A navigation device and method in which quantification measurement errors are modeled as states and a computer-readable storage medium.

Abstract

NAVIGATION DEVICE AND METHOD IN WHICH QUANTIZATION MEASUREMENT ERRORS ARE MODELED AS STATES. The present invention relates to a navigation device that determines an estimated position of an object. Navigation information, which includes a sequence of quantized measurements, is received. Each quantized measurement has a corresponding quantization error that is negatively correlated with the quantization error of a previous quantized measurement. The device iteratively performs a navigation update operation that includes determining a state and a covariance matrix of the object for a current iteration based on a state and covariance matrix in a previous iteration. The state includes a final quantization error and an initial quantization error. The covariance matrix includes final and initial covariance values ​​corresponding to the final and initial quantization errors, respectively.Determining the state and covariance matrix for the current iteration involves replacing the initial quantization error with the final quantization error determined in a previous iteration, and updating the state and covariance matrix through a Kalman filter update operation.

Description

1 / 43 “NAVIGATION DEVICE AND METHOD IN WHICH QUANTIZATION MEASUREMENT ERRORS ARE MODELED AS STATES AND COMPUTER-READABLE STORAGE MEDIUM” Related Orders

[0001] This application is a continuation of U.S. Patent Application No. 17 / 235,856, filed April 20, 2021, which claims priority to U.S. Provisional Patent Application No. 63 / 019,213, filed May 1, 2020, each of which is incorporated herein by reference in its entirety. Field of Invention

[0002] The described embodiments generally refer to navigation systems that determine the position and, optionally, one or more of the speed, attitude and / or acceleration of navigation systems or moving objects associated with navigation systems, and update positioning information in navigation systems using measurements or navigation information having quantization errors. More particularly, the described embodiments refer to modeling quantization errors used to update positioning information in navigation systems. Fundamentals of the Invention

[0003] Receivers in global navigation satellite systems (GNSS), such as the Global Positioning System (GPS), use range measurements that are based on line-of-sight signals from satellites. The signal strength received by GPS receivers is weak, approximately 10-16 watts, and therefore susceptible to interference caused by a variety of environmental factors, including natural obstructions (e.g., trees, canyons), man-made physical obstructions (e.g., buildings, bridges), and electromagnetic interference (e.g., signal jammers, electronic equipment emitting broadband or narrowband noise). These reliability factors, along with accuracy limitations, are the main concerns limiting the adoption of GPS, or requiring GPS supplementation, in markets such as the growing automotive industry. Petition 870260032634, dated 08 / 04 / 2026, page 9 / 109 2 / 43 commercial tonomes. Note that all references to “GPS” systems and navigation methodologies in this document are understood to include all GNSS-based systems and navigation systems.

[0004] One approach to improving position information for a moving object is to use one or more measurement receivers that acquire a sequence of quantized measurements from one or more measurement sensors (e.g., encoder, odometer, GNSS signal receiver to measure quantized GNSS signal phases) to estimate and update the position information of a moving object. In some cases, quantized measurements are used independently of GPS signals (e.g., for short periods or time, or in parallel with GPS-based updates) to determine and update the estimated position of a moving object, allowing accurate position estimates of the moving object even when GPS signals become obstructed. Summary of the Invention

[0005] To address reliability and accuracy issues, navigation information from a measurement receiver can be used to update the positioning information of a moving object using a Kalman filter. Position updates for a moving object using measurements from a measurement receiver can be useful when GPS signals are weak or unreliable, as well as to provide position updates between updates based on GPS signals. Note that the term “position updates” in this document is understood to include all relevant updates of the state vector and therefore, where applicable, as including speed and / or acceleration updates and, optionally, other state values ​​such as attitude or sensor errors, of a navigation device or of a moving object associated with the navigation device.In this document, a mobile object associated with the navigation device is understood to be a mobile object in which the navigation device is embedded, a mobile object to which the navigation device is connected, or a mobile object in communication (e.g., using wireless communications) with the navigation device. Petition 870260032634, dated 08 / 04 / 2026, page 10 / 109 3 / 43

[0006] Position updates for the moving object based on information received from a measurement receiver can be calculated independently and / or in parallel with the position updates of the moving object that are based on GPS signals. In some cases, and typically, the information received at the measurement receivers are quantized measurements that have quantization errors. The quantization errors can be modeled as part of the estimated state (e.g., estimated state of the moving object), sometimes here called the Kalman state or Kalman filter state, thus providing greater accuracy in the estimates of the moving object's positioning information and, optionally, providing an enhanced figure of merit, such as an estimate of the accuracy or reliability of the estimated positioning information.By modeling the quantization error of quantized measurements in the estimated state rather than in a noise measurement matrix, the accumulated error associated with a sequence of quantized measurements can be accurately modeled and estimated.

[0007] In contrast, when updating the estimated position of the moving object according to a traditional Kalman filter update operation, the quantization error is modeled using a noise measurement matrix. This traditional method results in an increase in the accumulated error as the number of measurements or number of update iterations increases (e.g., the estimated error of updated position information increases with an increase in the number of update iterations). This apparent growth in the accumulated error does not accurately reflect the total quantity error of the measurements received from the measurement acquisition mechanism. However, the total error in the quantized measurements across multiple measurements does not grow with time or the number of measurements and is instead limited by a fixed amount associated with the quantization error of any given measurement.Thus, the methods, systems, and devices described here allow the quantization errors of quantization measurements to be estimated more accurately compared to traditional Kalman filtering methods, avoiding or mitigating the inaccurate result that the cumulative error associated with a sequence of received quantized measurements increases with the number of measurements or the number of calculations. Petition 870260032634, dated 08 / 04 / 2026, page 11 / 109 4 / 43 position update.

[0008] Some embodiments provide a system (e.g., a navigation device), a computer-readable storage medium storing instructions, or a method for determining an estimated position of an object based on navigation information from one or more measuring receivers.

[0009] (A1) In the method for determining an estimated position of an object, a navigation device receives navigation information corresponding to a position or change in the position of the object from one or more measurement receivers. A first measurement receiver of one or more measurement receivers receives, over time, navigation information comprising a sequence of quantized measurements (e.g., measurements distinct from pseudorange estimates produced by a GPS or GNSS system or subsystem). Each of the quantized measurements has a corresponding quantization error.In some embodiments, the quantization error of each quantized measurement in the sequence of quantized measurements is negatively correlated with the quantization error of the previous quantized measurement, or described somewhat differently, the quantization error of each quantized measurement is negatively correlated with the quantization error of previous quantized measurements in the sequence of quantized measurements. The method generates estimated positions of the object from the quantized measurements. The method includes iteratively performing a navigation update operation at a sequence of times separated by time intervals.The navigation update operation involves determining an estimated object state and an estimated object state covariance matrix for a current time (e.g., current iteration, a kaiteration) in the time sequence based on an estimated state and estimated object state covariance matrix for a previous time (e.g., previous iteration, a (k-1)aiteration) in the time sequence. The estimated object state includes a first part that includes at least estimated position values ​​and estimated velocity values, and a second part that includes an end-of-interval quantization error value and a start-of-interval quantization error value. The estimated object state covariance matrix includes a... Petition 870260032634, dated 08 / 04 / 2026, page 12 / 109 5 / 43 The first part includes covariance values ​​for the first part of the estimated state, and the second part includes final and initial covariance values ​​corresponding to the end-of-interval and start-of-interval quantization error values, respectively, in the second part of the estimated state. The navigation update operation also includes determining the estimated state of the object and the object's estimated state covariance matrix for the current time (e.g., current iteration, the kaiteration). The method includes, in each iteration, replacing the start-of-interval quantization error value in the estimated state with the end-of-interval quantization error value determined in a previous iteration of the navigation update operation.The method also includes updating the estimated state and the estimated state covariance matrix according to a predefined Kalman filter update operation to generate the object's estimated state and the object's estimated state covariance matrix for the current time (e.g., current iteration, kaiteration).

[0010] (A1A) In some embodiments, the estimated state and the estimated state covariance matrix are updated, using the predefined Kalman filter update operation, according to a respective quantized measurement in the sequence of quantized measurements (for example, a quantized measurement received since a last iteration of the navigation update operation was performed).

[0011] (A2) In some embodiments of the A1 or A1A method, determining the estimated state of the object and the estimated state covariance matrix of the object for the current time (e.g., current iteration, kaiteration) involves replacing an element of the estimated state covariance matrix corresponding to the end-of-interval quantization error value with a first predefined value. In some embodiments, the first predefined value is determined based on a largest quantization error (e.g., a largest predefined quantization error) of any respective quantized measurement in the sequence of quantized measurements (e.g., the amount of change in a quantized measurement if only the least significant bit (LSB) value of the quantized measurement is changed). Petition 870260032634, dated 08 / 04 / 2026, page 13 / 109 6 / 43

[0012] (A3) In some embodiments of the A2 method, determining the estimated state of the object and the estimated state covariance matrix of the object for the current time (e.g., current iteration, kaiteration) includes replacing an element of the estimated state covariance matrix corresponding to the interval start quantization error value with a covariance value determined in a previous iteration of the navigation update operation.

[0013] (A4) In some embodiments of the method of either A2 - A3, determining the estimated state of the object and the estimated state covariance matrix of the object for the current time (e.g., current iteration, kaiteration) includes replacing the end-of-interval quantization error value in the estimated state with a second predefined value (e.g., zero).

[0014] (A5) In some embodiments of the method of either A2 - A4, updating the estimated state and the estimated state covariance matrix according to the predefined Kalman filter update operation includes using a Kalman gain function that includes a noise measurement matrix that is equal to zero.

[0015] (A6) In some embodiments of the method of any of A2 - A5, updating the estimated state and the estimated state covariance matrix according to the predefined Kalman filter update operation includes using a Kalman gain function that includes a sensitivity matrix. The sensitivity matrix includes an end-of-interval quantization error sensitivity value that corresponds to the end-of-interval quantization error value and a start-of-interval quantization error sensitivity value that corresponds to the start-of-interval quantization error value. The end-of-interval quantization error sensitivity value and the start-of-interval quantization error sensitivity value are equal in magnitude and opposite in sign.

