Robotics Localization

SBE 403 B: Bioelectronic Systems (Biomedical Robotics)
Lecture 08
Localization
Muhammad Rushdi
mrushdi@eng1.cu.edu.eg

Main Problems in Robotics
• What are the basic issues to be resolved and what must we
learn in order to be able to program a robot to perform its
tasks?
• Problem 1: Forward Kinematics
• Problem 2: Inverse Kinematics
• Problem 3: Velocity Kinematics Robot localization: The• Problem 3: Velocity Kinematics
• Problem 4: Path Planning
• Problem 5: Vision
• Problem 6: Dynamics
• Problem 7: Position Control
• Problem 8: Force Control
2
process by which a robot
determines its current
position given its previous
position and sensor
information.

Textbook
• Ch. 6 Localization
Peter Corke:
Robotics, Vision andRobotics, Vision and
Control - Fundamental
Algorithms in MATLAB®.
http://www.petercorke.com/R
VC/
3

Classical Location Estimation
4

Location Estimation with Uncertainty
High UncertaintyLow Uncertainty
5
Multiple Hypotheses

• Determining the PDF based on knowledge of how the
vehicle moves and its observations of the world is a problem
in estimation which can be defined as:
“the process of inferring the value of some quantity of
interest, x, by processing data that is in some way
dependent on x.”dependent on x.”
• For our robot localization problem the true and estimated
states are vector quantities so uncertainty will be
represented as a covariance matrix. The diagonal elements
represent uncertainty of corresponding states, and the off-
diagonal elements represent correlations between states.
6

• Example: A ship’s navigator
estimates location by
measuring the bearing angles
to known landmarks or
celestial objects, and a GPS
receiver estimates latitudereceiver estimates latitude
and longitude by observing
the time delay from moving
satellites whose location is
known.
7
http://pubs.usgs.gov/ds/675/html/position_nav.html

Estimation Scenarios
1. Dead Reckoning (Find a robot location with
odometer information)
2. Using a Map (Find a robot location with
odometer information and a map)
3. Creating a Map (Given the robot location,3. Creating a Map (Given the robot location,
create a map)
4. Localization and Mapping (Find the robot
location and create a map simultaneously)
5. Monte-Carlo Methods (Multiple hypothses
and non-Gaussian distributions allowed) 8

1. Dead Reckoning
• Dead reckoning is the estimation of a robot’s location
based on its estimated speed, direction and time of travel
with respect to a previous estimate.
9

1. Dead Reckoning: Modeling the Vehicle
• The first step in estimating the robot’s position is to write a
function, f(·), that describes how the vehicle’s configuration
changes from one time step to the next.
• Eq. 4.2 describes evolution of the robot’s configuration as a
function of its control inputs,
however for real robots we rarely have access to these
control inputs.
• Most robotic platforms have proprietary motion control
systems that accept motion commands from the user
(speed and direction) and report odometry information.
10

1. Dead Reckoning: Odometry
• An odometer is a sensor that measures distance travelled,
typically by measuring the angular rotation of the wheels.
• The direction of travel can be measured using an electronic
compass, or the change in heading can be measured using
a gyroscope or differential odometry.
11
http://people.tamu.edu/~animodor/research.html

1. Dead Reckoning: Odometry
• These sensors are imperfect due to systematic errors such
an incorrect wheel radius or gyroscope bias, and random
errors such as slip between wheels and the ground, or the
effect of turning. We consider odometry to comprise both
distance and heading information.
12http://act.rasip.fer.hr/groups_amor_odometry.php

1. Dead Reckoning: A discrete-time model
• Instead of using Eq. 4.2 directly we will write a discrete-time
model for the evolution of configuration based on odometry
where δ<k>=(δd, δθ) is the distance travelled and change in
heading over the preceding interval, and k is the time step.
• The initial pose is represented in SE(2) as
13.

• We assume that motion over the time interval is small so the
order of applying the displacements is not significant. We
choose to move forward in the vehicle x-direction by δd, and
then rotate by δθ giving the new configuration
14.

• …or as a 3-vector:
which gives the new configuration in terms of the previous
configuration and the odometry.
• However this assumes that odometry is perfect, which is not
realistic. To model the error in odometry we add continuous
random variables vd and vθ to δd and δθ respectively.
15.

• The robot’s configuration at the next time step, including the
odometry error, is
which is the required function f(·)
where k is the time step, δ<k> is the odometry
measurement and v<k> the random measurement noise
over the preceding interval. 16.

• In the absence of any information to the contrary we model
the odometry noise as v =(vd, vθ) ∼ N(0, V), a zero-mean
Gaussian process with variance
• This matrix, the covariance matrix, is diagonal which means
that the errors in distance and heading are independent.
Choosing a value for V is not always easy but we can
conduct experiments or make some reasonable engineering
assumptions.
17.

