A closed-form method of calculating Input Shaping sequences for two modes of vibration is presented. The new method eliminates the optimization routines previously required to find the same solutions. Input Shaping is a feed forward method of reducing residual vibrations in flexible structures by convolving an Input Shaping sequence with a command profile. The two-mode sequences are installed on a four-axis robot used in the manufacture of silicon wafers — the Cassette Management System. The new sequences are found to significantly improve the performance of the system. In standard throughput tests, speed increases of 15%–25% were obtained on each axis while vibrations were simultaneously reduced by 20%–90%.

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