Ensuring safety in advanced robot control methods such as Model Predictive Control (MPC) and Reinforcement Learning remains an open challenge. A common approach relies on control-invariant sets, yet their exact computation for nonlinear systems is generally intractable, and numerical approximations compromise theoretical guarantees. In this work, we show that safety does not need to rely exclusively on control invariance. We introduce novel classes of safe sets, each broader than classical control-invariant sets, and analyze corresponding MPC schemes that guarantee recursive feasibility by constraining different nodes of the prediction horizon rather than only the terminal state. Furthermore, when the safe set is an inner approximation of a suitable control-invariant set, our methods can ensure safety through a safe-abort strategy, even in the absence of recursive feasibility. We validate our approaches in extensive simulations on a 6-joint robot manipulator, showing fewer constraint violations than state-of-the-art methods while maintaining comparable tracking performance and computational efficiency. Hardware experiments further demonstrate the practicality of our schemes for real-world robotic applications.

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