###### Abstract
For a 200 mm comprehensive tool mounted to a rigid tine, working at depths of 35, 100, and 150 mm in light sand with a soil density of 2510 kg\(\cdot\)m\({}^{-3}\), the relative error of the prediction of draught force based on [PERSON] model was 24.6%, 27.0%, and 32.0%, respectively (the model was overestimated) [24]. Despite much research on draught forces, less focus has been placed on vertical forces, which play a crucial role in understanding the tool's ability to penetrate and lift soil [25]. Vertical forces are fundamental in tools such as duckfoot, where the wings are designed to lift and crack the soil, and failure to take these forces into account can result in an inaccurate prediction of the performance of the loosening tool [26]. While previous studies have provided valuable information on the effects of individual factors--such as soil moisture, depth, width, and speed--on the forces acting on tillage tools [27], there remains a significant gap in understanding how these factors interact consistently with complex tools, such as duckfoot attached to tines with varying stiffness. Most models in the literature focus on more straightforward tools, such as narrow chisels with simple geometry [28] and wide duckfoots that cut, lift, and crack soil. The lack of empirical data on vertical forces limits the accuracy of existing models, especially in the context of tools that work at shallow depths [29]. The research problem is the identified gap in the current state of knowledge about the combined effects of soil moisture, depth, tool width, working speed, and, especially, tine stiffness on both draught and vertical forces acting on comprehensive duckfoot tools. From the current knowledge, the ratio of depth to tool width is more important than their absolute values [30], and will be used in the developed mathematical models. There is, therefore, a need to create comprehensive empirical models based on which it will be possible to evaluate these forces under different field conditions and use these models to optimise the performance of tools operating in various field environments. Considering variable factors, interactions between the tillage tool and the soil are essential for minimising agronomic and environmental problems ([PERSON] and [PERSON], 2022) [31]. Based on the research problem, the following scientific hypotheses were formulated: (1) Soil moisture, the ratio of depth to width of the duckfoot, the movement speed, and the stiffness of the tine flexibility significantly impact both the draught and vertical forces acting on duckfoots. An increased tool depth-to-width ratio and movement speed will result in greater straight forces, and a lower tine stiffness will have a measurable effect on reducing horizontal and vertical forces. It will be incorporated into the developed models, improving accuracy compared to existing models. (2) Mathematical empirical models developed in this study will allow for predicting forces in a two-dimensional (2D) system with different soil conditions and tool configurations. Empirical models focus mainly on linear relationships and interactions and include nonlinear variables. Models will be validated for optimal conditions based on relative errors. The scientific novelty of this study lies in developing comprehensive mathematical, empirical models that will allow for the simultaneous evaluation of both draught and vertical forces for duckfoot under different operating conditions. While previous research has primarily focused on draught forces and simpler tool geometries, this study will fill the gap by addressing vertical forces, which are critical to understanding the behaviour of tools for penetration, lifting, crushing, cracking, and soil spreading. The models presented in the manuscript consider the existing essential soil and implement parameters, such as soil moisture and movement speed, as well as the duckfoot depth-to-width ratio and, in particular, tine stiffness.