Mobile robot localization Francesco Martinelli University of Rome Tor Vergata Roma, Italy...
-
Upload
maura-truelove -
Category
Documents
-
view
232 -
download
2
Transcript of Mobile robot localization Francesco Martinelli University of Rome Tor Vergata Roma, Italy...
1/106
Mobile robot localization
Francesco MartinelliUniversity of Rome “Tor Vergata”
Roma, [email protected]
2/106
Objective:
Determination of the pose (= position + orientation)of a mobile robot in a known environment in order to succesfully perform a given task
3/106
Summary:
• Robot localization taxonomy (4-5)
• Robot localization as an estimation problem: the Bayes filter (6-17)
• Observability issue (18-23)
• Robot description (24) Motion model: the differential drive (25-34) Measurement model: rangefinder or landmarks (35-39)
• Robot localization algorithms (40-42): Gaussian filters: EKF, UKF (43-60) Non parametric filters: HF, PF (61-77)
• Examples (78-104)• Conclusions (105)• References (106)• Video
4/106
Robot Localization Taxonomy:
• Position tracking• Global localization• Kidnapped problem
Increasing degreeof difficulty
Other classifications include:• type of environment (static or dynamic; completely or partially known)• active or passive localization• single or multi robot localization.
5/106
Robot Localization Taxonomy:
• Position tracking• Global localization• Kidnapped problem
Increasing degreeof difficulty
Other classifications include:• type of environment (static or dynamic; completely or partially known)• active or passive localization• single or multi robot localization.
6/106
Localization as an estimation problem
The robot must infer its pose from available data
Data (noisy):
• Motion information:
Proprioceptive sensors (e.g. encoders, accelerometers, etc.)
• Environment Measurements
Exteroceptive sensors (e.g. laser, sonar, IR, GPS, camera, RFID, etc.)
A filtering approach is required to fuse all information
7/106
Localization as an estimation problem
Notation
Robot pose
x
y
Robot poses from time 0 to time t
Robot exteroceptive measurements from time 1 to time t
Motion commands (or proprioceptive measurements) from time 0 to time t
8/106
Localization as an estimation problem
Notation
Belief of the robot at time t: probability density function (pdf) describing the information the robot has regarding its pose at time t, based on all available data (exteroceptive measurements and motion commands):
Prior belief of the robot at time t: pdf before acquiring the last measurement zt:
9/106
Localization as an estimation problem
Notation
The robot motion model is the pdf of the robot pose at time t+1 given the robot pose and the motion action at time t. It takes into account the noise characterizing the proprioceptive sensors:
The measurement model describes the probability of observing at time t a given measurement zt when the robot pose is xr,t. It takes into account the noise characterizing the exteroceptive sensors:
10/106
Localization as an estimation problem
The Bayes Filter
Based on the total probability theorem:
where Bi, i=1,2,... is a partition of . In the continuous case:
(discrete case)
Prediction
Motion modelRobot pose space
11/106
Illustration of the prediction step in an ideal case
Localization as an estimation problem
Assume a robot moving along a unidimensional space, i.e. xr = x, and assume an ideal motion model:
x
The Bayes Filter - Prediction
12/106
Localization as an estimation problem
The Bayes Filter - Prediction
Illustration of the prediction step in a realistic case
Still a unidimensional space but consider a more realistic motion model:
x
where the noise
13/106
Based on the Bayes Rule:
Correction or Measurement Update
Measurement modelNormalizing factor
Taking:
We have:
i.e. also:
Localization as an estimation problem
The Bayes Filter
14/106
Illustration of the correction step in an ideal case
Assume a robot moving along a unidimensional space, i.e. xr = x, and measuringthe distance to a landmark placed in the origin. Under an ideal measurement model:
x
zt = z = x
Localization as an estimation problem
The Bayes Filter - Correction
15/106x
zt = z = x+
Illustration of the correction step in a realistic caseStill a unidimensional space, still the robot is measuring the distance to a landmark placed in the origin. Under a realistic measurement model:
where the noise
Localization as an estimation problem
The Bayes Filter - Correction
16/106
Localization as an estimation problem
The Bayes Filter
Motion model
Measurement model
For all xr do the following:
It is a recursive algorithm. At time t, given the belief at time t-1 belt-1(xr), the last motion control ut-1 and the last measurement zt, determine thenew belief belt(xr) as follows:
17/106
Localization as an estimation problem
The Bayes Filter - Observations
1. The Bayes filter is not a practical algorithm
Gaussian Filters: Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF),Extended Information Filter (EIF)
Non Parametric Filters: Histogram Filter (HF), Particle Filter (PF)
