The routine may be called by the names g13ejf or nagf_tsa_kalman_unscented_state_revcom.
3Description
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:
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 ${\Sigma}_{x}$, and the measurement noise by ${u}_{t}$, which is assumed to have mean zero and covariance structure ${\Sigma}_{y}$.
3.1Unscented 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):
The function $F$ is assumed to accept the ${m}_{x}\times n$ matrix, ${\mathcal{X}}_{t}$ and return an ${m}_{x}\times n$ matrix, ${\mathcal{F}}_{t}$. The columns of both ${\mathcal{X}}_{t}$ and ${\mathcal{F}}_{t}$ correspond to different possible states. The notation ${\mathcal{F}}_{t,i}$ is used to denote the $i$th column of ${\mathcal{F}}_{t}$, hence the result of applying $F$ to the $i$th possible state.
The function $H$ is assumed to accept the ${m}_{x}\times n$ matrix, ${\mathcal{Y}}_{t}$ and return an ${m}_{y}\times n$ matrix, ${\mathcal{H}}_{t}$. The columns of both ${\mathcal{Y}}_{t}$ and ${\mathcal{H}}_{t}$ correspond to different possible states. As above ${\mathcal{H}}_{t,i}$ is used to denote the $i$th column of ${\mathcal{H}}_{t}$.
Here ${\mathcal{K}}_{t}$ 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).
3.2Sigma 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 $\mu $ and covariance matrix $\Sigma $, the sigma points are usually constructed as:
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, $\gamma $, are constructed as follows:
where, usually $L=m$ and $\alpha ,\beta $ and $\kappa $ are constants. The total number of sigma points, $n$, is given by $2L+1$. The constant $\alpha $ is usually set to somewhere in the range ${10}^{-4}\le \alpha \le 1$ and for a Gaussian distribution, the optimal values of $\kappa $ and $\beta $ 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:
Augmenting the sigma points in this manner requires setting $L$ to $2L$ (and hence $n$ to $2n-1$) 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.
4References
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
5Arguments
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 thanfxt must remain unchanged.
1: $\mathbf{irevcm}$ – IntegerInput/Output
On initial entry: must be set to $0$ or $3$.
If ${\mathbf{irevcm}}=0$, it is assumed that $t=0$, otherwise it is assumed that $t\ne 0$ and that g13ejf has been called at least once before at an earlier time step.
On intermediate exit:
${\mathbf{irevcm}}=1$ or $2$. 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: ${\mathbf{irevcm}}=3$
Constraint:
${\mathbf{irevcm}}=0$, $1$, $2$ or $3$.
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: $\mathbf{mx}$ – IntegerInput
On entry: ${m}_{x}$, the number of state variables.
Constraint:
${\mathbf{mx}}\ge 1$.
3: $\mathbf{my}$ – IntegerInput
On entry: ${m}_{y}$, the number of observed variables.
Constraint:
${\mathbf{my}}\ge 1$.
4: $\mathbf{y}\left({\mathbf{my}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${y}_{t}$, the observed data at the current time point.
5: $\mathbf{lx}({\mathbf{ldlx}},*)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array lx
must be at least
${\mathbf{mx}}$.
On entry: ${L}_{x}$, such that ${L}_{x}{L}_{x}^{\mathrm{T}}={\Sigma}_{x}$, 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.
If ${\mathbf{ldlx}}=0$, there is no process noise (${v}_{t}=0$ for all $t$) and lx is not referenced.
If ${\Sigma}_{x}$ is time dependent, the value supplied should be for time $t$.
6: $\mathbf{ldlx}$ – IntegerInput
On entry: the first dimension of the array lx as declared in the (sub)program from which g13ejf is called.
Constraint:
${\mathbf{ldlx}}=0$ or ${\mathbf{ldlx}}\ge {\mathbf{mx}}$.
7: $\mathbf{ly}({\mathbf{ldly}},*)$ – Real (Kind=nag_wp) arrayInput
Note: the second dimension of the array ly
must be at least
${\mathbf{my}}$.