[0016] (A7) In some embodiments of the method of any of A1 - A6, the method further includes generating one or more figures of merit, for example, a figure of merit corresponding to the estimated value determined for the current time. Each figure of merit is based on one or more diagonal elements of the covariance matrix of Petition 870260032634, dated 08 / 04 / 2026, page 14 / 109 7 / 43 estimated state (for example, one or more diagonal elements of the estimated state covariance matrix determined for the current time, or a linear combination of one or more diagonal elements and one or more off-diagonal elements of the estimated state covariance matrix).

[0017] (A8) In some embodiments of the method of any of A1 - A7, the estimated state covariance matrix also includes covariance values ​​for any, or one or more of: sensor bias errors, object attitude, and sensor errors.

[0018] (A9) In some embodiments of the method of any of A1 - A8, the method includes reporting at least part of the estimated state (e.g., position and speed, or position, attitude and speed) of the object to a reporting apparatus (e.g., the user interface or user interface subsystem of a vehicle comprising the object; or a computer system external to the navigation apparatus) or other external system, external to the navigation apparatus. Optionally, a figure of merit is reported together with the estimated state part to the reporting apparatus or external system.

[0019] (A10) In some embodiments of the method of any of A1 - A9, the object is a movable object.

[0020] (A11) In some embodiments of the method of any of A1 - A10, the time intervals separating the sequence of times have a predetermined value.

[0021] (A12) In some embodiments of the method of any of A1 - A11, the object includes at least one of one or more measuring receivers.

[0022] (A14) In another aspect, in some embodiments, a navigation device for an object comprises: one or more processors; one or more measurement receivers for receiving navigation information corresponding to a position or change of position of the object; and memory storing one or more programs that include instructions for iteratively performing a navigation update operation in a sequence of times separated by time intervals. A first measurement receiver of one or more measurement receivers receives, along Petition 870260032634, dated 08 / 04 / 2026, page 15 / 109 8 / 43 of the time, navigation information comprising a sequence of quantized measurements. Each of the quantized measurements has a corresponding quantization error. In some embodiments, the quantization error of each quantized measurement in the sequence of quantized measurements is negatively correlated with the quantization error of the previous quantized measurement, or described somewhat differently, the quantization error of each quantized measurement is negatively correlated with the quantization error of previous quantized measurements in the sequence of quantized measurements.The navigation update operation involves determining an estimated object state and an estimated object state covariance matrix for a current time (e.g., current iteration, a(k-1)iteration) in the time sequence based on an estimated state and estimated object state covariance matrix for a previous time (e.g., previous iteration, a(k-1)iteration) in the time sequence. The estimated object state includes a first part that includes at least estimated position values ​​and estimated velocity values, and a second part that includes an end-of-interval quantization error value and a start-of-interval quantization error value.The object's estimated state covariance matrix includes a first part containing covariance values ​​for the first part of the estimated state, and a second part containing initial and final covariance values ​​corresponding to the end-of-interval and start-of-interval quantization error values, respectively, in the second part of the estimated state. The navigation update operation includes determining the object's estimated state and the object's estimated state covariance matrix for the current time (e.g., current iteration, kaiteration). The navigation update operation further includes, in each iteration, replacing the start-of-interval quantization error value in the estimated state with the end-of-interval quantization error value determined in a previous iteration of the navigation update operation.The navigation update operation also includes updating the estimated state and the estimated state covariance matrix according to a predefined Kalman filter update operation to generate the object's estimated state and matrix. Petition 870260032634, dated 08 / 04 / 2026, page 16 / 109 9 / 43 of the estimated state covariance of the object for the current time (e.g., current iteration, kaiteration).

[0023] In some modes of the A14 navigation device, one or more programs include instructions for performing the method of any of the A1 to A13.

[0024] (A15) In another aspect, in some embodiments, a computer-readable storage medium stores one or more programs, one or more programs including instructions which, when executed by one or more processors of a system (for example, a navigation device or navigation system) to navigate a moving object according to signals from a plurality of satellites, execute a method comprising: receiving navigation information corresponding to a position or change of position of the object, including receiving, over time, navigation information comprising a sequence of quantized measurements. Each of the quantized measurements has a corresponding quantization error.In some embodiments, the quantization error of each quantized measurement in the sequence of quantized measurements is negatively correlated with the quantization error of the previous quantized measurement, or described somewhat differently, the quantization error of each quantized measurement is negatively correlated with the quantization error of previous quantized measurements in the sequence of quantized measurements. The method also includes iteratively performing a navigation update operation on a sequence of times separated by time intervals. The navigation update operation includes determining an estimated state of the object and an estimated state covariance matrix of the object for a current time (e.g., current iteration, a(k-1)iteration) in the sequence of times based on an estimated state and estimated state covariance matrix of the object for a previous time (e.g., previous iteration, a(k-1)iteration) in the sequence of times.The estimated state of the object includes a first part that includes at least estimated position values ​​and estimated velocity values, and a second part that includes an end-of-range quantization error value and a start-of-range quantization error value. The matrix. Petition 870260032634, dated 08 / 04 / 2026, page 17 / 109 The 10 / 43 estimated state covariance matrix of the object includes a first part that includes covariance values ​​for the first part of the estimated state, and a second part that includes initial and final covariance values ​​corresponding to the start-of-interval and end-of-interval quantization error values, respectively, in the second part of the estimated state. The navigation update operation further includes replacing the start-of-interval quantization error value in the estimated state with the end-of-interval quantization error value determined in a previous iteration of the navigation update operation. The navigation update operation also includes updating the estimated state and the estimated state covariance matrix according to a predefined Kalman filter update operation to generate the estimated state of the object and the estimated state covariance matrix for the current time (e.g., current iteration, the kaiteration).

[0025] In some embodiments of one or more computer-readable storage media of A15, one or more programs include instructions that, when executed by one or more processors of the system, cause the system to execute the method of any one from A1 to A13. Brief Description of the Drawings

[0026] Figure 1 is a block diagram illustrating a navigation system, according to some modalities.

[0027] Figure 2 is a block diagram illustrating a navigation device associated with a moving object from Figure 1, according to some embodiments.

[0028] Figure 3 is a block diagram illustrating a moving object from Figure 1, according to some embodiments.

[0029] Figure 4 is a flowchart that illustrates a process for updating the position of a moving object using a Kalman filter, according to some modalities.

[0030] Figure 5A is an example of a data structure for the estimated state of a moving object, according to some modalities.

[0031] Figure 5B is an example of a Kalman gain function, according to some embodiments. Petition 870260032634, dated 08 / 04 / 2026, p. 18 / 109 11 / 43

[0032] Figure 5C is an example of a covariance matrix corresponding to the Kalman gain function in Figure 5B, according to some embodiments.

[0033] Figure 5D is an example of an estimated initial state for a moving object, according to some embodiments.

[0034] Figure 5E is an example of an updated covariance matrix corresponding to an initial state of Figure 5D, according to some embodiments.

[0035] Figures 6A - 6D are flowcharts of a method for estimating the position of a moving object, according to some modalities. Detailed Description of the Invention

[0036] Figure 1 is a block diagram illustrating a navigation system 100, according to some embodiments. The navigation system 100 allows the position, or more precisely an estimate of the position, of a moving object 110 (for example, a rover, such as a motor vehicle, for example, a car, truck or tractor, or an object, such as a trailer, being transported or otherwise moved by a vehicle or other object) to be determined at any point in time. The moving object 110 is sometimes called a mobile object, such as in embodiments in which the moving object 100 has wheels but no propulsion system.The navigation system 100 includes one or more measuring receivers 112 that receive information from one or more measuring sensors 118 (e.g., an odometer or an encoder, see Figure 2), and optionally includes another measuring device, such as one or more satellite signal receivers (e.g., GPS signal receiver, satellite antenna) that are configured to receive satellite navigation signals (e.g., GPS signals) from a plurality of more satellites (e.g., satellites 114-1 to 114-S, where S is an integer, typically having a value of at least 4) in view of the moving object, herein collectively described as satellites 114). The navigation system 100 optionally includes the moving object 110.

[0037] The satellite navigation signals received in the navigation system are typically Global Navigation Satellite System (GNSS) signals, such as Global Positioning System (GPS) signals in the L1 signal frequency of Petition 870260032634, dated 08 / 04 / 2026, page 19 / 109 12 / 43 1575.42 MHz and, optionally, GPS signals at the L2 signal frequency of 1227.6 MHz, or other navigation signals at other frequencies. The mobile object 110, or a navigation device associated with the mobile object 110 (for example, a navigation device coupled, embedded or otherwise associated with the mobile object 110), may be equipped with one or more of the measuring receivers 112 and, optionally, the other measuring device, such as one or more satellite signal receivers.In some embodiments, the mobile object 110 or its associated navigation device is also equipped with one or more communication interfaces, such as one or more communication ports for wired communication with other devices, and / or wireless communication interfaces (e.g., including a radio transmitter and receiver) for wireless communication with external systems (e.g., to transmit position information, such as a current position of the mobile object 110, or relative position information regarding the relative position of the mobile object 110 in relation to a previous position of the mobile object 110).

[0038] An estimated position of the moving object 110 is determined based, at least in part, on quantized measurements received at one or more measuring receivers 112. In some embodiments, the estimated position of the moving object 110 may also be determined based on satellite signals received from one or more satellites 114. In some embodiments, the estimated position is that the moving object 110 is at an absolute position. In some embodiments, the estimated position of the moving object 110 is represented by a differential position value, such as a relative position vector. In the following discussion, and throughout this document, the term “estimated position” means either an absolute position of the moving object 110 or a relative position of the moving object 110 with respect to a previous position or a predetermined base position (e.g., an initial position).

[0039] In some embodiments, a navigation device (e.g., navigation device 200, Figure 2) associated with the moving object 110 is configured to determine or generate an estimated position for the moving object 110 based on quantized measurements that are received at predefined times or a sequence of times separated by intervals (e.g., at 0.1-second intervals). In Petition 870260032634, dated 08 / 04 / 2026, page 20 / 109 13 / 43 In some modes, the intervals separating the time sequence have a predetermined value. In some modes, the navigation device can also determine an estimated position of the moving object 110 based on other forms of measurement, such as received satellite signals.

