Systems, methods, and media for calculating an overbound distribution, from a base mixture distribution, that can be used with a solution-separation RAIM algorithm

Patent metadata

Field	Value
Publication number	US-12504543-B2
Application number	US-202418586936-A
Country	US
Kind code	B2
Filing date	Feb 26, 2024
Priority date	Apr 12, 2023
Publication date	Dec 23, 2025
Grant date	Dec 23, 2025

Abstract

An overbound distribution is calculated from a base mixture distribution. For a bounded region, the base distribution is lower-bound at an evaluation point with a second-order polynomial of the base distribution and upper-bound at the evaluation point with a first-order polynomial of a single distribution with a standard deviation value calculated from the base distribution. If the step size from the evaluation point to an intersection of the lower and upper bounds is less than a threshold, the standard deviation value can be iteratively increased until the step size exceeds the threshold. The process is performed for additional portions of the base distribution up to a critical value to determine a final adjusted standard deviation value for the single distribution that is tightly bound to the base distribution and that can be used by a solution algorithm to determine a solution (used to seed filter states for a navigation filter).

First claim

What is claimed is:

1 . A navigation system, comprising:
a module executed by a processor; and for each of a plurality of different regions that define a base distribution that is a mixture of a plurality of components, the module when executed by the processor configured to:
identify a selected evaluation point of the base distribution;
lower-bound the base distribution at the selected evaluation point utilizing a second-order polynomial of the base distribution thereby defining a lower-bound function, wherein the second-order polynomial is determined utilizing a component mean value and a component standard deviation value for each of the plurality of components associated with the base distribution,
upper-bound the base distribution at the evaluation point utilizing a first-order polynomial of an overbound distribution thereby defining an upper-bound function, wherein the first-order polynomial is determined utilizing a selected mean value and a selected standard deviation value;
determine an intersection of the lower-bound function and the upper-bound function;
increment the selected standard deviation value to an incremented overbound standard deviation value when a step size from the selected evaluation point to the intersection is less than or equal to a threshold value; and
determine, for the base distribution, the overbound distribution based on the selected mean value and the incremented overbound standard deviation value.

2 . The navigation system of claim 1 , wherein the base distribution is a Gaussian mixture and the overbound distribution is a single Gaussian.

3 . The navigation system of claim 1 , wherein the selected mean value is a median of the base distribution.

4 . The navigation system of claim 1 , wherein the second-order polynomial includes (1) a first coefficient that is based on a first value of the base distribution at the selected evaluation point, (2) a second coefficient that is based on a second value of a first order derivative, of the base distribution, at the selected evaluation point, and (3) a third coefficient that is based on a minimum value of a second order derivative, of the base distribution, over a selected region of the base distribution from the selected evaluation point to a different evaluation point.

5 . The navigation system of claim 1 , wherein the first-order polynomial includes (1) a first coefficient that is based on a first value of the overbound distribution at the selected evaluation point, and (2) a second coefficient that is based on a second value of a first order derivative, of the overbound distribution, at the selected evaluation point.

6 . The navigation system of claim 1 , wherein the module when executed by the processor further configured to:
determine that the incremented overbound standard deviation value is valid when the step size from the evaluation point to the intersection is greater than the threshold value; and determine that the overbound distribution overbounds the base distribution for a selected region of the base distribution defined by the step size.

7 . The navigation system of claim 1 , wherein the module when executed by the processor further configured to:
define a next evaluation point of the base distribution by adding the selected evaluation point to a minimum of (1) a predetermined maximum step size and (2) the step size from the selected evaluation point to the intersection.

8 . The navigation system of claim 1 , further comprising a filter configured to:
execute a solution algorithm; and use the overbound distribution as an estimate of a filter state during execution of the solution algorithm.

9 . A method for determining an overbound distribution for a base distribution representing a mixture of a plurality of components, the method comprising:
for each of a plurality of different regions that define the base distribution:
identifying a selected evaluation point of the base distribution;
determining a second-order polynomial of the base distribution utilizing a component mean value and a component standard deviation value for each of the plurality of components associated with the base distribution;
lower-bounding the base distribution at the selected evaluation point utilizing the second-order polynomial thereby defining a lower-bound function;
determining a first-order polynomial of the overbound distribution utilizing a selected mean value and a selected standard deviation value;
upper-bounding the base distribution at the evaluation point utilizing the first-order polynomial thereby defining an upper-bound function;
determining an intersection of the lower-bound function and the upper-bound function;
incrementing the selected standard deviation value to an incremented overbound standard deviation value when a step size from the selected evaluation point to the intersection is less than or equal to a threshold value; and
determining, for the base distribution, the overbound distribution based on the selected mean value and the incremented overbound standard deviation value.

10 . The method of claim 9 , wherein the base distribution is a Gaussian mixture and the overbound distribution is a single Gaussian.

11 . The method of claim 9 , wherein the selected mean value is a median of the base distribution.

12 . The method of claim 9 , wherein the second-order polynomial includes (1) a first coefficient that is based on a first value of the base distribution at the selected evaluation point, (2) a second coefficient that is based on a second value of a first order derivative, of the base distribution, at the selected evaluation point, and (3) a third coefficient that is based on a minimum value of a second order derivative, of the base distribution, over a selected region of the base distribution from the selected evaluation point to a different evaluation point.

13 . The method of claim 9 , wherein the first-order polynomial includes (1) a first coefficient that is based on a first value of the overbound distribution at the selected evaluation point, and (2) a second coefficient that is based on a second value of a first order derivative, of the overbound distribution, at the selected evaluation point.

14 . The method of claim 9 , further comprising:
determining that the incremented overbound standard deviation value is valid when the step size from the evaluation point to the intersection is greater than the threshold value; and determining that the overbound distribution overbounds the base distribution for a selected region of the base distribution defined by the step size.

15 . The method of claim 9 , further comprising:
defining a next evaluation point of the base distribution by adding the selected evaluation point to a minimum of (1) a predetermined maximum step size and (2) the step size from the selected evaluation point to the intersection.

16 . The method of claim 9 , further comprising:
executing a solution algorithm; and using the overbound distribution as an estimate of a filter state during execution of the solution algorithm.

17 . A non-transitory computer readable medium having software encoded thereon, the software when executed by one or more computing devices operable to:
for each of a plurality of different regions that define a base distribution:
identify a selected evaluation point of the base distribution;
determine a second-order polynomial of the base distribution utilizing a component mean value and a component standard deviation value for each of a plurality of components associated with the

Assignees

• Novatel Inc

Inventors

• Garner Graeme

Classifications

• G01S19/47

  the supplementary measurement being an inertial measurement, e.g. tightly coupled inertial · CPC title

• G01S19/49

  whereby the further system is an inertial position system, e.g. loosely-coupled · CPC title

• G01S19/20 Primary

  Integrity monitoring, fault detection or fault isolation of space segment · CPC title