Integer, Intent (In)	::	mx, my, ldlx, ldly, ldst, ldxt, ldfxt, lropt, licomm, lrcomm
Integer, Intent (Inout)	::	irevcm, n, icomm(licomm), ifail
Real (Kind=nag_wp), Intent (In)	::	y(my), lx(ldlx,), ly(ldly,), ropt(lropt)
Real (Kind=nag_wp), Intent (Inout)	::	x(mx), st(ldst,), xt(ldxt,), fxt(ldfxt,*), rcomm(lrcomm)

C Header Interface

#include <nag.h>

void

g13ejf_ (Integer *irevcm, const Integer *mx, const Integer *my, const double y[], const double lx[], const Integer *ldlx, const double ly[], const Integer *ldly, double x[], double st[], const Integer *ldst, Integer *n, double xt[], const Integer *ldxt, double fxt[], const Integer *ldfxt, const double ropt[], const Integer *lropt, Integer icomm[], const Integer *licomm, double rcomm[], const Integer *lrcomm, Integer *ifail)

The routine may be called by the names g13ejf or nagf_tsa_kalman_unscented_state_revcom.

3 Description

g13ejf 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:

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

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

$Y_{t}$ $= [{\hat{x}}_{\hat{t}} {\hat{x}}_{\hat{t}} + γ \sqrt{P_{\bar{t}}} {\hat{x}}_{\hat{t}} - γ \sqrt{P_{\bar{t}}}]$ (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)
$P_{x y_{t}}$	$= \sum_{i = 1}^{n} W_{i}^{c} (F_{t, i} - {\hat{x}}_{\hat{t}}) {(H_{t, i} - {\hat{y}}_{t})}^{T}$	(9)
$K_{t}$	$= P_{x y_{t}} P_{y y_{t}}^{- 1}$	(10)
${\hat{x}}_{t}$	$= {\hat{x}}_{\hat{t}} + K_{t} (y_{t} - {\hat{y}}_{t})$	(11)
$P_{t}$	$= P_{\bar{t}} - K_{t} P_{y y_{t}} K_{t}^{T}$	(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 g13ejf 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 calculating 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.

Rather than redrawing another set of sigma points in (d) of the UKF an alternative method can be used where the sigma points used in (a) are augmented to take into account the process noise. This involves replacing equation (5) with:

Y_{t}

= [X_{t} X_{t, 1} + γ \sqrt{Σ_{x}} X_{t, 1} - γ \sqrt{Σ_{x}}]

(13)

Augmenting the sigma points in this manner requires setting

L

2 L

(and hence

n

2 n - 1

) and recalculating the weights. These new values are then used for the rest of the algorithm. The advantage of augmenting the sigma points is that it keeps any odd-moments information captured by the original propagated sigma points, at the cost of using a larger number of points.

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 AeroSense '97, 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 Arguments

Note: this routine uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the argument irevcm. Between intermediate exits and re-entries, all arguments other than fxt must remain unchanged.

1: $irevcm$ – Integer Input/Output

On initial entry: must be set to

0

3

irevcm = 0

, it is assumed that

t = 0

, otherwise it is assumed that

t \neq 0

and that g13ejf has been called at least once before at an earlier time step.

On intermediate exit:

irevcm = 1

2

. The value of irevcm specifies what intermediate values are returned by this routine and what values the calling program must assign to arguments of g13ejf before re-entering the routine. Details of the output and required input are given in the individual argument descriptions.

On intermediate re-entry: irevcm must remain unchanged.

On final exit:

irevcm = 3

Constraint:

irevcm = 0

1

2

3

Note: any values you return to g13ejf as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by g13ejf. If your code does inadvertently return any NaNs or infinities, g13ejf is likely to produce unexpected results.

2: $mx$ – Integer Input

On entry:

m_{x}

, the number of state variables.

Constraint:

mx \geq 1

3: $my$ – Integer Input

On entry:

m_{y}

, the number of observed variables.

Constraint:

my \geq 1

4: $y (my)$ – Real (Kind=nag_wp) array Input

On entry:

y_{t}

, the observed data at the current time point.

