1. Dynamics of PKM Prof. Rosario Sinatra Dipartimento di Ingegneria Industriale e Meccanica Università degli Studi di Catania March 27, 2007 EURON07 WINTER SCHOOL PARALLEL ROBOTS: Theory and Applications 2nd International UMH Robotics Winter School Flamingo Oasis Hotel, Benidorm, Spain March 26 - 30, 2007

2. Physical System S P U BP MP Mechanical and Iconic Models Physical Laws Mathematical Model Analysis

3. Multibody Dynamic Systems Software for Multibody System Simulation ADAMSADAMS by MSC Software, United States alaskaalaska, by Technical University of Chemnitz, Germany AUTOLEVAUTOLEV, by OnLine Dynamics Inc., United States AutoSimAutoSim by Mechanical Simulation Corp., United States COMPAMMCOMPAMM by CEIT, Spain DynawizDynawiz by Concurrent Dynamics International DynaFlexProDynaFlexPro by MotionPro Inc, Canada HyperviewHyperview and Motionview by Altair Engineering, United StatesMotionview LMS Virtual.Lab MotionLMS Virtual.Lab Motion by LMS, Belgium MECANOMECANO by Samtech, Belgium MBDynMBDyn by Politecnico di Milano, Italy MBSoftMBSoft by Universite Catholique de Louvain, Belgium NEWEULNEWEUL by University of Stuttgart, Germany RecurDynRecurDyn by Function Bay Inc., Korea RobotranRobotran by Universite Catholique de Louvain, Belgium SAMSAM by Artas Engineering Software, The Netherlands SD/FASTSD/FAST by PTC, United States SimCreatorSimCreator by Realtime Technologies Inc., United States SimMechanicsSimMechanics by The Mathworks, United States SIMPACK by INTEC GmbH, Germany SIMPACK SPACARSPACAR by University of Twente, The Netherlands Universal Mechanism by Bryansk State Technical University, Russia Universal Mechanism Working ModelWorking Model by Knowledge Revolution, United States http://real.uwaterloo.ca/~mbody/

4. THE NATURAL ORTHOGONAL COMPLEMENT METHOD First introduced by Jorge Angeles and Sangkoo Lee [1, 2] Preliminary Definitions twist of i-th rigid body: (1) wrench of i-th rigid body: (2) (3) Figure 1: notations

5. the 6 × 6 extend angular velocity matrix W i and matrix extend mass M i Newton-Euler equation for i-th body: angular velocity matrix of i-th rigid body: (4) (5) (6)

6. UNCONSTRAINED DYNAMICAL EQUATIONS where: N-E equations: (7) (8) (9) (10) (11) (12) Figure 2: notations

7. KINEMATIC CONSTRAINTS where K is p×6r constrains matrix: where T is 6r×n twist shaping matrix: the power developed by the constraint wrench w C is zero: the external wrench is: the matrix T is an orthogonal complement of K (13) (14) (15) (16) (17) (18) (19) (20)

8. CONSTRAINED DYNAMICAL EQUATIONS where Euler-Lagrange equations of the system: (21) (22) (23) (24)

9. DERIVATION OF CONSTRAINT EQUATIONS AND TWIST-SHAPE RELATIONS OF THE SYSTEM WITH SIMPLE KINEMATIC-CHAIN STRUCTURE Let be E i the cross product matrix of vector; for the i-th revolute joint and ; Figure 3: a revolute joint (25) (26)

10. if the first link is inertial: where Matrix K is: (27) (28) (29) (30)

11. Next, the link twists are expressed as linear combinations of the joint-rate vector Figure 4: kinematic subchain; links J,J+1,…i when (31) (32)

12. then the twist-shape matrix T is : (34) twist t i of i-th link as linear combination of the ith joint: (33)

13. Figure 5: a prismatic joint for the i-th revolute joint If i-th is a prismatic joint : We introduce a definition below: (35) (36) (37)

14. eq. (40) can be rewritten: Multiplication by E i a 6-dimensional linear homogeneous equation in the t i : (38) (39) (40)

15. where if the first joint is prismatic: where if the kth joint is prismatic, with 1<k <i: (41) (42) (43) (44) (45)

16. DERIVATION OF MATRIX for the i-th revolute joint where for the k-th pair is prismatic and 1<k <i (46) (47) (48)

17. DYNAMICS of Parallel Manipulators Figure 6: J-leg of a simple platform parallel manipulator

18. Euler’s formula for Graphs (Harary, 1972) we label the legs with Roman numerals J = I, II, ….,VI (49) Figure 7: the free-body diagram of M