2. Important to check if we have enough information to localize the robot (observability issue).
Sometimes the observability can be evaluated through the observability rank condition (see Hermann&Krener, TAC 1977)
18/106
Observability issueThe robot may have not enough sensors to determine its pose, even if assuming perfect (i.e. without noise) measurements.
x
yThe robot is equipped with encoder readings and can measure the distance r to a landmark located in the origin.
t1t2
r1
r2
d12
d23
t3r3
2
ri = range measurement at time ti
dij = distance covered (and measured) by the robot in time interval [ti,tj]
2 = robot turn at time t2 (known
from odometry)
In the figure we represent two different robot paths agreeing with measurements. The robot pose is not completely observable: there is a radial symmetry around the origin
19/106
Observability issueCase the robot measures (with noise) the distance from two landmarks (in addition to the odometry). The robot pose is not completely observable.
True path
Path reconstructedthrough the EKF(position tracking)
The noise makes the estimated pose jump to the specular path.
20/106
Observability issue
In the nonlinear case, the observability depends on the particular control input considered.
Hence, the possibility of localizing a mobile robot may depend on the particular motion executed which in any case affects the localization performance Active Localization
x
t1
t2d12
d23
t3
y
The robot is equipped withodometry and measures theangle (a non symmetric landmark is located in the origin).
dij = distance covered (and measured) by the robot in time interval [ti,tj]
Under this motion (radial) thedistance of the robot from the origin(hence x and y) cannot be determinedbut only the orientation
21/106
Observability issue
A different kind of motion would make the pose observable (active localization):
x
t1t2d12
d23
t3
y
The robot is equipped withodometry and measures theangle (a non symmetric landmark is located in the origin).
Under this motion all the pose of the robot can be determined
dij = distance covered (and measured) by the robot in time interval [ti,tj]
22/106
Often localization is performed under a generic motion (passive localization): the robot tries to localize itself while performing its mission (once a satisfactory pose estimation has been reached) or under a random motion away from obstacles (at the beginning, when the pose is still completely unknown).
Observability issue
23/106
This is because usually mobile robots are equipped with several sensors: they can often localize themselves even if not moving (e.g. if measuring the distance to at least 3 non collinear landmarks).
Observability issue
Nevertheless...problems may arise with symmetrical environments (the robot has odometry + rangefinder, e.g. sonar or laser).
A proper robot motion may disambiguate in some cases the pose estimation.
24/106
Robot description
• Motion model (proprioceptive sensors)
• Measurement model (exteroceptive sensors)
From Thrun Burgard Fox, Probabilistic Robotics, MIT Press 2006
25/106
Motion model (proprioceptive sensors)
Assume:
• xr,t robot pose at time t • ut control input (i.e. motion command) at time t
Which is the pose at time t+1?
Due to sensor noise, xr,t+1 is available only through a probability density function.
