A Compact Formula for the Derivative of a 3-D Rotation in Exponential Coordinates

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• 作者：Guillermo Gallego (1) (2)
  Anthony Yezzi (3)
  
  1. Grupo de Tratamiento de Im谩genes ; Universidad Polit茅cnica de Madrid ; Madrid ; 28040 ; Spain
  2. Robotics and Perception Group ; AI Laboratory ; University of Z眉rich ; 8001 ; Z眉rich ; Switzerland
  3. School of Electrical and Computer Engineering ; Georgia Institute of Technology ; Atlanta ; GA ; 30332 ; USA
  
• 关键词：Rotation ; Lie group ; Exponential map ; Derivative of rotation ; Cross ; product matrix ; Rodrigues parameters ; Rotation vector
• 刊名：Journal of Mathematical Imaging and Vision
• 出版年：2015
• 出版时间：March 2015
• 年：2015
• 卷：51
• 期：3
• 页码：378-384
• 全文大小：180 KB
• 参考文献：1. Altmann, S (1986) Rotations, Quaternions, And Double Groups. Clarendon, Oxford
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• 刊物类别：Computer Science
• 刊物主题：Computer Imaging, Vision, Pattern Recognition and Graphics
  Image Processing and Computer Vision
  Artificial Intelligence and Robotics
  Automation and Robotics
  
• 出版者：Springer Netherlands
• ISSN：1573-7683

文摘

We present a compact formula for the derivative of a 3-D rotation matrix with respect to its exponential coordinates. A geometric interpretation of the resulting expression is provided, as well as its agreement with other less-compact but better-known formulas. To the best of our knowledge, this simpler formula does not appear anywhere in the literature. We hope by providing this more compact expression to alleviate the common pressure to reluctantly resort to alternative representations in various computational applications simply as a means to avoid the complexity of differential analysis in exponential coordinates.