ABSTRACT Recent innovative drives in hydraulics could introduce very competitive hybrid hydraulic vehicles (HHV). These drives has been considered and analyzed only in the serial HHV architecture. The series-parallel transmission architecture, also called power-split or e-CVT is highlighted as the most popular concept for full (strong) hybrid electric vehicles (HEV). The examples are one-mode power-split in Toyota Prius and two-mode (compound) power-split in GM-Allison EVT. Ambitions to make the hybrid hydraulic power trains better and more efficient would certainly require deeper analysis of more complex power-split (series-parallel) HHV transmission structures and related optimal controls. This paper presents bond graph based mathematical model of kinematics of a one-mode and a two-mode power-split hybrid hydraulic vehicle transmissions which are based on their hybrid electrical counterpart. Such models serve as a basis for the analysis of the steady-state behavior of two representative power-split HHV transmissions. The analysis encloses the power train power flow. Related consideration of limitations and possible energy recuperation is given, too. Obtained kinematics mathematical models provide the basis for their upgrade with dynamics effects using bond-graphs, and the example is given in the paper, as well. INTRODUCTION Recently, there are several innovative hydraulic drives that could considerably affect the application of hydraulics. That could concern particularly to hybrid hydraulic vehicles (HHV). The floating cup hydrostatic unit and hydraulic transformer from Innas B.V. [ 1]; digital displacement hydraulic unit from Artemis Intelligent Power Ltd. [ 2], or similar emerging digital hydraulic technologies (like one described in [ 3]) have excellent efficiency of hydrostatic units even at partial load operating conditions. Energy required for their control is considerably smaller compared to classicalvariable displacement units (analysis is given in [ 1] for Innas hydraulic transformer); it is declared that their noise is smaller, as well. At last, the cost of these innovative drives is expected to be within the frame of classical hydraulic unit costs, or even better. Such characteristics could inaugurate very competitive hybrid hydraulic vehicles (HHV), what could extend possible application of HHV not only on heavy vehicles, but also on smaller trucks or personal vehicles. Generally, the advantages of HHV with respect to hybrid electrical vehicles (HEV) can be cost, power to weight ratio and the high power density of hydro-pneumatic accumulators, but on the other side their low energy density can be the critical limitation. These innovative hydraulic drives has been already considered and analyzed in the HHV applications, but only in the serial hybrid vehicle architecture ([ 4,5,6]). It is important to note that these applications are at the personal vehicles, and presented results are really encouraging. The serial hybrid architecture is also very often example of application of the classical hydraulic drives. However, heavy commercial vehicles are usual application exemplars, and there are many references on that topic. The classical hydraulic drives have been regarded in the parallel hybrid architecture mostly just for retrofitting or after-market applications for commercial heavy vehicles ([ 7 and 8]). Worth of notion is the parallel hybrid hydraulic system called Hydraulic Launch Assist from Ford, which was presented on the Ford Mighty F-350 Tonka concept truck debuting at the 2002 North American International Auto Show (NAIAS), with the aim to improve fuel efficiency of large pick-ups. The hydraulic power-split architecture, yet without energy accumulation, has been applied just for slow and heavy vehicles. Usually, the hydraulic power-split has been applied at the technologically most advanced agricultural tractors since mid 90-ies ([ 9 and 10]). They have a standard four-wheel drive and an infinite gearbox