This document presents a mathematical model of forward and inverse kinematics for a manipulator composed of joints with one degree of freedom al-lowing only the rotation of a Cartesian coordinate system around one axis. The manipulator is controlled by means of internal ties running from the point of con-nection with the controlled arm to the place of connection with the mechanism regulating the length of the individual tie. This length is the basis in derived equa-tions, and its change is the direct cause of the movement of the described robot. In contrast to the commonly used arm drives, the articulated variable in derived kinematic equations is the length of the tie, which affects the change in the angle between the arms, while it is not the same value of this angle. In addition, these joints are arranged parallel to each other, which results in the working space of the manipulator in one plane. The kinematic chain, which is the object of the described research, consisted of three series connected joints enabling rotation around one axis of the coordinate system. However, dependencies and formulas enabling their application to the description of forward kinematics and inverse kinematic chains built from a larger number of similarly arranged joints have been demonstrated.