NAG Library Routine Document

G13EKF

G13EKF applies the Unscented Kalman Filter (UKF), as described in Julier and Uhlmann (1997b) to a nonlinear state space model, with additive noise, which, at time

t

, can be described by:

\begin{matrix} x_{t + 1} & = F (x_{t}) + v_{t} \\ y_{t} & = H (x_{t}) + u_{t} \end{matrix}

where

x_{t}

represents the unobserved state vector of length

m_{x}

and

y_{t}

the observed measurement vector of length

m_{y}

. The process noise is denoted

v_{t}

, which is assumed to have mean zero and covariance structure

Σ_{x}

, and the measurement noise by

u_{t}

, which is assumed to have mean zero and covariance structure

Σ_{y}

3.1 Unscented Kalman Filter Algorithm

Given

{\hat{x}}_{0}

, an initial estimate of the state and

P_{0}

and initial estimate of the state covariance matrix, the UKF can be described as follows:

(a)

Generate a set of sigma points (see Section 3.2):

X_{t}

= [{\hat{x}}_{t - 1} {\hat{x}}_{t - 1} + γ \sqrt{P_{t - 1}} {\hat{x}}_{t - 1} - γ \sqrt{P_{t - 1}}]

(1)

(b)

Evaluate the known model function

F

F_{t}

= F (X_{t})

(2)

The function

F

is assumed to accept the

m_{x} \times n

matrix,

X_{t}

and return an

m_{x} \times n

matrix,

F_{t}

. The columns of both

X_{t}

and

F_{t}

correspond to different possible states. The notation

F_{t, i}

is used to denote the

i

th column of

F_{t}

, hence the result of applying

F

to the

i

th possible state.

(c)

Time Update:

${\hat{x}}_{t}^{_}$	$= \sum_{i = 1}^{n} W_{i}^{m} F_{t, i}$	(3)
$P_{t}^{_}$	$= \sum_{i = 1}^{n} W_{i}^{c} (F_{t, i} - {\hat{x}}_{t}^{_}) {(F_{t, i} - {\hat{x}}_{t}^{_})}^{T} + Σ_{x}$	(4)

(d)

Redraw another set of sigma points (see Section 3.2):

Y_{t}

= [{\hat{x}}_{t}^{_} {\hat{x}}_{t}^{_} + γ \sqrt{P_{t}^{_}} {\hat{x}}_{t}^{_} - γ \sqrt{P_{t}^{_}}]

(5)

(e)

Evaluate the known model function

H

H_{t}

= H (Y_{t})

(6)

The function

H

is assumed to accept the

m_{x} \times n

matrix,

Y_{t}

and return an

m_{y} \times n

matrix,

H_{t}

. The columns of both

Y_{t}

and

H_{t}

correspond to different possible states. As above

H_{t, i}

is used to denote the

i

th column of

H_{t}

(f)

Measurement Update:

${\hat{y}}_{t}$	$= \sum_{i = 1}^{n} W_{i}^{m} H_{t, i}$	(7)
$P_{y y_{t}}$	$= \sum_{i = 1}^{n} W_{i}^{c} (H_{t, i} - {\hat{y}}_{t}) {(H_{t, i} - {\hat{y}}_{t})}^{T} + Σ_{y}$	(8)
$P_{x y_{t}}$	$= \sum_{i = 1}^{n} W_{i}^{c} (F_{t, i} - {\hat{x}}_{t}^{_}) {(H_{t, i} - {\hat{y}}_{t})}^{T}$	(9)
$K_{t}$	$= P_{x y_{t}} P_{y y_{t}}^{- 1}$	(10)
${\hat{x}}_{t}$	$= {\hat{x}}_{t}^{_} + K_{t} (y_{t} - {\hat{y}}_{t})$	(11)
$P_{t}$	$= P_{t}^{_} - K_{t} P_{y y_{t}} K_{t}^{T}$	(12)

Here

K_{t}

is the Kalman gain matrix,

{\hat{x}}_{t}

is the estimated state vector at time

t

and

P_{t}

the corresponding covariance matrix. Rather than implementing the standard UKF as stated above G13EKF uses the square-root form described in the Haykin (2001).

