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

APPENDIX B
RECURSIVE SYMBOLIC CALCULATION ALGORITHMS
IN MATHCAD
Recurrent equations to be used in kinematic analysis are developed in Chapter 2 and
the solution for a six DOF spatial parallel manipulator is given in Chapter 4. We need to find
the components of E4 and E5 for each branch, to find their intersection points. Derivation of
screw equations using recurrent equations is a straight forward algorithm. However, using
computer to develop the equation greatly saves time and minimizes the possibility of making
calculation errors. In what follows is the derivation of screw equations from recurrent
equation using MathCAD software.
B.1 Derivation of Unit Vectors
To find the unit vectors of the screws e k ( Lk
, M k , Nk
) , we input the two initial unit
vectors e1 and e2 as given in Section 4.1.2. The second step is to use a loop to calculate the
unit vectors of the unknown screws using equation (2.28). Using MathCAD the algorithm is
written as in figure B.1.
B.2 Derivation of Moments of Unit Vectors
o
Deriving the equations for moments of unit vectors ek ( Pk
, Qk
, Rk
) is more complicated
since these depend on e k ( Lk
, M k , Nk
) as given in (2.30). To find the unit vector of the screws,
we input the two initial screws e1, e ,e2, e . Similarly, a loop is used to calculate the unit
o
1
o
2
vectors of the unknown screws using equation (2.30). Using MathCAD, the algorithm is
written as in figure B.2. The final results are compiled in (4.1). Note that, MathCAD cannot
fully simplify the trigonometric equations, final touches are made manually.
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