Modern Inertial Technology

Lawrence, Anthony

doi:10.1007/978-1-4684-0444-9

Cited by 113 publications

(45 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…When (1) is interpreted as an Ito SDE, the FokkerPlanck equation is written explicitly as (2) We present the result without proof. The methodology was first employed by Fokker 53 and Planck.…”

Section: Fokker-planck Equationsmentioning

confidence: 99%

“…In their most general form, the results of this paper are as follows, where G is a unimodular Lie group (e.g., SO(3), SE (2), SE(3)). Given functions f i (g) for i = 1, 2 which are square integrable with respect to a bi-invariant integration measure dg on a the group (G, •), we can define the convolution product and the Fourier transform where U(·, λ) is a unitary matrix function (called an irreducible matrix representation) for each value of the parameter λ (where the set of all values of λ is called the dual of the group, and is denoted as Ĝ).…”

Section: Thenmentioning

confidence: 99%

“…We develop explicit expressions for U(g, λ) and u(X i , λ) using the exponential map and corresponding parameterizations for the groups SO(3), SE (2) and SE (3). 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%

“…These u mn (X k , p) are infinite-dimensional sparse skew-Hermitian matrices, which when truncated to a finite range −d ≤ m, n ≤ d results in (2d + 1) × (2d + 1) matrices. The truncation does not change their skew-Hermitian nature, and exponentiation of these matrices results in unitary matrices that approximate the IURs for SE (2). This approximation becomes better as d becomes large.…”

Section: Example 2: the Se(2) Case-the Sementioning

confidence: 99%

“…(2) results in: (21) Of course, this system is evolving on the group of rigid-body motions of the plane, SE (2), which has group elements that can be described in the form…”

Section: Pose Estimation For the Stochastic Cartmentioning

confidence: 99%

See 4 more Smart Citations

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

Park¹,

Liu²,

et al. 2008

Robotica

View full text Add to dashboard Cite

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.

show abstract

Section: Fokker-planck Equationsmentioning

confidence: 99%

Section: Thenmentioning

confidence: 99%

Section: Thenmentioning

confidence: 99%

Section: Example 2: the Se(2) Case-the Sementioning

confidence: 99%

“…(2) results in: (21) Of course, this system is evolving on the group of rigid-body motions of the plane, SE (2), which has group elements that can be described in the form…”

Section: Pose Estimation For the Stochastic Cartmentioning

confidence: 99%

See 3 more Smart Citations

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

Park¹,

Liu²,

et al. 2008

Robotica

View full text Add to dashboard Cite

show abstract

Initial calibration and alignment of low-cost inertial navigation units for land vehicle applications

Nebot

Durrant‐Whyte

1999

J. Robotic Syst.

View full text Add to dashboard Cite

This work presents an efficient initial calibration and alignment algorithm for a six-degree of freedom inertial measurement unit (IMU) to be used in land vehicle applications. Error models for the gyros and accelerometers are presented with a study of their perturbation in trajectory prediction. A full inertial error model is also presented to determine the sensors needed for full observability of the different perturbation parameters. Finally, dead reckoning experimental results are presented based on the initial alignment and calibration parameters obtained with the algorithms presented. The results show that the algorithms proposed provide accurate position and velocity information for extended period of time using non-aided IMU.

show abstract