[0040] For example, the navigation device (e.g., using measurement update module 226 of navigation device 200, Figure 2) can generate updated position estimates (or more generally, estimated states) for the moving object 110 at a first set of intervals, such as 0.1-second intervals, occurring at a first rate (e.g., 10 Hz), based on quantized measurements (e.g., measurements of the amount of rotation of a wheel of the moving object) and (e.g., using GPS or another update module 224 of navigation device 200, Figure 2) at a second set of intervals, such as 1.0-second intervals, occurring at a second rate (e.g., 1 Hz) different from the first rate, based on received satellite signals, using two distinct position update processes or modules (e.g., modules 224 and 226, Figure 2) that update the same position estimate (or estimated state) for the moving object. 110.

[0041] In some embodiments, the estimated position for moving object 110 is determined by combining the estimated position for moving object 110 at a previous time (e.g., an initial time, or a time that is prior to the current time) with the change in position of moving object 110 between the current time and the previous time. In some implementations, the navigation device is configured to generate updated estimated positions for moving object 110 at a rate greater than or equal to 10 Hz (e.g., an updated estimated position is generated every 100 milliseconds when the update rate is 10 Hz, 40 milliseconds when the update rate is 25 Hz, or every 20 milliseconds when the update rate is 50 Hz).

[0042] In some embodiments, the navigation device reports an estimated position of the moving object 110 to an intelligent system, sometimes called an external system, external to the navigation device. For example, an intelligent system Petition 870260032634, dated 08 / 04 / 2026, page 21 / 109 14 / 43 can be a server or a client system (e.g., a desktop, a laptop, a mobile phone, a tablet, a personal digital assistant (PDA), etc.). In some embodiments, the intelligent system is located on the mobile object 110 (e.g., a user interface subsystem that presents maps and other information to a user of the mobile object) or is coupled to the mobile object 110. The intelligent system is optionally connected to a network, such as the Internet and / or a local area network within the mobile object 110. Optionally, the intelligent system is configured to control the movement of the mobile object 110 (e.g., controlling the steering and / or propulsion systems of the mobile object 110) so as to maintain a predefined distance or relative position between the mobile object 110 and a predefined position or another object, such as a second mobile object or a stationary object.In some embodiments, the navigation device for moving object 110 does not include an intelligent system, while in some other embodiments, the navigation device for moving object 110 communicates with the intelligent system only intermittently. In some embodiments, the navigation system 100 is configured to report updated estimated positions for moving object 110 at a rate that is less than or equal to the rate at which the updated estimated positions are generated. For example, the updated estimated positions may be generated at 10 Hz, but the navigation system 100 may report updated estimated positions at a rate of 0.5 Hz.

[0043] Figure 2 is a block diagram illustrating a navigation device 200 that is part of the navigation system 100 of Figure 1, according to some embodiments. The navigation device 200 typically includes one or more processors (CPUs) 202 to execute programs or instructions; one or more measurement receivers 204 to receive measurements from one or more measurement sensors 118 (such as encoders, odometers, GNSS signal receiver to measure quantized GNSS signal phases, etc.), although one or more of the measurement receivers 204 may be external to the navigation device 200 (for example, coupled to the navigation device by a wired or wireless connection 211); optionally it includes a satellite signal receiver 208 to receive satellite navigation signals (by Petition 870260032634, dated 08 / 04 / 2026, page 22 / 109 15 / 43 For example, if external to the navigation device, the satellite signal receiver 208 is coupled by a wired connection 215 to the navigation device 200); one or more communication interfaces 207; memory 210; and one or more communication buses 205 to interconnect these components. The navigation device 200 optionally includes a user interface 209 comprising a display device and one or more input devices (e.g., one or more keyboards, mice, touch screens, keyboards, etc.). In some embodiments, the user interface 209 is a system or subsystem external to the navigation device 200, coupled to the navigation system by a wired or wireless connection 219. One or more communication buses 205 may include sets of circuits (sometimes called a chipset) that interconnect and control communications between the system components.

[0044] The communication interface 207 (for example, a transmitter or transceiver, such as a radio transmitter or transceiver, or a wired communication transmitter or transceiver) is used by the navigation device 200 to receive communications, such as communications from an intelligent system, as described above, or another system (internal to or external to the moving object 110, navigation device 200 or navigation system 100) that is in communication with the navigation device 200. The communication interface 207, if included in the navigation device 200, is also used by the navigation device 200 to send signals from the navigation device 200 to an intelligent system and / or report information corresponding to the position of the moving object 110. In some embodiments, the communication interface 207 is a single transceiver, while in other embodiments, the communication interface 207 includes separate transceivers or separate communication interfaces.In some embodiments, the communication interface 207 is a wireless communication interface for connecting the navigation device 200 to an intelligent system via the internet. In some embodiments, the navigation device 200 includes one or more additional receivers or communication interfaces for receiving or exchanging information with other systems, for example, to receive information from a wide-area differential global positioning system. Petition 870260032634, dated 08 / 04 / 2026, page 23 / 109 16 / 43

[0045] In some embodiments, one or more communication interfaces 207 of the navigation device 200 include a reporting device 206 (for example, a transmitter or transceiver, such as a radio transmitter or transceiver, or a wired communication transmitter or transceiver) that is used by the navigation device 200 to send information, such as information corresponding to the position of the moving object 110, from the navigation device 200 to one or more external systems 144.Examples of external systems that receive information from the reporting device 206 and that are embedded in or coupled to the mobile object 110 are a user interface subsystem (e.g., a user interface subsystem of the mobile object 110 for displaying maps, such as presenting a map showing the location of the mobile object 110 and optionally providing other mapping and navigation functions for a user of the mobile object 110) and a steering and / or propulsion system 144 of the mobile object. In some embodiments, an external system to which the reporting device 206 sends information is external to the mobile object 110. In some embodiments, the reporting device 206 is a single transceiver, while in other embodiments, the reporting device 206 includes two or more separate transceivers or separate communication interfaces.In some embodiments, the reporting device 206 is or includes a wireless communication interface for connecting the navigation device 200 to one or more external systems 144 via the internet or other wide area network. In some embodiments, the reporting device 206, whether internal to or external to the navigation device 200, is or includes a “user-level navigation system” that displays maps, shows the location of the moving object 110 on a map, and provides other mapping and navigation functions for a user of the moving object 110.

[0046] In some embodiments, an example of which is shown in Figure 3, the reporting device 206 is separated from one or more communication interfaces 207 and is optionally external to the navigation device 200. In some of these embodiments, the reporting device 206 receives information from the navigation device 200 through a respective communication interface from one or more communication interfaces 207. Petition 870260032634, dated 08 / 04 / 2026, p. 24 / 109 17 / 43

[0047] Memory 210 includes high-speed random-access memory, such as DRAM, SRAM, DDR RAM, or other random-access solid-state memory devices; and may include non-volatile memory, such as one or more magnetic disk storage devices, optical disk storage devices, flash memory devices, or other non-volatile solid-state storage devices. Memory 210 optionally includes one or more storage devices located remotely from the CPU(s) 202. Memory 210, or alternatively the non-volatile memory device(s) within memory 210, comprises a computer-readable storage medium. In some embodiments, memory 210 or the computer-readable storage medium of memory 210 stores the following programs, modules, and data structures, or a subset thereof:

[0048] An operating system 212 that includes procedures for handling various basic system services and for performing hardware-dependent tasks.

[0049] One or more communication modules 214 that operate in conjunction with one or more communication interfaces 207 (for example, one or more transceivers and / or communication ports) to handle communications between the navigation device 200 and other systems. In some embodiments, the communication modules 214 include drivers that, when executed by one or more processors 202, enable various hardware modules on one or more communication interfaces 207 to receive and / or send messages to external systems, external to the navigation device 200.

[0050] Data received from one or more measurement receivers 216 (for example, one or more recently received datasets, optionally implemented as a local database), which includes data (for example, data or information corresponding to one or more measurement sensors 118, such as an odometer or encoder) associated with the position or relative position (for example, an encoder that counts the number of revolutions of a wheel, or odometer) of the moving object 110. In some embodiments, one of the measurement receivers 216 receives information from a wheel encoder, which produces N “clicks” (for example, 100 Petition 870260032634, dated 08 / 04 / 2026, p. 25 / 109 18 / 43 clicks) per revolution of a wheel of the moving object's propulsion system, where each “click” corresponds to a change in the least significant bit (LSB) of the measurement carried by the encoder to the measurement receiver 216. In some embodiments, the data received by one or more measurement receivers 216 are used by the Kalman module(s) 222, as described in more detail below, to help determine the location of the moving object 110.

[0051] Navigation application 218, which in some embodiments includes instructions that enable the navigation device 200 to perform navigation functions, such as directing the moving object 110 along a path (for example, on a road or in an agricultural field), or maintaining the moving object 110 at a fixed attitude and distance relative to another moving object.

[0052] Steering or propulsion control module 220 for steering or controlling the propulsion of the moving object 110.

[0053] One or more Kalman modules 222 (sometimes called navigation modules) which, when executed by CPU 202, determine a moving object position 110 or a relative moving object position 110 using satellite navigation data received from satellites (e.g., GPS or GNSS satellite) 114 by satellite receiver 208, and / or data received from measurement receiver(s) 112.

[0054] In some implementations, the Kalman 222 module(s) includes one or more GPS and / or other 224 update module(s) and a 226 measurement update module, as described below.

[0055] In some embodiments, one or more GPS and / or other update module(s) 224 process received satellite navigation signals to determine satellite navigation data and estimated positions for the moving object 110 in a sequence of times, sometimes called epoch timeouts, and optionally sometimes between epoch timeouts. This processing involves measuring or determining measurements of the received satellite navigation signals. For example, the measurements may include, for each satellite from which navigation signals are received, a pseudorange between the moving object and the satellite, and Petition 870260032634, dated 08 / 04 / 2026, page 26 / 109 19 / 43 Phase measurements at one or more predefined frequencies, such as the L1 frequency, or the L1 and L2 frequencies. One or more GPS and / or other update module(s) 224 use the satellite navigation signals and provide estimated positions of the moving object 110 that are updated using a Kalman filter. The term Kalman filter here means any of a wide range of Kalman filters, including not only conventional Kalman filters, but also extended Kalman filters, unscented Kalman filters, and other estimation techniques (e.g., linear quadratic estimation techniques) that use a series of measurements observed over time, which may include statistical noise and other inaccuracies, and a recursive update process to make estimates of the current state of a system or object.