19. The twist of M (50) N-E equations of M: (51) (52)

20. The dynamics model is: (53) (54) : 6×6 inertial matrix of the manipulator; :6×6 matrix coefficient of the inertia terms; : 6-dimensional vector of joint variables; :6-dimensional vector of joint torques; for J-leg: (55)

21. (56) if k-joint is prismatic: if k-joint is revolute: mapping (57) from eq.(75): (58)

22. N-E equations of the moving platform free of constraints (59) NOC method : (60) which upon differentiation with respect to time: (61) then: (62)

23. which upon differentiation with respect to time Model in terms of actuated joint: where (63) (64) (65) (66) Final step is to formulate the model in terms only of actuated joint variables (67)

24. where, (68) If : (36×36 matrix) (6×36 matrix) (6-dimensional vector)

25. and hence The mathematical model takes a form: (69) (70) For purposes of the Inverse dynamics: For purposes of the direct dynamics: (71) (72)

26. Figure 8: Hexapod with fixed-length legs HEXAPOD with fixed-length legs

27. Dynamic Modeling N-E equation for each body (73) (74) (75) assembled system dynamics equations are given as: (76) (77)

28. (78) (Angeles and Lee 1988) kinematic constraints (79) (80)

29. (81) (82) (83) (84) (85)

30. Inverse Dynamics generalized twist: (86) (87) (88) (89) (90)

31. DYNAMIC ISOTROPY AND PERFORMANCES isotropy dynamic: In the literature there are many performance indices: - ASADA: generalized inertia ellipsoid (GIE); -YOSHIKAWA: dynamic manipulability ellipsoid ; -WIENS et al.: indices for measure of non linear inertia forces; -KHATIB and BURDICK: Isotropy acceleration; -MA and ANGELES: isotropy dynamic and dynamic conditioning index. dynamic conditioning index : (91) (92) (93)

32. 2 DOF POINTING SYSTEM Figure 9 Figure 10

33. Workspace

34. Spherical Parallel Manipulator Figure 11: spherical parallel manipulator (94)

35. twist (i = I, II, III h = 1, 2) Twist-shaping matrix (i=I,II,III h=1,2) with M = platform matrix (95) (96) (97) (98)

36. (i = I, II, III h=1,2) generalized mass matrix M and generalized angular velocity matrix W: (99) (100) (101) (102)

37. References J. Angeles, 2002, Fundamental of Robotic Mechanical Systems, Springer. J. Angeles, and S. Lee, 1988, “The Formulation of Dynamical Equations of Holonomic Mechanical Systems Using a Natural Orthogonal Complement”, ASME Journal of Applied Mechanics, Vol. 55, pp. 243-244. J. Angeles, and S. Lee, 1989, “The modelling of Holonomic Mechanical Systems Using a Natural orthogonal Complement”, Trans. Canadian Society of Mechanical Engineers, vol. 13, pp. 81-89. K. E. Zanganesh, R. Sinatra and J. Angeles, 1997, Kinematics and Dynamics of a Six-Degree-of-Freedom Parallel Manipulator with Revolute Legs, ROBOTICA International Journal, Vol. 15, pp. 385-394. O. Ma and J. Angeles, “The concept of dynamic isotropy and its applications to inverse kinematics and trajectory planning”, Proc. Of ICRA, Cincinnati, USA, 1990, pp. 481-486. F. Xi, R. Sinatra, and W. Han, 2001, Effect of Leg Inertia on Dynamics of Sliding-Leg Hexapods. ASME Journal of Dynamics, Measurement and Control, Vol. 123, pp. 265-271. F. Xi and R. Sinatra, 2002, Inverse Dynamics of Hexapods using the Natural Orthogonal Complement Method, Journal of Manufacturing Systems, Vol. 21, No 2, pp. 73-82. F. Xi, O. Angelico and R. Sinatra, 2005. Tripod Dynamics and Its Inertia Effect. ASME Journal of Mechanical Design, Vol. 127/1, pp. 144-149. A. Cammarata and R. Sinatra, 2005, Dynamics of a two-dof parallel pointing mechanism, Fifth ASME International Conference on Multibody Systems, Nonlinear Dynamics and Control, Sept. 24-28, 2005, Long Beach, California, USA. R. Di Gregorio, A. Cammarata and R. Sinatra, 2005, On The Dynamic Isotropy Of 2-Dof Mechanisms, Fifth ASME International Conference on Multibody Systems, Nonlinear Dynamics and Control, Sept. 24-28, 2005, Long Beach, California, USA.