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

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Figure 4.2 - Connection points of branches and platform Now, we can solve the set of equations (4.4) for the 12 unknown variables using an iterative approach. Accurate solutions are achived using constrained nonlinear function optimization, using MathCAD. Please see Appendix C for details. It should be noted that setting the initial values of the unknown joint angles and lengths equal to the inputs of their respective neighboring branch ( α = α and a i 35 i+ 1 v = a , i = 1, 3, 5) and setting all α = π / 2 reduces computation time and increases the 35 possibility of finding a reasonable solution. 4.2 Forward Displacement Analysis Using Numerical Integration and Verification of Results 24 Solution of forward displacement problem is accomplished using a software called Visual NASTRAN Desktop. For the details on numerical formulas and a brief overview of NASTRAN please see Appendix C. In the following sections, the results acquired using screw algebra and function optimization is compared with the solutions acquired using NASTRAN. For the analysis in the software, Kutta - Merson numerical integration formulas are used with a time step of 0.0001 s. A typical solution of 3 real-time seconds therefore comprises 30,000 integrations. For the manipulator discussed in this thesis, a typical simulation takes about four hours to complete on a Intel Pentium 4 1.4 Ghz system. i+ 1 13 i 13 42
The best way of visualizing the discrepancies between two solutions is to plot the graphs of corresponding data. However for the parallel manipulator, there are six input variables and six output variables, for which there isn’t a convenient way to show all data in a single graph. For this reason the orientation and location of the platform center are plotted on separate sheets. Also for the cases where two or more actuators are used simultaneously, the values of those are set equal (i.e. linear speeds of pistons are the same where as the angular velocities of the motors are the same). Finally for comparison of the results, the main criterion used is the distance of the platform centroid from the origin. 4.2.1 Results for the Actuation of the First Rotary Actuator The first rotary actuator corresponds to Input( α ) in figure 4.1. Figures 4.3, 4.4 and 4.5 shows the graphs of the results. For figure 4.3, Eavg = 1.35, %Eavg = 0.83. The maximum error occurs at near singularity configuration, Platform Centroid Distance (mm) 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 Optimization Numerical Integration 1 13 1 α 13 =90º, Emax = 112.56, %Emax = 8.71. 0 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 Input Actuator Angle (deg) Figure 4.3 - Comparison of results for a single actuated rotary actuators 43
Page 1 and 2: Kinematic and Dynamic Analysis of S
Page 3 and 4: Research is what I'm doing when I d
Page 5 and 6: ÖZ Bu tez yeni bir tip uzaysal alt
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 and 40: Order of a structural group is the
Page 41 and 42: 3.3 Geometrical Structural Synthesi
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: v v v P4 = yv N 4 − zv M 4 v v v
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
Page 81 and 82: A.1.1 CASSoM Source Code Like all v
Page 83 and 84: in the 'Input Parameters' section.
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
Page 91 and 92: PP := L1 ← 0 M1 ← 0 N1 ← 1 L2
Page 93 and 94: Preleminary function definitions In
Page 95 and 96: C.2 Visual NASTRAN Desktop NASTRAN

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