Passive Decomposition of Mechanical Systems With Coordination Requirement

We show the fundamental passive decomposition property of general mechanical systems on a
n
-dim. configuration manifold
M
, i.e., when endowed with a submersion
h
:
M
→
N
, where
N
is a
m
-dim. manifold (
m
≤
n
), their Lagrangian dynamics with the kinetic energy as the Lagrangian can always be decomposed into: 1) shape system, describing the
m
-dim. dynamics of
h
(
q
) on
N
; 2) locked system, representing the (
n
-
m
)-dim. dynamics along the level set of
h
; and 3) energetically-conservative coupling between them. The locked and shape systems also individually inherit the Lagrangian structure and passivity of the original dynamics. We exhibit and analyze geometric and energetic properties of the passive decomposition in a coordinate-free manner. An illustrative example on SO(3) is also provided.