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Safety is a crucial property of every robotic platform: any control policy should always comply with actuator limits and avoid collisions with the environment and humans. In reinforcement learning, safety is even more fundamental for exploring an environment without causing any damage. While there are many proposed solutions to the safe exploration problem, only a few of them can deal with the complexity of the real world. This paper introduces a new formulation of safe exploration for reinforcement learning of various robotic tasks. Our approach applies to a wide class of robotic platforms and enforces safety even under complex collision constraints learned from data by exploring the tangent space of the constraint manifold. Our proposed approach achieves state-of-the-art performance in simulated high-dimensional and dynamic tasks while avoiding collisions with the environment. We show safe real-world deployment of our learned controller on a TIAGo++ robot, achieving remarkable performance in manipulation and human-robot interaction tasks.