3.2 Sigma Points

A nonlinear state space model involves propagating a vector of random variables through a nonlinear system and we are interested in what happens to the mean and covariance matrix of those variables. Rather than trying to directly propagate the mean and covariance matrix, the UKF uses a set of carefully chosen sample points, referred to as sigma points, and propagates these through the system of interest. An estimate of the propagated mean and covariance matrix is then obtained via the weighted sample mean and covariance matrix.

For a vector of

m

random variables,

x

, with mean

μ

and covariance matrix

Σ

, the sigma points are usually constructed as:

X_{t} = [μ μ + γ \sqrt{Σ} μ - γ \sqrt{Σ}]

When calculating the weighted sample mean and covariance matrix two sets of weights are required, one used when calculating the weighted sample mean, denoted

W^{m}

and one used when calculated the weighted sample covariance matrix, denoted

W^{c}

. The weights and multiplier,

γ

, are constructed as follows:

\begin{matrix} λ & = α^{2} (L + κ) - L \\ γ & = \sqrt{L + λ} \\ W_{i}^{m} & = \{\begin{matrix} \frac{λ}{L + λ} & i = 1 \\ \frac{1}{2 (L + λ)} & i = 2, 3, \dots, 2 L + 1 \end{matrix} \\ W_{i}^{c} & = \{\begin{matrix} \frac{λ}{L + λ} + 1 - α^{2} + β & i = 1 \\ \frac{1}{2 (L + λ)} & i = 2, 3, \dots, 2 L + 1 \end{matrix} \end{matrix}

where, usually

L = m

and

α, β

and

κ

are constants. The total number of sigma points,

n

, is given by

2 L + 1

. The constant

α

is usually set to somewhere in the range

10^{- 4} \leq α \leq 1

and for a Gaussian distribution, the optimal values of

κ

and

β

are

3 - L

and

2

respectively.

The constants,

κ

α

and

β

are given by

κ = 3 - m_{x}

α = 1.0

and

β = 2

. If more control is required over the construction of the sigma points then the reverse communication routine, G13EJF, can be used instead.

4 References

Haykin S (2001) Kalman Filtering and Neural Networks John Wiley and Sons

Julier S J (2002) The scaled unscented transformation Proceedings of the 2002 American Control Conference (Volume 6) 4555–4559

Julier S J and Uhlmann J K (1997a) A consistent, debiased method for converting between polar and Cartesian coordinate systems Proceedings of AeroSense97, International Society for Optics and Phonotonics 110–121

Julier S J and Uhlmann J K (1997b) A new extension of the Kalman Filter to nonlinear systems International Symposium for Aerospace/Defense, Sensing, Simulation and Controls (Volume 3) 26

5 Parameters

1: $MX$ – INTEGERInput

On entry:

m_{x}

, the number of state variables.

Constraint:

MX \geq 1

2: $MY$ – INTEGERInput

On entry:

m_{y}

, the number of observed variables.

Constraint:

MY \geq 1

3: $Y (MY)$ – REAL (KIND=nag_wp) arrayInput

On entry:

y_{t}

, the observed data at the current time point.

4: $LX (MX, MX)$ – REAL (KIND=nag_wp) arrayInput

On entry:

L_{x}

, such that

L_{x} L_{x}^{T} = Σ_{x}

, i.e., the lower triangular part of a Cholesky decomposition of the process noise covariance structure. Only the lower triangular part of LX is referenced.

Σ_{x}

is time dependent, then the value supplied should be for time

t

5: $LY (MY, MY)$ – REAL (KIND=nag_wp) arrayInput

On entry:

L_{y}

, such that

L_{y} L_{y}^{T} = Σ_{y}

, i.e., the lower triangular part of a Cholesky decomposition of the observation noise covariance structure. Only the lower triangular part of LY is referenced.

Σ_{y}

is time dependent, then the value supplied should be for time

t

6: $F$ – SUBROUTINE, supplied by the user.External Procedure

The state function,

F

as described in (b).

