The function may be called by the names: g13ekc, nag_tsa_kalman_unscented_state or nag_kalman_unscented_state.
3Description
g13ekc 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 , can be described by:
where represents the unobserved state vector of length and the observed measurement vector of length . The process noise is denoted , which is assumed to have mean zero and covariance structure , and the measurement noise by , which is assumed to have mean zero and covariance structure .
3.1Unscented Kalman Filter Algorithm
Given , an initial estimate of the state and 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):
(1)
(b)Evaluate the known model function :
(2)
The function is assumed to accept the matrix, and return an matrix, . The columns of both and correspond to different possible states. The notation is used to denote the th column of , hence the result of applying to the th possible state.
(c)Time Update:
(3)
(4)
(d)Redraw another set of sigma points (see Section 3.2):
(5)
(e)Evaluate the known model function :
(6)
The function is assumed to accept the matrix, and return an matrix, . The columns of both and correspond to different possible states. As above is used to denote the th column of .
(f)Measurement Update:
(7)
(8)
(9)
(10)
(11)
(12)
Here is the Kalman gain matrix, is the estimated state vector at time and the corresponding covariance matrix. Rather than implementing the standard UKF as stated above g13ekc 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 random variables, , with mean and covariance matrix , 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 and one used when calculating the weighted sample covariance matrix, denoted . The weights and multiplier, , are constructed as follows:
where, usually and and are constants. The total number of sigma points, , is given by . The constant is usually set to somewhere in the range and for a Gaussian distribution, the optimal values of and are and respectively.
The constants, , and are given by , and . If more control is required over the construction of the sigma points then the reverse communication function, g13ejc, can be used instead.
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
1: – IntegerInput
On entry: , the number of state variables.
Constraint:
.
2: – IntegerInput
On entry: , the number of observed variables.
Constraint:
.
3: – const doubleInput
On entry: , the observed data at the current time point.
4: – const doubleInput
Note: the th element of the matrix is stored in .
On entry: , such that , 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 is time dependent, the value supplied should be for time .
5: – const doubleInput
Note: the th element of the matrix is stored in .
On entry: , such that , 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 is time dependent, the value supplied should be for time .
6: – function, supplied by the userExternal Function
f (Integer mx,Integer n,const double xt[],double fxt[],Nag_Comm *comm,Integer *info)
1: – IntegerInput
On entry: , the number of state variables.
2: – IntegerInput
On entry: , the number of sigma points.
3: – const doubleInput
On entry: , the sigma points generated in (a). For the th sigma point, the value for the th parameter is held in
, for and .
4: – doubleOutput
On exit: .
For the th sigma point the value for the th parameter should be held in
, for and .
5: – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to f.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling g13ekc you may allocate memory and initialize these pointers with various quantities for use by f when called from g13ekc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
6: – Integer *Input/Output
On entry: .
On exit: set info to a nonzero value if you wish g13ekc to terminate with NE_USER_STOP.
Note:f should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g13ekc. If your code inadvertently does return any NaNs or infinities, g13ekc is likely to produce unexpected results.
7: – function, supplied by the userExternal Function
h (Integer mx,Integer my,Integer n,const double yt[],double hyt[],Nag_Comm *comm,Integer *info)
1: – IntegerInput
On entry: , the number of state variables.
2: – IntegerInput
On entry: , the number of observed variables.
3: – IntegerInput
On entry: , the number of sigma points.
4: – const doubleInput
On entry: , the sigma points generated in (d). For the th sigma point, the value for the th parameter is held in
, for and , where is the number of state variables and is the number of sigma points.
5: – doubleOutput
On exit: .
For the th sigma point the value for the th parameter should be held in
, for and .
6: – Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to h.
user – double *
iuser – Integer *
p – Pointer
The type Pointer will be void *. Before calling g13ekc you may allocate memory and initialize these pointers with various quantities for use by h when called from g13ekc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
7: – Integer *Input/Output
On entry: .
On exit: set info to a nonzero value if you wish g13ekc to terminate with NE_USER_STOP.
Note:h should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by g13ekc. If your code inadvertently does return any NaNs or infinities, g13ekc is likely to produce unexpected results.
8: – doubleInput/Output
On entry: the state vector for the previous time point.
On exit: the updated state vector.
9: – doubleInput/Output
Note: the th element of the matrix is stored in .
On entry: , such that , 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.
On exit: , the lower triangular part of a Cholesky factorization of the updated state covariance matrix.
10: – Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
11: – NagError *Input/Output
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_BAD_PARAM
On entry, argument had an illegal value.
NE_INT
On entry, . Constraint: .
On entry, . Constraint: .
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_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.
Background information to multithreading can be found in the Multithreading documentation.
g13ekc 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
None.
10Example
This example implements the following nonlinear state space model, with the state vector and state update function given by:
where and are known constants and and are time-dependent knowns. The measurement vector and measurement function is given by:
where and are known constants. The initial values, and , are given by
and the Cholesky factorizations of the error covariance matrices, and by
The example described above can be thought of relating to the movement of a hypothetical robot. The unknown state, , is the position of the robot (with respect to a reference frame) and facing, with giving the and coordinates and the angle (with respect to the -axis) that the robot is facing. The robot has two drive wheels, of radius on an axle of length . During time period the right wheel is believed to rotate at a velocity of and the left at a velocity of . In this example, these velocities are fixed with and . The state update function, , 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 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, , from the origin and its angle, , from the -axis. The robot has a sensor that is able to measure , with being the distance to the wall and the angle to the wall. The measurement function gives the expected distance and angle to the wall if the robot's position is given by . Therefore, the state space model allows the robot to incorporate the sensor information to update the estimate of its position.