A micro Lie theory for state estimation in robotics

Abstract

A Lie group is an old mathematical abstract object dating back to the XIX century, when mathematician Sophus Lie laid the foundations of the theory of continuous transformation groups. Its influence has spread over diverse areas of science and technology many years later. In robotics, we are recently experiencing an important trend in its usage, at least in the fields of estimation, and particularly in motion estimation for navigation. Yet for a vast majority of roboticians, Lie groups are highly abstract constructions and therefore difficult to understand and to use.In estimation for robotics it is often not necessary to exploit the full capacity of the theory, and therefore an effort of selection of materials is required. In this paper, we will walk through the most basic principles of the Lie theory, with the aim of conveying clear and useful ideas, and leave a significant corpus of the Lie theory behind. Even with this mutilation, the material included here has proven to be extremely useful in modern estimation algorithms for robotics, especially in the fields of SLAM, visual odometry, and the like.Alongside this micro Lie theory, we provide a chapter with a few application examples, and a vast reference of formulas for the major Lie groups used in robotics, including most jacobian matrices and the way to easily manipulate them. We also present a new C++ template-only library implementing all the functionality described here.

Jacobian Computation for Cumulative B-Splines on SE(3) and Application to Continuous-Time Object Tracking

“…where R wo ∈ SO(3) and t ∈ R 3 encode respectively, the orientation and translation of {o} with respect to {w}. SE(3) is both a group and a smooth manifold, implying that at each point, T ∈ SE(3), exists a unique tangent space called Lie Algebra or se(3), which can be defined locally at T, and at the identity I [38]. An element τ ∧ ∈ se(3) has the following form:…”

“…where we have used the capitalized notation of [38]. This way, we can compose a transformation matrix T wo with another parameterized in the local tangent space: T wo Exp(τ o ), or in the tangent space defined at the identity: Exp(τ w )T wo .…”

“…Lastly, due to the its importance in the derivation of the Jacobians (Sec. IV-B), we introduce the left Jacobian J l of SE(3) [38]. It is defined as:…”

Low-Cost Inertial Aiding for Deep-Urban Tightly-Coupled Multi-Antenna Precise GNSS

“…This paper adopts the conventions and notation of Solá et al [35], which appeals to Lie theory to unify and generalize the various methods so that the filtering equations are agnostic to the specific choice of attitude parameterization. The filter state x k is a point on the composite manifold X ∆ " R 3 R3 ˆSOp3q ˆR3 ˆR3 , which has N x " 15 independent degrees of freedom.…”

“…where arctan denotes the 4-quadrant arctangent (i.e., atan2), cx denotes cospxq, and sx denotes sinpxq. These attitude parameterizations could, of course, easily be substituted with alternate parameterizations such as unit quaternions and axisangle rotation vectors, with appropriate redefinition of the Exp and Log maps following [35].…”