Integrating learning-based techniques, especially reinforcement learning, into robotics is promising for solving complex problems in unstructured environments. However, most existing approaches are trained in well-tuned simulators and subsequently deployed on real robots without online fine-tuning. In this setting, extensive engineering is required to mitigate the sim-to-real gap, which can be challenging for complex systems. Instead, learning with real-world interaction data offers a promising alternative: it not only eliminates the need for a fine-tuned simulator but also applies to a broader range of tasks where accurate modeling is unfeasible. One major problem for on-robot reinforcement learning is ensuring safety, as uncontrolled exploration can cause catastrophic damage to the robot or the environment. Indeed, safety specifications, often represented as constraints, can be complex and non-linear, making safety challenging to guarantee in learning systems. In this paper, we show how we can impose complex safety constraints on learning-based robotics systems in a principled manner, both from theoretical and practical points of view. Our approach is based on the concept of the Constraint Manifold, representing the set of safe robot configurations. Exploiting differential geometry techniques, i.e., the tangent space, we can construct a safe action space, allowing learning agents to sample arbitrary actions while ensuring safety. We demonstrate the method's effectiveness in a real-world Robot Air Hockey task, showing that our method can handle high-dimensional tasks with complex constraints. Videos of the real robot experiments are available on the project website (https://puzeliu.github.io/TRO-ATACOM).

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