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A tractor-trailer combination or a tractor with a towed implement is a common form for agricultural machines, such as seed drillers, harrows, fertilizers or sprayers. When optimizing field traffic, it is most significant to minimize the time spent in the headlands. Therefore the behavior of tractor-trailer combination in a headland is studied in this paper.

The tractor-trailer combination is modelled as a differential equation system or dynamic model with six states and two inputs. The movement is restricted into 2D-plane. The supplementary requirements for the model are the ability to reverse driving, allowing the switch-back turning and the standard formulation of optimal control problem. The dynamic model is parametrized in terms of mechanical parameters of the vehicle.

Optimal control theory and tools are reviewed. The minimum-time optimal control problem of headland turning with tractor-trailer vehicle is presented. Vehicle's mechanical and field's geometrical constraints are considered. The optimal control problem is highly nonlinear so it cannot be solved analytically.

Numerical algorithms for solving this kind of optimal control problems demand high computational cost and cannot be solved in real-time even with modern desktop computers. The optimal control problem is solved with fixed vehicle parameters and varying headland parameters, headland angle and width. In order to use optimal control problem solutions in real-time applications, the solutions in variable headland cases are approximated with simple trajectories. The rules for selecting a turning method with fixed vehicle parameters and certain headland parameters are presented.