1. Dead Reckoning: Estimating Pose
• The problem we face is how to best estimate our new pose
given the previous pose and noisy odometry.
• The mathematical tool that we will use is the Kalman filter
which provides the optimal estimate for a system in which
the noise is zero-mean and Gaussian.
• In this application the state of the Kalman filter is the• In this application the state of the Kalman filter is the
estimated configuration of the robot. The filter is a recursive
algorithm that updates, at each time step, the optimal
estimate of the unknown true configuration and the
uncertainty associated with that estimate. That is, it provides
the best estimate of where we are and how certain we are
about that.
18.

1. Dead Reckoning: Kalman Filtering
• The Kalman filter is formulated for linear systems but our
model of the vehicle’s motion Eq. 6.3 is non-linear. We
create a local linear approximation or linearization (with a
truncated Taylor series) of the function x^<k> by
with respect to the current state estimate x^<k>. The terms
Fx and Fv are Jacobians which are vector versions of
derivatives which are sometimes written as ∂f/∂x and ∂f/∂v
respectively. This approach to estimation of a non-linear
system is known as the extended Kalman filter (EKF).
19.

• The Jacobians are obtained by differentiating
Eq. 6.2 and evaluating them for v= 0
• giving
20

21

22

23

2. Using a Map: Observe Known Features
• We have seen how uncertainty in position grows without
bound using dead-reckoning alone.
• The solution is to bring in new information from
observations of known features (landmarks whose position
is known) in the world.is known) in the world.
• The robot is equipped with a sensor that provides
observations of the features with respect to the robot as
described by
where xv the vehicle state, xf is the known location of the
observed feature in the world frame and w is a random
variable that models errors in the sensor. 24

2. Using a Map: Laser Rangefinders
• To make this tangible we will consider a common type
of sensor that measures the range and bearing angle
to a landmark in the environment, for instance a radar
or a scanning-laser rangefinder.
25
https://msdn.microsoft.com/en-us/library/bb483042.aspx

2. Using a Map: Noisy Observations
• The sensor is mounted onboard the robot so the
observation of the ith feature xfi= (xi, yi) is
• where z =(r, β)T and r is the range, β the bearing
Something wrong?
• where z =(r, β)T and r is the range, β the bearing
angle, and the measurement noise is
• The diagonal covariance matrix indicates that range
and bearing errors are independent. 26

2. Using a Map: Noisy Observations
• Using Eq. 6.9 the robot can estimate the range and
bearing angle to the feature based on its own estimated
position and the known position of the feature from the
map.
• Any difference between the observation and the estimated
observation indicates an error in the robot’s positionobservation indicates an error in the robot’s position
estimate – it isn’t where it thought it was. This difference
is key to the operation of the Kalman filter. It is called the
innovation since it represents new information. The
Kalman filter uses the innovation to correct the state
estimate and update the uncertainty estimate P<k>.
27

2. Using a Map: Linearized Observation
Model
• We linearize the observation Eq. 6.8 and write
where the Jacobians are obtained by differentiating
Eq. 6.9 yieldingEq. 6.9 yielding
28

2. Using a Map: Kalman Update
• Now we use the innovation to update the predicted
state computed earlier using Eq. 6.6 and Eq. 6.7
Kalman gain matrix
which are the Kalman filter update equations.
These take the predicted values for the next time
step denoted k + 1|k and apply information from
time step k +1 to compute values denoted k +1|k
+1.
29
Subtraction from the covariance provides a means for
covariance to decrease which was not possible for the
dead-reckoning case of Eq. 6.7.

2. Using a Map: Kalman Update
• The innovation has been added to the
estimated state after multiplying by the
Kalman gain matrix K which is defined as
where W^ is the estimated covariance of the
sensor noise.
30

2. Using a Map: EKF Simulation
31

2. Using a Map: EKF Simulation
32

2. Using a Map: Flexibility of EKF
• This EKF framework allows data from many and varied
sensors to update the state which is why the estimation
problem is also referred to as sensor fusion.
• For example heading angle from a compass, yaw rate
from a gyroscope, target bearing angle from a camera,
position from GPS could all be used to update the state.position from GPS could all be used to update the state.
• For each sensor we need only to provide the observation
function h(·), the Jacobians Hx and Hw and some estimate
of the sensor output covariance W. The function h(·) can
be non-linear and even non-invertible – the EKF will do the
rest.
33

