Simplified decision making in the belief space using belief sparsification

In this work, we introduce a new and efficient solution approach for the problem of decision making under uncertainty, which can be formulated as decision making in a belief space, over a possibly high-dimensional state space. Typically, to solve a decision problem, one should identify the optimal action from a set of candidates, according to some objective. We claim that one can often generate and solve an analogous yet simplified decision problem, which can be solved more efficiently. A wise simplification method can lead to the same action selection, or one for which the maximal loss in optimality can be guaranteed. Furthermore, such simplification is separated from the state inference and does not compromise its accuracy, as the selected action would finally be applied on the original state. First, we present the concept for general decision problems and provide a theoretical framework for a coherent formulation of the approach. We then practically apply these ideas to decision problems in the belief space, which can be simplified by considering a sparse approximation of their initial belief. The scalable belief sparsification algorithm we provide is able to yield solutions which are guaranteed to be consistent with the original problem. We demonstrate the benefits of the approach in the solution of a realistic active-SLAM problem and manage to significantly reduce computation time, with no loss in the quality of solution. This work is both fundamental and practical and holds numerous possible extensions.

Reliable Graphs for SLAM

Estimation-over-graphs (EoG) is a class of estimation problems that admit a natural graphical representation. Several key problems in robotics and sensor networks, including sensor network localization, synchronization over a group, and simultaneous localization and mapping (SLAM) fall into this category. We pursue two main goals in this work. First, we aim to characterize the impact of the graphical structure of SLAM and related problems on estimation reliability. We draw connections between several notions of graph connectivity and various properties of the underlying estimation problem. In particular, we establish results on the impact of the weighted number of spanning trees on the D-optimality criterion in 2D SLAM. These results enable agents to evaluate estimation reliability based only on the graphical representation of the EoG problem. We then use our findings and study the problem of designing sparse SLAM problems that lead to reliable maximum likelihood estimates through the synthesis of sparse graphs with the maximum weighted tree connectivity. Characterizing graphs with the maximum number of spanning trees is an open problem in general. To tackle this problem, we establish several new theoretical results, including the monotone log-submodularity of the weighted number of spanning trees. We exploit these structures and design a complementary greedy–convex pair of efficient approximation algorithms with provable guarantees. The proposed synthesis framework is applied to various forms of the measurement selection problem in resource-constrained SLAM. Our algorithms and theoretical findings are validated using random graphs, existing and new synthetic SLAM benchmarks, and publicly available real pose-graph SLAM datasets.

Exactly sparse delayed state filter on Lie groups for long-term pose graph SLAM

In this paper we propose a simultaneous localization and mapping (SLAM) back-end solution called the exactly sparse delayed state filter on Lie groups (LG-ESDSF). We derive LG-ESDSF and demonstrate that it retains all the good characteristics of the classic Euclidean ESDSF, the main advantage being the exact sparsity of the information matrix. The key advantage of LG-ESDSF in comparison with the classic ESDSF lies in the ability to respect the state space geometry by negotiating uncertainties and employing filtering equations directly on Lie groups. We also exploit the special structure of the information matrix in order to allow long-term operation while the robot is moving repeatedly through the same environment. To prove the effectiveness of the proposed SLAM solution, we conducted extensive experiments on two different publicly available datasets, namely the KITTI and EuRoC datasets, using two front-ends: one based on the stereo camera and the other on the 3D LIDAR. We compare LG-ESDSF with the general graph optimization framework ([Formula: see text]) when coupled with the same front-ends. Similarly to [Formula: see text] the proposed LG-ESDSF is front-end agnostic and the comparison demonstrates that our solution can match the accuracy of [Formula: see text], while maintaining faster computation times. Furthermore, the proposed back-end coupled with the stereo camera front-end forms a complete visual SLAM solution dubbed LG-SLAM. Finally, we evaluated LG-SLAM using the online KITTI protocol and at the time of writing it achieved the second best result among the stereo odometry solutions and the best result among the tested SLAM algorithms.

