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Many industrial applications of machining require a bounded tracking error during transient and steady processes. Traditional control architectures in machining are unable to explicitly bound tracking errors, and therefore conservative operation is required to ensure satisfactory performance. In this paper, we propose a model-predictive-based approach to guarantee bounded error tracking for the family of systems with available end effector position measurements. The state and input constraints and the bounded error requirements are satisfied by a model predictive controller (MPC) that enforces the system and reference states to remain in a polyhedral robust control invariant (RCI) set. We propose an algorithm for calculating this RCI set in a finite number of computation steps and give the formulation of the MPC. The superiority of the proposed control approach over a conventional tracking controller is demonstrated via simulation of a laboratory machine.