Dual Quaternions, Augumented Quaternions and Dual Quaternion Laplacian


2:00 PM - 3:00 PM



  • Time: 2 pm -3 pm
  • Date: Wednesday, January 24, 2024
  • Venue: MB441
  • Speaker:Prof. Liqun Qi (Hong Kong Polytechnic University)


In this talk, I will first report our work on eigenvalues of Hermitian dual quaternion matrices. This work extends the classical theory of eigenvalues of Hermitian complex matrices and Zhang’s 1997 result on eigenvalues of Hermitian quaternion matrices. Then we apply this result to the formation control study, while formation control is very important in the UAV research.
Then I will report our work on augmented quaternions, and formulate hand-eye calibration, which is a basic problem in robotic research, and SLAM (Simultaneous Location and Mapping), which is a very hot topic in robotic research, as equality constrained augmented dual quaternion optimization problems. This approach reduces the size of the problem and keep the smoothness of the model.
Finally, I will report our recent work on dual quaternion Laplacian. The dual quaternion Laplacian matrix of desired relative configurations in multi-agent formation control is similar to the classical unweighted Laplacian matrix via a dual quaternion diagonal unitary matrix. Its eigenvalues are all positive numbers except one zero eigenvalue. A unit dual quaternion vector is a desired formation vector if and only if it is in the null space of this dual quaternion Laplacian matrix.


Professor Qi Liqun graduated from Tsinghua University in 1968, received a master’s degree in computer science from the University of Wisconsin in 1981, and a doctorate from the University of Wisconsin in 1984. He is a Chair Professor in the Department of Applied Mathematics at the Hong Kong Polytechnic University. He is a foreign academician of the Petrovskaya Academy of Sciences and Arts in Russia. In 2003, he was listed in the ISI ranking list of the international authoritative citation database, and was hailed as one of the most influential mathematics in the world from 1981 to 1999; from 2003 to 2004, he won the President’s Award of the Hong Kong Polytechnic University for his outstanding work in scientific research and academic activities; In 2010, he won the first prize of the Science and Technology Award of the Second Chinese Operations Research Society. Professor Qi Liqun was listed as one of the most highly cited mathematicians in the world within 10 years from 2003 to 2010. Recently, he was selected as the Global Highly Cited Scientist in 2018 and is an editor and editorial board member of 10 international magazines.
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