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

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to these problems are far from completeness and a systematic approach for design of parallel manipulators is still inexistent. The different structures proposed are mainly flourishing from intuition. From kinematic and dynamic point of view, analysis of parallel manipulators are much more complicated then serial manipulators and most solutions lack analytical results. The complexity of the governing equations inevitably increases the computation time. To generate a desired motion, inverse kinematics of a manipulator has to be solved. For serial manipulators inverse kinematics leads to multiple, and real solutions that the controller has to choose the most suitable. For a parallel manipulator, inverse kinematic solution is unique and generally a simple analytical solution can be found. However, a parallel manipulator is not able to attain every position because of singularity configurations and link interference. Therefore, a designer has to investigate the problem thoroughly. 1.4 Singular Configurations of Parallel Manipulators Let the joint variables denoted by a vector q and the location of the moving platform be described by a vector x. Then the kinematic constraints imposed by the branches can be written in the general form: f ( q( t ), x( t)) = 0 (1.1) where f is an n-dimensional implicit function of q and x, 0 is an n-dimensional zero vector. Differentiating Eqn. (1.1) with respect to time, a relationship between the input joint rates and the end-effector output velocity as follows: where . . J x x = J q q (1.2) ⎡∂f ⎤ ⎡ ∂f ⎤ J x = ⎢ ⎣∂x ⎥ and J q = ⎢ ⎥ ⎦ ⎣∂q ⎦ Above derivation leads to two separate Jacobian matrices. Hence the overall Jacobian matrix, J, can be written as 6
−1 where J = J q J x . . . q = J x (1.3) Due to existence of two Jacobian matrices, a parallel manipulator is said to be at a singular configuration when either Jx or Jq or both are singular. Three different types of singularities can be identified. 1.4.1 Singularity of First Kind Singularity of first kind occurs when the determinant of Jx is equal to zero, namely, ( J ) 0 det = x Assuming that in the presence of such a singular condition the null space of Jx is not empty, there exist some nonzero . x vectors that result in zero . q vectors. That is, the moving platform can possess infinitesimal motion in some directions while all the actuators are completely locked. Hence the moving platform gains one or more degrees of freedom. This is in contradiction with a serial manipulator, which loses one or more degrees of freedom (Waldron and Hunt, 1988). In other words, at a second kind of singular configuration, the manipulator cannot resist forces or moments in some directions. 1.4.2 Singularity of Second Kind Singularity of first kind occurs when the determinant of Jq is equal to zero, namely, ( J ) 0 det = q In the presence of such a singular condition the null space of Jx is not empty, there exist some nonzero . q vectors that result in zero . x vectors. Infinitesimal motion of platform along certain directions cannot be accomplished. Hence the manipulator loses one or more degrees of freedom. Second type of singularity generally occurs at the workspace boundary. 7
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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: Generally, the actuators of a seria
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 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
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Thus, according to Lagrange, the re
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J is the 6x6 Jacobian matrix that
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Chapter 6 DISCUSSION AND CONCLUSION
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REFERENCES [1] D. Stewart, A platfo
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[19] Seung-Bok Choi, Dong-Won Park,
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[38] Kotelnikov A. P., Screw calcul
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[60] R. I. Alizade, Investigation o
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in the 'Input Parameters' section.
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Figure A.2 - a Screenshot of iMIDAS
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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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