Construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body |
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Authors: | O V Zelepukina Yu N Chelnokov |
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Institution: | 1.Institute for Precision Mechanics and Control Problems,Saratov,Russia |
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Abstract: | We consider the problem of construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric
rigid body so as to ensure the transition of the rigid body from an arbitrary initial angular position to the required final
angular position. For the functionals to be minimized, we use combined performance functionals, one of which characterizes
the expenditure of time and of the squared modulus of the angular momentum vector in a given proportion, while the other characterizes
the expenditure of time and momentum of the modulus of the angular momentum vector necessary to change the rigid body orientation.
The control (the vector of the rigid body angular momentum) is assumed to be bounded in the modulus. The problem is solved
by using Pontryagin’s maximum principle and the quaternion differential equation 1, 2] relating the vector of the dynamically
symmetric rigid body angular momentum to the quaternion of orientation of the coordinate system rotating with respect to the
rigid body about its dynamical symmetry axis at an angular velocity proportional to the angular momentum vector projection
on the axis. The use of such a model of rotational motion leads to the problem of optimal control with the moving right end
of the trajectory and significantly simplifies the analytic study of the problem of construction of optimal laws of variation
in the angular momentum vector, because this model explicitly exploits the body angular momentum quaternion (control) instead
of the rigid body absolute angular velocity quaternion. We construct general analytic solutions of the differential equations
for the boundary-value problems which form systems of nine nonlinear differential equations. It is shown that the process
of solving the differential boundary-value problems is reduced to solving two scalar algebraic transcendental equations. |
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