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Factorization of Dual Quaternion Polynomials Without Study's Condition
Johannes Siegele1, Martin Pfurner1, Hans-Peter Schröcker1
1Department of Basic Sciences in Engineering Sciences, University of Innsbruck, Techikerstr. 13, 6020 Innsbruck, Austria.
Advances in Applied Clifford Algebras
|March 22, 2021
Summary
This study explores polynomial factorization over dual quaternions, enabling new mechanism designs. The research introduces novel "vertical Darboux joints" and investigates their potential replacement with cylindrical joints.
Area of Science:
- Mathematics
- Mechanical Engineering
- Robotics
Background:
- Polynomial factorization is crucial for kinematic synthesis.
- Existing methods for dual quaternion polynomials are limited to "motion polynomials" with real norm polynomials.
- Novel factorization approaches are needed for broader mechanism design.
Purpose of the Study:
- To investigate factorizations of polynomials over dual quaternions into linear factors without the "motion polynomial" constraint.
- To develop an algorithm for computing such factorizations.
- To construct new mechanisms using these factorizations and address their limitations.
Main Methods:
- Investigating polynomial factorization over the ring of dual quaternions.
- Developing an algorithm for computing factorizations based on specific polynomial properties.
- Applying the computed factorizations to kinematic synthesis of mechanisms.
Main Results:
- Established a necessary and sufficient condition for the existence of factorizations.
- Developed a novel algorithm for computing factorizations of dual quaternion polynomials.
- Constructed new mechanisms featuring "vertical Darboux joints" not achievable with prior methods.
Conclusions:
- The developed factorization method expands the scope of mechanism synthesis beyond "motion polynomials".
- The new mechanisms with "vertical Darboux joints" present unique kinematic capabilities but require further refinement.
- Future work focuses on replacing "vertical Darboux joints" with cylindrical joints to overcome mechanical deficiencies while maintaining constraint.
Keywords:
Construction of mechanismsDual quaternion polynomialsDual quaternionsFactorizationRational motionSkew polynomial ringMore Related Videos
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