scholarly journals Dot Product Equality Constrained Attitude Determination from Two Vector Observations: Theory and Astronautical Applications

Aerospace ◽  
2019 ◽  
Vol 6 (9) ◽  
pp. 102 ◽  
Author(s):  
Jin Wu ◽  
Shangqiu Shan
Keyword(s):  
Analytical Solutions ◽  
Perturbation Analysis ◽  
Attitude Determination ◽  
Equality Constraint ◽  
Q Method ◽  
Maximum Eigenvalue ◽  
Engineering Problems ◽  
Dot Product

In this paper, the attitude determination problem from two vector observations is revisited, incorporating the redundant equality constraint obtained by the dot product of vector observations. Analytical solutions to this constrained attitude determination problem are derived. It is found out that the studied two-vector attitude determination problem by Davenport q-method under the dot product constraint has deterministic maximum eigenvalue, which leads to its advantage in error/perturbation analysis and covariance determination. The proposed dot product constrained two-vector attitude solution is applied then to solve several engineering problems. Detailed simulations on spacecrafts attitude determination indicate the efficiency of the proposed theory.

scholarly journals Eigenvalues and Eigenvectors for 3×3 Symmetric Matrices: An Analytical Approach

2020 ◽  
pp. 106-118
Author(s):  
Abu Bakar Siddique ◽  
Tariq A. Khraishi
Keyword(s):  
Analytical Solutions ◽  
Analytical Approach ◽  
Numerical Solutions ◽  
Linear Equations ◽  
Symmetric Matrices ◽  
Matrix Equations ◽  
Eigenvalues And Eigenvectors ◽  
Engineering Problems ◽  
The Matrix ◽  
Research Problems

Research problems are often modeled using sets of linear equations and presented as matrix equations. Eigenvalues and eigenvectors of those coupling matrices provide vital information about the dynamics/flow of the problems and so needs to be calculated accurately. Analytical solutions are advantageous over numerical solutions because numerical solutions are approximate in nature, whereas analytical solutions are exact. In many engineering problems, the dimension of the problem matrix is 3 and the matrix is symmetric. In this paper, the theory behind finding eigenvalues and eigenvectors for order 3×3 symmetric matrices is presented. This is followed by the development of analytical solutions for the eigenvalues and eigenvectors, depending on patterns of the sparsity of the matrix. The developed solutions are tested against some examples with numerical solutions.

Singular Perturbation Analysis of Speed Controlled Reciprocating Compressors

1989 ◽  
Vol 111 (2) ◽  
pp. 313-321 ◽  
Author(s):  
A. Liakopoulos ◽  
W. H. Boykin
Keyword(s):  
Steady State ◽  
Singular Perturbation ◽  
Transient Response ◽  
Analytical Solutions ◽  
Perturbation Analysis ◽  
System Parameters ◽  
Singular Perturbation Analysis ◽  
Frequency Domains ◽  
Perturbed Systems ◽  
Reaction Forces

A singular perturbation analysis of speed controlled reciprocating compressors is presented. For weakly perturbed systems, analytical solutions for the steady-state and transient response are given. For strongly perturbed systems numerical results in both time and frequency domains are presented. The analytical solutions are useful in calculating reaction forces and torques applied on inertially stabilized platforms, in designing feed forward compensators and in simplifying parameter identification procedures. Furthermore, they clearly exhibit the dependence of response to system parameters.

The Method of Rotated Solutions: A Highly Efficient Procedure for Code Verification

2020 ◽  
Author(s):  
Marc Horner
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Source Code ◽  
Numerical Algorithms ◽  
Coordinate Transformations ◽  
Code Verification ◽  
Efficient Procedure ◽  
One Dimensional ◽  
Engineering Problems

Abstract Code verification provides mathematical evidence that the source code of a scientific computing software platform is free of bugs and that the numerical algorithms are consistent. The most stringent form of code verification requires the user to demonstrate agreement between the formal and observed orders of accuracy. The observed order is based on a determination of the discretization error, and therefore requires the existence of an analytical solution. One drawback of analytical solutions based on traditional engineering problems is that most derivatives are identically zero, which limits their scope during code verification. The Method of Rotated Solutions is introduced herein as a methodology that utilizes coordinate transformations to generate additional non-zero derivatives in the numerical and analytical solutions. These transformations extend the utility of even the simplest one-dimensional solutions to be able to perform more thorough evaluations. This paper outlines the rotated solutions methodology and provides an example that demonstrates and confirms the utility of this new technique.

