IEEE Robotics & Automation Magazine - June 2014 - 29

extraction) uncertainty, spurious measurements, and spatial uncertainty. The ability to account for all of these errors
in a joint and principled manner has a huge impact on FBSLAM and provides the motivation for remodeling FBSLAM under a set-based framework.
Remodeling SLAM to Account for
Detection and Spatial Uncertainty
Feature Measurements Modeled as RFSs
In the visual, laser-, and radar-based measurement examples of Figure 2, it is clear that, in general, a measurement
consists of a random number of detections, each with spatial uncertainty. This implies that the result of any realistic
feature detection process is a measurement that is not a
vector. For example, for the point feature radar detections
in Figure 2(a), resulting in measured range and bearing values r and i, respectively, let the collected detections be
z 1 = [r 1 i 1] T , f, z z = [r z i z] T , which could be false or otherwise. A vector model for this measurement [z 1 f z z] T
has a fixed dimension z, but the number of detections can
vary from zero to some arbitrarily large number due to the
possibility of missed detections and multiple false alarms.
Another problem is that the components of the vector
z 1, f, z z have a fixed order, but the actual detections
z 1, f, z z have no inherent physical order. Therefore, a more
precise model of the measurement is a finite observation
set, which, by definition, has no fixed order and has elements comprising the individual detections
Z = {z 1, f, z z} = {[r 1

i

1 T

] , f, [r z

i

z T

] }.

(1)

Therefore, at any instant, a sensor can be considered to collect
a finite set Z = {z 1, g, z z} of measurements z 1, f, z z from a
measurement space Z 0 as follows:
Z=
4
(no features detected)
Z=
{z 1}
(one feature z 1 detected)
Z = {z 1, z 2} (two features z 1, z 2 detected)
h
h
h
Z = {z 1, f, z z} (z features z 1, f, z z detected) .

(2)

Since the number of feature detections in Z as well as the
values of the individual detections z i are random in nature,
Z is referred to as an RFS.
Vehicle State Modeled as a Random Vector
In SLAM formulations, the vehicle's current pose state is
typically modeled as a time-varying vector X k , containing
its 2-D x k, y k position and its orientation z k at time k.
The three-dimensional (3-D) vehicle states are also possible, containing the six-degrees-of-freedom state variables,
x k, y k, z k, as well as the vehicle's roll pitch and yaw angles
and possibly more states containing velocity, acceleration,
and higher-order variables. Irrespective of the complexity

of the chosen vehicle state, its dimensions are fixed as
time progresses, and the order of the variables in the vector remain the same, i.e., unlike the map feature estimates
and measurements, in single-robot SLAM, the state
related to the robot's position has no dimensional uncertainty. Therefore, the robot vehicle state is adequately
modeled as a random vector, which, in its general form, is
propagated forward in discrete time according to the state
transition equation
X k = f veh (X k -1, U k -1, v k -1),

(3)

where f veh ($) is the (generally nonlinear) state transition
function, a specific example of which will be demonstrated in
the marine application in (11), U k -1 are any deterministic
inputs applied to the vehicle at time k - 1, and v k -1 models
the assumed zero mean random noise modeling the uncertainty of the function f veh ($) .
Map State Space Modeled as an RFS
The number of features in the map state can vary from zero to
some arbitrarily large number. Ideally, it should grow monotonically as features enter the FoV of the sensor(s). This further justifies the need for a set-based map representation containing individual feature states as follows:
M=
4
(no features present)
M=
{m 1}
(one feature with state m 1 present)
M = {m 1, m 2}
(two features m 1 ! m 2 present)
h
h
M = {m 1, f, m m} (m features m 1 ! g ! m m present) .
Relating RFS Measurements to the SLAM State
To encapsulate detection uncertainty as well as spatial measurement noise, the detected features from a vehicle with
pose X k at time k can be mathematically modeled by an RFS
Z k . This is formed by the union of a set of features expected
to be generated under the current map estimate and a set of
false detections. Importantly, each set encapsulates the aforementioned detection and spatial uncertainties, and hence
Zk
=
7
All
Features

'

D k (m, X k) , C k (X k),
1 44 2 44 3
>
False
Expected
Features
Features

m ! Mk

(4)

where D k (m, X k) is the RFS of measurements generated by a
feature at location m and C k (X k) is the RFS of the spurious
measurements at time k, which may depend on the vehicle
z
pose X k . Z k = {z 1k, z 2k, f, z kk} consists of a random number,
z k of spatial measurements z ik , whose order of appearance
has no physical significance with respect to the estimated
map of features. For each feature, m ! M k and z ik ! Z k,
June 2014

*

IEEE ROBOTICS & AUTOMATION MAGAZINE

*

29



Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - June 2014

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https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_december2021
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https://www.nxtbook.com/nxtbooks/ieee/roboticsautomation_june2021
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