  We are interested in the problem of time-optimal control of omnidirectional
robots with bounded acceleration (TOC-ORBA). While there exist approximate
solutions for such robots, and exact solutions with unbounded acceleration,
exact solvers to the TOC-ORBA problem have remained elusive until now. In this
paper, we present a real-time solver for true time-optimal control of
omnidirectional robots with bounded acceleration. We first derive the general
parameterized form of the solution to the TOC-ORBA problem by application of
Pontryagin's maximum principle. We then frame the boundary value problem of
TOC-ORBA as an optimization problem over the parametrized control space. To
overcome local minima and poor initial guesses to the optimization problem, we
introduce a two-stage optimal control solver (TSOCS): The first stage computes
an upper bound to the total time for the TOC-ORBA problem and holds the time
constant while optimizing the parameters of the trajectory to approach the
boundary value conditions. The second stage uses the parameters found by the
first stage, and relaxes the constraint on the total time to solve for the
parameters of the complete TOC-ORBA problem. We further implement TSOCS as a
closed loop controller to overcome actuation errors on real robots in
real-time. We empirically demonstrate the effectiveness of TSOCS in simulation
and on real robots, showing that 1) it runs in real time, generating solutions
in less than 0.5ms on average; 2) it generates faster trajectories compared to
an approximate solver; and 3) it is able to solve TOC-ORBA problems with
non-zero final velocities that were previously unsolvable in real-time.
