We present formalism and analysis of three active alignment reconstruction techniques applied to the Advanced Technology Solar Telescope. The three reconstructors generate optical control signals that are a matrix product of a wavefront-sensing signal and the reconstructors themselves. The optical control signals are fed to the six rigid body degrees of freedom of the telescope secondary mirror and the eight bending modes of the primary mirror. The resulting aligned state is a least-squares alignment of the telescope subject to perturbations that result from thermal and gravitational flexures. Two of the reconstructors utilize a single guide object in the telescope field of view and the third reconstructor utilizes three guide objects. One of the single-guide-object reconstructors is developed with an explicit minimum force constraint that minimizes the actuator forces exerted on the telescope primary mirror during active alignment. The force optimized reconstructor also achieves close to the minimum residual wavefront error. Simulation results, optical control analysis, and a discussion of the reconstructor methods and properties are presented.
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