A tractor speeds across stubble, carving a circle. At the same instant, on a monitor, its exact shadow repeats the maneuver with a tiny difference. Engineers in Munich built that shadow, a virtual twin of the actual machine, to teach the tractor to work without a human.

The shadow is no mere dummy. It’s a living model assembled in the numerical simulation environment Mevea. The body, wheels, axles — all broken into separate pieces. Their masses came after Digital Image Processing of a 3D model from the game Farming Simulator 25. The total tractor mass is about 4 tons, like two passenger cars. The parts were linked with joints, and for each one, differential equations of motion were written — tying forces and acceleration just as Newton taught, and rotation according to Euler's laws. This whole agile Dynamical Systems obeyed commands: the real tractor sent them via wires, and the computer signal processing relayed them to the shadow. The result was a single organism — a physical machine and its digital shadow, living in the same rhythm.

Tests showed: in turns, the shadow barely lags — the error never exceeds 10%. Even the nonlinear behavior of the tires and random ground irregularities (nonlinear dynamics) didn’t spoil the accuracy. However, acceleration fell short: the real tractor smoothly varies its gear ratio (CVT), while the shadow used a simplified gearbox. Stochastic modeling of uncertainties in mass and inertia parameters added scatter. Now the shadow itself traces circles on virtual stubble, honing control theory. With each maneuver, it brings closer the day when smart tractors roll onto fields without drivers.