On entry: ${L}_{y}$, such that ${L}_{y}{L}_{y}^{\mathrm{T}}={\Sigma}_{y}$, 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.
If ${\Sigma}_{y}$ is time dependent, the value supplied should be for time $t$.
8: $\mathbf{ldly}$ – IntegerInput
On entry: the first dimension of the array ly as declared in the (sub)program from which g13ejf is called.
Constraint:
${\mathbf{ldly}}\ge {\mathbf{my}}$.
9: $\mathbf{x}\left({\mathbf{mx}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On initial entry: ${\hat{x}}_{t-1}$ the state vector for the previous time point.
On intermediate re-entry: x must remain unchanged.
On final exit: ${\hat{x}}_{t}$ the updated state vector.
10: $\mathbf{st}({\mathbf{ldst}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array st
must be at least
${\mathbf{mx}}$.
On initial entry: ${S}_{t}$, such that ${S}_{t-1}{S}_{t-1}^{\mathrm{T}}={P}_{t-1}$, 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.
${S}_{\overline{t}}$, the lower triangular part of a Cholesky factorization of ${P}_{\overline{t}}$.
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: $\mathbf{ldst}$ – IntegerInput
On entry: the first dimension of the array st as declared in the (sub)program from which g13ejf is called.
Constraint:
${\mathbf{ldst}}\ge {\mathbf{mx}}$.
12: $\mathbf{n}$ – IntegerInput/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:
if ${\mathbf{irevcm}}=0$ or $3$,
if redrawn sigma points are used, ${\mathbf{n}}\ge 2\times {\mathbf{mx}}+1$;
13: $\mathbf{xt}({\mathbf{ldxt}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array xt
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{my}},{\mathbf{n}})$.
On initial entry: need not be set.
On intermediate exit:
${\mathcal{X}}_{t}$ when ${\mathbf{irevcm}}=1$, otherwise ${\mathcal{Y}}_{t}$.
For the $j$th sigma point, the value for the $i$th parameter is held in
${\mathbf{xt}}(\mathit{i},\mathit{j})$, for $\mathit{i}=1,2,\dots ,{\mathbf{mx}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{n}}$.
On intermediate re-entry: xt must remain unchanged.
On entry: the first dimension of the array xt as declared in the (sub)program from which g13ejf is called.
Constraint:
${\mathbf{ldxt}}\ge {\mathbf{mx}}$.
15: $\mathbf{fxt}({\mathbf{ldfxt}},*)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the second dimension of the array fxt
must be at least
${\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{mx}},{\mathbf{my}})$.
On initial entry: need not be set.
On intermediate exit:
the contents of fxt are undefined.
On intermediate re-entry: $F\left({\mathcal{X}}_{t}\right)$ when ${\mathbf{irevcm}}=1$, otherwise $H\left({\mathcal{Y}}_{t}\right)$ for the values of ${\mathcal{X}}_{t}$ and ${\mathcal{Y}}_{t}$ held in xt.
For the $j$th sigma point the value for the $i$th parameter should be held in
${\mathbf{fxt}}(i,\mathit{j})$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$. When ${\mathbf{irevcm}}=1$, $i=1,2,\dots ,{\mathbf{mx}}$ and when ${\mathbf{irevcm}}=2$, $i=1,2,\dots ,{\mathbf{my}}$.
17: $\mathbf{ropt}\left({\mathbf{lropt}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: optional parameters. The default value will be used for ${\mathbf{ropt}}\left(i\right)$ if ${\mathbf{lropt}}<i$. Setting ${\mathbf{lropt}}=0$ will use the default values for all optional parameters and ropt need not be set.
${\mathbf{ropt}}\left(1\right)$
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., ${\mathbf{ropt}}\left(1\right)=1$)
${\mathbf{ropt}}\left(2\right)$
${\kappa}_{x}$, value of $\kappa $ used when constructing the first set of sigma points, ${\mathcal{X}}_{t}$.
