205 TECHNICAL UNIVERSITY OF CLUJ-NAPOCA ACTA TECHNICA NAPOCENSIS Series: Applied Mathematics, Mechanics, and Engineering Vol. 57, Issue II, June, 2014 1 i O − 0 O 1 i O + n O P ( ) n T ( ) 0 T ( ) 1 i T + ( ) 1 i T − ( ) 1 n T + n n ε i q element i-1 element i i q & i q & & i i k i i v i i ε i i ω i i a ( ) i T i O n n ω n n a n n v Fig.1. The kinematic structure of a robot with (n) degrees of freedom THE KINEMATIC MODEL OF THE TRTR MODULAR SERIAL ROBOT Ramona-Maria GUI (LUNG), Ovidiu-Aurelian DETEŞAN, Viorel ISPAS, Virgil ISPAS Abstract: The paper has three parts in its structure: theoretical approach, the kinematic modeling of the TRTR modular serial robot and conclusions, followed by a reference list. After a brief presentation of the iterative method for the kinematic modeling, the mechanical structure of the TRTR robot is mentioned, according to [4]. The iterative method is then applied to the mechanical structure of TRTR robot, based on certain geometrical expressions previously determined. The kinematic parameters are thus obtained, which are the operational angular and linear velocities and accelerations of the gripper with respect to the mobile frame (T 5 ). Applying some transformation equations, these kinematic parameters are eventually expressed with respect to the fixed frame (T 0 ) from the robot base. Key words: modular serial robot, kinematic model, iterative method. 1. THEORETICAL APPROACH The iterative method is one of the study methods frequently used in the kinematic modeling, according to [1]. This method is based on considering the position vectors, the rotation matrices and it makes use of iterative matrix computation in order to determine the kinematic parameters. According to [1] and [2], the figure 1 presents the kinematic structure of a robot having (n) degrees of freedom (DOF), consisting in (n) links considered as rigid bodies, which can perform movements determined by the active kinematic joints of the 5 th class, considered to be mechanically perfect. The iterative method consists in crossing the robot kinematic chain from the fixed base 0 to the gripper (n+1) and determining the following kinematic parameters by consecutive iterations: { } . 1 , , , , , n i a v k i i i i i i i i i i ÷ = ε ω (1) The mentioned kinematic parameters describe the motion of each link i (i = 1÷n) with respect to the fixed frame (T 0 ) from the robot base. These parameters are expressed into the frame (T i ) and they have the following meaning: i i k is the i th order axis versor; i i i i ε ω , represent the angular velocity and acceleration of the rotation of link i around its point O i , the origin of (T i ) frame, expressed with respect to (T 0 ); i i i i a v , are the linear velocity and acceleration