The discussion of Prof. Angeles suggests the use of the four Euler parameters instead of the three Rodriguez parameters to describe the platform orientation and shows that the maximum number of solutions to the problem (b) is eight. This conclusion is obtained by evaluating the Bezout number of a closure equation system of four quadratic equations in four unknowns: the platform Euler parameters. Prof. Angeles then proposes to find the solutions to his system either by the continuation method or by the computer algebra.

Regarding the orientation parameterization, we recognize that the Rodriguez parameter may give rise to computational problems very close to their (representation) singularity, which occurs when the rotation angle is equal to ±180 degrees. Indeed, it is known that the orientation parameterization with only three parameters has representation singularities while the use of four parameters, in general, eliminates the representation singularities.

However...

1.
Di Gregorio
,
R.
, and
Parenti-Castelli
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V.
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2001
, “
Position Analysis in Analytical Form of the 3-PSP Mechanism
,”
ASME J. Mech. Des.
,
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, pp.
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