• Biomedical Engineering, Semester 2, position 4
• Automatic Control, Semester 2, position 4
• Transportno inženjerstvo, konstrukcije i logistika, Semester 2, position 4

Introduce students to basic concepts of kinematics and dynamics of robotic systems. It is possible to solve direct and inverse kinematics and dynamics of the robot system (RS) using modern theory based on Rodriguez transformation matrix as well as the theory of finite rotations. Determination (simulation) models of RS - i.e. differential equations of motion of the RS, which are important in practical problems of the RS.Practical simulations RS using Cyberbotics Webots software package and students work with laboratory robot NEUROARM.

•Determine the number of degrees of freedom of robotic system (RS) •Define the matrix of transformation, in the case of (Euler angles, Rezal angles, Hamilton-Rodrigues parameters, ...) • Forming expressions to determine the basic kinematic characteristics RS using Rodrigues approach: characteristic position vectors of RS, speed and acceleration of the center of inertia of the robot segments (RSE), angular velocity and angular acceleration RSE, speed and acceleration of the robotic gripper • Forming a kinematic model RS and solve direct and inverse kinematics task of RS • Analyze singular cases in solving the task of kinematics RS • Formed terms of linear momentum, angular momentum and kinetic energy of arbitrary segment RS • Determine the kinetic energy of the whole RS, the basic metric tensor RS, the corresponding generalized forces, Christoffel symbols of the first kind for given RS • Forming the differential equations of motion using the RS covariant form of Lagrange equations of the second kind and solve other types of direct and inverse task of dynamics • Numerical simulate the previously formed kinematic / dynamic models using programming environment (MATLAB, Mathematica, etc.) • Forming the differential equations of motion RS for the case of RS: which is given in the form of a kinematic chain with branching , RS given in the form of a closed kinematic chain. • Set additional constraint equations in the case of constrained robotic gripper movements • Distinguish non-redundant and redundant RS and determine the degree of redundancy RS • Distinguish the basic concepts of control of RS

Basic concepts, definition of robot system (RS). Orthogonal transformation of coordinates.Rodriguez formula and the transformation matrix (MT), arbitrary and reference configuration of RS.Complex MT of coordinates. Position vectors that define the configuration of the RS, internal and external coordinates of RS. Velocity ​​and acceleration of the center of inertia of an arbitrary robot segment(RSE). Angular velocity and angular acceleration of an arbitrary RSE. Velocity of gripper tip of RS. Direct and inverse kinematics of robot task-as well as singular cases. Constraints of RS. Momentum, angular momentum, kinetic energy of arbitrary robot segment of RS. Kinetic energy and the metric tensor of RS. Generalized forces and the principle of ideality RS-different cases. Differential equations (DIFE)of motion of RS. (DIFE) of motion of the RS in covariant form. Other methods of forming (DIFE) of motion of RS. DIFE of motion of RS given in the form of kinematic chain with the structure of topological three; DIFE of motion of RS given in the form of closed-kinematic chain. Additional equations of contraints. Constrained motion of robotic gripper. Equations of motion of RS with Langrange multipliers. Redundant RS. Basic concepts of control RS.

Examples of determining the number of degrees of motion of the RS; Calculation the transformation matrix(MT)- in case of Euler angles, and Hamilton-Rodriguez parameters; Determination of kinematic characteristics of the robot segment (RSE): angular velocity and angular acceleration RSE, velocity and acceleration of the observed point-RSE cases of Rezales and Euler angles.Application of Rodriguez transformation matrix, determine position vectors which define the configuration of the RS-in MATLAB environment. Kinematic characteristics of the i-th robot segment. Solving the direct and inverse kinematic task of RS. Determination of (planar) inertia tensor RSE, RS. Obtaining momentum and angular momentum, kinetic energy, the coefficient of the metric tensor RS, generalized forces, Christoffel symbols of the first kind. Solving the direct and inverse dynamics task of the RS. Examples of DIFE of RS simulation in MATLAB-GUI, MATHEMATICA environment, an example of a redundant RS. An example of simulation RS using Cyberbotics Webots package. Example of control of the RS-laboratory robot NeuroArm with 7 degrees of freedom in the MATLAB environment.

desirable courses: Mechanics 1, Mechanics 2 Mechanics 3,

1.Čović M. V. Lazarević, Mechanics of Robot, MF Belgrade,2021.(Book) 2.Lazarević M. Exercises in mechanics of robot, MF Belgrade,2006.(ZZD) 3.Wittenburg J., Dynamics of Systems of Rigid Bodies, Teubner, Stuttgart, 1977. (XJ) 4.Craig J., Introduction to Robotics, Mechanics and Control, Pearson, 2017. 5.Written abstracts from the lectures (Handouts) 6.Cyberbotics Webots - software package 7.NeuroArm-laboratory robot with 7 degrees of freedom. 8.MATLAB,MATHEMATICA-mathematics software packages

Total assigned hours: 75

New material: 20
Elaboration and examples (recapitulation): 10

Auditory exercises: 10
Laboratory exercises: 6
Calculation tasks: 5
Seminar paper: 0
Project: 9
Consultations: 0
Discussion/workshop: 0
Research study work: 0

Review and grading of calculation tasks: 0
Review and grading of lab reports: 0
Review and grading of seminar papers: 0
Review and grading of the project: 4
Test: 6
Test: 0
Final exam: 5

Activity during lectures: 0
Test/test: 40
Laboratory practice: 0
Calculation tasks: 0
Seminar paper: 0
Project: 20
Final exam: 40
Requirement for taking the exam (required number of points): 30

Bruno Siciliano, Oussama Khatib, Springer Handbook of Robotics,Springer-Verlag Berlin Heidelberg 2008.; Thomas R. Kurfess.,Robotics and automation handbook,CRC Press LLC, Boca Raton, Florida,2005; Ahmed A. Shabana, Dynamics of Multibody Systems,Cambridge University Press The Edinburgh Building, Cambridge , UK,2020; M.W. Spong, M. Vidyasagar: Robot Dynamics and Control (Wiley, New York 1991); R. Paul: Robot Manipulators: Mathematics, Programming and Control (MIT Press, Cambridge 1982)