No belief propagation required: Belief space planning in high-dimensional state spaces via factor graphs, the matrix determinant lemma, and re-use of calculation

We develop a computationally efficient approach for evaluating the information-theoretic term within belief space planning (BSP), where during belief propagation the state vector can be constant or augmented. We consider both unfocused and focused problem settings, whereas uncertainty reduction of the entire system or only of chosen variables is of interest, respectively. State-of-the-art approaches typically propagate the belief state, for each candidate action, through calculation of the posterior information (or covariance) matrix and subsequently compute its determinant (required for entropy). In contrast, our approach reduces runtime complexity by avoiding these calculations. We formulate the problem in terms of factor graphs and show that belief propagation is not needed, requiring instead a one-time calculation that depends on (the increasing with time) state dimensionality, and per-candidate calculations that are independent of the latter. To that end, we develop an augmented version of the matrix determinant lemma, and show that computations can be re-used when evaluating impact of different candidate actions. These two key ingredients and the factor graph representation of the problem result in a computationally efficient (augmented) BSP approach that accounts for different sources of uncertainty and can be used with various sensing modalities. We examine the unfocused and focused instances of our approach, and compare it with the state of the art, in simulation and using real-world data, considering problems such as autonomous navigation in unknown environments, measurement selection and sensor deployment. We show that our approach significantly reduces running time without any compromise in performance.

SLAM++-A highly efficient and temporally scalable incremental SLAM framework

The most common way to deal with the uncertainty present in noisy sensorial perception and action is to model the problem with a probabilistic framework. Maximum likelihood estimation is a well-known estimation method used in many robotic and computer vision applications. Under Gaussian assumption, the maximum likelihood estimation converts to a nonlinear least squares problem. Efficient solutions to nonlinear least squares exist and they are based on iteratively solving sparse linear systems until convergence. In general, the existing solutions provide only an estimation of the mean state vector, the resulting covariance being computationally too expensive to recover. Nevertheless, in many simultaneous localization and mapping (SLAM) applications, knowing only the mean vector is not enough. Data association, obtaining reduced state representations, active decisions and next best view are only a few of the applications that require fast state covariance recovery. Furthermore, computer vision and robotic applications are in general performed online. In this case, the state is updated and recomputed every step and its size is continuously growing, therefore, the estimation process may become highly computationally demanding.

This paper introduces a general framework for incremental maximum likelihood estimation called SLAM++, which fully benefits from the incremental nature of the online applications, and provides efficient estimation of both the mean and the covariance of the estimate. Based on that, we propose a strategy for maintaining a sparse and scalable state representation for large scale mapping, which uses information theory measures to integrate only informative and non-redundant contributions to the state representation. SLAM++ differs from existing implementations by performing all the matrix operations by blocks. This led to extremely fast matrix manipulation and arithmetic operations used in nonlinear least squares. Even though this paper tests SLAM++ efficiency on SLAM problems, its applicability remains general.

Nonlinear factor recovery for long-term SLAM

For long-term operations, graph-based simultaneous localization and mapping (SLAM) approaches require nodes to be marginalized in order to control the computational cost. In this paper, we present a method to recover a set of nonlinear factors that best represents the marginal distribution in terms of Kullback–Leibler divergence. The proposed method, which we call nonlinear factor recovery (NFR), estimates both the mean and the information matrix of the set of nonlinear factors, where the recovery of the latter is equivalent to solving a convex optimization problem. NFR is able to provide either the dense distribution or a sparse approximation of it. In contrast to previous algorithms, our method does not necessarily require a global linearization point and can be used with any nonlinear measurement function. Moreover, we are not restricted to only using tree-based sparse approximations and binary factors, but we can include any topology and correlations between measurements. Experiments performed on several publicly available datasets demonstrate that our method outperforms the state of the art with respect to the Kullback–Leibler divergence and the sparsity of the solution.