This paper presents a method of polynomial transition curve application for making agricultural aggregate movement paths during headland turn drives as well as within the field. Four types of agricultural aggregate paths in five different variant designs are discussed. Each path is composed of only two curves, making the so-called transition bi-curve. The curvature described by the linear function as well as the third, fifth, seventh, and ninth degree polynomials was designated. Moreover, a trajectory planning algorithm in which the movement proceeds along two transition curves composing the so-called bi-curve was proposed. The simulation was carried out applying the MATLAB program in which the 4th order Runge–Kutta method was used. The results were presented by means of figures showing the proposed paths and kinematic quantity courses in the displacement function. The obtained trajectories were compared regarding the size and kinematic quantities. The trajectories, whose curvature is described by the 3° polynomial, were found to possess the smallest absolute values of maximal acceleration and jerk and to lack jerk discontinuity. The proposed solutions can be applied for planning trajectory of not only agriculture machines and aggregates but also autonomous vehicles or mobile robots.

The paper presents the PR-APT method (Planning Rectilinear-Arc Polynomial Trajectory) for planning a trajectory of the manipulator end-effector. In the described method, a path consists of two rectilinear segments of intersecting directions combined with the blending arc of a set radius. The arc can make a fragment of an avoidable obstacle. he algorithm and the implemented method on the example of tomato harvest in the greenhouse was presented. The developed algorithm facilitates the trajectory planning in only few steps and therefore, it is supremely effective. Notably, the proposed algorithm ensures the continuity of displacement, velocity and tangential acceleration for the planned motion of the end-effector. The numerical example includes the computational results and the courses of displacement, speed and acceleration for the planned trajectory.

This paper presents planning of smooth trajectories using the asymmetric profiles of displacement, velocity, acceleration and jerk of described higher degree polynomials. The presented way of determination of polynomial and asymmetry coefficients as well as motion time includes the procedure at both constraints of single kinematic quantities, i.e., velocity, acceleration, jerk and introducing a few constraints simultaneously. Moreover, the paper presents a method of planning trajectories with intermediate points consisting in concatenation of the analyzed polynomials. The proposed method of trajectory planning was compared to that of using the S-curve. The results in the form of kinematic quantity courses are presented graphically. The discussed method guarantees continuity of displacement, velocity, acceleration and jerk. It can be used for planning trajectories of various technical objects (e.g., manipulators, mobile robots, CNC machine tools, optical disk drives, autonomous vehicles, etc.).

The purpose of the study was to use reverse engineering to model biological products, especially sugar beet root. In the process of creating the solid model, the appropriate tools available in the 3D design environment were applied. The 3D scan of the beet, in the form of a spatial point cloud, was used to project the root geometry. This was, in turn, used to construct a triangulation grid that includes nodal points of triangles. The subsequent steps presented the process of creating a solid model using the Interpolation Spline tool. Attention has been paid to the possibility of modifying the geometry by inserting additional points into the existing interpolation spline and changing angular position as well as the distance of the structural planes. Geometry mapping error values were determined with regard to the reference model depending on the spread value of the Structural Planes. Error courses are non-linear with a logarithmic line trend (surface field error) and a linear line trend (volume error). The effects demonstrated the usefulness of geometry projection and its applicability to support the strength testing of biological materials, with particular emphasis on dynamic tests using whole roots.