This paper presents an analytical geometry method for kinematics and efficiency of planetary gear trains (PGTs). The novel method which is capable of evolution and contrast analysis of mechanism kinematics, can be applied to any typical one-and two-degree-of-freedom plane PGTs containing any number of simple, compound or complex-compound planetary gear sets. The efficiency analysis of this method features a systematized and programmed process and its independence of the speed ratio. The primary contribution of this work lies in the integration of quantitative calculation, qualitative evolution and comparative analysis of kinematics of PGTs into one diagram, and in the integration of kinematics and efficiency analysis into a single method system. First, the analytical geometry method is defined, its basic properties are given, and the systematization procedure to perform kinematic analysis is demonstrated. As an application, analytical geometry diagrams of common PGTs are exhibited in the form of a list, whose kinematic characteristics and general evolution tendency are discussed. Then, with the mapping of PGTs onto the angular speed plane, the efficiency formula of analytical geometry, which has an extremely concise form, and a simple method for power flow estimation are put forward. Moreover, a general procedure is provided to analyze the efficiency and power flow. Finally, four numerical examples including a complicated eleven-link differential PGTs are given to illustrate the simpleness and intuitiveness of the analytical geometry method.
