ProcessNoise.hpp File Reference

General process Noise definitions. More...

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Functions
Eigen::Matrix3d	NAV::G_GaussMarkov1 (const Eigen::Vector3d &sigma2, const Eigen::Vector3d &beta)
	Submatrix 𝐆_a of the noise input matrix 𝐆
Eigen::Matrix3d	NAV::G_RandomWalk (const Eigen::Vector3d &sigma2)
	Random walk noise input matrix 𝐆
Eigen::Matrix3d	NAV::ie_Q_dr_df (const Eigen::Vector3d &S_bad, const Eigen::Matrix3d &ie_Dcm_b, const double &tau_s)
	Submatrix 𝐐_34 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::ie_Q_dr_domega (const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &ie_F_21, const Eigen::Matrix3d &ie_Dcm_b, const double &tau_s)
	Submatrix 𝐐_35 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::ie_Q_dr_dr (const Eigen::Vector3d &S_ra, const Eigen::Vector3d &S_bad, const Eigen::Vector3d &S_rg, const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &ie_F_21, const double &tau_s)
	Submatrix 𝐐_33 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::ie_Q_dr_dv (const Eigen::Vector3d &S_ra, const Eigen::Vector3d &S_bad, const Eigen::Vector3d &S_rg, const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &ie_F_21, const double &tau_s)
	Submatrix 𝐐_32 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::ie_Q_dr_psi (const Eigen::Vector3d &S_rg, const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &ie_F_21, const double &tau_s)
	Submatrix 𝐐_31 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::ien_Q_dv_domega (const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &ien_F_21, const Eigen::Matrix3d &ien_Dcm_b, const double &tau_s)
	Submatrix 𝐐_25 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::ien_Q_dv_dv (const Eigen::Vector3d &S_ra, const Eigen::Vector3d &S_bad, const Eigen::Vector3d &S_rg, const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &ien_F_21, const double &tau_s)
	Submatrix 𝐐_22 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::ien_Q_dv_psi (const Eigen::Vector3d &S_rg, const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &ien_F_21, const double &tau_s)
	Submatrix 𝐐_21 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::n_Q_dr_df (const Eigen::Vector3d &S_bad, const Eigen::Matrix3d &T_rn_p, const Eigen::Matrix3d &n_Dcm_b, const double &tau_s)
	Submatrix 𝐐_34 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::n_Q_dr_domega (const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &n_F_21, const Eigen::Matrix3d &T_rn_p, const Eigen::Matrix3d &n_Dcm_b, const double &tau_s)
	Submatrix 𝐐_35 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::n_Q_dr_dr (const Eigen::Vector3d &S_ra, const Eigen::Vector3d &S_bad, const Eigen::Vector3d &S_rg, const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &n_F_21, const Eigen::Matrix3d &T_rn_p, const double &tau_s)
	Submatrix 𝐐_33 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::n_Q_dr_dv (const Eigen::Vector3d &S_ra, const Eigen::Vector3d &S_bad, const Eigen::Vector3d &S_rg, const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &n_F_21, const Eigen::Matrix3d &T_rn_p, const double &tau_s)
	Submatrix 𝐐_32 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::n_Q_dr_psi (const Eigen::Vector3d &S_rg, const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &n_F_21, const Eigen::Matrix3d &T_rn_p, const double &tau_s)
	Submatrix 𝐐_31 of the system noise covariance matrix 𝐐
Eigen::Vector3d	NAV::psdBiasGaussMarkov (const Eigen::Vector3d &sigma2_bd, const Eigen::Vector3d &tau_bd)
	u_bias Power Spectral Density of the bias for a Gauss-Markov random process
Eigen::Matrix3d	NAV::Q_df_df (const Eigen::Vector3d &S_bad, const double &tau_s)
	Submatrix 𝐐_44 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::Q_df_dv (const Eigen::Vector3d &S_bad, const Eigen::Matrix3d &b_Dcm_ien, const double &tau_s)
	Submatrix 𝐐_42 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::Q_domega_domega (const Eigen::Vector3d &S_bgd, const double &tau_s)
	Submatrix 𝐐_55 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::Q_domega_psi (const Eigen::Vector3d &S_bgd, const Eigen::Matrix3d &b_Dcm_ien, const double &tau_s)
	Submatrix 𝐐_51 of the system noise covariance matrix 𝐐
Eigen::Matrix2d	NAV::Q_gnss (const double &S_cPhi, const double &S_cf, const double &tau_s)
	Submatrix 𝐐_66 of the system noise covariance matrix 𝐐
Eigen::Matrix3d	NAV::Q_psi_psi (const Eigen::Vector3d &S_rg, const Eigen::Vector3d &S_bgd, const double &tau_s)
	Submatrix 𝐐_11 of the system noise covariance matrix 𝐐

Detailed Description

General process Noise definitions.

