1. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 ROBOT DYNAMICS T. Bajd and M. Mihelj

2. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 In contrast to kinematics, dynamics represents the part of mechanics, which is interested into the forces and torques which are producing the motion of a mechanism. The analysis of robot dynamics enables us to consider –the torques necessary to compensate the gravity forces of robot segments, –the differences in moments of inertia occurring during the robot motion, –dynamic couplings caused by simultaneous movements of all robot segments. Robot dynamics

3. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Forward and inverse dynamics Applied torquesJoint motions

4. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 The dynamic analysis of a robot is based on a two- segment robot mechanism. The motion of the robot manipulator with two rotational joints occurs in the vertical plane. Both segments are of equal length. The dynamic model is simplified by assuming that the whole mass of each segment is concentrated in its center of mass. Such a pair of segments appears both in the anthropomorphic and in the SCARA robot structures. The robot trajectory is denoted by the two joint angles. The robot is placed into the fixed reference frame with z axis aligned with the axis of the first joint. Two-segment robot mechanism

5. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Position, velocity and acceleration of the center of mass of the second segment Torque in the second joint

6. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 The motion of the second segment mass is given by Newton’s law In addition to the force of gravity, the mass is acted upon by the force, transmitted by the massless segment Torque in the second joint

7. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Robot segments and are rigid, thus Center of mass acceleration Centripetal accelerationTangential acceleration

8. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 The torque in the second joint is or Torque in the second joint

9. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Considering the torque in the second joint is With Torque in the second joint Inertial couplingInertialCentrifugalGravitational

10. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Relation between the total torque of external forces and the time derivative of the angular momentum The sum of the torques produced by the external forces Torque in the first joint

11. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 The angular momentum of the mass equals with The angular momentum of the mass equals with Angular momentum

12. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 With Torque in the first joint Inertial coupling Inertial Centrifugal Gravitational Coriolis

13. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 The torques in the robot joints can be written in the following general form where Dynamic model in matrix form

14. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Inertial matrix b11b12 b21 b22

15. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Coriolis and centrifugal terms c11 c12 c21

16. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Gravitational terms g1 g2

17. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Inverse dynamic model with friction (diagonal matrix of the joint friction coefficients ) Forward dynamic model with friction Forward and inverse dynamic model

18. T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 Forward dynamic model block scheme