The specification of F is:

SUBROUTINE F (

MX, N, XT, FXT, IUSER, RUSER, INFO)

INTEGER	MX, N, IUSER(*), INFO
REAL (KIND=nag_wp)	XT(MX,N), FXT(MX,N), RUSER(*)

1: $MX$ – INTEGERInput: On entry: $m_{x}$ , the number of state variables.
2: $N$ – INTEGERInput: On entry: $n$ , the number of sigma points.
3: $XT (MX, N)$ – REAL (KIND=nag_wp) arrayInput: On entry: $X_{t}$ , the sigma points generated in (a). For the $j$ th sigma point, the value for the $i$ th parameter is held in $XT (i, j)$ , for $i = 1, 2, \dots, m_{x}$ and $j = 1, 2, \dots, n$ .
4: $FXT (MX, N)$ – REAL (KIND=nag_wp) arrayOutput: On exit: $F (X_{t})$ .
For the $j$ th sigma point the value for the $i$ th parameter should be held in $FXT (i, j)$ , for $i = 1, 2, \dots, m_{x}$ and $j = 1, 2, \dots, n$ .
5: $IUSER (*)$ – INTEGER arrayUser Workspace
6: $RUSER (*)$ – REAL (KIND=nag_wp) arrayUser Workspace: F is called with the parameters IUSER and RUSER as supplied to G13EKF. You are free to use the arrays IUSER and RUSER to supply information to F as an alternative to using COMMON global variables.
7: $INFO$ – INTEGERInput/Output: On entry: $INFO = 0$ .

On exit: set INFO to a nonzero value if you wish G13EKF to terminate with $IFAIL = 61$ .

F must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which G13EKF is called. Parameters denoted as Input must not be changed by this procedure.

7: $H$ – SUBROUTINE, supplied by the user.External Procedure

The measurement function,

H

as described in (e).

The specification of H is:

SUBROUTINE H (

MX, MY, N, YT, HYT, IUSER, RUSER, INFO)

INTEGER	MX, MY, N, IUSER(*), INFO
REAL (KIND=nag_wp)	YT(MX,N), HYT(MY,N), RUSER(*)

1: $MX$ – INTEGERInput: On entry: $m_{x}$ , the number of state variables.
2: $MY$ – INTEGERInput: On entry: $m_{y}$ , the number of observed variables.
3: $N$ – INTEGERInput: On entry: $n$ , the number of sigma points.
4: $YT (MX, N)$ – REAL (KIND=nag_wp) arrayInput: On entry: $Y_{t}$ , the sigma points generated in (d). For the $j$ th sigma point, the value for the $i$ th parameter is held in $YT (i, j)$ , for $i = 1, 2, \dots, m_{x}$ and $j = 1, 2, \dots, n$ , where $m_{x}$ is the number of state variables and $n$ is the number of sigma points.
5: $HYT (MY, N)$ – REAL (KIND=nag_wp) arrayOutput: On exit: $H (Y_{t})$ .
For the $j$ th sigma point the value for the $i$ th parameter should be held in $HYT (i, j)$ , for $i = 1, 2, \dots, m_{y}$ and $j = 1, 2, \dots, n$ .
6: $IUSER (*)$ – INTEGER arrayUser Workspace
7: $RUSER (*)$ – REAL (KIND=nag_wp) arrayUser Workspace: H is called with the parameters IUSER and RUSER as supplied to G13EKF. You are free to use the arrays IUSER and RUSER to supply information to H as an alternative to using COMMON global variables.
8: $INFO$ – INTEGERInput/Output: On entry: $INFO = 0$ .

On exit: set INFO to a nonzero value if you wish G13EKF to terminate with $IFAIL = 71$ .

H must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which G13EKF is called. Parameters denoted as Input must not be changed by this procedure.

8: $X (MX)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry:

{\hat{x}}_{t - 1}

the state vector for the previous time point.

On exit:

{\hat{x}}_{t}

the updated state vector.

9: $ST (MX, MX)$ – REAL (KIND=nag_wp) arrayInput/Output

On entry:

S_{t}

, such that

S_{t - 1} S_{t - 1}^{T} = P_{t - 1}

, i.e., the lower triangular part of a Cholesky decomposition of the state covariance matrix at the previous time point. Only the lower triangular part of ST is referenced.

On exit:

S_{t}

, the lower triangular part of a Cholesky factorization of the updated state covariance matrix.

10: $IUSER (*)$ – INTEGER arrayUser Workspace

11: $RUSER (*)$ – REAL (KIND=nag_wp) arrayUser Workspace

IUSER and RUSER are not used by G13EKF, but are passed directly to F and H and may be used to pass information to these routines as an alternative to using COMMON global variables.

12: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 11$: On entry, $MX = 〈value〉$ .
Constraint: $MX \geq 1$ .

