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We study a class of entangling gates for trapped atomic ions and demonstrate the use of numeric optimization techniques to create a wide range of fast, error-robust gate constructions. Our approach introduces a framework for numeric optimization using individually addressed, amplitude and phase modulated controls targeting maximally and partially entangling operations on ion pairs, complete multi-ion registers, multi-ion subsets of large registers, and parallel operations within a single register. Our calculations and simulations demonstrate that the inclusion of modulation of the difference phase for the bichromatic drive used in the Mølmer-Sørensen gate permits approximately time-optimal control across a range of gate configurations, and when suitably combined with analytic constraints can also provide robustness against key experimental sources of error. We further demonstrate the impact of experimental constraints such as bounds on coupling rates or modulation band-limits on achievable performance. Using a custom optimization engine based on TensorFlow we also demonstrate time-to-solution for optimizations on ion registers up to 20 ions of order tens of minutes using a local-instance laptop, allowing computational access to system-scales relevant to near-term trapped-ion devices.