Post on 17-Dec-2015
4-1
Probabilistic Robotics: Motion and Sensing
Slide credits: Wolfram Burgard, Dieter Fox, Cyrill Stachniss, Giorgio Grisetti, Maren Bennewitz, Christian Plagemann, Dirk Haehnel, Mike Montemerlo, Nick Roy, Kai Arras, Patrick Pfaff and others
Sebastian Thrun & Alex Teichman
Stanford Artificial Intelligence Lab
4-2
Bayes Filters
111 )(),|()|()( tttttttt dxxBelxuxPxzPxBel
4-3
111 )(),|()|()( tttttttt dxxBelxuxPxzPxBel
ActionSensing
Bayes Filters
4-4
Probabilistic Motion Models
4-5
111 )(),|()|()( tttttttt dxxBelxuxPxzPxBel
Action: p(x’ | u, x)
Bayes Filters
4-6
Robot Motion
Robot motion is inherently uncertain.
How can we model this uncertainty?
4-7
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.
4-8
Locomotion of Wheeled Robots
Locomotion (Oxford Dict.): Power of motion from place to place
• Differential drive (AmigoBot, Pioneer 2-DX)• Car drive (Ackerman steering)• Synchronous drive (B21)• Mecanum wheels, XR4000
y
roll
z motion
x
y
4-9
Instantaneous Center of Curvature
ICC
For rolling motion to occur, each wheel has to move along its y-axis
4-10
Differential Drive
R
ICCw
(x,y)
y
l/2
qx
vl
vr
l
vv
vv
vvlR
vlR
vlR
lr
lr
rl
l
r
)(
)(
2
)2/(
)2/(
]cos,sin[ICC RyRx
4-11
Differential Drive: Forward Kinematics
ICC
R
P(t)
P(t+dt)
t
y
x
tt
tt
y
x
y
x
y
x
ICC
ICC
ICC
ICC
100
0)cos()sin(
0)sin()cos(
'
'
'
')'()(
')]'(sin[)'()(
')]'(cos[)'()(
0
0
0
t
t
t
dttt
dtttvty
dtttvtx
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Differential Drive: Forward Kinematics
ICC
R
P(t)
P(t+dt)
t
y
x
tt
tt
y
x
y
x
y
x
ICC
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ICC
100
0)cos()sin(
0)sin()cos(
'
'
'
')]'()'([1
)(
')]'(sin[)]'()'([2
1)(
')]'(cos[)]'()'([2
1)(
0
0
0
t
lr
t
lr
t
lr
dttvtvl
t
dtttvtvty
dtttvtvtx
4-13
Mecanum Wheels
4
4
4
4
3210
3210
3210
3210
/)vvvv(v
/)vvvv(v
/)vvvv(v
/)vvvv(v
error
x
y
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Example
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XR4000 Drive
q
y
x
vi(t)
wi(t)
')'()(
')]'(sin[)'()(
')]'(cos[)'()(
0
0
0
t
t
t
dttt
dtttvty
dtttvtx
ICC
4-16
XR4000
[courtesy by Oliver Brock & Oussama Khatib]
4-17
Tracked Vehicle: Urban Robot
4-18
Tracked Vehicle: OmniTread
[courtesy by Johann Borenstein]
4-19
Odometry
4-20
Reasons for Motion Errors
bump
ideal case different wheeldiameters
carpetand many more …
4-21
Coordinate Systems
• 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,).
4-22
Typical Motion Models
• In practice, one often finds two types of motion models:• 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.
4-23
Example Wheel EncodersThese 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.
Source: http://www.active-robots.com/
4-24
Dead Reckoning
• Derived from “deduced reckoning.”• Mathematical procedure for
determining the present location of a vehicle.
• Achieved by calculating the current pose of the vehicle based on its velocities and the time elapsed.
Odometry Model
22 )'()'( yyxxtrans
)','(atan21 xxyyrot
12 ' rotrot
Robot moves from to .
Odometry information .
,, yx ',',' yx
transrotrotu ,, 21
trans1rot
2rot
,, yx
',',' yx
4-26
The atan2 Function
Extends the inverse tangent and correctly copes with the signs of x and y.
Noise Model for Odometry
• The measured motion is given by the true motion corrupted with noise.
||||11 211
ˆtransrotrotrot
||||22 221
ˆtransrotrotrot
|||| 2143
ˆrotrottranstranstrans
Typical Distributions for Probabilistic Motion Models
2
2
22
1
22
1)(
x
ex
2
2
2
6
||6
6|x|if0)(2
xx
Normal distribution Triangular distribution
4-29
Calculating the Probability (zero-centered)
• For a normal distribution
• For a triangular distribution
1. Algorithm prob_normal_distribution(a,b):
2. return
1. Algorithm prob_triangular_distribution(a,b):
2. return
4-30
Calculating the Posterior Given x, x’, and u
22 )'()'( yyxxtrans )','(atan21 xxyyrot
12 ' rotrot 22 )'()'(ˆ yyxxtrans )','(atan2ˆ
1 xxyyrot
12ˆ'ˆrotrot
)ˆ|ˆ|,ˆ(prob trans21rot11rot1rot1 p|))ˆ||ˆ(|ˆ,ˆ(prob rot2rot14trans3transtrans2 p
)ˆ|ˆ|,ˆ(prob trans22rot12rot2rot3 p
1. Algorithm motion_model_odometry(x,x’,u)
2.