[0056] The measurement update module 226 processes the data received from the measurement receiver(s) 112 to provide estimated positions for the moving object 110 in a sequence of times separated by time intervals. This processing involves receiving the measurements 216 (e.g., quantized measurements) from one or more measurement sensors 118, and the measurement update module 226 uses the received measurements 216 to provide estimated positions of the moving object 110 that are updated using a Kalman filter. In some embodiments, the measurement update module 226 includes an error update module 228 (or instruction set) and a covariance matrix update module 230 (or instruction set).The measurement update module 226 models information about the moving object 110, such as position, velocity, acceleration, and attitude (e.g., yaw, roll, pitch), as well as quantization error associated with received measurements (e.g., an interval start quantization error value and an interval end quantization error value corresponding to quantization error values ​​at the beginning and end of an interval, respectively) as elements of an estimated state of the moving object. In some embodiments, the estimated state may also include elements corresponding to sensor bias errors or sensor errors, representing persistent errors (or slow variation) in received measurements or navigation signals. Petition 870260032634, dated 08 / 04 / 2026, page 27 / 109 20 / 43

[0057] As described above with reference to Figure 1, in some embodiments, the navigation system 200 updates the estimated state 232 of the moving object 110 in two different but overlapping time sequences, including, in a first time sequence, based on quantized measurements from one or more measuring sensors 118, and in a second time sequence, based on GPS, GNSS or other satellite signals received using a satellite receiver 208. For example, in some embodiments, updates (e.g., performed by the measurement update module 226) to the estimated state based on received quantized measurements are performed in a first time sequence, tstart + ΔT1 X i, for i = 0 to ilasten, while updates (e.g., using GPS or another update module 224 of the navigation device 200, Figure 2) to the estimated state 232 based on satellite signals (e.g.,(GPS or GNSS signals) are performed in a second sequence of times, tstart + ΔF + ΔT2X j, for j = 0 to jlast where tstart represents a start time, such as when the navigation device first begins generating and updating the estimated state or, alternatively, represents any point in time during which the navigation device is actively updating the estimated state; ΔT1 and ΔT2 are different non-zero time intervals, such as 0.1 seconds and 1 second, respectively, in which the estimated state is updated by the two update operations (e.g., one based on received quantized measurements and the other based on received satellite signals, respectively); ΔF is a non-zero offset time, e.g., 0.01 or 0.05 seconds, to ensure that the two sets of updates occur in different but overlapping time sequences; and jlast and jlast corresponding to the last, or most recent,Updates or iterations of the two sets of update operations.

[0058] In some embodiments, the error update module 228 updates (e.g., replaces) the interval start quantization error value in a Petition 870260032634, dated 08 / 04 / 2026, page 28 / 109 21 / 43 initial state (e.g., an estimated initial state) for a current interval (e.g., a time interval corresponding to the kaiteration of a navigation update operation) with the end-of-interval quantization error value of the most recent previous interval (e.g., the previous iteration, the (k-1)iteration), since the end of an interval corresponds to (e.g., is the same as) the beginning of a next interval (e.g., successive). Details on updating an estimated state of the moving object are described below in relation to Figure 5D. The covariance matrix update module 230 updates (e.g., replaces) elements of a covariance matrix for a next interval (e.g., a time interval corresponding to the (k+1) iteration of a navigation update operation). Details on the ways in which a covariance matrix is ​​updated are described below in relation to Figure 5E.The measurement update module 226 calculates an updated estimated state 232 and an estimated covariance matrix 234, based on a most recently received measurement 216 (e.g., the last measurement received for the current interval or update iteration). In some embodiments, the information corresponding to the estimated state 232 is reported to an external system via the communication interface 207. For example, in some embodiments, the information corresponding to the estimated state 232 is transmitted to a steering and / or propulsion system 144 via the reporting device 206.

[0059] The operating system 212 and each of the modules and applications identified above correspond to a set of instructions for performing a function described above. The instruction set can be executed by one or more processors 202 of the navigation device 200. The modules, applications, or programs identified above (i.e., instruction sets) need not be implemented as separate software programs, procedures, or modules, and therefore, various subsets of these modules can be combined or otherwise rearranged in various embodiments. In some embodiments, memory 210 stores a subset of the modules and data structures identified above. In addition, memory 210 optionally stores additional modules and data structures. Petition 870260032634, dated 08 / 04 / 2026, page 29 / 109 22 / 43 not described above.

[0060] Figure 2 is intended more as a functional description of the various features that may be present in a navigation device 200 than as a structural scheme of the modalities described herein. In practice, and as recognized by those skilled in the art, the items shown separately may be combined and some items may be separated. For example, some items shown separately in Figure 2 may be combined into a single module or component, and single items may be implemented using two or more modules or components. For example, the GPS and / or other update module(s) 224 and the measurement update module 226 may be implemented by different computers or by different processors, each processor having its own memory to store programs for execution by that processor, which are part of the navigation device 200.The actual number of modules and components, and how resources are allocated among them, varies from one implementation to another. In another set of examples, in some embodiments, one or more of the components or subsystems shown in Figure 2 are external to the navigation device 200. For example, in some embodiments, one or more of the measurement receivers 112 (204 in Figure 2) are external to the navigation device 200, coupled to the navigation device 200 by a wired or wireless connection 211; in some embodiments, the reporting device 206 is external to the navigation device 200, coupled to the navigation device 200 by a wired or wireless connection 217; and in some embodiments, the satellite receiver 208 is external to the navigation device 200, coupled to the navigation device 200 by a wired connection 215.

[0061] Figure 3 is a block diagram illustrating the mobile object 110 in Figure 1 in some embodiments. In some embodiments, one or more measuring receivers 112 and the navigation device 200 are both part of (e.g., fitted, embedded, or coupled to) the mobile object 110. For example, measuring receiver(s) 112, such as a receiver for an encoder or odometer, and navigation device 200 are mounted on a rover vehicle (sometimes called a mobile object and illustrated as a tractor in Figure 1). The mobile object 110 may also include Petition 870260032634, dated 08 / 04 / 2026, p. 30 / 109 23 / 43 any of the communication apparatus 206 and steering and / or propulsion system 144. For example, the communication apparatus 206 may be a communication device that is part of the moving object 110 that facilitates communication between the navigation apparatus 200 and the steering and / or propulsion system 144. In some embodiments, the reporting apparatus 206 and / or steering and / or propulsion system 144 may be external to the moving object 110 and in communication with the navigation apparatus 200. Furthermore, in some embodiments, as shown in Figure 2, one or more of the measuring receivers 112 (204 in Figure 2) are part of the navigation apparatus 200, and in some embodiments, the reporting apparatus 206 is part of the navigation apparatus 200. However, in some other embodiments, one or more of the measuring receivers 112 (204 in Figure 2) are external to the navigation apparatus. 200 and, in some forms, the reporting device 206 is external to the navigation device 200.

[0062] The measurement receiver(s) 112 receives measurements (e.g., quantized measurements) from one or more measurement sensors 118 (e.g., encoder, odometer, or GNSS signal receiver for measuring quantized GNSS signal phases) and transmits the received information (e.g., the received measurement) to the navigation device 200. The navigation device 200 determines an estimated state (e.g., estimated position, speed, acceleration, attitude, etc.) of the moving object 110 using the received information in a Kalman filter update operation, as described in more detail below. Optionally, the information computed from the estimated state (e.g., position, estimated speed, etc.) is transmitted to the steering and / or propulsion system 144, optionally via the reporting device 206. The steering and / or propulsion system 144, if provided, provides instructions for navigating and / or controlling the movements of the moving object 110.Alternatively, the information computed from the estimated state (e.g., position, estimated speed, etc.) is provided or transmitted to the reporting device 206, which may be internal or external to the navigation device 200.

[0063] Figure 4 is a flowchart that illustrates a 400 process of updating a Petition 870260032634, dated 08 / 04 / 2026, page 31 / 109 24 / 43 Estimated state of a moving object 110 using a Kalman filter update operation. When a measurement (such as a quantized measurement) is recorded, the navigation device 200 updates an estimated state (e.g., position, speed, attitude) of the moving object 110 using the measurement(s) 216 (e.g., data, information, quantized measurements) received from one or more measurement receivers 112. In some embodiments, the estimated state of the moving object 110 is updated in response to receiving a measurement 216 or is updated for each measurement received 216. The parts of the process 400 that correspond to the iterative update of the estimated state of the moving object 110 are shown in a dashed box 402 in Figure 4.

[0064] At the beginning of process 400, (operation 410) the navigation information (e.g., position, velocity, acceleration, attitude) for the moving object 110 is represented by an initial state. For example, the initial state may show an initial position (e.g., from start) (e.g., x, y, z representing a recently determined position for the moving object, or a last determined position for the moving object) for the moving object 110 as well as any one of an initial velocity, an initial acceleration, and / or an initial attitude (e.g., pitch, orientation, roll). In some embodiments, process 400 optionally includes (operation 420) calculating a current state (e.g., a state for the kaiteration) for the moving object using one or more dynamic models (e.g., physical models, kinematic equations, etc.). In the first iteration, the navigation information from the initial state (from operation 410) is used in the calculations of operation 420.In iterations following the first iteration, an initial state from a previous (e.g., most recent) iteration is used in the calculations of operation 420. In some embodiments, as shown in Figure 4, the calculation of a state using dynamic model(s) (e.g., operation 420) is performed before (operation 440) receives measurement 216. Alternatively, (operation 420) the calculation of a state using the dynamic model(s) can be performed simultaneously with or after (operation 440) receives measurement 216. Petition 870260032634, dated 08 / 04 / 2026, page 32 / 109 25 / 43

[0065] The update of the estimated state of the moving object 110 normally begins after a measurement 216 is received (operation 440) in one or more measurement receivers 112. Before being received, the measurement is optionally recorded (operation 430), for example, by sensor 118 or another device external to the navigation device 200. After receiving (operation 440) the measurement and optionally computing (operation 420) a state using the dynamic model(s), the process 400 includes the update operation (450) of the state of the moving object 110 based on the received measurement 216 using a Kalman filter update operation.

[0066] In some embodiments, operations 420-460 are performed at a predefined rate, whether or not any new measurements are recorded and received. This predefined rate is typically in the range of 1000 times per second to 1 time per second inclusive, with 10 times per second (at 0.1 second intervals) being a typical example. Optionally, in some embodiments, if no new measurement is received during the current time period (e.g., because the moving object did not move during that time interval or the amount of movement detected by the sensor or encoder was less than a predefined minimum quantized distance (e.g., corresponding to a single bit change in the quantized measurement value)), the recording, receiving, and estimated state determination operations 430, 440, and 450 are ignored.But in some other modes, if the determined state (operation 420) using dynamic models causes any change in the state of the moving object, then the estimated state determination operation 450 is performed even if no new measurement is received during the current time interval.