Motion model
Usually, ut is obtained through proprioceptive sensors (e.g. encoders, gyroscopes, etc) Odometry motion model
From Thrun Burgard Fox, Probabilistic Robotics, MIT Press 2006
26/106
Bump (casual source of noise)
Ideal case Different wheeldiameters (systematic source of noise)
Carpet (casual source of noise)
and many more (e.g. distance between the wheels) …
From Thrun Burgard Fox, Probabilistic Robotics, MIT Press 2006
Possible sources of noiseMotion model
27/106
Accumulation of the pose estimation error under the robot motion(only proprioceptive measurements)
Motion model
From Thrun Burgard Fox, Probabilistic Robotics, MIT Press 2006
28/106
Motion model
Discretized unicycle model:
Robot shift in time interval [t,t+1]
Robot rotation in time interval [t,t+1]
xr,t
t
t
xr,t+1
x
y
29/106
Discretized unicycle model: the differential drive case
Motion model
Minibot
30/106
Motion model
uR,t = distance covered by the Right wheel in [t,t+1]uL,t = distance covered by the Left wheel in [t,t+1]d = distance between the wheels
If the sampling time is small enough:
Differential drive
xr,t
t
t
xr,t+1
x
y
t
D
d
31/106
Motion model
Differential drive
There is a bijective relation between () and (uR, uL):
Mismatch between real (uR,t, uL,t) and measured (uR,te, uL,t
e) distances covered by the wheels of the robot. The encoder readings are relatedto the real distances covered by the two wheels as follows:
(casual source of noise)
32/106
Motion model
Differential drive
Estimation (red points) of the path covered by a differential drive robot. Blue = real position in each step.
Starting point
33/106
Motion model
Differential drive
Given:
• xr,t = [x,y,^T robot pose at time t,
• encoder reading ut = [uRe, uL
e]^T (i.e.result of the control input) relative to the interval [t, t+1],
we want to estimate the probability that the robot pose at time t+1 will bea given xr,t+1 = [x’,y’,’]^T (under the differential drive model considered), i.e.:
34/106
We proceed as follows:
Actual robot displacement if we want to arrive at [x’,y’,’]^T from [x,y,]^T:
Corresponding distances covered by the two wheels, while the actual encoder readings are (uR
e, uLe):
The probability of arriving in [x’,y’,’]^T from [x,y,]^T is the probability of reading (uR
e, uLe) when the actual wheel shifts are (uR, uL)
Motion model
Differential drive
35/106
Measurement model (exteroceptive sensors)
Assume:
• xr,t robot pose at time t
Which is the reading zt of a sensor at time t?
Due to sensor noise, zt is available only through a probability density function.
36/106
Two typical settings are considered:
Range Finder (e.g. laser or sonar readings).
The environment is known and isrepresented by line segments (e.g. the walls). The robot senses the distance from them at some given directions.
Landmark measurements.
The environment is characterized by features (artificial or natural landmarks), whose position is known and whose identity can be known (e.g. RFID tags) or must be estimated (data association problem). The robot measures the distance and/or the bearing to these fixed points.
Measurement model (exteroceptive sensors)
37/106
Measurement model: the Range Finder case
Measurement given by beam i
Distance of sensor i from the reflecting obstacle (ideal reading of the sensor)
Each measurement zi,t is a range measurementtipically based on the time of flight of the beam (e.g. of a sonar or a laser) generated by sensor i and reflected by an obstacle
38/106
Measurement model: the Range Finder case
Given:
• xr,t = [x,y,^T robot pose at time t,
we want to estimate the probability that the measurement of sensor i (with orientation i with respect to the robot motion direction) at time t is a given zt.
x
y
i
yp
xp
P
• First compute the distance hi(xrt) between sensor i and the reflecting point P (this is possible given the robot pose since the environment is assumed known)
• Then:
39/106
Measurement model: landmark measurements
x
y
i,t
yi
xi
Li
r i,t
Assuming known the identities (and the position) of the landmarks, for each landmark Li = (xi,yi) there is a measurement
where
Let xr,t = [x,y,]^T be the robot pose at time t
and
40/106
Robot Localization Algorithms
The application of the Bayes filter is referred to as Markov Localization:
Motion model
Measurement model
For all xr do the following:
… in the case of Global Localization
41/106L
Robot Localization Algorithms
Example of Markov Localization (here xr = x)
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
z0: door detected
u0: the robot moves
z1: door detected again
u1: the robot moves again
42/106
The Bayes filter is not a practical algorithm
Gaussian Filters: Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF),Extended Information Filter (EIF)
Non Parametric Filters: Histogram Filter (HF), Particle Filter (PF)
Robot Localization Algorithms
43/106
Gaussian filters (EKF, UKF)Robot Localization Algorithms
Tractable implementations of the Bayes Filter obtained by means of a Gaussian approximation of the pdf
Unimodal approximations completely defined through mean m and covariance P
OK for Position Tracking or in the global case if measurements allow to restrict the estimation to a small region (e.g. RFID)
The robot detects a corner and sees onlyfree corridors Bimodal pdf.A bimodal pdf is not well approximatedthrough a Gaussian density Multi-Hypothesis Gaussian (also in the case of unknown data association)
44/106
Robot Localization Algorithms
GaussianExtended Kalman Filter (EKF)
In the linear-gaussian case, all steps of the Bayes filter result in gaussian pdf the Kalman Filter (KF) is optimal for these systems.
