Simultaneous localization and mapping: part I
Summary (3 min read)
History of the SLAM Problem
- A key element of this work was to show that there must be a high degree of correlation between estimates of the location of different landmarks in a map and that, indeed, these correlations would grow with successive observations.
- These two strands of research had much in common and resulted soon after in the landmark paper by Smith et al. [40] .
Probabilistic SLAM
- This probability distribution describes the joint posterior density of the landmark locations and vehicle state (at time k) given the recorded observations and control inputs up to and including time k together with the initial state of the vehicle.
- This computation requires that a state transition model and an observation model are defined describing the effect of the control input and observation respectively.
- The observation model describes the probability of making an observation z k when the vehicle location and landmark locations are known and is generally described in the form EQUATION.
- It is reasonable to assume that once the vehicle location and map are defined, observations are conditionally independent given the map and the current vehicle state.
- This assumes that the landmark locations are known with certainty, and the objective is to compute an estimate of vehicle location with respect to these landmarks.
Structure of Probabilistic SLAM
- The SLAM problem has more structure than is immediately obvious from these equations.
- (These results have only been proved for the linear Gaussian case [14] .
- The relative location of observed landmarks is clearly independent of the coordinate frame of the vehicle, and successive observations from this fixed location would yield further independent measurements of the relative relationship between landmarks.
- This occurs because the two landmarks are highly correlated (their relative location is well known) from previous measurements.
- In the theoretical limit, robot relative location accuracy becomes equal to the localization accuracy achievable with a given map.
Solutions to the SLAM Problem
- Solutions to the probabilistic SLAM problem involve finding an appropriate representation for both the observation model ( 2) and motion model (3) that allows efficient and consistent computation of the prior and posterior distributions in ( 4) and (5) .
- By far, the most common representation is in the form of a state-space model with additive Gaussian noise, leading to the use of the extended Kalman filter (EKF) to solve the SLAM problem.
- This leads to the use of the Rao-Blackwellized particle filter, or FastSLAM algorithm, to solve the SLAM problem.
- While EKF-SLAM and FastSLAM are the two most important solution methods, newer alternatives, which offer much potential, have been proposed, including the use of the information-state form [43] .
Figure 2. Spring network analogy. The landmarks are connected by springs describing correlations between landmarks. As the vehicle moves back and forth through the environment, spring stiffness or correlations increase (red links become thicker).
- As landmarks are observed and estimated locations are corrected, these changes are propagated through the spring network.
- Note, the robot itself is correlated to the map.
Convergence
- In the EKF-SLAM problem, convergence of the map is manifest in the monotonic convergence of the determinant of the map covariance matrix P mm,k and all landmark pair submatrices, toward zero.
- The individual landmark variances converge toward a lower bound determined by initial uncertainties in robot position and observations.
Computational Effort
- The observation update step requires that all landmarks and the joint covariance matrix be updated every time an observation is made.
- Naively, this means computation grows quadratically with the number of landmarks.
- There has been a great deal of work undertaken in developing efficient variants of the EKF-SLAM solution and real-time implementations with many thousands of landmarks have been demonstrated [21] , [29] .
- Efficient variants of the EKF-SLAM algorithm are discussed in Part II of this tutorial.
Figure 3. The convergence in landmark uncertainty. The plot shows a time history of standard deviations of a set of landmark locations. A landmark is initially observed with uncertainty inherited from the robot location and observation.
- Over time, the standard deviations reduce monotonically to a lower bound.
- New landmarks are acquired during motion (from [14] ).
Data Association
- The loop-closure problem, when a robot returns to reobserve landmarks after a large traverse, is especially difficult.
- The association problem is compounded in environments where landmarks are not simple points and indeed look different from different viewpoints.
- Nonlinearity EKF-SLAM employs linearized models of nonlinear motion and observation models and so inherits many caveats.
- Convergence and consistency can only be guaranteed in the linear case.
Implementation of SLAM
- Practical realizations of probabilistic SLAM have become increasingly impressive in recent years, covering larger areas in more challenging environments.
- Here the authors discuss two representative implementations and mention other notable applications.
- For the return trip, the robot plans a path and returns without human intervention.
- Guivant and Nebot [21] pioneered the application of SLAM in very large outdoor environments .
- There are different sensing modalities such as bearing only [13] and range only [30] .
Conclusions
- This article has described the SLAM problem and the essential methods for solving the SLAM problem and has summarized key implementations and demonstrations of the method.
- While there are still many practical issues to overcome, especially in more complex outdoor environments, the general SLAM method is now a well understood and established part of robotics.
- Part II of this tutorial will summarize more recent work in addressing some of the remaining issues in SLAM, including computation, feature representation, and data association.
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Cites background or methods from "Simultaneous localization and mappi..."
...oses a constraint on a pair of poses. MAP estimation has been proven to be more accurate and efficient than original approaches for SLAM, based on nonlinear filtering. We refer the reader to the survey [13, 86] for an overview on filtering approaches, and to [264] for a comparison between filtering and smoothing approaches. However, we remark that some SLAM systems based on EKF have also been demonstrated to ...
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...that robot operation is possible in absence of an ad-hoc localization infrastructure. A thorough historical review of the first 20 years of the SLAM problem is discussed by Durrant-Whyte and Bailey in [13, 86]. The surveys [13, 86] mainly cover what we call the classical age (1986-2004); the classical age saw the introduction of the main probabilistic formulations for SLAM, including approaches based on Ex...
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...nd control theory, and cross-fertilization among these fields is a necessary condition to enable fast progress. For the non-expert reader, we recommend to read DurrantWhyte and Bailey’s SLAM tutorials [13, 86] before delving in this position paper. The more experienced researcher can jump directly to the section of interest, where he/she will find a self-contained overview of the state of the art and open p...
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...Mobile Sensors: Setting future arXiv:1606.05830v1 [cs.RO] 19 Jun 2016 2 TABLE I: Surveying the surveys Year Topic Reference 2006 Probabilistic approaches and data association Durrant-Whyte and Bailey [13, 86] 2008 Filtering approaches Aulinas et al. [12] 2008 Visual SLAM Neira et al. (special issue) [200] 2011 SLAM back-end Grisetti et al. [117] 2011 Observability, Consistency, Convergence Dissanayake et ...
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References
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"Simultaneous localization and mappi..." refers background or methods in this paper
...SLAM 1.0 [ 32 ] and FastSLAM 2.0 [33], differ only in terms of the form of their proposal distribution (Step 1) and, consequently, in their importance weight (Step 2). FastSLAM 2.0 is by far the more efficient solution....
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...The FastSLAM algorithm, introduced by Montemerlo et al. [ 32 ], marked a fundamental conceptual shift in the design of recursive probabilistic SLAM....
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