  This paper proposes a design methodology for non-linear state observers for
invariant kinematic systems posed on finite dimensional connected Lie groups,
and studies the associated fundamental system structure. The concept of
synchrony of two dynamical systems is specialised to systems on Lie groups. For
invariant systems this leads to a general factorisation theorem of a nonlinear
observer into a synchronous (internal model) term and an innovation term. The
synchronous term is fully specified by the system model. We propose a design
methodology for the innovation term based on gradient-like terms derived from
invariant or non-invariant cost functions. The resulting nonlinear observers
have strong (almost) global convergence properties and examples are used to
demonstrate the relevance of the proposed approach.
