We present a novel data structure, the Bayes tree, that provides an algorithmic foundation enabling a better understanding of existing graphical model inference algorithms and their connection to sparse matrix factorization methods. Similar to a clique tree, a Bayes tree encodes a factored probability density, but unlike the clique tree it is directed and maps more naturally to the square root information matrix of the simultaneous localization and mapping (SLAM) problem. In this paper, we highlight three insights provided by our new data structure. First, the Bayes tree provides a better understanding of the matrix factorization in terms of probability densities. Second, we show how the fairly abstract updates to a matrix factorization translate to a simple editing of the Bayes tree and its conditional densities. Third, we apply the Bayes tree to obtain a completely novel algorithm for sparse nonlinear incremental optimization, named iSAM2, which achieves improvements in efficiency through incremental variable re-ordering and fluid relinearization, eliminating the need for periodic batch steps. We analyze various properties of iSAM2 in detail, and show on a range of real and simulated datasets that our algorithm compares favorably with other recent mapping algorithms in both quality and efficiency.

Abstract-Pose SLAM is the variant of simultaneous localization and map building (SLAM) is the variant of SLAM, in which only the robot trajectory is estimated and where landmarks are only used to produce relative constraints between robot poses. To reduce the computational cost of the information filter form of Pose SLAM and, at the same time, to delay inconsistency as much as possible, we introduce an approach that takes into account only highly informative loop-closure links and nonredundant poses. This approach includes constant time procedures to compute the distance between poses, the expected information gain for each potential link, and the exact marginal covariances while moving in open loop, as well as a procedure to recover the state after a loop closure that, in practical situations, scales linearly in terms of both time and memory. Using these procedures, the robot operates most of the time in open loop, and the cost of the loop closure is amortized over long trajectories. This way, the computational bottleneck shifts to data association, which is the search over the set of previously visited poses to determine good candidates for sensor registration. To speed up data association, we introduce a method to search for neighboring poses whose complexity ranges from logarithmic in the usual case to linear in degenerate situations. The method is based on organizing the pose information in a balanced tree whose internal levels are defined using interval arithmetic. The proposed Pose-SLAM approach is validated through simulations, real mapping sessions, and experiments using standard SLAM data sets.Index Terms-Information filter, information gain, interval arithmetic, Pose SLAM, state recovery, tree-based data association.

Abstract-We present iSAM2, a fully incremental, graphbased version of incremental smoothing and mapping (iSAM). iSAM2 is based on a novel graphical model-based interpretation of incremental sparse matrix factorization methods, afforded by the recently introduced Bayes tree data structure. The original iSAM algorithm incrementally maintains the square root information matrix by applying matrix factorization updates. We analyze the matrix updates as simple editing operations on the Bayes tree and the conditional densities represented by its cliques. Based on that insight, we present a new method to incrementally change the variable ordering which has a large effect on efficiency. The efficiency and accuracy of the new method is based on fluid relinearization, the concept of selectively relinearizing variables as needed. This allows us to obtain a fully incremental algorithm without any need for periodic batch steps. We analyze the properties of the resulting algorithm in detail, and show on various real and simulated datasets that the iSAM2 algorithm compares favorably with other recent mapping algorithms in both quality and efficiency.

We present a novel data structure, the Bayes tree, that provides an algorithmic foundation enabling a better understanding of existing graphical model inference algorithms and their connection to sparse matrix factorization methods. Similar to a clique tree, a Bayes tree encodes a factored probability density, but unlike the clique tree it is directed and maps more naturally to the square root information matrix of the simultaneous localization and mapping (SLAM) problem. In this paper, we highlight three insights provided by our new data structure. First, the Bayes tree provides a better understanding of batch matrix factorization in terms of probability densities. Second, we show how the fairly abstract updates to a matrix factorization translate to a simple editing of the Bayes tree and its conditional densities. Third, we apply the Bayes tree to obtain a completely novel algorithm for sparse nonlinear incremental optimization, that combines incremental updates with fluid relinearization of a reduced set of variables for efficiency, combined with fast convergence to the exact solution. We also present a novel strategy for incremental variable reordering to retain sparsity. We evaluate our algorithm on standard datasets in both landmark and pose SLAM settings.

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.

Abstract-Efficiently solving nonlinear least squares (NLS) problems is crucial for many applications in robotics. In online applications, solving the associated nolinear systems every step may become very expensive. This paper introduces online, incremental solutions, which take full advantage of the sparseblock structure of the problems in robotics. In general, the solution of the nonlinear system is approximated by incrementally solving a series of linearized problems. The most computationally demanding part is to assemble and solve the linearized system at each iteration. In our solution, this is mitigated by incrementally updating the factorized form of the linear system and changing the linearization point only if needed. The incremental updates are done using a resumed factorization only on the parts affected by the new information added to the system at every step. The sparsity of the factorized form directly affects the efficiency. In order to obtain an incremental factorization with persistent reduced fill-in, a new incremental ordering scheme is proposed. Furthermore, the implementation exploits the block structure of the problems and offers efficient solutions to manipulate block matrices, including a highly efficient Cholesky factorization on sparse block matrices. In this work, we focus our efforts on testing the method on SLAM applications, but the applicability of the technique remains general. The experimental results show that our implementation outperforms the state of the art SLAM implementations on all the tested datasets.

The scene rigidity assumption, also known as the static world assumption, is common in SLAM algorithms. Most existing algorithms operating in complex dynamic environments simplify the problem by removing moving objects from consideration or tracking them separately. Such strong assumptions limit the deployment of autonomous mobile robotic systems in a wide range of important real world applications involving highly dynamic and unstructured environments. This paper presents VDO-SLAM, a robust object-aware dynamic SLAM system that exploits semantic information to enable motion estimation of rigid objects in the scene without any prior knowledge of the objects shape or motion models. The proposed approach integrates dynamic and static structures in the environment into a unified estimation framework resulting in accurate robot pose and spatio-temporal map estimation. We provide a way to extract velocity estimates from object pose change of moving objects in the scene providing an important functionality for navigation in complex dynamic environments. We demonstrate the performance of the proposed system on a number of real indoor and outdoor datasets. Results show consistent and substantial improvements over state-of-the-art algorithms. An open-source version of the source code is available * .

Many estimation problems in robotics rely on efficiently solving nonlinear least squares (NLS). For example, it is well known that the simultaneous localisation and mapping (SLAM) problem can be formulated as a maximum likelihood estimation (MLE) and solved using NLS, yielding a mean state vector. However, for many applications recovering only the mean vector is not enough. Data association, active decisions, next best view, are only few of the applications that require fast state covariance recovery. The problem is not simple since, in general, the covariance is obtained by inverting the system matrix and the result is dense.The main contribution of this paper is a novel algorithm for fast incremental covariance update, complemented by a highly efficient implementation of the covariance recovery. This combination yields to two orders of magnitude reduction in computation time, compared to the other state of the art solutions. The proposed algorithm is applicable to any NLS solver implementation, and does not depend on incremental strategies described in our previous papers, which are not a subject of this paper.