5: $lx (ldlx, *)$ – Real (Kind=nag_wp) array Input

Note: the second dimension of the array lx must be at least

mx

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.

ldlx = 0

, there is no process noise (

v_{t} = 0

for all

t

) and lx is not referenced.

Σ_{x}

is time dependent, the value supplied should be for time

t

6: $ldlx$ – Integer Input

On entry: the first dimension of the array lx as declared in the (sub)program from which g13ejf is called.

Constraint:

ldlx = 0

ldlx \geq mx

7: $ly (ldly, *)$ – Real (Kind=nag_wp) array Input

Note: the second dimension of the array ly must be at least

my

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, the value supplied should be for time

t

8: $ldly$ – Integer Input

On entry: the first dimension of the array ly as declared in the (sub)program from which g13ejf is called.

Constraint:

ldly \geq my

9: $x (mx)$ – Real (Kind=nag_wp) array Input/Output

On initial entry:

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

the state vector for the previous time point.

On intermediate exit: when

$irevcm = 1$: x is unchanged.
$irevcm = 2$: ${\hat{x}}_{\bar{t}}$ .

On intermediate re-entry: x must remain unchanged.

On final exit:

{\hat{x}}_{t}

the updated state vector.

10: $st (ldst, *)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array st must be at least

mx

On initial 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 intermediate exit: when

$irevcm = 1$: st is unchanged.
$irevcm = 2$: $S_{\bar{t}}$ , the lower triangular part of a Cholesky factorization of $P_{\bar{t}}$ .

On intermediate re-entry: st must remain unchanged.

On final exit:

S_{t}

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

11: $ldst$ – Integer Input

On entry: the first dimension of the array st as declared in the (sub)program from which g13ejf is called.

Constraint:

ldst \geq mx

12: $n$ – Integer Input/Output

On initial entry: the value used in the sizing of the fxt and xt arrays. The value of n supplied must be at least as big as the maximum number of sigma points that the algorithm will use. g13ejf allows sigma points to be calculated in two ways during the measurement update; they can be redrawn or augmented. Which is used is controlled by ropt.

If redrawn sigma points are used, then the maximum number of sigma points will be

2 m_{x} + 1

, otherwise the maximum number of sigma points will be

4 m_{x} + 1

On intermediate exit: the number of sigma points actually being used.

On intermediate re-entry: n must remain unchanged.

On final exit: reset to its value on initial entry.

Constraints:

irevcm = 0

3

if redrawn sigma points are used, $n \geq 2 \times mx + 1$ ;
otherwise $n \geq 4 \times mx + 1$ .

13: $xt (ldxt, *)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array xt must be at least

\max (my, n)

On initial entry: need not be set.

On intermediate exit:

X_{t}

when

irevcm = 1

, otherwise

Y_{t}

For the

j

th sigma point, the value for the

i

th parameter is held in

xt (i, j)

, for

i = 1, 2, \dots, mx

and

j = 1, 2, \dots, n

On intermediate re-entry: xt must remain unchanged.

On final exit: the contents of xt are undefined.

14: $ldxt$ – Integer Input

On entry: the first dimension of the array xt as declared in the (sub)program from which g13ejf is called.

Constraint:

ldxt \geq mx

15: $fxt (ldfxt, *)$ – Real (Kind=nag_wp) array Input/Output

Note: the second dimension of the array fxt must be at least

n + \max (mx, my)

On initial entry: need not be set.

On intermediate exit: the contents of fxt are undefined.

On intermediate re-entry:

F (X_{t})

when

irevcm = 1

, otherwise

H (Y_{t})

for the values of

X_{t}

and

Y_{t}

held in xt.

For the

j

th sigma point the value for the

i

th parameter should be held in

fxt (i, j)

, for

j = 1, 2, \dots, n

. When

irevcm = 1

i = 1, 2, \dots, mx

and when

irevcm = 2

i = 1, 2, \dots, my

On final exit: the contents of fxt are undefined.

16: $ldfxt$ – Integer Input

On entry: the first dimension of the array fxt as declared in the (sub)program from which g13ejf is called.