3.
4.
5.
6.
7.
8.
9.
10.
11. return p1 · p2 · p3
odometry values (u)
values of interest (x,x’)
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Examples
Application
• Repeated application of the sensor model for short movements.
• Typical banana-shaped distributions obtained for 2d-projection of 3d posterior.
x’u
p(x|u,x’)
u
x’
Sample-based Density Representation
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Sample-based Density Representation
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How to Sample from Normal or Triangular Distributions?
• Sampling from a normal distribution
• Sampling from a triangular distribution
1. Algorithm sample_normal_distribution(b):
2. return
1. Algorithm sample_triangular_distribution(b):
2. return
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Normally Distributed Samples
106 samples
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For Triangular Distribution
103 samples 104 samples
106 samples105 samples
Sample Odometry Motion Model1. Algorithm sample_motion_model(u, x):
2.
3.
4.
5.
6.
7.
8. Return
)||sample(ˆ21111 transrotrotrot
|))||(|sample(ˆ2143 rotrottranstranstrans
)||sample(ˆ22122 transrotrotrot
)ˆcos(ˆ' 1rottransxx )ˆsin(ˆ' 1rottransyy
21ˆˆ' rotrot
',',' yx
,,,,, 21 yxxu transrotrot
sample_normal_distribution
Sampling from Our Motion Model
Start
Examples
Map-Consistent Motion Model
)',|( xuxp ),',|( mxuxp
)',|()|(),',|( xuxpmxpmxuxp Approximation:
4-42
Summary
• We discussed motion models for odometry-based motion (see book for velocity-based systems)
• We discussed ways to calculate the posterior probability p(x| x’, u).
• We also described how to sample from p(x| x’, u).• Typically the calculations are done in fixed time
intervals t.• In practice, the parameters of the models have to
be learned.• We also discussed an extended motion model that
takes the map into account.
4-43
Probabilistic Sensor Models
4-44
111 )(),|()|()( tttttttt dxxBelxuxPxzPxBel
Sensing: p(z | x)
Bayes Filters
4-45
Sensors for Mobile Robots
• Contact sensors: Bumpers, Whiskers
• Internal sensors• Accelerometers (spring-mounted masses)• Gyroscopes (spinning mass, laser light)• Compasses, inclinometers (earth magnetic field, gravity)
• Proximity sensors• Sonar (time of flight)• Radar (phase and frequency)• Laser range-finders (triangulation, tof, phase)• Infrared (intensity)• Stereo
• Visual sensors: Cameras, Stereo
• Satellite-based sensors: GPS
4-46
Ultrasound Sensors
• Emit an ultrasound signal• Wait until they receive the echo• Time of flight sensor
Polaroyd 6500
4-47
Time of Flight sensors
v: speed of the signalt: time elapsed between broadcast of signal
and reception of the echo.
2/tvd
emitter
object
4-48
Properties of Ultrasounds
• Signal profile [Polaroid]
4-49
Sources of Error
• Opening angle• Crosstalk• Specular reflection
4-50
Typical Ultrasound Scan
4-51
Laser Range Scanner
4-52
Properties
• High precision• Wide field of view • Approved security for collision
detection
4-53
Robots Equipped with Laser Scanners
Herbert:Zora: Groundhog:
4-54
Typical Scans
4-55
Proximity Sensors
• The central task is to determine P(z|x), i.e., the probability of a measurement z given that the robot is at position x.
• Question: Where do the probabilities come from?• Approach: Let’s try to explain a measurement.
4-56
Beam-based Sensor Model
• Scan z consists of K measurements.
• Individual measurements are independent given the robot position.
},...,,{ 21 Kzzzz
K
kk mxzPmxzP
1
),|(),|(
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Beam-based Sensor Model
K
kk mxzPmxzP
1
),|(),|(
4-58
Typical Measurement Errors of an Range Measurements
1. Beams reflected by obstacles
2. Beams reflected by persons / caused by crosstalk
3. Random measurements
4. Maximum range measurements
4-59
Proximity Measurement
• Measurement can be caused by …• a known obstacle.• cross-talk.• an unexpected obstacle (people, furniture, …).• missing all obstacles (total reflection, glass, …).
• Noise is due to uncertainty …• in measuring distance to known obstacle.• in position of known obstacles.• in position of additional obstacles.• whether obstacle is missed.
