Kinematic and Dynamic Analysis of Spatial Six Degree of Freedom ...

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Parameters of Structural Synthesis Structural Classification λ B N bv C b L j jb Class Kind Type Order Mod. 6 3 16 6 5 11 8 48 4(4) 3 5 4,6,6 6 1 λ B N bv C b L j jb Class Kind Type Order Mod. 5 1 4 4 0 4 3 15 3(3) 1 0 4 4 1 λ B N bv C b L j jb Class Kind Type Order Mod. 3 5 15 5 5 10 5 15 1(5) 5 5 3,3,3,3,3 5 2 Figure 3.1 – Examples of structural synthesis of some parallel manipulators operating in space and subspaces. 2 30
3.3 Geometrical Structural Synthesis of Parallel Manipulators The purpose of geometrical structural synthesis is creating the foundation to discover the particular geometrical features and optimum structures by: 1) Generating set of main branches of platforms and structural groups. 2) Linking structural groups to the vacant branches of the manipulator. 3) Creating modular systems with multi DOF using successive layers of parallel manipulators. 3.3.1 Generating Set of Main Branches of Platforms and Structural Groups Consider the task of geometrical structural synthesis by generating a set of branches for a parallel manipulator. Definition: Position and orientation of ‘rigid body’ and its subsets ‘plane or line’, ‘cone surface’, ‘spherical or plane motion’ in space can be described by six, five, four and three independent parameters respectively. A branch, taken as individual, must also has as many DOF as independent parameters to be able to describe a rigid body or its subsets. Let’s consider lower kinematic pairs only: one mobility p1, two mobility p2(p1- p1), and three mobility p3(p1- p1- p1) where p1 = R (revolute) or P (prismatic) or H(screw), p2 = C(cylindrical) or U(Universal), p3 = S(spherical) joints. On the base of interchangeability of kinematic pairs we get common number of modification of platform branches (Table 3.1). 3.3.2 Linking Structural Groups to the Vacant Branches In Figure 3.2a, a spatial parallel manipulator with a triangular platform is shown. Each of three branches has two actuators. To place the actuators on the fixed base, a R-R-R structural group is added to each branch. 31
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Page 7 and 8: Chapter 4 KINEMATIC ANALYSIS ......
Page 9 and 10: Figure 4.15 - Comparison of results
Page 11 and 12: Chapter 1 INTRODUCTION The introduc
Page 13 and 14: Figure 1.4 - The first flight simul
Page 15 and 16: Generally, the actuators of a seria
Page 17 and 18: −1 where J = J q J x . . . q = J
Page 19 and 20: Chapter 2 SCREW KINEMATICS In this
Page 21 and 22: From equation (2.6) we have; or x z
Page 23 and 24: Using (2.7) and (2.12), one can fin
Page 25 and 26: LP + MQ + NR = 0 L 1 2 2 + M 2 1 LP
Page 27 and 28: Following Crammer’s method, the u
Page 29 and 30: That’s, the variable angle α ki
Page 31 and 32: ~ ~ ~ ~ ~ ~ Let Ei ( Li , M i, Ni )
Page 33 and 34: Qk = (NiPj + LjRi - LiRj - NjPi - a
Page 35 and 36: The methods reported by F. Freudens
Page 37 and 38: Using equations (3.4) and (3.6), we
Page 39: Order of a structural group is the
Page 43 and 44: 3.4 Kinematic Structural Synthesis
Page 45 and 46: Let’s consider a 5 DOF parallel m
Page 47 and 48: Chapter 4 KINEMATIC ANALYSIS Kinema
Page 49 and 50: Figure 4.1 - The six degree of free
Page 51 and 52: v v v P4 = yv N 4 − zv M 4 v v v
Page 53 and 54: The best way of visualizing the dis
Page 55 and 56: 4.2.2 Results for the Actuation of
Page 63 and 64: Figure 4.17 - Corresponding piston
Page 65 and 66: Chapter 5 DYNAMIC ANALYSIS Dynamics
Page 67 and 68: Thus, according to Lagrange, the re
Page 69 and 70: J is the 6x6 Jacobian matrix that
Page 71 and 72: Chapter 6 DISCUSSION AND CONCLUSION
Page 73 and 74: REFERENCES [1] D. Stewart, A platfo
Page 75 and 76: [19] Seung-Bok Choi, Dong-Won Park,
Page 77 and 78: [38] Kotelnikov A. P., Screw calcul
Page 79 and 80: [60] R. I. Alizade, Investigation o
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Page 85 and 86: pty:=roy+dy; ptz:=roz+dz; end; func
Page 87 and 88: Figure A.2 - a Screenshot of iMIDAS
Page 89 and 90: APPENDIX B RECURSIVE SYMBOLIC CALCU
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PP := L1 ← 0 M1 ← 0 N1 ← 1 L2
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Preleminary function definitions In
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C.2 Visual NASTRAN Desktop NASTRAN
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