# Article

**Keywords:**

robot trajectory; transformation matrix; Euler angle of rotation; orthogonal group

**Summary:**

This article is focused on calculating the trajectory of an industrial robot in the production of composites for the automotive industry. The production technology is based on the winding of carbon fibres on a polyurethane frame. The frame is fastened to the end-effector of the robot arm (i.e. robot-end-effector, REE). The passage of the frame through the fibre processing head is determined by the REE trajectory. The position of the fibre processing head is fixed and is composed of three fibre guide wheels with coils of carbon fibres. The fibre processing head winds three layers of filaments onto the frame. The polyurethane frame is determined by the local Euclidean coordinate system $E_{3}$, which has its origin in the REE. We use a mathematical model and matrix calculus to compute the trajectory of the REE to guarantee the desired passage of the frame through the fibre processing head. The translation and rotation matrices of the local coordinate system (of the REE) are calculated with respect to the base coordinate system of the robot.