2. Using a Map:
Target Association Problem
• We assumed that the sensor provides information about the position
of the target with respect to the robot and also the identity of the
target. In practice most landmarks are anonymous, that is, we do not
know their identity and hence do not know their true location in the
world. We therefore need to solve the correspondence or target
association problem – given the map and the observation,
determine which feature is the most likely to have been seen. Errorsdetermine which feature is the most likely to have been seen. Errors
in this step can lead rapidly to failure of the estimator – the system
sees target i but thinks it is target j.
• An alternative is a multi-hypothesis estimator, such as the particle
filter which can model the possibility of observing landmark A or
landmark B, and future observations will reinforce one hypothesis
and weaken the others. The extended Kalman filter uses a Gaussian
probability model, with just one peak, which limits it to holding only a
single hypothesis about location. 34

3. Creating a Map
• Consider the problem of a robot moving in an
environment with landmarks and creating a map of
their locations.
• As before we have a range and bearing sensor
mounted on the robot which measures, imperfectly,
the position of features with respect to the robot.
There are a total of N features in the environment
and as for the previous example we assume that
the sensor can determine the identity of each
observed feature. However for this case we
assume that the robot knows its own location
perfectly – it has ideal localization. 35

3. Creating a Map: Estimating Landmark
Coordinates
• Since the vehicle pose is known perfectly we do not need
to estimate it, but we do need to estimate the coordinates
of the landmarks. For this problem the state vector
comprises the estimated coordinates of the M landmarks
that have been observed so far:
and has 2M elements. The corresponding estimated
covariance P^ will be a 2M× 2M matrix. The state vector
has a variable length since we do not know in advance
how many landmarks exist in the environment. Initially
M=0 and is incremented every time a previously unseen
feature is observed. 36

3. Creating a Map: EKF Simulation
37

4. Localization and Mapping
• Finally we tackle the problem of determining our
position and creating a map at the same time. This
is an old problem in marine navigation and
cartography – incrementally extending maps while
also using the map for navigation.
• In robotics this problem is known as simultaneous• In robotics this problem is known as simultaneous
localization and mapping (SLAM) or concurrent
mapping and localization (CML). This is
sometimes referred to as a “chicken and egg”
problem.
38

4. Localization and Mapping
• The state vector comprises the vehicle configuration and the
coordinates of the M landmarks that have been observed so far
and has 2M+ 3 elements. The covariance is a (2M+3) × (2M+and has 2M+ 3 elements. The covariance is a (2M+3) × (2M+
3) matrix and has the structure
where Pvv is the covariance of the vehicle state, Pmm the
covariance of the map features, and Pvm is the correlation
between vehicle and map states.
39

4. Localization and Mapping: EKF
Simulation
40

4. Localization and Mapping: EKF
Simulation
41

5. Monte-Carlo Estimation
• The estimation examples so far have assumed that the error in
sensors such odometry and landmark range and bearing have
a Gaussian probability density function. In practice we might
find that a sensor has a one sided distribution (like a Poisson
distribution) or a multimodal distribution with several peaks.
• The functions we used in the Kalman filter such as Eq. 6.3 and• The functions we used in the Kalman filter such as Eq. 6.3 and
Eq. 6.8 are strongly non-linear which means that sensor noise
with a Gaussian distribution will not result in a Gaussian error
distribution on the value of the function.
• The probability density function associated with a robot’s
configuration may have multiple peaks to reflect several
hypotheses that equally well explain the data from the sensors.
42

• The Monte-Carlo estimator makes no assumptions about
the distribution of errors. It can also handle multiple
hypotheses for the state of the system.
• The basic idea is simple. We maintain many different
versions of the vehicle’s state vector. When a new
measurement is available we score how well each versionmeasurement is available we score how well each version
of the state explains the data. We keep the best fitting
states and randomly perturb them to form a new
generation of states. Collectively these many possible
states and their scores approximate a probability density
function for the state we are trying to estimate.
43

• For the Monte-Carlo estimator, there is never any
assumption about Gaussian distributions nor any need
to linearize the system. While computationally
expensive it is quite feasible to use this technique withexpensive it is quite feasible to use this technique with
standard desktop computers.
• If we plot these state vectors as points we have a
cloud of particles hence this type of estimator is often
referred to as a particle filter.
44

• For the Monte-Carlo estimator, there is never any
assumption about Gaussian distributions nor any need
to linearize the system. While computationally
expensive it is quite feasible to use this technique withexpensive it is quite feasible to use this technique with
standard desktop computers.
• If we plot these state vectors as points we have a
cloud of particles hence this type of estimator is often
referred to as a particle filter.
45

5. Monte-Carlo Estimation: Simulation
46

5. Monte-Carlo Estimation: Simulation
47

Robotics Localization

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