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

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
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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