## 1 Introduction  
The multi degrees-of-freedom gear drive is a key component in hybrid vehicles. Its capability of managing power flows from different sources is of paramount importance for energy saving. For these reasons, the mechanical efficiency of planetary multi degrees-of-freedom planetary gear trains is a topic that received many contributions in recent times (e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]). The analysis carried out herein offers a reliable approach toward a synthetic description of the relationship between kinematics conditions and gear train mechanical efficiency. In this paper, the kinematic conditions governing the relationship between the algebraic signs of actual and virtual or potential powers are elucidated for the first time. Such an analysis has a significant influence on the casting of physically reliable and consistent formulas for the computation of mechanical efficiency in a two dof PGU (Planetary Gear Unit). Subsequently, two general formulas for mechanical efficiency computation in a two dof PGU are proposed. The differential unit is the building block of more complex planetary gear trains. Thus, the understanding of the two dof PGU basic mechanics influences the design of the entire planetary gear drive [15]. The approach adopted herein is based on the concept of _virtual_ or _potential_ power, at the base of thoughtful contributions on the mechanical efficiency of planetary gear trains [5, 16, 17, 18, 19, 20, 21, 22]. The _virtual power_, a term coined by [PERSON] and [PERSON] [17], or _potential power_, coined by [PERSON] [5], is the power measured by an observer on any arbitrary moving frame [23, 24]. However, the use of kinematic inversion for torque and efficiency analysis on gear trains dates back to [PERSON] [16]. In particular, he recognized that the torques acting on the PGU links and power losses are independent of the observer's motion. The following quote is taken from [16]:  
_...our analysis is based upon an important principle relating to torques and the power lost in friction; this is the fact that magnitudes of the torques acting upon the various members of the gear are quite independent of the motion of the observer who measures them. In addition, the power lost, being determined solely by the internal torques and the relative motions of the wheels within the gear, is also independent of the observer's motion._  
Although [PERSON]'s analysis was limited to one dof PGU, the importance of kinematic inversion in the gear trains mechanical efficiency analysis is well represented. The present paper was stimulated by the discovery that, under certain kinematic conditions, the classic Radzimovsky formulas [15; 25; 26; 27] for the computation of mechanical efficiency in a two dof gear unit are not valid anymore [28]. This situation may lead to an estimation of the overall mechanical efficiency flawed by errors. As discussed by [PERSON] & [PERSON] [15], the Radzimovsky formulas consider the two dof epicyclic gear train as two single dof devices in parallel and crossed by a power-flow. These formulas are deduced after application of the general mechanical efficiency formula of a mechanical system composed of two parallel devices with a given mechanical efficiency and power-flow direction. However, with the introduction of the concept of _virtual_ or _potential_ power, the mechanical efficiency of an epicyclic gear train is reduced to that of an ordinary gear train. For this purpose, a kinematic inversion is introduced such that the observed motion of the gear carrier is canceled. The orientation of power flow observed under such kinematic inversion may not be the same as one of epicyclic arrangement. This situation requires special attention and the algebraic expression of the overall mechanical efficiency should be set in a consistent manner. The new proposed formulas herein are of a general nature and do not suffer any of the mentioned pitfalls.