ABSTRACT
To begin, we observe that every vector function of x, say f(x), corresponds to a field of vectors in n-dimensional space with vector f(x) emanating from point x. Thus following the terminology employed in differential geometry, a vector function, f(x), is a vector field in Rn. The Lie derivatives of h(x) in the direction of a vector field f(x) are denoted as Lfh(x). Explicitly, the Lie derivative is the directional derivative of h(x) in the direction fi(x) and is
L h h x
f hf i
i i xx x x f x( ) = ¶¶ ( ) = Ñ( )
( ) ( ). (3.1)
The adjoint operator or Lie brackets are defined as
ad adf f x x0 0s x s s x f s s f f s( ) = ( ) = éë ùû = Ñ( ) Ñ( ), , - , (3.2)
and the higher operators are defined recursively as
ad adfi fi+ ( ) = ( )éë ùû
1s x f s x, . (3.3)
Two vector fields g1 and g2 are said to be involutive if and only if there exist scalar functions α(x) and β(x) such that the Lie bracket
g g x g x g1 2 1 2,éë ùû = ( ) + ( )a b . (3.4)
A Lie algebra L, named in honour of Sophus Lie, pronounced as ‘lee’, a Norwegian mathematician who pioneered the study of these mathematical objects, is a vector space over some
field together with a bilinear multiplication called the bracket. A non-associative algebra obeyed by objects such as the Lie bracket and Poisson bracket is a Lie algebra. Elements f, g and h of a Lie algebra satisfy
f f f g h f h g h, , , , ,éë ùû = +éë ùû = éë ùû + éë ùû0 . (3.5)
The Jacobi identity is defined by
f h g h f h f g, , ,g, , , éë ùûéë ùû + éë ùûéë ùû + éë ùûéë ùû = 0. (3.6)
The relation [ f, f] = 0 implies [ f, g] = −[g, f]. The binary operation of a Lie algebra is the bracket
fg h f g h f h g, , ,éë ùû = éë ùû + éë ùû . (3.7)
Lie algebras permit the recursive formulation of rigid body dynamics for a general class of mechanisms. In particular, the Lie algebra formulation allows one to easily derive the analytic gradient of the recursive dynamics transformation. They may then be employed to develop optimal solutions for a class of mechanisms. The Lie algebra should be thought of as a set of special vector fields, and the brackets associated with it play a key role in its application.
