Engineering Applications of Noncommutative Harmonic Analysis

Kyatkin, Alexander B.; Chirikjian, Gregory S.

doi:10.1201/9781420041767

Cited by 205 publications

(275 citation statements)

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“…Examples of (5) will be demonstrated in subsequent sections in the context of gyroscopes, kinematic carts, and flexible needle steering. Focusing on the "r" case, when h and H are constant, we can write (6) where When considering stochastic differential equations and the corresponding Fokker-Planck equations, it is usually important to specify whether the Ito or Stratanovich form is used. Without getting into too much detail, it suffices to say that the Ito form has been assumed in the derivation of both of the above Fokker-Planck equations.…”

Section: Fokker-planck Equationsmentioning

confidence: 99%

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Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map

Park¹,

Liu²,

et al. 2008

Robotica

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SUMMARYA nonholonomic system subjected to external noise from the environment, or internal noise in its own actuators, will evolve in a stochastic manner described by an ensemble of trajectories. This ensemble of trajectories is equivalent to the solution of a Fokker-Planck equation that typically evolves on a Lie group. If the most likely state of such a system is to be estimated, and plans for subsequent motions from the current state are to be made so as to move the system to a desired state with high probability, then modeling how the probability density of the system evolves is critical. Methods for solving Fokker-Planck equations that evolve on Lie groups then become important. Such equations can be solved using the operational properties of group Fourier transforms in which irreducible unitary representation (IUR) matrices play a critical role. Therefore, we develop a simple approach for the numerical approximation of all the IUR matrices for two of the groups of most interest in robotics: the rotation group in three-dimensional space, SO(3), and the Euclidean motion group of the plane, SE(2). This approach uses the exponential mapping from the Lie algebras of these groups, and takes advantage of the sparse nature of the Lie algebra representation matrices. Other techniques for density estimation on groups are also explored. The computed densities are applied in the context of probabilistic path planning for kinematic cart in the plane and flexible needle steering in three-dimensional space. In these examples the injection of artificial noise into the computational models (rather than noise in the actual physical systems) serves as a tool to search the configuration spaces and plan paths. Finally, we illustrate how density estimation problems arise in the characterization of physical noise in orientational sensors such as gyroscopes.

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Section: Fokker-planck Equationsmentioning

confidence: 99%

“…In particular, given the scalar parameters x i ∈ IR, the linearity of the Lie algebra allows one to write and a famous theorem states that 7,14,6 :…”

Section: Thenmentioning

confidence: 99%

Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map

Park¹,

Liu²,

et al. 2008

Robotica

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“…In our previous works 28, 51 , a methodology for solving equations such as (3) was developed using the technique of the Fourier transform for SE (3). The matrix elements of this transform are 51, 61 (4) where for each r ∈ Z and p ∈ ℜ + , U r (a,R;p) is the irreducible unitary representation matrix of SE (3), p is the frequency factor introduced by the transform, and dRda is the invariant integration measure for SE (3).…”

Section: Review Of the General Inextensible Semi-flexible Macromolecumentioning

confidence: 92%

“…4). Therefore, in general, denoting f(β h ) as the probability distribution of the bending angle β h at the hinge, the PDF of rigid-body motion on SE(3) at this hinge can be obtained as a convolution (15) where α ∈ [0,2π], β ∈ [0, π], and γ ∈ [0,2π], and Then the Fourier transform of f j (a,R) on SE(3) can be obtained as (18) where (19) (20) where δ i,j is the Kronecker delta function, and is the generalized Legendre function 51,62 .…”

Section: The Hinge Casementioning

confidence: 99%

“…For some basic polymer models, closed-form formulas have been derived for the probability distributions of loop length 38 , segmental orientations 39 , trajectories of a segment 40 , radius of gyration 41 , end-to-end position and distance [42][43][44][45][46][47][48][49] , and moments 46,49,50 . In our previous work, we showed that the probability density function (PDF) of the end-to-end pose for the most general model of the inextensible semi-flexible macromolecular chain can be obtained by either solving a diffusion equation or convolving PDFs for short segments of the chain 28,[51][52][53] .…”

Section: Introductionmentioning

confidence: 99%

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Conformational Statistics of Semiflexible Macromolecular Chains with Internal Joints

Chirikjian

2006

Macromolecules

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Fluctuations in the bending angles at internal irregularities of DNA and RNA (such as symmetric loops, bulges, and nicks/gaps) have been observed from various experiments. However, little effort has been made to computationally predict and explain the statistical behavior of semiflexible chains with internal defects. In this paper, we describe the general structure of these macromolecular chains as inextensible elastic chains with one or more internal joints which have limited ranges of rotation, and propose a method to compute the probability density functions of the end-to-end pose of these macromolecular chains. Our method takes advantage of the operational properties of the non-commutative Fourier transform for the group of rigid-body motions in three-dimensional space, SE(3). Two representative types of joints, the hinge for planar rotation and the ball joint for spatial rotation, are discussed in detail. The proposed method applies to various stiffness models of semi-flexible chain-like macromolecules. Examples are calculated using the Kratky-Porod model with specified stiffness, angular fluctuation, and joint locations. Entropic effects associated with internal angular fluctuations of semi-flexible macromolecular chains with internal joints can be computed using this formulation. Our method also provides a potential tool to detect the existence of internal irregularities.

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Data analysis onnonstandardspaces

Huckemann

Eltzner

2020

WIREs Computational Stats

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The task to write on data analysis on nonstandard spaces is quite substantial, with a huge body of literature to cover, from parametric to nonparametrics, from shape spaces to Wasserstein spaces. In this survey we convey simple (e.g., Fréchet means) and more complicated ideas (e.g., empirical process theory), common to many approaches with focus on their interaction with one‐another. Indeed, this field is fast growing and it is imperative to develop a mathematical view point, drawing power, and diversity from a higher level of abstraction, for example, by introducing generalized Fréchet means. While many problems have found ingenious solutions (e.g., Procrustes analysis for principal component analysis [PCA] extensions on shape spaces and diffusion on the frame bundle to mimic anisotropic Gaussians), more problems emerge, often more difficult (e.g., topology and geometry influencing limiting rates and defining generic intrinsic PCA extensions). Along this survey, we point out some open problems, that will, as it seems, keep mathematicians, statisticians, computer and data scientists busy for a while. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Analysis of High Dimensional Data

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Engineering Applications of Noncommutative Harmonic Analysis

Cited by 205 publications

References 0 publications

Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map

Kinematic state estimation and motion planning for stochastic nonholonomic systems using the exponential map

Conformational Statistics of Semiflexible Macromolecular Chains with Internal Joints

Data analysis onnonstandardspaces

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