Auton Robot DOI 10.1007/s10514-015-9447-y Ellipsoid SLAM: a novel set membership method for simultaneous localization and mapping Wen Yu 1 · Erik Zamora 1 · Alberto Soria 1 Received: 30 September 2013 / Accepted: 19 June 2015 © Springer Science+Business Media New York 2015 Abstract The extended Kalman filter (EKF) simultaneous localization and mapping (SLAM) requires the uncertainty to be Gaussian noise. This assumption can be relaxed to bounded noise by the set membership SLAM. However, the published set membership SLAMs are not suitable for large- scale and online problems. In this paper, we use ellipsoid algorithm for solving SLAM problem. The proposed ellip- soid SLAM has advantages over EKF SLAM and the other set membership SLAMs, in noise condition, online realization, and large-scale problem. By bounded ellipsoid technique, we analyze the convergence and stability of the ellipsoid SLAM. Simulation and experimental results show that the proposed ellipsoid SLAM is effective for online and large-scale prob- lems such as Victoria Park dataset. Keywords SLAM · Ellipsoid · Set membership · Convergence · Stability 1 Introduction When autonomous mobile robots are in new environments, such as rescuing in collapsed structures and exploration in unknown environments, intelligent methods are needed to find the routes in the environments. The robots should know their current locations and the environment maps. This is the simultaneous localization and mapping (SLAM) prob- lem (Dissanayake et al. 2001). The objective of SLAM is to localize the robot from any initial location and create a model of the unknown environment at the same time. In B Wen Yu yuw@ctrl.cinvestav.mx 1 Departamento de Control Automatico, CINVESTAV-IPN (National Polytechnic Institute), 07360 Mexico City, Mexico the last decade, many SLAMs have been developed. They can be categorized as feature-based SLAM (Thrun et al. 2004), pose-based SLAM (Folkesson and Christensen 2007), appearance-based SLAM (Ho and Newman 2006), and other variants (Nieto et al. 2006). In the SLAM problem, the robot needs a map to localize itself, and the map needs the localization of robot to update itself. Therefore many SLAM solutions are recursive. There are uncertainties in determining the robot position and the map, such as sensor noises and environment model errors. In order to compensate these errors, some statistical techniques are used to estimate the states (robot and landmark positions). Extended Kalman filter (EKF) is the most popular method for solving the SLAM problem (Dissanayake et al. 2001; Thrun et al. 2004). It represents the robot and environment in a state space model with Gaussian noises. Some modified EKFs can improve SLAM accuracy. For example, extended information filter (EIF) (Thrun et al. 2004; Wang et al. 2007), particle filters (Montemerlo and Thrun 2007; Mullane et al. 2011), Bayes filter (Kaess et al. 2012), and sliding window filter (Sibley et al. 2010). However, the above EKF SLAMs do not solve the fundamental limitation for the model noise, i.e., the probability distributions of the state and observations are Gaussian (Montemerlo and Thrun 2007). The set membership technique is an effective tool to solve the state estimation with bounded disturbances (Scholte and Campbell 2003). There are two common methods: polytopes and ellipsoids. The polytope set method is easy and sim- ple (Garulli and Vicino 2001). It can be applied to SLAM directly as in Marco et al. (2004) and Jaulin (2009, 2011). In Jaulin (2009, 2011), the interval analysis, contraction tech- nique and polytope set memberships are used to find the minimal envelope of robot trajectory. The algorithms can only work off-line due to the large computational cost of contraction. In Marco et al. (2004), an online algorithm with 123