Motion polynomials (polynomials over the dual quaternions with nonzero real norm) describe rational motions. We present a necessary and sufficient condition for reduced bounded motion polynomials to admit factorizations into monic linear factors, and we give an algorithm to compute them. We can use those linear factors to construct mechanisms because the factorization corresponds to the decomposition of the rational motion into simple rotations or translations. Bounded motion polynomials always admit a factorization into linear factors after multiplying with a suitable real or quaternion polynomial. Our criterion for factorizability allows us to improve on earlier algorithms to compute a suitable real or quaternion polynomial co-factor.

Publication:

SIAM J. APPL. ALGEBRA GEOMETRY Vol. 9, No. 1, pp. 186--210

http://dx.doi.org/10.1137/22M1520670

Author:

Zijia Li 

Contact the author

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China website https://people.ucas.ac.cn/∼zijia/.

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Hans-Peter Schröcker

Department of Basic Sciences in Engineering Sciences, University of Innsbruck, Innsbruck, Austria 

Mikhail Skopenkov

CEMSE, King Abdullah University of Science and Technology, Thuwal, Mecca Province, Saudi Arabia

Daniel F. Scharler

Daniel F. Scharler greatly contributed to an earlier version of this manuscript. He tragically passed away on April 12, 2022, at the age of only 29.