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Tanygin S.,Analytical Graphics Inc.
Journal of Guidance, Control, and Dynamics | Year: 2015

Anew algorithm for designing attitude maneuver paths in the presence of multiple three-axis attitude constraints is presented. The algorithm projects the full three-axis attitude representation into a three-dimensional space using a new distortion-minimizing transformation. The projected representation is discretized on a three-dimensional Cartesian grid and the admissible regions emerge in the projected space from the discretized representation via interpolation of the constraint satisfaction across the grid points. The points are enumerated and graph search algorithms are employed to find minimum cost paths using the appropriately selected cost definitions. The paths are then smoothed and visualized alongside with the admissible regions in three dimensions. Ultimately, the paths can serve as targets for attitude tracking control algorithms. © 2015 by Sergei Tanygin. Published by the American Institute of Aeronautics and Astronautics, Inc. Source


Tanygin S.,Analytical Graphics Inc.
Journal of Guidance, Control, and Dynamics | Year: 2013

This paper examines the projective geometry of three-parameter attitude representations that are constructed by projecting a four-parameter unit quaternion representation from its unit hypersphere onto a three-dimensional hyperplane. Using this geometrical perspective, the paper demonstrates how kinematics of relative attitude motion characteristic of attitude tracking problems follow naturally from comparing projections from two different reference directions. The paper also demonstrates that among a continuum of possible projection pole placements there exist optimal distances for which resulting projected parameterizations yield magnitudes that accurately approximate all practical rotation angles. These parameterizations referred to in this paper as proxy-rotation vectors can be custom tuned for any expected range of rotation angles. They and their kinematics are free from trigonometric functions and do not require special handling if the rotation angle approaches zero. It is shown that this computational simplicity of the proxy-rotation vectors can be advantageous for linear feedback attitude controls and for certain classes of time-efficient attitude steering laws, where they can replace the more computationally cumbersome true rotation vector. The paper studies how kinematic singularities affect closed-loop stability and demonstrates that redesigning control laws to ensure closed-loop convergence toward the nearest equilibrium is equivalent to augmenting attitude parameterizations with their shadow counterparts. Copyright © 2012 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Source


Tanygin S.,Analytical Graphics Inc.
Journal of Guidance, Control, and Dynamics | Year: 2012

A class of vectorial attitude parameterizations that are formulated as a product of the unit rotation vector and various functions of the rotation angle is examined. When related to a four-dimensional unit quaternion, these vectorial parameterizations are shown to be analogous to higher-dimensional azimuthal projections from a threedimensional unit hypersphere. Several types of these projections are examined. Singularities are identified and numerical accuracy is evaluated based on the singular value decomposition of the attitude kinematics. It is shown how shadow parameterizations can be constructed in order to alleviate the kinematical singularities. It is also shown that the kinematical passivity and optimality of the Rodrigues and modified Rodrigues parameters are special cases of the more general result that holds for a wider range of vectorial parameterizations. This result is used to formulate and compare passivity-based control laws using various parameterizations. Copyright © 2011 by Sergei Tanygin. Published by the American Institute of Aeronautics and Astronautics, Inc. Source


Coppola V.T.,Analytical Graphics Inc.
Advances in the Astronautical Sciences | Year: 2012

While there has been much research on computing the probability of collision between space objects, there is little work on incorporating velocity uncertainty into the computation. We derive the formula from first principles, including both position and velocity uncertainty. Moreover, trajectories will evolve according to differential equations and not by approximating the relative motion. The end result is a 3-dimensional integral over time on the surface of a sphere. We show that the formula recovers the classic formula in the limit as the velocity uncertainty approaches zero. Finally, the results produced using the new formula will be compared to the results of Monte Carlo simulations. Source


Coppola V.T.,Analytical Graphics Inc.
Advances in the Astronautical Sciences | Year: 2012

The formula for the probability of collision for space objects results from many assumptions concerning the motion of the objects, not least of which is that the encounter duration is short. We develop a formula that characterizes the encounter duration for the conjunction of two space objects and then compute it for every conjunction in an all-on-all assessment of the public catalog. We then introduce the concept of a short-term encounter validity interval that characterizes the total encounter time under which the short-term assumptions are assumed met. This metric provides the means for assessing whether a conjunction satisfies the short encounter assumption so that the standard collision probability metric is valid. Source

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