Mathematical review of the attitude control mechanism for a spacecraft

The issue of inertial pointing for a spacecraft with magnetic actuators is considered and a practical global response to this problem is obtained by static attitude and speed feedback methods. A local solution dependent on dynamic attitude feedback is additionally introduced. The simulation results show the practical applicability of the proposed approach. The issue of attitude regulation of rigid spacecraft, i.e., spacecraft demonstrated by the Euler's conditions and by an appropriate parameterization of the attitude, has been broadly concentrated as of late. As a matter of first importance, it is beyond the realm of imagination by methods for magnetic actuators to give three autonomous control torques at each time instant. Moreover, the conduct of these actuators is characteristically time-varying, as the control instrument relies on the varieties of the Earth magnetic field along the spacecraft orbit. In any case, demeanor adjustment is conceivable in light of the fact that on normal the framework has solid controllability properties for a wide range of orbit inclinations. A lot of work has been devoted as of late to the issues of examination and structure of attractive control laws in the straight case, i.e., nominal operation of a satellite near its equilibrium attitude. Specifically, ostensible and vigorous solidness and execution have been contemplated, utilizing either devices from occasional control hypothesis misusing the (quasi) intermittent conduct of the framework close to an equilibrium or other techniques aiming at developing suitable time-varying controllers.

Multi-dimensional connectivity: a conceptual and mathematical review

The estimation of functional connectivity between regions of the brain, for example based on statistical dependencies between the time series of activity in each region, has become increasingly important in neuroimaging. Typically, multiple time series (e.g. from each voxel in fMRI data) are first reduced to a single time series that summarises the activity in a region of interest, e.g. by averaging across voxels or by taking the first principal component; an approach we call one-dimensional connectivity. However, this summary approach ignores potential multi-dimensional connectivity between two regions, and a number of recent methods have been proposed to capture such complex dependencies. Here we review the most common multi-dimensional connectivity methods, from an intuitive perspective, from a formal (mathematical) point of view, and through a number of simulated and real (fMRI and MEG) data examples that illustrate the strengths and weaknesses of each method. The paper is accompanied with both functions and scripts, which implement each method and reproduce all the examples.

Multi-dimensional connectivity: a conceptual and mathematical review

The estimation of functional connectivity between regions of the brain, for example based on statistical dependencies between the time series of activity in each region, has become increasingly important in neuroimaging. Typically, multiple time series (e.g. from each voxel in fMRI data) are first reduced to a single time series that summarises the activity in a region of interest, e.g. by averaging across voxels or by taking the first principal component; an approach we call one-dimensional connectivity. However, this summary approach ignores potential multi-dimensional connectivity between two regions, and a number of recent methods have been proposed to capture such complex dependencies. Here we review the most common multi-dimensional connectivity methods, from an intuitive perspective, from a formal (mathematical) point of view, and through a number of simulated and real (fMRI and MEG) data examples that illustrate the strengths and weaknesses of each method. The paper is accompanied with both functions and scripts, which implement each method and reproduce all the examples.