Reinforcement learning (RL) enables robots to learn skills from interactions with the real world. In practice, the unstructured step-based exploration used in Deep RL – often very successful in simulation – leads to jerky motion patterns on real robots. Consequences of the resulting shaky behavior are poor exploration, or even damage to the robot. We address these issues by adapting state-dependent exploration (SDE) to current Deep RL algorithms. To enable this adaptation, we propose three extensions to the original SDE, which leads to a new exploration method generalized state-dependent exploration (gSDE). We evaluate gSDE both in simulation, on PyBullet continuous control tasks, and directly on a tendon-driven elastic robot. gSDE yields competitive results in simulation but outperforms the unstructured exploration on the real robot.