Author

T. Topp (topp@.nosp@m.ins..nosp@m.uni-s.nosp@m.tutt.nosp@m.gart..nosp@m.de)

Date

2022-01-31

Function Documentation

G_GaussMarkov1()

Eigen::Matrix3d NAV::G_GaussMarkov1	(	const Eigen::Vector3d &	sigma2,
		const Eigen::Vector3d &	beta )

Submatrix 𝐆_a of the noise input matrix 𝐆

Parameters

[in]	sigma2	Variance of the noise on the measurements
[in]	beta	Gauss-Markov constant 𝛽 = 1 / 𝜏 (𝜏 correlation length)

Note

See T. Hobiger (2021) Inertialnavigation V06 - equation (6.3)

G_RandomWalk()

Eigen::Matrix3d NAV::G_RandomWalk

(

const Eigen::Vector3d &

sigma2

)

Random walk noise input matrix 𝐆

Parameters

[in]

sigma2

Variance of the noise on the measurements

Note

See T. Hobiger (2021) Inertialnavigation V06 - equation (6.3)

ie_Q_dr_df()

Eigen::Matrix3d NAV::ie_Q_dr_df	(	const Eigen::Vector3d &	S_bad,
		const Eigen::Matrix3d &	ie_Dcm_b,
		const double &	tau_s )

Submatrix 𝐐_34 of the system noise covariance matrix 𝐐

Parameters

[in]	S_bad	Power Spectral Density of the accelerometer bias variation
[in]	ie_Dcm_b	Direction Cosine Matrix from body to {i,e} coordinates
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_34

Note

See Groves (2013) equation (14.81)

ie_Q_dr_domega()

Eigen::Matrix3d NAV::ie_Q_dr_domega	(	const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	ie_F_21,
		const Eigen::Matrix3d &	ie_Dcm_b,
		const double &	tau_s )

Submatrix 𝐐_35 of the system noise covariance matrix 𝐐

Parameters

[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	ie_F_21	Submatrix 𝐅_21 of the system matrix 𝐅
[in]	ie_Dcm_b	Direction Cosine Matrix from body to {i,e} coordinates
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_35

Note

See Groves (2013) equation (14.81)

ie_Q_dr_dr()

Eigen::Matrix3d NAV::ie_Q_dr_dr	(	const Eigen::Vector3d &	S_ra,
		const Eigen::Vector3d &	S_bad,
		const Eigen::Vector3d &	S_rg,
		const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	ie_F_21,
		const double &	tau_s )

Submatrix 𝐐_33 of the system noise covariance matrix 𝐐

Parameters

[in]	S_ra	Power Spectral Density of the accelerometer random noise
[in]	S_bad	Power Spectral Density of the accelerometer bias variation
[in]	S_rg	Power Spectral Density of the gyroscope random noise
[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	ie_F_21	Submatrix 𝐅_21 of the system matrix 𝐅 in {i,e} frame
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_33

Note

See Groves (2013) equation (14.81)

ie_Q_dr_dv()

Eigen::Matrix3d NAV::ie_Q_dr_dv	(	const Eigen::Vector3d &	S_ra,
		const Eigen::Vector3d &	S_bad,
		const Eigen::Vector3d &	S_rg,
		const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	ie_F_21,
		const double &	tau_s )

Submatrix 𝐐_32 of the system noise covariance matrix 𝐐

Parameters

[in]	S_ra	Power Spectral Density of the accelerometer random noise
[in]	S_bad	Power Spectral Density of the accelerometer bias variation
[in]	S_rg	Power Spectral Density of the gyroscope random noise
[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	ie_F_21	Submatrix 𝐅_21 of the system matrix 𝐅 in {i,e} frame
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_32

Note

See Groves (2013) equation (14.81)

ie_Q_dr_psi()

Eigen::Matrix3d NAV::ie_Q_dr_psi	(	const Eigen::Vector3d &	S_rg,
		const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	ie_F_21,
		const double &	tau_s )

Submatrix 𝐐_31 of the system noise covariance matrix 𝐐

Parameters

[in]	S_rg	Power Spectral Density of the gyroscope random noise
[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	ie_F_21	Submatrix 𝐅_21 of the system matrix 𝐅 in {i,e} frame
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_31

Note

See Groves (2013) equation (14.81)

ien_Q_dv_domega()

