# A Non-recursive Solution Method for Fixed-interval Smoothing Problems Applied to Short-term Inertial Navigation

##### Nieminen, Tuukka (2013)

Nieminen, Tuukka

Tampere University of Technology

2013

Teknis-taloudellinen tiedekunta - Faculty of Business and Technology Management

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**Julkaisun pysyvä osoite on**

https://urn.fi/URN:ISBN:978-952-15-3055-5

##### Tiivistelmä

This study focuses on short-term inertial navigation performed within a fixed time interval; one which is already over before the gathered data is processed. This yields a fixed-interval smoothing problem. The time interval is assumed to be short in order to simplify the equations related to inertial navigation without causing excessive errors to the estimates of attitude, velocity, and position, these values being the solutions to the problem. The aim is to develop a new solution method for applications of inertial navigation, particularly in sports, where the objective is often to ensure that the hardware can be integrated with the relevant equipment. This obviously imposes serious constraints on the size and mass of the used navigation system. Therefore, this study focuses on the use of consumer-grade sensors, and new calibration methods are also presented to improve the performance of such sensors.

The traditional approach to fixed-interval smoothing problems is based on the principle of combining two recursive filters, which are run forwards and backwards in time. This study, however, uses a non-recursive solution method. The advantages of this approach are best described with a single word: flexibility. Firstly, with this solution method there is no need to decide whether the fixed-interval smoothing problem is based on initial or boundary values, i.e. whether the ordinary differential equation describing the time evolution of the system is posed as an initial value problem or a boundary value problem. Secondly, it allows many forms of additional information to be used, which can be related to an arbitrary number of time instances. And thirdly, this solution method produces accurate results in the absence of any detailed knowledge of the involved errors.

The proposed non-recursive solution method uses a specific combination of the constructed state and observation equations in order to find a solution to the problem. The problem itself is expressed as a Tikhonov regularization problem, which allows one to obtain accurate results without detailed knowledge of the involved errors. When the problem is linear and the errors fulfill certain assumptions, the resulting solution is known to be the best linear unbiased estimator.

The main objective of this study is to construct a new solution method for fixed-interval smoothing problems; one which can be readily used in practical applications, where detailed knowledge of the involved errors is not available. The proposed solution method is presented in a detailed enough level to be implemented in a high-level environment such as Matlab. Therefore, the thesis also presents a reference implementation of an algorithm designed to solve linear fixed-interval smoothing problems. This thesis concludes by applying the proposed solution method to two sports in which such technology has not been used before.

The traditional approach to fixed-interval smoothing problems is based on the principle of combining two recursive filters, which are run forwards and backwards in time. This study, however, uses a non-recursive solution method. The advantages of this approach are best described with a single word: flexibility. Firstly, with this solution method there is no need to decide whether the fixed-interval smoothing problem is based on initial or boundary values, i.e. whether the ordinary differential equation describing the time evolution of the system is posed as an initial value problem or a boundary value problem. Secondly, it allows many forms of additional information to be used, which can be related to an arbitrary number of time instances. And thirdly, this solution method produces accurate results in the absence of any detailed knowledge of the involved errors.

The proposed non-recursive solution method uses a specific combination of the constructed state and observation equations in order to find a solution to the problem. The problem itself is expressed as a Tikhonov regularization problem, which allows one to obtain accurate results without detailed knowledge of the involved errors. When the problem is linear and the errors fulfill certain assumptions, the resulting solution is known to be the best linear unbiased estimator.

The main objective of this study is to construct a new solution method for fixed-interval smoothing problems; one which can be readily used in practical applications, where detailed knowledge of the involved errors is not available. The proposed solution method is presented in a detailed enough level to be implemented in a high-level environment such as Matlab. Therefore, the thesis also presents a reference implementation of an algorithm designed to solve linear fixed-interval smoothing problems. This thesis concludes by applying the proposed solution method to two sports in which such technology has not been used before.

##### Kokoelmat

- Väitöskirjat [4885]