Quaternion Dynamics Research Articles

Quaternion-based non-singular nonlinear impact angle guidance for three-dimensional engagements

This paper proposes a three-dimensional impact angle-constrained nonlinear guidance strategy against different types of targets. To avoid the occurrence of singularities, in Euler angle-based approaches, the development of this guidance strategy is solely based on quaternion dynamics. An explicit relationship between the quaternions representing impact orientation and line-of-sight orientation at the time of collision is derived. Based on the obtained relation, a guidance law is derived using a sliding mode control technique to track the desired line-of-sight orientation for target interception. The performance of the proposed guidance strategy is evaluated for constant-speed and different time-varying-speed interceptor models accounting for the effect of varying aerodynamic parameters on interceptor speed. The numerical simulations performed show satisfactory results for all engagements, validating the efficacy and robustness of the proposed guidance law.

Cobra aerobatic flight for quadrotors using θ-D nonlinear control

The control design for the Cobra maneuver of quadrotors is presented based on quaternion dynamics. Cobra maneuver commands the quadrotor in ± π / 2 (rad) pitch angle such that the conventional hovering assumption for quadrotor flight is not applicable. The finite-time θ-D method is proposed to control the Cobra maneuver. Based on this technique, the intractable Hamilton–Jacobi–Bellman equation can be approximated by a differential Riccati equation and a series of state-dependent Lyapunov equations, which can be solved recursively and lead to the closed-form state feedback control law . This control law can achieve semi-global asymptomatic stability of the closed-loop system. Numerical simulations verify that the proposed technique is compared favourably with the finite-time state-dependent Riccati equation (SDRE) technique in terms of computation efficiency and control effort, and also shows faster recovery of the quadrotor after the Cobra maneuver compared with conventional dynamics without the derivative of the rotation matrix.

Comments on “quaternion models of a rigid body rotation motion and their application for spacecraft attitude control”

The commented paper discusses the attitude dynamics representation using quaternions. The main idea of the paper is to eliminate angular velocity from the dynamics equations. The obtained equations are interesting. However, all applications presented in the discussed paper have plain errors in the equations derivations. This affects the main results and conclusions of the paper, which require significant corrections. • Errors in commented paper are outlined. • Quaternion dynamics equations application for the attitude control is discussed. • Control singularity is shown.

Adaptive Sliding Mode Control of a Perturbed Satellite in a Formation Antenna Array

Control of a perturbed satellite formation antenna array, with dual quaternion dynamics, is considered. The perturbations and their derivatives are assumed bounded with unknown bounds. The objective of this article is to employ continuous control algorithms which feature adaptive gain nonoverestimation. In this work, three novel dual quaternion-based adaptive continuous sliding mode control algorithms to control a perturbed satellite antenna array are assessed. These methods of adaptive sliding mode control are caspable of handling coupled perturbed systems, such as dual quaternion models. The efficacy of the proposed algorithms is verified via simulations and comparison to the available traditional PD Plus controller and classical sliding mode control methods.

Commutativity of quaternion‐matrix–valued functions and quaternion matrix dynamic equations on time scales

AbstractIn this paper, we obtain some basic results of quaternion algorithms and quaternion calculus on time scales. Based on this, a Liouville formula and some related properties are derived for quaternion dynamic equations on time scales through conjugate transposed matrix algorithms. Moreover, we introduce the quaternion matrix exponential function by homogeneous quaternion matrix dynamic equations. Also a corresponding existence and uniqueness theorem is proved. In addition, the commutativity of quaternion‐matrix–valued functions is investigated and some sufficient and necessary conditions of commutativity and noncommutativity are established on time scales. Also the fundamental solution matrices of some basic quaternion matrix dynamic equations are obtained. Examples are provided to illustrate the results, which are completely new on hybrid domains particularly when the time scales are the quantum case and the discrete case ; , both of which are significant for the study of quaternion q‐dynamic equations and quaternion difference dynamic equations. Finally, we present several applications including multidimensional rotations and transformations of the submarine, the gyroscope, and the planet whose dynamical behaviors are depicted by quaternion dynamics on time scales and the corresponding iteration numerical solution for homogeneous quaternion dynamic equations are provided on various time scales.