4-60
Beam-based Proximity Model
Measurement noise
zexp zmax0
b
zz
hit eb
mxzP
2exp )(
2
1
2
1),|(
otherwise
zzmxzP
z
0
e),|( exp
unexp
Unexpected obstacles
zexp zmax0
4-61
Beam-based Proximity Model
Random measurement Max range
max
1),|(
zmxzPrand
smallzmxzP
1),|(max
zexp zmax0zexp zmax
0
4-62
Resulting Mixture Density
),|(
),|(
),|(
),|(
),|(
rand
max
unexp
hit
rand
max
unexp
hit
mxzP
mxzP
mxzP
mxzP
mxzP
T
How can we determine the model parameters?
4-63
Raw Sensor DataMeasured distances for expected distance of 300 cm.
Sonar Laser
4-64
Approximation
• Maximize log likelihood of the data
• Search space of n-1 parameters.• Hill climbing• Gradient descent• Genetic algorithms• …
• Deterministically compute the n-th parameter to satisfy normalization constraint.
)|( expzzP
4-65
Approximation Results
Sonar
Laser
300cm 400cm
4-66
Example
z P(z|x,m)
4-67
Discrete Model of Proximity Sensors
• Instead of densities, consider discrete steps along the sensor beam.
• Consider dependencies between different cases.
Laser sensor Sonar sensor
4-68
Approximation Results
Laser
Sonar
4-69
"sonar-0"
0 10 20 30 40 50 60 70 010
2030
4050
6070
00.05
0.10.15
0.20.25
Influence of Angle to Obstacle
4-70
"sonar-1"
0 10 20 30 40 50 60 70 010
2030
4050
6070
00.05
0.10.15
0.20.25
0.3
Influence of Angle to Obstacle
4-71
"sonar-2"
0 10 20 30 40 50 60 70 010
2030
4050
6070
00.05
0.10.15
0.20.25
0.3
Influence of Angle to Obstacle
4-72
"sonar-3"
0 10 20 30 40 50 60 70 010
2030
4050
6070
00.05
0.10.15
0.20.25
Influence of Angle to Obstacle
4-73
Summary Beam-based Model
• Assumes independence between beams.• Justification?• Overconfident!
• Models physical causes for measurements.• Mixture of densities for these causes.• Assumes independence between causes. Problem?
• Implementation• Learn parameters based on real data.• Different models should be learned for different angles at
which the sensor beam hits the obstacle.• Determine expected distances by ray-tracing.• Expected distances can be pre-processed.
4-74
Scan-based Model
• Beam-based model is …• not smooth for small obstacles and at
edges.• not very efficient.
• Idea: Instead of following along the beam, just check the end point.
4-75
Scan-based Model
• Probability is a mixture of …• a Gaussian distribution with mean at
distance to closest obstacle,• a uniform distribution for random
measurements, and • a small uniform distribution for max
range measurements.• Again, independence between
different components is assumed.
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Example
P(z|x,m)
Map m
Likelihood field
4-77
San Jose Tech Museum
Occupancy grid map Likelihood field
4-78
Scan Matching
• Extract likelihood field from scan and use it to match different scan.
4-79
Scan Matching
• Extract likelihood field from first scan and use it to match second scan.
~0.01 sec
4-80
Properties of Scan-based Model
• Highly efficient, uses 2D tables only.• Smooth w.r.t. to small changes in robot
position.
• Allows gradient descent, scan matching.
• Ignores physical properties of beams.
• Will it work for ultrasound sensors?
4-81
Landmarks
• Active beacons (e.g., radio, GPS)• Passive (e.g., visual, retro-reflective)• Standard approach is triangulation
• Sensor provides• distance, or• bearing, or• distance and bearing.
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Distance and Bearing
4-83
Probabilistic Model
1. Algorithm landmark_detection_model(z,x,m):
2.
3.
4.
5. Return
22 ))(())((ˆ yimximd yx
),ˆprob(),ˆprob(det dddp
,,,,, yxxdiz
))(,)(atan2(ˆ ximyima xy
),|(uniformfpdetdet mxzPzpz
4-84
Distributions
4-85
Distances OnlyNo Uncertainty
P1 P2
d1 d2
x
X’
a
)(
2/)(22
1
22
21
2
xdy
addax
P1=(0,0)
P2=(a,0)
4-86
P1
P2
D1
z1
z2a
P3
D2
bz3
D3
Bearings OnlyNo Uncertainty
P1
P2
D1
z1
z2
a
a
cos2 2122
21
21 zzzzD
)cos(2
)cos(2
)cos(2
2123
21
23
2123
22
22
2122
21
21
zzzzD
zzzzD
zzzzDLaw of cosine
4-87
Bearings Only With Uncertainty
P1
P2
P3
P1
P2
Most approaches attempt to find estimation mean.
4-88
Summary of Sensor Models
• Explicitly modeling uncertainty in sensing is key to robustness.
• In many cases, good models can be found by the following approach:
1. Determine parametric model of noise free measurement.2. Analyze sources of noise.3. Add adequate noise to parameters (eventually mix in densities
for noise).4. Learn (and verify) parameters by fitting model to data.5. Likelihood of measurement is given by “probabilistically
comparing” the actual with the expected measurement.• This holds for motion models as well.• It is extremely important to be aware of the underlying
assumptions!