[0067] In some embodiments, in the Kalman filter update operation (operation 450), an estimated state is updated based on equation (1), below. *k =xk +Kk(zk-f(Xk)) (1)

[0068] In equation (1), xk is the estimated state (such as the estimated state 510, described below in relation to Figure 5A) for a moving object 110, x£ is an updated estimated state for the moving object generated by the Kalman filter update operation, / (¾) is the estimated value of the measurement, based on the estimated current state. Petition 870260032634, dated 08 / 04 / 2026, page 33 / 109 26 / 43 χκ for moving object 110, which is calculated using dynamic models (e.g., calculated in operation 420), Kk is the Kalman gain, sometimes called the Kalman gain function, such as Kalman gain function 530, described below in relation to Figure 5B), zk is the measurement (e.g., quantized measurement) received in operation 440, and the subscript k refers to a current time or computational iteration (e.g., in equation (1), the kaiteration usually represents the current iteration). The Kalman gain can generally be thought of as a function that determines how much weight should be given to the estimated value of the measurement, based on the estimated current state x^ (as calculated using dynamic models in operation 420) versus the received measurement zk (as recorded in operation 430 and received in operation 440) in the calculation to determine the estimated state of moving object 110.Additional details regarding the 450 update operation are provided below in relation to Figures 5A - 5C, including details on the estimated state (described in relation to Figure 5A) and the Kalman gain equation (described in relation to Figures 5B and 5C).

[0069] The estimated state of the navigation device or moving object for the current time, produced by the current iteration (e.g., kaiteration) of the estimated state update process, is output in operation 460 and used in operation 470 to update an initial state for the next iteration (e.g., the (k+1)aiteration) of operations 420-450. Operation 470 occurs after the Kalman filter update process (e.g., operation 450) of the current iteration and before the Kalman filter update process (e.g., operation 450) of the next iteration. In other words, operation 470 is performed between operation 450 of successive iterations. Although operation 470 is shown in Figure 4 as being performed after the exit operation 460 for a kaiteration and before operation 420 for a (k+1) iteration, in other embodiments, operation 470 may be performed after operation 420 and before operation 450.Additional details regarding operation 470 are described below in relation to Figures 5D and 5E.

[0070] In some embodiments, the estimated state for the moving object 110 that is generated (e.g., calculated, updated) in operation 450 is reported to a reporting device 206 and / or steering and / or propulsion system 144. Petition 870260032634, dated 08 / 04 / 2026, page 34 / 109 27 / 43

[0071] Process 400, as described above, can be repeated any number of times (e.g., iterations) in order to update the estimated state of moving object 110 over time. In some embodiments, the number of iterations or the time period over which process 400 is repeated is predetermined.

[0072] Figure 5A is an example of an estimated state 510 (e.g., corresponding to estimated state 232 in Figure 2, and xk in equation (1)) for a moving object 110, according to some embodiments. The estimated state 510 is determined (e.g., generated, calculated, computed) based on received measurements (e.g., received measurements 216) which are quantized measurements. The estimated state 510 is a 1 by L matrix (also designated 1 x L) that includes L elements. Each element is represented as ai, for i = 1 to L, where i (or indicating) is the row of the 1 x L matrix that contains that element. For example, the first element of the estimated state 510 is a1, the second element is a2, and so on. The estimated state 510 includes a first part 512 and a second part 514. The first part 512 of the estimated state 510 includes L-2 elements. The elements in the first part 512 of the estimated state 510 correspond to attributes related to the position of the moving object 110.For example, in some embodiments, each of the first, second, and third elements a1, a2, and a3 corresponds to a position of the moving object 110 along the x, y, and z directions, respectively. Each of the fourth, fifth, and sixth elements a4, a5, and a6 corresponds to a velocity of the moving object 110 along the x, y, and z directions, respectively. The first part 512 of the estimated state 510 may also include other elements corresponding to various other attributes related to the position of the moving object 110, such as attitude (pitch, orientation, roll) (sometimes called direction of movement) and acceleration. In some embodiments, the first part 512 of the estimated state 510 may also include one or more elements that correspond to sensor bias errors or sensor errors, representing persistent errors (or slow variation) in received measurements or navigation signals.

[0073] The second part 514 of the estimated state 510 includes two elements a(L-1) and aL. The elements of the second part 514 of the estimated state 510 correspond to a Petition 870260032634, dated 08 / 04 / 2026, page 35 / 109 28 / 43 end-of-interval quantization error value, εend (e.g., the quantization error value at the end of an interval), and an interval start quantization error value, Estart (e.g., the quantization error value at the beginning of an interval). It is understood that the elements of the estimated state 510, as described above, can be arranged in any order and do not need to follow the order shown in Figure 5A.

[0074] Figure 5B is an example of a Kalman gain function 530 (e.g., Kk in equation (1)), sometimes called Kalman gain, used in the Kalman filter update operation (e.g., operation 450 in Figure 4) to update the estimated state 510 (shown in Figure 5A). The Kalman gain function 530 includes the covariance matrix 520 (represented by P and described below in relation to Figure 5C), a sensitivity matrix 540 (represented by H in Figure 5B), and a noise measurement matrix 550 (represented by R in Figure 5B). The sensitivity matrix 540 is a model of how the received measurement (e.g., measurement 216, quantized measurement, zk in equation (1)) affects the current state (e.g., xk in equation (1)). The 550 noise measurement matrix is ​​a model of the known or expected noise in a measurement technique used to acquire (e.g., record, capture) the measurement (e.g., 216 measurement, quantized measurement, zk in equation (1)).

[0075] The sensitivity matrix 540 (H), sometimes called the design matrix, is a 1 by L matrix that includes the same number of elements as a corresponding estimated state 510 and, similarly, includes a first part 542 corresponding to the first part 512 of the estimated state 510 and a second part 544 corresponding to the second part 514 of the estimated state 510. The elements of the sensitivity matrix 540 are represented as ci with i being (or indicating) the row of the matrix 540 of each element. In the first part of the sensitivity matrix 540, the elements ci to C(l-2), inclusive, correspond to the effect of the received measurement (e.g., measurement 216, quantized measurement, zk in equation (1)) on the current state (e.g., x^ in equation (1)).

[0076] In addition, the second part 544 of the 540 sensitivity matrix includes a Petition 870260032634, dated 08 / 04 / 2026, p. 36 / 109 29 / 43 element, for example, C(Li), which represents an end-of-interval quantization error sensitivity value and corresponds to the end-of-interval quantization error value, εend, in the estimated state 510, and includes another element, for example, cl, which represents a start-of-interval quantization error sensitivity value and corresponds to the start-of-interval quantization error value, Estart, of the estimated state, and these two elements, sometimes called quantization error sensitivity value elements, are in positions of the sensitivity matrix 540 that correspond to the positions of the corresponding quantization error values ​​in the estimated state 510.In the example shown in Figures 5A-5B, the value of the end-of-interval quantization error, εend, in the estimated state 510, and the corresponding end-of-interval quantization error sensitivity value is the (L-1)th element of both matrices, and the value of the start-of-interval quantization error, Estart, in the estimated state 510, and the corresponding start-of-interval quantization error sensitivity value are the Loelement of both matrices.

[0077] In some embodiments, such as embodiments in which successive received measurements (e.g., different from the first received measurement) are quantized measurements, in a sequence of received quantized measurements, each with a negative correlation with at least one previous measurement (e.g., a preceding measurement, a closer previous measurement) in the sequence of received quantized measurements, the end-of-interval quantization error sensitivity value and the beginning-of-interval quantization error sensitivity value are equal in magnitude and opposite in sign (e.g., equal to 1 and -1, respectively), representing (e.g., reflecting, indicating) that the measurement for a given interval is formed from the quantized value at the end of the interval minus the quantized value at the beginning of the interval.

[0078] The 550 noise measurement matrix is ​​a noise measurement model in the measurement technique used to obtain the measurements (e.g., 216 measurements, quantized measurement, zk in equation (1)) used in updating the Kalman filter. Thus, the 550 noise measurement matrix will differ for different measurement types or different measurement mechanisms. For example, a noise measurement matrix Petition 870260032634, dated 08 / 04 / 2026, page 37 / 109 30 / 43 for a measurement corresponding to GPS signals will be different from a noise measurement matrix for a measurement corresponding to encoder readings from a wheel encoder. In another example, noise measurement matrices corresponding to GPS signals, odometer readings, and radar signals will be different from each other.

[0079] In some embodiments, such as when modeling the state (e.g., position, velocity, attitude, etc.) of a moving object 110 based on quantized measurements and using the estimated state 510 as described in relation to Figure 5A), the measurement error of the quantized measurements, including any non-quantization errors associated with the measurement, is modeled by the quantization error in the estimated state 510 and its corresponding covariance matrix elements. In these cases, the remaining measurement noise (in the measurement technique used to obtain the quantized measurements) is zero and therefore the noise measurement matrix 550 is zero.

[0080] An example of the 520 covariance matrix (e.g., covariance matrix P in Figure 5B) of the 530 Kalman gain function is shown in Figure 5C. The 520 covariance matrix is ​​an L by L matrix that includes L2 elements (e.g., the number of elements in the covariance matrix is ​​equal to L2). Each element is represented as bm,n, with m being (or indicating) the row of the element and n being (or indicating) the column of the element. The diagonal elements of the 520 covariance matrix represent squared values. The 520 covariance matrix includes a first part 522 and a second part 524. The first part 522 of the 520 covariance matrix includes (L-2)2 elements. The elements in the first part 522 of the covariance matrix 520 correspond to elements in the first part 512 of the estimated state 510 (for example, elements representing attributes related to the position of the moving object 110).The elements in the second part 524 of the covariance matrix 520 correspond to elements in the second part 514 of the estimated state 510 (for example, elements representing quantization errors at the beginning and end of an interval). In the example shown in Figure 5C, the second part 524 of the covariance matrix 520 includes the last two rows (also called the last and second-to-last row) and the last two columns (also called the last and second-to-last column) of the... Petition 870260032634, dated 08 / 04 / 2026, page 38 / 109 31 / 43 covariance matrix 520 in the example shown in Figure 5C, and more generally includes two rows and two columns corresponding to the elements in the second part of the estimated state 510.

