  Non-linear Trajectory Optimisation (TO) methods require good initial guesses
to converge to a locally optimal solution. A feasible guess can often be
obtained by allocating a large amount of time for the trajectory to complete.
However for unstable dynamical systems such as humanoid robots, this
quasi-static assumption does not always hold.
  We propose a conservative formulation of the TO problem that simultaneously
computes a feasible path and its time allocation. The problem is solved as an
efficient convex optimisation problem guaranteed to converge to a locally
optimal solution.
  The interest of the approach is illustrated with the computation of feasible
trajectories that traverse sequentially a sequence of polytopes. We demonstrate
that on instances of the problem where the quasi static solutions are not
admissible, our approach is able to find a feasible solution with a success
rate above $80 \%$ in all the scenarios considered, in less than 10ms for
problems involving traversing less than 5 polytopes and less than 1s for
problems involving 20 polytopes, thus demonstrating its ability to reliably
provide initial guesses to advanced non linear solvers.