EKF: extension of KF to non linear systems (no more optimal)
Linear-gaussian system:Process noise
Measurement noise
Assume:
45/106
Prediction step (Bayes filter)
Robot Localization Algorithms
Gaussian - EKF
... is the convolution of two Gaussian distributions still Gaussian!
The Kalman Filter results in the update of mean m and covariance P:
where:
Prediction Step KF
46/106
Correction step (Bayes Filter)
Robot Localization Algorithms
Gaussian - EKF
... is the product of two Gaussian distributions still Gaussian!
The Kalman Filter results in the update of mean m and covariance P:
where:
Kalman Gain
CorrectionStep KF
innovation
47/106
Extension to non linear systems (EKF)
Robot Localization Algorithms
Gaussian - EKF
Non linear - Gaussian system:Process noise
Measurement noise
Assume:
48/106
Prediction Step EKF
Robot Localization Algorithms
Gaussian - EKFPrediction Step KF
Correction Step EKF
Correction Step KF
49/106
Robot Localization Algorithms
Gaussian - EKFApplication to the localization problem:
50/106
Robot Localization Algorithms
GaussianUnscented Kalman Filter (UKF)
Specific for non linear systems: avoids jacobians and derives the pdf ofa random variable transformed through a non linear function through thetransformation of a set of Sigma Points. Same numerical complexity of EKF. Generally more performant, but still not optimal!
Let
a scalar gaussian random variable. Let
be a non linear function. Clearly y is not gaussian. If using Gaussian filters, y is approximated through a Gaussian random variable, i.e.:
51/106
Robot Localization Algorithms
The EKF:
The UKF:
1. Generate a set of Sigma Points i and corresponding weights wi.2. Compute the transformation of each Sigma Point:
3. Determine mean and covariance of the transformed variable y by:
Gaussian - UKF
52/106
Robot Localization AlgorithmsLinear transformation of a Gaussian variable
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
Gaussian
53/106
Robot Localization Algorithms
Non Linear transformation of a Gaussian variable
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
Gaussian
54/106
EKF approximationRobot Localization Algorithms
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
Gaussian
55/106
UKF approximationRobot Localization Algorithms
Gaussian
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
56/106
EKF vs UKF approximationRobot Localization Algorithms
UKF
EKF
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
Gaussian
57/106
EKF vs UKF approximation (2)Robot Localization Algorithms
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
Gaussian
58/106
UKF
EKF
Robot Localization Algorithms
EKF vs UKF approximation (2)
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
Gaussian
59/106
Robot Localization Algorithms
EKF vs UKF approximation (3)
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
Gaussian
60/106
Prediction quality of EKF vs UKF
EKF UKF
From Thrun Burgard Fox, Probabilistic Robotics, MIT Press 2006
Robot Localization Algorithms
Gaussian
61/106
Non parametric filters (HF, PF)
• Numerical implementation of the Bayes Filter
• Suitable also for global localization problems
• Precision dependent on the available computational time
Robot Localization Algorithms
62/106
Histogram filter: grid representation of the pdf The environment is partitioned into a set of cells and the pdf is numerically represented by the probability of each cell
Particle filter: sample (particles) representation of the pdf The pdf is represented by a set of particles: the larger the pdf in a given region, more particles in that region
Non parametric filters
Robot Localization Algorithms
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
63/106
Histogram filterNon parametric filters - HF
Robot Localization Algorithms
Grid representation of the belief:
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
64/106
Prediction step (Bayes filter)
Prediction Step HFFor all discretized ckD do:
Non parametric filters - HF
Robot Localization Algorithms
For all xr do the following:
65/106
Non parametric filters - HF
Robot Localization Algorithms
Correction step (Bayes filter)
Correction Step HFFor all discretized ckD do:
For all xr do the following:
