  We present Ruckig, an algorithm for Online Trajectory Generation (OTG)
respecting third-order constraints and complete kinematic target states. Given
any initial state of a system with multiple Degrees of Freedom (DoFs), Ruckig
calculates a time-optimal trajectory to an arbitrary target state defined by
its position, velocity, and acceleration limited by velocity, acceleration, and
jerk constraints. The proposed algorithm and implementation allows three
contributions: (1) To the best of our knowledge, we derive the first
time-optimal OTG algorithm for arbitrary, multi-dimensional target states, in
particular including non-zero target acceleration. (2) This is the first
open-source prototype of time-optimal OTG with limited jerk and complete time
synchronization for multiple DoFs. (3) Ruckig allows for directional velocity
and acceleration limits, enabling robots to better use their dynamical
resources. We evaluate the robustness and real-time capability of the proposed
algorithm on a test suite with over 1,000,000,000 random trajectories as well
as in real-world applications.
