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We introduce a new methodology and experimental implementations for real-time wave-based robot navigation in a complex, dynamically changing environment. The main idea behind the approach is to consider the robot arena as an excitable medium, in which moving objects-obstacles and the target-are represented by sites of autowave generation: the target generates attractive waves, while the obstacles repulsive ones. The moving robot detects traveling and colliding wave fronts and uses the information about dynamics of the autowaves to adapt its direction of collision-free motion toward the target. This approach allows us to achieve a highly adaptive robot behavior and thus an optimal path along which the robot reaches the target while avoiding obstacles. At the computational and experimental levels, we adopt principles of computation in reaction-diffusion (RD) nonlinear active media. Nonlinear media where autowaves are used for information processing purposes can therefore be considered as RD computing devices. In this paper, we design and experiment with three types of RD processors: experimental and computational Belousov-Zhabotinsky chemical processor, computational CNN processor, and experimental RD-CNN very large-scale integration chip-the complex analog and logic computing engine (CACE1k). We demonstrate how to experimentally implement robot navigation using space-time snapshots of active chemical medium and how to overcome low-speed limitation of this "wetware" implementation in CNN-based silicon processors.