Eigen::Matrix3d NAV::ien_Q_dv_domega	(	const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	ien_F_21,
		const Eigen::Matrix3d &	ien_Dcm_b,
		const double &	tau_s )

Submatrix 𝐐_25 of the system noise covariance matrix 𝐐

Parameters

[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	ien_F_21	Submatrix 𝐅_21 of the system matrix 𝐅
[in]	ien_Dcm_b	Direction Cosine Matrix from body to {i,e,n} frame
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_25

Note

See Groves (2013) equation (14.80)

ien_Q_dv_dv()

Eigen::Matrix3d NAV::ien_Q_dv_dv	(	const Eigen::Vector3d &	S_ra,
		const Eigen::Vector3d &	S_bad,
		const Eigen::Vector3d &	S_rg,
		const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	ien_F_21,
		const double &	tau_s )

Submatrix 𝐐_22 of the system noise covariance matrix 𝐐

Parameters

[in]	S_ra	Power Spectral Density of the accelerometer random noise
[in]	S_bad	Power Spectral Density of the accelerometer bias variation
[in]	S_rg	Power Spectral Density of the gyroscope random noise
[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	ien_F_21	Submatrix 𝐅_21 of the system matrix 𝐅 in {i,e,n} frame
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_22

Note

See Groves (2013) equation (14.81)

ien_Q_dv_psi()

Eigen::Matrix3d NAV::ien_Q_dv_psi	(	const Eigen::Vector3d &	S_rg,
		const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	ien_F_21,
		const double &	tau_s )

Submatrix 𝐐_21 of the system noise covariance matrix 𝐐

Parameters

[in]	S_rg	Power Spectral Density of the gyroscope random noise
[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	ien_F_21	Submatrix 𝐅_21 of the system matrix 𝐅 in {i,e,n} frame
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_21

Note

See Groves (2013) equation (14.81)

n_Q_dr_df()

Eigen::Matrix3d NAV::n_Q_dr_df	(	const Eigen::Vector3d &	S_bad,
		const Eigen::Matrix3d &	T_rn_p,
		const Eigen::Matrix3d &	n_Dcm_b,
		const double &	tau_s )

Submatrix 𝐐_34 of the system noise covariance matrix 𝐐

Parameters

[in]	S_bad	Power Spectral Density of the accelerometer bias variation
[in]	T_rn_p	Conversion matrix between cartesian and curvilinear perturbations to the position
[in]	n_Dcm_b	Direction Cosine Matrix from body to navigation coordinates
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_34

Note

See Groves (2013) equation (14.81)

n_Q_dr_domega()

Eigen::Matrix3d NAV::n_Q_dr_domega	(	const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	n_F_21,
		const Eigen::Matrix3d &	T_rn_p,
		const Eigen::Matrix3d &	n_Dcm_b,
		const double &	tau_s )

Submatrix 𝐐_35 of the system noise covariance matrix 𝐐

Parameters

[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	n_F_21	Submatrix 𝐅_21 of the system matrix 𝐅
[in]	T_rn_p	Conversion matrix between cartesian and curvilinear perturbations to the position
[in]	n_Dcm_b	Direction Cosine Matrix from body to navigation coordinates
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_35

Note

See Groves (2013) equation (14.81)

n_Q_dr_dr()

Eigen::Matrix3d NAV::n_Q_dr_dr	(	const Eigen::Vector3d &	S_ra,
		const Eigen::Vector3d &	S_bad,
		const Eigen::Vector3d &	S_rg,
		const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	n_F_21,
		const Eigen::Matrix3d &	T_rn_p,
		const double &	tau_s )

Submatrix 𝐐_33 of the system noise covariance matrix 𝐐

Parameters

[in]	S_ra	Power Spectral Density of the accelerometer random noise
[in]	S_bad	Power Spectral Density of the accelerometer bias variation
[in]	S_rg	Power Spectral Density of the gyroscope random noise
[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	n_F_21	Submatrix 𝐅_21 of the system matrix 𝐅
[in]	T_rn_p	Conversion matrix between cartesian and curvilinear perturbations to the position
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_33

Note

See Groves (2013) equation (14.81)

n_Q_dr_dv()

Eigen::Matrix3d NAV::n_Q_dr_dv	(	const Eigen::Vector3d &	S_ra,
		const Eigen::Vector3d &	S_bad,
		const Eigen::Vector3d &	S_rg,
		const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	n_F_21,
		const Eigen::Matrix3d &	T_rn_p,
		const double &	tau_s )

