We propose a robust two-dimensional polynomial beamformer design method, formulated as a convex optimization problem, which allows for flexible steering of a previously proposed data-independent robust beamformer in both azimuth and elevation direction. As an exemplary application, the proposed twodimensional polynomial beamformer design is applied to a twelveelement microphone array, integrated into the head of a humanoid robot. To account for the effects of the robot's head on the sound field, measured head-related transfer functions are integrated into the optimization problem as steering vectors. The two-dimensional polynomial beamformer design is evaluated using signal-independent and signal-dependent measures. The results confirm that the proposed polynomial beamformer design approximates the original fixed beamformer design very accurately, which makes it an attractive approach for robust real-time dataindependent beamforming.Index Terms-Robust superdirective beamforming, polynomial beamforming, white noise gain, robot audition