Passive Global Localisation of Mobile Robot via 2D Fourier-Mellin Invariant Matching

Abstract

Passive global localisation is defined as locating a robot on a map, under global pose uncertainty, without prescribing motion controls. The majority of current solutions either assume structured environments or require tuning of parameters relevant to establishing correspondences between sensor measurements and segments of the map. This article advocates for a solution that dispenses with both in order to achieve greater portability and universality across disparate static environments. A single 2D panoramic LIght Detection And Ranging (LIDAR) sensor is used as the measurement device, this way reducing computational and investment costs. The proposed method disperses pose hypotheses on the map of the robot’s environment and then captures virtual scans from each of them. Subsequently, each virtual scan is matched against the one derived from the physical sensor. Angular alignment is performed via 2D Fourier-Mellin Invariant (FMI) matching; positional alignment is performed via feedback of the position estimation error. In order to deduce the robot’s pose the method sifts through hypotheses by using measures extracted from FMI. Simulations and experiments illustrate the efficacy of the proposed global localisation solution in realistic surroundings and scenarios. In addition, the proposed method is pitted against the most effective Iterative Closest Point (ICP) variant under the same task, and three conclusions are drawn. The first is that the proposed method is effective in both structured and unstructured environments. The second is that it concludes to fewer false positives. The third is that the two methods are largely equivalent in terms of pose error.

Appendix

Let us illustrate the methodology introduced with an example: consider again the map depicted in Fig. 3, and let p_a(11.56, 12.20, 0.0) [m, m, rad] be the robot’s true pose and a pose hypothesis p_c(7.56, 11.20,π/4) [m, m, rad] be disposed by (− 4.0,− 1.0,π/4). At the rotation estimation stage the range scan captured from the robot’s true pose, $\mathcal {S}_{r}^{a}$, and the virtual range scan captured from the hypothesis, $\mathcal {S}_{\text {v}}^{c}$, are projected to the x − y plane as if each was captured from (0, 0, 0). Figure 27a shows the projected range scan points from p_a, $\mathcal {P}_{r}^{a}$, while Fig. 27b shows those from p_c, $\mathcal {P}_{\text {v}}^{c}$. Notice that these connected point-sets consist of the surroundings of the real and virtual range scan sensors from their local reference frame perspective. These point-sets are then converted into 2D grids via discretisation, inputted to FMI-SPOMF, and the rotation angle between them is used to align the orientation of p_c with respect to that of p_a.

Fig. 27

Illustration of orientational and then positional alignment of candidate pose p_c with respect to true pose p_a in environment CORRIDOR (Fig. 3)

Once the orientation of the hypothesis is corrected, a new map scan is captured from the renewed hypothesis $\boldsymbol {p}_{c}^{\prime }$. Then the centroids of $\mathcal {P}_{r}^{a}$ and the point set of the newly projected map scan $\mathcal {P}_{\text {v}}^{c\prime }$ are computed. Figure 27c depicts $\mathcal {P}_{r}^{a}$ and its centroid C_a(− 3.57,− 0.78) [m, m], while Fig. 27d depicts $\mathcal {P}_{\text {v}}^{c\prime }$ and its corresponding centroid C_c(0.42, 0.09) [m, m]. Notice how the two shapes are almost identical, but differ in terms of their position in the x − y plane. Notice also the discrepancy between these two points sets at the left-hand side: due to the offset between the positions of p_a and $\boldsymbol {p}_{c}^{\prime }$, a larger proportion of the map is visible from the latter, and therefore the difference between their corresponding points-sets’ centroids C_a −C_c = [3.99, 0.87]^⊤ does not correspond exactly with the difference in position between the two poses, which is [4.0, 1.0]^⊤. Adding, however, C_a −C_c to $\boldsymbol {p}_{c}^{\prime }$ and repeating the same translation estimation process makes $\mathcal {P}_{\text {v}}^{c\prime }$ converge to $\mathcal {P}_{r}^{a}$, and therefore $\boldsymbol {p}_{c}^{\prime }$ to p_a. Figure 27e shows the final point-set $\mathcal {P}_{\text {v}}^{c\prime }$, which is overlaid in Fig. 27f (coloured with red) on top of $\mathcal {P}_{r}^{a}$ (black).

Let us now examine the possible outcomes of the same process for a false candidate pose, for instance p_b. In theory p_b would be either discarded at the end of the angular estimation process due to the extraction of an (viewed externally) arbitrary scaling factor $\sigma \in (-\infty , \underline {\sigma }] \cup [\overline {\sigma }, +\infty )$, or accepted for position estimation, whereupon the position of the hypothesis would in all probability be moved divergently from the robot’s true pose. If $\boldsymbol {p}_{c} \in {\mathscr{H}}$ then FMT-SPOMF would report a higher similarity degree w_c > w_b, and p_b would be filtered out as a true negative. Consequently, if no pose hypothesis resided in the vicinity of p_a, the projected range scan images captured from every hypothesis would not be able to be angularly aligned by FMI-SPOMF, and a false hypothesis would be erroneously reported as the system’s pose estimate. This leads to the formulation of the following observation:

Remark 1

A pose hypothesis $\boldsymbol {h} \in {\mathscr{H}}$ that resides in the vicinity of the robot’s true pose is a necessary condition for a correct solution to problem P (in the sense of Remark 1) when approached by a scan–to–map-scan method. In consequence: the supplied number of pose hypotheses $|{\mathscr{H}}|$ should be proportional to the area of M.¹

In more complex environments, where the environment and its map feature repetitive structures, it may be the case that ambiguity cannot be resolved at all regardless of the number of pose hypotheses. In others, wide open spaces may result to missing information due to sensor maximum range limits. The effects of these conditions may be so pronounced that higher similarity is established between an incorrect pose and the robot’s true pose than between the robot’s true pose and a pose residing near it. The first issue plagues all global localisation methods as, even for a human, its solution is undecidable, and maximum sensitivity is of paramount importance in such conditions. The second issue is also uncontrollable, as it manifests itself as a limitation imposed by the combination of environment and robot equipment limits.