$IFAIL = 21$: On entry, $MY = 〈value〉$ .
Constraint: $MY \geq 1$ .

$IFAIL = 61$: User requested termination in F.

$IFAIL = 71$: User requested termination in H.

$IFAIL = 301$: A weight was negative and it was not possible to downdate the Cholesky factorization.

$IFAIL = 302$: Unable to calculate the Kalman gain matrix.

$IFAIL = 303$: Unable to calculate the Cholesky factorization of the updated state covariance matrix.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

G13EKF is not threaded by NAG in any implementation.

G13EKF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example implements the following nonlinear state space model, with the state vector

x

and state update function

F

given by:

\begin{matrix} m_{x} & = 3 \\ x_{t + 1} & = {(\begin{matrix} ξ_{t + 1} & η_{t + 1} & θ_{t + 1} \end{matrix})}^{T} \\ = F (x_{t}) + v_{t} \\ = x_{t} + (\begin{array}{l} \cos θ_{t} & - \sin θ_{t} & 0 \\ \sin θ_{t} & \cos θ_{t} & 0 \\ 0 & 0 & 1 \end{array}) (\begin{array}{l} 0.5 r & 0.5 r \\ 0 & 0 \\ r / d & - r / d \end{array}) (\begin{array}{l} ϕ_{R t} \\ ϕ_{L t} \end{array}) + v_{t} \end{matrix}

where

r

and

d

are known constants and

ϕ_{R t}

and

ϕ_{L t}

are time-dependent knowns. The measurement vector

y

and measurement function

H

is given by:

\begin{matrix} m_{y} & = 2 \\ y_{t} & = {(δ_{t}, α_{t})}^{T} \\ = H (x_{t}) + u_{t} \\ = (\begin{matrix} Δ - ξ_{t} \cos A - η_{t} \sin A \\ θ_{t} - A \end{matrix}) + u_{t} \end{matrix}

where

A

and

Δ

are known constants. The initial values,

x_{0}

and

P_{0}

, are given by

\begin{matrix} x_{0} & = & (\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}), & P_{0} & = & (\begin{matrix} 0.1 & 0 & 0 \\ 0 & 0.1 & 0 \\ 0 & 0 & 0.1 \end{matrix}) \end{matrix}

and the Cholesky factorizations of the error covariance matrices,

L_{x}

and

L_{x}

\begin{matrix} \begin{matrix} L_{x} & = & (\begin{matrix} 0.1 & 0 & 0 \\ 0 & 0.1 & 0 \\ 0 & 0 & 0.1 \end{matrix}) \end{matrix}, & \begin{matrix} L_{y} & = & (\begin{matrix} 0.01 & 0 \\ 0 & 0.01 \end{matrix}) \end{matrix} \end{matrix}

The example described above can be thought of relating to the movement of a hypothetical robot. The unknown state,

x

, is the position of the robot (with respect to a reference frame) and facing, with

(ξ, η)

giving the

x

and

y

coordinates and

θ

the angle (with respect to the

x

-axis) that the robot is facing. The robot has two drive wheels, of radius

r

on an axle of length

d

. During time period

t

the right wheel is believed to rotate at a velocity of

ϕ_{R t}

and the left at a velocity of

ϕ_{L t}

. In this example, these velocities are fixed with

ϕ_{R t} = 0.4

and

ϕ_{L t} = 0.1

. The state update function,

F

, calculates where the robot should be at each time point, given its previous position. However, in reality, there is some random fluctuation in the velocity of the wheels, for example, due to slippage. Therefore the actual position of the robot and the position given by equation

F

will differ.

In the area that the robot is moving there is a single wall. The position of the wall is known and defined by its distance,

Δ

, from the origin and its angle,

A

, from the

x

-axis. The robot has a sensor that is able to measure

y

, with

δ

being the distance to the wall and

α

the angle to the wall. The measurement function

H

gives the expected distance and angle to the wall if the robot's position is given by

x_{t}

. Therefore the state space model allows the robot to incorporate the sensor information to update the estimate of its position.

G13EKF (PDF version)

G13 Chapter Contents

G13 Chapter Introduction

NAG Library Manual

NAG Library Routine DocumentG13EKF

▸▿ Contents

1 Purpose

2 Specification

3 Description

3.1 Unscented Kalman Filter Algorithm

3.2 Sigma Points

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Routine Document

G13EKF