The Fixed Point Iteration-based Adaptive Control design methodology is an alternative
to the Lyapunov function-based technology. It contains higher-order feedback terms
than the standard resolved acceleration rate control. This design approach strictly
separates the kinematic and dynamic issues. At first, a purely kinematic prescription
is formulated for driving the components of the tracking error to zero. Then an available
approximate dynamic model is used to calculate the approximated necessary control
forces. Before exerting on the controlled system, these forces are adaptively deformed
in order to precisely obtain the prescribed kinematic behavior. The necessary deformation
is iteratively found by the use of a contractive map that results in a sequence that
converges to the unique fixed point of this map. In the case of underactuated systems,
when the relative order of the control task also increases, the highest-order time-derivative
depends on the lower-order ones according to the dynamic model of the system. This
makes it impossible to realize the arbitrarily constructed kinematic design. In the
paper, a resolution to this discrepancy is proposed. The method is demonstrated using
two non-linear paradigms, a three-degree-of-freedom robot arm, and a two-degree-of-freedom
system, i.e., two coupled non-linear springs. The operation of the method was investigated
via simulations made by the use of Julia language and simple sequential programs.
It was found that the suggested solution could be considered as a new variant of the
fixed point iteration-based model reference adaptive control that is applicable for
underactuated systems even if the relative order of the task is increased.