  Providing smooth reference trajectories can effectively increase performance
and accuracy of tracking control applications while overshoot and unwanted
vibrations are reduced. Trajectory planning computations can often be
simplified significantly by transforming the system dynamics into decoupled
integrator chains using methods such as feedback linearization, differential
flatness or the controller canonical form. We present an efficient method to
plan time optimal trajectories for integrator chains subject to derivative
bound constraints. Therefore, an algebraic precomputation algorithm formulates
the necessary conditions for time optimality in form of a set of polynomial
systems, followed by a symbolic polynomial elimination using Gr\"obner bases. A
fast online algorithm then plans the trajectories by calculating the roots of
the decomposed polynomial systems. These roots describe the switching time
instants of the input signal and the full trajectory simply follows by multiple
integration. This method presents a systematic way to compute time optimal
trajectories exactly via algebraic calculations without numerical approximation
iterations. It is applied to various trajectory types with different continuity
order, asymmetric derivative bounds and non-rest initial and final states.