Submatrix 𝐐_32 of the system noise covariance matrix 𝐐

Parameters

[in]	S_ra	Power Spectral Density of the accelerometer random noise
[in]	S_bad	Power Spectral Density of the accelerometer bias variation
[in]	S_rg	Power Spectral Density of the gyroscope random noise
[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	n_F_21	Submatrix 𝐅_21 of the system matrix 𝐅
[in]	T_rn_p	Conversion matrix between cartesian and curvilinear perturbations to the position
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_32

Note

See Groves (2013) equation (14.81)

n_Q_dr_psi()

Eigen::Matrix3d NAV::n_Q_dr_psi	(	const Eigen::Vector3d &	S_rg,
		const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	n_F_21,
		const Eigen::Matrix3d &	T_rn_p,
		const double &	tau_s )

Submatrix 𝐐_31 of the system noise covariance matrix 𝐐

Parameters

[in]	S_rg	Power Spectral Density of the gyroscope random noise
[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	n_F_21	Submatrix 𝐅_21 of the system matrix 𝐅
[in]	T_rn_p	Conversion matrix between cartesian and curvilinear perturbations to the position
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_31

Note

See Groves (2013) equation (14.81)

psdBiasGaussMarkov()

Eigen::Vector3d NAV::psdBiasGaussMarkov	(	const Eigen::Vector3d &	sigma2_bd,
		const Eigen::Vector3d &	tau_bd )

u_bias Power Spectral Density of the bias for a Gauss-Markov random process

Parameters

[in]	sigma2_bd	𝜎²_bd standard deviation of the bias noise
[in]	tau_bd	𝜏 Correlation length in [s]

Note

See Brown & Hwang (2011) - Introduction to Random Signals and Applied Kalman Filtering (example 9.6)

Q_df_df()

Eigen::Matrix3d NAV::Q_df_df	(	const Eigen::Vector3d &	S_bad,
		const double &	tau_s )

Submatrix 𝐐_44 of the system noise covariance matrix 𝐐

Parameters

[in]	S_bad	Power Spectral Density of the accelerometer bias variation
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_44

Note

See Groves (2013) equation (14.80)

Q_df_dv()

Eigen::Matrix3d NAV::Q_df_dv	(	const Eigen::Vector3d &	S_bad,
		const Eigen::Matrix3d &	b_Dcm_ien,
		const double &	tau_s )

Submatrix 𝐐_42 of the system noise covariance matrix 𝐐

Parameters

[in]	S_bad	Power Spectral Density of the accelerometer bias variation
[in]	b_Dcm_ien	Direction Cosine Matrix from {i,e,n} to body coordinates
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_42

Note

See Groves (2013) equation (14.80)

Q_domega_domega()

Eigen::Matrix3d NAV::Q_domega_domega	(	const Eigen::Vector3d &	S_bgd,
		const double &	tau_s )

Submatrix 𝐐_55 of the system noise covariance matrix 𝐐

Parameters

[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_55

Note

See Groves (2013) equation (14.80)

Q_domega_psi()

Eigen::Matrix3d NAV::Q_domega_psi	(	const Eigen::Vector3d &	S_bgd,
		const Eigen::Matrix3d &	b_Dcm_ien,
		const double &	tau_s )

Submatrix 𝐐_51 of the system noise covariance matrix 𝐐

Parameters

[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	b_Dcm_ien	Direction Cosine Matrix from {i,e,n} to body coordinates
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_51

Note

See Groves (2013) equation (14.80)

Q_gnss()

Eigen::Matrix2d NAV::Q_gnss	(	const double &	S_cPhi,
		const double &	S_cf,
		const double &	tau_s )

Submatrix 𝐐_66 of the system noise covariance matrix 𝐐

Parameters

[in]	S_cPhi	Power Spectral Density of the receiver clock phase drift in [m^2 s^-1]
[in]	S_cf	Power Spectral Density of the receiver clock frequency-drift [m^2 s^-3]
[in]	tau_s	Time interval in [s]

Returns

See Groves (2013) equation (9.152)

Q_psi_psi()

Eigen::Matrix3d NAV::Q_psi_psi	(	const Eigen::Vector3d &	S_rg,
		const Eigen::Vector3d &	S_bgd,
		const double &	tau_s )

Submatrix 𝐐_11 of the system noise covariance matrix 𝐐

Parameters

[in]	S_rg	Power Spectral Density of the gyroscope random noise
[in]	S_bgd	Power Spectral Density of the gyroscope bias variation
[in]	tau_s	Time interval in [s]

Returns

The 3x3 matrix 𝐐_11

Note

See Groves (2013) equation (14.81)