This paper introduces a new method for motion-trajectory generation of mechanical systems with multiple degrees of freedom (DOFs). The key feature of this new concept is that motion trajectories are generated online, i.e., within every control cycle, typically every millisecond. This enables systems to react instantaneously to unforeseen and unpredictable (sensor) events at any time instant and in any state of motion. As a consequence, (multi)sensor integration in robotics, in particular the development of control systems enabling sensor-guided and sensor-guarded motions, becomes greatly simplified. We introduce a class of online trajectory-generation algorithms and present the mathematical basics of this new approach. The algorithms presented here consist of three steps: calculation of the minimum synchronization time for all DOFs, synchronization of all DOFs, and calculation of output values. The theory is followed by real-world experimental results indicating new possibilities in robot-motion control.