Lie algebra rank that used in studying the controllability in mobile robot.
The book "A Mathematical Introduction to Robotic Manipulation" by Li, Lurray and Sastry is a reference. it includes chapters on non-holonomic behavior and motion planning with controllability issues.
Check these books:
1) Geometric Fundamentals of Robotics - J Selig ( Provides the most comprehensive treatment of this subject)
2) A Mathematical Introduction to Robotic Manipulation - Murray, Li , Sastry ( an awesome book! Provides ample understanding of lie algebra)
You may want to check out these notes
http://www.pages.drexel.edu/~hgk22/courses/MEM636_638/MEM636_638.htm
especially Lectures 2 and 3.
Please see the attachment, which describes the dual number basic concepts.
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