Sequential sampling strategies have been developed for managing complexity when using computationally expensive computer simulations in engineering design. However, much of the literature has focused on objective-oriented sequential sampling methods for deterministic optimization. These methods cannot be directly applied to robust design which must account for uncontrollable variations in certain input variables (i.e., noise variables). Obtaining a robust design that is insensitive to variations in the noise variables is more challenging. Even though methods exist for sequential sampling in design under uncertainty, the majority of the existing literature does not systematically take into account the interpolation uncertainty that results from limitations on the number of simulation runs, the effect of which is inherently more severe than in deterministic design. In this paper, we develop a systematic objective-oriented sequential sampling approach to robust design with consideration of both noise variable uncertainty and interpolation uncertainty. The method uses Gaussian processes to model the costly simulator and quantify the interpolation uncertainty within a robust design objective. We examine several criteria, including our own proposed criteria, for sampling the design and noise variables and provide insight into their performance behaviors. We show that for both of the examples considered in this paper the proposed sequential algorithm is more efficient in finding the robust design solution than a one-shot space filling design.