To avoid the general mathematical difficulties related to the use of Lyapunov's "direct" method in adaptive control in the present paper an alternative approach, the use of "Robust Fixed Point Transformation (RFPT)" is applied in a decentralized adaptive control of two dynamically coupled mechanical systems. Each system is a cart plus double pendulum system provided with a local controller for which a rough dynamic model of the cart and one of the pendulums is available. No any information is available on the existence and the physical state of the second pendulum and on the existence of and dynamic connection to the other cart+pendulums system. It is shown by convincing simulation results that via observing and controlling the state propagation only of the modeled degrees of freedom, the uncorrelated controllers can precisely track their prescribed trajectories. The present solution is a more precise and far simpler tackling of the same task for which special symplectic matrices were used in the past. While the stability of the older solution required the use of special ancillary scaling methods, the present one can do without them. It is also shown that the present approach can also cooperate with a transformation reduction technique that successfully was used in the earlier case, too.