Emrys::Kinematic< Scalar_, Paths > Class Template Reference

A chain of two or more derivatives whose last one is driven by white noise: constant velocity and constant acceleration. More...

#include <EmrysFilter/Models/ProcessModel.h>

Inheritance diagram for Emrys::Kinematic< Scalar_, Paths >:

Public Types
using	Scalar = Scalar_
	the scalar type of the spectral density
using	SpectralDensityMatrix = Eigen::Matrix<Scalar_, Dimension, Dimension>
	The spectral density's type: \( d \times d \) over the last derivative's coordinates.

Public Member Functions
	Kinematic (Scalar_ spectralDensity)
	Spectral density spectralDensity on every axis, independently.
	Kinematic (SpectralDensityMatrix const &spectralDensity)
	Spectral density matrix spectralDensity.
SpectralDensityMatrix const &	spectralDensity () const

Static Public Attributes
static constexpr int	Order = int(sizeof...(Paths))
	Number of blocks in the chain.
static constexpr int	Dimension = First::FixedValueDimension
	Dimension of each block.

Detailed Description

template<RealScalar Scalar_, typename... Paths>
class Emrys::Kinematic< Scalar_, Paths >

A chain of two or more derivatives whose last one is driven by white noise: constant velocity and constant acceleration.

Paths name the chain in order, such as position and then velocity. Each is a tag or a TagPath to a fixed-size block, and all have the same dimension \( d \). A chain adds one block's value to another's, so each block's differences must have the size of its values. A single block is a RandomWalk, which adds nothing and takes any block. Over a step \( \Delta t \), block \( i \) of an \( N \)-block chain moves by \( x_i' = \sum_{j \ge i} \frac{\Delta t^{j-i}}{(j-i)!} x_j \), and the noise is

\[ Q_{ij} = \frac{\Delta t^{k}}{k \, (N-1-i)! \, (N-1-j)!} \, Q_c, \qquad k = 2N - 1 - i - j, \]

the exact discretization of white noise with spectral density \( Q_c \) (a \( d \times d \) matrix) on the last derivative. The model is affine in its blocks and returns its Jacobian, so Linearized predicts it exactly without differentiating.

These are the continuous white noise acceleration and jerk models of Bar-Shalom, Li, and Kirubarajan (2001), for any number of derivatives: the closed form of the exact discretization (as given in Discretization), \( \int_0^{\Delta t} e^{A s} G Q_c G^T e^{A^T s}\,ds \) for the chain's nilpotent \( A \).

Scalar_ is the type of \( Q_c \), and comes before the paths, as a type ending in a pack takes it; the aliases take it last, double by default. The step computes in the scalar of the step it is given, the filter's, converting \( Q_c \) to it where the two differ.

See also

RandomWalk, ConstantVelocity, ConstantAcceleration

Member Function Documentation

spectralDensity()

template<RealScalar Scalar_, typename... Paths>

SpectralDensityMatrix const & Emrys::Kinematic< Scalar_, Paths >::spectralDensity ( ) const

inlinenodiscard

Returns

the spectral density \( Q_c \) of the white noise on the last derivative

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The documentation for this class was generated from the following file:

• EmrysFilter/Models/ProcessModel.h