66/106
Non parametric filters - HF
Robot Localization Algorithms
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
67/106
Particle filter Non parametric filters
Robot Localization Algorithms
Particle representation of the belief:
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
68/106
Prediction step (Bayes filter)
Prediction Step PF
Non parametric filters - PF
Robot Localization Algorithms
For all m = 1,2,...,M sample the new position of particle m accordingto the motion model:
Motion noise for particle m
(weights wtm unchanged)
For all xr do the following:
69/106
Motion model
Non parametric filters - PF
Robot Localization Algorithms
70/106
Non parametric filters - PF
Robot Localization Algorithms
Correction step (Bayes filter)
Correction Step PF
For all m = 1,2,...,M update particle weights according to the lastmeasurement (particle poses remain unchanged in this step):
For all xr do the following:
71/106
Non parametric filters - PF
Robot Localization Algorithms
Resampling (performed periodically, possibly at all steps,to regenerate the set of particles):
For M times:
1. Draw particle i with probability
2. Add the particle to the set of particles of time t+1
3. The weights are re-initialized:
72/106
Belief at time 0: Non parametric filters - PF
Robot Localization Algorithms
From Thrun Burgard Fox, Probabilistic Robotics, MIT Press 2006
Global localization problem:
• Particles spread uniformly in the environment and
•
73/106
Sensor Information: Importance SamplingNon parametric filters - PF
Robot Localization Algorithms
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
74/106
Resampling
Non parametric filters - PF
Robot Localization Algorithms
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
75/106
Robot motionNon parametric filters - PF
Robot Localization Algorithms
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
76/106
Sensor Information: Importance SamplingNon parametric filters - PF
Robot Localization Algorithms
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
77/106
Resampling + Robot MotionNon parametric filters - PF
Robot Localization Algorithms
Fro
m T
hru
n B
urg
ard
Fo
x, P
roba
bili
stic
Ro
bot
ics,
MIT
Pre
ss 2
00
6
78/106
ExamplesThe presentation ends with a simulative comparison of the approachesdescribed so far under different scenarios.
The environment considered is shown in the figure, together with the robot path (blue-dashed). Initial robot pose
It will be reported a comparisonbetween the estimation obtainedthrough Gaussian approaches (EKF, UKF) and the one obtained through non-parametric approaches (PF), in the case of:• global localization problems• or position tracking problems;considering a robot endowed with odometry and using, as exteroceptive measurements, either • sonar/laser readings (range finder)• range to a set of landmarks.
79/106
Examples
Simulation data
Motion model (in the simulation we consider a synchro drive system):
Measurement model (range measurements):
with:
with:
80/106
ExamplesSimulation Data
Standard deviation for the estimate of the initial pose in the position trackingcase: 0.01 m for x and y and 0.02 rad for
We assume that the exteroceptive meaurements (laser or landmark readings)are available every Ns steps, with Ns ≥ 1, while odometry is available at all steps
81/106
Examples
To evaluate the performance we consider an average position estimation error:
real position of the robot at time t
estimated position of the robot at time t
82/106
Odometry estimation = integration of the encoder readings:
while the real robot pose is given by:
Examples
83/106
Examples
Odometry estimation:
J = 4.9 m
84/106
Examples
Range finder case: 20 beams equally spaced in the robot motion direction
85/106
What to do if a measurement is very different from the expected one?It is possible that this does not depend on a bad pose estimation....
Examples
Actual reading
Expected reading
Select a threshold and reject the measurements which give an innovation larger than this threshold(this results in a reduction of the number of available measurements but makes the filter more robust).
86/106
... consider the "second" nearest intersection...
