Lagrangian duality in 3D SLAM: Verification techniques and optimal solutions
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Lagrangian-Duality-in-3D-SLAM.pdf
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Author(s)
Carlone, Luca
Calafiore, Giuseppe
Dellaert, Frank
Rosen, David Matthew
Leonard, John J
Date Issued
December 2015
Journal
Proceedings of the 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Carlone, Luca et al. “Lagrangian Duality in 3D SLAM: Verification Techniques and Optimal Solutions.” IEEE, 2015. 125–132.
Version
Author's final manuscript
Abstract
State-of-the-art techniques for simultaneous localization and mapping (SLAM) employ iterative nonlinear optimization methods to compute an estimate for robot poses. While these techniques often work well in practice, they do not provide guarantees on the quality of the estimate. This paper shows that Lagrangian duality is a powerful tool to assess the quality of a given candidate solution. Our contribution is threefold. First, we discuss a revised formulation of the SLAM inference problem. We show that this formulation is probabilistically grounded and has the advantage of leading to an optimization problem with quadratic objective. The second contribution is the derivation of the corresponding Lagrangian dual problem. The SLAM dual problem is a (convex) semidefinite program, which can be solved reliably and globally by off-the-shelf solvers. The third contribution is to discuss the relation between the original SLAM problem and its dual. We show that from the dual problem, one can evaluate the quality (i.e., the suboptimality gap) of a candidate SLAM solution, and ultimately provide a certificate of optimality. Moreover, when the duality gap is zero, one can compute a guaranteed optimal SLAM solution from the dual problem, circumventing non-convex optimization. We present extensive (real and simulated) experiments supporting our claims and discuss practical relevance and open problems.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
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Creative Commons Attribution-Noncommercial-Share Alike
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DOI of Published Version
https://doi.org/10.1109/IROS.2015.7353364
