SLAM by Durrant 2006 Part 1

Notes on the Abstract

There appear to be several reasons why this paper is an especial industry favorite as it clearly opens with the following hooks:

targets small, mobile, embedded and low power systems of the time
and discusses implementation in a ==tutorial== format

Discourse

Introduction

Starting with a definition, the Simultaneous Localisation and Mapping (SLAM) problem/problem scenario (more on this later) requires and the checks the possibility of a "mobile robot" incrementally building a map of its vicinity while also determining its location in the course, when placed into an unknown and un-cached environment. It is of an intrinsic essence of said problem to have very influential solution in the field of robotics, should they be found. Needless to say, this is a fundamental problem of robotic motion.

The Problem

Defining variables

In the image above, the color filled icons represent what the robot perceives using its sensors or LiDAR instruments, while the icons without a fill represent true vectors and landmarks. Directed triangles represent the path the robot is taking, while all forms of lines represent motion vectors fitting a curve. The indicated variable k represent the time instance at which the labeled vector is being considered. The unexplained variable i is an iterator of the the closest landmark given a k^{th} vector.

\vec{x_k} is the state vector of the robot at time k
\vec{u_k} is the change vector applied at k-1 to change \vec{x_{k-1}} to \vec{x_k}
\vec{m_i} is a offset/erorr vector from the estimated to true location of a landmark given landmark of identity i, assuming the true location is time invariant
z_{i,k} is an observation pertaining to the ==location== of the i^{th} landmark at time k. When referring to a set or history of such measurements, it is conventional short hand to denote them as z_k

Set definitions

X_{0:k} = {\lbrace x_0,x_1,...,x_k \rbrace}= {\lbrace X_{1:k-1},x_k \rbrace} \rightarrow The history of robot locations
U_{1:k} = {\lbrace u_1,u_2,...,u_k \rbrace}= {\lbrace U_{1:k-1},u_k \rbrace} \rightarrow The history of control vectors
m is the set of all landmarks
Z_{1:k} = {\lbrace z_1,z_2,...,z_k \rbrace}= {\lbrace Z_{1:k-1},z_k \rbrace} \rightarrow The historical set of all landmark observations

Probabilistic SLAM

The probability distribution given time k:


P(x_{k},m \vert Z_{1:k},U_{1:k},x_0)

This distribution describes the joint end density of the coplanar measurements of landmark location and vehicle state (losing a dimension or two of vehicle state vector.) Since a vehicle control vector is improbable to be perfectly implemented in a real-world situation, and the change in measurements, temporally recursive computations are desirable. It is important to understand that this distribution is possible when given the set U_{0:k}, the control vectors including u_{k}. This can be read as the probability of a random \vec{p_k} taking on the desired value of \vec{x_k} in conjunction with i_n number of random vectors taking on the values in set m given observations and adjustment control vectors and a initial positional state vector x_0. Importantly, this is a computation of the probability of both observing the correct landmarks in the map, and possessing the desired state x_k. In direct corollary, there exist four vital probabilistic models:

Model	Distribution
Observation	`P(z_{k}\vert x_k,m)`
Motion	`P(x_{k}\vert x_{k-1},u_k)`
Time-update	`P(x_k,m\vert Z_{1:k-1},U_{1:k},x_{0)}=\int{P(x_k\vert x_{k-1},u_k)}{P(x_{k-1},m\vert Z_{1:k-1},U_{1:k-1},x_0)}dx_{k-1}`
And finally Measurement Update:


P(x_k,m\vert Z_{1:k},U_{1:k},x_0)
=\frac{P(z_k\vert x_k,m)*P(x_k,m\vert Z_{1:k-1},U_{1:k},x_0)}{P(z_k\vert Z_{1:k-1},U_{1:k})}

As seen the last two are both recursive over a new piece of information each. Leaving the math here, the author will refrain it for the rest of this paper, as the rest of the matter depends highly on model variant and implementation.

Core Concepts

Sparing you the the drivel:

A large portion of error across multiple landmark observations z_k is common to all landmarks when compared to m
This is a high monotonous correlation in landmark location estimates constructed using Z
Therefore, relative location is easier to discern than absolute location for all entities
m_i-m_j may be known with high accuracy even with a uncertain m_i (i and j not relating with previous definitions)
In probabilistic terms, P(m_i,m_j) has a higher density than P(m_i)
Correlations between landmark estimates increase monotonically
P(m) progressively becomes more peaked
Even if a landmark m_i is not observed in the current frame of sense, relative changes in position in perspective of x_k propagate back to landmark m_i
A robot usually has a circular field of vision, like our car in F1Tenth

Solutions

Solutions to the probabilistic SLAM problem necessitate a hunt for an appropriate algorithmic and thorough mathematical model to facilitate the computation of the Motion, Observation, and Time-Update posterior probability density models. The first solution discussed in this paper is EKF-SLAM, which will not be discussed here, in favor of a deeper dive in another note:

Citation

H. Durrant-Whyte and T. Bailey, "Simultaneous localization and mapping: part I," in IEEE Robotics & Automation Magazine, vol. 13, no. 2, pp. 99-110, June 2006, doi: 10.1109/MRA.2006.1638022.

Foundational Wikis

Baye's Factor Posterior probability

Credits

Author of this note: Krishna Ayyalasomayajula

#paper-review #SLAM #krishna #literature-review

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