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Approximation Filters for Nonlinear Systems
Published in Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala, Nonlinear Filtering, 2017
Jitendra R. Raol, Girija Gopalratnam, Bhekisipho Twala
The point-mass approximation (PMA) may be regarded as a special case of the GSA where the different pdfs at the grid points are Dirac impulses with given areas called the point masses, and hence the resulting density is always positive. We consider the problem of state estimation on manifolds, which is commonly performed by embedding the manifold in a linear space of higher dimension, by combining estimation techniques for linear spaces with some projection scheme [23]. The main idea is to examine how robust state estimation on compact manifolds of low dimension can be performed while still respecting the geometric nature of the problem. The main aspect in utilizing non-parametric filtering in curved space is computational cost compared to the linear space framework. The equidistant meshes do not exist. The idea is then to cover the whole manifold by a mesh with finitely many nodes. Also, the idea is to dynamically adapt the mesh to enhance the degree of detail in regions of interest, especially where the probability density is high. The point-mass approach for filtering in curved space has three components: (1) compute (and update) a tessellation of the manifold; (2) implementation of the time propagation and measurement update, requiring a system model which cannot have additive noise on the state; and (3) obtain a point estimate. The manifold here is the differentiable Riemannian manifold. The curve called geodesic on the manifold locally connects points along the shortest path between points, and the ‘exponential map’ maps the vectors to points on the manifold. A geodesic complete map has the exponential map defined for all vectors. A tessellation of the manifold is a set of subsets of the manifold, such that there is neither an overlay nor any gap between regions. That is, the union of all the regions would be the whole manifold; the intersection of any two regions would have zero measure.
Consensus-based formation control for nonholonomic vehicles with parallel desired formations
Published in International Journal of Control, 2021
The inverse of exponential map is named logarithm map . Provided , , and , is defined as where , , and . Then are referred to the exponential coordinates of group element g. Note that for the case of , is a two valued function corresponding to or π. Thus, we can specify its value as needed, which leads to more choices when designing the control law.
Logarithmic control, trajectory tracking, and formation for nonholonomic vehicles on Lie group SE(2)
Published in International Journal of Control, 2019
Adopting the notation from Bullo and Murray (1995), we develop the exponential coordinates of SE(2) and its time derivative using the logarithm map. Since SE(2) is a matrix Lie group, the exponential map coincides with the matrix exponential as The inverse map is also given by the matrix logarithm as provided trace(g) ≠ −1 where q = (qx, qy)T = A−1(θ)p, , and . We refer to as the exponential coordinates of g which is parameterised by X = (θ, qx, qy)T. Note that even though and are both the Lie algebra elements corresponding to g but they represent velocity and configuration, respectively.