The issue of implementing nonlinear model predictive control (NMPC) on mechanical systems evolving on special orthogonal group (SO(3)) is taken into consideration in the first place. Necessary conditions of optimality are extracted based on Lie group variational integrators, leading to a two-point boundary value problem (TPBVP) which is solved using sensitivity derivatives and indirect shooting methods. Fast Newton-like methods referred to as fast solvers which are commonly used to solve the TPBVP are established based on the repetition of a nonlinear process. The numerical schemes employed to alleviate the computation burden consist of eliminating some constraint-related but non-essential terms in the trend of sensitivity derivatives calculation and for solving the TPBVP equations. As another claim, assuming that a first attempt to resolve the NMPC problem is accessible, the problem subjected to some changes in its initial conditions (due to some re-planning schemes) can be resolved cost-effectively based on it. Instead of solving the whole optimization process from the scratch, the optimal control inputs and states of the system are updated based on the neighboring extremal (NE) method. For this purpose, two approaches are considered: applying NE method on the first solution that leads to a neighboring optimal solution, or assisting this latter by updating the NMPC-related optimization using exact TPBVP equations at some predefined intermediate steps. It is shown through an example that the first method is not accurate enough due to error accumulations. In contrast, the second method preserves the accuracy while reducing the computation time significantly.

Discovering community structure in complex networks is a mature field since a tremendous number of community detection methods have been introduced in the literature. Nevertheless, it is still very challenging for practitioners to determine which method would be suitable to get insights into the structural information of the networks they study. Many recent efforts have been devoted to investigating various quality scores of the community structure, but the problem of distinguishing between different types of communities is still open. In this paper, we propose a comparative, extensive, and empirical study to investigate what types of communities many state-of-the-art and well-known community detection methods are producing. Specifically, we provide comprehensive analyses on computation time, community size distribution, a comparative evaluation of methods according to their optimization schemes as well as a comparison of their partitioning strategy through validation metrics. We process our analyses on a very large corpus of hundreds of networks from five different network categories and propose ways to classify community detection methods, helping a potential user to navigate the complex landscape of community detection.

To achieve fast satellite selection for a multi-Global Navigation Satellite System (GNSS), thereby reducing the burden on a receiver's processing element and the cost of hardware, and improving the utilisation ratio of receiver signal channels, the relationship between the number of satellites and Geometric Dilution Of Precision (GDOP), the number of satellites selected and the computation time is analysed. A fast rotating partition algorithm for satellite selection based on equal distribution of the sky is proposed. The algorithm divides the satellite selection process into two parts: rough selection and detailed selection. Unhealthy satellites, according to a health identifier, and low elevation angle satellites with a large troposphere delay are eliminated during the rough selection process. During the detailed satellite selection process, the satellite sky is divided and rotated to match satellites based on the average angle distance between the satellite and central partition line. Static data from the International GNSS Service (IGS) station and dynamic data collected at China University of Mining and Technology were used to verify the algorithm, and the results demonstrated that an inverse matrix could be avoided to reduce computation complexity. Additionally, the new satellite selection algorithm has the merit that there is little effect on the computation when the selected satellites and number of satellites in the field increased. A single system of the Global Positioning System (GPS) and double system of GPS/Globalnaya Navigazionnaya Sputnikovaya Sistema (GLONASS) both passed the hypothesis test for each epoch. By including BeiDou Navigation Satellite System (BDS) data, data utilisation increased to more than 95% using the rotating partition algorithm. Also, the GDOP and positioning performance of a rotating partition algorithm and an optimal Dilution Of Precision (DOP) algorithm are compared in this paper, and the analysis result shows that both of the algorithms have only a small difference of GDOP and have comparable positioning performance.