[0081] In some embodiments, a figure of merit for an attribute (e.g., an attribute related to position) of the moving object may be calculated based on one or more diagonal elements of the covariance matrix 520, and provided (e.g., reported) along with at least a portion of the estimated state of the moving object 110, to a reporting device (e.g., a device or system external to the navigation device 200). Optionally, the figure of merit for a moving object attribute may be calculated based on one or more diagonal elements of the covariance matrix 520, as well as one or more off-diagonal elements of the covariance matrix 520. Providing a figure of merit to the reporting device allows the reporting device to determine the level of precision that should be assigned to the estimated state, or to the portion of the estimated state (e.g., position) that corresponds to the figure of merit. For example,If the figure of merit indicates that the reported position is likely accurate with a first margin (or range) of error (for example, plus or minus 1 meter), the position may be represented by the reporting device to a user or other system in a manner consistent with that first margin (or range) of error. If the figure of merit indicates that the reported position is likely accurate with a second margin (or range) of error (for example, plus or minus 10 meters), different from the first margin of error, the position may be represented by the reporting device to a user or other system in a manner consistent with the second margin (or range) of error, different from the manner consistent with the first margin (or range) of error (for example, by representing the current position differently (for example, with a larger circle or object representation on a displayed map).or with fewer non-zero digits) than when representing the position in a manner consistent with the first margin (or range) of error.

[0082] As noted above, the diagonal elements of the 520 covariance matrix represent squared values. In some forms, a figure of Petition 870260032634, dated 08 / 04 / 2026, page 39 / 109 32 / 43 merit corresponding to a position of the moving object 110 corresponds to the square root of the sum of the diagonal elements corresponding to the position elements of the estimated state. For example, the element b1,1 may correspond to the variance in the calculated position of the moving object 110 along the x direction (e.g., ^b1t1 or 4Ãx corresponds to the variance in a1), the element b2,2 may correspond to the variance in the calculated position of the moving object 110 along the y direction (e.g., jb2,2 or JÃy corresponds to the variance in a2), the element b3,3 may correspond to the variance in the calculated position of the moving object 110 along the z direction (e.g., ^h3,3 or VÃz corresponds to the variance in a3), and so on. Thus, a figure of merit (FOM) corresponding to a position (in x, y, z) of the moving object 110 can be determined as follows: FOM = ^b1t1+ h2,2+ b3>3.Thus, the figure of merit for an estimated three-dimensional position is similar to a "standard deviation" for the three-dimensional position, where the "standard deviation" quantifies the extent to which the estimated position lacks reliability.

[0083] Two diagonal elements (e.g., the last two elements) of the covariance matrix 520 correspond to the variance in the end-of-interval quantization error value, Δεend, and the variance in the start-of-interval quantization error value, Δεstart. For example, when the quantization error of an interval is a random error assumed to have a normal distribution, the variance in the end-of-interval quantization error value, Δεend, has a value of (LSB2) / 12, where LSB is the distance (or other unit of measurement) corresponding to the least significant bit in the measurement (e.g., the smallest possible non-zero value difference (e.g., distance) between any two measurement values). The elements of the covariance matrix 520 are determined based on the elements of the estimated state 510.Thus, if the elements of an estimated state differ (either by having different elements, or by elements being presented in a different order) from the estimated state 510, the corresponding covariance matrix would consequently differ from the covariance matrix 520 shown in Figure 5C.

[0084] After the estimated state 510 is generated (e.g., computed, issued) Petition 870260032634, dated 08 / 04 / 2026, page 40 / 109 33 / 43 for a current time (e.g., kaiteration of the Kalman filter update operation) and before the next Kalman filter update operation (e.g., operation 450 for the (k+1)iteration) begins, an initial state 560 and an updated Kalman gain are determined for the next iteration (e.g., the (k+1)iteration). The initial state 560 for the next iteration (e.g., the (k+1)iteration) is determined (e.g., updated) based on the estimated state 510 determined during the current iteration. The Kalman gain for the next iteration (e.g., the (k+1) iteration) is updated by updating the covariance matrix in the Kalman gain function.

[0085] Figure 5D illustrates how the initial state 560 (e.g., an initial state for the (k+1) iteration) is determined (e.g., updated) based on the estimated state 510 (e.g., an estimated state that is generated using a Kalman filter update operation in the kaiteration). The process of determining the initial state 560 corresponds to operation 470 in Figure 4. The elements in the estimated state 510 are indicated with the superscript k to indicate that the elements of the estimated state 510 correspond to (e.g., is calculated as part of, is generated and,) the kaiteration. Similarly, the elements in the initial state 560, represented as di (with i being or indicating the element's row), are denoted with the superscript k+1 to indicate that the elements of the initial state 560 correspond to (e.g., are used in) the (k+1) kaiteration. Similar to the estimated state 510, the initial state 560 includes a first part 562 and a second part 564.The first part 562 of the initial state 560 includes L-2 elements, and the elements in the first part 562 of the initial state 560 correspond to attributes related to the position of the moving object 110 in the (k+1)th iteration. The second part 564 of the initial state 560 includes two elements, ά^_\) and df+1. The elements of the second part 562 of the initial state 560 correspond to the end-of-interval quantization error value, ε^1, and the start-of-interval quantization error value, s^t, for the (k+1)th interval (for example, the (k+1)th iteration).

[0086] Since the end of the kointerval corresponds to (occurs at the same time as) the beginning of the (k+1) interval, the value of the interval start quantization error Petition 870260032634, dated 08 / 04 / 2026, page 41 / 109 34 / 43 for the (k+1) interval is the same as the end-of-interval quantization error value for the k interval. Thus, the start-of-interval quantization error value for the (k+1) interval, £start, is updated to have the same value as (for example, is replaced by) the end-of-interval quantization error value of the k interval, c^ (ΐΊίΊΓ ovomnln «fc+1 — r.& m· om lix / Qlontomonto c^ —X pfc+1 \£end(for example, £start =£end>or equivalently, £end^ Estart).

[0087] In some embodiments, the interval start quantization error value for the (k+1) interval, eS^alt, is updated (e.g., replaced) with a predefined value. In some embodiments, the predefined value is zero. Once any updates (e.g., replacements) to the initial state elements 560 have been made based on the estimated state 510, determined during the previous iteration (e.g., the kaiteration) of the update process, the initial state 560 can be used in the Kalman filter update operation of the next (e.g., successive) iteration (e.g., the initial state 560 is used as an input to equation (1) in the (k+1)ateration and will result in a new estimated state being determined based on the initial state 560 and a new measurement received).

[0088] Figure 5E shows an updated covariance matrix 570 for a Kalman gain (e.g., Kalman gain function 530) used in the next iteration (e.g., (k+1) iteration) of the Kalman filter update process. The updated covariance matrix 570 is determined, for example, during operation 470 (Figure 4), based on the covariance matrix 520 generated in the Kalman filter update process for the current iteration (e.g., the kaiteration). The elements in the updated covariance matrix 570 are represented as em,n with m being or indicating the row of the element and en being or indicating the column of the element in the updated covariance matrix 570. Similarly to the covariance matrix 520, the updated covariance matrix 570 includes a first part 572 and a second part 574.The first part 572 of the updated covariance matrix 570 includes (L-2)2 elements, and elements in the first part 572 of the updated covariance matrix 570 correspond to elements in the first part 562 of the initial state. Petition 870260032634, dated 08 / 04 / 2026, page 42 / 109 35 / 43 560. The second part 574 of the updated covariance matrix 570 includes the last two rows (also here called the last and second-to-last row) and the last two columns (also here called the last and second-to-last column) of the updated covariance matrix 570, and the elements in the second part 574 of the updated covariance matrix 570 correspond to elements in the second part 564 of the initial state 560.

[0089] The elements of the updated covariance matrix 570 are determined by a similarity transformation, shown in equation (2), below. Pk+i = APkAT(2)

[0090] In equation (2), the covariance matrix 520 is represented as Pk, denoted with the subscript k to indicate that the covariance matrix 520 corresponds to (i.e., is part of) the Kalman gain function 530 used in the Kalman filter update operation of the iteration. The updated covariance matrix 570 is represented as Pk+i, denoted with the subscript k+1 to indicate that the updated covariance matrix 570 corresponds to (i.e., is part of) the Kalman gain function 530 used in the Kalman filter update operation of the (k+1)iteration. The transformation matrix A has the same dimensions (and therefore the same number of elements) as the covariance matrix 520 and the updated covariance matrix 570.For example, when an estimated state has 4 elements (e.g., position, velocity, end-of-range quantization error value, and start-of-range quantization error value), the corresponding covariance matrix (e.g., the covariance matrix used in the Kalman gain equation) and therefore the transformation matrix A, are 4 x 4 matrices. In this example, the transformation matrix... 1 0 0 0 A looks like: 0 1 0 0 0 0 . In some forms, the transformation .0 0 1 0 The covariance matrix from the k-th iteration to the (k+1)th iteration includes additional terms that define the transition of the state vector over time. This transformation matrix only considers the update associated with changes in the received measurements. Petition 870260032634, dated 08 / 04 / 2026, page 43 / 109 36 / 43

[0091] The similarity transformation shown in equation (2) (e.g., operation 470 in Figure 4) is performed between two consecutive Kalman filter update operations (e.g., operation 450 of two consecutive iterations). The similarity transformation updates the covariance matrix of the Kalman gain function 530 and therefore updates the Kalman gain function 530, for use in a subsequent iteration (e.g., the Kalman gain function 530, which includes the covariance matrix 520, is used in equation (1) during the kaiteration and an updated Kalman gain function 530 which includes the updated covariance matrix 570 is used in equation (1) during the (k+1)iteration).

[0092] After applying the similarity transformation shown in equation (2), the updated covariance matrix 570 (e.g., updated covariance matrix Pk+i) has the following characteristics:

[0093] The elements in the second to last row e(Li),n of the updated covariance matrix 570 and the elements in the second to last column em,(Li) of the updated covariance matrix 570 are equal to zero.

[0094] The elements of the last column eL,n of the updated covariance matrix 570 are equal to the elements of the second to last column of the covariance matrix 520, except for the element eL,(Li) being zero, as mentioned earlier.