Dual quaternion based dynamics modeling for electromagnetic collocated satellites of diffraction imaging on geostationary orbit

This paper proposes an innovated approach to model an electromagnetic force which is an ideal control force without plume and light pollution for optical imaging system. The main scope is modeling the 3-axis coupled electromagnetic force using dual quaternion for its advantages in describing spiral motion with translation and rotation motion simultaneously. The derivation procedure relies on a far field model of the electromagnetic force, and also, the force is re-formulized. Final equations show the dual quaternion dynamics can express the relative motion caused by the electromagnetic force acting on the position and attitude integrally. A new mission concept of diffraction imaging system in GEO is discussed as application scenario of the new dynamics model. Two cases with different mass were simulated to verify the new model, and the results display the electromagnetic force is significant for a small mass spacecraft in GEO.

Robust rotational-velocity-Verlet integration methods

Two rotational integration algorithms for rigid-body dynamics are proposed in velocity-Verlet formulation. The first method uses quaternion dynamics and was derived from the original rotational leap-frog method by Svanberg [Mol. Phys. 92, 1085 (1997)]; it produces time consistent positions and momenta. The second method is also formulated in terms of quaternions but it is not quaternion specific and can be easily adapted for any other orientational representation. Both the methods are tested extensively and compared to existing rotational integrators. The proposed integrators demonstrated performance at least at the level of previously reported rotational algorithms. The choice of simulation parameters is also discussed.

Global Magnetometer-Based Spacecraft Attitude and Rate Estimation

A globally self-initializing attitude determination filter has been developed to estimate three-axis attitude and attitude rate from magnetometer data only. This filter can serve as the main attitude determination system for a low-weight, low-budget, low-accuracy mission or as a backup system for other missions. The filter uses a new attitude representation that is based on the minimum quaternion to align the body-axis and reference magnetic field vectors. It uses implicit quaternion dynamics and implicit Euler dynamics to propagate its attitude and attitude-rate estimates. It is based on a generalization of the iterated extended Kalman-filter concept. Its global initialization strategy employs a number of filters that cover a two-dimensional box in which the principal initial condition uncertainties lie. Hypothesis testing is used to eliminate all but one of the filters after the best one has reached steady state. Flight data from two missions have been used to test the new system. Convergence without the need for initial attitude or rate knowledge has been demonstrated in all cases, and per-axis 3-σ accuracies on the order 5 deg in attitude and 0.03 deg/s in rate have been achieved.

Classical molecular simulations of complex, industrially-important systems on the intel paragon

Advances in parallel supercomputing now make possible molecular-based engineering and science calculations that will soon revolutionize many technologies, such as those involving polymers and those involving aqueous electrolytes. We have developed a suite of message-passing codes for classical molecular simulation of these industrially-important systems on the Intel Paragon and summarize some of the results. Nonequilibrium, multiple time step molecular dynamics lets us investigate the rheology of molecular fluids. Chain molecule Monte Carlo simulations in the Gibbs ensemble permit calculation of phase equilibrium of long-chain molecular systems. Complementary equilibrium molecular dynamics yields fundamental insight into the technologically-important problem of liquid-liquid phase separation in polymer blends. Parallel codes for quaternion dynamics using techniques for handling long-range Coulombic forces allow study of ion pairing in supercritical aqueous electrolyte solutions.

On the numerical integration of motion for rigid polyatomics: The modified quaternion approach

A revised version of the quaternion approach for numerical integration of the equations of motion for rigid polyatomic molecules is proposed. The modified approach is based on a formulation of the quaternion dynamics with constraints. This allows one to resolve the rigidity problem rigorously using constraint forces. It is shown that the procedure for preservation of molecular rigidity can be realized particularly simply within the Verlet algorithm in velocity form. We demonstrate that the method presented leads to an improved numerical stability with respect to the usual quaternion rescaling scheme and it is roughly as good as the cumbersome atomic-constraint technique. © 1998 American Institute of Physics.

The One-Equator Property

We show the existence of points in the Mandelbrot cardioid that have the one-equator property, a property useful for the study of quaternionic dynamics. The question whether the Julia set is homeomorphic to a codimension-one sphere becomes a good deal more subtle in quaternionic dynamics.

Superweak CP nonconservation arising from an underlying quarternionic quantum dynamics.

We show that the hypothesis of an underlying quaternionic dynamics for particle physics naturally implies an effective complex $\mathrm{CP}$-nonconserving interaction of superweak form, but arising in second order in the quaternionic perturbation. The energy scale for the appearance of strong quaternionic effects is estimated to be 20 TeV.