<p dir="ltr">In mobile manipulator motion planning, typical sampling-based methods have difficulty in (1) solving the planning problem fast; (2) finding a path with high quality; (3) maintaining the non-holonomic constraint along the path. This paper firstly proposes to solve the general non-holonomic planning problem based on sampling with various connection paths between the samples with constraint satisfied, which enables typical planners such as RRT, RRT*, RRT-Connect, PRM, etc., to be able to solve mobile manipulation motion planning with differential drive constraints. Meanwhile, with proper sampling and activating alternative paths added to connect state pairs, it is proved that the rate of valid connectable state samples significantly increases and this also contributes in improving the planner efficiency and success rate. Simulation experiments have been conducted through MoveIt! with differential-drive based UR5e robot in environments with obstacles. Results show that the proposed method has the highest success rates against four existing planners. In general, the planner generates faster solutions with better path quality in all five experiment scenarios, useful in achieving responsive real-time mobile manipulation behavior in daily applications.</p>