The function may be called by the names: g13ejc, nag_tsa_kalman_unscented_state_revcom or nag_kalman_unscented_state_revcom.
3Description
g13ejc 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 g13ejc 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 ${10}^{\mathrm{-4}}\le \alpha \le 1$ 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 function 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: $\mathbf{irevcm}$ – Integer *Input/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 g13ejc 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 function and what values the calling program must assign to arguments of g13ejc 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 g13ejc 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 g13ejc. If your code inadvertently does return any NaNs or infinities, g13ejc 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.
Note: the dimension, dim, of the array lx
must be at least
${\mathbf{pdlx}}\times {\mathbf{mx}}$.
The $(i,j)$th element of the matrix is stored in ${\mathbf{lx}}\left[(j-1)\times {\mathbf{pdlx}}+i-1\right]$.
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 the matrix stored in lx is referenced.
If ${\mathbf{pdlx}}=0$, there is no process noise (${v}_{t}=0$ for all $t$) and lx is not referenced and may be NULL.
If ${\Sigma}_{x}$ is time dependent, the value supplied should be for time $t$.
6: $\mathbf{pdlx}$ – IntegerInput
On entry: the stride separating matrix row elements in the array lx.
Constraint:
${\mathbf{pdlx}}=0$ or ${\mathbf{pdlx}}\ge {\mathbf{mx}}$.
Note: the dimension, dim, of the array ly
must be at least
${\mathbf{pdly}}\times {\mathbf{my}}$.
The $(i,j)$th element of the matrix is stored in ${\mathbf{ly}}\left[(j-1)\times {\mathbf{pdly}}+i-1\right]$.
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 the matrix stored in ly is referenced.
If ${\Sigma}_{y}$ is time dependent, the value supplied should be for time $t$.
8: $\mathbf{pdly}$ – IntegerInput
On entry: the stride separating matrix row elements in the array ly.
Note: the dimension, dim, of the array st
must be at least
${\mathbf{pdst}}\times {\mathbf{mx}}$.
The $(i,j)$th element of the matrix is stored in ${\mathbf{st}}\left[(j-1)\times {\mathbf{pdst}}+i-1\right]$.
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 the matrix stored in 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: $\mathbf{pdst}$ – IntegerInput
On entry: the stride separating matrix row elements in the array st.
Constraint:
${\mathbf{pdst}}\ge {\mathbf{mx}}$.
12: $\mathbf{n}$ – 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. g13ejc 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, ${\mathbf{n}}\ge 2\times {\mathbf{mx}}+1$;
Note: the dimension, dim, of the array xt
must be at least
${\mathbf{pdxt}}\times \mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{my}},{\mathbf{n}})$.
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
${\mathbf{xt}}\left[\left(\mathit{j}-1\right)\times {\mathbf{pdxt}}+\mathit{i}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{mx}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{n}}$.
On intermediate re-entry: xt must remain unchanged.
Note: the dimension, dim, of the array fxt
must be at least
${\mathbf{pdfxt}}\times ({\mathbf{n}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}({\mathbf{mx}},{\mathbf{my}}))$.
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
${\mathbf{fxt}}\left[\left(\mathit{j}-1\right)\times {\mathbf{pdfxt}}+i-1\right]$, for $\mathit{j}=1,2,\dots ,{\mathbf{n}}$. When ${\mathbf{irevcm}}=1$, $i=1,2,\dots ,{\mathbf{mx}}$ and when ${\mathbf{irevcm}}=2$, $i=1,2,\dots ,{\mathbf{my}}$.
On entry: optional parameters. The default value will be used for ${\mathbf{ropt}}\left[i-1\right]$ if ${\mathbf{lropt}}<i$. Setting ${\mathbf{lropt}}=0$ will use the default values for all optional parameters and ropt need not be set and may be NULL.
${\mathbf{ropt}}\left[0\right]$
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., ${\mathbf{ropt}}\left[0\right]=1$)
${\mathbf{ropt}}\left[1\right]$
${\kappa}_{x}$, value of $\kappa $ used when constructing the first set of sigma points, ${\mathcal{X}}_{t}$.
Defaults to $3-{\mathbf{mx}}$.
${\mathbf{ropt}}\left[2\right]$
${\alpha}_{x}$, value of $\alpha $ used when constructing the first set of sigma points, ${\mathcal{X}}_{t}$.
Defaults to $1$.
${\mathbf{ropt}}\left[3\right]$
${\beta}_{x}$, value of $\beta $ used when constructing the first set of sigma points, ${\mathcal{X}}_{t}$.
Defaults to $2$.
${\mathbf{ropt}}\left[4\right]$
Value of $\kappa $ used when constructing the second set of sigma points, ${\mathcal{Y}}_{t}$.
