
similar problem with vanilla HMM's
? aren't plain pf's = vanilla HMM's?
SLAM
errors in robot localization
-> systematic errors in feature localization
absence of initial map makes it impossible
to localize robot during mapping
using MCL algorithms
? more possible with KF?
data association problem:
if two features observed at diff times are same feature
space of all maps (?) is hundreds of thousdands dimensional
dimension of map unknown at beginning
ekf's address all of these problems except data association
assume statice environment - features not moving
time-invariant feature locations
time-variant robot pose
(2) integral only involves robot pose, not map
main problems
1) complexity = O(N^2) -even wo data assoc problem
scaling issues:
can't manage > 1000 features in rt
research into:
heirarchical map representations
maps decomposed recursively
into submaps
still O(N^2), but constant smaller
2) cannot incorporate negative information
false negatives
when failing to see a feature
that was expected
neg info leads to non-gaussian posteriors
3) no soln to data assoc problem
max likelihood estimator
thresholding outliers
new features placed on provisional list
problem: false positives
-> catastrophic erros
Rao-Blackwellized pf's
significantly more efficient than ekf for slam
1) O(MlogN)
M can be constant in case of bounded uncertaintly
SLAM ekf's only work with bounded uncertainty anyway
2) can deal with neg info
3) exp results, not proven, say better at data assoc
assume, first, no data assoc problem
independence property ( ?)
(not exploited by EKF soln's)
knowledge of robot path
-> feature locations cond indep (3)
fig 4 - shaded region is robot path
leads to (4) - more efficient version of (2)
estimates post on robot paths x^t
rather than xt
p(x^t|z^t,u^t)
each particle = posterior over entire path
p(yn|x^t,z^t)
feature posteriors
in FastSLAM
one EKF for each feature, dim = 2
pf's for robot paths
as in MCL for localization
feature ekfs exact, robot pose pfs approximate
O(NM)
N = number of features, M = num particles
because of (3), conditional independence
in FastSLAM, exploit fact that robot can only
observe finite number of features at any
point in time
features in tree structures (?)
becomes O(MlogN)
data assoc
since FastSLAM has an EKF for each feature,
can keep multiple hypotheses,
instead of just keeping best one
can incorporate neg information
by modifying importance factors
of indiv particles (?)
ekf more computationally expensive
question: (?) for FastSLAM, why can't you just use 3d ekf
for robot pose, and have no pf's?
fig 5 - FastSLAM with outdoor tree detection w lasers
fig 6 - lazy version of FastSLAM
each particle uses max likelihood data assoc
? particle for robot pose or feature?
feature locations calculated in a lazy way
for the mose likely particle only
? again, particles for robot pose
or feature?
point estimates for features (no variance)
similar to FastSLAM
pf's for estimates over robot paths
very good for indoor mapping
tracking
moving features
tracking people, service robotics
1) factored pf
features associated with indep pfs
using maximum likelihood
2) maps to detect people
differences between previous map
similar to SLAM
simultaneous localization of robots and people
similar to FastSLAM - exploits indep property
5) Discussion
tradicionally, pfs mostly used for low-dim localization problems
recently, higher dim problems
hybrid representations that exploit structure
conditional independences
further research: control
above only looks at perception
robot needs to do the right thing