Exponential Fourier Densities on $SO( 3 )$ and Optimal Estimation and Detection for Rotational Processes

Lo, James Ting-Ho; Eshleman, Linda R.

doi:10.1137/0136007

Cited by 33 publications

(24 citation statements)

References 6 publications

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“…22,24 Abstract estimation problems on SO(3) and compact Lie groups in general have been reported elsewhere. 36,37 Both of the variations of the problem described above can be formulated as follows:…”

Section: Fourier-based Estimation On Lie Groups and Applicationsmentioning

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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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: Fourier-based Estimation On Lie Groups and Applicationsmentioning

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 this section, we develop a computational method to propagate this density along the 47th IEEE CDC, Cancun, Mexico, Dec. [9][10][11]2008 TuA02.5…”

Section: Global Symplectic Uncertainty Propagationmentioning

confidence: 99%

“…Substituting (10), we can compute the Fourier spectrum of the propagated probability density function. The propagated distribution can be reconstructed by (11). This is a global particle-based method to construct the propagated probability density function on SO(3) × R 3 .…”

Section: B Noncommutative Harmonic Analysis On So(3)mentioning

confidence: 99%

“…Although it is not widely known in the engineering community, the theories of probability and stochastic processes on manifolds have been developed by theoretical statisticians [8], [9], [10]. Earlier works on attitude estimation on SO(3) include [11], where a probability density function is expressed using noncommutative harmonic analysis. This idea of using Fourier analysis on manifolds has been applied in [12], [13], [14], [15], and recently, results in [11] are extended to include the effects of process noise and sensor parameters in [16].…”

Section: Introductionmentioning

confidence: 99%

“…Earlier works on attitude estimation on SO(3) include [11], where a probability density function is expressed using noncommutative harmonic analysis. This idea of using Fourier analysis on manifolds has been applied in [12], [13], [14], [15], and recently, results in [11] are extended to include the effects of process noise and sensor parameters in [16]. The use of noncommutative harmonic analysis allows a probability density function to be expressed globally on SO(3) without needing partition of unity methods on multiple coordinate charts.…”

Section: Introductionmentioning

confidence: 99%

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Global symplectic uncertainty propagation on SO(3)

Lee

Leok

McClamroch

2008

2008 47th IEEE Conference on Decision and Control

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Abstract-This paper introduces a global uncertainty propagation scheme for the attitude dynamics of a rigid body, through a combination of numerical parametric uncertainty techniques, noncommutative harmonic analysis, and geometric numerical integration. This method is distinguished from prior approaches, as it allows one to consider probability densities that are global, and are not supported on only a single coordinate chart on the manifold. It propagates a global probability density through the full attitude dynamics, instead of replacing angular velocity dynamics with a gyro bias model. The use of Lie group variational integrators, that are symplectic and remain on the Lie group, as the underlying numerical propagator ensures that the advected probability densities respect the geometric properties of uncertainty propagation in Hamiltonian systems, which arise as consequence of the Gromov nonsqueezing theorem from symplectic geometry. We also describe how the global uncertainty propagation scheme can be applied to the problem of global attitude estimation.

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Spherical Deconvolution

Healy

Hendriks

Kim

1998

Journal of Multivariate Analysis

104

107

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This paper proposes nonparametric deconvolution density estimation over S 2 . Here we would think of the S 2 elements of interest being corrupted by random SO(3) elements (rotations). The resulting density on the observations would be a convolution of the SO(3) density with the true S 2 density. Consequently, the methodology, as in the Euclidean case, would be to use Fourier analysis on SO(3) and S 2 , involving rotational and spherical harmonics, respectively. We especially consider the case where the deconvolution operator is a bounded operator lowering the Sobolev order by a finite amount. Consistency results are obtained with rates of convergence calculated under the expected L 2 and Sobolev square norms that are proportionally inverse to some power of the sample size. As an example we introduce the rotational version of the Laplace distribution.1998 Academic Press

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Exponential Fourier Densities on $SO( 3 )$ and Optimal Estimation and Detection for Rotational Processes

Cited by 33 publications

References 6 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

Global symplectic uncertainty propagation on SO(3)

Spherical Deconvolution

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