Can expansion of space cause the motion of galaxies, does expansion of space do effect motion of our sun around center and earth around Sun? I'm just wondering that expansion is anyway related to in effecting motion of a star under the galaxies ? Planets are currently revolving around sun in helix motion, if galaxies has motion as well reason to expansion of space then it is possible that, that can effect anyway on that planetary motion around stars ?
No. The solar system is a gravitationally bound system so it is unaffected by the large scale expansion of the universe - see this Wikipedia article for further explanation. Similarly, galaxies like the Milky Way, and even groups of galaxies such as the Local Group, which includes the Milky Way and Andromeda, are gravitationally bound systems and are also unaffected.