Defaults to $3-{\mathbf{mx}}$.
${\mathbf{ropt}}\left(3\right)$
${\alpha}_{x}$, value of $\alpha $ used when constructing the first set of sigma points, ${\mathcal{X}}_{t}$.
Defaults to $1$.
${\mathbf{ropt}}\left(4\right)$
${\beta}_{x}$, value of $\beta $ used when constructing the first set of sigma points, ${\mathcal{X}}_{t}$.
Defaults to $2$.
${\mathbf{ropt}}\left(5\right)$
Value of $\kappa $ used when constructing the second set of sigma points, ${\mathcal{Y}}_{t}$.
Defaults to $3-2\times {\mathbf{mx}}$ when ${\mathbf{ldlx}}\ne 0$ and the second set of sigma points are augmented and ${\kappa}_{x}$ otherwise.
${\mathbf{ropt}}\left(6\right)$
Value of $\alpha $ used when constructing the second set of sigma points, ${\mathcal{Y}}_{t}$.
Defaults to ${\alpha}_{x}$.
${\mathbf{ropt}}\left(7\right)$
Value of $\beta $ used when constructing the second set of sigma points, ${\mathcal{Y}}_{t}$.
Defaults to ${\beta}_{x}$.
Constraints:
${\mathbf{ropt}}\left(1\right)=1$ or $2$;
${\mathbf{ropt}}\left(2\right)>-{\mathbf{mx}}$;
${\mathbf{ropt}}\left(5\right)>-2\times {\mathbf{mx}}$ when ${\mathbf{ldly}}\ne 0$ and the second set of sigma points are augmented, otherwise${\mathbf{ropt}}\left(5\right)>-{\mathbf{mx}}$;
${\mathbf{ropt}}\left(\mathit{i}\right)>0$, for $\mathit{i}=3,4,5,6$.
On entry: the length of the array icomm. If licomm is too small and ${\mathbf{licomm}}\ge 2$ then ${\mathbf{ifail}}={\mathbf{201}}$ is returned and the minimum value for licomm and lrcomm are given by ${\mathbf{icomm}}\left(1\right)$ and ${\mathbf{icomm}}\left(2\right)$ respectively.
Constraint:
${\mathbf{licomm}}\ge 30$.
21: $\mathbf{rcomm}\left({\mathbf{lrcomm}}\right)$ – Real (Kind=nag_wp) arrayCommunication Array
On entry: the length of the array rcomm. If lrcomm is too small and ${\mathbf{licomm}}\ge 2$ then ${\mathbf{ifail}}={\mathbf{202}}$ is returned and the minimum value for licomm and lrcomm are given by ${\mathbf{icomm}}\left(1\right)$ and ${\mathbf{icomm}}\left(2\right)$ respectively.
Suggested value:
${\mathbf{lrcomm}}=30+{\mathbf{my}}+{\mathbf{mx}}\times {\mathbf{my}}+(1+\mathit{nb})\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{mx}},{\mathbf{my}})$, where $\mathit{nb}$ is the optimal block size. In most cases a block size of $128$ will be sufficient.
On initial entry: ifail must be set to $0$, $-1$ or $1$ 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 $-1$ or $1$ 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 ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On final exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=11$
On entry, ${\mathbf{irevcm}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{irevcm}}=0$, $1$, $2$ or $3$.
${\mathbf{ifail}}=21$
On entry, ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{mx}}\ge 1$.
${\mathbf{ifail}}=22$
mx has changed between calls. On intermediate entry, ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. On initial entry, ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=31$
On entry, ${\mathbf{my}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{my}}\ge 1$.
${\mathbf{ifail}}=32$
my has changed between calls. On intermediate entry, ${\mathbf{my}}=\u27e8\mathit{\text{value}}\u27e9$. On initial entry, ${\mathbf{my}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=61$
On entry, ${\mathbf{ldlx}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{ldlx}}=0$ or ${\mathbf{ldlx}}\ge {\mathbf{mx}}$.
