Statistical filtering is a desirable mathematical framework for the estimation of the internal state variables of a dynamical system in the light of various sensors’ measurements. Since the advent of the space age, it has played a pivotal role in guidance and navigation problems in aeronautics and robotics. Its workhorse, the Kalman filter – albeit usable in a nonlinear context by linearizing about the estimated state trajectory – deeply builds upon the specificity of linear systems. As a result, key theoretical properties are lost in nonlinear contexts, in particular when dealing with challenging nonlinear problems related to the navigation of autonomous systems. However, it turns out that many such estimation problems bear a structure akin to linear systems, after a proper embedding of the state space into a matrix group has been found. Essentially, by replacing vector addition with matrix multiplication, linear observer design (or linear filter design) carries over, as well as a number of convergence and consistency guarantees discovered over the past fifteen years. We illustrate this perspective by addressing various problems, from the design of high performance industrial inertial navigation systems to robot simultaneous localization and mapping. For the latter, the geometric approach resolves problems connected to the notion of observability that have long impeded the use of the classical extended Kalman filter.
Silvère Bonnabel is Professor at University of New Caledonia and Mines ParisTech, France. He received the engineering (M. Sc) degree in applied mathematics and the Ph.D. degree in mathematics and control from Mines ParisTech in 2004 and 2007, respectively, and the Habilitation in mathematics from Sorbonne University in 2014. He was a postdoctoral fellow at the University of Liège, Belgium, in 2008, and was hired as a permanent faculty member at Mines ParisTech in 2009. In 2017, he was an Invited Fellow of Sidney Sussex College at the University of Cambridge, UK. His primary research interests revolve around systems and control, robotics, with a focus on state estimation and geometric methods. He has contributed to developing products in the aeronautics industry. He serves as an Associate Editor for IEEE Control Systems Magazine, and he has also served as an Associate Editor for Systems & Control Letters. He is a recipient of various awards including the joint IEEE & SEE Glavieux Award in 2015 and the Automatica Paper Prize in 2020.
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