Defaults to $3-2\times {\mathbf{mx}}$ when ${\mathbf{pdlx}}\ne 0$ and the second set of sigma points are augmented and ${\kappa}_{x}$ otherwise.
${\mathbf{ropt}}\left[5\right]$
Value of $\alpha $ used when constructing the second set of sigma points, ${\mathcal{Y}}_{t}$.
Defaults to ${\alpha}_{x}$.
${\mathbf{ropt}}\left[6\right]$
Value of $\beta $ used when constructing the second set of sigma points, ${\mathcal{Y}}_{t}$.
Defaults to ${\beta}_{x}$.
Constraints:
${\mathbf{ropt}}\left[0\right]=1$ or $2$;
${\mathbf{ropt}}\left[1\right]>-{\mathbf{mx}}$;
${\mathbf{ropt}}\left[4\right]>-2\times {\mathbf{mx}}$ when ${\mathbf{pdly}}\ne 0$ and the second set of sigma points are augmented, otherwise${\mathbf{ropt}}\left[4\right]>-{\mathbf{mx}}$;
${\mathbf{ropt}}\left[\mathit{i}-1\right]>0$, 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 ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NE_TOO_SMALL is returned and the minimum value for licomm and lrcomm are given by ${\mathbf{icomm}}\left[0\right]$ and ${\mathbf{icomm}}\left[1\right]$ respectively.
On entry: the length of the array rcomm. If lrcomm is too small and ${\mathbf{licomm}}\ge 2$ then ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$NW_INT is returned and the minimum value for licomm and lrcomm are given by ${\mathbf{icomm}}\left[0\right]$ and ${\mathbf{icomm}}\left[1\right]$ 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.
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE
On entry, ${\mathbf{pdfxt}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: if ${\mathbf{irevcm}}=1$, ${\mathbf{pdfxt}}\ge {\mathbf{mx}}$.
On entry, ${\mathbf{pdfxt}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{my}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: if ${\mathbf{irevcm}}=2$, ${\mathbf{pdfxt}}\ge {\mathbf{my}}$.
On entry, ${\mathbf{pdlx}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{pdlx}}=0$ or ${\mathbf{pdlx}}\ge {\mathbf{mx}}$.
On entry, ${\mathbf{pdly}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{my}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{pdly}}\ge {\mathbf{my}}$.
On entry, ${\mathbf{pdst}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{pdst}}\ge {\mathbf{mx}}$.
On entry, ${\mathbf{pdxt}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{pdxt}}\ge {\mathbf{mx}}$.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
On entry, ${\mathbf{lropt}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: $0\le {\mathbf{lropt}}\le 7$.
On entry, ${\mathbf{irevcm}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{irevcm}}=0$, $1$, $2$ or $3$.
On entry, ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{mx}}\ge 1$.
On entry, ${\mathbf{my}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{my}}\ge 1$.
On entry, augmented sigma points requested, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{n}}\ge \u27e8\mathit{\text{value}}\u27e9$.
On entry, redrawn sigma points requested, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{n}}\ge \u27e8\mathit{\text{value}}\u27e9$.
NE_INT_CHANGED
mx has changed between calls. On intermediate entry, ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$. On initial entry, ${\mathbf{mx}}=\u27e8\mathit{\text{value}}\u27e9$.
my has changed between calls. On intermediate entry, ${\mathbf{my}}=\u27e8\mathit{\text{value}}\u27e9$. On initial entry, ${\mathbf{my}}=\u27e8\mathit{\text{value}}\u27e9$.
n has changed between calls. On intermediate entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$. On intermediate exit, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_OPTION
On entry, ${\mathbf{ropt}}\left[0\right]=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{ropt}}\left[0\right]=1$ or $2$.
On entry, ${\mathbf{ropt}}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$. Constraint: $\alpha >0$.
On entry, ${\mathbf{ropt}}\left[\u27e8\mathit{\text{value}}\u27e9\right]=\u27e8\mathit{\text{value}}\u27e9$. Constraint: $\kappa >\u27e8\mathit{\text{value}}\u27e9$.
NE_MAT_NOT_POS_DEF
A weight was negative and it was not possible to downdate the Cholesky factorization.
Unable to calculate the Cholesky factorization of the updated state covariance matrix.
Unable to calculate the Kalman gain matrix.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_TOO_SMALL
On entry, ${\mathbf{licomm}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{licomm}}\ge 2$. icomm is too small to return the required array sizes.
NW_INT
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 ${\mathbf{icomm}}\left[0\right]$ and ${\mathbf{icomm}}\left[1\right]$ respectively.
7Accuracy
Not applicable.
8Parallelism and Performance
g13ejc 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 function. 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, g13ejc can also be used to apply the Unscented Transform (see Julier (2002)) to the function $F$, by setting ${\mathbf{pdlx}}=0$ 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.