Design of Optimized Time-Shifted Sine Motion Profiles for High-Speed, Low-Vibration Motion

High-speed precision positioning systems require motion profiles that achieve rapid transfer while suppressing motion-induced vibration. Conventional time-optimal trajectories often minimize travel time at the expense of residual vibration, which prolongs settling and degrades positioning accuracy. This paper proposes a systematic framework for designing optimized time-shifted sine motion profiles that explicitly incorporate vibration suppression in the frequency domain. By integrating time-domain profile construction with Laplace-domain analysis, motion profiles are derived in a unified manner from 1st-order to generalized nth-order forms. A key theoretical result shows that the residual vibration amplitude after motion completion is proportional to the magnitude of |sX(s)| evaluated at the system poles, providing a clear analytical basis for a closed-form zero placement strategy. Explicit algebraic design conditions are obtained without iterative numerical optimization. Simulation-based case studies demonstrate that the proposed approach drastically reduces transient and residual vibrations while maintaining competitive motion completion times compared with time-optimal designs. Robustness is quantitatively evaluated using insensitivity and high-frequency roll-off metrics, revealing that increasing the profile order improves uncertainty tolerance by approximately −20 dB/decade per order. Furthermore, a short-stroke scenario shows that lower-order sine profiles can be advantageous under moderate uncertainty. The proposed framework provides a practical guideline for vibration-aware high-speed motion control.