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This paper presents a robustness analysis and design approach for impedance control of a flexible structure mounted manipulator (FSMM) performing contact tasks. In the analysis, the dynamics of the flexible structure are treated as uncertain. The environment is also treated as uncertain, but with force-displacement characteristics satisfying a sector-bound condition. First, a one dimensional model for an end-effector under impedance control is considered. The effect of base flexibility on stability under contact conditions is then assessed via the Popov criterion. Based on these results, an approach to robust impedance control is proposed. It is shown that by augmenting low-order filters within a standard impedance control law, robust global stability in the presence of base flexibility can be achieved. The controller is shown to perform successfully in experiments on a two-link lab-scale FSMM.