Gaussian approximation of non-linear measurement models on Lie groups

Abstract

Extended Kalman filters on Lie groups arise naturally in the context of pose estimation and more generally in robot localization and mapping. Typically in such settings one deals with nonlinear measurement models that are handled through linearization and linearized uncertainty transformation. To circumvent the loss of accuracy resulting from the typical coordinate-based linearization, this paper develops a method for accurately describing the probability density associated with nonlinear measurement models by a second-order approximation of a distribution defined directly on the Lie group configuration space. We show that, like the case of linearized measurement models, this density can be described well as a Gaussian distribution in exponential coordinates (though with different mean and covariance than those that result from linearized measurement models). And therefore previously developed methods for propagation of uncertainty and fusion of measurements can be applied to this generalized formulation without the a priori assumption of linearized measurement. A case study using a range-bearing model in planar robot localization is presented to demonstrate the method.

A Mathematical Framework for IMU Error Propagation with Applications to Preintegration

“…Finally green dots are generated using the endpoint distribution computed by a standard (multiplicative) EKF: we see linearization implies the assumed dispersion lies within a horizontal plane. However, the true distribution (black) is "banana" shaped in 3D, as already observed mainly in the case of poses in 2D for wheeled robots [26,11,1,12,2], and (30) captures this effect and agrees with ground truth.…”

“…Using the exponential map of SE(3) to describe statistical dispersion of poses has been often advocated. In the robotics community, early attempts date back to [30], and references [11,1,12,2,33,14,3,22,29,26,32] revolve around those ideas. Gaussians in exponential coordinates are also referred to as concentrated Gaussians [8].…”

“…Geometric approaches to associating uncertainty with poses, essentially for mobile robot localization, have been quite successful in the robotics community over the past decade. Since the discovery that mobile robots dispersion under the effect of sensor noise resembles more a "banana" than a standard Gaussian ellipse, which can be traced back to [31], studies have evidenced the fact that the Lie group structure of the configuration space SE(3) plays a prominent role in probabilistic robotics, see [1,12,2,33,14,3,22,29]. In particular, Gaussian distributions in (Lie) exponential coordinates provide accurate approximations of banana distributions as first advocated by [26].…”

Covid-19 Incidence Rate Evolution Modeling using Dual Wave Gaussian-Lorentzian Composite Functions

“…The CLT states that when an observed random variable is the sum of many random processes, the resulting distribution of the random variable is a Gaussian distribution. Further, the Gaussian function has been used for non-linear approximation problems due to its simplicity and mathematical transformative properties [18, 19]. To account for the uncertainties during the start, growth, decay and the end of the Covid-19 incidence rates, the Lorentzian function is chosen.…”

Global unscented attitude estimation via the matrix Fisher distributions on SO(3)

“…Recent literature is rich with filtering techniques and measurement models developed in terms of exponential coordinates [17], [18], [19], [20]. This is perhaps the most natural approach to develop an estimator formally on an abstract Lie group, while taking advantages of the fact that the lie algebra is a linear space.…”