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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.

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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.