Period-1 to Period-2 Motions in a Periodically Forced Nonlinear Spring Pendulum

2019 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yaoguang Yuan
Keyword(s):  
Analytical Solutions ◽  
Perturbation Analysis ◽  
Mapping Method ◽  
Analytical Prediction ◽  
Pendulum System ◽  
Nonlinear Spring ◽  
Period 2 ◽  
Stability And Bifurcations ◽  
The Stability

Abstract In this paper, bifurcation trees of period-1 to period-2 motions in a periodically forced, nonlinear spring pendulum system are predicted analytically through the discrete mapping method. The stability and bifurcations of period-1 to period-2 motions on the bifurcation trees are presented as well. From the analytical prediction, numerical illustrations of period-1 and period-2 motions are completed for comparison of numerical and analytical solutions. The results presented in this paper is totally different from the traditional perturbation analysis.

Computational Mechanics

1985 ◽  
Vol 38 (10) ◽  
pp. 1282-1283 ◽  
Author(s):  
A. Needleman
Keyword(s):  
Mechanical Behavior ◽  
Computational Methods ◽  
Analytical Solutions ◽  
Solid Mechanics ◽  
Computational Mechanics ◽  
Mechanical Systems ◽  
Engineering Practice ◽  
Computational Capability ◽  
Engineering Problems ◽  
Complex Engineering

Computational methods play a key role in solid mechanics, as a way of modelling fundamental aspects of mechanical behavior, as a vehicle for transferring this improved modelling capability into new engineering tools, and as a means of utilizing these tools in engineering practice. Modern computational methods enable realistic models of mechanical systems to be formulated without regard as to whether or not analytical solutions are feasible. Increased computational capability is also an incentive for developing more accurate theories, since it becomes possible to use such theories to solve complex engineering problems.

Frustrations and Phase Transitions in Ising Model on 2D Lattices

2010 ◽  
Vol 168-169 ◽  
pp. 435-438 ◽  
Author(s):  
Felix A. Kassan-Ogly ◽  
B.N. Filippov ◽  
V.V. Men’shenin ◽  
Akai K. Murtazaev ◽  
M.K. Ramazanov ◽  
...  
Keyword(s):  
Phase Transition ◽  
Monte Carlo ◽  
Phase Transitions ◽  
Monte Carlo Method ◽  
Ising Model ◽  
Analytical Solutions ◽  
Nearest Neighbors ◽  
Square Lattice ◽  
Maximum Eigenvalue ◽  
Exact Analytical Solutions

The problem of frustrations and phase transition appearance or suppression is studied on the base of exact analytical solutions for maximum eigenvalue of Kramers-Wannier transfer matrix in the Ising model on a square lattice [1], a triangular and honeycomb lattices [2], and kagome lattice [3] with allowance for the interactions between nearest J is studied. We also studied these phenomena by the numerical “replica Monte Carlo method”, taking also into account the interactions between next-nearest neighbors J'.

The long jump record revisited

1986 ◽  
Vol 28 (2) ◽  
pp. 246-259 ◽  
Author(s):  
Neville de mestre
Keyword(s):  
Mathematical Model ◽  
Analytical Solutions ◽  
Perturbation Analysis ◽  
Computer Time ◽  
Centre Of Mass ◽  
Numerical Schemes ◽  
Modified Model ◽  
Athletic Coaches ◽  
Approximate Analytical Solutions ◽  
Long Jump

AbstractA mathematical model is presented in which the long Jump is treated as the motion of a projectile under gravity with slight drag. The first two terms of a perturbation solution are obtained and are shown to be more accurate than earlier approximate analytical solutions. Results from the perturbation analysis are just as accurate as results from various numerical schemes, and require far less computer time.The model is modified to include the observation that a long-jumper's centre of mass is forward of his feet at take-off and behind his feet on landing.The modified model is used to determine the take-off angle for the current world long jump record, resulting in several interesting observations for athletic coaches.

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