For an arbitrary countable discrete infinite group G, non-singular rank-one actions are introduced. It is shown that the class of non-singular rank-one actions coincides with the class of non-singular $(C,F)$-actions. Given a decreasing sequence of cofinite subgroups in G with $\bigcap _{n=1}^\infty \bigcap _{g\in G}g\Gamma _ng^{-1}=\{1_G\}$, the projective limit of the homogeneous G-spaces $G/\Gamma _n$ as $n\to \infty $ is a G-space. Endowing this G-space with an ergodic non-singular non-atomic measure, we obtain a dynamical system which is called a non-singular odometer. Necessary and sufficient conditions are found for a rank-one non-singular G-action to have a finite factor and a non-singular odometer factor in terms of the underlying $(C,F)$-parameters. Similar conditions are also found for a rank-one non-singular G-action to be isomorphic to an odometer. Minimal Radon uniquely ergodic locally compact Cantor models are constructed for the non-singular rank-one extensions of odometers. Several concrete examples are constructed and several facts are proved that illustrate a sharp difference of the non-singular non-commutative case from the classical finite measure preserving one: odometer actions which are not of rank-one and factors of rank-one systems which are not of rank one; however, each probability preserving odometer is a factor of an infinite measure preserving rank-one system, etc.

We characterize the stabilized automorphism group for odometers and Toeplitz subshifts, and then prove an invariance property of the stabilized automorphism group of these dynamical systems. Namely, we prove the isomorphism invariance of the primes for which the p-adic valuation of the period structure tends to infinity. A particular case of interest is that for torsion-free odometers, the stabilized automorphism group is a full isomorphism invariant.

Foreman and Weiss [Measure preserving diffeomorphisms of the torus are unclassifiable. Preprint, 2020, arXiv:1705.04414] obtained an anti-classification result for smooth ergodic diffeomorphisms, up to measure isomorphism, by using a functor $\mathcal {F}$ (see [Foreman and Weiss, From odometers to circular systems: a global structure theorem. J. Mod. Dyn.15 (2019), 345–423]) mapping odometer-based systems, $\mathcal {OB}$ , to circular systems, $\mathcal {CB}$ . This functor transfers the classification problem from $\mathcal {OB}$ to $\mathcal {CB}$ , and it preserves weakly mixing extensions, compact extensions, factor maps, the rank-one property, and certain types of isomorphisms. Thus it is natural to ask whether $\mathcal {F}$ preserves other dynamical properties. We show that $\mathcal {F}$ does not preserve the loosely Bernoulli property by providing positive and zero-entropy examples of loosely Bernoulli odometer-based systems whose corresponding circular systems are not loosely Bernoulli. We also construct a loosely Bernoulli circular system whose corresponding odometer-based system has zero entropy and is not loosely Bernoulli.

In this paper, we study the boundary quotient $\text{C}^{\ast }$-algebras associated with products of odometers. One of our main results shows that the boundary quotient $\text{C}^{\ast }$-algebra of the standard product of $k$ odometers over $n_{i}$-letter alphabets $(1\leqslant i\leqslant k)$ is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_{i}:1\leqslant i\leqslant k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph $\text{C}^{\ast }$-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz–Pimsner $\text{C}^{\ast }$-algebra is isomorphic to the boundary quotient $\text{C}^{\ast }$-algebra. Some relations between the boundary quotient $\text{C}^{\ast }$-algebra and the $\text{C}^{\ast }$-algebra $\text{Q}_{\mathbb{N}}$ introduced by Cuntz are also investigated.

In Strapdown Inertial Navigation System (SINS)/Odometer (OD) integrated navigation systems, OD scale factor errors change with roadways and vehicle loads. In addition, the random noises of gyros and accelerometers tend to vary with time. These factors may cause the Kalman filter to be degraded or even diverge. To address this problem and reduce the computation load, an Adaptive Two-stage Kalman Filter (ATKF) for SINS/OD integrated navigation systems is proposed. In the Two-stage Kalman Filter (TKF), only the innovation in the bias estimator is a white noise sequence with zero-mean while the innovation in the bias-free estimator is not zero-mean. Based on this fact, a novel algorithm for computing adaptive factors is presented. The proposed ATKF is evaluated in a SINS/OD integrated navigation system, and the simulation results show that it is effective in estimating the change of the OD scale factor error and robust to the varying process noises. A real experiment is carried out to further validate the performance of the proposed algorithm.

The integration of Global Positioning Systems (GPS) with Inertial Navigation Systems (INS) has been very actively studied and widely applied for many years. Some sensors and artificial intelligence methods have been applied to handle GPS outages in GPS/INS integrated navigation. However, the integrated system using the above method still results in seriously degraded navigation solutions over long GPS outages. To deal with the problem, this paper presents a GPS/INS/odometer integrated system using a fuzzy neural network (FNN) for land vehicle navigation applications. Provided that the measurement type of GPS and odometer is the same, the topology of a FNN used in a GPS/INS/odometer integrated system is constructed. The information from GPS, odometer and IMU is input into a FNN system for network training during signal availability, while the FNN model receives the observations from IMU and odometer to generate odometer velocity correction to enhance resolution accuracy over long GPS outages. An actual experiment was performed to validate the new algorithm. The results indicate that the proposed method can improve the position, velocity and attitude accuracy of the integrated system, especially the position parameters, over long GPS outages.