[0095] The elements in the last row of the updated covariance matrix 570 are equal to the elements in the second to last row of the covariance matrix 520, except for the element e(Li),L being zero, as mentioned earlier.

[0096] As a result of the similarity transformation, the element eL,L of the updated covariance matrix 570 has the same value as ob(Li),(Li) of the covariance matrix 520.

[0097] Furthermore, the element e(Li),(Li) of the updated covariance matrix 570 is updated to be equal to (e.g., replaced by) a predefined value. In some embodiments, the predefined value corresponds to an error associated with a technique for acquiring the quantized measurements (e.g., LSB2 / i2 for a quantization error that is expected to follow a normal distribution).

[0098] Figures 6A - 6D represent a flowchart of a method 600 for Petition 870260032634, dated 08 / 04 / 2026, page 44 / 109 37 / 43 determine an estimated position of an object 110 (e.g., moving object 110, Figure 1), according to some embodiments. In some embodiments, the method 600 is implemented by the navigation device 200, under the control of instructions stored in memory 210 (Figure 2) that are executed by one or more processors (202, Figure 2) of the navigation device 200. Each of the operations shown in Figures 6A - 6D corresponds to computer-readable instructions stored in a computer-readable memory storage medium 210 in an information processing device or system, such as the navigation device 200. The computer-readable instructions are in source code, assembly language code, object code, or other instruction format that is interpreted and / or executed by one or more processors 202 of the navigation device 200.

[0099] The navigation device 200 (Figure 2) receives (610), over time, navigation information which includes a sequence of quantized measurements. The navigation information corresponds to a position or change of position of the object 110 (e.g., moving object 110). In some embodiments, each quantized measurement has a corresponding quantization error and the quantization error of each quantized measurement is negatively correlated with the quantization error of one or more previous quantized measurements in the sequence of quantized measurements (see part 544 of the sensitivity matrix 540 in Figure 5B). For example, the quantized measurements may be “clicks” received from a wheel encoder that produces N “clicks” (e.g., 100 clicks) per revolution of a wheel of the moving object’s propulsion system, and each “click” corresponds to a change in the least significant bit (LSB) of the measurement transmitted by the encoder to the measurement receiver 216.

[0100] In a sequence of times that are separated by time intervals, navigation device 200 (e.g., measurement update module 226 (Figure 2) of navigation device 200) iteratively performs (620) a navigation update operation, including: determining (630) an estimated state 510 (Figure 5A) of object 110 and an estimated state covariance matrix 520 (Figure Petition 870260032634, dated 08 / 04 / 2026, page 45 / 109 38 / 43 5C) of object 110 for a current time (e.g., a k0 time, corresponding to a kaiteration of the navigation update operation) in the sequence of times based on an estimated state and an estimated state covariance matrix of the object for a previous time (e.g., a (k-1)0 time, corresponding to a (k-1)ateration of the navigation update operation) in the sequence of times. As described above, the estimated state 510 of object 110 includes a first part 512 (Figure 5A) that includes at least estimated position values ​​(e.g., x, y, z) and estimated velocity values ​​(e.g., vx, vy, vz), and a second part 514 (Figure 5A) that includes an end-of-interval quantization error value (e.g., εend, Figure 5A) and a start-of-interval quantization error (e.g., Estart, Figure 5A).The estimated state covariance matrix 520 of object 110 includes a first part 522 (Figure 5C) which includes covariance values ​​for the first part of the estimated state and a second part 524 (Figure 5C) which includes final and initial covariance values ​​(e.g., Δεend and Δε^, respectively, Figure 5C) corresponding to the end-of-interval and beginning-of-interval quantization error values, respectively, in the second part of the estimated state. Details about the elements of the covariance matrix 520 are provided above in relation to Figure 5C.

[0101] As part of the navigation update operation (620), navigation device 200 (e.g., error update module 228 (Figure 2) of navigation device 200) replaces (640) the interval start quantization error value in the estimated state with the interval end quantization error value determined in a previous iteration (e.g., the immediately preceding iteration) of the navigation update operation. An example is described above in relation to Figure 5D, where the interval start quantization error value for the (k+1) iteration (e.g., Estãrt in estimated state 560) is replaced by the interval end quantization error value determined in the kaiteration (e.g., £gnd in estimated state 510).

[0102] As part of the navigation update operation (620), the navigation device 200 (measurement update module 226 (Figure 2) of the device Petition 870260032634, dated 08 / 04 / 2026, page 46 / 10939 / 43 navigation 200) also updates (650) the estimated state 510 and the estimated state covariance matrix 520 according to a predefined Kalman filter update operation (e.g., using the Kalman gain function 530, Figure 5B) to generate an updated estimated state 560 of the object 110 and an updated estimated state covariance matrix 570 of the object 110 for the current time (e.g., the (k+1) iteration). An example is described above in relation to Figures 5D and 5E. The updated estimated state 560 of the object 110 for the (k+1) iteration is shown in Figure 5D and the updated estimated covariance matrix 570 for the object 110 for the (k+1) iteration is shown in Figure 5E. The estimated state 560 for the (k+1)iteration is determined based on the estimated state 510 for the kaiteration.The elements of the updated covariance matrix 570 for the (k+1)th iteration are determined by a similarity transformation, shown in equation (2) and the estimated covariance matrix 520 for the ka iteration.

[0103] In some embodiments, as described above in relation to Figure 4, updating (650) the estimated state and the estimated state covariance matrix includes updating (652) the estimated state and the estimated state covariance matrix, using the predefined Kalman filter update operation, according to a more recent quantized measurement in the sequence of quantized measurements.

[0104] In some embodiments, the covariance matrix update module 230 of the navigation device 200 replaces (660) an element of the estimated state covariance matrix corresponding to the end-of-interval quantization error value with a first predefined value. An example is described above in relation to Figure 5E: the element e(L-1),(L-1) of the covariance matrix 570 for the (k+1)iteration is replaced by a predefined value. In some embodiments, the predefined value corresponds to an error associated with a technique for acquiring the quantized measurements (e.g., LSB2 / 12 for a quantization error that is expected to follow a normal distribution).

[0105] In some embodiments, in addition to replacing (640) the interval start quantization error value in the estimated state with the interval end quantization error value determined in a previous iteration of the update operation Petition 870260032634, dated 08 / 04 / 2026, page 47 / 109 40 / 43 navigation, the navigation device 200 (e.g., covariance matrix update module 230 of navigation device 200) replaces (670) an element of the estimated state covariance matrix corresponding to the interval start quantization error value with a covariance value determined in a previous iteration of the navigation update operation. An example is described above with respect to Figure 5E: the element eL,L of the covariance matrix 570 for the (k+1)iteration is updated to have the same value as b(Li),(Li) of the covariance matrix 520 for the kaiteration.

[0106] In some embodiments, navigation device 200 (e.g., navigation device error update module 228) replaces (680) the end-of-range quantization error value in the estimated state with a second predefined value. In some embodiments, the second predefined value is zero. For example, the end-of-range quantization error value in the estimated state 560 (e.g., e£+J, Figure 5D) is replaced by a predefined value. In some cases, the end-of-range quantization error value in the estimated state 560 is replaced by a value of zero (e.g., replaced by zero), indicating that the end-of-range quantization error value in the estimated state 560 should be calculated according to a Kalman filter update operation for the (k+1) iteration.

[0107] In some embodiments, navigation device 200 (e.g., measurement update module 226 of navigation device 200) uses (654) a Kalman gain function (e.g., Kalman gain function 530, Figure 5B) which includes a noise measurement matrix 550 (Figure 5B) corresponding to the first measurement receiver (e.g., from one or more measurement receivers 216, a received measurement configured to receive quantized measurements), and the noise measurement matrix 550 corresponding to the first measurement receiver is zero. An example is given above with respect to Figure 5B.

[0108] In some embodiments, the navigation device 200 (for example, measurement update module 226 of the navigation device 200) uses (656) a Kalman gain function (for example, Kalman gain function 530, Figure 5B) which includes a sensitivity matrix 540 (Figure 5B). The sensitivity matrix Petition 870260032634, dated 08 / 04 / 2026, p. 48 / 109 41 / 43 540 includes an end-of-interval quantization error sensitivity value corresponding to the end-of-interval quantization error value and a start-of-interval quantization error sensitivity value corresponding to the start-of-interval quantization error value. The end-of-interval quantization error sensitivity value and the start-of-interval quantization error sensitivity value are equal in magnitude and opposite in sign. For example, as shown in Figure 5B, the 544 part of the 540 sensitivity matrix shows that the end-of-interval quantization error sensitivity value is equal to 1 and the start-of-interval quantization error sensitivity value is equal to -1.

[0109] In some embodiments, the navigation device 200 (e.g., measurement update module 226 of the navigation device 200) generates (690) one or more figures of merit, and each figure of merit is based on one or more diagonal elements of the estimated state covariance matrix. For example, as described above in relation to Figure 5C, a figure of merit for an attribute (e.g., an attribute related to position) of the moving object 110 can be calculated based on one or more diagonal elements of the covariance matrix 520. In some embodiments, the figure of merit is similar to a “standard deviation” that quantifies the extent to which the estimated position lacks reliability. In some embodiments, as discussed above, the figure of merit is reported, along with at least part of the estimated state, to a reporting device internal or external to the navigation device 200.

[0110] In some embodiments, navigation device 200 (e.g., measurement update module 226 of navigation device 200) reports (692) at least part of the estimated state of the object to a reporting device 206 (Figure 2). For example, once measurement update module 226 determines or generates the estimated state 232 or 510 (Figures 2 and 5A, respectively), at least part of the estimated state 510, such as elements of the estimated state corresponding to a position-related attribute (e.g., one or more elements corresponding to a position, velocity, attitude, and / or acceleration) of the object 110, is reported to the reporting device 206. In one specific example, after Petition 870260032634, dated 08 / 04 / 2026, page 49 / 109 42 / 43 Performing an iteration of the navigation update process, measurement update module 226 reports the position (e.g., using x, yez, or latitude, longitude, and altitude) of object 110 to reporting device 206. In a second example, measurement update module 226 reports the velocity (in x, yez, or latitude, longitude, and altitude) of object 110 to reporting device 206 at predetermined time intervals (e.g., every 10 iterations of the navigation update process, or at intervals of half a second, 1 second, 2 seconds, 5 seconds, or 60 seconds).