Constraint:

ldfxt \geq \max (mx, my)

17: $ropt (lropt)$ – Real (Kind=nag_wp) array Input

On entry: optional parameters. The default value will be used for

ropt (i)

lropt < i

. Setting

lropt = 0

will use the default values for all optional parameters and ropt need not be set.

$ropt (1)$: If set to $1$ then the second set of sigma points are redrawn, as given by equation (5). If set to $2$ then the second set of sigma points are generated via augmentation, as given by equation (13).
Default is for the sigma points to be redrawn (i.e., $ropt (1) = 1$ )
$ropt (2)$: $κ_{x}$ , value of $κ$ used when constructing the first set of sigma points, $X_{t}$ .
Defaults to $3 - mx$ .
$ropt (3)$: $α_{x}$ , value of $α$ used when constructing the first set of sigma points, $X_{t}$ .
Defaults to $1$ .
$ropt (4)$: $β_{x}$ , value of $β$ used when constructing the first set of sigma points, $X_{t}$ .
Defaults to $2$ .
$ropt (5)$: Value of $κ$ used when constructing the second set of sigma points, $Y_{t}$ .
Defaults to $3 - 2 \times mx$ when $ldlx \neq 0$ and the second set of sigma points are augmented and $κ_{x}$ otherwise.
$ropt (6)$: Value of $α$ used when constructing the second set of sigma points, $Y_{t}$ .
Defaults to $α_{x}$ .
$ropt (7)$: Value of $β$ used when constructing the second set of sigma points, $Y_{t}$ .
Defaults to $β_{x}$ .

Constraints:

$ropt (1) = 1$ or $2$ ;
$ropt (2) > - mx$ ;
$ropt (5) > - 2 \times mx$ when $ldly \neq 0$ and the second set of sigma points are augmented, otherwise $ropt (5) > - mx$ ;
$ropt (i) > 0$ , for $i = 3, 4, 5, 6$ .

18: $lropt$ – Integer Input

On entry: length of the options array ropt.

Constraint:

0 \leq lropt \leq 7

19: $icomm (licomm)$ – Integer array Communication Array

On initial entry: icomm need not be set.

On intermediate exit: icomm is used for storage between calls to g13ejf.

On intermediate re-entry: icomm must remain unchanged.

On final exit: icomm is not defined.

20: $licomm$ – Integer Input

On entry: the length of the array icomm. If licomm is too small and

licomm \geq 2

then

ifail = 201

is returned and the minimum value for licomm and lrcomm are given by

icomm (1)

and

icomm (2)

respectively.

Constraint:

licomm \geq 30

21: $rcomm (lrcomm)$ – Real (Kind=nag_wp) array Communication Array

On initial entry: rcomm need not be set.

On intermediate exit: rcomm is used for storage between calls to g13ejf.

On intermediate re-entry: rcomm must remain unchanged.

On final exit: rcomm is not defined.

22: $lrcomm$ – Integer Input

On entry: the length of the array rcomm. If lrcomm is too small and

licomm \geq 2

then

ifail = 202

is returned and the minimum value for licomm and lrcomm are given by

icomm (1)

and

icomm (2)

respectively.

Suggested value:

lrcomm = 30 + my + mx \times my + (1 + nb) \times \max (mx, my)

, where

nb

is the optimal block size. In most cases a block size of

128

will be sufficient.

Constraint:

lrcomm \geq 30 + my + mx \times my + 2 \times \max (mx, my)

23: $ifail$ – Integer Input/Output

On initial entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

−1

is recommended since useful values can be provided in some output arguments even when

ifail \neq 0

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

On final 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, $irevcm = ⟨ value ⟩$ .
Constraint: $irevcm = 0$ , $1$ , $2$ or $3$ .

$ifail = 21$: On entry, $mx = ⟨ value ⟩$ .
Constraint: $mx \geq 1$ .

$ifail = 22$: mx has changed between calls.
On intermediate entry, $mx = ⟨ value ⟩$ .
On initial entry, $mx = ⟨ value ⟩$ .

$ifail = 31$: On entry, $my = ⟨ value ⟩$ .
Constraint: $my \geq 1$ .

