  This paper proposes a two-stage approach to formulate the time-optimal
point-to-point motion planning problem, involving a first stage with a fixed
time grid and a second stage with a variable time grid. The proposed approach
brings benefits through its straightforward optimal control problem formulation
with a fixed and low number of control steps for manageable computational
complexity and the avoidance of interpolation errors associated with time
scaling, especially when aiming to reach a distant goal. Additionally, an
asynchronous nonlinear model predictive control (NMPC) update scheme is
integrated with this two-stage approach to address delayed and fluctuating
computation times, facilitating online replanning. The effectiveness of the
proposed two-stage approach and NMPC implementation is demonstrated through
numerical examples centered on autonomous navigation with collision avoidance.
