This paper addresses the problem of motion planning for differential drive micro-mobility platforms. This class of vehicle is designed to perform small-distance transportation of passengers and goods in structured environments. Our approach leverages mixed-integer linear programming (MILP) to compute global optimal collision-free trajectories taking into account the kinematics and dynamics of the vehicle. We propose novel constraints for intersample collision avoidance and demonstrate its effectiveness using pick-up and delivery missions and statistical analysis of Monte Carlo simulations. The results show that the novel formulation provides the best trajectories in terms of time expenditure and control effort when compared to two state-of-the-art approaches.