[0111] In some embodiments, performing (620) the navigation update operation includes generating (e.g., calculating, determining, computing) a current state (e.g., x^ shown in equation (1)) based on an estimated state determined in a previous iteration (e.g., previous iteration, most recent iteration) and one or more dynamic models (e.g., physical equations) of the moving object 110 (see operation 420 in Figure 4). The navigation update operation also includes receiving a quantized measurement in a measurement receiver from one or more measurement receivers 216 (see operation 440 in Figure 4) and generating (e.g., calculating, determining, computing) the estimated state 510 of the moving object 110 based on the current state (e.g., x^ shown in equation (1)), the quantized measurement, and a Kalman gain function 530 (see equation (1), described above in relation to Figure 4).The navigation update operation also includes transmitting the estimated state 510 of the moving object 110 (see operation 460 in Figure 4), for example, to a reporting device internal or external to the navigation device 200.

[0112] As described above, the navigation device 200 performs (620) a navigation update operation (e.g., a Kalman filter update) using quantized measurements to update the estimated position of the object 110. In some embodiments, additional information about the position of the object 110, such as GPS signals or radar information, can also be used to determine the estimated position of the object 110. Thus, the estimated position of the object 110 can be determined using two or more measurement methods (e.g., using an encoder that provides quantized information and using GPS signals). Petition 870260032634, dated 08 / 04 / 2026, page 50 / 109 43 / 43 In such embodiments, the estimated position of object 110 is determined using quantized measurements as described above in relation to Figures 6A-6D, where the quantization error associated with the quantized measurements is modeled in the estimated state (e.g., Kalman state), and the estimated position of object 110 determined using other measurements (e.g., non-quantized measurements, such as GPS signals or radar information) is determined according to a traditional Kalman update operation. In some embodiments, the navigation update operation performed using quantized measurements can be performed independently of the traditional Kalman update operation.In some embodiments, the navigation update operation performed using quantized measurements can be carried out in parallel with the traditional Kalman update operation so that the estimated position determined using quantized measurements can continue to provide updates to the estimated position of the object between estimated position updates using GPS signals or when GPS signals are weak, for example.

[0113] The preceding description, for explanatory purposes, has been described with reference to specific embodiments. However, the above illustrative discussions are not intended to be exhaustive or to limit the invention to the precise forms described. Many modifications and variations are possible in view of the above teachings. The embodiments have been chosen and described to better explain the principles of the invention and its practical applications, thus enabling others skilled in the art to better utilize the invention and various embodiments with various modifications suitable for the particular use considered. Petition 870260032634, dated 08 / 04 / 2026, page 51 / 109

Claims

1 / 6 CLAIMS 1. Navigation apparatus (200) for determining an estimated position of an object (110), comprising: one or more measuring receivers (204) for receiving navigation information corresponding to a position or change in the position of the object (110), wherein a first measuring receiver of one or more measuring receivers receives, over time, navigation information comprising a sequence of quantized measurements, each having a corresponding quantization error; one or more processors (202);memory (210) storing one or more programs, characterized in that one or more programs include instructions for iteratively, in a sequence of times separated by time intervals, performing a navigation update operation, including: determining (610) an estimated state of the object and an estimated state covariance matrix of the object for a current time in the sequence of times based on an estimated state and estimated state covariance matrix of the object for a previous time in the sequence of times, wherein the estimated state of the object includes a first part that includes at least estimated position values ​​and estimated velocity values, and a second part that includes an end-of-interval quantization error value and a start-of-interval quantization error value;the object's estimated state covariance matrix includes a first part that includes covariance values ​​for the first part of the estimated state, and a second part that includes final and initial covariance values ​​corresponding to the end-of-interval and start-of-interval quantization error values, respectively, in the second part of the estimated state; and determining (630) the object's estimated state and the object's estimated state covariance matrix for the current time comprises: replacing (640) the start-of-interval quantization error value in the estimated state with the end-of-interval quantization error value determined in Petition 870260032634, dated 08 / 04 / 2026, page 52 / 109 2 / 6 a previous iteration of the navigation update operation;and update (650) the estimated state and the estimated state covariance matrix according to a predefined Kalman filter update operation to generate the object's estimated state and the object's estimated state covariance matrix for the current time.; 2. Navigation apparatus, according to claim 1, characterized in that the quantization error of each quantized measurement is negatively correlated with the quantization error of one or more previous quantized measurements in the sequence of quantized measurements.

3. Navigation device, according to claim 1, characterized in that updating the estimated state and the estimated state covariance matrix comprises updating the estimated state and the estimated state covariance matrix, using the predefined Kalman filter update operation, according to a more recent quantized measurement in the sequence of quantized measurements.

4. Navigation device, according to any one of claims 1 to 3, characterized in that determining the estimated state of the object and the estimated state covariance matrix of the object for the current time includes replacing an element of the estimated state covariance matrix corresponding to the end-of-interval quantization error value with a first predefined value.

5. Navigation apparatus, according to any one of claims 1 to 4, characterized in that determining the estimated state of the object and the estimated state covariance matrix of the object for the current time includes replacing an element of the estimated state covariance matrix corresponding to the interval start quantization error value with a covariance value determined in a previous iteration of the navigation update operation.

6. Navigation apparatus, according to any one of claims 1 to 5, characterized in that determining the estimated state of the object and the covariance matrix of the estimated state of the object for the current time includes replacing the end-of-interval quantization error value in the estimated state with a second predefined value.

7. Navigation device, according to any one of claims 1 to 6, characterized in that: updating the estimated state and the estimated state covariance matrix according to the predefined Kalman filter update operation includes using a Kalman gain function that includes a noise measurement matrix corresponding to the first measurement receiver; and the noise measurement matrix corresponding to the first measurement receiver is zero.

8. Navigation apparatus, according to any one of claims 1 to 7, characterized in that: updating the estimated state and the estimated state covariance matrix according to the predefined Kalman filter update operation includes using a Kalman gain function that includes a sensitivity matrix; the sensitivity matrix includes an end-of-range quantization error sensitivity value corresponding to the end-of-range quantization error value and a start-of-range quantization error sensitivity value corresponding to the start-of-range quantization error value; and the end-of-range quantization error sensitivity value and the start-of-range quantization error sensitivity value are equal in magnitude and opposite in sign.

9. Navigation device, according to any one of claims 1 to 8, characterized in that one or more programs include instructions for generating one or more figures of merit, each figure of merit based on one or more diagonal elements of the estimated state covariance matrix.

10. Navigation device, according to any one of claims 1 to 9, characterized in that: the estimated state covariance matrix additionally includes covariance values ​​for any of: sensor bias errors, object attitude, and sensor errors. Petition 870260032634, dated 08 / 04 / 2026, p. 54 / 109 4 / 6 11. Navigation apparatus, according to any one of claims 1 to 10, characterized in that one or more programs include instructions for reporting at least part of the estimated state of the object to a reporting apparatus.

12. Navigation apparatus, according to any one of claims 1 to 11, characterized in that the object is a movable object.

13. Navigation apparatus, according to any one of claims 1 to 12, characterized in that the time intervals separating the sequence of times have a predetermined value.

14. Navigation apparatus, according to any one of claims 1 to 13, characterized in that the object includes at least one or more measuring receivers.

15. Method (600) for determining an estimated position of an object (110), the method comprising: in a navigation device (200) for determining an estimated position of an object (110): receiving (610) navigation information corresponding to a position or change in the position of the object, including receiving, over time, navigation information comprising a sequence of quantized measurements, each having a corresponding quantization error, wherein the method is characterized by, in a sequence of times separated by time intervals, iteratively performing a navigation update operation, including: determining (630) an estimated state of the object and an estimated state covariance matrix of the object for a current time in the sequence of times based on an estimated state and estimated state covariance matrix of the object for a previous time in the sequence of times,where the estimated state of the object includes a first part that includes at least estimated position values ​​and estimated velocity values, and a second part that includes an end-of-interval quantization error value and a start-of-interval quantization error value; the object's estimated state covariance matrix includes a first part that includes covariance values ​​for the first part of the estimated state, and a second part that includes final and initial covariance values ​​corresponding to the end-of-interval and start-of-interval quantization error values, respectively,in the second part of the estimated state; and determining the estimated state of the object and the estimated state covariance matrix of the object for the current time comprises: replacing (640) the interval start quantization error value in the estimated state with the interval end quantization error value determined in a previous iteration of the navigation update operation; and updating (650) the estimated state and the estimated state covariance matrix according to a predefined Kalman filter update operation to generate the estimated state of the object and the estimated state covariance matrix of the object for the current time.

16. Method according to claim 15, characterized in that it further comprises the operations as described in any one of claims 2 to 14.

17. Computer-readable storage medium that stores one or more programs, including instructions which, when executed by one or more processors of a navigation device to determine an estimated position of an object (110), cause the navigation device (200) to perform operations comprising: receiving (610) navigation information corresponding to a position or change in the position of the object, including receiving, over time, navigation information comprising a sequence of quantized measurements, each having a corresponding quantization error;characterized by, in a sequence of times separated by time intervals, iteratively performing a navigation update operation, including: Petition 870260032634, dated 08 / 04 / 2026, page 56 / 109 6 / 6 determining (630) an estimated state of the object and an estimated state covariance matrix of the object for a current time in the sequence of times based on an estimated state and estimated state covariance matrix of the object for a previous time in the sequence of times, wherein the estimated state of the object includes a first part that includes at least estimated position values ​​and estimated velocity values, and a second part that includes an end-of-interval quantization error value and a start-of-interval quantization error value;The object's estimated state covariance matrix includes a first part that includes covariance values ​​for the first part of the estimated state, and a second part that includes final and initial covariance values ​​corresponding to the end-of-interval and start-of-interval quantization error values, respectively, in the second part of the estimated state; and determining the object's estimated state and the object's estimated state covariance matrix for the current time comprises: replacing (640) the start-of-interval quantization error value in the estimated state with the end-of-interval quantization error value determined in a previous iteration of the navigation update operation; and updating (650) the estimated state and the estimated state covariance matrix according to a predefined Kalman filter update operation to generate the object's estimated state and the object's estimated state covariance matrix for the current time.

18. Computer-readable storage medium according to claim 17, characterized in that one or more programs include instructions that, when executed by one or more processors of the navigation device, cause the navigation device to perform the operations as described in any one of claims 2 to 14. Petition 870260032634, dated 08 / 04 / 2026, pp. 57 / 109

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