${\mathbf{ifail}}=81$
On entry, ${\mathbf{ldly}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{my}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{ldly}}\ge {\mathbf{my}}$.
${\mathbf{ifail}}=111$
On entry, ${\mathbf{ldst}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{ldst}}\ge {\mathbf{mx}}$.
${\mathbf{ifail}}=121$
On entry, augmented sigma points requested, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{n}}\ge \u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=122$
On entry, redrawn sigma points requested, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{n}}\ge \u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=123$
n has changed between calls. On intermediate entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$. On intermediate exit, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=141$
On entry, ${\mathbf{ldxt}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{ldxt}}\ge {\mathbf{mx}}$.
${\mathbf{ifail}}=161$
On entry, ${\mathbf{ldfxt}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: if ${\mathbf{irevcm}}=1$, ${\mathbf{ldfxt}}\ge {\mathbf{mx}}$.
${\mathbf{ifail}}=162$
On entry, ${\mathbf{ldfxt}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{my}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: if ${\mathbf{irevcm}}=2$, ${\mathbf{ldfxt}}\ge {\mathbf{my}}$.
${\mathbf{ifail}}=171$
On entry, ${\mathbf{ropt}}\left(1\right)=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{ropt}}\left(1\right)=1$ or $2$.
${\mathbf{ifail}}=172$
On entry, ${\mathbf{ropt}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$. Constraint: $\kappa >\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=173$
On entry, ${\mathbf{ropt}}\left(\u27e8\mathit{\text{value}}\u27e9\right)=\u27e8\mathit{\text{value}}\u27e9$. Constraint: $\alpha >0$.
${\mathbf{ifail}}=181$
On entry, ${\mathbf{lropt}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: $0\le {\mathbf{lropt}}\le 7$.
On entry, ${\mathbf{licomm}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{licomm}}\ge 2$. icomm is too small to return the required array sizes.
${\mathbf{ifail}}=202$
On entry, ${\mathbf{licomm}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{lrcomm}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{licomm}}\ge 30$ and ${\mathbf{lrcomm}}\ge 30+{\mathbf{my}}+{\mathbf{mx}}\times {\mathbf{my}}+2\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{mx}},{\mathbf{my}})$. The minimum required values for licomm and lrcomm are returned in ${\mathbf{icomm}}\left(1\right)$ and ${\mathbf{icomm}}\left(2\right)$ respectively.
A weight was negative and it was not possible to downdate the Cholesky factorization.
${\mathbf{ifail}}=302$
Unable to calculate the Kalman gain matrix.
${\mathbf{ifail}}=303$
Unable to calculate the Cholesky factorization of the updated state covariance matrix.
${\mathbf{ifail}}=-99$
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.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
Not applicable.
8Parallelism 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.
9Further 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 ${\mathbf{ldlx}}=0$ and terminating the calling sequence when ${\mathbf{irevcm}}=2$ rather than ${\mathbf{irevcm}}=3$. 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 ${\mathbf{irevcm}}=2$) they would hold the mean and Cholesky factorization of the covariance matrix of the transformed sample.
10Example
This example implements the following nonlinear state space model, with the state vector $x$ and state update function $F$ given by:
where $r$ and $d$ are known constants and ${\varphi}_{Rt}$ and ${\varphi}_{Lt}$ are time-dependent knowns. The measurement vector $y$ and measurement function $H$ is given by:
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 $(\xi ,\eta )$ giving the $x$ and $y$ coordinates and $\theta $ 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 ${\varphi}_{Rt}$ and the left at a velocity of ${\varphi}_{Lt}$. In this example, these velocities are fixed with ${\varphi}_{Rt}=0.4$ and ${\varphi}_{Lt}=0.1$. 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, $\Delta $, from the origin and its angle, $A$, from the $x$-axis. The robot has a sensor that is able to measure $y$, with $\delta $ being the distance to the wall and $\alpha $ 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.