$ifail = 32$: my has changed between calls.
On intermediate entry, $my = ⟨ value ⟩$ .
On initial entry, $my = ⟨ value ⟩$ .

$ifail = 61$: On entry, $ldlx = ⟨ value ⟩$ and $mx = ⟨ value ⟩$ .
Constraint: $ldlx = 0$ or $ldlx \geq mx$ .

$ifail = 81$: On entry, $ldly = ⟨ value ⟩$ and $my = ⟨ value ⟩$ .
Constraint: $ldly \geq my$ .

$ifail = 111$: On entry, $ldst = ⟨ value ⟩$ and $mx = ⟨ value ⟩$ .
Constraint: $ldst \geq mx$ .

$ifail = 121$: On entry, augmented sigma points requested, $n = ⟨ value ⟩$ and $mx = ⟨ value ⟩$ .
Constraint: $n \geq ⟨ value ⟩$ .

$ifail = 122$: On entry, redrawn sigma points requested, $n = ⟨ value ⟩$ and $mx = ⟨ value ⟩$ .
Constraint: $n \geq ⟨ value ⟩$ .

$ifail = 123$: n has changed between calls.
On intermediate entry, $n = ⟨ value ⟩$ .
On intermediate exit, $n = ⟨ value ⟩$ .

$ifail = 141$: On entry, $ldxt = ⟨ value ⟩$ and $mx = ⟨ value ⟩$ .
Constraint: $ldxt \geq mx$ .

$ifail = 161$: On entry, $ldfxt = ⟨ value ⟩$ and $mx = ⟨ value ⟩$ .
Constraint: if $irevcm = 1$ , $ldfxt \geq mx$ .

$ifail = 162$: On entry, $ldfxt = ⟨ value ⟩$ and $my = ⟨ value ⟩$ .
Constraint: if $irevcm = 2$ , $ldfxt \geq my$ .

$ifail = 171$: On entry, $ropt (1) = ⟨ value ⟩$ .
Constraint: $ropt (1) = 1$ or $2$ .

$ifail = 172$: On entry, $ropt (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: $κ > ⟨ value ⟩$ .

$ifail = 173$: On entry, $ropt (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: $α > 0$ .

$ifail = 181$: On entry, $lropt = ⟨ value ⟩$ .
Constraint: $0 \leq lropt \leq 7$ .

$ifail = 191$: icomm has been corrupted between calls.

$ifail = 201$: On entry, $licomm = ⟨ value ⟩$ .
Constraint: $licomm \geq 2$ .
icomm is too small to return the required array sizes.

$ifail = 202$: On entry, $licomm = ⟨ value ⟩$ and $lrcomm = ⟨ value ⟩$ .
Constraint: $licomm \geq 30$ and $lrcomm \geq 30 + my + mx \times my + 2 \times \max (mx, my)$ .
The minimum required values for licomm and lrcomm are returned in $icomm (1)$ and $icomm (2)$ respectively.

$ifail = 211$: rcomm has been corrupted between calls.

$ifail = 301$: 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 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g13ejf 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

As well as implementing the Unscented Kalman Filter, g13ejf can also be used to apply the Unscented Transform (see Julier (2002)) to the function

F

, by setting

ldlx = 0

and terminating the calling sequence when

irevcm = 2

rather than

irevcm = 3

. In this situation, on initial entry, x and st would hold the mean and Cholesky factorization of the covariance matrix of the untransformed sample and on exit (when

irevcm = 2

) they would hold the mean and Cholesky factorization of the covariance matrix of the transformed sample.

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 as 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.

NAG Library Manual, Mark 27.3

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

G13 (Tsa) Chapter Contents

G13 (Tsa) Chapter Introduction

g13ej: FL CL CPP AD

NAG FL Interfaceg13ejf (kalman_​unscented_​state_​revcom)

▸▿ Contents

1 Purpose

2 Specification

3 Description

3.1 Unscented Kalman Filter Algorithm

3.2 Sigma Points

4 References

5 Arguments

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 FL Interface
g13ejf (kalman_unscented_state_revcom)