OK
NO
Examples
What to do if the estimate (in the EKF case) or a Sigma Point (in the UKF case) falls outside the map?
87/106
Examples
EKF estimation, Ns = 10:
Ns = number of steps between two consecutive exteroceptive measurements, while odometry is assumed available at all 5000 steps
J = 0.15 m
88/106
Examples
EKF estimation, Ns = 200:
J = 0.87 m
89/106
Examples
UKF estimation, Ns = 200:
J = 0.45 m
90/106
Examples
PF estimation, Mp = 100 particles, Ns = 200:
J = 0.24 m
91/106
ExamplesA comparison between EKF, UKF and PF is reported below for the range finder case (position tracking), after removing some outliers (Jodo=7.8m):
Zoom next page
92/106
Examples
93/106
Examples
The percentage of outliers (over 100 independent simulation runs, 10 for PF1000). Jk is considered an outlier if:
94/106
Examples
Computational times (for Ns=1):
• EKF: 3 s• UKF: 15 s• PF, 100 particles: 127 s• PF, 1000 particles: 1198 s
0
200
400
600
800
1000
1200
EKF PF_100
Time (s)
UKF PF_1000
95/106
The EKF can not be applied to global localization problems (Ns = 10):Examples
96/106
Examples
Similarly the UKF can not be applied to global localization problems (Ns = 10):
97/106
Examples
The EKF may work in the global case for suitable initial conditions of the estimation (left: the estimation of the initial orientation allows to recover the correct trajectory). Here Ns = 100.
98/106
Examples
The PF can be applied to global localization problems if a sufficient number of particles is adopted (here Mp = 1000 and Ns = 10):
99/106
Examples
In the landmark case the EKF and the UKF provide identical perfomances
A comparison between EKF and UKF is reported below for the landmark case (position tracking):
Landmark (range) case:
100/106
Examples
A comparison between PF100 and PF1000 is reported below for the landmark case (position tracking):
101/106
Examples
All the filters show similar performances…
102/106
A comparison between the range finder and the landmark case (position tracking):
• in the landmark case all filters provide similar performance;
• in the landmark case the performance is higher
Examples
Landmark case
Range finder case
103/106
The percentage of outliers in the landmark case (over 100 independent simulation runs, 10 for PF1000) is 0 for EKF, UKF and PF1000. There is only 1% of outliers for PF100.
In the landmark case the EKF and the UKF can also be applied to solve a global localization problem, unlike in the laser reading case (see also videos).
Examples
EKF estimation, Ns = 100
104/106
UKF estimation (global), Ns = 10 (left) and 100 (right)
Examples
105/106
Conclusions
• The localization of a mobile robot in a known environment can be cast as a filtering problem.
• The robot must be equipped with enough sensors to guarantee observability. Symmetric environment can also be problematic.
• The theoretical solution of the problem is the Bayes filter which usually cannot be implemented.
• Gaussian or non parametric implementations should be considered according to the considered problem (global vs position tracking, computational time).
• Active localization allows to improve localization capabilities of the robot.
• The basic problem can be extended to: dynamic environments, multi-robot, partially known environments, etc.
106/106
References
1. S. Thrun, W. Burgard and D. Fox. ''Probablistic Robotics''. MIT Press, 2006 (and references therein)
Robot motion models (Ch. 5), measurement models (Ch. 6), Localization(Ch. 7-8), Filtering (Ch. 2-4), Active localization (Ch. 17).
2. Observability issues:
• R. Herman and A. Krener. ''Nonlinear controllability and observability''. IEEE Transactions on Automatic Control, n. 22(5), p. 728-740, 1977
• A. Martinelli and R. Siegwart. ''Observability analysis for mobile robot localization''. Intelligent Robots and Systems, 2005. (IROS 2005), p. 1471 - 1476. Edmonton, Alberta, Canada. August 2005
3. A. Censi. ''On achievable accuracy for range-finder localization''. 2007 IEEE International Conference on Robotics and Automation, (ICRA 2007), p. 4170 - 4175. Roma, Italy. April 2007