scholarly journals Solving the Coriolis Vibratory Gyroscope Motion Equations by Means of the Angular Rate B-Spline Approximation

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 292
Author(s):  
Mikhail Basarab ◽  
Boris Lunin
Keyword(s):  
Piecewise Linear ◽  
Angular Rate ◽  
Spline Approximation ◽  
Piecewise Linear Approximation ◽  
Finite Difference Schemes ◽  
Parabolic Cylinder ◽  
Sampling Step ◽  
Linear Solution ◽  
Vibratory Gyroscope ◽  
Asymptotic Representations

The exact solution of the movement equation of the Coriolis vibratory gyroscope (CVG) with a linear law of variation of the angular rate of rotation of the base is given. The solution is expressed in terms of the Weber functions (the parabolic cylinder functions) and their asymptotic representations. On the basis of the obtained solution, an analytical solution to the equation of the ring dynamics in the case of piecewise linear approximation of an arbitrary angular velocity profile on a time grid is derived. The piecewise linear solution is compared with the more rough piecewise constant solution and the dependence of the error of such approximations on the sampling step in time is estimated numerically. The results obtained make it possible to significantly reduce the number of operations when it is necessary to study long-range dynamics of oscillations of the system, as well as quantitatively and qualitatively control the convergence of finite-difference schemes for solving the movement equations of the Coriolis vibratory gyroscope.

Analytical Solution of the Dynamics Equations for a Wave Solid-State Gyroscope Using the Angular Rate Linear Approximation

2021 ◽  
pp. 17-32
Author(s):  
M.A. Basarab ◽  
B.S. Lunin ◽  
I.P. Ivanov
Keyword(s):  
Solid State ◽  
Analytical Solution ◽  
Linear Approximation ◽  
Angular Rate ◽  
Piecewise Linear Approximation ◽  
Finite Difference Schemes ◽  
Parabolic Cylinder ◽  
Mathieu Function ◽  
Weber Function ◽  
Angular Rotation

The exact solution is provided of the dynamics equation for an elastic inextensible ring being the basic model of a wave solid-state gyroscope with the linear law of the base angular rotation rate alteration. This solution is presented in terms of the parabolic cylinder functions (Weber function). Asymptotic approximations are used in the device certain operating modes. On the basis of the solution obtained, the analytical solution to the equation of the ring dynamics in case of piecewise linear approximation of the angular rate arbitrary profile on a time grid is derived. This significantly expands the class of angular rate dependences, for which the solution could be written down analytically. Earlier, in addition to the simplest case of constant angular rate, solutions were obtained for angular rate varying according to the square root law with time (Airy function), as well as according to the harmonic law (Mathieu function). Error dependence of such approximation on the discretization step in time is estimated numerically. Results obtained make it possible to reduce the number of operations, when it is necessary to study long-term evolutions of the dynamic system oscillations, as well as to quantitatively and qualitatively control convergence of finite-difference schemes in solving dynamics equations for a wave solid-state gyroscope with the ring resonator

scholarly journals Actor-critic learning-based energy optimization for UAV access and backhaul networks

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yaxiong Yuan ◽  
Lei Lei ◽  
Thang X. Vu ◽  
Symeon Chatzinotas ◽  
Sumei Sun ◽  
...  
Keyword(s):  
Power Allocation ◽  
Heuristic Algorithms ◽  
Piecewise Linear ◽  
Access Network ◽  
Computation Time ◽  
Combinatorial Problem ◽  
Base Station ◽  
Piecewise Linear Approximation ◽  
Group Scheduling ◽  
User Group

AbstractIn unmanned aerial vehicle (UAV)-assisted networks, UAV acts as an aerial base station which acquires the requested data via backhaul link and then serves ground users (GUs) through an access network. In this paper, we investigate an energy minimization problem with a limited power supply for both backhaul and access links. The difficulties for solving such a non-convex and combinatorial problem lie at the high computational complexity/time. In solution development, we consider the approaches from both actor-critic deep reinforcement learning (AC-DRL) and optimization perspectives. First, two offline non-learning algorithms, i.e., an optimal and a heuristic algorithms, based on piecewise linear approximation and relaxation are developed as benchmarks. Second, toward real-time decision-making, we improve the conventional AC-DRL and propose two learning schemes: AC-based user group scheduling and backhaul power allocation (ACGP), and joint AC-based user group scheduling and optimization-based backhaul power allocation (ACGOP). Numerical results show that the computation time of both ACGP and ACGOP is reduced tenfold to hundredfold compared to the offline approaches, and ACGOP is better than ACGP in energy savings. The results also verify the superiority of proposed learning solutions in terms of guaranteeing the feasibility and minimizing the system energy compared to the conventional AC-DRL.

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