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I am curious about the relationship between the Lie algebra/group approach to robot kinematics given by Murray (et al.) in "An Introduction to Mathematical Robotic Manipulation" and Lynch (et al.) in "Modern Robotics", and the spatial vector approach given by Featherstone in "Rigid Body Dynamics Algorithms."

Spatial vectors (screws/twists) are also used extensively by Murray and Lynch, but Featherstone defines many more mathematical operations on the vectors (spatial cross product, scalar product) which are not found in Murray or Lynch. Also, Featherstone does not use the product of exponentials formula for forward kinematics (he instead uses Denavit-Hartenberg parameters) and does not discuss Lie groups. It was surprising to me that while both approaches use spatial vectors, they explain them (and use them) in very different ways. I had previously assumed that the spatial vector approach was synonymous with an approach based on Lie groups and transformation matrices.

Are the differences primarily on an abstract mathematical level, or are there advantages/disadvantages to each approach for specific applications? If so, what are they? Any more general information on the relationship between these approaches would be appreciated as well.

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