OASIS: Optimal Arrangements for Sensing in SLAM

TL;DR

This work shows how to formalize the sensor arrangement problem as a form of subset selection under the E-optimality performance criterion, and shows that a combination of greedy sensor selection and fast convex relaxation-based post-hoc verification enables the efficient recovery of certifiably optimal sensor designs in practice.

Abstract

The number and arrangement of sensors on mobile robot dramatically influence its perception capabilities. Ensuring that sensors are mounted in a manner that enables accurate detection, localization, and mapping is essential for the success of downstream control tasks. However, when designing a new robotic platform, researchers and practitioners alike usually mimic standard configurations or maximize simple heuristics like field-of-view (FOV) coverage to decide where to place exteroceptive sensors. In this work, we conduct an information-theoretic investigation of this overlooked element of robotic perception in the context of simultaneous localization and mapping (SLAM). We show how to formalize the sensor arrangement problem as a form of subset selection under the E-optimality performance criterion. While this formulation is NP-hard in general, we show that a combination of greedy sensor selection and fast convex relaxation-based post-hoc verification enables the efficient recovery of certifiably optimal sensor designs in practice. Results from synthetic experiments reveal that sensors placed with OASIS outperform benchmarks in terms of mean squared error of visual SLAM estimates.

Figures (4)

• Figure 1: Overview of OASIS. At each pose $x _i$, sensor $c_j$ observes some subset of the landmarks. OASIS maximizes the minimum eigenvalue of the joint Fisher information matrix, which is composed of sub-matrices $\mathcal{I}_j$ from each sensor $c_j$. In this example, the sensor "budget" limits us to selecting two out of the three candidate sensors. Finally, note that the discrete binary variables indicating sensor selection have been relaxed to a convex superset.
• Figure 2: Experimental Results I: Quantitative results of optimal placement on the synthetic dataset of 50 simulations of a random motion of the sensor rig with varying number of selected cameras, $k$. (a) and (b) Show the trend of the optimized score of the objective function $\lambda_1$ and median RMSE of the translational component of the pose estimates computed from the graph resulting from the optimal camera selection with respect to the ground truth computed across simulations. (c) Gives a closer look at the mean and standard deviation of the objective scores of greedy, convex relaxation solutions before and after k-max rounding across simulations. The greedy optimization results in a near-optimal solution, as demonstrated by its closeness to the score of the unrounded convex relaxation approach, which gives the upper bound on the optimal value, especially for $k > 2$. (d) Shows that we achieve a very tight relative sub-optimality gap, $\gamma^* = f(w^*) - x_g/x_g$ asserting the effectiveness of submodular greedy optimization.
• Figure 3: Experimental Results II: (a) The synthetic data collection setup. A simulated room-like environment with landmarks and random trajectories from the top view. (b-d) Quantitative results showing median score and RMSE of the benchmarking algorithms for Circular, Forward and Lateral motions.
• Figure 4: Experimental Results III. Visualization of the greedy optimal selection across multiple experiments overlayed for a candidate pool (a) lying on a linear array configuration and (b) regularly spaced in both translation and orientation. Deeper/darker colors indicate a higher frequency of selection. (c) Shows that the score $\lambda_1$ is inversely related to the RMSE of SLAM pose estimates for both greedy and convex relaxation approaches. Thus, E-optimality improves SLAM performance. (d) Run time comparison of greedy optimization with the convex relaxation approach. Time complexity of the greedy method increases linearly with the number of cameras, while there is not much effect on the convex relaxation approach.

Theorems & Definitions (1)

• Proposition 1: Concavity of $f_{\mathrm{E}}$