Gradient flow of magnetic inclination and intensity as a GPS-free guidance mechanism

It is well established that some animals, such as sea turtles, sense the Earth's magnetic field and use it as a primary cue for long-distance navigation across the vast and featureless ocean. This remarkable ability has inspired the development of bio-inspired magnetic navigation (BIMN), which seeks to mimic these animals to develop robust and frugal navigation methods. BIMN exploits the Earth's magnetic field, sometimes described as ''Nature's GPS'', as a natural navigation infrastructure, providing a potential alternative to vulnerable satellite navigation systems. However, unlike geographic coordinates, magnetic coordinates extracted from the Earth's magnetic field do not provide a one-to-one mapping to geographic locations. Different locations can exhibit similar magnetic signatures, creating ambiguity and making BIMN inherently more challenging. Despite this fundamental challenge, we demonstrate that, within predefined regions of the Earth, magnetic inclination and intensity provide sufficient information to navigate to any destination. By formulating BIMN as a gradient flow control and guidance problem, we show that the resulting gradient flow act as current that carries the system and can be controlled to guide it through intermediate waypoints toward the desired destination. The resulting framework provides a GPS-free navigation mechanism with potential applications in both engineering and biology. In engineering, it can be used to establish a geomagnetic navigation infrastructure through which prescribed routes can be defined and followed, while in biology, it can be used to analyse the migration routes of animals known to rely on magnetoreception.