Although sampling-based planning algorithms have been extensively used to approximately solve motion planning problems with differential constraints, gaps usually appear in their solution paths due to various factors. These gaps often cause that higher precision solutions come at the expense of dramatically increased computation time. In this paper, we propose a gap reduc- tion technique that substantially improves the performance of sampling-based algorithms by perturbing the trajectory and eliminating large gaps in solution path candidates. By exploiting group symmetries of the system, gap reduction is greatly accelerated by avoiding unnecessary costly numerical integrations and naturally maintaining local constraints on the system. The improvement is demonstrated for unidirectional, bi-directional, and PRM-based sampling-based algorithms in solving problems with a nonholonomic system and a system with dynamics. [PDF]