A compact formula for the derivative of a 3-D rotation in exponential coordinates
This work addresses a computational bottleneck in robotics and computer graphics by simplifying differential analysis, though it is incremental as it builds on known mathematical frameworks.
The authors derived a compact formula for the derivative of a 3-D rotation matrix with respect to exponential coordinates, providing a geometric interpretation and verifying its agreement with existing formulas.
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.